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|---|---|---|---|---|---|---|
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
p : FormalMultilinearSeries 𝕜 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
⊢ HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ (z : 𝕜) in 𝓝 0, HasSum (fun n => z ^ n • coeff p n) (f (z₀ + z))
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Logic.Equiv.Fin
import Mathlib.Topology.Algebra.InfiniteSum.Module
#align_import analysis.analytic.basic from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
/-!
# Analytic functions
A function is analytic in one dimension around `0` if it can be written as a converging power series
`Σ pₙ zⁿ`. This definition can be extended to any dimension (even in infinite dimension) by
requiring that `pₙ` is a continuous `n`-multilinear map. In general, `pₙ` is not unique (in two
dimensions, taking `p₂ (x, y) (x', y') = x y'` or `y x'` gives the same map when applied to a
vector `(x, y) (x, y)`). A way to guarantee uniqueness is to take a symmetric `pₙ`, but this is not
always possible in nonzero characteristic (in characteristic 2, the previous example has no
symmetric representative). Therefore, we do not insist on symmetry or uniqueness in the definition,
and we only require the existence of a converging series.
The general framework is important to say that the exponential map on bounded operators on a Banach
space is analytic, as well as the inverse on invertible operators.
## Main definitions
Let `p` be a formal multilinear series from `E` to `F`, i.e., `p n` is a multilinear map on `E^n`
for `n : ℕ`.
* `p.radius`: the largest `r : ℝ≥0∞` such that `‖p n‖ * r^n` grows subexponentially.
* `p.le_radius_of_bound`, `p.le_radius_of_bound_nnreal`, `p.le_radius_of_isBigO`: if `‖p n‖ * r ^ n`
is bounded above, then `r ≤ p.radius`;
* `p.isLittleO_of_lt_radius`, `p.norm_mul_pow_le_mul_pow_of_lt_radius`,
`p.isLittleO_one_of_lt_radius`,
`p.norm_mul_pow_le_of_lt_radius`, `p.nnnorm_mul_pow_le_of_lt_radius`: if `r < p.radius`, then
`‖p n‖ * r ^ n` tends to zero exponentially;
* `p.lt_radius_of_isBigO`: if `r ≠ 0` and `‖p n‖ * r ^ n = O(a ^ n)` for some `-1 < a < 1`, then
`r < p.radius`;
* `p.partialSum n x`: the sum `∑_{i = 0}^{n-1} pᵢ xⁱ`.
* `p.sum x`: the sum `∑'_{i = 0}^{∞} pᵢ xⁱ`.
Additionally, let `f` be a function from `E` to `F`.
* `HasFPowerSeriesOnBall f p x r`: on the ball of center `x` with radius `r`,
`f (x + y) = ∑'_n pₙ yⁿ`.
* `HasFPowerSeriesAt f p x`: on some ball of center `x` with positive radius, holds
`HasFPowerSeriesOnBall f p x r`.
* `AnalyticAt 𝕜 f x`: there exists a power series `p` such that holds `HasFPowerSeriesAt f p x`.
* `AnalyticOn 𝕜 f s`: the function `f` is analytic at every point of `s`.
We develop the basic properties of these notions, notably:
* If a function admits a power series, it is continuous (see
`HasFPowerSeriesOnBall.continuousOn` and `HasFPowerSeriesAt.continuousAt` and
`AnalyticAt.continuousAt`).
* In a complete space, the sum of a formal power series with positive radius is well defined on the
disk of convergence, see `FormalMultilinearSeries.hasFPowerSeriesOnBall`.
* If a function admits a power series in a ball, then it is analytic at any point `y` of this ball,
and the power series there can be expressed in terms of the initial power series `p` as
`p.changeOrigin y`. See `HasFPowerSeriesOnBall.changeOrigin`. It follows in particular that
the set of points at which a given function is analytic is open, see `isOpen_analyticAt`.
## Implementation details
We only introduce the radius of convergence of a power series, as `p.radius`.
For a power series in finitely many dimensions, there is a finer (directional, coordinate-dependent)
notion, describing the polydisk of convergence. This notion is more specific, and not necessary to
build the general theory. We do not define it here.
-/
noncomputable section
variable {𝕜 E F G : Type*}
open Topology Classical BigOperators NNReal Filter ENNReal
open Set Filter Asymptotics
namespace FormalMultilinearSeries
variable [Ring 𝕜] [AddCommGroup E] [AddCommGroup F] [Module 𝕜 E] [Module 𝕜 F]
variable [TopologicalSpace E] [TopologicalSpace F]
variable [TopologicalAddGroup E] [TopologicalAddGroup F]
variable [ContinuousConstSMul 𝕜 E] [ContinuousConstSMul 𝕜 F]
/-- Given a formal multilinear series `p` and a vector `x`, then `p.sum x` is the sum `Σ pₙ xⁿ`. A
priori, it only behaves well when `‖x‖ < p.radius`. -/
protected def sum (p : FormalMultilinearSeries 𝕜 E F) (x : E) : F :=
∑' n : ℕ, p n fun _ => x
#align formal_multilinear_series.sum FormalMultilinearSeries.sum
/-- Given a formal multilinear series `p` and a vector `x`, then `p.partialSum n x` is the sum
`Σ pₖ xᵏ` for `k ∈ {0,..., n-1}`. -/
def partialSum (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) (x : E) : F :=
∑ k in Finset.range n, p k fun _ : Fin k => x
#align formal_multilinear_series.partial_sum FormalMultilinearSeries.partialSum
/-- The partial sums of a formal multilinear series are continuous. -/
theorem partialSum_continuous (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) :
Continuous (p.partialSum n) := by
unfold partialSum -- Porting note: added
continuity
#align formal_multilinear_series.partial_sum_continuous FormalMultilinearSeries.partialSum_continuous
end FormalMultilinearSeries
/-! ### The radius of a formal multilinear series -/
variable [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F]
[NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G]
namespace FormalMultilinearSeries
variable (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
/-- The radius of a formal multilinear series is the largest `r` such that the sum `Σ ‖pₙ‖ ‖y‖ⁿ`
converges for all `‖y‖ < r`. This implies that `Σ pₙ yⁿ` converges for all `‖y‖ < r`, but these
definitions are *not* equivalent in general. -/
def radius (p : FormalMultilinearSeries 𝕜 E F) : ℝ≥0∞ :=
⨆ (r : ℝ≥0) (C : ℝ) (_ : ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C), (r : ℝ≥0∞)
#align formal_multilinear_series.radius FormalMultilinearSeries.radius
/-- If `‖pₙ‖ rⁿ` is bounded in `n`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_bound (C : ℝ) {r : ℝ≥0} (h : ∀ n : ℕ, ‖p n‖ * (r : ℝ) ^ n ≤ C) :
(r : ℝ≥0∞) ≤ p.radius :=
le_iSup_of_le r <| le_iSup_of_le C <| le_iSup (fun _ => (r : ℝ≥0∞)) h
#align formal_multilinear_series.le_radius_of_bound FormalMultilinearSeries.le_radius_of_bound
/-- If `‖pₙ‖ rⁿ` is bounded in `n`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_bound_nnreal (C : ℝ≥0) {r : ℝ≥0} (h : ∀ n : ℕ, ‖p n‖₊ * r ^ n ≤ C) :
(r : ℝ≥0∞) ≤ p.radius :=
p.le_radius_of_bound C fun n => mod_cast h n
#align formal_multilinear_series.le_radius_of_bound_nnreal FormalMultilinearSeries.le_radius_of_bound_nnreal
/-- If `‖pₙ‖ rⁿ = O(1)`, as `n → ∞`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_isBigO (h : (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] fun _ => (1 : ℝ)) :
↑r ≤ p.radius :=
Exists.elim (isBigO_one_nat_atTop_iff.1 h) fun C hC =>
p.le_radius_of_bound C fun n => (le_abs_self _).trans (hC n)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.le_radius_of_is_O FormalMultilinearSeries.le_radius_of_isBigO
theorem le_radius_of_eventually_le (C) (h : ∀ᶠ n in atTop, ‖p n‖ * (r : ℝ) ^ n ≤ C) :
↑r ≤ p.radius :=
p.le_radius_of_isBigO <| IsBigO.of_bound C <| h.mono fun n hn => by simpa
#align formal_multilinear_series.le_radius_of_eventually_le FormalMultilinearSeries.le_radius_of_eventually_le
theorem le_radius_of_summable_nnnorm (h : Summable fun n => ‖p n‖₊ * r ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_bound_nnreal (∑' n, ‖p n‖₊ * r ^ n) fun _ => le_tsum' h _
#align formal_multilinear_series.le_radius_of_summable_nnnorm FormalMultilinearSeries.le_radius_of_summable_nnnorm
theorem le_radius_of_summable (h : Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_summable_nnnorm <| by
simp only [← coe_nnnorm] at h
exact mod_cast h
#align formal_multilinear_series.le_radius_of_summable FormalMultilinearSeries.le_radius_of_summable
theorem radius_eq_top_of_forall_nnreal_isBigO
(h : ∀ r : ℝ≥0, (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] fun _ => (1 : ℝ)) : p.radius = ∞ :=
ENNReal.eq_top_of_forall_nnreal_le fun r => p.le_radius_of_isBigO (h r)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.radius_eq_top_of_forall_nnreal_is_O FormalMultilinearSeries.radius_eq_top_of_forall_nnreal_isBigO
theorem radius_eq_top_of_eventually_eq_zero (h : ∀ᶠ n in atTop, p n = 0) : p.radius = ∞ :=
p.radius_eq_top_of_forall_nnreal_isBigO fun r =>
(isBigO_zero _ _).congr' (h.mono fun n hn => by simp [hn]) EventuallyEq.rfl
#align formal_multilinear_series.radius_eq_top_of_eventually_eq_zero FormalMultilinearSeries.radius_eq_top_of_eventually_eq_zero
theorem radius_eq_top_of_forall_image_add_eq_zero (n : ℕ) (hn : ∀ m, p (m + n) = 0) :
p.radius = ∞ :=
p.radius_eq_top_of_eventually_eq_zero <|
mem_atTop_sets.2 ⟨n, fun _ hk => tsub_add_cancel_of_le hk ▸ hn _⟩
#align formal_multilinear_series.radius_eq_top_of_forall_image_add_eq_zero FormalMultilinearSeries.radius_eq_top_of_forall_image_add_eq_zero
@[simp]
theorem constFormalMultilinearSeries_radius {v : F} :
(constFormalMultilinearSeries 𝕜 E v).radius = ⊤ :=
(constFormalMultilinearSeries 𝕜 E v).radius_eq_top_of_forall_image_add_eq_zero 1
(by simp [constFormalMultilinearSeries])
#align formal_multilinear_series.const_formal_multilinear_series_radius FormalMultilinearSeries.constFormalMultilinearSeries_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` tends to zero exponentially:
for some `0 < a < 1`, `‖p n‖ rⁿ = o(aⁿ)`. -/
theorem isLittleO_of_lt_radius (h : ↑r < p.radius) :
∃ a ∈ Ioo (0 : ℝ) 1, (fun n => ‖p n‖ * (r : ℝ) ^ n) =o[atTop] (a ^ ·) := by
have := (TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 4
rw [this]
-- Porting note: was
-- rw [(TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 4]
simp only [radius, lt_iSup_iff] at h
rcases h with ⟨t, C, hC, rt⟩
rw [ENNReal.coe_lt_coe, ← NNReal.coe_lt_coe] at rt
have : 0 < (t : ℝ) := r.coe_nonneg.trans_lt rt
rw [← div_lt_one this] at rt
refine' ⟨_, rt, C, Or.inr zero_lt_one, fun n => _⟩
calc
|‖p n‖ * (r : ℝ) ^ n| = ‖p n‖ * (t : ℝ) ^ n * (r / t : ℝ) ^ n := by
field_simp [mul_right_comm, abs_mul]
_ ≤ C * (r / t : ℝ) ^ n := by gcongr; apply hC
#align formal_multilinear_series.is_o_of_lt_radius FormalMultilinearSeries.isLittleO_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ = o(1)`. -/
theorem isLittleO_one_of_lt_radius (h : ↑r < p.radius) :
(fun n => ‖p n‖ * (r : ℝ) ^ n) =o[atTop] (fun _ => 1 : ℕ → ℝ) :=
let ⟨_, ha, hp⟩ := p.isLittleO_of_lt_radius h
hp.trans <| (isLittleO_pow_pow_of_lt_left ha.1.le ha.2).congr (fun _ => rfl) one_pow
#align formal_multilinear_series.is_o_one_of_lt_radius FormalMultilinearSeries.isLittleO_one_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` tends to zero exponentially:
for some `0 < a < 1` and `C > 0`, `‖p n‖ * r ^ n ≤ C * a ^ n`. -/
theorem norm_mul_pow_le_mul_pow_of_lt_radius (h : ↑r < p.radius) :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C * a ^ n := by
-- Porting note: moved out of `rcases`
have := ((TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 5).mp
(p.isLittleO_of_lt_radius h)
rcases this with ⟨a, ha, C, hC, H⟩
exact ⟨a, ha, C, hC, fun n => (le_abs_self _).trans (H n)⟩
#align formal_multilinear_series.norm_mul_pow_le_mul_pow_of_lt_radius FormalMultilinearSeries.norm_mul_pow_le_mul_pow_of_lt_radius
/-- If `r ≠ 0` and `‖pₙ‖ rⁿ = O(aⁿ)` for some `-1 < a < 1`, then `r < p.radius`. -/
theorem lt_radius_of_isBigO (h₀ : r ≠ 0) {a : ℝ} (ha : a ∈ Ioo (-1 : ℝ) 1)
(hp : (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] (a ^ ·)) : ↑r < p.radius := by
-- Porting note: moved out of `rcases`
have := ((TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 2 5)
rcases this.mp ⟨a, ha, hp⟩ with ⟨a, ha, C, hC, hp⟩
rw [← pos_iff_ne_zero, ← NNReal.coe_pos] at h₀
lift a to ℝ≥0 using ha.1.le
have : (r : ℝ) < r / a := by
simpa only [div_one] using (div_lt_div_left h₀ zero_lt_one ha.1).2 ha.2
norm_cast at this
rw [← ENNReal.coe_lt_coe] at this
refine' this.trans_le (p.le_radius_of_bound C fun n => _)
rw [NNReal.coe_div, div_pow, ← mul_div_assoc, div_le_iff (pow_pos ha.1 n)]
exact (le_abs_self _).trans (hp n)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.lt_radius_of_is_O FormalMultilinearSeries.lt_radius_of_isBigO
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem norm_mul_pow_le_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C :=
let ⟨_, ha, C, hC, h⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius h
⟨C, hC, fun n => (h n).trans <| mul_le_of_le_one_right hC.lt.le (pow_le_one _ ha.1.le ha.2.le)⟩
#align formal_multilinear_series.norm_mul_pow_le_of_lt_radius FormalMultilinearSeries.norm_mul_pow_le_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem norm_le_div_pow_of_pos_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h0 : 0 < r) (h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖ ≤ C / (r : ℝ) ^ n :=
let ⟨C, hC, hp⟩ := p.norm_mul_pow_le_of_lt_radius h
⟨C, hC, fun n => Iff.mpr (le_div_iff (pow_pos h0 _)) (hp n)⟩
#align formal_multilinear_series.norm_le_div_pow_of_pos_of_lt_radius FormalMultilinearSeries.norm_le_div_pow_of_pos_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem nnnorm_mul_pow_le_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖₊ * r ^ n ≤ C :=
let ⟨C, hC, hp⟩ := p.norm_mul_pow_le_of_lt_radius h
⟨⟨C, hC.lt.le⟩, hC, mod_cast hp⟩
#align formal_multilinear_series.nnnorm_mul_pow_le_of_lt_radius FormalMultilinearSeries.nnnorm_mul_pow_le_of_lt_radius
theorem le_radius_of_tendsto (p : FormalMultilinearSeries 𝕜 E F) {l : ℝ}
(h : Tendsto (fun n => ‖p n‖ * (r : ℝ) ^ n) atTop (𝓝 l)) : ↑r ≤ p.radius :=
p.le_radius_of_isBigO (h.isBigO_one _)
#align formal_multilinear_series.le_radius_of_tendsto FormalMultilinearSeries.le_radius_of_tendsto
theorem le_radius_of_summable_norm (p : FormalMultilinearSeries 𝕜 E F)
(hs : Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_tendsto hs.tendsto_atTop_zero
#align formal_multilinear_series.le_radius_of_summable_norm FormalMultilinearSeries.le_radius_of_summable_norm
theorem not_summable_norm_of_radius_lt_nnnorm (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(h : p.radius < ‖x‖₊) : ¬Summable fun n => ‖p n‖ * ‖x‖ ^ n :=
fun hs => not_le_of_lt h (p.le_radius_of_summable_norm hs)
#align formal_multilinear_series.not_summable_norm_of_radius_lt_nnnorm FormalMultilinearSeries.not_summable_norm_of_radius_lt_nnnorm
theorem summable_norm_mul_pow (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0} (h : ↑r < p.radius) :
Summable fun n : ℕ => ‖p n‖ * (r : ℝ) ^ n := by
obtain ⟨a, ha : a ∈ Ioo (0 : ℝ) 1, C, - : 0 < C, hp⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius h
exact .of_nonneg_of_le (fun n => mul_nonneg (norm_nonneg _) (pow_nonneg r.coe_nonneg _))
hp ((summable_geometric_of_lt_1 ha.1.le ha.2).mul_left _)
#align formal_multilinear_series.summable_norm_mul_pow FormalMultilinearSeries.summable_norm_mul_pow
theorem summable_norm_apply (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : Summable fun n : ℕ => ‖p n fun _ => x‖ := by
rw [mem_emetric_ball_zero_iff] at hx
refine' .of_nonneg_of_le
(fun _ => norm_nonneg _) (fun n => ((p n).le_op_norm _).trans_eq _) (p.summable_norm_mul_pow hx)
simp
#align formal_multilinear_series.summable_norm_apply FormalMultilinearSeries.summable_norm_apply
theorem summable_nnnorm_mul_pow (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0} (h : ↑r < p.radius) :
Summable fun n : ℕ => ‖p n‖₊ * r ^ n := by
rw [← NNReal.summable_coe]
push_cast
exact p.summable_norm_mul_pow h
#align formal_multilinear_series.summable_nnnorm_mul_pow FormalMultilinearSeries.summable_nnnorm_mul_pow
protected theorem summable [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : Summable fun n : ℕ => p n fun _ => x :=
(p.summable_norm_apply hx).of_norm
#align formal_multilinear_series.summable FormalMultilinearSeries.summable
theorem radius_eq_top_of_summable_norm (p : FormalMultilinearSeries 𝕜 E F)
(hs : ∀ r : ℝ≥0, Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : p.radius = ∞ :=
ENNReal.eq_top_of_forall_nnreal_le fun r => p.le_radius_of_summable_norm (hs r)
#align formal_multilinear_series.radius_eq_top_of_summable_norm FormalMultilinearSeries.radius_eq_top_of_summable_norm
theorem radius_eq_top_iff_summable_norm (p : FormalMultilinearSeries 𝕜 E F) :
p.radius = ∞ ↔ ∀ r : ℝ≥0, Summable fun n => ‖p n‖ * (r : ℝ) ^ n := by
constructor
· intro h r
obtain ⟨a, ha : a ∈ Ioo (0 : ℝ) 1, C, - : 0 < C, hp⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius
(show (r : ℝ≥0∞) < p.radius from h.symm ▸ ENNReal.coe_lt_top)
refine' .of_norm_bounded
(fun n => (C : ℝ) * a ^ n) ((summable_geometric_of_lt_1 ha.1.le ha.2).mul_left _) fun n => _
specialize hp n
rwa [Real.norm_of_nonneg (mul_nonneg (norm_nonneg _) (pow_nonneg r.coe_nonneg n))]
· exact p.radius_eq_top_of_summable_norm
#align formal_multilinear_series.radius_eq_top_iff_summable_norm FormalMultilinearSeries.radius_eq_top_iff_summable_norm
/-- If the radius of `p` is positive, then `‖pₙ‖` grows at most geometrically. -/
theorem le_mul_pow_of_radius_pos (p : FormalMultilinearSeries 𝕜 E F) (h : 0 < p.radius) :
∃ (C r : _) (hC : 0 < C) (_ : 0 < r), ∀ n, ‖p n‖ ≤ C * r ^ n := by
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 h with ⟨r, r0, rlt⟩
have rpos : 0 < (r : ℝ) := by simp [ENNReal.coe_pos.1 r0]
rcases norm_le_div_pow_of_pos_of_lt_radius p rpos rlt with ⟨C, Cpos, hCp⟩
refine' ⟨C, r⁻¹, Cpos, by simp only [inv_pos, rpos], fun n => _⟩
-- Porting note: was `convert`
rw [inv_pow, ← div_eq_mul_inv]
exact hCp n
#align formal_multilinear_series.le_mul_pow_of_radius_pos FormalMultilinearSeries.le_mul_pow_of_radius_pos
/-- The radius of the sum of two formal series is at least the minimum of their two radii. -/
theorem min_radius_le_radius_add (p q : FormalMultilinearSeries 𝕜 E F) :
min p.radius q.radius ≤ (p + q).radius := by
refine' ENNReal.le_of_forall_nnreal_lt fun r hr => _
rw [lt_min_iff] at hr
have := ((p.isLittleO_one_of_lt_radius hr.1).add (q.isLittleO_one_of_lt_radius hr.2)).isBigO
refine' (p + q).le_radius_of_isBigO ((isBigO_of_le _ fun n => _).trans this)
rw [← add_mul, norm_mul, norm_mul, norm_norm]
exact mul_le_mul_of_nonneg_right ((norm_add_le _ _).trans (le_abs_self _)) (norm_nonneg _)
#align formal_multilinear_series.min_radius_le_radius_add FormalMultilinearSeries.min_radius_le_radius_add
@[simp]
theorem radius_neg (p : FormalMultilinearSeries 𝕜 E F) : (-p).radius = p.radius := by
simp only [radius, neg_apply, norm_neg]
#align formal_multilinear_series.radius_neg FormalMultilinearSeries.radius_neg
protected theorem hasSum [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : HasSum (fun n : ℕ => p n fun _ => x) (p.sum x) :=
(p.summable hx).hasSum
#align formal_multilinear_series.has_sum FormalMultilinearSeries.hasSum
theorem radius_le_radius_continuousLinearMap_comp (p : FormalMultilinearSeries 𝕜 E F)
(f : F →L[𝕜] G) : p.radius ≤ (f.compFormalMultilinearSeries p).radius := by
refine' ENNReal.le_of_forall_nnreal_lt fun r hr => _
apply le_radius_of_isBigO
apply (IsBigO.trans_isLittleO _ (p.isLittleO_one_of_lt_radius hr)).isBigO
refine' IsBigO.mul (@IsBigOWith.isBigO _ _ _ _ _ ‖f‖ _ _ _ _) (isBigO_refl _ _)
refine IsBigOWith.of_bound (eventually_of_forall fun n => ?_)
simpa only [norm_norm] using f.norm_compContinuousMultilinearMap_le (p n)
#align formal_multilinear_series.radius_le_radius_continuous_linear_map_comp FormalMultilinearSeries.radius_le_radius_continuousLinearMap_comp
end FormalMultilinearSeries
/-! ### Expanding a function as a power series -/
section
variable {f g : E → F} {p pf pg : FormalMultilinearSeries 𝕜 E F} {x : E} {r r' : ℝ≥0∞}
/-- Given a function `f : E → F` and a formal multilinear series `p`, we say that `f` has `p` as
a power series on the ball of radius `r > 0` around `x` if `f (x + y) = ∑' pₙ yⁿ` for all `‖y‖ < r`.
-/
structure HasFPowerSeriesOnBall (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (x : E) (r : ℝ≥0∞) :
Prop where
r_le : r ≤ p.radius
r_pos : 0 < r
hasSum :
∀ {y}, y ∈ EMetric.ball (0 : E) r → HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y))
#align has_fpower_series_on_ball HasFPowerSeriesOnBall
/-- Given a function `f : E → F` and a formal multilinear series `p`, we say that `f` has `p` as
a power series around `x` if `f (x + y) = ∑' pₙ yⁿ` for all `y` in a neighborhood of `0`. -/
def HasFPowerSeriesAt (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (x : E) :=
∃ r, HasFPowerSeriesOnBall f p x r
#align has_fpower_series_at HasFPowerSeriesAt
variable (𝕜)
/-- Given a function `f : E → F`, we say that `f` is analytic at `x` if it admits a convergent power
series expansion around `x`. -/
def AnalyticAt (f : E → F) (x : E) :=
∃ p : FormalMultilinearSeries 𝕜 E F, HasFPowerSeriesAt f p x
#align analytic_at AnalyticAt
/-- Given a function `f : E → F`, we say that `f` is analytic on a set `s` if it is analytic around
every point of `s`. -/
def AnalyticOn (f : E → F) (s : Set E) :=
∀ x, x ∈ s → AnalyticAt 𝕜 f x
#align analytic_on AnalyticOn
variable {𝕜}
theorem HasFPowerSeriesOnBall.hasFPowerSeriesAt (hf : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesAt f p x :=
⟨r, hf⟩
#align has_fpower_series_on_ball.has_fpower_series_at HasFPowerSeriesOnBall.hasFPowerSeriesAt
theorem HasFPowerSeriesAt.analyticAt (hf : HasFPowerSeriesAt f p x) : AnalyticAt 𝕜 f x :=
⟨p, hf⟩
#align has_fpower_series_at.analytic_at HasFPowerSeriesAt.analyticAt
theorem HasFPowerSeriesOnBall.analyticAt (hf : HasFPowerSeriesOnBall f p x r) : AnalyticAt 𝕜 f x :=
hf.hasFPowerSeriesAt.analyticAt
#align has_fpower_series_on_ball.analytic_at HasFPowerSeriesOnBall.analyticAt
theorem HasFPowerSeriesOnBall.congr (hf : HasFPowerSeriesOnBall f p x r)
(hg : EqOn f g (EMetric.ball x r)) : HasFPowerSeriesOnBall g p x r :=
{ r_le := hf.r_le
r_pos := hf.r_pos
hasSum := fun {y} hy => by
convert hf.hasSum hy using 1
apply hg.symm
simpa [edist_eq_coe_nnnorm_sub] using hy }
#align has_fpower_series_on_ball.congr HasFPowerSeriesOnBall.congr
/-- If a function `f` has a power series `p` around `x`, then the function `z ↦ f (z - y)` has the
same power series around `x + y`. -/
theorem HasFPowerSeriesOnBall.comp_sub (hf : HasFPowerSeriesOnBall f p x r) (y : E) :
HasFPowerSeriesOnBall (fun z => f (z - y)) p (x + y) r :=
{ r_le := hf.r_le
r_pos := hf.r_pos
hasSum := fun {z} hz => by
convert hf.hasSum hz using 2
abel }
#align has_fpower_series_on_ball.comp_sub HasFPowerSeriesOnBall.comp_sub
theorem HasFPowerSeriesOnBall.hasSum_sub (hf : HasFPowerSeriesOnBall f p x r) {y : E}
(hy : y ∈ EMetric.ball x r) : HasSum (fun n : ℕ => p n fun _ => y - x) (f y) := by
have : y - x ∈ EMetric.ball (0 : E) r := by simpa [edist_eq_coe_nnnorm_sub] using hy
simpa only [add_sub_cancel'_right] using hf.hasSum this
#align has_fpower_series_on_ball.has_sum_sub HasFPowerSeriesOnBall.hasSum_sub
theorem HasFPowerSeriesOnBall.radius_pos (hf : HasFPowerSeriesOnBall f p x r) : 0 < p.radius :=
lt_of_lt_of_le hf.r_pos hf.r_le
#align has_fpower_series_on_ball.radius_pos HasFPowerSeriesOnBall.radius_pos
theorem HasFPowerSeriesAt.radius_pos (hf : HasFPowerSeriesAt f p x) : 0 < p.radius :=
let ⟨_, hr⟩ := hf
hr.radius_pos
#align has_fpower_series_at.radius_pos HasFPowerSeriesAt.radius_pos
theorem HasFPowerSeriesOnBall.mono (hf : HasFPowerSeriesOnBall f p x r) (r'_pos : 0 < r')
(hr : r' ≤ r) : HasFPowerSeriesOnBall f p x r' :=
⟨le_trans hr hf.1, r'_pos, fun hy => hf.hasSum (EMetric.ball_subset_ball hr hy)⟩
#align has_fpower_series_on_ball.mono HasFPowerSeriesOnBall.mono
theorem HasFPowerSeriesAt.congr (hf : HasFPowerSeriesAt f p x) (hg : f =ᶠ[𝓝 x] g) :
HasFPowerSeriesAt g p x := by
rcases hf with ⟨r₁, h₁⟩
rcases EMetric.mem_nhds_iff.mp hg with ⟨r₂, h₂pos, h₂⟩
exact ⟨min r₁ r₂,
(h₁.mono (lt_min h₁.r_pos h₂pos) inf_le_left).congr
fun y hy => h₂ (EMetric.ball_subset_ball inf_le_right hy)⟩
#align has_fpower_series_at.congr HasFPowerSeriesAt.congr
protected theorem HasFPowerSeriesAt.eventually (hf : HasFPowerSeriesAt f p x) :
∀ᶠ r : ℝ≥0∞ in 𝓝[>] 0, HasFPowerSeriesOnBall f p x r :=
let ⟨_, hr⟩ := hf
mem_of_superset (Ioo_mem_nhdsWithin_Ioi (left_mem_Ico.2 hr.r_pos)) fun _ hr' =>
hr.mono hr'.1 hr'.2.le
#align has_fpower_series_at.eventually HasFPowerSeriesAt.eventually
theorem HasFPowerSeriesOnBall.eventually_hasSum (hf : HasFPowerSeriesOnBall f p x r) :
∀ᶠ y in 𝓝 0, HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y)) := by
filter_upwards [EMetric.ball_mem_nhds (0 : E) hf.r_pos] using fun _ => hf.hasSum
#align has_fpower_series_on_ball.eventually_has_sum HasFPowerSeriesOnBall.eventually_hasSum
theorem HasFPowerSeriesAt.eventually_hasSum (hf : HasFPowerSeriesAt f p x) :
∀ᶠ y in 𝓝 0, HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y)) :=
let ⟨_, hr⟩ := hf
hr.eventually_hasSum
#align has_fpower_series_at.eventually_has_sum HasFPowerSeriesAt.eventually_hasSum
theorem HasFPowerSeriesOnBall.eventually_hasSum_sub (hf : HasFPowerSeriesOnBall f p x r) :
∀ᶠ y in 𝓝 x, HasSum (fun n : ℕ => p n fun _ : Fin n => y - x) (f y) := by
filter_upwards [EMetric.ball_mem_nhds x hf.r_pos] with y using hf.hasSum_sub
#align has_fpower_series_on_ball.eventually_has_sum_sub HasFPowerSeriesOnBall.eventually_hasSum_sub
theorem HasFPowerSeriesAt.eventually_hasSum_sub (hf : HasFPowerSeriesAt f p x) :
∀ᶠ y in 𝓝 x, HasSum (fun n : ℕ => p n fun _ : Fin n => y - x) (f y) :=
let ⟨_, hr⟩ := hf
hr.eventually_hasSum_sub
#align has_fpower_series_at.eventually_has_sum_sub HasFPowerSeriesAt.eventually_hasSum_sub
theorem HasFPowerSeriesOnBall.eventually_eq_zero
(hf : HasFPowerSeriesOnBall f (0 : FormalMultilinearSeries 𝕜 E F) x r) :
∀ᶠ z in 𝓝 x, f z = 0 := by
filter_upwards [hf.eventually_hasSum_sub] with z hz using hz.unique hasSum_zero
#align has_fpower_series_on_ball.eventually_eq_zero HasFPowerSeriesOnBall.eventually_eq_zero
theorem HasFPowerSeriesAt.eventually_eq_zero
(hf : HasFPowerSeriesAt f (0 : FormalMultilinearSeries 𝕜 E F) x) : ∀ᶠ z in 𝓝 x, f z = 0 :=
let ⟨_, hr⟩ := hf
hr.eventually_eq_zero
#align has_fpower_series_at.eventually_eq_zero HasFPowerSeriesAt.eventually_eq_zero
theorem hasFPowerSeriesOnBall_const {c : F} {e : E} :
HasFPowerSeriesOnBall (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e ⊤ := by
refine' ⟨by simp, WithTop.zero_lt_top, fun _ => hasSum_single 0 fun n hn => _⟩
simp [constFormalMultilinearSeries_apply hn]
#align has_fpower_series_on_ball_const hasFPowerSeriesOnBall_const
theorem hasFPowerSeriesAt_const {c : F} {e : E} :
HasFPowerSeriesAt (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e :=
⟨⊤, hasFPowerSeriesOnBall_const⟩
#align has_fpower_series_at_const hasFPowerSeriesAt_const
theorem analyticAt_const {v : F} : AnalyticAt 𝕜 (fun _ => v) x :=
⟨constFormalMultilinearSeries 𝕜 E v, hasFPowerSeriesAt_const⟩
#align analytic_at_const analyticAt_const
theorem analyticOn_const {v : F} {s : Set E} : AnalyticOn 𝕜 (fun _ => v) s :=
fun _ _ => analyticAt_const
#align analytic_on_const analyticOn_const
theorem HasFPowerSeriesOnBall.add (hf : HasFPowerSeriesOnBall f pf x r)
(hg : HasFPowerSeriesOnBall g pg x r) : HasFPowerSeriesOnBall (f + g) (pf + pg) x r :=
{ r_le := le_trans (le_min_iff.2 ⟨hf.r_le, hg.r_le⟩) (pf.min_radius_le_radius_add pg)
r_pos := hf.r_pos
hasSum := fun hy => (hf.hasSum hy).add (hg.hasSum hy) }
#align has_fpower_series_on_ball.add HasFPowerSeriesOnBall.add
theorem HasFPowerSeriesAt.add (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x) :
HasFPowerSeriesAt (f + g) (pf + pg) x := by
rcases (hf.eventually.and hg.eventually).exists with ⟨r, hr⟩
exact ⟨r, hr.1.add hr.2⟩
#align has_fpower_series_at.add HasFPowerSeriesAt.add
theorem AnalyticAt.congr (hf : AnalyticAt 𝕜 f x) (hg : f =ᶠ[𝓝 x] g) : AnalyticAt 𝕜 g x :=
let ⟨_, hpf⟩ := hf
(hpf.congr hg).analyticAt
theorem analyticAt_congr (h : f =ᶠ[𝓝 x] g) : AnalyticAt 𝕜 f x ↔ AnalyticAt 𝕜 g x :=
⟨fun hf ↦ hf.congr h, fun hg ↦ hg.congr h.symm⟩
theorem AnalyticAt.add (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) : AnalyticAt 𝕜 (f + g) x :=
let ⟨_, hpf⟩ := hf
let ⟨_, hqf⟩ := hg
(hpf.add hqf).analyticAt
#align analytic_at.add AnalyticAt.add
theorem HasFPowerSeriesOnBall.neg (hf : HasFPowerSeriesOnBall f pf x r) :
HasFPowerSeriesOnBall (-f) (-pf) x r :=
{ r_le := by
rw [pf.radius_neg]
exact hf.r_le
r_pos := hf.r_pos
hasSum := fun hy => (hf.hasSum hy).neg }
#align has_fpower_series_on_ball.neg HasFPowerSeriesOnBall.neg
theorem HasFPowerSeriesAt.neg (hf : HasFPowerSeriesAt f pf x) : HasFPowerSeriesAt (-f) (-pf) x :=
let ⟨_, hrf⟩ := hf
hrf.neg.hasFPowerSeriesAt
#align has_fpower_series_at.neg HasFPowerSeriesAt.neg
theorem AnalyticAt.neg (hf : AnalyticAt 𝕜 f x) : AnalyticAt 𝕜 (-f) x :=
let ⟨_, hpf⟩ := hf
hpf.neg.analyticAt
#align analytic_at.neg AnalyticAt.neg
theorem HasFPowerSeriesOnBall.sub (hf : HasFPowerSeriesOnBall f pf x r)
(hg : HasFPowerSeriesOnBall g pg x r) : HasFPowerSeriesOnBall (f - g) (pf - pg) x r := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align has_fpower_series_on_ball.sub HasFPowerSeriesOnBall.sub
theorem HasFPowerSeriesAt.sub (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x) :
HasFPowerSeriesAt (f - g) (pf - pg) x := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align has_fpower_series_at.sub HasFPowerSeriesAt.sub
theorem AnalyticAt.sub (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) :
AnalyticAt 𝕜 (f - g) x := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align analytic_at.sub AnalyticAt.sub
theorem AnalyticOn.mono {s t : Set E} (hf : AnalyticOn 𝕜 f t) (hst : s ⊆ t) : AnalyticOn 𝕜 f s :=
fun z hz => hf z (hst hz)
#align analytic_on.mono AnalyticOn.mono
theorem AnalyticOn.congr' {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : f =ᶠ[𝓝ˢ s] g) :
AnalyticOn 𝕜 g s :=
fun z hz => (hf z hz).congr (mem_nhdsSet_iff_forall.mp hg z hz)
theorem analyticOn_congr' {s : Set E} (h : f =ᶠ[𝓝ˢ s] g) : AnalyticOn 𝕜 f s ↔ AnalyticOn 𝕜 g s :=
⟨fun hf => hf.congr' h, fun hg => hg.congr' h.symm⟩
theorem AnalyticOn.congr {s : Set E} (hs : IsOpen s) (hf : AnalyticOn 𝕜 f s) (hg : s.EqOn f g) :
AnalyticOn 𝕜 g s :=
hf.congr' $ mem_nhdsSet_iff_forall.mpr
(fun _ hz => eventuallyEq_iff_exists_mem.mpr ⟨s, hs.mem_nhds hz, hg⟩)
theorem analyticOn_congr {s : Set E} (hs : IsOpen s) (h : s.EqOn f g) : AnalyticOn 𝕜 f s ↔
AnalyticOn 𝕜 g s := ⟨fun hf => hf.congr hs h, fun hg => hg.congr hs h.symm⟩
theorem AnalyticOn.add {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
AnalyticOn 𝕜 (f + g) s :=
fun z hz => (hf z hz).add (hg z hz)
#align analytic_on.add AnalyticOn.add
theorem AnalyticOn.sub {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
AnalyticOn 𝕜 (f - g) s :=
fun z hz => (hf z hz).sub (hg z hz)
#align analytic_on.sub AnalyticOn.sub
theorem HasFPowerSeriesOnBall.coeff_zero (hf : HasFPowerSeriesOnBall f pf x r) (v : Fin 0 → E) :
pf 0 v = f x := by
have v_eq : v = fun i => 0 := Subsingleton.elim _ _
have zero_mem : (0 : E) ∈ EMetric.ball (0 : E) r := by simp [hf.r_pos]
have : ∀ i, i ≠ 0 → (pf i fun j => 0) = 0 := by
intro i hi
have : 0 < i := pos_iff_ne_zero.2 hi
exact ContinuousMultilinearMap.map_coord_zero _ (⟨0, this⟩ : Fin i) rfl
have A := (hf.hasSum zero_mem).unique (hasSum_single _ this)
simpa [v_eq] using A.symm
#align has_fpower_series_on_ball.coeff_zero HasFPowerSeriesOnBall.coeff_zero
theorem HasFPowerSeriesAt.coeff_zero (hf : HasFPowerSeriesAt f pf x) (v : Fin 0 → E) :
pf 0 v = f x :=
let ⟨_, hrf⟩ := hf
hrf.coeff_zero v
#align has_fpower_series_at.coeff_zero HasFPowerSeriesAt.coeff_zero
/-- If a function `f` has a power series `p` on a ball and `g` is linear, then `g ∘ f` has the
power series `g ∘ p` on the same ball. -/
theorem ContinuousLinearMap.comp_hasFPowerSeriesOnBall (g : F →L[𝕜] G)
(h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (g ∘ f) (g.compFormalMultilinearSeries p) x r :=
{ r_le := h.r_le.trans (p.radius_le_radius_continuousLinearMap_comp _)
r_pos := h.r_pos
hasSum := fun hy => by
simpa only [ContinuousLinearMap.compFormalMultilinearSeries_apply,
ContinuousLinearMap.compContinuousMultilinearMap_coe, Function.comp_apply] using
g.hasSum (h.hasSum hy) }
#align continuous_linear_map.comp_has_fpower_series_on_ball ContinuousLinearMap.comp_hasFPowerSeriesOnBall
/-- If a function `f` is analytic on a set `s` and `g` is linear, then `g ∘ f` is analytic
on `s`. -/
theorem ContinuousLinearMap.comp_analyticOn {s : Set E} (g : F →L[𝕜] G) (h : AnalyticOn 𝕜 f s) :
AnalyticOn 𝕜 (g ∘ f) s := by
rintro x hx
rcases h x hx with ⟨p, r, hp⟩
exact ⟨g.compFormalMultilinearSeries p, r, g.comp_hasFPowerSeriesOnBall hp⟩
#align continuous_linear_map.comp_analytic_on ContinuousLinearMap.comp_analyticOn
/-- If a function admits a power series expansion, then it is exponentially close to the partial
sums of this power series on strict subdisks of the disk of convergence.
This version provides an upper estimate that decreases both in `‖y‖` and `n`. See also
`HasFPowerSeriesOnBall.uniform_geometric_approx` for a weaker version. -/
theorem HasFPowerSeriesOnBall.uniform_geometric_approx' {r' : ℝ≥0}
(hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n := by
obtain ⟨a, ha, C, hC, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n, ‖p n‖ * (r' : ℝ) ^ n ≤ C * a ^ n :=
p.norm_mul_pow_le_mul_pow_of_lt_radius (h.trans_le hf.r_le)
refine' ⟨a, ha, C / (1 - a), div_pos hC (sub_pos.2 ha.2), fun y hy n => _⟩
have yr' : ‖y‖ < r' := by
rw [ball_zero_eq] at hy
exact hy
have hr'0 : 0 < (r' : ℝ) := (norm_nonneg _).trans_lt yr'
have : y ∈ EMetric.ball (0 : E) r := by
refine' mem_emetric_ball_zero_iff.2 (lt_trans _ h)
exact mod_cast yr'
rw [norm_sub_rev, ← mul_div_right_comm]
have ya : a * (‖y‖ / ↑r') ≤ a :=
mul_le_of_le_one_right ha.1.le (div_le_one_of_le yr'.le r'.coe_nonneg)
suffices ‖p.partialSum n y - f (x + y)‖ ≤ C * (a * (‖y‖ / r')) ^ n / (1 - a * (‖y‖ / r')) by
refine' this.trans _
have : 0 < a := ha.1
gcongr
apply_rules [sub_pos.2, ha.2]
apply norm_sub_le_of_geometric_bound_of_hasSum (ya.trans_lt ha.2) _ (hf.hasSum this)
intro n
calc
‖(p n) fun _ : Fin n => y‖
_ ≤ ‖p n‖ * ∏ _i : Fin n, ‖y‖ := ContinuousMultilinearMap.le_op_norm _ _
_ = ‖p n‖ * (r' : ℝ) ^ n * (‖y‖ / r') ^ n := by field_simp [mul_right_comm]
_ ≤ C * a ^ n * (‖y‖ / r') ^ n := by gcongr ?_ * _; apply hp
_ ≤ C * (a * (‖y‖ / r')) ^ n := by rw [mul_pow, mul_assoc]
#align has_fpower_series_on_ball.uniform_geometric_approx' HasFPowerSeriesOnBall.uniform_geometric_approx'
/-- If a function admits a power series expansion, then it is exponentially close to the partial
sums of this power series on strict subdisks of the disk of convergence. -/
theorem HasFPowerSeriesOnBall.uniform_geometric_approx {r' : ℝ≥0}
(hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
∃ a ∈ Ioo (0 : ℝ) 1,
∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n, ‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n := by
obtain ⟨a, ha, C, hC, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n :=
hf.uniform_geometric_approx' h
refine' ⟨a, ha, C, hC, fun y hy n => (hp y hy n).trans _⟩
have yr' : ‖y‖ < r' := by rwa [ball_zero_eq] at hy
gcongr
exacts [mul_nonneg ha.1.le (div_nonneg (norm_nonneg y) r'.coe_nonneg),
mul_le_of_le_one_right ha.1.le (div_le_one_of_le yr'.le r'.coe_nonneg)]
#align has_fpower_series_on_ball.uniform_geometric_approx HasFPowerSeriesOnBall.uniform_geometric_approx
/-- Taylor formula for an analytic function, `IsBigO` version. -/
theorem HasFPowerSeriesAt.isBigO_sub_partialSum_pow (hf : HasFPowerSeriesAt f p x) (n : ℕ) :
(fun y : E => f (x + y) - p.partialSum n y) =O[𝓝 0] fun y => ‖y‖ ^ n := by
rcases hf with ⟨r, hf⟩
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hf.r_pos with ⟨r', r'0, h⟩
obtain ⟨a, -, C, -, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n :=
hf.uniform_geometric_approx' h
refine' isBigO_iff.2 ⟨C * (a / r') ^ n, _⟩
replace r'0 : 0 < (r' : ℝ); · exact mod_cast r'0
filter_upwards [Metric.ball_mem_nhds (0 : E) r'0] with y hy
simpa [mul_pow, mul_div_assoc, mul_assoc, div_mul_eq_mul_div] using hp y hy n
set_option linter.uppercaseLean3 false in
#align has_fpower_series_at.is_O_sub_partial_sum_pow HasFPowerSeriesAt.isBigO_sub_partialSum_pow
/-- If `f` has formal power series `∑ n, pₙ` on a ball of radius `r`, then for `y, z` in any smaller
ball, the norm of the difference `f y - f z - p 1 (fun _ ↦ y - z)` is bounded above by
`C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖`. This lemma formulates this property using `IsBigO` and
`Filter.principal` on `E × E`. -/
theorem HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal
(hf : HasFPowerSeriesOnBall f p x r) (hr : r' < r) :
(fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) =O[𝓟 (EMetric.ball (x, x) r')]
fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖ := by
lift r' to ℝ≥0 using ne_top_of_lt hr
rcases (zero_le r').eq_or_lt with (rfl | hr'0)
· simp only [isBigO_bot, EMetric.ball_zero, principal_empty, ENNReal.coe_zero]
obtain ⟨a, ha, C, hC : 0 < C, hp⟩ :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n : ℕ, ‖p n‖ * (r' : ℝ) ^ n ≤ C * a ^ n
exact p.norm_mul_pow_le_mul_pow_of_lt_radius (hr.trans_le hf.r_le)
simp only [← le_div_iff (pow_pos (NNReal.coe_pos.2 hr'0) _)] at hp
set L : E × E → ℝ := fun y =>
C * (a / r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * (a / (1 - a) ^ 2 + 2 / (1 - a))
have hL : ∀ y ∈ EMetric.ball (x, x) r', ‖f y.1 - f y.2 - p 1 fun _ => y.1 - y.2‖ ≤ L y := by
intro y hy'
have hy : y ∈ EMetric.ball x r ×ˢ EMetric.ball x r := by
rw [EMetric.ball_prod_same]
exact EMetric.ball_subset_ball hr.le hy'
set A : ℕ → F := fun n => (p n fun _ => y.1 - x) - p n fun _ => y.2 - x
have hA : HasSum (fun n => A (n + 2)) (f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) := by
convert (hasSum_nat_add_iff' 2).2 ((hf.hasSum_sub hy.1).sub (hf.hasSum_sub hy.2)) using 1
rw [Finset.sum_range_succ, Finset.sum_range_one, hf.coeff_zero, hf.coeff_zero, sub_self,
zero_add, ← Subsingleton.pi_single_eq (0 : Fin 1) (y.1 - x), Pi.single,
← Subsingleton.pi_single_eq (0 : Fin 1) (y.2 - x), Pi.single, ← (p 1).map_sub, ← Pi.single,
Subsingleton.pi_single_eq, sub_sub_sub_cancel_right]
rw [EMetric.mem_ball, edist_eq_coe_nnnorm_sub, ENNReal.coe_lt_coe] at hy'
set B : ℕ → ℝ := fun n => C * (a / r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * ((n + 2) * a ^ n)
have hAB : ∀ n, ‖A (n + 2)‖ ≤ B n := fun n =>
calc
‖A (n + 2)‖ ≤ ‖p (n + 2)‖ * ↑(n + 2) * ‖y - (x, x)‖ ^ (n + 1) * ‖y.1 - y.2‖ := by
-- porting note: `pi_norm_const` was `pi_norm_const (_ : E)`
simpa only [Fintype.card_fin, pi_norm_const, Prod.norm_def, Pi.sub_def,
Prod.fst_sub, Prod.snd_sub, sub_sub_sub_cancel_right] using
(p <| n + 2).norm_image_sub_le (fun _ => y.1 - x) fun _ => y.2 - x
_ = ‖p (n + 2)‖ * ‖y - (x, x)‖ ^ n * (↑(n + 2) * ‖y - (x, x)‖ * ‖y.1 - y.2‖) := by
rw [pow_succ ‖y - (x, x)‖]
ring
-- porting note: the two `↑` in `↑r'` are new, without them, Lean fails to synthesize
-- instances `HDiv ℝ ℝ≥0 ?m` or `HMul ℝ ℝ≥0 ?m`
_ ≤ C * a ^ (n + 2) / ↑r' ^ (n + 2)
* ↑r' ^ n * (↑(n + 2) * ‖y - (x, x)‖ * ‖y.1 - y.2‖) := by
have : 0 < a := ha.1
gcongr
· apply hp
· apply hy'.le
_ = B n := by
-- porting note: in the original, `B` was in the `field_simp`, but now Lean does not
-- accept it. The current proof works in Lean 4, but does not in Lean 3.
field_simp [pow_succ]
simp only [mul_assoc, mul_comm, mul_left_comm]
have hBL : HasSum B (L y) := by
apply HasSum.mul_left
simp only [add_mul]
have : ‖a‖ < 1 := by simp only [Real.norm_eq_abs, abs_of_pos ha.1, ha.2]
rw [div_eq_mul_inv, div_eq_mul_inv]
exact (hasSum_coe_mul_geometric_of_norm_lt_1 this).add -- porting note: was `convert`!
((hasSum_geometric_of_norm_lt_1 this).mul_left 2)
exact hA.norm_le_of_bounded hBL hAB
suffices L =O[𝓟 (EMetric.ball (x, x) r')] fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖ by
refine' (IsBigO.of_bound 1 (eventually_principal.2 fun y hy => _)).trans this
rw [one_mul]
exact (hL y hy).trans (le_abs_self _)
simp_rw [mul_right_comm _ (_ * _)] -- porting note: there was an `L` inside the `simp_rw`.
exact (isBigO_refl _ _).const_mul_left _
set_option linter.uppercaseLean3 false in
#align has_fpower_series_on_ball.is_O_image_sub_image_sub_deriv_principal HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal
/-- If `f` has formal power series `∑ n, pₙ` on a ball of radius `r`, then for `y, z` in any smaller
ball, the norm of the difference `f y - f z - p 1 (fun _ ↦ y - z)` is bounded above by
`C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖`. -/
theorem HasFPowerSeriesOnBall.image_sub_sub_deriv_le (hf : HasFPowerSeriesOnBall f p x r)
(hr : r' < r) :
∃ C, ∀ᵉ (y ∈ EMetric.ball x r') (z ∈ EMetric.ball x r'),
‖f y - f z - p 1 fun _ => y - z‖ ≤ C * max ‖y - x‖ ‖z - x‖ * ‖y - z‖ := by
simpa only [isBigO_principal, mul_assoc, norm_mul, norm_norm, Prod.forall, EMetric.mem_ball,
Prod.edist_eq, max_lt_iff, and_imp, @forall_swap (_ < _) E] using
hf.isBigO_image_sub_image_sub_deriv_principal hr
#align has_fpower_series_on_ball.image_sub_sub_deriv_le HasFPowerSeriesOnBall.image_sub_sub_deriv_le
/-- If `f` has formal power series `∑ n, pₙ` at `x`, then
`f y - f z - p 1 (fun _ ↦ y - z) = O(‖(y, z) - (x, x)‖ * ‖y - z‖)` as `(y, z) → (x, x)`.
In particular, `f` is strictly differentiable at `x`. -/
theorem HasFPowerSeriesAt.isBigO_image_sub_norm_mul_norm_sub (hf : HasFPowerSeriesAt f p x) :
(fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) =O[𝓝 (x, x)] fun y =>
‖y - (x, x)‖ * ‖y.1 - y.2‖ := by
rcases hf with ⟨r, hf⟩
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hf.r_pos with ⟨r', r'0, h⟩
refine' (hf.isBigO_image_sub_image_sub_deriv_principal h).mono _
exact le_principal_iff.2 (EMetric.ball_mem_nhds _ r'0)
set_option linter.uppercaseLean3 false in
#align has_fpower_series_at.is_O_image_sub_norm_mul_norm_sub HasFPowerSeriesAt.isBigO_image_sub_norm_mul_norm_sub
/-- If a function admits a power series expansion at `x`, then it is the uniform limit of the
partial sums of this power series on strict subdisks of the disk of convergence, i.e., `f (x + y)`
is the uniform limit of `p.partialSum n y` there. -/
theorem HasFPowerSeriesOnBall.tendstoUniformlyOn {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r)
(h : (r' : ℝ≥0∞) < r) :
TendstoUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
(Metric.ball (0 : E) r') := by
obtain ⟨a, ha, C, -, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n
exact hf.uniform_geometric_approx h
refine' Metric.tendstoUniformlyOn_iff.2 fun ε εpos => _
have L : Tendsto (fun n => (C : ℝ) * a ^ n) atTop (𝓝 ((C : ℝ) * 0)) :=
tendsto_const_nhds.mul (tendsto_pow_atTop_nhds_0_of_lt_1 ha.1.le ha.2)
rw [mul_zero] at L
refine' (L.eventually (gt_mem_nhds εpos)).mono fun n hn y hy => _
rw [dist_eq_norm]
exact (hp y hy n).trans_lt hn
#align has_fpower_series_on_ball.tendsto_uniformly_on HasFPowerSeriesOnBall.tendstoUniformlyOn
/-- If a function admits a power series expansion at `x`, then it is the locally uniform limit of
the partial sums of this power series on the disk of convergence, i.e., `f (x + y)`
is the locally uniform limit of `p.partialSum n y` there. -/
theorem HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn (hf : HasFPowerSeriesOnBall f p x r) :
TendstoLocallyUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
(EMetric.ball (0 : E) r) := by
intro u hu x hx
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hx with ⟨r', xr', hr'⟩
have : EMetric.ball (0 : E) r' ∈ 𝓝 x := IsOpen.mem_nhds EMetric.isOpen_ball xr'
refine' ⟨EMetric.ball (0 : E) r', mem_nhdsWithin_of_mem_nhds this, _⟩
simpa [Metric.emetric_ball_nnreal] using hf.tendstoUniformlyOn hr' u hu
#align has_fpower_series_on_ball.tendsto_locally_uniformly_on HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn
/-- If a function admits a power series expansion at `x`, then it is the uniform limit of the
partial sums of this power series on strict subdisks of the disk of convergence, i.e., `f y`
is the uniform limit of `p.partialSum n (y - x)` there. -/
theorem HasFPowerSeriesOnBall.tendstoUniformlyOn' {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r)
(h : (r' : ℝ≥0∞) < r) :
TendstoUniformlyOn (fun n y => p.partialSum n (y - x)) f atTop (Metric.ball (x : E) r') := by
convert (hf.tendstoUniformlyOn h).comp fun y => y - x using 1
· simp [(· ∘ ·)]
· ext z
simp [dist_eq_norm]
#align has_fpower_series_on_ball.tendsto_uniformly_on' HasFPowerSeriesOnBall.tendstoUniformlyOn'
/-- If a function admits a power series expansion at `x`, then it is the locally uniform limit of
the partial sums of this power series on the disk of convergence, i.e., `f y`
is the locally uniform limit of `p.partialSum n (y - x)` there. -/
theorem HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn' (hf : HasFPowerSeriesOnBall f p x r) :
TendstoLocallyUniformlyOn (fun n y => p.partialSum n (y - x)) f atTop
(EMetric.ball (x : E) r) := by
have A : ContinuousOn (fun y : E => y - x) (EMetric.ball (x : E) r) :=
(continuous_id.sub continuous_const).continuousOn
convert hf.tendstoLocallyUniformlyOn.comp (fun y : E => y - x) _ A using 1
· ext z
simp
· intro z
simp [edist_eq_coe_nnnorm, edist_eq_coe_nnnorm_sub]
#align has_fpower_series_on_ball.tendsto_locally_uniformly_on' HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn'
/-- If a function admits a power series expansion on a disk, then it is continuous there. -/
protected theorem HasFPowerSeriesOnBall.continuousOn (hf : HasFPowerSeriesOnBall f p x r) :
ContinuousOn f (EMetric.ball x r) :=
hf.tendstoLocallyUniformlyOn'.continuousOn <|
eventually_of_forall fun n =>
((p.partialSum_continuous n).comp (continuous_id.sub continuous_const)).continuousOn
#align has_fpower_series_on_ball.continuous_on HasFPowerSeriesOnBall.continuousOn
protected theorem HasFPowerSeriesAt.continuousAt (hf : HasFPowerSeriesAt f p x) :
ContinuousAt f x :=
let ⟨_, hr⟩ := hf
hr.continuousOn.continuousAt (EMetric.ball_mem_nhds x hr.r_pos)
#align has_fpower_series_at.continuous_at HasFPowerSeriesAt.continuousAt
protected theorem AnalyticAt.continuousAt (hf : AnalyticAt 𝕜 f x) : ContinuousAt f x :=
let ⟨_, hp⟩ := hf
hp.continuousAt
#align analytic_at.continuous_at AnalyticAt.continuousAt
protected theorem AnalyticOn.continuousOn {s : Set E} (hf : AnalyticOn 𝕜 f s) : ContinuousOn f s :=
fun x hx => (hf x hx).continuousAt.continuousWithinAt
#align analytic_on.continuous_on AnalyticOn.continuousOn
/-- Analytic everywhere implies continuous -/
theorem AnalyticOn.continuous {f : E → F} (fa : AnalyticOn 𝕜 f univ) : Continuous f := by
rw [continuous_iff_continuousOn_univ]; exact fa.continuousOn
/-- In a complete space, the sum of a converging power series `p` admits `p` as a power series.
This is not totally obvious as we need to check the convergence of the series. -/
protected theorem FormalMultilinearSeries.hasFPowerSeriesOnBall [CompleteSpace F]
(p : FormalMultilinearSeries 𝕜 E F) (h : 0 < p.radius) :
HasFPowerSeriesOnBall p.sum p 0 p.radius :=
{ r_le := le_rfl
r_pos := h
hasSum := fun hy => by
rw [zero_add]
exact p.hasSum hy }
#align formal_multilinear_series.has_fpower_series_on_ball FormalMultilinearSeries.hasFPowerSeriesOnBall
theorem HasFPowerSeriesOnBall.sum (h : HasFPowerSeriesOnBall f p x r) {y : E}
(hy : y ∈ EMetric.ball (0 : E) r) : f (x + y) = p.sum y :=
(h.hasSum hy).tsum_eq.symm
#align has_fpower_series_on_ball.sum HasFPowerSeriesOnBall.sum
/-- The sum of a converging power series is continuous in its disk of convergence. -/
protected theorem FormalMultilinearSeries.continuousOn [CompleteSpace F] :
ContinuousOn p.sum (EMetric.ball 0 p.radius) := by
rcases (zero_le p.radius).eq_or_lt with h | h
· simp [← h, continuousOn_empty]
· exact (p.hasFPowerSeriesOnBall h).continuousOn
#align formal_multilinear_series.continuous_on FormalMultilinearSeries.continuousOn
end
/-!
### Uniqueness of power series
If a function `f : E → F` has two representations as power series at a point `x : E`, corresponding
to formal multilinear series `p₁` and `p₂`, then these representations agree term-by-term. That is,
for any `n : ℕ` and `y : E`, `p₁ n (fun i ↦ y) = p₂ n (fun i ↦ y)`. In the one-dimensional case,
when `f : 𝕜 → E`, the continuous multilinear maps `p₁ n` and `p₂ n` are given by
`ContinuousMultilinearMap.mkPiField`, and hence are determined completely by the value of
`p₁ n (fun i ↦ 1)`, so `p₁ = p₂`. Consequently, the radius of convergence for one series can be
transferred to the other.
-/
section Uniqueness
open ContinuousMultilinearMap
theorem Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero {n : ℕ} {p : E[×n]→L[𝕜] F}
(h : (fun y => p fun _ => y) =O[𝓝 0] fun y => ‖y‖ ^ (n + 1)) (y : E) : (p fun _ => y) = 0 := by
obtain ⟨c, c_pos, hc⟩ := h.exists_pos
obtain ⟨t, ht, t_open, z_mem⟩ := eventually_nhds_iff.mp (isBigOWith_iff.mp hc)
obtain ⟨δ, δ_pos, δε⟩ := (Metric.isOpen_iff.mp t_open) 0 z_mem
clear h hc z_mem
cases' n with n
· exact norm_eq_zero.mp (by
-- porting note: the symmetric difference of the `simpa only` sets:
-- added `Nat.zero_eq, zero_add, pow_one`
-- removed `zero_pow', Ne.def, Nat.one_ne_zero, not_false_iff`
simpa only [Nat.zero_eq, fin0_apply_norm, norm_eq_zero, norm_zero, zero_add, pow_one,
mul_zero, norm_le_zero_iff] using ht 0 (δε (Metric.mem_ball_self δ_pos)))
· refine' Or.elim (Classical.em (y = 0))
(fun hy => by simpa only [hy] using p.map_zero) fun hy => _
replace hy := norm_pos_iff.mpr hy
refine' norm_eq_zero.mp (le_antisymm (le_of_forall_pos_le_add fun ε ε_pos => _) (norm_nonneg _))
have h₀ := _root_.mul_pos c_pos (pow_pos hy (n.succ + 1))
obtain ⟨k, k_pos, k_norm⟩ := NormedField.exists_norm_lt 𝕜
(lt_min (mul_pos δ_pos (inv_pos.mpr hy)) (mul_pos ε_pos (inv_pos.mpr h₀)))
have h₁ : ‖k • y‖ < δ := by
rw [norm_smul]
exact inv_mul_cancel_right₀ hy.ne.symm δ ▸
mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_left _ _)) hy
have h₂ :=
calc
‖p fun _ => k • y‖ ≤ c * ‖k • y‖ ^ (n.succ + 1) := by
-- porting note: now Lean wants `_root_.`
simpa only [norm_pow, _root_.norm_norm] using ht (k • y) (δε (mem_ball_zero_iff.mpr h₁))
--simpa only [norm_pow, norm_norm] using ht (k • y) (δε (mem_ball_zero_iff.mpr h₁))
_ = ‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))) := by
-- porting note: added `Nat.succ_eq_add_one` since otherwise `ring` does not conclude.
simp only [norm_smul, mul_pow, Nat.succ_eq_add_one]
-- porting note: removed `rw [pow_succ]`, since it now becomes superfluous.
ring
have h₃ : ‖k‖ * (c * ‖y‖ ^ (n.succ + 1)) < ε :=
inv_mul_cancel_right₀ h₀.ne.symm ε ▸
mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_right _ _)) h₀
calc
‖p fun _ => y‖ = ‖k⁻¹ ^ n.succ‖ * ‖p fun _ => k • y‖ := by
simpa only [inv_smul_smul₀ (norm_pos_iff.mp k_pos), norm_smul, Finset.prod_const,
Finset.card_fin] using
congr_arg norm (p.map_smul_univ (fun _ : Fin n.succ => k⁻¹) fun _ : Fin n.succ => k • y)
_ ≤ ‖k⁻¹ ^ n.succ‖ * (‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1)))) := by gcongr
_ = ‖(k⁻¹ * k) ^ n.succ‖ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))) := by
rw [← mul_assoc]
simp [norm_mul, mul_pow]
_ ≤ 0 + ε := by
rw [inv_mul_cancel (norm_pos_iff.mp k_pos)]
simpa using h₃.le
set_option linter.uppercaseLean3 false in
#align asymptotics.is_O.continuous_multilinear_map_apply_eq_zero Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero
/-- If a formal multilinear series `p` represents the zero function at `x : E`, then the
terms `p n (fun i ↦ y)` appearing in the sum are zero for any `n : ℕ`, `y : E`. -/
theorem HasFPowerSeriesAt.apply_eq_zero {p : FormalMultilinearSeries 𝕜 E F} {x : E}
(h : HasFPowerSeriesAt 0 p x) (n : ℕ) : ∀ y : E, (p n fun _ => y) = 0 := by
refine' Nat.strong_induction_on n fun k hk => _
have psum_eq : p.partialSum (k + 1) = fun y => p k fun _ => y := by
funext z
refine' Finset.sum_eq_single _ (fun b hb hnb => _) fun hn => _
· have := Finset.mem_range_succ_iff.mp hb
simp only [hk b (this.lt_of_ne hnb), Pi.zero_apply]
· exact False.elim (hn (Finset.mem_range.mpr (lt_add_one k)))
replace h := h.isBigO_sub_partialSum_pow k.succ
simp only [psum_eq, zero_sub, Pi.zero_apply, Asymptotics.isBigO_neg_left] at h
exact h.continuousMultilinearMap_apply_eq_zero
#align has_fpower_series_at.apply_eq_zero HasFPowerSeriesAt.apply_eq_zero
/-- A one-dimensional formal multilinear series representing the zero function is zero. -/
theorem HasFPowerSeriesAt.eq_zero {p : FormalMultilinearSeries 𝕜 𝕜 E} {x : 𝕜}
(h : HasFPowerSeriesAt 0 p x) : p = 0 := by
-- porting note: `funext; ext` was `ext (n x)`
funext n
ext x
rw [← mkPiField_apply_one_eq_self (p n)]
-- porting note: nasty hack, was `simp [h.apply_eq_zero n 1]`
have := Or.intro_right ?_ (h.apply_eq_zero n 1)
simpa using this
#align has_fpower_series_at.eq_zero HasFPowerSeriesAt.eq_zero
/-- One-dimensional formal multilinear series representing the same function are equal. -/
theorem HasFPowerSeriesAt.eq_formalMultilinearSeries {p₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E}
{f : 𝕜 → E} {x : 𝕜} (h₁ : HasFPowerSeriesAt f p₁ x) (h₂ : HasFPowerSeriesAt f p₂ x) : p₁ = p₂ :=
sub_eq_zero.mp (HasFPowerSeriesAt.eq_zero (by simpa only [sub_self] using h₁.sub h₂))
#align has_fpower_series_at.eq_formal_multilinear_series HasFPowerSeriesAt.eq_formalMultilinearSeries
theorem HasFPowerSeriesAt.eq_formalMultilinearSeries_of_eventually
{p q : FormalMultilinearSeries 𝕜 𝕜 E} {f g : 𝕜 → E} {x : 𝕜} (hp : HasFPowerSeriesAt f p x)
(hq : HasFPowerSeriesAt g q x) (heq : ∀ᶠ z in 𝓝 x, f z = g z) : p = q :=
(hp.congr heq).eq_formalMultilinearSeries hq
#align has_fpower_series_at.eq_formal_multilinear_series_of_eventually HasFPowerSeriesAt.eq_formalMultilinearSeries_of_eventually
/-- A one-dimensional formal multilinear series representing a locally zero function is zero. -/
theorem HasFPowerSeriesAt.eq_zero_of_eventually {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E}
{x : 𝕜} (hp : HasFPowerSeriesAt f p x) (hf : f =ᶠ[𝓝 x] 0) : p = 0 :=
(hp.congr hf).eq_zero
#align has_fpower_series_at.eq_zero_of_eventually HasFPowerSeriesAt.eq_zero_of_eventually
/-- If a function `f : 𝕜 → E` has two power series representations at `x`, then the given radii in
which convergence is guaranteed may be interchanged. This can be useful when the formal multilinear
series in one representation has a particularly nice form, but the other has a larger radius. -/
theorem HasFPowerSeriesOnBall.exchange_radius {p₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E}
{r₁ r₂ : ℝ≥0∞} {x : 𝕜} (h₁ : HasFPowerSeriesOnBall f p₁ x r₁)
(h₂ : HasFPowerSeriesOnBall f p₂ x r₂) : HasFPowerSeriesOnBall f p₁ x r₂ :=
h₂.hasFPowerSeriesAt.eq_formalMultilinearSeries h₁.hasFPowerSeriesAt ▸ h₂
#align has_fpower_series_on_ball.exchange_radius HasFPowerSeriesOnBall.exchange_radius
/-- If a function `f : 𝕜 → E` has power series representation `p` on a ball of some radius and for
each positive radius it has some power series representation, then `p` converges to `f` on the whole
`𝕜`. -/
theorem HasFPowerSeriesOnBall.r_eq_top_of_exists {f : 𝕜 → E} {r : ℝ≥0∞} {x : 𝕜}
{p : FormalMultilinearSeries 𝕜 𝕜 E} (h : HasFPowerSeriesOnBall f p x r)
(h' : ∀ (r' : ℝ≥0) (_ : 0 < r'), ∃ p' : FormalMultilinearSeries 𝕜 𝕜 E,
HasFPowerSeriesOnBall f p' x r') :
HasFPowerSeriesOnBall f p x ∞ :=
{ r_le := ENNReal.le_of_forall_pos_nnreal_lt fun r hr _ =>
let ⟨_, hp'⟩ := h' r hr
(h.exchange_radius hp').r_le
r_pos := ENNReal.coe_lt_top
hasSum := fun {y} _ =>
let ⟨r', hr'⟩ := exists_gt ‖y‖₊
let ⟨_, hp'⟩ := h' r' hr'.ne_bot.bot_lt
(h.exchange_radius hp').hasSum <| mem_emetric_ball_zero_iff.mpr (ENNReal.coe_lt_coe.2 hr') }
#align has_fpower_series_on_ball.r_eq_top_of_exists HasFPowerSeriesOnBall.r_eq_top_of_exists
end Uniqueness
/-!
### Changing origin in a power series
If a function is analytic in a disk `D(x, R)`, then it is analytic in any disk contained in that
one. Indeed, one can write
$$
f (x + y + z) = \sum_{n} p_n (y + z)^n = \sum_{n, k} \binom{n}{k} p_n y^{n-k} z^k
= \sum_{k} \Bigl(\sum_{n} \binom{n}{k} p_n y^{n-k}\Bigr) z^k.
$$
The corresponding power series has thus a `k`-th coefficient equal to
$\sum_{n} \binom{n}{k} p_n y^{n-k}$. In the general case where `pₙ` is a multilinear map, this has
to be interpreted suitably: instead of having a binomial coefficient, one should sum over all
possible subsets `s` of `Fin n` of cardinal `k`, and attribute `z` to the indices in `s` and
`y` to the indices outside of `s`.
In this paragraph, we implement this. The new power series is called `p.changeOrigin y`. Then, we
check its convergence and the fact that its sum coincides with the original sum. The outcome of this
discussion is that the set of points where a function is analytic is open.
-/
namespace FormalMultilinearSeries
section
variable (p : FormalMultilinearSeries 𝕜 E F) {x y : E} {r R : ℝ≥0}
/-- A term of `FormalMultilinearSeries.changeOriginSeries`.
Given a formal multilinear series `p` and a point `x` in its ball of convergence,
`p.changeOrigin x` is a formal multilinear series such that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense. Each term of `p.changeOrigin x`
is itself an analytic function of `x` given by the series `p.changeOriginSeries`. Each term in
`changeOriginSeries` is the sum of `changeOriginSeriesTerm`'s over all `s` of cardinality `l`.
The definition is such that `p.changeOriginSeriesTerm k l s hs (fun _ ↦ x) (fun _ ↦ y) =
p (k + l) (s.piecewise (fun _ ↦ x) (fun _ ↦ y))`
-/
def changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
E[×l]→L[𝕜] E[×k]→L[𝕜] F := by
let a := ContinuousMultilinearMap.curryFinFinset 𝕜 E F hs
(by erw [Finset.card_compl, Fintype.card_fin, hs, add_tsub_cancel_right])
exact a (p (k + l))
#align formal_multilinear_series.change_origin_series_term FormalMultilinearSeries.changeOriginSeriesTerm
theorem changeOriginSeriesTerm_apply (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l)
(x y : E) :
(p.changeOriginSeriesTerm k l s hs (fun _ => x) fun _ => y) =
p (k + l) (s.piecewise (fun _ => x) fun _ => y) :=
ContinuousMultilinearMap.curryFinFinset_apply_const _ _ _ _ _
#align formal_multilinear_series.change_origin_series_term_apply FormalMultilinearSeries.changeOriginSeriesTerm_apply
@[simp]
theorem norm_changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
‖p.changeOriginSeriesTerm k l s hs‖ = ‖p (k + l)‖ := by
simp only [changeOriginSeriesTerm, LinearIsometryEquiv.norm_map]
#align formal_multilinear_series.norm_change_origin_series_term FormalMultilinearSeries.norm_changeOriginSeriesTerm
@[simp]
theorem nnnorm_changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
‖p.changeOriginSeriesTerm k l s hs‖₊ = ‖p (k + l)‖₊ := by
simp only [changeOriginSeriesTerm, LinearIsometryEquiv.nnnorm_map]
#align formal_multilinear_series.nnnorm_change_origin_series_term FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm
theorem nnnorm_changeOriginSeriesTerm_apply_le (k l : ℕ) (s : Finset (Fin (k + l)))
(hs : s.card = l) (x y : E) :
‖p.changeOriginSeriesTerm k l s hs (fun _ => x) fun _ => y‖₊ ≤
‖p (k + l)‖₊ * ‖x‖₊ ^ l * ‖y‖₊ ^ k := by
rw [← p.nnnorm_changeOriginSeriesTerm k l s hs, ← Fin.prod_const, ← Fin.prod_const]
apply ContinuousMultilinearMap.le_of_op_nnnorm_le
apply ContinuousMultilinearMap.le_op_nnnorm
#align formal_multilinear_series.nnnorm_change_origin_series_term_apply_le FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm_apply_le
/-- The power series for `f.changeOrigin k`.
Given a formal multilinear series `p` and a point `x` in its ball of convergence,
`p.changeOrigin x` is a formal multilinear series such that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense. Its `k`-th term is the sum of
the series `p.changeOriginSeries k`. -/
def changeOriginSeries (k : ℕ) : FormalMultilinearSeries 𝕜 E (E[×k]→L[𝕜] F) := fun l =>
∑ s : { s : Finset (Fin (k + l)) // Finset.card s = l }, p.changeOriginSeriesTerm k l s s.2
#align formal_multilinear_series.change_origin_series FormalMultilinearSeries.changeOriginSeries
theorem nnnorm_changeOriginSeries_le_tsum (k l : ℕ) :
‖p.changeOriginSeries k l‖₊ ≤
∑' _ : { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + l)‖₊ :=
(nnnorm_sum_le _ (fun t => changeOriginSeriesTerm p k l (Subtype.val t) t.prop)).trans_eq <| by
simp_rw [tsum_fintype, nnnorm_changeOriginSeriesTerm (p := p) (k := k) (l := l)]
#align formal_multilinear_series.nnnorm_change_origin_series_le_tsum FormalMultilinearSeries.nnnorm_changeOriginSeries_le_tsum
theorem nnnorm_changeOriginSeries_apply_le_tsum (k l : ℕ) (x : E) :
‖p.changeOriginSeries k l fun _ => x‖₊ ≤
∑' _ : { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + l)‖₊ * ‖x‖₊ ^ l := by
rw [NNReal.tsum_mul_right, ← Fin.prod_const]
exact (p.changeOriginSeries k l).le_of_op_nnnorm_le _ (p.nnnorm_changeOriginSeries_le_tsum _ _)
#align formal_multilinear_series.nnnorm_change_origin_series_apply_le_tsum FormalMultilinearSeries.nnnorm_changeOriginSeries_apply_le_tsum
/-- Changing the origin of a formal multilinear series `p`, so that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense.
-/
def changeOrigin (x : E) : FormalMultilinearSeries 𝕜 E F :=
fun k => (p.changeOriginSeries k).sum x
#align formal_multilinear_series.change_origin FormalMultilinearSeries.changeOrigin
/-- An auxiliary equivalence useful in the proofs about
`FormalMultilinearSeries.changeOriginSeries`: the set of triples `(k, l, s)`, where `s` is a
`Finset (Fin (k + l))` of cardinality `l` is equivalent to the set of pairs `(n, s)`, where `s` is a
`Finset (Fin n)`.
The forward map sends `(k, l, s)` to `(k + l, s)` and the inverse map sends `(n, s)` to
`(n - Finset.card s, Finset.card s, s)`. The actual definition is less readable because of problems
with non-definitional equalities. -/
@[simps]
def changeOriginIndexEquiv :
(Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) ≃ Σn : ℕ, Finset (Fin n) where
toFun s := ⟨s.1 + s.2.1, s.2.2⟩
invFun s :=
⟨s.1 - s.2.card, s.2.card,
⟨s.2.map
(Fin.castIso <| (tsub_add_cancel_of_le <| card_finset_fin_le s.2).symm).toEquiv.toEmbedding,
Finset.card_map _⟩⟩
left_inv := by
rintro ⟨k, l, ⟨s : Finset (Fin <| k + l), hs : s.card = l⟩⟩
dsimp only [Subtype.coe_mk]
-- Lean can't automatically generalize `k' = k + l - s.card`, `l' = s.card`, so we explicitly
-- formulate the generalized goal
suffices ∀ k' l', k' = k → l' = l → ∀ (hkl : k + l = k' + l') (hs'),
(⟨k', l', ⟨Finset.map (Fin.castIso hkl).toEquiv.toEmbedding s, hs'⟩⟩ :
Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) = ⟨k, l, ⟨s, hs⟩⟩ by
apply this <;> simp only [hs, add_tsub_cancel_right]
rintro _ _ rfl rfl hkl hs'
simp only [Equiv.refl_toEmbedding, Fin.castIso_refl, Finset.map_refl, eq_self_iff_true,
OrderIso.refl_toEquiv, and_self_iff, heq_iff_eq]
right_inv := by
rintro ⟨n, s⟩
simp [tsub_add_cancel_of_le (card_finset_fin_le s), Fin.castIso_to_equiv]
#align formal_multilinear_series.change_origin_index_equiv FormalMultilinearSeries.changeOriginIndexEquiv
theorem changeOriginSeries_summable_aux₁ {r r' : ℝ≥0} (hr : (r + r' : ℝ≥0∞) < p.radius) :
Summable fun s : Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l } =>
‖p (s.1 + s.2.1)‖₊ * r ^ s.2.1 * r' ^ s.1 := by
rw [← changeOriginIndexEquiv.symm.summable_iff]
dsimp only [Function.comp_def, changeOriginIndexEquiv_symm_apply_fst,
changeOriginIndexEquiv_symm_apply_snd_fst]
have : ∀ n : ℕ,
HasSum (fun s : Finset (Fin n) => ‖p (n - s.card + s.card)‖₊ * r ^ s.card * r' ^ (n - s.card))
(‖p n‖₊ * (r + r') ^ n) := by
intro n
-- TODO: why `simp only [tsub_add_cancel_of_le (card_finset_fin_le _)]` fails?
convert_to HasSum (fun s : Finset (Fin n) => ‖p n‖₊ * (r ^ s.card * r' ^ (n - s.card))) _
· ext1 s
rw [tsub_add_cancel_of_le (card_finset_fin_le _), mul_assoc]
rw [← Fin.sum_pow_mul_eq_add_pow]
exact (hasSum_fintype _).mul_left _
refine' NNReal.summable_sigma.2 ⟨fun n => (this n).summable, _⟩
simp only [(this _).tsum_eq]
exact p.summable_nnnorm_mul_pow hr
#align formal_multilinear_series.change_origin_series_summable_aux₁ FormalMultilinearSeries.changeOriginSeries_summable_aux₁
theorem changeOriginSeries_summable_aux₂ (hr : (r : ℝ≥0∞) < p.radius) (k : ℕ) :
Summable fun s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l } =>
‖p (k + s.1)‖₊ * r ^ s.1 := by
rcases ENNReal.lt_iff_exists_add_pos_lt.1 hr with ⟨r', h0, hr'⟩
simpa only [mul_inv_cancel_right₀ (pow_pos h0 _).ne'] using
((NNReal.summable_sigma.1 (p.changeOriginSeries_summable_aux₁ hr')).1 k).mul_right (r' ^ k)⁻¹
#align formal_multilinear_series.change_origin_series_summable_aux₂ FormalMultilinearSeries.changeOriginSeries_summable_aux₂
theorem changeOriginSeries_summable_aux₃ {r : ℝ≥0} (hr : ↑r < p.radius) (k : ℕ) :
Summable fun l : ℕ => ‖p.changeOriginSeries k l‖₊ * r ^ l := by
refine' NNReal.summable_of_le
(fun n => _) (NNReal.summable_sigma.1 <| p.changeOriginSeries_summable_aux₂ hr k).2
simp only [NNReal.tsum_mul_right]
exact mul_le_mul' (p.nnnorm_changeOriginSeries_le_tsum _ _) le_rfl
#align formal_multilinear_series.change_origin_series_summable_aux₃ FormalMultilinearSeries.changeOriginSeries_summable_aux₃
theorem le_changeOriginSeries_radius (k : ℕ) : p.radius ≤ (p.changeOriginSeries k).radius :=
ENNReal.le_of_forall_nnreal_lt fun _r hr =>
le_radius_of_summable_nnnorm _ (p.changeOriginSeries_summable_aux₃ hr k)
#align formal_multilinear_series.le_change_origin_series_radius FormalMultilinearSeries.le_changeOriginSeries_radius
theorem nnnorm_changeOrigin_le (k : ℕ) (h : (‖x‖₊ : ℝ≥0∞) < p.radius) :
‖p.changeOrigin x k‖₊ ≤
∑' s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + s.1)‖₊ * ‖x‖₊ ^ s.1 := by
refine' tsum_of_nnnorm_bounded _ fun l => p.nnnorm_changeOriginSeries_apply_le_tsum k l x
have := p.changeOriginSeries_summable_aux₂ h k
refine' HasSum.sigma this.hasSum fun l => _
exact ((NNReal.summable_sigma.1 this).1 l).hasSum
#align formal_multilinear_series.nnnorm_change_origin_le FormalMultilinearSeries.nnnorm_changeOrigin_le
/-- The radius of convergence of `p.changeOrigin x` is at least `p.radius - ‖x‖`. In other words,
`p.changeOrigin x` is well defined on the largest ball contained in the original ball of
convergence. -/
theorem changeOrigin_radius : p.radius - ‖x‖₊ ≤ (p.changeOrigin x).radius := by
refine' ENNReal.le_of_forall_pos_nnreal_lt fun r _h0 hr => _
rw [lt_tsub_iff_right, add_comm] at hr
have hr' : (‖x‖₊ : ℝ≥0∞) < p.radius := (le_add_right le_rfl).trans_lt hr
apply le_radius_of_summable_nnnorm
have : ∀ k : ℕ,
‖p.changeOrigin x k‖₊ * r ^ k ≤
(∑' s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + s.1)‖₊ * ‖x‖₊ ^ s.1) *
r ^ k :=
fun k => mul_le_mul_right' (p.nnnorm_changeOrigin_le k hr') (r ^ k)
refine' NNReal.summable_of_le this _
simpa only [← NNReal.tsum_mul_right] using
(NNReal.summable_sigma.1 (p.changeOriginSeries_summable_aux₁ hr)).2
#align formal_multilinear_series.change_origin_radius FormalMultilinearSeries.changeOrigin_radius
end
-- From this point on, assume that the space is complete, to make sure that series that converge
-- in norm also converge in `F`.
variable [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x y : E} {r R : ℝ≥0}
theorem hasFPowerSeriesOnBall_changeOrigin (k : ℕ) (hr : 0 < p.radius) :
HasFPowerSeriesOnBall (fun x => p.changeOrigin x k) (p.changeOriginSeries k) 0 p.radius :=
have := p.le_changeOriginSeries_radius k
((p.changeOriginSeries k).hasFPowerSeriesOnBall (hr.trans_le this)).mono hr this
#align formal_multilinear_series.has_fpower_series_on_ball_change_origin FormalMultilinearSeries.hasFPowerSeriesOnBall_changeOrigin
/-- Summing the series `p.changeOrigin x` at a point `y` gives back `p (x + y)`. -/
theorem changeOrigin_eval (h : (‖x‖₊ + ‖y‖₊ : ℝ≥0∞) < p.radius) :
(p.changeOrigin x).sum y = p.sum (x + y) := by
have radius_pos : 0 < p.radius := lt_of_le_of_lt (zero_le _) h
have x_mem_ball : x ∈ EMetric.ball (0 : E) p.radius :=
mem_emetric_ball_zero_iff.2 ((le_add_right le_rfl).trans_lt h)
have y_mem_ball : y ∈ EMetric.ball (0 : E) (p.changeOrigin x).radius := by
refine' mem_emetric_ball_zero_iff.2 (lt_of_lt_of_le _ p.changeOrigin_radius)
rwa [lt_tsub_iff_right, add_comm]
have x_add_y_mem_ball : x + y ∈ EMetric.ball (0 : E) p.radius := by
refine' mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt _ h)
exact mod_cast nnnorm_add_le x y
set f : (Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) → F := fun s =>
p.changeOriginSeriesTerm s.1 s.2.1 s.2.2 s.2.2.2 (fun _ => x) fun _ => y
have hsf : Summable f := by
refine' .of_nnnorm_bounded _ (p.changeOriginSeries_summable_aux₁ h) _
rintro ⟨k, l, s, hs⟩
dsimp only [Subtype.coe_mk]
exact p.nnnorm_changeOriginSeriesTerm_apply_le _ _ _ _ _ _
have hf : HasSum f ((p.changeOrigin x).sum y) := by
refine' HasSum.sigma_of_hasSum ((p.changeOrigin x).summable y_mem_ball).hasSum (fun k => _) hsf
· dsimp only
refine' ContinuousMultilinearMap.hasSum_eval _ _
have := (p.hasFPowerSeriesOnBall_changeOrigin k radius_pos).hasSum x_mem_ball
rw [zero_add] at this
refine' HasSum.sigma_of_hasSum this (fun l => _) _
· simp only [changeOriginSeries, ContinuousMultilinearMap.sum_apply]
apply hasSum_fintype
· refine' .of_nnnorm_bounded _
(p.changeOriginSeries_summable_aux₂ (mem_emetric_ball_zero_iff.1 x_mem_ball) k) fun s => _
refine' (ContinuousMultilinearMap.le_op_nnnorm _ _).trans_eq _
simp
refine' hf.unique (changeOriginIndexEquiv.symm.hasSum_iff.1 _)
refine' HasSum.sigma_of_hasSum
(p.hasSum x_add_y_mem_ball) (fun n => _) (changeOriginIndexEquiv.symm.summable_iff.2 hsf)
erw [(p n).map_add_univ (fun _ => x) fun _ => y]
-- porting note: added explicit function
convert hasSum_fintype (fun c : Finset (Fin n) => f (changeOriginIndexEquiv.symm ⟨n, c⟩))
rename_i s _
dsimp only [changeOriginSeriesTerm, (· ∘ ·), changeOriginIndexEquiv_symm_apply_fst,
changeOriginIndexEquiv_symm_apply_snd_fst, changeOriginIndexEquiv_symm_apply_snd_snd_coe]
rw [ContinuousMultilinearMap.curryFinFinset_apply_const]
have : ∀ (m) (hm : n = m), p n (s.piecewise (fun _ => x) fun _ => y) =
p m ((s.map (Fin.castIso hm).toEquiv.toEmbedding).piecewise (fun _ => x) fun _ => y) := by
rintro m rfl
simp (config := { unfoldPartialApp := true }) [Finset.piecewise]
apply this
#align formal_multilinear_series.change_origin_eval FormalMultilinearSeries.changeOrigin_eval
/-- Power series terms are analytic as we vary the origin -/
theorem analyticAt_changeOrigin (p : FormalMultilinearSeries 𝕜 E F) (rp : p.radius > 0) (n : ℕ) :
AnalyticAt 𝕜 (fun x ↦ p.changeOrigin x n) 0 :=
(FormalMultilinearSeries.hasFPowerSeriesOnBall_changeOrigin p n rp).analyticAt
end FormalMultilinearSeries
section
variable [CompleteSpace F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x y : E} {r : ℝ≥0∞}
/-- If a function admits a power series expansion `p` on a ball `B (x, r)`, then it also admits a
power series on any subball of this ball (even with a different center), given by `p.changeOrigin`.
-/
theorem HasFPowerSeriesOnBall.changeOrigin (hf : HasFPowerSeriesOnBall f p x r)
(h : (‖y‖₊ : ℝ≥0∞) < r) : HasFPowerSeriesOnBall f (p.changeOrigin y) (x + y) (r - ‖y‖₊) :=
{ r_le := by
apply le_trans _ p.changeOrigin_radius
exact tsub_le_tsub hf.r_le le_rfl
r_pos := by simp [h]
hasSum := fun {z} hz => by
have : f (x + y + z) =
FormalMultilinearSeries.sum (FormalMultilinearSeries.changeOrigin p y) z := by
rw [mem_emetric_ball_zero_iff, lt_tsub_iff_right, add_comm] at hz
rw [p.changeOrigin_eval (hz.trans_le hf.r_le), add_assoc, hf.sum]
refine' mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt _ hz)
exact mod_cast nnnorm_add_le y z
rw [this]
apply (p.changeOrigin y).hasSum
refine' EMetric.ball_subset_ball (le_trans _ p.changeOrigin_radius) hz
exact tsub_le_tsub hf.r_le le_rfl }
#align has_fpower_series_on_ball.change_origin HasFPowerSeriesOnBall.changeOrigin
/-- If a function admits a power series expansion `p` on an open ball `B (x, r)`, then
it is analytic at every point of this ball. -/
theorem HasFPowerSeriesOnBall.analyticAt_of_mem (hf : HasFPowerSeriesOnBall f p x r)
(h : y ∈ EMetric.ball x r) : AnalyticAt 𝕜 f y := by
have : (‖y - x‖₊ : ℝ≥0∞) < r := by simpa [edist_eq_coe_nnnorm_sub] using h
have := hf.changeOrigin this
rw [add_sub_cancel'_right] at this
exact this.analyticAt
#align has_fpower_series_on_ball.analytic_at_of_mem HasFPowerSeriesOnBall.analyticAt_of_mem
theorem HasFPowerSeriesOnBall.analyticOn (hf : HasFPowerSeriesOnBall f p x r) :
AnalyticOn 𝕜 f (EMetric.ball x r) :=
fun _y hy => hf.analyticAt_of_mem hy
#align has_fpower_series_on_ball.analytic_on HasFPowerSeriesOnBall.analyticOn
variable (𝕜 f)
/-- For any function `f` from a normed vector space to a Banach space, the set of points `x` such
that `f` is analytic at `x` is open. -/
theorem isOpen_analyticAt : IsOpen { x | AnalyticAt 𝕜 f x } := by
rw [isOpen_iff_mem_nhds]
rintro x ⟨p, r, hr⟩
exact mem_of_superset (EMetric.ball_mem_nhds _ hr.r_pos) fun y hy => hr.analyticAt_of_mem hy
#align is_open_analytic_at isOpen_analyticAt
variable {𝕜}
theorem AnalyticAt.eventually_analyticAt {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∀ᶠ y in 𝓝 x, AnalyticAt 𝕜 f y :=
(isOpen_analyticAt 𝕜 f).mem_nhds h
theorem AnalyticAt.exists_mem_nhds_analyticOn {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∃ s ∈ 𝓝 x, AnalyticOn 𝕜 f s :=
h.eventually_analyticAt.exists_mem
/-- If we're analytic at a point, we're analytic in a nonempty ball -/
theorem AnalyticAt.exists_ball_analyticOn {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∃ r : ℝ, 0 < r ∧ AnalyticOn 𝕜 f (Metric.ball x r) :=
Metric.isOpen_iff.mp (isOpen_analyticAt _ _) _ h
end
section
open FormalMultilinearSeries
variable {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E} {z₀ : 𝕜}
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z)) := by
|
refine' ⟨fun ⟨r, _, r_pos, h⟩ =>
eventually_of_mem (EMetric.ball_mem_nhds 0 r_pos) fun _ => by simpa using h, _⟩
|
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z)) := by
|
Mathlib.Analysis.Analytic.Basic.1430_0.jQw1fRSE1vGpOll
|
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z))
|
Mathlib_Analysis_Analytic_Basic
|
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
p : FormalMultilinearSeries 𝕜 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
x✝¹ : HasFPowerSeriesAt f p z₀
r : ℝ≥0∞
r_le✝ : r ≤ radius p
r_pos : 0 < r
h : ∀ {y : 𝕜}, y ∈ EMetric.ball 0 r → HasSum (fun n => (p n) fun x => y) (f (z₀ + y))
x✝ : 𝕜
⊢ x✝ ∈ EMetric.ball 0 r → HasSum (fun n => x✝ ^ n • coeff p n) (f (z₀ + x✝))
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Logic.Equiv.Fin
import Mathlib.Topology.Algebra.InfiniteSum.Module
#align_import analysis.analytic.basic from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
/-!
# Analytic functions
A function is analytic in one dimension around `0` if it can be written as a converging power series
`Σ pₙ zⁿ`. This definition can be extended to any dimension (even in infinite dimension) by
requiring that `pₙ` is a continuous `n`-multilinear map. In general, `pₙ` is not unique (in two
dimensions, taking `p₂ (x, y) (x', y') = x y'` or `y x'` gives the same map when applied to a
vector `(x, y) (x, y)`). A way to guarantee uniqueness is to take a symmetric `pₙ`, but this is not
always possible in nonzero characteristic (in characteristic 2, the previous example has no
symmetric representative). Therefore, we do not insist on symmetry or uniqueness in the definition,
and we only require the existence of a converging series.
The general framework is important to say that the exponential map on bounded operators on a Banach
space is analytic, as well as the inverse on invertible operators.
## Main definitions
Let `p` be a formal multilinear series from `E` to `F`, i.e., `p n` is a multilinear map on `E^n`
for `n : ℕ`.
* `p.radius`: the largest `r : ℝ≥0∞` such that `‖p n‖ * r^n` grows subexponentially.
* `p.le_radius_of_bound`, `p.le_radius_of_bound_nnreal`, `p.le_radius_of_isBigO`: if `‖p n‖ * r ^ n`
is bounded above, then `r ≤ p.radius`;
* `p.isLittleO_of_lt_radius`, `p.norm_mul_pow_le_mul_pow_of_lt_radius`,
`p.isLittleO_one_of_lt_radius`,
`p.norm_mul_pow_le_of_lt_radius`, `p.nnnorm_mul_pow_le_of_lt_radius`: if `r < p.radius`, then
`‖p n‖ * r ^ n` tends to zero exponentially;
* `p.lt_radius_of_isBigO`: if `r ≠ 0` and `‖p n‖ * r ^ n = O(a ^ n)` for some `-1 < a < 1`, then
`r < p.radius`;
* `p.partialSum n x`: the sum `∑_{i = 0}^{n-1} pᵢ xⁱ`.
* `p.sum x`: the sum `∑'_{i = 0}^{∞} pᵢ xⁱ`.
Additionally, let `f` be a function from `E` to `F`.
* `HasFPowerSeriesOnBall f p x r`: on the ball of center `x` with radius `r`,
`f (x + y) = ∑'_n pₙ yⁿ`.
* `HasFPowerSeriesAt f p x`: on some ball of center `x` with positive radius, holds
`HasFPowerSeriesOnBall f p x r`.
* `AnalyticAt 𝕜 f x`: there exists a power series `p` such that holds `HasFPowerSeriesAt f p x`.
* `AnalyticOn 𝕜 f s`: the function `f` is analytic at every point of `s`.
We develop the basic properties of these notions, notably:
* If a function admits a power series, it is continuous (see
`HasFPowerSeriesOnBall.continuousOn` and `HasFPowerSeriesAt.continuousAt` and
`AnalyticAt.continuousAt`).
* In a complete space, the sum of a formal power series with positive radius is well defined on the
disk of convergence, see `FormalMultilinearSeries.hasFPowerSeriesOnBall`.
* If a function admits a power series in a ball, then it is analytic at any point `y` of this ball,
and the power series there can be expressed in terms of the initial power series `p` as
`p.changeOrigin y`. See `HasFPowerSeriesOnBall.changeOrigin`. It follows in particular that
the set of points at which a given function is analytic is open, see `isOpen_analyticAt`.
## Implementation details
We only introduce the radius of convergence of a power series, as `p.radius`.
For a power series in finitely many dimensions, there is a finer (directional, coordinate-dependent)
notion, describing the polydisk of convergence. This notion is more specific, and not necessary to
build the general theory. We do not define it here.
-/
noncomputable section
variable {𝕜 E F G : Type*}
open Topology Classical BigOperators NNReal Filter ENNReal
open Set Filter Asymptotics
namespace FormalMultilinearSeries
variable [Ring 𝕜] [AddCommGroup E] [AddCommGroup F] [Module 𝕜 E] [Module 𝕜 F]
variable [TopologicalSpace E] [TopologicalSpace F]
variable [TopologicalAddGroup E] [TopologicalAddGroup F]
variable [ContinuousConstSMul 𝕜 E] [ContinuousConstSMul 𝕜 F]
/-- Given a formal multilinear series `p` and a vector `x`, then `p.sum x` is the sum `Σ pₙ xⁿ`. A
priori, it only behaves well when `‖x‖ < p.radius`. -/
protected def sum (p : FormalMultilinearSeries 𝕜 E F) (x : E) : F :=
∑' n : ℕ, p n fun _ => x
#align formal_multilinear_series.sum FormalMultilinearSeries.sum
/-- Given a formal multilinear series `p` and a vector `x`, then `p.partialSum n x` is the sum
`Σ pₖ xᵏ` for `k ∈ {0,..., n-1}`. -/
def partialSum (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) (x : E) : F :=
∑ k in Finset.range n, p k fun _ : Fin k => x
#align formal_multilinear_series.partial_sum FormalMultilinearSeries.partialSum
/-- The partial sums of a formal multilinear series are continuous. -/
theorem partialSum_continuous (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) :
Continuous (p.partialSum n) := by
unfold partialSum -- Porting note: added
continuity
#align formal_multilinear_series.partial_sum_continuous FormalMultilinearSeries.partialSum_continuous
end FormalMultilinearSeries
/-! ### The radius of a formal multilinear series -/
variable [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F]
[NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G]
namespace FormalMultilinearSeries
variable (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
/-- The radius of a formal multilinear series is the largest `r` such that the sum `Σ ‖pₙ‖ ‖y‖ⁿ`
converges for all `‖y‖ < r`. This implies that `Σ pₙ yⁿ` converges for all `‖y‖ < r`, but these
definitions are *not* equivalent in general. -/
def radius (p : FormalMultilinearSeries 𝕜 E F) : ℝ≥0∞ :=
⨆ (r : ℝ≥0) (C : ℝ) (_ : ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C), (r : ℝ≥0∞)
#align formal_multilinear_series.radius FormalMultilinearSeries.radius
/-- If `‖pₙ‖ rⁿ` is bounded in `n`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_bound (C : ℝ) {r : ℝ≥0} (h : ∀ n : ℕ, ‖p n‖ * (r : ℝ) ^ n ≤ C) :
(r : ℝ≥0∞) ≤ p.radius :=
le_iSup_of_le r <| le_iSup_of_le C <| le_iSup (fun _ => (r : ℝ≥0∞)) h
#align formal_multilinear_series.le_radius_of_bound FormalMultilinearSeries.le_radius_of_bound
/-- If `‖pₙ‖ rⁿ` is bounded in `n`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_bound_nnreal (C : ℝ≥0) {r : ℝ≥0} (h : ∀ n : ℕ, ‖p n‖₊ * r ^ n ≤ C) :
(r : ℝ≥0∞) ≤ p.radius :=
p.le_radius_of_bound C fun n => mod_cast h n
#align formal_multilinear_series.le_radius_of_bound_nnreal FormalMultilinearSeries.le_radius_of_bound_nnreal
/-- If `‖pₙ‖ rⁿ = O(1)`, as `n → ∞`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_isBigO (h : (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] fun _ => (1 : ℝ)) :
↑r ≤ p.radius :=
Exists.elim (isBigO_one_nat_atTop_iff.1 h) fun C hC =>
p.le_radius_of_bound C fun n => (le_abs_self _).trans (hC n)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.le_radius_of_is_O FormalMultilinearSeries.le_radius_of_isBigO
theorem le_radius_of_eventually_le (C) (h : ∀ᶠ n in atTop, ‖p n‖ * (r : ℝ) ^ n ≤ C) :
↑r ≤ p.radius :=
p.le_radius_of_isBigO <| IsBigO.of_bound C <| h.mono fun n hn => by simpa
#align formal_multilinear_series.le_radius_of_eventually_le FormalMultilinearSeries.le_radius_of_eventually_le
theorem le_radius_of_summable_nnnorm (h : Summable fun n => ‖p n‖₊ * r ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_bound_nnreal (∑' n, ‖p n‖₊ * r ^ n) fun _ => le_tsum' h _
#align formal_multilinear_series.le_radius_of_summable_nnnorm FormalMultilinearSeries.le_radius_of_summable_nnnorm
theorem le_radius_of_summable (h : Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_summable_nnnorm <| by
simp only [← coe_nnnorm] at h
exact mod_cast h
#align formal_multilinear_series.le_radius_of_summable FormalMultilinearSeries.le_radius_of_summable
theorem radius_eq_top_of_forall_nnreal_isBigO
(h : ∀ r : ℝ≥0, (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] fun _ => (1 : ℝ)) : p.radius = ∞ :=
ENNReal.eq_top_of_forall_nnreal_le fun r => p.le_radius_of_isBigO (h r)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.radius_eq_top_of_forall_nnreal_is_O FormalMultilinearSeries.radius_eq_top_of_forall_nnreal_isBigO
theorem radius_eq_top_of_eventually_eq_zero (h : ∀ᶠ n in atTop, p n = 0) : p.radius = ∞ :=
p.radius_eq_top_of_forall_nnreal_isBigO fun r =>
(isBigO_zero _ _).congr' (h.mono fun n hn => by simp [hn]) EventuallyEq.rfl
#align formal_multilinear_series.radius_eq_top_of_eventually_eq_zero FormalMultilinearSeries.radius_eq_top_of_eventually_eq_zero
theorem radius_eq_top_of_forall_image_add_eq_zero (n : ℕ) (hn : ∀ m, p (m + n) = 0) :
p.radius = ∞ :=
p.radius_eq_top_of_eventually_eq_zero <|
mem_atTop_sets.2 ⟨n, fun _ hk => tsub_add_cancel_of_le hk ▸ hn _⟩
#align formal_multilinear_series.radius_eq_top_of_forall_image_add_eq_zero FormalMultilinearSeries.radius_eq_top_of_forall_image_add_eq_zero
@[simp]
theorem constFormalMultilinearSeries_radius {v : F} :
(constFormalMultilinearSeries 𝕜 E v).radius = ⊤ :=
(constFormalMultilinearSeries 𝕜 E v).radius_eq_top_of_forall_image_add_eq_zero 1
(by simp [constFormalMultilinearSeries])
#align formal_multilinear_series.const_formal_multilinear_series_radius FormalMultilinearSeries.constFormalMultilinearSeries_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` tends to zero exponentially:
for some `0 < a < 1`, `‖p n‖ rⁿ = o(aⁿ)`. -/
theorem isLittleO_of_lt_radius (h : ↑r < p.radius) :
∃ a ∈ Ioo (0 : ℝ) 1, (fun n => ‖p n‖ * (r : ℝ) ^ n) =o[atTop] (a ^ ·) := by
have := (TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 4
rw [this]
-- Porting note: was
-- rw [(TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 4]
simp only [radius, lt_iSup_iff] at h
rcases h with ⟨t, C, hC, rt⟩
rw [ENNReal.coe_lt_coe, ← NNReal.coe_lt_coe] at rt
have : 0 < (t : ℝ) := r.coe_nonneg.trans_lt rt
rw [← div_lt_one this] at rt
refine' ⟨_, rt, C, Or.inr zero_lt_one, fun n => _⟩
calc
|‖p n‖ * (r : ℝ) ^ n| = ‖p n‖ * (t : ℝ) ^ n * (r / t : ℝ) ^ n := by
field_simp [mul_right_comm, abs_mul]
_ ≤ C * (r / t : ℝ) ^ n := by gcongr; apply hC
#align formal_multilinear_series.is_o_of_lt_radius FormalMultilinearSeries.isLittleO_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ = o(1)`. -/
theorem isLittleO_one_of_lt_radius (h : ↑r < p.radius) :
(fun n => ‖p n‖ * (r : ℝ) ^ n) =o[atTop] (fun _ => 1 : ℕ → ℝ) :=
let ⟨_, ha, hp⟩ := p.isLittleO_of_lt_radius h
hp.trans <| (isLittleO_pow_pow_of_lt_left ha.1.le ha.2).congr (fun _ => rfl) one_pow
#align formal_multilinear_series.is_o_one_of_lt_radius FormalMultilinearSeries.isLittleO_one_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` tends to zero exponentially:
for some `0 < a < 1` and `C > 0`, `‖p n‖ * r ^ n ≤ C * a ^ n`. -/
theorem norm_mul_pow_le_mul_pow_of_lt_radius (h : ↑r < p.radius) :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C * a ^ n := by
-- Porting note: moved out of `rcases`
have := ((TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 5).mp
(p.isLittleO_of_lt_radius h)
rcases this with ⟨a, ha, C, hC, H⟩
exact ⟨a, ha, C, hC, fun n => (le_abs_self _).trans (H n)⟩
#align formal_multilinear_series.norm_mul_pow_le_mul_pow_of_lt_radius FormalMultilinearSeries.norm_mul_pow_le_mul_pow_of_lt_radius
/-- If `r ≠ 0` and `‖pₙ‖ rⁿ = O(aⁿ)` for some `-1 < a < 1`, then `r < p.radius`. -/
theorem lt_radius_of_isBigO (h₀ : r ≠ 0) {a : ℝ} (ha : a ∈ Ioo (-1 : ℝ) 1)
(hp : (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] (a ^ ·)) : ↑r < p.radius := by
-- Porting note: moved out of `rcases`
have := ((TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 2 5)
rcases this.mp ⟨a, ha, hp⟩ with ⟨a, ha, C, hC, hp⟩
rw [← pos_iff_ne_zero, ← NNReal.coe_pos] at h₀
lift a to ℝ≥0 using ha.1.le
have : (r : ℝ) < r / a := by
simpa only [div_one] using (div_lt_div_left h₀ zero_lt_one ha.1).2 ha.2
norm_cast at this
rw [← ENNReal.coe_lt_coe] at this
refine' this.trans_le (p.le_radius_of_bound C fun n => _)
rw [NNReal.coe_div, div_pow, ← mul_div_assoc, div_le_iff (pow_pos ha.1 n)]
exact (le_abs_self _).trans (hp n)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.lt_radius_of_is_O FormalMultilinearSeries.lt_radius_of_isBigO
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem norm_mul_pow_le_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C :=
let ⟨_, ha, C, hC, h⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius h
⟨C, hC, fun n => (h n).trans <| mul_le_of_le_one_right hC.lt.le (pow_le_one _ ha.1.le ha.2.le)⟩
#align formal_multilinear_series.norm_mul_pow_le_of_lt_radius FormalMultilinearSeries.norm_mul_pow_le_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem norm_le_div_pow_of_pos_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h0 : 0 < r) (h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖ ≤ C / (r : ℝ) ^ n :=
let ⟨C, hC, hp⟩ := p.norm_mul_pow_le_of_lt_radius h
⟨C, hC, fun n => Iff.mpr (le_div_iff (pow_pos h0 _)) (hp n)⟩
#align formal_multilinear_series.norm_le_div_pow_of_pos_of_lt_radius FormalMultilinearSeries.norm_le_div_pow_of_pos_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem nnnorm_mul_pow_le_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖₊ * r ^ n ≤ C :=
let ⟨C, hC, hp⟩ := p.norm_mul_pow_le_of_lt_radius h
⟨⟨C, hC.lt.le⟩, hC, mod_cast hp⟩
#align formal_multilinear_series.nnnorm_mul_pow_le_of_lt_radius FormalMultilinearSeries.nnnorm_mul_pow_le_of_lt_radius
theorem le_radius_of_tendsto (p : FormalMultilinearSeries 𝕜 E F) {l : ℝ}
(h : Tendsto (fun n => ‖p n‖ * (r : ℝ) ^ n) atTop (𝓝 l)) : ↑r ≤ p.radius :=
p.le_radius_of_isBigO (h.isBigO_one _)
#align formal_multilinear_series.le_radius_of_tendsto FormalMultilinearSeries.le_radius_of_tendsto
theorem le_radius_of_summable_norm (p : FormalMultilinearSeries 𝕜 E F)
(hs : Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_tendsto hs.tendsto_atTop_zero
#align formal_multilinear_series.le_radius_of_summable_norm FormalMultilinearSeries.le_radius_of_summable_norm
theorem not_summable_norm_of_radius_lt_nnnorm (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(h : p.radius < ‖x‖₊) : ¬Summable fun n => ‖p n‖ * ‖x‖ ^ n :=
fun hs => not_le_of_lt h (p.le_radius_of_summable_norm hs)
#align formal_multilinear_series.not_summable_norm_of_radius_lt_nnnorm FormalMultilinearSeries.not_summable_norm_of_radius_lt_nnnorm
theorem summable_norm_mul_pow (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0} (h : ↑r < p.radius) :
Summable fun n : ℕ => ‖p n‖ * (r : ℝ) ^ n := by
obtain ⟨a, ha : a ∈ Ioo (0 : ℝ) 1, C, - : 0 < C, hp⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius h
exact .of_nonneg_of_le (fun n => mul_nonneg (norm_nonneg _) (pow_nonneg r.coe_nonneg _))
hp ((summable_geometric_of_lt_1 ha.1.le ha.2).mul_left _)
#align formal_multilinear_series.summable_norm_mul_pow FormalMultilinearSeries.summable_norm_mul_pow
theorem summable_norm_apply (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : Summable fun n : ℕ => ‖p n fun _ => x‖ := by
rw [mem_emetric_ball_zero_iff] at hx
refine' .of_nonneg_of_le
(fun _ => norm_nonneg _) (fun n => ((p n).le_op_norm _).trans_eq _) (p.summable_norm_mul_pow hx)
simp
#align formal_multilinear_series.summable_norm_apply FormalMultilinearSeries.summable_norm_apply
theorem summable_nnnorm_mul_pow (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0} (h : ↑r < p.radius) :
Summable fun n : ℕ => ‖p n‖₊ * r ^ n := by
rw [← NNReal.summable_coe]
push_cast
exact p.summable_norm_mul_pow h
#align formal_multilinear_series.summable_nnnorm_mul_pow FormalMultilinearSeries.summable_nnnorm_mul_pow
protected theorem summable [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : Summable fun n : ℕ => p n fun _ => x :=
(p.summable_norm_apply hx).of_norm
#align formal_multilinear_series.summable FormalMultilinearSeries.summable
theorem radius_eq_top_of_summable_norm (p : FormalMultilinearSeries 𝕜 E F)
(hs : ∀ r : ℝ≥0, Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : p.radius = ∞ :=
ENNReal.eq_top_of_forall_nnreal_le fun r => p.le_radius_of_summable_norm (hs r)
#align formal_multilinear_series.radius_eq_top_of_summable_norm FormalMultilinearSeries.radius_eq_top_of_summable_norm
theorem radius_eq_top_iff_summable_norm (p : FormalMultilinearSeries 𝕜 E F) :
p.radius = ∞ ↔ ∀ r : ℝ≥0, Summable fun n => ‖p n‖ * (r : ℝ) ^ n := by
constructor
· intro h r
obtain ⟨a, ha : a ∈ Ioo (0 : ℝ) 1, C, - : 0 < C, hp⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius
(show (r : ℝ≥0∞) < p.radius from h.symm ▸ ENNReal.coe_lt_top)
refine' .of_norm_bounded
(fun n => (C : ℝ) * a ^ n) ((summable_geometric_of_lt_1 ha.1.le ha.2).mul_left _) fun n => _
specialize hp n
rwa [Real.norm_of_nonneg (mul_nonneg (norm_nonneg _) (pow_nonneg r.coe_nonneg n))]
· exact p.radius_eq_top_of_summable_norm
#align formal_multilinear_series.radius_eq_top_iff_summable_norm FormalMultilinearSeries.radius_eq_top_iff_summable_norm
/-- If the radius of `p` is positive, then `‖pₙ‖` grows at most geometrically. -/
theorem le_mul_pow_of_radius_pos (p : FormalMultilinearSeries 𝕜 E F) (h : 0 < p.radius) :
∃ (C r : _) (hC : 0 < C) (_ : 0 < r), ∀ n, ‖p n‖ ≤ C * r ^ n := by
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 h with ⟨r, r0, rlt⟩
have rpos : 0 < (r : ℝ) := by simp [ENNReal.coe_pos.1 r0]
rcases norm_le_div_pow_of_pos_of_lt_radius p rpos rlt with ⟨C, Cpos, hCp⟩
refine' ⟨C, r⁻¹, Cpos, by simp only [inv_pos, rpos], fun n => _⟩
-- Porting note: was `convert`
rw [inv_pow, ← div_eq_mul_inv]
exact hCp n
#align formal_multilinear_series.le_mul_pow_of_radius_pos FormalMultilinearSeries.le_mul_pow_of_radius_pos
/-- The radius of the sum of two formal series is at least the minimum of their two radii. -/
theorem min_radius_le_radius_add (p q : FormalMultilinearSeries 𝕜 E F) :
min p.radius q.radius ≤ (p + q).radius := by
refine' ENNReal.le_of_forall_nnreal_lt fun r hr => _
rw [lt_min_iff] at hr
have := ((p.isLittleO_one_of_lt_radius hr.1).add (q.isLittleO_one_of_lt_radius hr.2)).isBigO
refine' (p + q).le_radius_of_isBigO ((isBigO_of_le _ fun n => _).trans this)
rw [← add_mul, norm_mul, norm_mul, norm_norm]
exact mul_le_mul_of_nonneg_right ((norm_add_le _ _).trans (le_abs_self _)) (norm_nonneg _)
#align formal_multilinear_series.min_radius_le_radius_add FormalMultilinearSeries.min_radius_le_radius_add
@[simp]
theorem radius_neg (p : FormalMultilinearSeries 𝕜 E F) : (-p).radius = p.radius := by
simp only [radius, neg_apply, norm_neg]
#align formal_multilinear_series.radius_neg FormalMultilinearSeries.radius_neg
protected theorem hasSum [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : HasSum (fun n : ℕ => p n fun _ => x) (p.sum x) :=
(p.summable hx).hasSum
#align formal_multilinear_series.has_sum FormalMultilinearSeries.hasSum
theorem radius_le_radius_continuousLinearMap_comp (p : FormalMultilinearSeries 𝕜 E F)
(f : F →L[𝕜] G) : p.radius ≤ (f.compFormalMultilinearSeries p).radius := by
refine' ENNReal.le_of_forall_nnreal_lt fun r hr => _
apply le_radius_of_isBigO
apply (IsBigO.trans_isLittleO _ (p.isLittleO_one_of_lt_radius hr)).isBigO
refine' IsBigO.mul (@IsBigOWith.isBigO _ _ _ _ _ ‖f‖ _ _ _ _) (isBigO_refl _ _)
refine IsBigOWith.of_bound (eventually_of_forall fun n => ?_)
simpa only [norm_norm] using f.norm_compContinuousMultilinearMap_le (p n)
#align formal_multilinear_series.radius_le_radius_continuous_linear_map_comp FormalMultilinearSeries.radius_le_radius_continuousLinearMap_comp
end FormalMultilinearSeries
/-! ### Expanding a function as a power series -/
section
variable {f g : E → F} {p pf pg : FormalMultilinearSeries 𝕜 E F} {x : E} {r r' : ℝ≥0∞}
/-- Given a function `f : E → F` and a formal multilinear series `p`, we say that `f` has `p` as
a power series on the ball of radius `r > 0` around `x` if `f (x + y) = ∑' pₙ yⁿ` for all `‖y‖ < r`.
-/
structure HasFPowerSeriesOnBall (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (x : E) (r : ℝ≥0∞) :
Prop where
r_le : r ≤ p.radius
r_pos : 0 < r
hasSum :
∀ {y}, y ∈ EMetric.ball (0 : E) r → HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y))
#align has_fpower_series_on_ball HasFPowerSeriesOnBall
/-- Given a function `f : E → F` and a formal multilinear series `p`, we say that `f` has `p` as
a power series around `x` if `f (x + y) = ∑' pₙ yⁿ` for all `y` in a neighborhood of `0`. -/
def HasFPowerSeriesAt (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (x : E) :=
∃ r, HasFPowerSeriesOnBall f p x r
#align has_fpower_series_at HasFPowerSeriesAt
variable (𝕜)
/-- Given a function `f : E → F`, we say that `f` is analytic at `x` if it admits a convergent power
series expansion around `x`. -/
def AnalyticAt (f : E → F) (x : E) :=
∃ p : FormalMultilinearSeries 𝕜 E F, HasFPowerSeriesAt f p x
#align analytic_at AnalyticAt
/-- Given a function `f : E → F`, we say that `f` is analytic on a set `s` if it is analytic around
every point of `s`. -/
def AnalyticOn (f : E → F) (s : Set E) :=
∀ x, x ∈ s → AnalyticAt 𝕜 f x
#align analytic_on AnalyticOn
variable {𝕜}
theorem HasFPowerSeriesOnBall.hasFPowerSeriesAt (hf : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesAt f p x :=
⟨r, hf⟩
#align has_fpower_series_on_ball.has_fpower_series_at HasFPowerSeriesOnBall.hasFPowerSeriesAt
theorem HasFPowerSeriesAt.analyticAt (hf : HasFPowerSeriesAt f p x) : AnalyticAt 𝕜 f x :=
⟨p, hf⟩
#align has_fpower_series_at.analytic_at HasFPowerSeriesAt.analyticAt
theorem HasFPowerSeriesOnBall.analyticAt (hf : HasFPowerSeriesOnBall f p x r) : AnalyticAt 𝕜 f x :=
hf.hasFPowerSeriesAt.analyticAt
#align has_fpower_series_on_ball.analytic_at HasFPowerSeriesOnBall.analyticAt
theorem HasFPowerSeriesOnBall.congr (hf : HasFPowerSeriesOnBall f p x r)
(hg : EqOn f g (EMetric.ball x r)) : HasFPowerSeriesOnBall g p x r :=
{ r_le := hf.r_le
r_pos := hf.r_pos
hasSum := fun {y} hy => by
convert hf.hasSum hy using 1
apply hg.symm
simpa [edist_eq_coe_nnnorm_sub] using hy }
#align has_fpower_series_on_ball.congr HasFPowerSeriesOnBall.congr
/-- If a function `f` has a power series `p` around `x`, then the function `z ↦ f (z - y)` has the
same power series around `x + y`. -/
theorem HasFPowerSeriesOnBall.comp_sub (hf : HasFPowerSeriesOnBall f p x r) (y : E) :
HasFPowerSeriesOnBall (fun z => f (z - y)) p (x + y) r :=
{ r_le := hf.r_le
r_pos := hf.r_pos
hasSum := fun {z} hz => by
convert hf.hasSum hz using 2
abel }
#align has_fpower_series_on_ball.comp_sub HasFPowerSeriesOnBall.comp_sub
theorem HasFPowerSeriesOnBall.hasSum_sub (hf : HasFPowerSeriesOnBall f p x r) {y : E}
(hy : y ∈ EMetric.ball x r) : HasSum (fun n : ℕ => p n fun _ => y - x) (f y) := by
have : y - x ∈ EMetric.ball (0 : E) r := by simpa [edist_eq_coe_nnnorm_sub] using hy
simpa only [add_sub_cancel'_right] using hf.hasSum this
#align has_fpower_series_on_ball.has_sum_sub HasFPowerSeriesOnBall.hasSum_sub
theorem HasFPowerSeriesOnBall.radius_pos (hf : HasFPowerSeriesOnBall f p x r) : 0 < p.radius :=
lt_of_lt_of_le hf.r_pos hf.r_le
#align has_fpower_series_on_ball.radius_pos HasFPowerSeriesOnBall.radius_pos
theorem HasFPowerSeriesAt.radius_pos (hf : HasFPowerSeriesAt f p x) : 0 < p.radius :=
let ⟨_, hr⟩ := hf
hr.radius_pos
#align has_fpower_series_at.radius_pos HasFPowerSeriesAt.radius_pos
theorem HasFPowerSeriesOnBall.mono (hf : HasFPowerSeriesOnBall f p x r) (r'_pos : 0 < r')
(hr : r' ≤ r) : HasFPowerSeriesOnBall f p x r' :=
⟨le_trans hr hf.1, r'_pos, fun hy => hf.hasSum (EMetric.ball_subset_ball hr hy)⟩
#align has_fpower_series_on_ball.mono HasFPowerSeriesOnBall.mono
theorem HasFPowerSeriesAt.congr (hf : HasFPowerSeriesAt f p x) (hg : f =ᶠ[𝓝 x] g) :
HasFPowerSeriesAt g p x := by
rcases hf with ⟨r₁, h₁⟩
rcases EMetric.mem_nhds_iff.mp hg with ⟨r₂, h₂pos, h₂⟩
exact ⟨min r₁ r₂,
(h₁.mono (lt_min h₁.r_pos h₂pos) inf_le_left).congr
fun y hy => h₂ (EMetric.ball_subset_ball inf_le_right hy)⟩
#align has_fpower_series_at.congr HasFPowerSeriesAt.congr
protected theorem HasFPowerSeriesAt.eventually (hf : HasFPowerSeriesAt f p x) :
∀ᶠ r : ℝ≥0∞ in 𝓝[>] 0, HasFPowerSeriesOnBall f p x r :=
let ⟨_, hr⟩ := hf
mem_of_superset (Ioo_mem_nhdsWithin_Ioi (left_mem_Ico.2 hr.r_pos)) fun _ hr' =>
hr.mono hr'.1 hr'.2.le
#align has_fpower_series_at.eventually HasFPowerSeriesAt.eventually
theorem HasFPowerSeriesOnBall.eventually_hasSum (hf : HasFPowerSeriesOnBall f p x r) :
∀ᶠ y in 𝓝 0, HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y)) := by
filter_upwards [EMetric.ball_mem_nhds (0 : E) hf.r_pos] using fun _ => hf.hasSum
#align has_fpower_series_on_ball.eventually_has_sum HasFPowerSeriesOnBall.eventually_hasSum
theorem HasFPowerSeriesAt.eventually_hasSum (hf : HasFPowerSeriesAt f p x) :
∀ᶠ y in 𝓝 0, HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y)) :=
let ⟨_, hr⟩ := hf
hr.eventually_hasSum
#align has_fpower_series_at.eventually_has_sum HasFPowerSeriesAt.eventually_hasSum
theorem HasFPowerSeriesOnBall.eventually_hasSum_sub (hf : HasFPowerSeriesOnBall f p x r) :
∀ᶠ y in 𝓝 x, HasSum (fun n : ℕ => p n fun _ : Fin n => y - x) (f y) := by
filter_upwards [EMetric.ball_mem_nhds x hf.r_pos] with y using hf.hasSum_sub
#align has_fpower_series_on_ball.eventually_has_sum_sub HasFPowerSeriesOnBall.eventually_hasSum_sub
theorem HasFPowerSeriesAt.eventually_hasSum_sub (hf : HasFPowerSeriesAt f p x) :
∀ᶠ y in 𝓝 x, HasSum (fun n : ℕ => p n fun _ : Fin n => y - x) (f y) :=
let ⟨_, hr⟩ := hf
hr.eventually_hasSum_sub
#align has_fpower_series_at.eventually_has_sum_sub HasFPowerSeriesAt.eventually_hasSum_sub
theorem HasFPowerSeriesOnBall.eventually_eq_zero
(hf : HasFPowerSeriesOnBall f (0 : FormalMultilinearSeries 𝕜 E F) x r) :
∀ᶠ z in 𝓝 x, f z = 0 := by
filter_upwards [hf.eventually_hasSum_sub] with z hz using hz.unique hasSum_zero
#align has_fpower_series_on_ball.eventually_eq_zero HasFPowerSeriesOnBall.eventually_eq_zero
theorem HasFPowerSeriesAt.eventually_eq_zero
(hf : HasFPowerSeriesAt f (0 : FormalMultilinearSeries 𝕜 E F) x) : ∀ᶠ z in 𝓝 x, f z = 0 :=
let ⟨_, hr⟩ := hf
hr.eventually_eq_zero
#align has_fpower_series_at.eventually_eq_zero HasFPowerSeriesAt.eventually_eq_zero
theorem hasFPowerSeriesOnBall_const {c : F} {e : E} :
HasFPowerSeriesOnBall (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e ⊤ := by
refine' ⟨by simp, WithTop.zero_lt_top, fun _ => hasSum_single 0 fun n hn => _⟩
simp [constFormalMultilinearSeries_apply hn]
#align has_fpower_series_on_ball_const hasFPowerSeriesOnBall_const
theorem hasFPowerSeriesAt_const {c : F} {e : E} :
HasFPowerSeriesAt (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e :=
⟨⊤, hasFPowerSeriesOnBall_const⟩
#align has_fpower_series_at_const hasFPowerSeriesAt_const
theorem analyticAt_const {v : F} : AnalyticAt 𝕜 (fun _ => v) x :=
⟨constFormalMultilinearSeries 𝕜 E v, hasFPowerSeriesAt_const⟩
#align analytic_at_const analyticAt_const
theorem analyticOn_const {v : F} {s : Set E} : AnalyticOn 𝕜 (fun _ => v) s :=
fun _ _ => analyticAt_const
#align analytic_on_const analyticOn_const
theorem HasFPowerSeriesOnBall.add (hf : HasFPowerSeriesOnBall f pf x r)
(hg : HasFPowerSeriesOnBall g pg x r) : HasFPowerSeriesOnBall (f + g) (pf + pg) x r :=
{ r_le := le_trans (le_min_iff.2 ⟨hf.r_le, hg.r_le⟩) (pf.min_radius_le_radius_add pg)
r_pos := hf.r_pos
hasSum := fun hy => (hf.hasSum hy).add (hg.hasSum hy) }
#align has_fpower_series_on_ball.add HasFPowerSeriesOnBall.add
theorem HasFPowerSeriesAt.add (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x) :
HasFPowerSeriesAt (f + g) (pf + pg) x := by
rcases (hf.eventually.and hg.eventually).exists with ⟨r, hr⟩
exact ⟨r, hr.1.add hr.2⟩
#align has_fpower_series_at.add HasFPowerSeriesAt.add
theorem AnalyticAt.congr (hf : AnalyticAt 𝕜 f x) (hg : f =ᶠ[𝓝 x] g) : AnalyticAt 𝕜 g x :=
let ⟨_, hpf⟩ := hf
(hpf.congr hg).analyticAt
theorem analyticAt_congr (h : f =ᶠ[𝓝 x] g) : AnalyticAt 𝕜 f x ↔ AnalyticAt 𝕜 g x :=
⟨fun hf ↦ hf.congr h, fun hg ↦ hg.congr h.symm⟩
theorem AnalyticAt.add (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) : AnalyticAt 𝕜 (f + g) x :=
let ⟨_, hpf⟩ := hf
let ⟨_, hqf⟩ := hg
(hpf.add hqf).analyticAt
#align analytic_at.add AnalyticAt.add
theorem HasFPowerSeriesOnBall.neg (hf : HasFPowerSeriesOnBall f pf x r) :
HasFPowerSeriesOnBall (-f) (-pf) x r :=
{ r_le := by
rw [pf.radius_neg]
exact hf.r_le
r_pos := hf.r_pos
hasSum := fun hy => (hf.hasSum hy).neg }
#align has_fpower_series_on_ball.neg HasFPowerSeriesOnBall.neg
theorem HasFPowerSeriesAt.neg (hf : HasFPowerSeriesAt f pf x) : HasFPowerSeriesAt (-f) (-pf) x :=
let ⟨_, hrf⟩ := hf
hrf.neg.hasFPowerSeriesAt
#align has_fpower_series_at.neg HasFPowerSeriesAt.neg
theorem AnalyticAt.neg (hf : AnalyticAt 𝕜 f x) : AnalyticAt 𝕜 (-f) x :=
let ⟨_, hpf⟩ := hf
hpf.neg.analyticAt
#align analytic_at.neg AnalyticAt.neg
theorem HasFPowerSeriesOnBall.sub (hf : HasFPowerSeriesOnBall f pf x r)
(hg : HasFPowerSeriesOnBall g pg x r) : HasFPowerSeriesOnBall (f - g) (pf - pg) x r := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align has_fpower_series_on_ball.sub HasFPowerSeriesOnBall.sub
theorem HasFPowerSeriesAt.sub (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x) :
HasFPowerSeriesAt (f - g) (pf - pg) x := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align has_fpower_series_at.sub HasFPowerSeriesAt.sub
theorem AnalyticAt.sub (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) :
AnalyticAt 𝕜 (f - g) x := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align analytic_at.sub AnalyticAt.sub
theorem AnalyticOn.mono {s t : Set E} (hf : AnalyticOn 𝕜 f t) (hst : s ⊆ t) : AnalyticOn 𝕜 f s :=
fun z hz => hf z (hst hz)
#align analytic_on.mono AnalyticOn.mono
theorem AnalyticOn.congr' {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : f =ᶠ[𝓝ˢ s] g) :
AnalyticOn 𝕜 g s :=
fun z hz => (hf z hz).congr (mem_nhdsSet_iff_forall.mp hg z hz)
theorem analyticOn_congr' {s : Set E} (h : f =ᶠ[𝓝ˢ s] g) : AnalyticOn 𝕜 f s ↔ AnalyticOn 𝕜 g s :=
⟨fun hf => hf.congr' h, fun hg => hg.congr' h.symm⟩
theorem AnalyticOn.congr {s : Set E} (hs : IsOpen s) (hf : AnalyticOn 𝕜 f s) (hg : s.EqOn f g) :
AnalyticOn 𝕜 g s :=
hf.congr' $ mem_nhdsSet_iff_forall.mpr
(fun _ hz => eventuallyEq_iff_exists_mem.mpr ⟨s, hs.mem_nhds hz, hg⟩)
theorem analyticOn_congr {s : Set E} (hs : IsOpen s) (h : s.EqOn f g) : AnalyticOn 𝕜 f s ↔
AnalyticOn 𝕜 g s := ⟨fun hf => hf.congr hs h, fun hg => hg.congr hs h.symm⟩
theorem AnalyticOn.add {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
AnalyticOn 𝕜 (f + g) s :=
fun z hz => (hf z hz).add (hg z hz)
#align analytic_on.add AnalyticOn.add
theorem AnalyticOn.sub {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
AnalyticOn 𝕜 (f - g) s :=
fun z hz => (hf z hz).sub (hg z hz)
#align analytic_on.sub AnalyticOn.sub
theorem HasFPowerSeriesOnBall.coeff_zero (hf : HasFPowerSeriesOnBall f pf x r) (v : Fin 0 → E) :
pf 0 v = f x := by
have v_eq : v = fun i => 0 := Subsingleton.elim _ _
have zero_mem : (0 : E) ∈ EMetric.ball (0 : E) r := by simp [hf.r_pos]
have : ∀ i, i ≠ 0 → (pf i fun j => 0) = 0 := by
intro i hi
have : 0 < i := pos_iff_ne_zero.2 hi
exact ContinuousMultilinearMap.map_coord_zero _ (⟨0, this⟩ : Fin i) rfl
have A := (hf.hasSum zero_mem).unique (hasSum_single _ this)
simpa [v_eq] using A.symm
#align has_fpower_series_on_ball.coeff_zero HasFPowerSeriesOnBall.coeff_zero
theorem HasFPowerSeriesAt.coeff_zero (hf : HasFPowerSeriesAt f pf x) (v : Fin 0 → E) :
pf 0 v = f x :=
let ⟨_, hrf⟩ := hf
hrf.coeff_zero v
#align has_fpower_series_at.coeff_zero HasFPowerSeriesAt.coeff_zero
/-- If a function `f` has a power series `p` on a ball and `g` is linear, then `g ∘ f` has the
power series `g ∘ p` on the same ball. -/
theorem ContinuousLinearMap.comp_hasFPowerSeriesOnBall (g : F →L[𝕜] G)
(h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (g ∘ f) (g.compFormalMultilinearSeries p) x r :=
{ r_le := h.r_le.trans (p.radius_le_radius_continuousLinearMap_comp _)
r_pos := h.r_pos
hasSum := fun hy => by
simpa only [ContinuousLinearMap.compFormalMultilinearSeries_apply,
ContinuousLinearMap.compContinuousMultilinearMap_coe, Function.comp_apply] using
g.hasSum (h.hasSum hy) }
#align continuous_linear_map.comp_has_fpower_series_on_ball ContinuousLinearMap.comp_hasFPowerSeriesOnBall
/-- If a function `f` is analytic on a set `s` and `g` is linear, then `g ∘ f` is analytic
on `s`. -/
theorem ContinuousLinearMap.comp_analyticOn {s : Set E} (g : F →L[𝕜] G) (h : AnalyticOn 𝕜 f s) :
AnalyticOn 𝕜 (g ∘ f) s := by
rintro x hx
rcases h x hx with ⟨p, r, hp⟩
exact ⟨g.compFormalMultilinearSeries p, r, g.comp_hasFPowerSeriesOnBall hp⟩
#align continuous_linear_map.comp_analytic_on ContinuousLinearMap.comp_analyticOn
/-- If a function admits a power series expansion, then it is exponentially close to the partial
sums of this power series on strict subdisks of the disk of convergence.
This version provides an upper estimate that decreases both in `‖y‖` and `n`. See also
`HasFPowerSeriesOnBall.uniform_geometric_approx` for a weaker version. -/
theorem HasFPowerSeriesOnBall.uniform_geometric_approx' {r' : ℝ≥0}
(hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n := by
obtain ⟨a, ha, C, hC, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n, ‖p n‖ * (r' : ℝ) ^ n ≤ C * a ^ n :=
p.norm_mul_pow_le_mul_pow_of_lt_radius (h.trans_le hf.r_le)
refine' ⟨a, ha, C / (1 - a), div_pos hC (sub_pos.2 ha.2), fun y hy n => _⟩
have yr' : ‖y‖ < r' := by
rw [ball_zero_eq] at hy
exact hy
have hr'0 : 0 < (r' : ℝ) := (norm_nonneg _).trans_lt yr'
have : y ∈ EMetric.ball (0 : E) r := by
refine' mem_emetric_ball_zero_iff.2 (lt_trans _ h)
exact mod_cast yr'
rw [norm_sub_rev, ← mul_div_right_comm]
have ya : a * (‖y‖ / ↑r') ≤ a :=
mul_le_of_le_one_right ha.1.le (div_le_one_of_le yr'.le r'.coe_nonneg)
suffices ‖p.partialSum n y - f (x + y)‖ ≤ C * (a * (‖y‖ / r')) ^ n / (1 - a * (‖y‖ / r')) by
refine' this.trans _
have : 0 < a := ha.1
gcongr
apply_rules [sub_pos.2, ha.2]
apply norm_sub_le_of_geometric_bound_of_hasSum (ya.trans_lt ha.2) _ (hf.hasSum this)
intro n
calc
‖(p n) fun _ : Fin n => y‖
_ ≤ ‖p n‖ * ∏ _i : Fin n, ‖y‖ := ContinuousMultilinearMap.le_op_norm _ _
_ = ‖p n‖ * (r' : ℝ) ^ n * (‖y‖ / r') ^ n := by field_simp [mul_right_comm]
_ ≤ C * a ^ n * (‖y‖ / r') ^ n := by gcongr ?_ * _; apply hp
_ ≤ C * (a * (‖y‖ / r')) ^ n := by rw [mul_pow, mul_assoc]
#align has_fpower_series_on_ball.uniform_geometric_approx' HasFPowerSeriesOnBall.uniform_geometric_approx'
/-- If a function admits a power series expansion, then it is exponentially close to the partial
sums of this power series on strict subdisks of the disk of convergence. -/
theorem HasFPowerSeriesOnBall.uniform_geometric_approx {r' : ℝ≥0}
(hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
∃ a ∈ Ioo (0 : ℝ) 1,
∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n, ‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n := by
obtain ⟨a, ha, C, hC, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n :=
hf.uniform_geometric_approx' h
refine' ⟨a, ha, C, hC, fun y hy n => (hp y hy n).trans _⟩
have yr' : ‖y‖ < r' := by rwa [ball_zero_eq] at hy
gcongr
exacts [mul_nonneg ha.1.le (div_nonneg (norm_nonneg y) r'.coe_nonneg),
mul_le_of_le_one_right ha.1.le (div_le_one_of_le yr'.le r'.coe_nonneg)]
#align has_fpower_series_on_ball.uniform_geometric_approx HasFPowerSeriesOnBall.uniform_geometric_approx
/-- Taylor formula for an analytic function, `IsBigO` version. -/
theorem HasFPowerSeriesAt.isBigO_sub_partialSum_pow (hf : HasFPowerSeriesAt f p x) (n : ℕ) :
(fun y : E => f (x + y) - p.partialSum n y) =O[𝓝 0] fun y => ‖y‖ ^ n := by
rcases hf with ⟨r, hf⟩
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hf.r_pos with ⟨r', r'0, h⟩
obtain ⟨a, -, C, -, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n :=
hf.uniform_geometric_approx' h
refine' isBigO_iff.2 ⟨C * (a / r') ^ n, _⟩
replace r'0 : 0 < (r' : ℝ); · exact mod_cast r'0
filter_upwards [Metric.ball_mem_nhds (0 : E) r'0] with y hy
simpa [mul_pow, mul_div_assoc, mul_assoc, div_mul_eq_mul_div] using hp y hy n
set_option linter.uppercaseLean3 false in
#align has_fpower_series_at.is_O_sub_partial_sum_pow HasFPowerSeriesAt.isBigO_sub_partialSum_pow
/-- If `f` has formal power series `∑ n, pₙ` on a ball of radius `r`, then for `y, z` in any smaller
ball, the norm of the difference `f y - f z - p 1 (fun _ ↦ y - z)` is bounded above by
`C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖`. This lemma formulates this property using `IsBigO` and
`Filter.principal` on `E × E`. -/
theorem HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal
(hf : HasFPowerSeriesOnBall f p x r) (hr : r' < r) :
(fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) =O[𝓟 (EMetric.ball (x, x) r')]
fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖ := by
lift r' to ℝ≥0 using ne_top_of_lt hr
rcases (zero_le r').eq_or_lt with (rfl | hr'0)
· simp only [isBigO_bot, EMetric.ball_zero, principal_empty, ENNReal.coe_zero]
obtain ⟨a, ha, C, hC : 0 < C, hp⟩ :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n : ℕ, ‖p n‖ * (r' : ℝ) ^ n ≤ C * a ^ n
exact p.norm_mul_pow_le_mul_pow_of_lt_radius (hr.trans_le hf.r_le)
simp only [← le_div_iff (pow_pos (NNReal.coe_pos.2 hr'0) _)] at hp
set L : E × E → ℝ := fun y =>
C * (a / r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * (a / (1 - a) ^ 2 + 2 / (1 - a))
have hL : ∀ y ∈ EMetric.ball (x, x) r', ‖f y.1 - f y.2 - p 1 fun _ => y.1 - y.2‖ ≤ L y := by
intro y hy'
have hy : y ∈ EMetric.ball x r ×ˢ EMetric.ball x r := by
rw [EMetric.ball_prod_same]
exact EMetric.ball_subset_ball hr.le hy'
set A : ℕ → F := fun n => (p n fun _ => y.1 - x) - p n fun _ => y.2 - x
have hA : HasSum (fun n => A (n + 2)) (f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) := by
convert (hasSum_nat_add_iff' 2).2 ((hf.hasSum_sub hy.1).sub (hf.hasSum_sub hy.2)) using 1
rw [Finset.sum_range_succ, Finset.sum_range_one, hf.coeff_zero, hf.coeff_zero, sub_self,
zero_add, ← Subsingleton.pi_single_eq (0 : Fin 1) (y.1 - x), Pi.single,
← Subsingleton.pi_single_eq (0 : Fin 1) (y.2 - x), Pi.single, ← (p 1).map_sub, ← Pi.single,
Subsingleton.pi_single_eq, sub_sub_sub_cancel_right]
rw [EMetric.mem_ball, edist_eq_coe_nnnorm_sub, ENNReal.coe_lt_coe] at hy'
set B : ℕ → ℝ := fun n => C * (a / r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * ((n + 2) * a ^ n)
have hAB : ∀ n, ‖A (n + 2)‖ ≤ B n := fun n =>
calc
‖A (n + 2)‖ ≤ ‖p (n + 2)‖ * ↑(n + 2) * ‖y - (x, x)‖ ^ (n + 1) * ‖y.1 - y.2‖ := by
-- porting note: `pi_norm_const` was `pi_norm_const (_ : E)`
simpa only [Fintype.card_fin, pi_norm_const, Prod.norm_def, Pi.sub_def,
Prod.fst_sub, Prod.snd_sub, sub_sub_sub_cancel_right] using
(p <| n + 2).norm_image_sub_le (fun _ => y.1 - x) fun _ => y.2 - x
_ = ‖p (n + 2)‖ * ‖y - (x, x)‖ ^ n * (↑(n + 2) * ‖y - (x, x)‖ * ‖y.1 - y.2‖) := by
rw [pow_succ ‖y - (x, x)‖]
ring
-- porting note: the two `↑` in `↑r'` are new, without them, Lean fails to synthesize
-- instances `HDiv ℝ ℝ≥0 ?m` or `HMul ℝ ℝ≥0 ?m`
_ ≤ C * a ^ (n + 2) / ↑r' ^ (n + 2)
* ↑r' ^ n * (↑(n + 2) * ‖y - (x, x)‖ * ‖y.1 - y.2‖) := by
have : 0 < a := ha.1
gcongr
· apply hp
· apply hy'.le
_ = B n := by
-- porting note: in the original, `B` was in the `field_simp`, but now Lean does not
-- accept it. The current proof works in Lean 4, but does not in Lean 3.
field_simp [pow_succ]
simp only [mul_assoc, mul_comm, mul_left_comm]
have hBL : HasSum B (L y) := by
apply HasSum.mul_left
simp only [add_mul]
have : ‖a‖ < 1 := by simp only [Real.norm_eq_abs, abs_of_pos ha.1, ha.2]
rw [div_eq_mul_inv, div_eq_mul_inv]
exact (hasSum_coe_mul_geometric_of_norm_lt_1 this).add -- porting note: was `convert`!
((hasSum_geometric_of_norm_lt_1 this).mul_left 2)
exact hA.norm_le_of_bounded hBL hAB
suffices L =O[𝓟 (EMetric.ball (x, x) r')] fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖ by
refine' (IsBigO.of_bound 1 (eventually_principal.2 fun y hy => _)).trans this
rw [one_mul]
exact (hL y hy).trans (le_abs_self _)
simp_rw [mul_right_comm _ (_ * _)] -- porting note: there was an `L` inside the `simp_rw`.
exact (isBigO_refl _ _).const_mul_left _
set_option linter.uppercaseLean3 false in
#align has_fpower_series_on_ball.is_O_image_sub_image_sub_deriv_principal HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal
/-- If `f` has formal power series `∑ n, pₙ` on a ball of radius `r`, then for `y, z` in any smaller
ball, the norm of the difference `f y - f z - p 1 (fun _ ↦ y - z)` is bounded above by
`C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖`. -/
theorem HasFPowerSeriesOnBall.image_sub_sub_deriv_le (hf : HasFPowerSeriesOnBall f p x r)
(hr : r' < r) :
∃ C, ∀ᵉ (y ∈ EMetric.ball x r') (z ∈ EMetric.ball x r'),
‖f y - f z - p 1 fun _ => y - z‖ ≤ C * max ‖y - x‖ ‖z - x‖ * ‖y - z‖ := by
simpa only [isBigO_principal, mul_assoc, norm_mul, norm_norm, Prod.forall, EMetric.mem_ball,
Prod.edist_eq, max_lt_iff, and_imp, @forall_swap (_ < _) E] using
hf.isBigO_image_sub_image_sub_deriv_principal hr
#align has_fpower_series_on_ball.image_sub_sub_deriv_le HasFPowerSeriesOnBall.image_sub_sub_deriv_le
/-- If `f` has formal power series `∑ n, pₙ` at `x`, then
`f y - f z - p 1 (fun _ ↦ y - z) = O(‖(y, z) - (x, x)‖ * ‖y - z‖)` as `(y, z) → (x, x)`.
In particular, `f` is strictly differentiable at `x`. -/
theorem HasFPowerSeriesAt.isBigO_image_sub_norm_mul_norm_sub (hf : HasFPowerSeriesAt f p x) :
(fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) =O[𝓝 (x, x)] fun y =>
‖y - (x, x)‖ * ‖y.1 - y.2‖ := by
rcases hf with ⟨r, hf⟩
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hf.r_pos with ⟨r', r'0, h⟩
refine' (hf.isBigO_image_sub_image_sub_deriv_principal h).mono _
exact le_principal_iff.2 (EMetric.ball_mem_nhds _ r'0)
set_option linter.uppercaseLean3 false in
#align has_fpower_series_at.is_O_image_sub_norm_mul_norm_sub HasFPowerSeriesAt.isBigO_image_sub_norm_mul_norm_sub
/-- If a function admits a power series expansion at `x`, then it is the uniform limit of the
partial sums of this power series on strict subdisks of the disk of convergence, i.e., `f (x + y)`
is the uniform limit of `p.partialSum n y` there. -/
theorem HasFPowerSeriesOnBall.tendstoUniformlyOn {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r)
(h : (r' : ℝ≥0∞) < r) :
TendstoUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
(Metric.ball (0 : E) r') := by
obtain ⟨a, ha, C, -, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n
exact hf.uniform_geometric_approx h
refine' Metric.tendstoUniformlyOn_iff.2 fun ε εpos => _
have L : Tendsto (fun n => (C : ℝ) * a ^ n) atTop (𝓝 ((C : ℝ) * 0)) :=
tendsto_const_nhds.mul (tendsto_pow_atTop_nhds_0_of_lt_1 ha.1.le ha.2)
rw [mul_zero] at L
refine' (L.eventually (gt_mem_nhds εpos)).mono fun n hn y hy => _
rw [dist_eq_norm]
exact (hp y hy n).trans_lt hn
#align has_fpower_series_on_ball.tendsto_uniformly_on HasFPowerSeriesOnBall.tendstoUniformlyOn
/-- If a function admits a power series expansion at `x`, then it is the locally uniform limit of
the partial sums of this power series on the disk of convergence, i.e., `f (x + y)`
is the locally uniform limit of `p.partialSum n y` there. -/
theorem HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn (hf : HasFPowerSeriesOnBall f p x r) :
TendstoLocallyUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
(EMetric.ball (0 : E) r) := by
intro u hu x hx
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hx with ⟨r', xr', hr'⟩
have : EMetric.ball (0 : E) r' ∈ 𝓝 x := IsOpen.mem_nhds EMetric.isOpen_ball xr'
refine' ⟨EMetric.ball (0 : E) r', mem_nhdsWithin_of_mem_nhds this, _⟩
simpa [Metric.emetric_ball_nnreal] using hf.tendstoUniformlyOn hr' u hu
#align has_fpower_series_on_ball.tendsto_locally_uniformly_on HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn
/-- If a function admits a power series expansion at `x`, then it is the uniform limit of the
partial sums of this power series on strict subdisks of the disk of convergence, i.e., `f y`
is the uniform limit of `p.partialSum n (y - x)` there. -/
theorem HasFPowerSeriesOnBall.tendstoUniformlyOn' {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r)
(h : (r' : ℝ≥0∞) < r) :
TendstoUniformlyOn (fun n y => p.partialSum n (y - x)) f atTop (Metric.ball (x : E) r') := by
convert (hf.tendstoUniformlyOn h).comp fun y => y - x using 1
· simp [(· ∘ ·)]
· ext z
simp [dist_eq_norm]
#align has_fpower_series_on_ball.tendsto_uniformly_on' HasFPowerSeriesOnBall.tendstoUniformlyOn'
/-- If a function admits a power series expansion at `x`, then it is the locally uniform limit of
the partial sums of this power series on the disk of convergence, i.e., `f y`
is the locally uniform limit of `p.partialSum n (y - x)` there. -/
theorem HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn' (hf : HasFPowerSeriesOnBall f p x r) :
TendstoLocallyUniformlyOn (fun n y => p.partialSum n (y - x)) f atTop
(EMetric.ball (x : E) r) := by
have A : ContinuousOn (fun y : E => y - x) (EMetric.ball (x : E) r) :=
(continuous_id.sub continuous_const).continuousOn
convert hf.tendstoLocallyUniformlyOn.comp (fun y : E => y - x) _ A using 1
· ext z
simp
· intro z
simp [edist_eq_coe_nnnorm, edist_eq_coe_nnnorm_sub]
#align has_fpower_series_on_ball.tendsto_locally_uniformly_on' HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn'
/-- If a function admits a power series expansion on a disk, then it is continuous there. -/
protected theorem HasFPowerSeriesOnBall.continuousOn (hf : HasFPowerSeriesOnBall f p x r) :
ContinuousOn f (EMetric.ball x r) :=
hf.tendstoLocallyUniformlyOn'.continuousOn <|
eventually_of_forall fun n =>
((p.partialSum_continuous n).comp (continuous_id.sub continuous_const)).continuousOn
#align has_fpower_series_on_ball.continuous_on HasFPowerSeriesOnBall.continuousOn
protected theorem HasFPowerSeriesAt.continuousAt (hf : HasFPowerSeriesAt f p x) :
ContinuousAt f x :=
let ⟨_, hr⟩ := hf
hr.continuousOn.continuousAt (EMetric.ball_mem_nhds x hr.r_pos)
#align has_fpower_series_at.continuous_at HasFPowerSeriesAt.continuousAt
protected theorem AnalyticAt.continuousAt (hf : AnalyticAt 𝕜 f x) : ContinuousAt f x :=
let ⟨_, hp⟩ := hf
hp.continuousAt
#align analytic_at.continuous_at AnalyticAt.continuousAt
protected theorem AnalyticOn.continuousOn {s : Set E} (hf : AnalyticOn 𝕜 f s) : ContinuousOn f s :=
fun x hx => (hf x hx).continuousAt.continuousWithinAt
#align analytic_on.continuous_on AnalyticOn.continuousOn
/-- Analytic everywhere implies continuous -/
theorem AnalyticOn.continuous {f : E → F} (fa : AnalyticOn 𝕜 f univ) : Continuous f := by
rw [continuous_iff_continuousOn_univ]; exact fa.continuousOn
/-- In a complete space, the sum of a converging power series `p` admits `p` as a power series.
This is not totally obvious as we need to check the convergence of the series. -/
protected theorem FormalMultilinearSeries.hasFPowerSeriesOnBall [CompleteSpace F]
(p : FormalMultilinearSeries 𝕜 E F) (h : 0 < p.radius) :
HasFPowerSeriesOnBall p.sum p 0 p.radius :=
{ r_le := le_rfl
r_pos := h
hasSum := fun hy => by
rw [zero_add]
exact p.hasSum hy }
#align formal_multilinear_series.has_fpower_series_on_ball FormalMultilinearSeries.hasFPowerSeriesOnBall
theorem HasFPowerSeriesOnBall.sum (h : HasFPowerSeriesOnBall f p x r) {y : E}
(hy : y ∈ EMetric.ball (0 : E) r) : f (x + y) = p.sum y :=
(h.hasSum hy).tsum_eq.symm
#align has_fpower_series_on_ball.sum HasFPowerSeriesOnBall.sum
/-- The sum of a converging power series is continuous in its disk of convergence. -/
protected theorem FormalMultilinearSeries.continuousOn [CompleteSpace F] :
ContinuousOn p.sum (EMetric.ball 0 p.radius) := by
rcases (zero_le p.radius).eq_or_lt with h | h
· simp [← h, continuousOn_empty]
· exact (p.hasFPowerSeriesOnBall h).continuousOn
#align formal_multilinear_series.continuous_on FormalMultilinearSeries.continuousOn
end
/-!
### Uniqueness of power series
If a function `f : E → F` has two representations as power series at a point `x : E`, corresponding
to formal multilinear series `p₁` and `p₂`, then these representations agree term-by-term. That is,
for any `n : ℕ` and `y : E`, `p₁ n (fun i ↦ y) = p₂ n (fun i ↦ y)`. In the one-dimensional case,
when `f : 𝕜 → E`, the continuous multilinear maps `p₁ n` and `p₂ n` are given by
`ContinuousMultilinearMap.mkPiField`, and hence are determined completely by the value of
`p₁ n (fun i ↦ 1)`, so `p₁ = p₂`. Consequently, the radius of convergence for one series can be
transferred to the other.
-/
section Uniqueness
open ContinuousMultilinearMap
theorem Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero {n : ℕ} {p : E[×n]→L[𝕜] F}
(h : (fun y => p fun _ => y) =O[𝓝 0] fun y => ‖y‖ ^ (n + 1)) (y : E) : (p fun _ => y) = 0 := by
obtain ⟨c, c_pos, hc⟩ := h.exists_pos
obtain ⟨t, ht, t_open, z_mem⟩ := eventually_nhds_iff.mp (isBigOWith_iff.mp hc)
obtain ⟨δ, δ_pos, δε⟩ := (Metric.isOpen_iff.mp t_open) 0 z_mem
clear h hc z_mem
cases' n with n
· exact norm_eq_zero.mp (by
-- porting note: the symmetric difference of the `simpa only` sets:
-- added `Nat.zero_eq, zero_add, pow_one`
-- removed `zero_pow', Ne.def, Nat.one_ne_zero, not_false_iff`
simpa only [Nat.zero_eq, fin0_apply_norm, norm_eq_zero, norm_zero, zero_add, pow_one,
mul_zero, norm_le_zero_iff] using ht 0 (δε (Metric.mem_ball_self δ_pos)))
· refine' Or.elim (Classical.em (y = 0))
(fun hy => by simpa only [hy] using p.map_zero) fun hy => _
replace hy := norm_pos_iff.mpr hy
refine' norm_eq_zero.mp (le_antisymm (le_of_forall_pos_le_add fun ε ε_pos => _) (norm_nonneg _))
have h₀ := _root_.mul_pos c_pos (pow_pos hy (n.succ + 1))
obtain ⟨k, k_pos, k_norm⟩ := NormedField.exists_norm_lt 𝕜
(lt_min (mul_pos δ_pos (inv_pos.mpr hy)) (mul_pos ε_pos (inv_pos.mpr h₀)))
have h₁ : ‖k • y‖ < δ := by
rw [norm_smul]
exact inv_mul_cancel_right₀ hy.ne.symm δ ▸
mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_left _ _)) hy
have h₂ :=
calc
‖p fun _ => k • y‖ ≤ c * ‖k • y‖ ^ (n.succ + 1) := by
-- porting note: now Lean wants `_root_.`
simpa only [norm_pow, _root_.norm_norm] using ht (k • y) (δε (mem_ball_zero_iff.mpr h₁))
--simpa only [norm_pow, norm_norm] using ht (k • y) (δε (mem_ball_zero_iff.mpr h₁))
_ = ‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))) := by
-- porting note: added `Nat.succ_eq_add_one` since otherwise `ring` does not conclude.
simp only [norm_smul, mul_pow, Nat.succ_eq_add_one]
-- porting note: removed `rw [pow_succ]`, since it now becomes superfluous.
ring
have h₃ : ‖k‖ * (c * ‖y‖ ^ (n.succ + 1)) < ε :=
inv_mul_cancel_right₀ h₀.ne.symm ε ▸
mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_right _ _)) h₀
calc
‖p fun _ => y‖ = ‖k⁻¹ ^ n.succ‖ * ‖p fun _ => k • y‖ := by
simpa only [inv_smul_smul₀ (norm_pos_iff.mp k_pos), norm_smul, Finset.prod_const,
Finset.card_fin] using
congr_arg norm (p.map_smul_univ (fun _ : Fin n.succ => k⁻¹) fun _ : Fin n.succ => k • y)
_ ≤ ‖k⁻¹ ^ n.succ‖ * (‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1)))) := by gcongr
_ = ‖(k⁻¹ * k) ^ n.succ‖ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))) := by
rw [← mul_assoc]
simp [norm_mul, mul_pow]
_ ≤ 0 + ε := by
rw [inv_mul_cancel (norm_pos_iff.mp k_pos)]
simpa using h₃.le
set_option linter.uppercaseLean3 false in
#align asymptotics.is_O.continuous_multilinear_map_apply_eq_zero Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero
/-- If a formal multilinear series `p` represents the zero function at `x : E`, then the
terms `p n (fun i ↦ y)` appearing in the sum are zero for any `n : ℕ`, `y : E`. -/
theorem HasFPowerSeriesAt.apply_eq_zero {p : FormalMultilinearSeries 𝕜 E F} {x : E}
(h : HasFPowerSeriesAt 0 p x) (n : ℕ) : ∀ y : E, (p n fun _ => y) = 0 := by
refine' Nat.strong_induction_on n fun k hk => _
have psum_eq : p.partialSum (k + 1) = fun y => p k fun _ => y := by
funext z
refine' Finset.sum_eq_single _ (fun b hb hnb => _) fun hn => _
· have := Finset.mem_range_succ_iff.mp hb
simp only [hk b (this.lt_of_ne hnb), Pi.zero_apply]
· exact False.elim (hn (Finset.mem_range.mpr (lt_add_one k)))
replace h := h.isBigO_sub_partialSum_pow k.succ
simp only [psum_eq, zero_sub, Pi.zero_apply, Asymptotics.isBigO_neg_left] at h
exact h.continuousMultilinearMap_apply_eq_zero
#align has_fpower_series_at.apply_eq_zero HasFPowerSeriesAt.apply_eq_zero
/-- A one-dimensional formal multilinear series representing the zero function is zero. -/
theorem HasFPowerSeriesAt.eq_zero {p : FormalMultilinearSeries 𝕜 𝕜 E} {x : 𝕜}
(h : HasFPowerSeriesAt 0 p x) : p = 0 := by
-- porting note: `funext; ext` was `ext (n x)`
funext n
ext x
rw [← mkPiField_apply_one_eq_self (p n)]
-- porting note: nasty hack, was `simp [h.apply_eq_zero n 1]`
have := Or.intro_right ?_ (h.apply_eq_zero n 1)
simpa using this
#align has_fpower_series_at.eq_zero HasFPowerSeriesAt.eq_zero
/-- One-dimensional formal multilinear series representing the same function are equal. -/
theorem HasFPowerSeriesAt.eq_formalMultilinearSeries {p₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E}
{f : 𝕜 → E} {x : 𝕜} (h₁ : HasFPowerSeriesAt f p₁ x) (h₂ : HasFPowerSeriesAt f p₂ x) : p₁ = p₂ :=
sub_eq_zero.mp (HasFPowerSeriesAt.eq_zero (by simpa only [sub_self] using h₁.sub h₂))
#align has_fpower_series_at.eq_formal_multilinear_series HasFPowerSeriesAt.eq_formalMultilinearSeries
theorem HasFPowerSeriesAt.eq_formalMultilinearSeries_of_eventually
{p q : FormalMultilinearSeries 𝕜 𝕜 E} {f g : 𝕜 → E} {x : 𝕜} (hp : HasFPowerSeriesAt f p x)
(hq : HasFPowerSeriesAt g q x) (heq : ∀ᶠ z in 𝓝 x, f z = g z) : p = q :=
(hp.congr heq).eq_formalMultilinearSeries hq
#align has_fpower_series_at.eq_formal_multilinear_series_of_eventually HasFPowerSeriesAt.eq_formalMultilinearSeries_of_eventually
/-- A one-dimensional formal multilinear series representing a locally zero function is zero. -/
theorem HasFPowerSeriesAt.eq_zero_of_eventually {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E}
{x : 𝕜} (hp : HasFPowerSeriesAt f p x) (hf : f =ᶠ[𝓝 x] 0) : p = 0 :=
(hp.congr hf).eq_zero
#align has_fpower_series_at.eq_zero_of_eventually HasFPowerSeriesAt.eq_zero_of_eventually
/-- If a function `f : 𝕜 → E` has two power series representations at `x`, then the given radii in
which convergence is guaranteed may be interchanged. This can be useful when the formal multilinear
series in one representation has a particularly nice form, but the other has a larger radius. -/
theorem HasFPowerSeriesOnBall.exchange_radius {p₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E}
{r₁ r₂ : ℝ≥0∞} {x : 𝕜} (h₁ : HasFPowerSeriesOnBall f p₁ x r₁)
(h₂ : HasFPowerSeriesOnBall f p₂ x r₂) : HasFPowerSeriesOnBall f p₁ x r₂ :=
h₂.hasFPowerSeriesAt.eq_formalMultilinearSeries h₁.hasFPowerSeriesAt ▸ h₂
#align has_fpower_series_on_ball.exchange_radius HasFPowerSeriesOnBall.exchange_radius
/-- If a function `f : 𝕜 → E` has power series representation `p` on a ball of some radius and for
each positive radius it has some power series representation, then `p` converges to `f` on the whole
`𝕜`. -/
theorem HasFPowerSeriesOnBall.r_eq_top_of_exists {f : 𝕜 → E} {r : ℝ≥0∞} {x : 𝕜}
{p : FormalMultilinearSeries 𝕜 𝕜 E} (h : HasFPowerSeriesOnBall f p x r)
(h' : ∀ (r' : ℝ≥0) (_ : 0 < r'), ∃ p' : FormalMultilinearSeries 𝕜 𝕜 E,
HasFPowerSeriesOnBall f p' x r') :
HasFPowerSeriesOnBall f p x ∞ :=
{ r_le := ENNReal.le_of_forall_pos_nnreal_lt fun r hr _ =>
let ⟨_, hp'⟩ := h' r hr
(h.exchange_radius hp').r_le
r_pos := ENNReal.coe_lt_top
hasSum := fun {y} _ =>
let ⟨r', hr'⟩ := exists_gt ‖y‖₊
let ⟨_, hp'⟩ := h' r' hr'.ne_bot.bot_lt
(h.exchange_radius hp').hasSum <| mem_emetric_ball_zero_iff.mpr (ENNReal.coe_lt_coe.2 hr') }
#align has_fpower_series_on_ball.r_eq_top_of_exists HasFPowerSeriesOnBall.r_eq_top_of_exists
end Uniqueness
/-!
### Changing origin in a power series
If a function is analytic in a disk `D(x, R)`, then it is analytic in any disk contained in that
one. Indeed, one can write
$$
f (x + y + z) = \sum_{n} p_n (y + z)^n = \sum_{n, k} \binom{n}{k} p_n y^{n-k} z^k
= \sum_{k} \Bigl(\sum_{n} \binom{n}{k} p_n y^{n-k}\Bigr) z^k.
$$
The corresponding power series has thus a `k`-th coefficient equal to
$\sum_{n} \binom{n}{k} p_n y^{n-k}$. In the general case where `pₙ` is a multilinear map, this has
to be interpreted suitably: instead of having a binomial coefficient, one should sum over all
possible subsets `s` of `Fin n` of cardinal `k`, and attribute `z` to the indices in `s` and
`y` to the indices outside of `s`.
In this paragraph, we implement this. The new power series is called `p.changeOrigin y`. Then, we
check its convergence and the fact that its sum coincides with the original sum. The outcome of this
discussion is that the set of points where a function is analytic is open.
-/
namespace FormalMultilinearSeries
section
variable (p : FormalMultilinearSeries 𝕜 E F) {x y : E} {r R : ℝ≥0}
/-- A term of `FormalMultilinearSeries.changeOriginSeries`.
Given a formal multilinear series `p` and a point `x` in its ball of convergence,
`p.changeOrigin x` is a formal multilinear series such that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense. Each term of `p.changeOrigin x`
is itself an analytic function of `x` given by the series `p.changeOriginSeries`. Each term in
`changeOriginSeries` is the sum of `changeOriginSeriesTerm`'s over all `s` of cardinality `l`.
The definition is such that `p.changeOriginSeriesTerm k l s hs (fun _ ↦ x) (fun _ ↦ y) =
p (k + l) (s.piecewise (fun _ ↦ x) (fun _ ↦ y))`
-/
def changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
E[×l]→L[𝕜] E[×k]→L[𝕜] F := by
let a := ContinuousMultilinearMap.curryFinFinset 𝕜 E F hs
(by erw [Finset.card_compl, Fintype.card_fin, hs, add_tsub_cancel_right])
exact a (p (k + l))
#align formal_multilinear_series.change_origin_series_term FormalMultilinearSeries.changeOriginSeriesTerm
theorem changeOriginSeriesTerm_apply (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l)
(x y : E) :
(p.changeOriginSeriesTerm k l s hs (fun _ => x) fun _ => y) =
p (k + l) (s.piecewise (fun _ => x) fun _ => y) :=
ContinuousMultilinearMap.curryFinFinset_apply_const _ _ _ _ _
#align formal_multilinear_series.change_origin_series_term_apply FormalMultilinearSeries.changeOriginSeriesTerm_apply
@[simp]
theorem norm_changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
‖p.changeOriginSeriesTerm k l s hs‖ = ‖p (k + l)‖ := by
simp only [changeOriginSeriesTerm, LinearIsometryEquiv.norm_map]
#align formal_multilinear_series.norm_change_origin_series_term FormalMultilinearSeries.norm_changeOriginSeriesTerm
@[simp]
theorem nnnorm_changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
‖p.changeOriginSeriesTerm k l s hs‖₊ = ‖p (k + l)‖₊ := by
simp only [changeOriginSeriesTerm, LinearIsometryEquiv.nnnorm_map]
#align formal_multilinear_series.nnnorm_change_origin_series_term FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm
theorem nnnorm_changeOriginSeriesTerm_apply_le (k l : ℕ) (s : Finset (Fin (k + l)))
(hs : s.card = l) (x y : E) :
‖p.changeOriginSeriesTerm k l s hs (fun _ => x) fun _ => y‖₊ ≤
‖p (k + l)‖₊ * ‖x‖₊ ^ l * ‖y‖₊ ^ k := by
rw [← p.nnnorm_changeOriginSeriesTerm k l s hs, ← Fin.prod_const, ← Fin.prod_const]
apply ContinuousMultilinearMap.le_of_op_nnnorm_le
apply ContinuousMultilinearMap.le_op_nnnorm
#align formal_multilinear_series.nnnorm_change_origin_series_term_apply_le FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm_apply_le
/-- The power series for `f.changeOrigin k`.
Given a formal multilinear series `p` and a point `x` in its ball of convergence,
`p.changeOrigin x` is a formal multilinear series such that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense. Its `k`-th term is the sum of
the series `p.changeOriginSeries k`. -/
def changeOriginSeries (k : ℕ) : FormalMultilinearSeries 𝕜 E (E[×k]→L[𝕜] F) := fun l =>
∑ s : { s : Finset (Fin (k + l)) // Finset.card s = l }, p.changeOriginSeriesTerm k l s s.2
#align formal_multilinear_series.change_origin_series FormalMultilinearSeries.changeOriginSeries
theorem nnnorm_changeOriginSeries_le_tsum (k l : ℕ) :
‖p.changeOriginSeries k l‖₊ ≤
∑' _ : { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + l)‖₊ :=
(nnnorm_sum_le _ (fun t => changeOriginSeriesTerm p k l (Subtype.val t) t.prop)).trans_eq <| by
simp_rw [tsum_fintype, nnnorm_changeOriginSeriesTerm (p := p) (k := k) (l := l)]
#align formal_multilinear_series.nnnorm_change_origin_series_le_tsum FormalMultilinearSeries.nnnorm_changeOriginSeries_le_tsum
theorem nnnorm_changeOriginSeries_apply_le_tsum (k l : ℕ) (x : E) :
‖p.changeOriginSeries k l fun _ => x‖₊ ≤
∑' _ : { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + l)‖₊ * ‖x‖₊ ^ l := by
rw [NNReal.tsum_mul_right, ← Fin.prod_const]
exact (p.changeOriginSeries k l).le_of_op_nnnorm_le _ (p.nnnorm_changeOriginSeries_le_tsum _ _)
#align formal_multilinear_series.nnnorm_change_origin_series_apply_le_tsum FormalMultilinearSeries.nnnorm_changeOriginSeries_apply_le_tsum
/-- Changing the origin of a formal multilinear series `p`, so that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense.
-/
def changeOrigin (x : E) : FormalMultilinearSeries 𝕜 E F :=
fun k => (p.changeOriginSeries k).sum x
#align formal_multilinear_series.change_origin FormalMultilinearSeries.changeOrigin
/-- An auxiliary equivalence useful in the proofs about
`FormalMultilinearSeries.changeOriginSeries`: the set of triples `(k, l, s)`, where `s` is a
`Finset (Fin (k + l))` of cardinality `l` is equivalent to the set of pairs `(n, s)`, where `s` is a
`Finset (Fin n)`.
The forward map sends `(k, l, s)` to `(k + l, s)` and the inverse map sends `(n, s)` to
`(n - Finset.card s, Finset.card s, s)`. The actual definition is less readable because of problems
with non-definitional equalities. -/
@[simps]
def changeOriginIndexEquiv :
(Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) ≃ Σn : ℕ, Finset (Fin n) where
toFun s := ⟨s.1 + s.2.1, s.2.2⟩
invFun s :=
⟨s.1 - s.2.card, s.2.card,
⟨s.2.map
(Fin.castIso <| (tsub_add_cancel_of_le <| card_finset_fin_le s.2).symm).toEquiv.toEmbedding,
Finset.card_map _⟩⟩
left_inv := by
rintro ⟨k, l, ⟨s : Finset (Fin <| k + l), hs : s.card = l⟩⟩
dsimp only [Subtype.coe_mk]
-- Lean can't automatically generalize `k' = k + l - s.card`, `l' = s.card`, so we explicitly
-- formulate the generalized goal
suffices ∀ k' l', k' = k → l' = l → ∀ (hkl : k + l = k' + l') (hs'),
(⟨k', l', ⟨Finset.map (Fin.castIso hkl).toEquiv.toEmbedding s, hs'⟩⟩ :
Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) = ⟨k, l, ⟨s, hs⟩⟩ by
apply this <;> simp only [hs, add_tsub_cancel_right]
rintro _ _ rfl rfl hkl hs'
simp only [Equiv.refl_toEmbedding, Fin.castIso_refl, Finset.map_refl, eq_self_iff_true,
OrderIso.refl_toEquiv, and_self_iff, heq_iff_eq]
right_inv := by
rintro ⟨n, s⟩
simp [tsub_add_cancel_of_le (card_finset_fin_le s), Fin.castIso_to_equiv]
#align formal_multilinear_series.change_origin_index_equiv FormalMultilinearSeries.changeOriginIndexEquiv
theorem changeOriginSeries_summable_aux₁ {r r' : ℝ≥0} (hr : (r + r' : ℝ≥0∞) < p.radius) :
Summable fun s : Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l } =>
‖p (s.1 + s.2.1)‖₊ * r ^ s.2.1 * r' ^ s.1 := by
rw [← changeOriginIndexEquiv.symm.summable_iff]
dsimp only [Function.comp_def, changeOriginIndexEquiv_symm_apply_fst,
changeOriginIndexEquiv_symm_apply_snd_fst]
have : ∀ n : ℕ,
HasSum (fun s : Finset (Fin n) => ‖p (n - s.card + s.card)‖₊ * r ^ s.card * r' ^ (n - s.card))
(‖p n‖₊ * (r + r') ^ n) := by
intro n
-- TODO: why `simp only [tsub_add_cancel_of_le (card_finset_fin_le _)]` fails?
convert_to HasSum (fun s : Finset (Fin n) => ‖p n‖₊ * (r ^ s.card * r' ^ (n - s.card))) _
· ext1 s
rw [tsub_add_cancel_of_le (card_finset_fin_le _), mul_assoc]
rw [← Fin.sum_pow_mul_eq_add_pow]
exact (hasSum_fintype _).mul_left _
refine' NNReal.summable_sigma.2 ⟨fun n => (this n).summable, _⟩
simp only [(this _).tsum_eq]
exact p.summable_nnnorm_mul_pow hr
#align formal_multilinear_series.change_origin_series_summable_aux₁ FormalMultilinearSeries.changeOriginSeries_summable_aux₁
theorem changeOriginSeries_summable_aux₂ (hr : (r : ℝ≥0∞) < p.radius) (k : ℕ) :
Summable fun s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l } =>
‖p (k + s.1)‖₊ * r ^ s.1 := by
rcases ENNReal.lt_iff_exists_add_pos_lt.1 hr with ⟨r', h0, hr'⟩
simpa only [mul_inv_cancel_right₀ (pow_pos h0 _).ne'] using
((NNReal.summable_sigma.1 (p.changeOriginSeries_summable_aux₁ hr')).1 k).mul_right (r' ^ k)⁻¹
#align formal_multilinear_series.change_origin_series_summable_aux₂ FormalMultilinearSeries.changeOriginSeries_summable_aux₂
theorem changeOriginSeries_summable_aux₃ {r : ℝ≥0} (hr : ↑r < p.radius) (k : ℕ) :
Summable fun l : ℕ => ‖p.changeOriginSeries k l‖₊ * r ^ l := by
refine' NNReal.summable_of_le
(fun n => _) (NNReal.summable_sigma.1 <| p.changeOriginSeries_summable_aux₂ hr k).2
simp only [NNReal.tsum_mul_right]
exact mul_le_mul' (p.nnnorm_changeOriginSeries_le_tsum _ _) le_rfl
#align formal_multilinear_series.change_origin_series_summable_aux₃ FormalMultilinearSeries.changeOriginSeries_summable_aux₃
theorem le_changeOriginSeries_radius (k : ℕ) : p.radius ≤ (p.changeOriginSeries k).radius :=
ENNReal.le_of_forall_nnreal_lt fun _r hr =>
le_radius_of_summable_nnnorm _ (p.changeOriginSeries_summable_aux₃ hr k)
#align formal_multilinear_series.le_change_origin_series_radius FormalMultilinearSeries.le_changeOriginSeries_radius
theorem nnnorm_changeOrigin_le (k : ℕ) (h : (‖x‖₊ : ℝ≥0∞) < p.radius) :
‖p.changeOrigin x k‖₊ ≤
∑' s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + s.1)‖₊ * ‖x‖₊ ^ s.1 := by
refine' tsum_of_nnnorm_bounded _ fun l => p.nnnorm_changeOriginSeries_apply_le_tsum k l x
have := p.changeOriginSeries_summable_aux₂ h k
refine' HasSum.sigma this.hasSum fun l => _
exact ((NNReal.summable_sigma.1 this).1 l).hasSum
#align formal_multilinear_series.nnnorm_change_origin_le FormalMultilinearSeries.nnnorm_changeOrigin_le
/-- The radius of convergence of `p.changeOrigin x` is at least `p.radius - ‖x‖`. In other words,
`p.changeOrigin x` is well defined on the largest ball contained in the original ball of
convergence. -/
theorem changeOrigin_radius : p.radius - ‖x‖₊ ≤ (p.changeOrigin x).radius := by
refine' ENNReal.le_of_forall_pos_nnreal_lt fun r _h0 hr => _
rw [lt_tsub_iff_right, add_comm] at hr
have hr' : (‖x‖₊ : ℝ≥0∞) < p.radius := (le_add_right le_rfl).trans_lt hr
apply le_radius_of_summable_nnnorm
have : ∀ k : ℕ,
‖p.changeOrigin x k‖₊ * r ^ k ≤
(∑' s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + s.1)‖₊ * ‖x‖₊ ^ s.1) *
r ^ k :=
fun k => mul_le_mul_right' (p.nnnorm_changeOrigin_le k hr') (r ^ k)
refine' NNReal.summable_of_le this _
simpa only [← NNReal.tsum_mul_right] using
(NNReal.summable_sigma.1 (p.changeOriginSeries_summable_aux₁ hr)).2
#align formal_multilinear_series.change_origin_radius FormalMultilinearSeries.changeOrigin_radius
end
-- From this point on, assume that the space is complete, to make sure that series that converge
-- in norm also converge in `F`.
variable [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x y : E} {r R : ℝ≥0}
theorem hasFPowerSeriesOnBall_changeOrigin (k : ℕ) (hr : 0 < p.radius) :
HasFPowerSeriesOnBall (fun x => p.changeOrigin x k) (p.changeOriginSeries k) 0 p.radius :=
have := p.le_changeOriginSeries_radius k
((p.changeOriginSeries k).hasFPowerSeriesOnBall (hr.trans_le this)).mono hr this
#align formal_multilinear_series.has_fpower_series_on_ball_change_origin FormalMultilinearSeries.hasFPowerSeriesOnBall_changeOrigin
/-- Summing the series `p.changeOrigin x` at a point `y` gives back `p (x + y)`. -/
theorem changeOrigin_eval (h : (‖x‖₊ + ‖y‖₊ : ℝ≥0∞) < p.radius) :
(p.changeOrigin x).sum y = p.sum (x + y) := by
have radius_pos : 0 < p.radius := lt_of_le_of_lt (zero_le _) h
have x_mem_ball : x ∈ EMetric.ball (0 : E) p.radius :=
mem_emetric_ball_zero_iff.2 ((le_add_right le_rfl).trans_lt h)
have y_mem_ball : y ∈ EMetric.ball (0 : E) (p.changeOrigin x).radius := by
refine' mem_emetric_ball_zero_iff.2 (lt_of_lt_of_le _ p.changeOrigin_radius)
rwa [lt_tsub_iff_right, add_comm]
have x_add_y_mem_ball : x + y ∈ EMetric.ball (0 : E) p.radius := by
refine' mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt _ h)
exact mod_cast nnnorm_add_le x y
set f : (Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) → F := fun s =>
p.changeOriginSeriesTerm s.1 s.2.1 s.2.2 s.2.2.2 (fun _ => x) fun _ => y
have hsf : Summable f := by
refine' .of_nnnorm_bounded _ (p.changeOriginSeries_summable_aux₁ h) _
rintro ⟨k, l, s, hs⟩
dsimp only [Subtype.coe_mk]
exact p.nnnorm_changeOriginSeriesTerm_apply_le _ _ _ _ _ _
have hf : HasSum f ((p.changeOrigin x).sum y) := by
refine' HasSum.sigma_of_hasSum ((p.changeOrigin x).summable y_mem_ball).hasSum (fun k => _) hsf
· dsimp only
refine' ContinuousMultilinearMap.hasSum_eval _ _
have := (p.hasFPowerSeriesOnBall_changeOrigin k radius_pos).hasSum x_mem_ball
rw [zero_add] at this
refine' HasSum.sigma_of_hasSum this (fun l => _) _
· simp only [changeOriginSeries, ContinuousMultilinearMap.sum_apply]
apply hasSum_fintype
· refine' .of_nnnorm_bounded _
(p.changeOriginSeries_summable_aux₂ (mem_emetric_ball_zero_iff.1 x_mem_ball) k) fun s => _
refine' (ContinuousMultilinearMap.le_op_nnnorm _ _).trans_eq _
simp
refine' hf.unique (changeOriginIndexEquiv.symm.hasSum_iff.1 _)
refine' HasSum.sigma_of_hasSum
(p.hasSum x_add_y_mem_ball) (fun n => _) (changeOriginIndexEquiv.symm.summable_iff.2 hsf)
erw [(p n).map_add_univ (fun _ => x) fun _ => y]
-- porting note: added explicit function
convert hasSum_fintype (fun c : Finset (Fin n) => f (changeOriginIndexEquiv.symm ⟨n, c⟩))
rename_i s _
dsimp only [changeOriginSeriesTerm, (· ∘ ·), changeOriginIndexEquiv_symm_apply_fst,
changeOriginIndexEquiv_symm_apply_snd_fst, changeOriginIndexEquiv_symm_apply_snd_snd_coe]
rw [ContinuousMultilinearMap.curryFinFinset_apply_const]
have : ∀ (m) (hm : n = m), p n (s.piecewise (fun _ => x) fun _ => y) =
p m ((s.map (Fin.castIso hm).toEquiv.toEmbedding).piecewise (fun _ => x) fun _ => y) := by
rintro m rfl
simp (config := { unfoldPartialApp := true }) [Finset.piecewise]
apply this
#align formal_multilinear_series.change_origin_eval FormalMultilinearSeries.changeOrigin_eval
/-- Power series terms are analytic as we vary the origin -/
theorem analyticAt_changeOrigin (p : FormalMultilinearSeries 𝕜 E F) (rp : p.radius > 0) (n : ℕ) :
AnalyticAt 𝕜 (fun x ↦ p.changeOrigin x n) 0 :=
(FormalMultilinearSeries.hasFPowerSeriesOnBall_changeOrigin p n rp).analyticAt
end FormalMultilinearSeries
section
variable [CompleteSpace F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x y : E} {r : ℝ≥0∞}
/-- If a function admits a power series expansion `p` on a ball `B (x, r)`, then it also admits a
power series on any subball of this ball (even with a different center), given by `p.changeOrigin`.
-/
theorem HasFPowerSeriesOnBall.changeOrigin (hf : HasFPowerSeriesOnBall f p x r)
(h : (‖y‖₊ : ℝ≥0∞) < r) : HasFPowerSeriesOnBall f (p.changeOrigin y) (x + y) (r - ‖y‖₊) :=
{ r_le := by
apply le_trans _ p.changeOrigin_radius
exact tsub_le_tsub hf.r_le le_rfl
r_pos := by simp [h]
hasSum := fun {z} hz => by
have : f (x + y + z) =
FormalMultilinearSeries.sum (FormalMultilinearSeries.changeOrigin p y) z := by
rw [mem_emetric_ball_zero_iff, lt_tsub_iff_right, add_comm] at hz
rw [p.changeOrigin_eval (hz.trans_le hf.r_le), add_assoc, hf.sum]
refine' mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt _ hz)
exact mod_cast nnnorm_add_le y z
rw [this]
apply (p.changeOrigin y).hasSum
refine' EMetric.ball_subset_ball (le_trans _ p.changeOrigin_radius) hz
exact tsub_le_tsub hf.r_le le_rfl }
#align has_fpower_series_on_ball.change_origin HasFPowerSeriesOnBall.changeOrigin
/-- If a function admits a power series expansion `p` on an open ball `B (x, r)`, then
it is analytic at every point of this ball. -/
theorem HasFPowerSeriesOnBall.analyticAt_of_mem (hf : HasFPowerSeriesOnBall f p x r)
(h : y ∈ EMetric.ball x r) : AnalyticAt 𝕜 f y := by
have : (‖y - x‖₊ : ℝ≥0∞) < r := by simpa [edist_eq_coe_nnnorm_sub] using h
have := hf.changeOrigin this
rw [add_sub_cancel'_right] at this
exact this.analyticAt
#align has_fpower_series_on_ball.analytic_at_of_mem HasFPowerSeriesOnBall.analyticAt_of_mem
theorem HasFPowerSeriesOnBall.analyticOn (hf : HasFPowerSeriesOnBall f p x r) :
AnalyticOn 𝕜 f (EMetric.ball x r) :=
fun _y hy => hf.analyticAt_of_mem hy
#align has_fpower_series_on_ball.analytic_on HasFPowerSeriesOnBall.analyticOn
variable (𝕜 f)
/-- For any function `f` from a normed vector space to a Banach space, the set of points `x` such
that `f` is analytic at `x` is open. -/
theorem isOpen_analyticAt : IsOpen { x | AnalyticAt 𝕜 f x } := by
rw [isOpen_iff_mem_nhds]
rintro x ⟨p, r, hr⟩
exact mem_of_superset (EMetric.ball_mem_nhds _ hr.r_pos) fun y hy => hr.analyticAt_of_mem hy
#align is_open_analytic_at isOpen_analyticAt
variable {𝕜}
theorem AnalyticAt.eventually_analyticAt {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∀ᶠ y in 𝓝 x, AnalyticAt 𝕜 f y :=
(isOpen_analyticAt 𝕜 f).mem_nhds h
theorem AnalyticAt.exists_mem_nhds_analyticOn {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∃ s ∈ 𝓝 x, AnalyticOn 𝕜 f s :=
h.eventually_analyticAt.exists_mem
/-- If we're analytic at a point, we're analytic in a nonempty ball -/
theorem AnalyticAt.exists_ball_analyticOn {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∃ r : ℝ, 0 < r ∧ AnalyticOn 𝕜 f (Metric.ball x r) :=
Metric.isOpen_iff.mp (isOpen_analyticAt _ _) _ h
end
section
open FormalMultilinearSeries
variable {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E} {z₀ : 𝕜}
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z)) := by
refine' ⟨fun ⟨r, _, r_pos, h⟩ =>
eventually_of_mem (EMetric.ball_mem_nhds 0 r_pos) fun _ => by
|
simpa using h
|
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z)) := by
refine' ⟨fun ⟨r, _, r_pos, h⟩ =>
eventually_of_mem (EMetric.ball_mem_nhds 0 r_pos) fun _ => by
|
Mathlib.Analysis.Analytic.Basic.1430_0.jQw1fRSE1vGpOll
|
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z))
|
Mathlib_Analysis_Analytic_Basic
|
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
p : FormalMultilinearSeries 𝕜 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
⊢ (∀ᶠ (z : 𝕜) in 𝓝 0, HasSum (fun n => z ^ n • coeff p n) (f (z₀ + z))) → HasFPowerSeriesAt f p z₀
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Logic.Equiv.Fin
import Mathlib.Topology.Algebra.InfiniteSum.Module
#align_import analysis.analytic.basic from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
/-!
# Analytic functions
A function is analytic in one dimension around `0` if it can be written as a converging power series
`Σ pₙ zⁿ`. This definition can be extended to any dimension (even in infinite dimension) by
requiring that `pₙ` is a continuous `n`-multilinear map. In general, `pₙ` is not unique (in two
dimensions, taking `p₂ (x, y) (x', y') = x y'` or `y x'` gives the same map when applied to a
vector `(x, y) (x, y)`). A way to guarantee uniqueness is to take a symmetric `pₙ`, but this is not
always possible in nonzero characteristic (in characteristic 2, the previous example has no
symmetric representative). Therefore, we do not insist on symmetry or uniqueness in the definition,
and we only require the existence of a converging series.
The general framework is important to say that the exponential map on bounded operators on a Banach
space is analytic, as well as the inverse on invertible operators.
## Main definitions
Let `p` be a formal multilinear series from `E` to `F`, i.e., `p n` is a multilinear map on `E^n`
for `n : ℕ`.
* `p.radius`: the largest `r : ℝ≥0∞` such that `‖p n‖ * r^n` grows subexponentially.
* `p.le_radius_of_bound`, `p.le_radius_of_bound_nnreal`, `p.le_radius_of_isBigO`: if `‖p n‖ * r ^ n`
is bounded above, then `r ≤ p.radius`;
* `p.isLittleO_of_lt_radius`, `p.norm_mul_pow_le_mul_pow_of_lt_radius`,
`p.isLittleO_one_of_lt_radius`,
`p.norm_mul_pow_le_of_lt_radius`, `p.nnnorm_mul_pow_le_of_lt_radius`: if `r < p.radius`, then
`‖p n‖ * r ^ n` tends to zero exponentially;
* `p.lt_radius_of_isBigO`: if `r ≠ 0` and `‖p n‖ * r ^ n = O(a ^ n)` for some `-1 < a < 1`, then
`r < p.radius`;
* `p.partialSum n x`: the sum `∑_{i = 0}^{n-1} pᵢ xⁱ`.
* `p.sum x`: the sum `∑'_{i = 0}^{∞} pᵢ xⁱ`.
Additionally, let `f` be a function from `E` to `F`.
* `HasFPowerSeriesOnBall f p x r`: on the ball of center `x` with radius `r`,
`f (x + y) = ∑'_n pₙ yⁿ`.
* `HasFPowerSeriesAt f p x`: on some ball of center `x` with positive radius, holds
`HasFPowerSeriesOnBall f p x r`.
* `AnalyticAt 𝕜 f x`: there exists a power series `p` such that holds `HasFPowerSeriesAt f p x`.
* `AnalyticOn 𝕜 f s`: the function `f` is analytic at every point of `s`.
We develop the basic properties of these notions, notably:
* If a function admits a power series, it is continuous (see
`HasFPowerSeriesOnBall.continuousOn` and `HasFPowerSeriesAt.continuousAt` and
`AnalyticAt.continuousAt`).
* In a complete space, the sum of a formal power series with positive radius is well defined on the
disk of convergence, see `FormalMultilinearSeries.hasFPowerSeriesOnBall`.
* If a function admits a power series in a ball, then it is analytic at any point `y` of this ball,
and the power series there can be expressed in terms of the initial power series `p` as
`p.changeOrigin y`. See `HasFPowerSeriesOnBall.changeOrigin`. It follows in particular that
the set of points at which a given function is analytic is open, see `isOpen_analyticAt`.
## Implementation details
We only introduce the radius of convergence of a power series, as `p.radius`.
For a power series in finitely many dimensions, there is a finer (directional, coordinate-dependent)
notion, describing the polydisk of convergence. This notion is more specific, and not necessary to
build the general theory. We do not define it here.
-/
noncomputable section
variable {𝕜 E F G : Type*}
open Topology Classical BigOperators NNReal Filter ENNReal
open Set Filter Asymptotics
namespace FormalMultilinearSeries
variable [Ring 𝕜] [AddCommGroup E] [AddCommGroup F] [Module 𝕜 E] [Module 𝕜 F]
variable [TopologicalSpace E] [TopologicalSpace F]
variable [TopologicalAddGroup E] [TopologicalAddGroup F]
variable [ContinuousConstSMul 𝕜 E] [ContinuousConstSMul 𝕜 F]
/-- Given a formal multilinear series `p` and a vector `x`, then `p.sum x` is the sum `Σ pₙ xⁿ`. A
priori, it only behaves well when `‖x‖ < p.radius`. -/
protected def sum (p : FormalMultilinearSeries 𝕜 E F) (x : E) : F :=
∑' n : ℕ, p n fun _ => x
#align formal_multilinear_series.sum FormalMultilinearSeries.sum
/-- Given a formal multilinear series `p` and a vector `x`, then `p.partialSum n x` is the sum
`Σ pₖ xᵏ` for `k ∈ {0,..., n-1}`. -/
def partialSum (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) (x : E) : F :=
∑ k in Finset.range n, p k fun _ : Fin k => x
#align formal_multilinear_series.partial_sum FormalMultilinearSeries.partialSum
/-- The partial sums of a formal multilinear series are continuous. -/
theorem partialSum_continuous (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) :
Continuous (p.partialSum n) := by
unfold partialSum -- Porting note: added
continuity
#align formal_multilinear_series.partial_sum_continuous FormalMultilinearSeries.partialSum_continuous
end FormalMultilinearSeries
/-! ### The radius of a formal multilinear series -/
variable [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F]
[NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G]
namespace FormalMultilinearSeries
variable (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
/-- The radius of a formal multilinear series is the largest `r` such that the sum `Σ ‖pₙ‖ ‖y‖ⁿ`
converges for all `‖y‖ < r`. This implies that `Σ pₙ yⁿ` converges for all `‖y‖ < r`, but these
definitions are *not* equivalent in general. -/
def radius (p : FormalMultilinearSeries 𝕜 E F) : ℝ≥0∞ :=
⨆ (r : ℝ≥0) (C : ℝ) (_ : ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C), (r : ℝ≥0∞)
#align formal_multilinear_series.radius FormalMultilinearSeries.radius
/-- If `‖pₙ‖ rⁿ` is bounded in `n`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_bound (C : ℝ) {r : ℝ≥0} (h : ∀ n : ℕ, ‖p n‖ * (r : ℝ) ^ n ≤ C) :
(r : ℝ≥0∞) ≤ p.radius :=
le_iSup_of_le r <| le_iSup_of_le C <| le_iSup (fun _ => (r : ℝ≥0∞)) h
#align formal_multilinear_series.le_radius_of_bound FormalMultilinearSeries.le_radius_of_bound
/-- If `‖pₙ‖ rⁿ` is bounded in `n`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_bound_nnreal (C : ℝ≥0) {r : ℝ≥0} (h : ∀ n : ℕ, ‖p n‖₊ * r ^ n ≤ C) :
(r : ℝ≥0∞) ≤ p.radius :=
p.le_radius_of_bound C fun n => mod_cast h n
#align formal_multilinear_series.le_radius_of_bound_nnreal FormalMultilinearSeries.le_radius_of_bound_nnreal
/-- If `‖pₙ‖ rⁿ = O(1)`, as `n → ∞`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_isBigO (h : (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] fun _ => (1 : ℝ)) :
↑r ≤ p.radius :=
Exists.elim (isBigO_one_nat_atTop_iff.1 h) fun C hC =>
p.le_radius_of_bound C fun n => (le_abs_self _).trans (hC n)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.le_radius_of_is_O FormalMultilinearSeries.le_radius_of_isBigO
theorem le_radius_of_eventually_le (C) (h : ∀ᶠ n in atTop, ‖p n‖ * (r : ℝ) ^ n ≤ C) :
↑r ≤ p.radius :=
p.le_radius_of_isBigO <| IsBigO.of_bound C <| h.mono fun n hn => by simpa
#align formal_multilinear_series.le_radius_of_eventually_le FormalMultilinearSeries.le_radius_of_eventually_le
theorem le_radius_of_summable_nnnorm (h : Summable fun n => ‖p n‖₊ * r ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_bound_nnreal (∑' n, ‖p n‖₊ * r ^ n) fun _ => le_tsum' h _
#align formal_multilinear_series.le_radius_of_summable_nnnorm FormalMultilinearSeries.le_radius_of_summable_nnnorm
theorem le_radius_of_summable (h : Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_summable_nnnorm <| by
simp only [← coe_nnnorm] at h
exact mod_cast h
#align formal_multilinear_series.le_radius_of_summable FormalMultilinearSeries.le_radius_of_summable
theorem radius_eq_top_of_forall_nnreal_isBigO
(h : ∀ r : ℝ≥0, (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] fun _ => (1 : ℝ)) : p.radius = ∞ :=
ENNReal.eq_top_of_forall_nnreal_le fun r => p.le_radius_of_isBigO (h r)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.radius_eq_top_of_forall_nnreal_is_O FormalMultilinearSeries.radius_eq_top_of_forall_nnreal_isBigO
theorem radius_eq_top_of_eventually_eq_zero (h : ∀ᶠ n in atTop, p n = 0) : p.radius = ∞ :=
p.radius_eq_top_of_forall_nnreal_isBigO fun r =>
(isBigO_zero _ _).congr' (h.mono fun n hn => by simp [hn]) EventuallyEq.rfl
#align formal_multilinear_series.radius_eq_top_of_eventually_eq_zero FormalMultilinearSeries.radius_eq_top_of_eventually_eq_zero
theorem radius_eq_top_of_forall_image_add_eq_zero (n : ℕ) (hn : ∀ m, p (m + n) = 0) :
p.radius = ∞ :=
p.radius_eq_top_of_eventually_eq_zero <|
mem_atTop_sets.2 ⟨n, fun _ hk => tsub_add_cancel_of_le hk ▸ hn _⟩
#align formal_multilinear_series.radius_eq_top_of_forall_image_add_eq_zero FormalMultilinearSeries.radius_eq_top_of_forall_image_add_eq_zero
@[simp]
theorem constFormalMultilinearSeries_radius {v : F} :
(constFormalMultilinearSeries 𝕜 E v).radius = ⊤ :=
(constFormalMultilinearSeries 𝕜 E v).radius_eq_top_of_forall_image_add_eq_zero 1
(by simp [constFormalMultilinearSeries])
#align formal_multilinear_series.const_formal_multilinear_series_radius FormalMultilinearSeries.constFormalMultilinearSeries_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` tends to zero exponentially:
for some `0 < a < 1`, `‖p n‖ rⁿ = o(aⁿ)`. -/
theorem isLittleO_of_lt_radius (h : ↑r < p.radius) :
∃ a ∈ Ioo (0 : ℝ) 1, (fun n => ‖p n‖ * (r : ℝ) ^ n) =o[atTop] (a ^ ·) := by
have := (TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 4
rw [this]
-- Porting note: was
-- rw [(TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 4]
simp only [radius, lt_iSup_iff] at h
rcases h with ⟨t, C, hC, rt⟩
rw [ENNReal.coe_lt_coe, ← NNReal.coe_lt_coe] at rt
have : 0 < (t : ℝ) := r.coe_nonneg.trans_lt rt
rw [← div_lt_one this] at rt
refine' ⟨_, rt, C, Or.inr zero_lt_one, fun n => _⟩
calc
|‖p n‖ * (r : ℝ) ^ n| = ‖p n‖ * (t : ℝ) ^ n * (r / t : ℝ) ^ n := by
field_simp [mul_right_comm, abs_mul]
_ ≤ C * (r / t : ℝ) ^ n := by gcongr; apply hC
#align formal_multilinear_series.is_o_of_lt_radius FormalMultilinearSeries.isLittleO_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ = o(1)`. -/
theorem isLittleO_one_of_lt_radius (h : ↑r < p.radius) :
(fun n => ‖p n‖ * (r : ℝ) ^ n) =o[atTop] (fun _ => 1 : ℕ → ℝ) :=
let ⟨_, ha, hp⟩ := p.isLittleO_of_lt_radius h
hp.trans <| (isLittleO_pow_pow_of_lt_left ha.1.le ha.2).congr (fun _ => rfl) one_pow
#align formal_multilinear_series.is_o_one_of_lt_radius FormalMultilinearSeries.isLittleO_one_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` tends to zero exponentially:
for some `0 < a < 1` and `C > 0`, `‖p n‖ * r ^ n ≤ C * a ^ n`. -/
theorem norm_mul_pow_le_mul_pow_of_lt_radius (h : ↑r < p.radius) :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C * a ^ n := by
-- Porting note: moved out of `rcases`
have := ((TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 5).mp
(p.isLittleO_of_lt_radius h)
rcases this with ⟨a, ha, C, hC, H⟩
exact ⟨a, ha, C, hC, fun n => (le_abs_self _).trans (H n)⟩
#align formal_multilinear_series.norm_mul_pow_le_mul_pow_of_lt_radius FormalMultilinearSeries.norm_mul_pow_le_mul_pow_of_lt_radius
/-- If `r ≠ 0` and `‖pₙ‖ rⁿ = O(aⁿ)` for some `-1 < a < 1`, then `r < p.radius`. -/
theorem lt_radius_of_isBigO (h₀ : r ≠ 0) {a : ℝ} (ha : a ∈ Ioo (-1 : ℝ) 1)
(hp : (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] (a ^ ·)) : ↑r < p.radius := by
-- Porting note: moved out of `rcases`
have := ((TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 2 5)
rcases this.mp ⟨a, ha, hp⟩ with ⟨a, ha, C, hC, hp⟩
rw [← pos_iff_ne_zero, ← NNReal.coe_pos] at h₀
lift a to ℝ≥0 using ha.1.le
have : (r : ℝ) < r / a := by
simpa only [div_one] using (div_lt_div_left h₀ zero_lt_one ha.1).2 ha.2
norm_cast at this
rw [← ENNReal.coe_lt_coe] at this
refine' this.trans_le (p.le_radius_of_bound C fun n => _)
rw [NNReal.coe_div, div_pow, ← mul_div_assoc, div_le_iff (pow_pos ha.1 n)]
exact (le_abs_self _).trans (hp n)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.lt_radius_of_is_O FormalMultilinearSeries.lt_radius_of_isBigO
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem norm_mul_pow_le_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C :=
let ⟨_, ha, C, hC, h⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius h
⟨C, hC, fun n => (h n).trans <| mul_le_of_le_one_right hC.lt.le (pow_le_one _ ha.1.le ha.2.le)⟩
#align formal_multilinear_series.norm_mul_pow_le_of_lt_radius FormalMultilinearSeries.norm_mul_pow_le_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem norm_le_div_pow_of_pos_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h0 : 0 < r) (h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖ ≤ C / (r : ℝ) ^ n :=
let ⟨C, hC, hp⟩ := p.norm_mul_pow_le_of_lt_radius h
⟨C, hC, fun n => Iff.mpr (le_div_iff (pow_pos h0 _)) (hp n)⟩
#align formal_multilinear_series.norm_le_div_pow_of_pos_of_lt_radius FormalMultilinearSeries.norm_le_div_pow_of_pos_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem nnnorm_mul_pow_le_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖₊ * r ^ n ≤ C :=
let ⟨C, hC, hp⟩ := p.norm_mul_pow_le_of_lt_radius h
⟨⟨C, hC.lt.le⟩, hC, mod_cast hp⟩
#align formal_multilinear_series.nnnorm_mul_pow_le_of_lt_radius FormalMultilinearSeries.nnnorm_mul_pow_le_of_lt_radius
theorem le_radius_of_tendsto (p : FormalMultilinearSeries 𝕜 E F) {l : ℝ}
(h : Tendsto (fun n => ‖p n‖ * (r : ℝ) ^ n) atTop (𝓝 l)) : ↑r ≤ p.radius :=
p.le_radius_of_isBigO (h.isBigO_one _)
#align formal_multilinear_series.le_radius_of_tendsto FormalMultilinearSeries.le_radius_of_tendsto
theorem le_radius_of_summable_norm (p : FormalMultilinearSeries 𝕜 E F)
(hs : Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_tendsto hs.tendsto_atTop_zero
#align formal_multilinear_series.le_radius_of_summable_norm FormalMultilinearSeries.le_radius_of_summable_norm
theorem not_summable_norm_of_radius_lt_nnnorm (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(h : p.radius < ‖x‖₊) : ¬Summable fun n => ‖p n‖ * ‖x‖ ^ n :=
fun hs => not_le_of_lt h (p.le_radius_of_summable_norm hs)
#align formal_multilinear_series.not_summable_norm_of_radius_lt_nnnorm FormalMultilinearSeries.not_summable_norm_of_radius_lt_nnnorm
theorem summable_norm_mul_pow (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0} (h : ↑r < p.radius) :
Summable fun n : ℕ => ‖p n‖ * (r : ℝ) ^ n := by
obtain ⟨a, ha : a ∈ Ioo (0 : ℝ) 1, C, - : 0 < C, hp⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius h
exact .of_nonneg_of_le (fun n => mul_nonneg (norm_nonneg _) (pow_nonneg r.coe_nonneg _))
hp ((summable_geometric_of_lt_1 ha.1.le ha.2).mul_left _)
#align formal_multilinear_series.summable_norm_mul_pow FormalMultilinearSeries.summable_norm_mul_pow
theorem summable_norm_apply (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : Summable fun n : ℕ => ‖p n fun _ => x‖ := by
rw [mem_emetric_ball_zero_iff] at hx
refine' .of_nonneg_of_le
(fun _ => norm_nonneg _) (fun n => ((p n).le_op_norm _).trans_eq _) (p.summable_norm_mul_pow hx)
simp
#align formal_multilinear_series.summable_norm_apply FormalMultilinearSeries.summable_norm_apply
theorem summable_nnnorm_mul_pow (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0} (h : ↑r < p.radius) :
Summable fun n : ℕ => ‖p n‖₊ * r ^ n := by
rw [← NNReal.summable_coe]
push_cast
exact p.summable_norm_mul_pow h
#align formal_multilinear_series.summable_nnnorm_mul_pow FormalMultilinearSeries.summable_nnnorm_mul_pow
protected theorem summable [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : Summable fun n : ℕ => p n fun _ => x :=
(p.summable_norm_apply hx).of_norm
#align formal_multilinear_series.summable FormalMultilinearSeries.summable
theorem radius_eq_top_of_summable_norm (p : FormalMultilinearSeries 𝕜 E F)
(hs : ∀ r : ℝ≥0, Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : p.radius = ∞ :=
ENNReal.eq_top_of_forall_nnreal_le fun r => p.le_radius_of_summable_norm (hs r)
#align formal_multilinear_series.radius_eq_top_of_summable_norm FormalMultilinearSeries.radius_eq_top_of_summable_norm
theorem radius_eq_top_iff_summable_norm (p : FormalMultilinearSeries 𝕜 E F) :
p.radius = ∞ ↔ ∀ r : ℝ≥0, Summable fun n => ‖p n‖ * (r : ℝ) ^ n := by
constructor
· intro h r
obtain ⟨a, ha : a ∈ Ioo (0 : ℝ) 1, C, - : 0 < C, hp⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius
(show (r : ℝ≥0∞) < p.radius from h.symm ▸ ENNReal.coe_lt_top)
refine' .of_norm_bounded
(fun n => (C : ℝ) * a ^ n) ((summable_geometric_of_lt_1 ha.1.le ha.2).mul_left _) fun n => _
specialize hp n
rwa [Real.norm_of_nonneg (mul_nonneg (norm_nonneg _) (pow_nonneg r.coe_nonneg n))]
· exact p.radius_eq_top_of_summable_norm
#align formal_multilinear_series.radius_eq_top_iff_summable_norm FormalMultilinearSeries.radius_eq_top_iff_summable_norm
/-- If the radius of `p` is positive, then `‖pₙ‖` grows at most geometrically. -/
theorem le_mul_pow_of_radius_pos (p : FormalMultilinearSeries 𝕜 E F) (h : 0 < p.radius) :
∃ (C r : _) (hC : 0 < C) (_ : 0 < r), ∀ n, ‖p n‖ ≤ C * r ^ n := by
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 h with ⟨r, r0, rlt⟩
have rpos : 0 < (r : ℝ) := by simp [ENNReal.coe_pos.1 r0]
rcases norm_le_div_pow_of_pos_of_lt_radius p rpos rlt with ⟨C, Cpos, hCp⟩
refine' ⟨C, r⁻¹, Cpos, by simp only [inv_pos, rpos], fun n => _⟩
-- Porting note: was `convert`
rw [inv_pow, ← div_eq_mul_inv]
exact hCp n
#align formal_multilinear_series.le_mul_pow_of_radius_pos FormalMultilinearSeries.le_mul_pow_of_radius_pos
/-- The radius of the sum of two formal series is at least the minimum of their two radii. -/
theorem min_radius_le_radius_add (p q : FormalMultilinearSeries 𝕜 E F) :
min p.radius q.radius ≤ (p + q).radius := by
refine' ENNReal.le_of_forall_nnreal_lt fun r hr => _
rw [lt_min_iff] at hr
have := ((p.isLittleO_one_of_lt_radius hr.1).add (q.isLittleO_one_of_lt_radius hr.2)).isBigO
refine' (p + q).le_radius_of_isBigO ((isBigO_of_le _ fun n => _).trans this)
rw [← add_mul, norm_mul, norm_mul, norm_norm]
exact mul_le_mul_of_nonneg_right ((norm_add_le _ _).trans (le_abs_self _)) (norm_nonneg _)
#align formal_multilinear_series.min_radius_le_radius_add FormalMultilinearSeries.min_radius_le_radius_add
@[simp]
theorem radius_neg (p : FormalMultilinearSeries 𝕜 E F) : (-p).radius = p.radius := by
simp only [radius, neg_apply, norm_neg]
#align formal_multilinear_series.radius_neg FormalMultilinearSeries.radius_neg
protected theorem hasSum [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : HasSum (fun n : ℕ => p n fun _ => x) (p.sum x) :=
(p.summable hx).hasSum
#align formal_multilinear_series.has_sum FormalMultilinearSeries.hasSum
theorem radius_le_radius_continuousLinearMap_comp (p : FormalMultilinearSeries 𝕜 E F)
(f : F →L[𝕜] G) : p.radius ≤ (f.compFormalMultilinearSeries p).radius := by
refine' ENNReal.le_of_forall_nnreal_lt fun r hr => _
apply le_radius_of_isBigO
apply (IsBigO.trans_isLittleO _ (p.isLittleO_one_of_lt_radius hr)).isBigO
refine' IsBigO.mul (@IsBigOWith.isBigO _ _ _ _ _ ‖f‖ _ _ _ _) (isBigO_refl _ _)
refine IsBigOWith.of_bound (eventually_of_forall fun n => ?_)
simpa only [norm_norm] using f.norm_compContinuousMultilinearMap_le (p n)
#align formal_multilinear_series.radius_le_radius_continuous_linear_map_comp FormalMultilinearSeries.radius_le_radius_continuousLinearMap_comp
end FormalMultilinearSeries
/-! ### Expanding a function as a power series -/
section
variable {f g : E → F} {p pf pg : FormalMultilinearSeries 𝕜 E F} {x : E} {r r' : ℝ≥0∞}
/-- Given a function `f : E → F` and a formal multilinear series `p`, we say that `f` has `p` as
a power series on the ball of radius `r > 0` around `x` if `f (x + y) = ∑' pₙ yⁿ` for all `‖y‖ < r`.
-/
structure HasFPowerSeriesOnBall (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (x : E) (r : ℝ≥0∞) :
Prop where
r_le : r ≤ p.radius
r_pos : 0 < r
hasSum :
∀ {y}, y ∈ EMetric.ball (0 : E) r → HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y))
#align has_fpower_series_on_ball HasFPowerSeriesOnBall
/-- Given a function `f : E → F` and a formal multilinear series `p`, we say that `f` has `p` as
a power series around `x` if `f (x + y) = ∑' pₙ yⁿ` for all `y` in a neighborhood of `0`. -/
def HasFPowerSeriesAt (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (x : E) :=
∃ r, HasFPowerSeriesOnBall f p x r
#align has_fpower_series_at HasFPowerSeriesAt
variable (𝕜)
/-- Given a function `f : E → F`, we say that `f` is analytic at `x` if it admits a convergent power
series expansion around `x`. -/
def AnalyticAt (f : E → F) (x : E) :=
∃ p : FormalMultilinearSeries 𝕜 E F, HasFPowerSeriesAt f p x
#align analytic_at AnalyticAt
/-- Given a function `f : E → F`, we say that `f` is analytic on a set `s` if it is analytic around
every point of `s`. -/
def AnalyticOn (f : E → F) (s : Set E) :=
∀ x, x ∈ s → AnalyticAt 𝕜 f x
#align analytic_on AnalyticOn
variable {𝕜}
theorem HasFPowerSeriesOnBall.hasFPowerSeriesAt (hf : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesAt f p x :=
⟨r, hf⟩
#align has_fpower_series_on_ball.has_fpower_series_at HasFPowerSeriesOnBall.hasFPowerSeriesAt
theorem HasFPowerSeriesAt.analyticAt (hf : HasFPowerSeriesAt f p x) : AnalyticAt 𝕜 f x :=
⟨p, hf⟩
#align has_fpower_series_at.analytic_at HasFPowerSeriesAt.analyticAt
theorem HasFPowerSeriesOnBall.analyticAt (hf : HasFPowerSeriesOnBall f p x r) : AnalyticAt 𝕜 f x :=
hf.hasFPowerSeriesAt.analyticAt
#align has_fpower_series_on_ball.analytic_at HasFPowerSeriesOnBall.analyticAt
theorem HasFPowerSeriesOnBall.congr (hf : HasFPowerSeriesOnBall f p x r)
(hg : EqOn f g (EMetric.ball x r)) : HasFPowerSeriesOnBall g p x r :=
{ r_le := hf.r_le
r_pos := hf.r_pos
hasSum := fun {y} hy => by
convert hf.hasSum hy using 1
apply hg.symm
simpa [edist_eq_coe_nnnorm_sub] using hy }
#align has_fpower_series_on_ball.congr HasFPowerSeriesOnBall.congr
/-- If a function `f` has a power series `p` around `x`, then the function `z ↦ f (z - y)` has the
same power series around `x + y`. -/
theorem HasFPowerSeriesOnBall.comp_sub (hf : HasFPowerSeriesOnBall f p x r) (y : E) :
HasFPowerSeriesOnBall (fun z => f (z - y)) p (x + y) r :=
{ r_le := hf.r_le
r_pos := hf.r_pos
hasSum := fun {z} hz => by
convert hf.hasSum hz using 2
abel }
#align has_fpower_series_on_ball.comp_sub HasFPowerSeriesOnBall.comp_sub
theorem HasFPowerSeriesOnBall.hasSum_sub (hf : HasFPowerSeriesOnBall f p x r) {y : E}
(hy : y ∈ EMetric.ball x r) : HasSum (fun n : ℕ => p n fun _ => y - x) (f y) := by
have : y - x ∈ EMetric.ball (0 : E) r := by simpa [edist_eq_coe_nnnorm_sub] using hy
simpa only [add_sub_cancel'_right] using hf.hasSum this
#align has_fpower_series_on_ball.has_sum_sub HasFPowerSeriesOnBall.hasSum_sub
theorem HasFPowerSeriesOnBall.radius_pos (hf : HasFPowerSeriesOnBall f p x r) : 0 < p.radius :=
lt_of_lt_of_le hf.r_pos hf.r_le
#align has_fpower_series_on_ball.radius_pos HasFPowerSeriesOnBall.radius_pos
theorem HasFPowerSeriesAt.radius_pos (hf : HasFPowerSeriesAt f p x) : 0 < p.radius :=
let ⟨_, hr⟩ := hf
hr.radius_pos
#align has_fpower_series_at.radius_pos HasFPowerSeriesAt.radius_pos
theorem HasFPowerSeriesOnBall.mono (hf : HasFPowerSeriesOnBall f p x r) (r'_pos : 0 < r')
(hr : r' ≤ r) : HasFPowerSeriesOnBall f p x r' :=
⟨le_trans hr hf.1, r'_pos, fun hy => hf.hasSum (EMetric.ball_subset_ball hr hy)⟩
#align has_fpower_series_on_ball.mono HasFPowerSeriesOnBall.mono
theorem HasFPowerSeriesAt.congr (hf : HasFPowerSeriesAt f p x) (hg : f =ᶠ[𝓝 x] g) :
HasFPowerSeriesAt g p x := by
rcases hf with ⟨r₁, h₁⟩
rcases EMetric.mem_nhds_iff.mp hg with ⟨r₂, h₂pos, h₂⟩
exact ⟨min r₁ r₂,
(h₁.mono (lt_min h₁.r_pos h₂pos) inf_le_left).congr
fun y hy => h₂ (EMetric.ball_subset_ball inf_le_right hy)⟩
#align has_fpower_series_at.congr HasFPowerSeriesAt.congr
protected theorem HasFPowerSeriesAt.eventually (hf : HasFPowerSeriesAt f p x) :
∀ᶠ r : ℝ≥0∞ in 𝓝[>] 0, HasFPowerSeriesOnBall f p x r :=
let ⟨_, hr⟩ := hf
mem_of_superset (Ioo_mem_nhdsWithin_Ioi (left_mem_Ico.2 hr.r_pos)) fun _ hr' =>
hr.mono hr'.1 hr'.2.le
#align has_fpower_series_at.eventually HasFPowerSeriesAt.eventually
theorem HasFPowerSeriesOnBall.eventually_hasSum (hf : HasFPowerSeriesOnBall f p x r) :
∀ᶠ y in 𝓝 0, HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y)) := by
filter_upwards [EMetric.ball_mem_nhds (0 : E) hf.r_pos] using fun _ => hf.hasSum
#align has_fpower_series_on_ball.eventually_has_sum HasFPowerSeriesOnBall.eventually_hasSum
theorem HasFPowerSeriesAt.eventually_hasSum (hf : HasFPowerSeriesAt f p x) :
∀ᶠ y in 𝓝 0, HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y)) :=
let ⟨_, hr⟩ := hf
hr.eventually_hasSum
#align has_fpower_series_at.eventually_has_sum HasFPowerSeriesAt.eventually_hasSum
theorem HasFPowerSeriesOnBall.eventually_hasSum_sub (hf : HasFPowerSeriesOnBall f p x r) :
∀ᶠ y in 𝓝 x, HasSum (fun n : ℕ => p n fun _ : Fin n => y - x) (f y) := by
filter_upwards [EMetric.ball_mem_nhds x hf.r_pos] with y using hf.hasSum_sub
#align has_fpower_series_on_ball.eventually_has_sum_sub HasFPowerSeriesOnBall.eventually_hasSum_sub
theorem HasFPowerSeriesAt.eventually_hasSum_sub (hf : HasFPowerSeriesAt f p x) :
∀ᶠ y in 𝓝 x, HasSum (fun n : ℕ => p n fun _ : Fin n => y - x) (f y) :=
let ⟨_, hr⟩ := hf
hr.eventually_hasSum_sub
#align has_fpower_series_at.eventually_has_sum_sub HasFPowerSeriesAt.eventually_hasSum_sub
theorem HasFPowerSeriesOnBall.eventually_eq_zero
(hf : HasFPowerSeriesOnBall f (0 : FormalMultilinearSeries 𝕜 E F) x r) :
∀ᶠ z in 𝓝 x, f z = 0 := by
filter_upwards [hf.eventually_hasSum_sub] with z hz using hz.unique hasSum_zero
#align has_fpower_series_on_ball.eventually_eq_zero HasFPowerSeriesOnBall.eventually_eq_zero
theorem HasFPowerSeriesAt.eventually_eq_zero
(hf : HasFPowerSeriesAt f (0 : FormalMultilinearSeries 𝕜 E F) x) : ∀ᶠ z in 𝓝 x, f z = 0 :=
let ⟨_, hr⟩ := hf
hr.eventually_eq_zero
#align has_fpower_series_at.eventually_eq_zero HasFPowerSeriesAt.eventually_eq_zero
theorem hasFPowerSeriesOnBall_const {c : F} {e : E} :
HasFPowerSeriesOnBall (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e ⊤ := by
refine' ⟨by simp, WithTop.zero_lt_top, fun _ => hasSum_single 0 fun n hn => _⟩
simp [constFormalMultilinearSeries_apply hn]
#align has_fpower_series_on_ball_const hasFPowerSeriesOnBall_const
theorem hasFPowerSeriesAt_const {c : F} {e : E} :
HasFPowerSeriesAt (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e :=
⟨⊤, hasFPowerSeriesOnBall_const⟩
#align has_fpower_series_at_const hasFPowerSeriesAt_const
theorem analyticAt_const {v : F} : AnalyticAt 𝕜 (fun _ => v) x :=
⟨constFormalMultilinearSeries 𝕜 E v, hasFPowerSeriesAt_const⟩
#align analytic_at_const analyticAt_const
theorem analyticOn_const {v : F} {s : Set E} : AnalyticOn 𝕜 (fun _ => v) s :=
fun _ _ => analyticAt_const
#align analytic_on_const analyticOn_const
theorem HasFPowerSeriesOnBall.add (hf : HasFPowerSeriesOnBall f pf x r)
(hg : HasFPowerSeriesOnBall g pg x r) : HasFPowerSeriesOnBall (f + g) (pf + pg) x r :=
{ r_le := le_trans (le_min_iff.2 ⟨hf.r_le, hg.r_le⟩) (pf.min_radius_le_radius_add pg)
r_pos := hf.r_pos
hasSum := fun hy => (hf.hasSum hy).add (hg.hasSum hy) }
#align has_fpower_series_on_ball.add HasFPowerSeriesOnBall.add
theorem HasFPowerSeriesAt.add (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x) :
HasFPowerSeriesAt (f + g) (pf + pg) x := by
rcases (hf.eventually.and hg.eventually).exists with ⟨r, hr⟩
exact ⟨r, hr.1.add hr.2⟩
#align has_fpower_series_at.add HasFPowerSeriesAt.add
theorem AnalyticAt.congr (hf : AnalyticAt 𝕜 f x) (hg : f =ᶠ[𝓝 x] g) : AnalyticAt 𝕜 g x :=
let ⟨_, hpf⟩ := hf
(hpf.congr hg).analyticAt
theorem analyticAt_congr (h : f =ᶠ[𝓝 x] g) : AnalyticAt 𝕜 f x ↔ AnalyticAt 𝕜 g x :=
⟨fun hf ↦ hf.congr h, fun hg ↦ hg.congr h.symm⟩
theorem AnalyticAt.add (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) : AnalyticAt 𝕜 (f + g) x :=
let ⟨_, hpf⟩ := hf
let ⟨_, hqf⟩ := hg
(hpf.add hqf).analyticAt
#align analytic_at.add AnalyticAt.add
theorem HasFPowerSeriesOnBall.neg (hf : HasFPowerSeriesOnBall f pf x r) :
HasFPowerSeriesOnBall (-f) (-pf) x r :=
{ r_le := by
rw [pf.radius_neg]
exact hf.r_le
r_pos := hf.r_pos
hasSum := fun hy => (hf.hasSum hy).neg }
#align has_fpower_series_on_ball.neg HasFPowerSeriesOnBall.neg
theorem HasFPowerSeriesAt.neg (hf : HasFPowerSeriesAt f pf x) : HasFPowerSeriesAt (-f) (-pf) x :=
let ⟨_, hrf⟩ := hf
hrf.neg.hasFPowerSeriesAt
#align has_fpower_series_at.neg HasFPowerSeriesAt.neg
theorem AnalyticAt.neg (hf : AnalyticAt 𝕜 f x) : AnalyticAt 𝕜 (-f) x :=
let ⟨_, hpf⟩ := hf
hpf.neg.analyticAt
#align analytic_at.neg AnalyticAt.neg
theorem HasFPowerSeriesOnBall.sub (hf : HasFPowerSeriesOnBall f pf x r)
(hg : HasFPowerSeriesOnBall g pg x r) : HasFPowerSeriesOnBall (f - g) (pf - pg) x r := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align has_fpower_series_on_ball.sub HasFPowerSeriesOnBall.sub
theorem HasFPowerSeriesAt.sub (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x) :
HasFPowerSeriesAt (f - g) (pf - pg) x := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align has_fpower_series_at.sub HasFPowerSeriesAt.sub
theorem AnalyticAt.sub (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) :
AnalyticAt 𝕜 (f - g) x := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align analytic_at.sub AnalyticAt.sub
theorem AnalyticOn.mono {s t : Set E} (hf : AnalyticOn 𝕜 f t) (hst : s ⊆ t) : AnalyticOn 𝕜 f s :=
fun z hz => hf z (hst hz)
#align analytic_on.mono AnalyticOn.mono
theorem AnalyticOn.congr' {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : f =ᶠ[𝓝ˢ s] g) :
AnalyticOn 𝕜 g s :=
fun z hz => (hf z hz).congr (mem_nhdsSet_iff_forall.mp hg z hz)
theorem analyticOn_congr' {s : Set E} (h : f =ᶠ[𝓝ˢ s] g) : AnalyticOn 𝕜 f s ↔ AnalyticOn 𝕜 g s :=
⟨fun hf => hf.congr' h, fun hg => hg.congr' h.symm⟩
theorem AnalyticOn.congr {s : Set E} (hs : IsOpen s) (hf : AnalyticOn 𝕜 f s) (hg : s.EqOn f g) :
AnalyticOn 𝕜 g s :=
hf.congr' $ mem_nhdsSet_iff_forall.mpr
(fun _ hz => eventuallyEq_iff_exists_mem.mpr ⟨s, hs.mem_nhds hz, hg⟩)
theorem analyticOn_congr {s : Set E} (hs : IsOpen s) (h : s.EqOn f g) : AnalyticOn 𝕜 f s ↔
AnalyticOn 𝕜 g s := ⟨fun hf => hf.congr hs h, fun hg => hg.congr hs h.symm⟩
theorem AnalyticOn.add {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
AnalyticOn 𝕜 (f + g) s :=
fun z hz => (hf z hz).add (hg z hz)
#align analytic_on.add AnalyticOn.add
theorem AnalyticOn.sub {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
AnalyticOn 𝕜 (f - g) s :=
fun z hz => (hf z hz).sub (hg z hz)
#align analytic_on.sub AnalyticOn.sub
theorem HasFPowerSeriesOnBall.coeff_zero (hf : HasFPowerSeriesOnBall f pf x r) (v : Fin 0 → E) :
pf 0 v = f x := by
have v_eq : v = fun i => 0 := Subsingleton.elim _ _
have zero_mem : (0 : E) ∈ EMetric.ball (0 : E) r := by simp [hf.r_pos]
have : ∀ i, i ≠ 0 → (pf i fun j => 0) = 0 := by
intro i hi
have : 0 < i := pos_iff_ne_zero.2 hi
exact ContinuousMultilinearMap.map_coord_zero _ (⟨0, this⟩ : Fin i) rfl
have A := (hf.hasSum zero_mem).unique (hasSum_single _ this)
simpa [v_eq] using A.symm
#align has_fpower_series_on_ball.coeff_zero HasFPowerSeriesOnBall.coeff_zero
theorem HasFPowerSeriesAt.coeff_zero (hf : HasFPowerSeriesAt f pf x) (v : Fin 0 → E) :
pf 0 v = f x :=
let ⟨_, hrf⟩ := hf
hrf.coeff_zero v
#align has_fpower_series_at.coeff_zero HasFPowerSeriesAt.coeff_zero
/-- If a function `f` has a power series `p` on a ball and `g` is linear, then `g ∘ f` has the
power series `g ∘ p` on the same ball. -/
theorem ContinuousLinearMap.comp_hasFPowerSeriesOnBall (g : F →L[𝕜] G)
(h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (g ∘ f) (g.compFormalMultilinearSeries p) x r :=
{ r_le := h.r_le.trans (p.radius_le_radius_continuousLinearMap_comp _)
r_pos := h.r_pos
hasSum := fun hy => by
simpa only [ContinuousLinearMap.compFormalMultilinearSeries_apply,
ContinuousLinearMap.compContinuousMultilinearMap_coe, Function.comp_apply] using
g.hasSum (h.hasSum hy) }
#align continuous_linear_map.comp_has_fpower_series_on_ball ContinuousLinearMap.comp_hasFPowerSeriesOnBall
/-- If a function `f` is analytic on a set `s` and `g` is linear, then `g ∘ f` is analytic
on `s`. -/
theorem ContinuousLinearMap.comp_analyticOn {s : Set E} (g : F →L[𝕜] G) (h : AnalyticOn 𝕜 f s) :
AnalyticOn 𝕜 (g ∘ f) s := by
rintro x hx
rcases h x hx with ⟨p, r, hp⟩
exact ⟨g.compFormalMultilinearSeries p, r, g.comp_hasFPowerSeriesOnBall hp⟩
#align continuous_linear_map.comp_analytic_on ContinuousLinearMap.comp_analyticOn
/-- If a function admits a power series expansion, then it is exponentially close to the partial
sums of this power series on strict subdisks of the disk of convergence.
This version provides an upper estimate that decreases both in `‖y‖` and `n`. See also
`HasFPowerSeriesOnBall.uniform_geometric_approx` for a weaker version. -/
theorem HasFPowerSeriesOnBall.uniform_geometric_approx' {r' : ℝ≥0}
(hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n := by
obtain ⟨a, ha, C, hC, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n, ‖p n‖ * (r' : ℝ) ^ n ≤ C * a ^ n :=
p.norm_mul_pow_le_mul_pow_of_lt_radius (h.trans_le hf.r_le)
refine' ⟨a, ha, C / (1 - a), div_pos hC (sub_pos.2 ha.2), fun y hy n => _⟩
have yr' : ‖y‖ < r' := by
rw [ball_zero_eq] at hy
exact hy
have hr'0 : 0 < (r' : ℝ) := (norm_nonneg _).trans_lt yr'
have : y ∈ EMetric.ball (0 : E) r := by
refine' mem_emetric_ball_zero_iff.2 (lt_trans _ h)
exact mod_cast yr'
rw [norm_sub_rev, ← mul_div_right_comm]
have ya : a * (‖y‖ / ↑r') ≤ a :=
mul_le_of_le_one_right ha.1.le (div_le_one_of_le yr'.le r'.coe_nonneg)
suffices ‖p.partialSum n y - f (x + y)‖ ≤ C * (a * (‖y‖ / r')) ^ n / (1 - a * (‖y‖ / r')) by
refine' this.trans _
have : 0 < a := ha.1
gcongr
apply_rules [sub_pos.2, ha.2]
apply norm_sub_le_of_geometric_bound_of_hasSum (ya.trans_lt ha.2) _ (hf.hasSum this)
intro n
calc
‖(p n) fun _ : Fin n => y‖
_ ≤ ‖p n‖ * ∏ _i : Fin n, ‖y‖ := ContinuousMultilinearMap.le_op_norm _ _
_ = ‖p n‖ * (r' : ℝ) ^ n * (‖y‖ / r') ^ n := by field_simp [mul_right_comm]
_ ≤ C * a ^ n * (‖y‖ / r') ^ n := by gcongr ?_ * _; apply hp
_ ≤ C * (a * (‖y‖ / r')) ^ n := by rw [mul_pow, mul_assoc]
#align has_fpower_series_on_ball.uniform_geometric_approx' HasFPowerSeriesOnBall.uniform_geometric_approx'
/-- If a function admits a power series expansion, then it is exponentially close to the partial
sums of this power series on strict subdisks of the disk of convergence. -/
theorem HasFPowerSeriesOnBall.uniform_geometric_approx {r' : ℝ≥0}
(hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
∃ a ∈ Ioo (0 : ℝ) 1,
∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n, ‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n := by
obtain ⟨a, ha, C, hC, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n :=
hf.uniform_geometric_approx' h
refine' ⟨a, ha, C, hC, fun y hy n => (hp y hy n).trans _⟩
have yr' : ‖y‖ < r' := by rwa [ball_zero_eq] at hy
gcongr
exacts [mul_nonneg ha.1.le (div_nonneg (norm_nonneg y) r'.coe_nonneg),
mul_le_of_le_one_right ha.1.le (div_le_one_of_le yr'.le r'.coe_nonneg)]
#align has_fpower_series_on_ball.uniform_geometric_approx HasFPowerSeriesOnBall.uniform_geometric_approx
/-- Taylor formula for an analytic function, `IsBigO` version. -/
theorem HasFPowerSeriesAt.isBigO_sub_partialSum_pow (hf : HasFPowerSeriesAt f p x) (n : ℕ) :
(fun y : E => f (x + y) - p.partialSum n y) =O[𝓝 0] fun y => ‖y‖ ^ n := by
rcases hf with ⟨r, hf⟩
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hf.r_pos with ⟨r', r'0, h⟩
obtain ⟨a, -, C, -, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n :=
hf.uniform_geometric_approx' h
refine' isBigO_iff.2 ⟨C * (a / r') ^ n, _⟩
replace r'0 : 0 < (r' : ℝ); · exact mod_cast r'0
filter_upwards [Metric.ball_mem_nhds (0 : E) r'0] with y hy
simpa [mul_pow, mul_div_assoc, mul_assoc, div_mul_eq_mul_div] using hp y hy n
set_option linter.uppercaseLean3 false in
#align has_fpower_series_at.is_O_sub_partial_sum_pow HasFPowerSeriesAt.isBigO_sub_partialSum_pow
/-- If `f` has formal power series `∑ n, pₙ` on a ball of radius `r`, then for `y, z` in any smaller
ball, the norm of the difference `f y - f z - p 1 (fun _ ↦ y - z)` is bounded above by
`C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖`. This lemma formulates this property using `IsBigO` and
`Filter.principal` on `E × E`. -/
theorem HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal
(hf : HasFPowerSeriesOnBall f p x r) (hr : r' < r) :
(fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) =O[𝓟 (EMetric.ball (x, x) r')]
fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖ := by
lift r' to ℝ≥0 using ne_top_of_lt hr
rcases (zero_le r').eq_or_lt with (rfl | hr'0)
· simp only [isBigO_bot, EMetric.ball_zero, principal_empty, ENNReal.coe_zero]
obtain ⟨a, ha, C, hC : 0 < C, hp⟩ :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n : ℕ, ‖p n‖ * (r' : ℝ) ^ n ≤ C * a ^ n
exact p.norm_mul_pow_le_mul_pow_of_lt_radius (hr.trans_le hf.r_le)
simp only [← le_div_iff (pow_pos (NNReal.coe_pos.2 hr'0) _)] at hp
set L : E × E → ℝ := fun y =>
C * (a / r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * (a / (1 - a) ^ 2 + 2 / (1 - a))
have hL : ∀ y ∈ EMetric.ball (x, x) r', ‖f y.1 - f y.2 - p 1 fun _ => y.1 - y.2‖ ≤ L y := by
intro y hy'
have hy : y ∈ EMetric.ball x r ×ˢ EMetric.ball x r := by
rw [EMetric.ball_prod_same]
exact EMetric.ball_subset_ball hr.le hy'
set A : ℕ → F := fun n => (p n fun _ => y.1 - x) - p n fun _ => y.2 - x
have hA : HasSum (fun n => A (n + 2)) (f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) := by
convert (hasSum_nat_add_iff' 2).2 ((hf.hasSum_sub hy.1).sub (hf.hasSum_sub hy.2)) using 1
rw [Finset.sum_range_succ, Finset.sum_range_one, hf.coeff_zero, hf.coeff_zero, sub_self,
zero_add, ← Subsingleton.pi_single_eq (0 : Fin 1) (y.1 - x), Pi.single,
← Subsingleton.pi_single_eq (0 : Fin 1) (y.2 - x), Pi.single, ← (p 1).map_sub, ← Pi.single,
Subsingleton.pi_single_eq, sub_sub_sub_cancel_right]
rw [EMetric.mem_ball, edist_eq_coe_nnnorm_sub, ENNReal.coe_lt_coe] at hy'
set B : ℕ → ℝ := fun n => C * (a / r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * ((n + 2) * a ^ n)
have hAB : ∀ n, ‖A (n + 2)‖ ≤ B n := fun n =>
calc
‖A (n + 2)‖ ≤ ‖p (n + 2)‖ * ↑(n + 2) * ‖y - (x, x)‖ ^ (n + 1) * ‖y.1 - y.2‖ := by
-- porting note: `pi_norm_const` was `pi_norm_const (_ : E)`
simpa only [Fintype.card_fin, pi_norm_const, Prod.norm_def, Pi.sub_def,
Prod.fst_sub, Prod.snd_sub, sub_sub_sub_cancel_right] using
(p <| n + 2).norm_image_sub_le (fun _ => y.1 - x) fun _ => y.2 - x
_ = ‖p (n + 2)‖ * ‖y - (x, x)‖ ^ n * (↑(n + 2) * ‖y - (x, x)‖ * ‖y.1 - y.2‖) := by
rw [pow_succ ‖y - (x, x)‖]
ring
-- porting note: the two `↑` in `↑r'` are new, without them, Lean fails to synthesize
-- instances `HDiv ℝ ℝ≥0 ?m` or `HMul ℝ ℝ≥0 ?m`
_ ≤ C * a ^ (n + 2) / ↑r' ^ (n + 2)
* ↑r' ^ n * (↑(n + 2) * ‖y - (x, x)‖ * ‖y.1 - y.2‖) := by
have : 0 < a := ha.1
gcongr
· apply hp
· apply hy'.le
_ = B n := by
-- porting note: in the original, `B` was in the `field_simp`, but now Lean does not
-- accept it. The current proof works in Lean 4, but does not in Lean 3.
field_simp [pow_succ]
simp only [mul_assoc, mul_comm, mul_left_comm]
have hBL : HasSum B (L y) := by
apply HasSum.mul_left
simp only [add_mul]
have : ‖a‖ < 1 := by simp only [Real.norm_eq_abs, abs_of_pos ha.1, ha.2]
rw [div_eq_mul_inv, div_eq_mul_inv]
exact (hasSum_coe_mul_geometric_of_norm_lt_1 this).add -- porting note: was `convert`!
((hasSum_geometric_of_norm_lt_1 this).mul_left 2)
exact hA.norm_le_of_bounded hBL hAB
suffices L =O[𝓟 (EMetric.ball (x, x) r')] fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖ by
refine' (IsBigO.of_bound 1 (eventually_principal.2 fun y hy => _)).trans this
rw [one_mul]
exact (hL y hy).trans (le_abs_self _)
simp_rw [mul_right_comm _ (_ * _)] -- porting note: there was an `L` inside the `simp_rw`.
exact (isBigO_refl _ _).const_mul_left _
set_option linter.uppercaseLean3 false in
#align has_fpower_series_on_ball.is_O_image_sub_image_sub_deriv_principal HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal
/-- If `f` has formal power series `∑ n, pₙ` on a ball of radius `r`, then for `y, z` in any smaller
ball, the norm of the difference `f y - f z - p 1 (fun _ ↦ y - z)` is bounded above by
`C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖`. -/
theorem HasFPowerSeriesOnBall.image_sub_sub_deriv_le (hf : HasFPowerSeriesOnBall f p x r)
(hr : r' < r) :
∃ C, ∀ᵉ (y ∈ EMetric.ball x r') (z ∈ EMetric.ball x r'),
‖f y - f z - p 1 fun _ => y - z‖ ≤ C * max ‖y - x‖ ‖z - x‖ * ‖y - z‖ := by
simpa only [isBigO_principal, mul_assoc, norm_mul, norm_norm, Prod.forall, EMetric.mem_ball,
Prod.edist_eq, max_lt_iff, and_imp, @forall_swap (_ < _) E] using
hf.isBigO_image_sub_image_sub_deriv_principal hr
#align has_fpower_series_on_ball.image_sub_sub_deriv_le HasFPowerSeriesOnBall.image_sub_sub_deriv_le
/-- If `f` has formal power series `∑ n, pₙ` at `x`, then
`f y - f z - p 1 (fun _ ↦ y - z) = O(‖(y, z) - (x, x)‖ * ‖y - z‖)` as `(y, z) → (x, x)`.
In particular, `f` is strictly differentiable at `x`. -/
theorem HasFPowerSeriesAt.isBigO_image_sub_norm_mul_norm_sub (hf : HasFPowerSeriesAt f p x) :
(fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) =O[𝓝 (x, x)] fun y =>
‖y - (x, x)‖ * ‖y.1 - y.2‖ := by
rcases hf with ⟨r, hf⟩
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hf.r_pos with ⟨r', r'0, h⟩
refine' (hf.isBigO_image_sub_image_sub_deriv_principal h).mono _
exact le_principal_iff.2 (EMetric.ball_mem_nhds _ r'0)
set_option linter.uppercaseLean3 false in
#align has_fpower_series_at.is_O_image_sub_norm_mul_norm_sub HasFPowerSeriesAt.isBigO_image_sub_norm_mul_norm_sub
/-- If a function admits a power series expansion at `x`, then it is the uniform limit of the
partial sums of this power series on strict subdisks of the disk of convergence, i.e., `f (x + y)`
is the uniform limit of `p.partialSum n y` there. -/
theorem HasFPowerSeriesOnBall.tendstoUniformlyOn {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r)
(h : (r' : ℝ≥0∞) < r) :
TendstoUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
(Metric.ball (0 : E) r') := by
obtain ⟨a, ha, C, -, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n
exact hf.uniform_geometric_approx h
refine' Metric.tendstoUniformlyOn_iff.2 fun ε εpos => _
have L : Tendsto (fun n => (C : ℝ) * a ^ n) atTop (𝓝 ((C : ℝ) * 0)) :=
tendsto_const_nhds.mul (tendsto_pow_atTop_nhds_0_of_lt_1 ha.1.le ha.2)
rw [mul_zero] at L
refine' (L.eventually (gt_mem_nhds εpos)).mono fun n hn y hy => _
rw [dist_eq_norm]
exact (hp y hy n).trans_lt hn
#align has_fpower_series_on_ball.tendsto_uniformly_on HasFPowerSeriesOnBall.tendstoUniformlyOn
/-- If a function admits a power series expansion at `x`, then it is the locally uniform limit of
the partial sums of this power series on the disk of convergence, i.e., `f (x + y)`
is the locally uniform limit of `p.partialSum n y` there. -/
theorem HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn (hf : HasFPowerSeriesOnBall f p x r) :
TendstoLocallyUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
(EMetric.ball (0 : E) r) := by
intro u hu x hx
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hx with ⟨r', xr', hr'⟩
have : EMetric.ball (0 : E) r' ∈ 𝓝 x := IsOpen.mem_nhds EMetric.isOpen_ball xr'
refine' ⟨EMetric.ball (0 : E) r', mem_nhdsWithin_of_mem_nhds this, _⟩
simpa [Metric.emetric_ball_nnreal] using hf.tendstoUniformlyOn hr' u hu
#align has_fpower_series_on_ball.tendsto_locally_uniformly_on HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn
/-- If a function admits a power series expansion at `x`, then it is the uniform limit of the
partial sums of this power series on strict subdisks of the disk of convergence, i.e., `f y`
is the uniform limit of `p.partialSum n (y - x)` there. -/
theorem HasFPowerSeriesOnBall.tendstoUniformlyOn' {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r)
(h : (r' : ℝ≥0∞) < r) :
TendstoUniformlyOn (fun n y => p.partialSum n (y - x)) f atTop (Metric.ball (x : E) r') := by
convert (hf.tendstoUniformlyOn h).comp fun y => y - x using 1
· simp [(· ∘ ·)]
· ext z
simp [dist_eq_norm]
#align has_fpower_series_on_ball.tendsto_uniformly_on' HasFPowerSeriesOnBall.tendstoUniformlyOn'
/-- If a function admits a power series expansion at `x`, then it is the locally uniform limit of
the partial sums of this power series on the disk of convergence, i.e., `f y`
is the locally uniform limit of `p.partialSum n (y - x)` there. -/
theorem HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn' (hf : HasFPowerSeriesOnBall f p x r) :
TendstoLocallyUniformlyOn (fun n y => p.partialSum n (y - x)) f atTop
(EMetric.ball (x : E) r) := by
have A : ContinuousOn (fun y : E => y - x) (EMetric.ball (x : E) r) :=
(continuous_id.sub continuous_const).continuousOn
convert hf.tendstoLocallyUniformlyOn.comp (fun y : E => y - x) _ A using 1
· ext z
simp
· intro z
simp [edist_eq_coe_nnnorm, edist_eq_coe_nnnorm_sub]
#align has_fpower_series_on_ball.tendsto_locally_uniformly_on' HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn'
/-- If a function admits a power series expansion on a disk, then it is continuous there. -/
protected theorem HasFPowerSeriesOnBall.continuousOn (hf : HasFPowerSeriesOnBall f p x r) :
ContinuousOn f (EMetric.ball x r) :=
hf.tendstoLocallyUniformlyOn'.continuousOn <|
eventually_of_forall fun n =>
((p.partialSum_continuous n).comp (continuous_id.sub continuous_const)).continuousOn
#align has_fpower_series_on_ball.continuous_on HasFPowerSeriesOnBall.continuousOn
protected theorem HasFPowerSeriesAt.continuousAt (hf : HasFPowerSeriesAt f p x) :
ContinuousAt f x :=
let ⟨_, hr⟩ := hf
hr.continuousOn.continuousAt (EMetric.ball_mem_nhds x hr.r_pos)
#align has_fpower_series_at.continuous_at HasFPowerSeriesAt.continuousAt
protected theorem AnalyticAt.continuousAt (hf : AnalyticAt 𝕜 f x) : ContinuousAt f x :=
let ⟨_, hp⟩ := hf
hp.continuousAt
#align analytic_at.continuous_at AnalyticAt.continuousAt
protected theorem AnalyticOn.continuousOn {s : Set E} (hf : AnalyticOn 𝕜 f s) : ContinuousOn f s :=
fun x hx => (hf x hx).continuousAt.continuousWithinAt
#align analytic_on.continuous_on AnalyticOn.continuousOn
/-- Analytic everywhere implies continuous -/
theorem AnalyticOn.continuous {f : E → F} (fa : AnalyticOn 𝕜 f univ) : Continuous f := by
rw [continuous_iff_continuousOn_univ]; exact fa.continuousOn
/-- In a complete space, the sum of a converging power series `p` admits `p` as a power series.
This is not totally obvious as we need to check the convergence of the series. -/
protected theorem FormalMultilinearSeries.hasFPowerSeriesOnBall [CompleteSpace F]
(p : FormalMultilinearSeries 𝕜 E F) (h : 0 < p.radius) :
HasFPowerSeriesOnBall p.sum p 0 p.radius :=
{ r_le := le_rfl
r_pos := h
hasSum := fun hy => by
rw [zero_add]
exact p.hasSum hy }
#align formal_multilinear_series.has_fpower_series_on_ball FormalMultilinearSeries.hasFPowerSeriesOnBall
theorem HasFPowerSeriesOnBall.sum (h : HasFPowerSeriesOnBall f p x r) {y : E}
(hy : y ∈ EMetric.ball (0 : E) r) : f (x + y) = p.sum y :=
(h.hasSum hy).tsum_eq.symm
#align has_fpower_series_on_ball.sum HasFPowerSeriesOnBall.sum
/-- The sum of a converging power series is continuous in its disk of convergence. -/
protected theorem FormalMultilinearSeries.continuousOn [CompleteSpace F] :
ContinuousOn p.sum (EMetric.ball 0 p.radius) := by
rcases (zero_le p.radius).eq_or_lt with h | h
· simp [← h, continuousOn_empty]
· exact (p.hasFPowerSeriesOnBall h).continuousOn
#align formal_multilinear_series.continuous_on FormalMultilinearSeries.continuousOn
end
/-!
### Uniqueness of power series
If a function `f : E → F` has two representations as power series at a point `x : E`, corresponding
to formal multilinear series `p₁` and `p₂`, then these representations agree term-by-term. That is,
for any `n : ℕ` and `y : E`, `p₁ n (fun i ↦ y) = p₂ n (fun i ↦ y)`. In the one-dimensional case,
when `f : 𝕜 → E`, the continuous multilinear maps `p₁ n` and `p₂ n` are given by
`ContinuousMultilinearMap.mkPiField`, and hence are determined completely by the value of
`p₁ n (fun i ↦ 1)`, so `p₁ = p₂`. Consequently, the radius of convergence for one series can be
transferred to the other.
-/
section Uniqueness
open ContinuousMultilinearMap
theorem Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero {n : ℕ} {p : E[×n]→L[𝕜] F}
(h : (fun y => p fun _ => y) =O[𝓝 0] fun y => ‖y‖ ^ (n + 1)) (y : E) : (p fun _ => y) = 0 := by
obtain ⟨c, c_pos, hc⟩ := h.exists_pos
obtain ⟨t, ht, t_open, z_mem⟩ := eventually_nhds_iff.mp (isBigOWith_iff.mp hc)
obtain ⟨δ, δ_pos, δε⟩ := (Metric.isOpen_iff.mp t_open) 0 z_mem
clear h hc z_mem
cases' n with n
· exact norm_eq_zero.mp (by
-- porting note: the symmetric difference of the `simpa only` sets:
-- added `Nat.zero_eq, zero_add, pow_one`
-- removed `zero_pow', Ne.def, Nat.one_ne_zero, not_false_iff`
simpa only [Nat.zero_eq, fin0_apply_norm, norm_eq_zero, norm_zero, zero_add, pow_one,
mul_zero, norm_le_zero_iff] using ht 0 (δε (Metric.mem_ball_self δ_pos)))
· refine' Or.elim (Classical.em (y = 0))
(fun hy => by simpa only [hy] using p.map_zero) fun hy => _
replace hy := norm_pos_iff.mpr hy
refine' norm_eq_zero.mp (le_antisymm (le_of_forall_pos_le_add fun ε ε_pos => _) (norm_nonneg _))
have h₀ := _root_.mul_pos c_pos (pow_pos hy (n.succ + 1))
obtain ⟨k, k_pos, k_norm⟩ := NormedField.exists_norm_lt 𝕜
(lt_min (mul_pos δ_pos (inv_pos.mpr hy)) (mul_pos ε_pos (inv_pos.mpr h₀)))
have h₁ : ‖k • y‖ < δ := by
rw [norm_smul]
exact inv_mul_cancel_right₀ hy.ne.symm δ ▸
mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_left _ _)) hy
have h₂ :=
calc
‖p fun _ => k • y‖ ≤ c * ‖k • y‖ ^ (n.succ + 1) := by
-- porting note: now Lean wants `_root_.`
simpa only [norm_pow, _root_.norm_norm] using ht (k • y) (δε (mem_ball_zero_iff.mpr h₁))
--simpa only [norm_pow, norm_norm] using ht (k • y) (δε (mem_ball_zero_iff.mpr h₁))
_ = ‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))) := by
-- porting note: added `Nat.succ_eq_add_one` since otherwise `ring` does not conclude.
simp only [norm_smul, mul_pow, Nat.succ_eq_add_one]
-- porting note: removed `rw [pow_succ]`, since it now becomes superfluous.
ring
have h₃ : ‖k‖ * (c * ‖y‖ ^ (n.succ + 1)) < ε :=
inv_mul_cancel_right₀ h₀.ne.symm ε ▸
mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_right _ _)) h₀
calc
‖p fun _ => y‖ = ‖k⁻¹ ^ n.succ‖ * ‖p fun _ => k • y‖ := by
simpa only [inv_smul_smul₀ (norm_pos_iff.mp k_pos), norm_smul, Finset.prod_const,
Finset.card_fin] using
congr_arg norm (p.map_smul_univ (fun _ : Fin n.succ => k⁻¹) fun _ : Fin n.succ => k • y)
_ ≤ ‖k⁻¹ ^ n.succ‖ * (‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1)))) := by gcongr
_ = ‖(k⁻¹ * k) ^ n.succ‖ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))) := by
rw [← mul_assoc]
simp [norm_mul, mul_pow]
_ ≤ 0 + ε := by
rw [inv_mul_cancel (norm_pos_iff.mp k_pos)]
simpa using h₃.le
set_option linter.uppercaseLean3 false in
#align asymptotics.is_O.continuous_multilinear_map_apply_eq_zero Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero
/-- If a formal multilinear series `p` represents the zero function at `x : E`, then the
terms `p n (fun i ↦ y)` appearing in the sum are zero for any `n : ℕ`, `y : E`. -/
theorem HasFPowerSeriesAt.apply_eq_zero {p : FormalMultilinearSeries 𝕜 E F} {x : E}
(h : HasFPowerSeriesAt 0 p x) (n : ℕ) : ∀ y : E, (p n fun _ => y) = 0 := by
refine' Nat.strong_induction_on n fun k hk => _
have psum_eq : p.partialSum (k + 1) = fun y => p k fun _ => y := by
funext z
refine' Finset.sum_eq_single _ (fun b hb hnb => _) fun hn => _
· have := Finset.mem_range_succ_iff.mp hb
simp only [hk b (this.lt_of_ne hnb), Pi.zero_apply]
· exact False.elim (hn (Finset.mem_range.mpr (lt_add_one k)))
replace h := h.isBigO_sub_partialSum_pow k.succ
simp only [psum_eq, zero_sub, Pi.zero_apply, Asymptotics.isBigO_neg_left] at h
exact h.continuousMultilinearMap_apply_eq_zero
#align has_fpower_series_at.apply_eq_zero HasFPowerSeriesAt.apply_eq_zero
/-- A one-dimensional formal multilinear series representing the zero function is zero. -/
theorem HasFPowerSeriesAt.eq_zero {p : FormalMultilinearSeries 𝕜 𝕜 E} {x : 𝕜}
(h : HasFPowerSeriesAt 0 p x) : p = 0 := by
-- porting note: `funext; ext` was `ext (n x)`
funext n
ext x
rw [← mkPiField_apply_one_eq_self (p n)]
-- porting note: nasty hack, was `simp [h.apply_eq_zero n 1]`
have := Or.intro_right ?_ (h.apply_eq_zero n 1)
simpa using this
#align has_fpower_series_at.eq_zero HasFPowerSeriesAt.eq_zero
/-- One-dimensional formal multilinear series representing the same function are equal. -/
theorem HasFPowerSeriesAt.eq_formalMultilinearSeries {p₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E}
{f : 𝕜 → E} {x : 𝕜} (h₁ : HasFPowerSeriesAt f p₁ x) (h₂ : HasFPowerSeriesAt f p₂ x) : p₁ = p₂ :=
sub_eq_zero.mp (HasFPowerSeriesAt.eq_zero (by simpa only [sub_self] using h₁.sub h₂))
#align has_fpower_series_at.eq_formal_multilinear_series HasFPowerSeriesAt.eq_formalMultilinearSeries
theorem HasFPowerSeriesAt.eq_formalMultilinearSeries_of_eventually
{p q : FormalMultilinearSeries 𝕜 𝕜 E} {f g : 𝕜 → E} {x : 𝕜} (hp : HasFPowerSeriesAt f p x)
(hq : HasFPowerSeriesAt g q x) (heq : ∀ᶠ z in 𝓝 x, f z = g z) : p = q :=
(hp.congr heq).eq_formalMultilinearSeries hq
#align has_fpower_series_at.eq_formal_multilinear_series_of_eventually HasFPowerSeriesAt.eq_formalMultilinearSeries_of_eventually
/-- A one-dimensional formal multilinear series representing a locally zero function is zero. -/
theorem HasFPowerSeriesAt.eq_zero_of_eventually {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E}
{x : 𝕜} (hp : HasFPowerSeriesAt f p x) (hf : f =ᶠ[𝓝 x] 0) : p = 0 :=
(hp.congr hf).eq_zero
#align has_fpower_series_at.eq_zero_of_eventually HasFPowerSeriesAt.eq_zero_of_eventually
/-- If a function `f : 𝕜 → E` has two power series representations at `x`, then the given radii in
which convergence is guaranteed may be interchanged. This can be useful when the formal multilinear
series in one representation has a particularly nice form, but the other has a larger radius. -/
theorem HasFPowerSeriesOnBall.exchange_radius {p₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E}
{r₁ r₂ : ℝ≥0∞} {x : 𝕜} (h₁ : HasFPowerSeriesOnBall f p₁ x r₁)
(h₂ : HasFPowerSeriesOnBall f p₂ x r₂) : HasFPowerSeriesOnBall f p₁ x r₂ :=
h₂.hasFPowerSeriesAt.eq_formalMultilinearSeries h₁.hasFPowerSeriesAt ▸ h₂
#align has_fpower_series_on_ball.exchange_radius HasFPowerSeriesOnBall.exchange_radius
/-- If a function `f : 𝕜 → E` has power series representation `p` on a ball of some radius and for
each positive radius it has some power series representation, then `p` converges to `f` on the whole
`𝕜`. -/
theorem HasFPowerSeriesOnBall.r_eq_top_of_exists {f : 𝕜 → E} {r : ℝ≥0∞} {x : 𝕜}
{p : FormalMultilinearSeries 𝕜 𝕜 E} (h : HasFPowerSeriesOnBall f p x r)
(h' : ∀ (r' : ℝ≥0) (_ : 0 < r'), ∃ p' : FormalMultilinearSeries 𝕜 𝕜 E,
HasFPowerSeriesOnBall f p' x r') :
HasFPowerSeriesOnBall f p x ∞ :=
{ r_le := ENNReal.le_of_forall_pos_nnreal_lt fun r hr _ =>
let ⟨_, hp'⟩ := h' r hr
(h.exchange_radius hp').r_le
r_pos := ENNReal.coe_lt_top
hasSum := fun {y} _ =>
let ⟨r', hr'⟩ := exists_gt ‖y‖₊
let ⟨_, hp'⟩ := h' r' hr'.ne_bot.bot_lt
(h.exchange_radius hp').hasSum <| mem_emetric_ball_zero_iff.mpr (ENNReal.coe_lt_coe.2 hr') }
#align has_fpower_series_on_ball.r_eq_top_of_exists HasFPowerSeriesOnBall.r_eq_top_of_exists
end Uniqueness
/-!
### Changing origin in a power series
If a function is analytic in a disk `D(x, R)`, then it is analytic in any disk contained in that
one. Indeed, one can write
$$
f (x + y + z) = \sum_{n} p_n (y + z)^n = \sum_{n, k} \binom{n}{k} p_n y^{n-k} z^k
= \sum_{k} \Bigl(\sum_{n} \binom{n}{k} p_n y^{n-k}\Bigr) z^k.
$$
The corresponding power series has thus a `k`-th coefficient equal to
$\sum_{n} \binom{n}{k} p_n y^{n-k}$. In the general case where `pₙ` is a multilinear map, this has
to be interpreted suitably: instead of having a binomial coefficient, one should sum over all
possible subsets `s` of `Fin n` of cardinal `k`, and attribute `z` to the indices in `s` and
`y` to the indices outside of `s`.
In this paragraph, we implement this. The new power series is called `p.changeOrigin y`. Then, we
check its convergence and the fact that its sum coincides with the original sum. The outcome of this
discussion is that the set of points where a function is analytic is open.
-/
namespace FormalMultilinearSeries
section
variable (p : FormalMultilinearSeries 𝕜 E F) {x y : E} {r R : ℝ≥0}
/-- A term of `FormalMultilinearSeries.changeOriginSeries`.
Given a formal multilinear series `p` and a point `x` in its ball of convergence,
`p.changeOrigin x` is a formal multilinear series such that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense. Each term of `p.changeOrigin x`
is itself an analytic function of `x` given by the series `p.changeOriginSeries`. Each term in
`changeOriginSeries` is the sum of `changeOriginSeriesTerm`'s over all `s` of cardinality `l`.
The definition is such that `p.changeOriginSeriesTerm k l s hs (fun _ ↦ x) (fun _ ↦ y) =
p (k + l) (s.piecewise (fun _ ↦ x) (fun _ ↦ y))`
-/
def changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
E[×l]→L[𝕜] E[×k]→L[𝕜] F := by
let a := ContinuousMultilinearMap.curryFinFinset 𝕜 E F hs
(by erw [Finset.card_compl, Fintype.card_fin, hs, add_tsub_cancel_right])
exact a (p (k + l))
#align formal_multilinear_series.change_origin_series_term FormalMultilinearSeries.changeOriginSeriesTerm
theorem changeOriginSeriesTerm_apply (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l)
(x y : E) :
(p.changeOriginSeriesTerm k l s hs (fun _ => x) fun _ => y) =
p (k + l) (s.piecewise (fun _ => x) fun _ => y) :=
ContinuousMultilinearMap.curryFinFinset_apply_const _ _ _ _ _
#align formal_multilinear_series.change_origin_series_term_apply FormalMultilinearSeries.changeOriginSeriesTerm_apply
@[simp]
theorem norm_changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
‖p.changeOriginSeriesTerm k l s hs‖ = ‖p (k + l)‖ := by
simp only [changeOriginSeriesTerm, LinearIsometryEquiv.norm_map]
#align formal_multilinear_series.norm_change_origin_series_term FormalMultilinearSeries.norm_changeOriginSeriesTerm
@[simp]
theorem nnnorm_changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
‖p.changeOriginSeriesTerm k l s hs‖₊ = ‖p (k + l)‖₊ := by
simp only [changeOriginSeriesTerm, LinearIsometryEquiv.nnnorm_map]
#align formal_multilinear_series.nnnorm_change_origin_series_term FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm
theorem nnnorm_changeOriginSeriesTerm_apply_le (k l : ℕ) (s : Finset (Fin (k + l)))
(hs : s.card = l) (x y : E) :
‖p.changeOriginSeriesTerm k l s hs (fun _ => x) fun _ => y‖₊ ≤
‖p (k + l)‖₊ * ‖x‖₊ ^ l * ‖y‖₊ ^ k := by
rw [← p.nnnorm_changeOriginSeriesTerm k l s hs, ← Fin.prod_const, ← Fin.prod_const]
apply ContinuousMultilinearMap.le_of_op_nnnorm_le
apply ContinuousMultilinearMap.le_op_nnnorm
#align formal_multilinear_series.nnnorm_change_origin_series_term_apply_le FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm_apply_le
/-- The power series for `f.changeOrigin k`.
Given a formal multilinear series `p` and a point `x` in its ball of convergence,
`p.changeOrigin x` is a formal multilinear series such that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense. Its `k`-th term is the sum of
the series `p.changeOriginSeries k`. -/
def changeOriginSeries (k : ℕ) : FormalMultilinearSeries 𝕜 E (E[×k]→L[𝕜] F) := fun l =>
∑ s : { s : Finset (Fin (k + l)) // Finset.card s = l }, p.changeOriginSeriesTerm k l s s.2
#align formal_multilinear_series.change_origin_series FormalMultilinearSeries.changeOriginSeries
theorem nnnorm_changeOriginSeries_le_tsum (k l : ℕ) :
‖p.changeOriginSeries k l‖₊ ≤
∑' _ : { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + l)‖₊ :=
(nnnorm_sum_le _ (fun t => changeOriginSeriesTerm p k l (Subtype.val t) t.prop)).trans_eq <| by
simp_rw [tsum_fintype, nnnorm_changeOriginSeriesTerm (p := p) (k := k) (l := l)]
#align formal_multilinear_series.nnnorm_change_origin_series_le_tsum FormalMultilinearSeries.nnnorm_changeOriginSeries_le_tsum
theorem nnnorm_changeOriginSeries_apply_le_tsum (k l : ℕ) (x : E) :
‖p.changeOriginSeries k l fun _ => x‖₊ ≤
∑' _ : { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + l)‖₊ * ‖x‖₊ ^ l := by
rw [NNReal.tsum_mul_right, ← Fin.prod_const]
exact (p.changeOriginSeries k l).le_of_op_nnnorm_le _ (p.nnnorm_changeOriginSeries_le_tsum _ _)
#align formal_multilinear_series.nnnorm_change_origin_series_apply_le_tsum FormalMultilinearSeries.nnnorm_changeOriginSeries_apply_le_tsum
/-- Changing the origin of a formal multilinear series `p`, so that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense.
-/
def changeOrigin (x : E) : FormalMultilinearSeries 𝕜 E F :=
fun k => (p.changeOriginSeries k).sum x
#align formal_multilinear_series.change_origin FormalMultilinearSeries.changeOrigin
/-- An auxiliary equivalence useful in the proofs about
`FormalMultilinearSeries.changeOriginSeries`: the set of triples `(k, l, s)`, where `s` is a
`Finset (Fin (k + l))` of cardinality `l` is equivalent to the set of pairs `(n, s)`, where `s` is a
`Finset (Fin n)`.
The forward map sends `(k, l, s)` to `(k + l, s)` and the inverse map sends `(n, s)` to
`(n - Finset.card s, Finset.card s, s)`. The actual definition is less readable because of problems
with non-definitional equalities. -/
@[simps]
def changeOriginIndexEquiv :
(Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) ≃ Σn : ℕ, Finset (Fin n) where
toFun s := ⟨s.1 + s.2.1, s.2.2⟩
invFun s :=
⟨s.1 - s.2.card, s.2.card,
⟨s.2.map
(Fin.castIso <| (tsub_add_cancel_of_le <| card_finset_fin_le s.2).symm).toEquiv.toEmbedding,
Finset.card_map _⟩⟩
left_inv := by
rintro ⟨k, l, ⟨s : Finset (Fin <| k + l), hs : s.card = l⟩⟩
dsimp only [Subtype.coe_mk]
-- Lean can't automatically generalize `k' = k + l - s.card`, `l' = s.card`, so we explicitly
-- formulate the generalized goal
suffices ∀ k' l', k' = k → l' = l → ∀ (hkl : k + l = k' + l') (hs'),
(⟨k', l', ⟨Finset.map (Fin.castIso hkl).toEquiv.toEmbedding s, hs'⟩⟩ :
Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) = ⟨k, l, ⟨s, hs⟩⟩ by
apply this <;> simp only [hs, add_tsub_cancel_right]
rintro _ _ rfl rfl hkl hs'
simp only [Equiv.refl_toEmbedding, Fin.castIso_refl, Finset.map_refl, eq_self_iff_true,
OrderIso.refl_toEquiv, and_self_iff, heq_iff_eq]
right_inv := by
rintro ⟨n, s⟩
simp [tsub_add_cancel_of_le (card_finset_fin_le s), Fin.castIso_to_equiv]
#align formal_multilinear_series.change_origin_index_equiv FormalMultilinearSeries.changeOriginIndexEquiv
theorem changeOriginSeries_summable_aux₁ {r r' : ℝ≥0} (hr : (r + r' : ℝ≥0∞) < p.radius) :
Summable fun s : Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l } =>
‖p (s.1 + s.2.1)‖₊ * r ^ s.2.1 * r' ^ s.1 := by
rw [← changeOriginIndexEquiv.symm.summable_iff]
dsimp only [Function.comp_def, changeOriginIndexEquiv_symm_apply_fst,
changeOriginIndexEquiv_symm_apply_snd_fst]
have : ∀ n : ℕ,
HasSum (fun s : Finset (Fin n) => ‖p (n - s.card + s.card)‖₊ * r ^ s.card * r' ^ (n - s.card))
(‖p n‖₊ * (r + r') ^ n) := by
intro n
-- TODO: why `simp only [tsub_add_cancel_of_le (card_finset_fin_le _)]` fails?
convert_to HasSum (fun s : Finset (Fin n) => ‖p n‖₊ * (r ^ s.card * r' ^ (n - s.card))) _
· ext1 s
rw [tsub_add_cancel_of_le (card_finset_fin_le _), mul_assoc]
rw [← Fin.sum_pow_mul_eq_add_pow]
exact (hasSum_fintype _).mul_left _
refine' NNReal.summable_sigma.2 ⟨fun n => (this n).summable, _⟩
simp only [(this _).tsum_eq]
exact p.summable_nnnorm_mul_pow hr
#align formal_multilinear_series.change_origin_series_summable_aux₁ FormalMultilinearSeries.changeOriginSeries_summable_aux₁
theorem changeOriginSeries_summable_aux₂ (hr : (r : ℝ≥0∞) < p.radius) (k : ℕ) :
Summable fun s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l } =>
‖p (k + s.1)‖₊ * r ^ s.1 := by
rcases ENNReal.lt_iff_exists_add_pos_lt.1 hr with ⟨r', h0, hr'⟩
simpa only [mul_inv_cancel_right₀ (pow_pos h0 _).ne'] using
((NNReal.summable_sigma.1 (p.changeOriginSeries_summable_aux₁ hr')).1 k).mul_right (r' ^ k)⁻¹
#align formal_multilinear_series.change_origin_series_summable_aux₂ FormalMultilinearSeries.changeOriginSeries_summable_aux₂
theorem changeOriginSeries_summable_aux₃ {r : ℝ≥0} (hr : ↑r < p.radius) (k : ℕ) :
Summable fun l : ℕ => ‖p.changeOriginSeries k l‖₊ * r ^ l := by
refine' NNReal.summable_of_le
(fun n => _) (NNReal.summable_sigma.1 <| p.changeOriginSeries_summable_aux₂ hr k).2
simp only [NNReal.tsum_mul_right]
exact mul_le_mul' (p.nnnorm_changeOriginSeries_le_tsum _ _) le_rfl
#align formal_multilinear_series.change_origin_series_summable_aux₃ FormalMultilinearSeries.changeOriginSeries_summable_aux₃
theorem le_changeOriginSeries_radius (k : ℕ) : p.radius ≤ (p.changeOriginSeries k).radius :=
ENNReal.le_of_forall_nnreal_lt fun _r hr =>
le_radius_of_summable_nnnorm _ (p.changeOriginSeries_summable_aux₃ hr k)
#align formal_multilinear_series.le_change_origin_series_radius FormalMultilinearSeries.le_changeOriginSeries_radius
theorem nnnorm_changeOrigin_le (k : ℕ) (h : (‖x‖₊ : ℝ≥0∞) < p.radius) :
‖p.changeOrigin x k‖₊ ≤
∑' s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + s.1)‖₊ * ‖x‖₊ ^ s.1 := by
refine' tsum_of_nnnorm_bounded _ fun l => p.nnnorm_changeOriginSeries_apply_le_tsum k l x
have := p.changeOriginSeries_summable_aux₂ h k
refine' HasSum.sigma this.hasSum fun l => _
exact ((NNReal.summable_sigma.1 this).1 l).hasSum
#align formal_multilinear_series.nnnorm_change_origin_le FormalMultilinearSeries.nnnorm_changeOrigin_le
/-- The radius of convergence of `p.changeOrigin x` is at least `p.radius - ‖x‖`. In other words,
`p.changeOrigin x` is well defined on the largest ball contained in the original ball of
convergence. -/
theorem changeOrigin_radius : p.radius - ‖x‖₊ ≤ (p.changeOrigin x).radius := by
refine' ENNReal.le_of_forall_pos_nnreal_lt fun r _h0 hr => _
rw [lt_tsub_iff_right, add_comm] at hr
have hr' : (‖x‖₊ : ℝ≥0∞) < p.radius := (le_add_right le_rfl).trans_lt hr
apply le_radius_of_summable_nnnorm
have : ∀ k : ℕ,
‖p.changeOrigin x k‖₊ * r ^ k ≤
(∑' s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + s.1)‖₊ * ‖x‖₊ ^ s.1) *
r ^ k :=
fun k => mul_le_mul_right' (p.nnnorm_changeOrigin_le k hr') (r ^ k)
refine' NNReal.summable_of_le this _
simpa only [← NNReal.tsum_mul_right] using
(NNReal.summable_sigma.1 (p.changeOriginSeries_summable_aux₁ hr)).2
#align formal_multilinear_series.change_origin_radius FormalMultilinearSeries.changeOrigin_radius
end
-- From this point on, assume that the space is complete, to make sure that series that converge
-- in norm also converge in `F`.
variable [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x y : E} {r R : ℝ≥0}
theorem hasFPowerSeriesOnBall_changeOrigin (k : ℕ) (hr : 0 < p.radius) :
HasFPowerSeriesOnBall (fun x => p.changeOrigin x k) (p.changeOriginSeries k) 0 p.radius :=
have := p.le_changeOriginSeries_radius k
((p.changeOriginSeries k).hasFPowerSeriesOnBall (hr.trans_le this)).mono hr this
#align formal_multilinear_series.has_fpower_series_on_ball_change_origin FormalMultilinearSeries.hasFPowerSeriesOnBall_changeOrigin
/-- Summing the series `p.changeOrigin x` at a point `y` gives back `p (x + y)`. -/
theorem changeOrigin_eval (h : (‖x‖₊ + ‖y‖₊ : ℝ≥0∞) < p.radius) :
(p.changeOrigin x).sum y = p.sum (x + y) := by
have radius_pos : 0 < p.radius := lt_of_le_of_lt (zero_le _) h
have x_mem_ball : x ∈ EMetric.ball (0 : E) p.radius :=
mem_emetric_ball_zero_iff.2 ((le_add_right le_rfl).trans_lt h)
have y_mem_ball : y ∈ EMetric.ball (0 : E) (p.changeOrigin x).radius := by
refine' mem_emetric_ball_zero_iff.2 (lt_of_lt_of_le _ p.changeOrigin_radius)
rwa [lt_tsub_iff_right, add_comm]
have x_add_y_mem_ball : x + y ∈ EMetric.ball (0 : E) p.radius := by
refine' mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt _ h)
exact mod_cast nnnorm_add_le x y
set f : (Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) → F := fun s =>
p.changeOriginSeriesTerm s.1 s.2.1 s.2.2 s.2.2.2 (fun _ => x) fun _ => y
have hsf : Summable f := by
refine' .of_nnnorm_bounded _ (p.changeOriginSeries_summable_aux₁ h) _
rintro ⟨k, l, s, hs⟩
dsimp only [Subtype.coe_mk]
exact p.nnnorm_changeOriginSeriesTerm_apply_le _ _ _ _ _ _
have hf : HasSum f ((p.changeOrigin x).sum y) := by
refine' HasSum.sigma_of_hasSum ((p.changeOrigin x).summable y_mem_ball).hasSum (fun k => _) hsf
· dsimp only
refine' ContinuousMultilinearMap.hasSum_eval _ _
have := (p.hasFPowerSeriesOnBall_changeOrigin k radius_pos).hasSum x_mem_ball
rw [zero_add] at this
refine' HasSum.sigma_of_hasSum this (fun l => _) _
· simp only [changeOriginSeries, ContinuousMultilinearMap.sum_apply]
apply hasSum_fintype
· refine' .of_nnnorm_bounded _
(p.changeOriginSeries_summable_aux₂ (mem_emetric_ball_zero_iff.1 x_mem_ball) k) fun s => _
refine' (ContinuousMultilinearMap.le_op_nnnorm _ _).trans_eq _
simp
refine' hf.unique (changeOriginIndexEquiv.symm.hasSum_iff.1 _)
refine' HasSum.sigma_of_hasSum
(p.hasSum x_add_y_mem_ball) (fun n => _) (changeOriginIndexEquiv.symm.summable_iff.2 hsf)
erw [(p n).map_add_univ (fun _ => x) fun _ => y]
-- porting note: added explicit function
convert hasSum_fintype (fun c : Finset (Fin n) => f (changeOriginIndexEquiv.symm ⟨n, c⟩))
rename_i s _
dsimp only [changeOriginSeriesTerm, (· ∘ ·), changeOriginIndexEquiv_symm_apply_fst,
changeOriginIndexEquiv_symm_apply_snd_fst, changeOriginIndexEquiv_symm_apply_snd_snd_coe]
rw [ContinuousMultilinearMap.curryFinFinset_apply_const]
have : ∀ (m) (hm : n = m), p n (s.piecewise (fun _ => x) fun _ => y) =
p m ((s.map (Fin.castIso hm).toEquiv.toEmbedding).piecewise (fun _ => x) fun _ => y) := by
rintro m rfl
simp (config := { unfoldPartialApp := true }) [Finset.piecewise]
apply this
#align formal_multilinear_series.change_origin_eval FormalMultilinearSeries.changeOrigin_eval
/-- Power series terms are analytic as we vary the origin -/
theorem analyticAt_changeOrigin (p : FormalMultilinearSeries 𝕜 E F) (rp : p.radius > 0) (n : ℕ) :
AnalyticAt 𝕜 (fun x ↦ p.changeOrigin x n) 0 :=
(FormalMultilinearSeries.hasFPowerSeriesOnBall_changeOrigin p n rp).analyticAt
end FormalMultilinearSeries
section
variable [CompleteSpace F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x y : E} {r : ℝ≥0∞}
/-- If a function admits a power series expansion `p` on a ball `B (x, r)`, then it also admits a
power series on any subball of this ball (even with a different center), given by `p.changeOrigin`.
-/
theorem HasFPowerSeriesOnBall.changeOrigin (hf : HasFPowerSeriesOnBall f p x r)
(h : (‖y‖₊ : ℝ≥0∞) < r) : HasFPowerSeriesOnBall f (p.changeOrigin y) (x + y) (r - ‖y‖₊) :=
{ r_le := by
apply le_trans _ p.changeOrigin_radius
exact tsub_le_tsub hf.r_le le_rfl
r_pos := by simp [h]
hasSum := fun {z} hz => by
have : f (x + y + z) =
FormalMultilinearSeries.sum (FormalMultilinearSeries.changeOrigin p y) z := by
rw [mem_emetric_ball_zero_iff, lt_tsub_iff_right, add_comm] at hz
rw [p.changeOrigin_eval (hz.trans_le hf.r_le), add_assoc, hf.sum]
refine' mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt _ hz)
exact mod_cast nnnorm_add_le y z
rw [this]
apply (p.changeOrigin y).hasSum
refine' EMetric.ball_subset_ball (le_trans _ p.changeOrigin_radius) hz
exact tsub_le_tsub hf.r_le le_rfl }
#align has_fpower_series_on_ball.change_origin HasFPowerSeriesOnBall.changeOrigin
/-- If a function admits a power series expansion `p` on an open ball `B (x, r)`, then
it is analytic at every point of this ball. -/
theorem HasFPowerSeriesOnBall.analyticAt_of_mem (hf : HasFPowerSeriesOnBall f p x r)
(h : y ∈ EMetric.ball x r) : AnalyticAt 𝕜 f y := by
have : (‖y - x‖₊ : ℝ≥0∞) < r := by simpa [edist_eq_coe_nnnorm_sub] using h
have := hf.changeOrigin this
rw [add_sub_cancel'_right] at this
exact this.analyticAt
#align has_fpower_series_on_ball.analytic_at_of_mem HasFPowerSeriesOnBall.analyticAt_of_mem
theorem HasFPowerSeriesOnBall.analyticOn (hf : HasFPowerSeriesOnBall f p x r) :
AnalyticOn 𝕜 f (EMetric.ball x r) :=
fun _y hy => hf.analyticAt_of_mem hy
#align has_fpower_series_on_ball.analytic_on HasFPowerSeriesOnBall.analyticOn
variable (𝕜 f)
/-- For any function `f` from a normed vector space to a Banach space, the set of points `x` such
that `f` is analytic at `x` is open. -/
theorem isOpen_analyticAt : IsOpen { x | AnalyticAt 𝕜 f x } := by
rw [isOpen_iff_mem_nhds]
rintro x ⟨p, r, hr⟩
exact mem_of_superset (EMetric.ball_mem_nhds _ hr.r_pos) fun y hy => hr.analyticAt_of_mem hy
#align is_open_analytic_at isOpen_analyticAt
variable {𝕜}
theorem AnalyticAt.eventually_analyticAt {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∀ᶠ y in 𝓝 x, AnalyticAt 𝕜 f y :=
(isOpen_analyticAt 𝕜 f).mem_nhds h
theorem AnalyticAt.exists_mem_nhds_analyticOn {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∃ s ∈ 𝓝 x, AnalyticOn 𝕜 f s :=
h.eventually_analyticAt.exists_mem
/-- If we're analytic at a point, we're analytic in a nonempty ball -/
theorem AnalyticAt.exists_ball_analyticOn {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∃ r : ℝ, 0 < r ∧ AnalyticOn 𝕜 f (Metric.ball x r) :=
Metric.isOpen_iff.mp (isOpen_analyticAt _ _) _ h
end
section
open FormalMultilinearSeries
variable {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E} {z₀ : 𝕜}
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z)) := by
refine' ⟨fun ⟨r, _, r_pos, h⟩ =>
eventually_of_mem (EMetric.ball_mem_nhds 0 r_pos) fun _ => by simpa using h, _⟩
|
simp only [Metric.eventually_nhds_iff]
|
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z)) := by
refine' ⟨fun ⟨r, _, r_pos, h⟩ =>
eventually_of_mem (EMetric.ball_mem_nhds 0 r_pos) fun _ => by simpa using h, _⟩
|
Mathlib.Analysis.Analytic.Basic.1430_0.jQw1fRSE1vGpOll
|
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z))
|
Mathlib_Analysis_Analytic_Basic
|
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
p : FormalMultilinearSeries 𝕜 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
⊢ (∃ ε > 0, ∀ ⦃y : 𝕜⦄, dist y 0 < ε → HasSum (fun n => y ^ n • coeff p n) (f (z₀ + y))) → HasFPowerSeriesAt f p z₀
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Logic.Equiv.Fin
import Mathlib.Topology.Algebra.InfiniteSum.Module
#align_import analysis.analytic.basic from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
/-!
# Analytic functions
A function is analytic in one dimension around `0` if it can be written as a converging power series
`Σ pₙ zⁿ`. This definition can be extended to any dimension (even in infinite dimension) by
requiring that `pₙ` is a continuous `n`-multilinear map. In general, `pₙ` is not unique (in two
dimensions, taking `p₂ (x, y) (x', y') = x y'` or `y x'` gives the same map when applied to a
vector `(x, y) (x, y)`). A way to guarantee uniqueness is to take a symmetric `pₙ`, but this is not
always possible in nonzero characteristic (in characteristic 2, the previous example has no
symmetric representative). Therefore, we do not insist on symmetry or uniqueness in the definition,
and we only require the existence of a converging series.
The general framework is important to say that the exponential map on bounded operators on a Banach
space is analytic, as well as the inverse on invertible operators.
## Main definitions
Let `p` be a formal multilinear series from `E` to `F`, i.e., `p n` is a multilinear map on `E^n`
for `n : ℕ`.
* `p.radius`: the largest `r : ℝ≥0∞` such that `‖p n‖ * r^n` grows subexponentially.
* `p.le_radius_of_bound`, `p.le_radius_of_bound_nnreal`, `p.le_radius_of_isBigO`: if `‖p n‖ * r ^ n`
is bounded above, then `r ≤ p.radius`;
* `p.isLittleO_of_lt_radius`, `p.norm_mul_pow_le_mul_pow_of_lt_radius`,
`p.isLittleO_one_of_lt_radius`,
`p.norm_mul_pow_le_of_lt_radius`, `p.nnnorm_mul_pow_le_of_lt_radius`: if `r < p.radius`, then
`‖p n‖ * r ^ n` tends to zero exponentially;
* `p.lt_radius_of_isBigO`: if `r ≠ 0` and `‖p n‖ * r ^ n = O(a ^ n)` for some `-1 < a < 1`, then
`r < p.radius`;
* `p.partialSum n x`: the sum `∑_{i = 0}^{n-1} pᵢ xⁱ`.
* `p.sum x`: the sum `∑'_{i = 0}^{∞} pᵢ xⁱ`.
Additionally, let `f` be a function from `E` to `F`.
* `HasFPowerSeriesOnBall f p x r`: on the ball of center `x` with radius `r`,
`f (x + y) = ∑'_n pₙ yⁿ`.
* `HasFPowerSeriesAt f p x`: on some ball of center `x` with positive radius, holds
`HasFPowerSeriesOnBall f p x r`.
* `AnalyticAt 𝕜 f x`: there exists a power series `p` such that holds `HasFPowerSeriesAt f p x`.
* `AnalyticOn 𝕜 f s`: the function `f` is analytic at every point of `s`.
We develop the basic properties of these notions, notably:
* If a function admits a power series, it is continuous (see
`HasFPowerSeriesOnBall.continuousOn` and `HasFPowerSeriesAt.continuousAt` and
`AnalyticAt.continuousAt`).
* In a complete space, the sum of a formal power series with positive radius is well defined on the
disk of convergence, see `FormalMultilinearSeries.hasFPowerSeriesOnBall`.
* If a function admits a power series in a ball, then it is analytic at any point `y` of this ball,
and the power series there can be expressed in terms of the initial power series `p` as
`p.changeOrigin y`. See `HasFPowerSeriesOnBall.changeOrigin`. It follows in particular that
the set of points at which a given function is analytic is open, see `isOpen_analyticAt`.
## Implementation details
We only introduce the radius of convergence of a power series, as `p.radius`.
For a power series in finitely many dimensions, there is a finer (directional, coordinate-dependent)
notion, describing the polydisk of convergence. This notion is more specific, and not necessary to
build the general theory. We do not define it here.
-/
noncomputable section
variable {𝕜 E F G : Type*}
open Topology Classical BigOperators NNReal Filter ENNReal
open Set Filter Asymptotics
namespace FormalMultilinearSeries
variable [Ring 𝕜] [AddCommGroup E] [AddCommGroup F] [Module 𝕜 E] [Module 𝕜 F]
variable [TopologicalSpace E] [TopologicalSpace F]
variable [TopologicalAddGroup E] [TopologicalAddGroup F]
variable [ContinuousConstSMul 𝕜 E] [ContinuousConstSMul 𝕜 F]
/-- Given a formal multilinear series `p` and a vector `x`, then `p.sum x` is the sum `Σ pₙ xⁿ`. A
priori, it only behaves well when `‖x‖ < p.radius`. -/
protected def sum (p : FormalMultilinearSeries 𝕜 E F) (x : E) : F :=
∑' n : ℕ, p n fun _ => x
#align formal_multilinear_series.sum FormalMultilinearSeries.sum
/-- Given a formal multilinear series `p` and a vector `x`, then `p.partialSum n x` is the sum
`Σ pₖ xᵏ` for `k ∈ {0,..., n-1}`. -/
def partialSum (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) (x : E) : F :=
∑ k in Finset.range n, p k fun _ : Fin k => x
#align formal_multilinear_series.partial_sum FormalMultilinearSeries.partialSum
/-- The partial sums of a formal multilinear series are continuous. -/
theorem partialSum_continuous (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) :
Continuous (p.partialSum n) := by
unfold partialSum -- Porting note: added
continuity
#align formal_multilinear_series.partial_sum_continuous FormalMultilinearSeries.partialSum_continuous
end FormalMultilinearSeries
/-! ### The radius of a formal multilinear series -/
variable [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F]
[NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G]
namespace FormalMultilinearSeries
variable (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
/-- The radius of a formal multilinear series is the largest `r` such that the sum `Σ ‖pₙ‖ ‖y‖ⁿ`
converges for all `‖y‖ < r`. This implies that `Σ pₙ yⁿ` converges for all `‖y‖ < r`, but these
definitions are *not* equivalent in general. -/
def radius (p : FormalMultilinearSeries 𝕜 E F) : ℝ≥0∞ :=
⨆ (r : ℝ≥0) (C : ℝ) (_ : ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C), (r : ℝ≥0∞)
#align formal_multilinear_series.radius FormalMultilinearSeries.radius
/-- If `‖pₙ‖ rⁿ` is bounded in `n`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_bound (C : ℝ) {r : ℝ≥0} (h : ∀ n : ℕ, ‖p n‖ * (r : ℝ) ^ n ≤ C) :
(r : ℝ≥0∞) ≤ p.radius :=
le_iSup_of_le r <| le_iSup_of_le C <| le_iSup (fun _ => (r : ℝ≥0∞)) h
#align formal_multilinear_series.le_radius_of_bound FormalMultilinearSeries.le_radius_of_bound
/-- If `‖pₙ‖ rⁿ` is bounded in `n`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_bound_nnreal (C : ℝ≥0) {r : ℝ≥0} (h : ∀ n : ℕ, ‖p n‖₊ * r ^ n ≤ C) :
(r : ℝ≥0∞) ≤ p.radius :=
p.le_radius_of_bound C fun n => mod_cast h n
#align formal_multilinear_series.le_radius_of_bound_nnreal FormalMultilinearSeries.le_radius_of_bound_nnreal
/-- If `‖pₙ‖ rⁿ = O(1)`, as `n → ∞`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_isBigO (h : (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] fun _ => (1 : ℝ)) :
↑r ≤ p.radius :=
Exists.elim (isBigO_one_nat_atTop_iff.1 h) fun C hC =>
p.le_radius_of_bound C fun n => (le_abs_self _).trans (hC n)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.le_radius_of_is_O FormalMultilinearSeries.le_radius_of_isBigO
theorem le_radius_of_eventually_le (C) (h : ∀ᶠ n in atTop, ‖p n‖ * (r : ℝ) ^ n ≤ C) :
↑r ≤ p.radius :=
p.le_radius_of_isBigO <| IsBigO.of_bound C <| h.mono fun n hn => by simpa
#align formal_multilinear_series.le_radius_of_eventually_le FormalMultilinearSeries.le_radius_of_eventually_le
theorem le_radius_of_summable_nnnorm (h : Summable fun n => ‖p n‖₊ * r ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_bound_nnreal (∑' n, ‖p n‖₊ * r ^ n) fun _ => le_tsum' h _
#align formal_multilinear_series.le_radius_of_summable_nnnorm FormalMultilinearSeries.le_radius_of_summable_nnnorm
theorem le_radius_of_summable (h : Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_summable_nnnorm <| by
simp only [← coe_nnnorm] at h
exact mod_cast h
#align formal_multilinear_series.le_radius_of_summable FormalMultilinearSeries.le_radius_of_summable
theorem radius_eq_top_of_forall_nnreal_isBigO
(h : ∀ r : ℝ≥0, (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] fun _ => (1 : ℝ)) : p.radius = ∞ :=
ENNReal.eq_top_of_forall_nnreal_le fun r => p.le_radius_of_isBigO (h r)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.radius_eq_top_of_forall_nnreal_is_O FormalMultilinearSeries.radius_eq_top_of_forall_nnreal_isBigO
theorem radius_eq_top_of_eventually_eq_zero (h : ∀ᶠ n in atTop, p n = 0) : p.radius = ∞ :=
p.radius_eq_top_of_forall_nnreal_isBigO fun r =>
(isBigO_zero _ _).congr' (h.mono fun n hn => by simp [hn]) EventuallyEq.rfl
#align formal_multilinear_series.radius_eq_top_of_eventually_eq_zero FormalMultilinearSeries.radius_eq_top_of_eventually_eq_zero
theorem radius_eq_top_of_forall_image_add_eq_zero (n : ℕ) (hn : ∀ m, p (m + n) = 0) :
p.radius = ∞ :=
p.radius_eq_top_of_eventually_eq_zero <|
mem_atTop_sets.2 ⟨n, fun _ hk => tsub_add_cancel_of_le hk ▸ hn _⟩
#align formal_multilinear_series.radius_eq_top_of_forall_image_add_eq_zero FormalMultilinearSeries.radius_eq_top_of_forall_image_add_eq_zero
@[simp]
theorem constFormalMultilinearSeries_radius {v : F} :
(constFormalMultilinearSeries 𝕜 E v).radius = ⊤ :=
(constFormalMultilinearSeries 𝕜 E v).radius_eq_top_of_forall_image_add_eq_zero 1
(by simp [constFormalMultilinearSeries])
#align formal_multilinear_series.const_formal_multilinear_series_radius FormalMultilinearSeries.constFormalMultilinearSeries_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` tends to zero exponentially:
for some `0 < a < 1`, `‖p n‖ rⁿ = o(aⁿ)`. -/
theorem isLittleO_of_lt_radius (h : ↑r < p.radius) :
∃ a ∈ Ioo (0 : ℝ) 1, (fun n => ‖p n‖ * (r : ℝ) ^ n) =o[atTop] (a ^ ·) := by
have := (TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 4
rw [this]
-- Porting note: was
-- rw [(TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 4]
simp only [radius, lt_iSup_iff] at h
rcases h with ⟨t, C, hC, rt⟩
rw [ENNReal.coe_lt_coe, ← NNReal.coe_lt_coe] at rt
have : 0 < (t : ℝ) := r.coe_nonneg.trans_lt rt
rw [← div_lt_one this] at rt
refine' ⟨_, rt, C, Or.inr zero_lt_one, fun n => _⟩
calc
|‖p n‖ * (r : ℝ) ^ n| = ‖p n‖ * (t : ℝ) ^ n * (r / t : ℝ) ^ n := by
field_simp [mul_right_comm, abs_mul]
_ ≤ C * (r / t : ℝ) ^ n := by gcongr; apply hC
#align formal_multilinear_series.is_o_of_lt_radius FormalMultilinearSeries.isLittleO_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ = o(1)`. -/
theorem isLittleO_one_of_lt_radius (h : ↑r < p.radius) :
(fun n => ‖p n‖ * (r : ℝ) ^ n) =o[atTop] (fun _ => 1 : ℕ → ℝ) :=
let ⟨_, ha, hp⟩ := p.isLittleO_of_lt_radius h
hp.trans <| (isLittleO_pow_pow_of_lt_left ha.1.le ha.2).congr (fun _ => rfl) one_pow
#align formal_multilinear_series.is_o_one_of_lt_radius FormalMultilinearSeries.isLittleO_one_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` tends to zero exponentially:
for some `0 < a < 1` and `C > 0`, `‖p n‖ * r ^ n ≤ C * a ^ n`. -/
theorem norm_mul_pow_le_mul_pow_of_lt_radius (h : ↑r < p.radius) :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C * a ^ n := by
-- Porting note: moved out of `rcases`
have := ((TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 5).mp
(p.isLittleO_of_lt_radius h)
rcases this with ⟨a, ha, C, hC, H⟩
exact ⟨a, ha, C, hC, fun n => (le_abs_self _).trans (H n)⟩
#align formal_multilinear_series.norm_mul_pow_le_mul_pow_of_lt_radius FormalMultilinearSeries.norm_mul_pow_le_mul_pow_of_lt_radius
/-- If `r ≠ 0` and `‖pₙ‖ rⁿ = O(aⁿ)` for some `-1 < a < 1`, then `r < p.radius`. -/
theorem lt_radius_of_isBigO (h₀ : r ≠ 0) {a : ℝ} (ha : a ∈ Ioo (-1 : ℝ) 1)
(hp : (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] (a ^ ·)) : ↑r < p.radius := by
-- Porting note: moved out of `rcases`
have := ((TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 2 5)
rcases this.mp ⟨a, ha, hp⟩ with ⟨a, ha, C, hC, hp⟩
rw [← pos_iff_ne_zero, ← NNReal.coe_pos] at h₀
lift a to ℝ≥0 using ha.1.le
have : (r : ℝ) < r / a := by
simpa only [div_one] using (div_lt_div_left h₀ zero_lt_one ha.1).2 ha.2
norm_cast at this
rw [← ENNReal.coe_lt_coe] at this
refine' this.trans_le (p.le_radius_of_bound C fun n => _)
rw [NNReal.coe_div, div_pow, ← mul_div_assoc, div_le_iff (pow_pos ha.1 n)]
exact (le_abs_self _).trans (hp n)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.lt_radius_of_is_O FormalMultilinearSeries.lt_radius_of_isBigO
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem norm_mul_pow_le_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C :=
let ⟨_, ha, C, hC, h⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius h
⟨C, hC, fun n => (h n).trans <| mul_le_of_le_one_right hC.lt.le (pow_le_one _ ha.1.le ha.2.le)⟩
#align formal_multilinear_series.norm_mul_pow_le_of_lt_radius FormalMultilinearSeries.norm_mul_pow_le_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem norm_le_div_pow_of_pos_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h0 : 0 < r) (h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖ ≤ C / (r : ℝ) ^ n :=
let ⟨C, hC, hp⟩ := p.norm_mul_pow_le_of_lt_radius h
⟨C, hC, fun n => Iff.mpr (le_div_iff (pow_pos h0 _)) (hp n)⟩
#align formal_multilinear_series.norm_le_div_pow_of_pos_of_lt_radius FormalMultilinearSeries.norm_le_div_pow_of_pos_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem nnnorm_mul_pow_le_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖₊ * r ^ n ≤ C :=
let ⟨C, hC, hp⟩ := p.norm_mul_pow_le_of_lt_radius h
⟨⟨C, hC.lt.le⟩, hC, mod_cast hp⟩
#align formal_multilinear_series.nnnorm_mul_pow_le_of_lt_radius FormalMultilinearSeries.nnnorm_mul_pow_le_of_lt_radius
theorem le_radius_of_tendsto (p : FormalMultilinearSeries 𝕜 E F) {l : ℝ}
(h : Tendsto (fun n => ‖p n‖ * (r : ℝ) ^ n) atTop (𝓝 l)) : ↑r ≤ p.radius :=
p.le_radius_of_isBigO (h.isBigO_one _)
#align formal_multilinear_series.le_radius_of_tendsto FormalMultilinearSeries.le_radius_of_tendsto
theorem le_radius_of_summable_norm (p : FormalMultilinearSeries 𝕜 E F)
(hs : Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_tendsto hs.tendsto_atTop_zero
#align formal_multilinear_series.le_radius_of_summable_norm FormalMultilinearSeries.le_radius_of_summable_norm
theorem not_summable_norm_of_radius_lt_nnnorm (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(h : p.radius < ‖x‖₊) : ¬Summable fun n => ‖p n‖ * ‖x‖ ^ n :=
fun hs => not_le_of_lt h (p.le_radius_of_summable_norm hs)
#align formal_multilinear_series.not_summable_norm_of_radius_lt_nnnorm FormalMultilinearSeries.not_summable_norm_of_radius_lt_nnnorm
theorem summable_norm_mul_pow (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0} (h : ↑r < p.radius) :
Summable fun n : ℕ => ‖p n‖ * (r : ℝ) ^ n := by
obtain ⟨a, ha : a ∈ Ioo (0 : ℝ) 1, C, - : 0 < C, hp⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius h
exact .of_nonneg_of_le (fun n => mul_nonneg (norm_nonneg _) (pow_nonneg r.coe_nonneg _))
hp ((summable_geometric_of_lt_1 ha.1.le ha.2).mul_left _)
#align formal_multilinear_series.summable_norm_mul_pow FormalMultilinearSeries.summable_norm_mul_pow
theorem summable_norm_apply (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : Summable fun n : ℕ => ‖p n fun _ => x‖ := by
rw [mem_emetric_ball_zero_iff] at hx
refine' .of_nonneg_of_le
(fun _ => norm_nonneg _) (fun n => ((p n).le_op_norm _).trans_eq _) (p.summable_norm_mul_pow hx)
simp
#align formal_multilinear_series.summable_norm_apply FormalMultilinearSeries.summable_norm_apply
theorem summable_nnnorm_mul_pow (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0} (h : ↑r < p.radius) :
Summable fun n : ℕ => ‖p n‖₊ * r ^ n := by
rw [← NNReal.summable_coe]
push_cast
exact p.summable_norm_mul_pow h
#align formal_multilinear_series.summable_nnnorm_mul_pow FormalMultilinearSeries.summable_nnnorm_mul_pow
protected theorem summable [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : Summable fun n : ℕ => p n fun _ => x :=
(p.summable_norm_apply hx).of_norm
#align formal_multilinear_series.summable FormalMultilinearSeries.summable
theorem radius_eq_top_of_summable_norm (p : FormalMultilinearSeries 𝕜 E F)
(hs : ∀ r : ℝ≥0, Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : p.radius = ∞ :=
ENNReal.eq_top_of_forall_nnreal_le fun r => p.le_radius_of_summable_norm (hs r)
#align formal_multilinear_series.radius_eq_top_of_summable_norm FormalMultilinearSeries.radius_eq_top_of_summable_norm
theorem radius_eq_top_iff_summable_norm (p : FormalMultilinearSeries 𝕜 E F) :
p.radius = ∞ ↔ ∀ r : ℝ≥0, Summable fun n => ‖p n‖ * (r : ℝ) ^ n := by
constructor
· intro h r
obtain ⟨a, ha : a ∈ Ioo (0 : ℝ) 1, C, - : 0 < C, hp⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius
(show (r : ℝ≥0∞) < p.radius from h.symm ▸ ENNReal.coe_lt_top)
refine' .of_norm_bounded
(fun n => (C : ℝ) * a ^ n) ((summable_geometric_of_lt_1 ha.1.le ha.2).mul_left _) fun n => _
specialize hp n
rwa [Real.norm_of_nonneg (mul_nonneg (norm_nonneg _) (pow_nonneg r.coe_nonneg n))]
· exact p.radius_eq_top_of_summable_norm
#align formal_multilinear_series.radius_eq_top_iff_summable_norm FormalMultilinearSeries.radius_eq_top_iff_summable_norm
/-- If the radius of `p` is positive, then `‖pₙ‖` grows at most geometrically. -/
theorem le_mul_pow_of_radius_pos (p : FormalMultilinearSeries 𝕜 E F) (h : 0 < p.radius) :
∃ (C r : _) (hC : 0 < C) (_ : 0 < r), ∀ n, ‖p n‖ ≤ C * r ^ n := by
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 h with ⟨r, r0, rlt⟩
have rpos : 0 < (r : ℝ) := by simp [ENNReal.coe_pos.1 r0]
rcases norm_le_div_pow_of_pos_of_lt_radius p rpos rlt with ⟨C, Cpos, hCp⟩
refine' ⟨C, r⁻¹, Cpos, by simp only [inv_pos, rpos], fun n => _⟩
-- Porting note: was `convert`
rw [inv_pow, ← div_eq_mul_inv]
exact hCp n
#align formal_multilinear_series.le_mul_pow_of_radius_pos FormalMultilinearSeries.le_mul_pow_of_radius_pos
/-- The radius of the sum of two formal series is at least the minimum of their two radii. -/
theorem min_radius_le_radius_add (p q : FormalMultilinearSeries 𝕜 E F) :
min p.radius q.radius ≤ (p + q).radius := by
refine' ENNReal.le_of_forall_nnreal_lt fun r hr => _
rw [lt_min_iff] at hr
have := ((p.isLittleO_one_of_lt_radius hr.1).add (q.isLittleO_one_of_lt_radius hr.2)).isBigO
refine' (p + q).le_radius_of_isBigO ((isBigO_of_le _ fun n => _).trans this)
rw [← add_mul, norm_mul, norm_mul, norm_norm]
exact mul_le_mul_of_nonneg_right ((norm_add_le _ _).trans (le_abs_self _)) (norm_nonneg _)
#align formal_multilinear_series.min_radius_le_radius_add FormalMultilinearSeries.min_radius_le_radius_add
@[simp]
theorem radius_neg (p : FormalMultilinearSeries 𝕜 E F) : (-p).radius = p.radius := by
simp only [radius, neg_apply, norm_neg]
#align formal_multilinear_series.radius_neg FormalMultilinearSeries.radius_neg
protected theorem hasSum [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : HasSum (fun n : ℕ => p n fun _ => x) (p.sum x) :=
(p.summable hx).hasSum
#align formal_multilinear_series.has_sum FormalMultilinearSeries.hasSum
theorem radius_le_radius_continuousLinearMap_comp (p : FormalMultilinearSeries 𝕜 E F)
(f : F →L[𝕜] G) : p.radius ≤ (f.compFormalMultilinearSeries p).radius := by
refine' ENNReal.le_of_forall_nnreal_lt fun r hr => _
apply le_radius_of_isBigO
apply (IsBigO.trans_isLittleO _ (p.isLittleO_one_of_lt_radius hr)).isBigO
refine' IsBigO.mul (@IsBigOWith.isBigO _ _ _ _ _ ‖f‖ _ _ _ _) (isBigO_refl _ _)
refine IsBigOWith.of_bound (eventually_of_forall fun n => ?_)
simpa only [norm_norm] using f.norm_compContinuousMultilinearMap_le (p n)
#align formal_multilinear_series.radius_le_radius_continuous_linear_map_comp FormalMultilinearSeries.radius_le_radius_continuousLinearMap_comp
end FormalMultilinearSeries
/-! ### Expanding a function as a power series -/
section
variable {f g : E → F} {p pf pg : FormalMultilinearSeries 𝕜 E F} {x : E} {r r' : ℝ≥0∞}
/-- Given a function `f : E → F` and a formal multilinear series `p`, we say that `f` has `p` as
a power series on the ball of radius `r > 0` around `x` if `f (x + y) = ∑' pₙ yⁿ` for all `‖y‖ < r`.
-/
structure HasFPowerSeriesOnBall (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (x : E) (r : ℝ≥0∞) :
Prop where
r_le : r ≤ p.radius
r_pos : 0 < r
hasSum :
∀ {y}, y ∈ EMetric.ball (0 : E) r → HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y))
#align has_fpower_series_on_ball HasFPowerSeriesOnBall
/-- Given a function `f : E → F` and a formal multilinear series `p`, we say that `f` has `p` as
a power series around `x` if `f (x + y) = ∑' pₙ yⁿ` for all `y` in a neighborhood of `0`. -/
def HasFPowerSeriesAt (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (x : E) :=
∃ r, HasFPowerSeriesOnBall f p x r
#align has_fpower_series_at HasFPowerSeriesAt
variable (𝕜)
/-- Given a function `f : E → F`, we say that `f` is analytic at `x` if it admits a convergent power
series expansion around `x`. -/
def AnalyticAt (f : E → F) (x : E) :=
∃ p : FormalMultilinearSeries 𝕜 E F, HasFPowerSeriesAt f p x
#align analytic_at AnalyticAt
/-- Given a function `f : E → F`, we say that `f` is analytic on a set `s` if it is analytic around
every point of `s`. -/
def AnalyticOn (f : E → F) (s : Set E) :=
∀ x, x ∈ s → AnalyticAt 𝕜 f x
#align analytic_on AnalyticOn
variable {𝕜}
theorem HasFPowerSeriesOnBall.hasFPowerSeriesAt (hf : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesAt f p x :=
⟨r, hf⟩
#align has_fpower_series_on_ball.has_fpower_series_at HasFPowerSeriesOnBall.hasFPowerSeriesAt
theorem HasFPowerSeriesAt.analyticAt (hf : HasFPowerSeriesAt f p x) : AnalyticAt 𝕜 f x :=
⟨p, hf⟩
#align has_fpower_series_at.analytic_at HasFPowerSeriesAt.analyticAt
theorem HasFPowerSeriesOnBall.analyticAt (hf : HasFPowerSeriesOnBall f p x r) : AnalyticAt 𝕜 f x :=
hf.hasFPowerSeriesAt.analyticAt
#align has_fpower_series_on_ball.analytic_at HasFPowerSeriesOnBall.analyticAt
theorem HasFPowerSeriesOnBall.congr (hf : HasFPowerSeriesOnBall f p x r)
(hg : EqOn f g (EMetric.ball x r)) : HasFPowerSeriesOnBall g p x r :=
{ r_le := hf.r_le
r_pos := hf.r_pos
hasSum := fun {y} hy => by
convert hf.hasSum hy using 1
apply hg.symm
simpa [edist_eq_coe_nnnorm_sub] using hy }
#align has_fpower_series_on_ball.congr HasFPowerSeriesOnBall.congr
/-- If a function `f` has a power series `p` around `x`, then the function `z ↦ f (z - y)` has the
same power series around `x + y`. -/
theorem HasFPowerSeriesOnBall.comp_sub (hf : HasFPowerSeriesOnBall f p x r) (y : E) :
HasFPowerSeriesOnBall (fun z => f (z - y)) p (x + y) r :=
{ r_le := hf.r_le
r_pos := hf.r_pos
hasSum := fun {z} hz => by
convert hf.hasSum hz using 2
abel }
#align has_fpower_series_on_ball.comp_sub HasFPowerSeriesOnBall.comp_sub
theorem HasFPowerSeriesOnBall.hasSum_sub (hf : HasFPowerSeriesOnBall f p x r) {y : E}
(hy : y ∈ EMetric.ball x r) : HasSum (fun n : ℕ => p n fun _ => y - x) (f y) := by
have : y - x ∈ EMetric.ball (0 : E) r := by simpa [edist_eq_coe_nnnorm_sub] using hy
simpa only [add_sub_cancel'_right] using hf.hasSum this
#align has_fpower_series_on_ball.has_sum_sub HasFPowerSeriesOnBall.hasSum_sub
theorem HasFPowerSeriesOnBall.radius_pos (hf : HasFPowerSeriesOnBall f p x r) : 0 < p.radius :=
lt_of_lt_of_le hf.r_pos hf.r_le
#align has_fpower_series_on_ball.radius_pos HasFPowerSeriesOnBall.radius_pos
theorem HasFPowerSeriesAt.radius_pos (hf : HasFPowerSeriesAt f p x) : 0 < p.radius :=
let ⟨_, hr⟩ := hf
hr.radius_pos
#align has_fpower_series_at.radius_pos HasFPowerSeriesAt.radius_pos
theorem HasFPowerSeriesOnBall.mono (hf : HasFPowerSeriesOnBall f p x r) (r'_pos : 0 < r')
(hr : r' ≤ r) : HasFPowerSeriesOnBall f p x r' :=
⟨le_trans hr hf.1, r'_pos, fun hy => hf.hasSum (EMetric.ball_subset_ball hr hy)⟩
#align has_fpower_series_on_ball.mono HasFPowerSeriesOnBall.mono
theorem HasFPowerSeriesAt.congr (hf : HasFPowerSeriesAt f p x) (hg : f =ᶠ[𝓝 x] g) :
HasFPowerSeriesAt g p x := by
rcases hf with ⟨r₁, h₁⟩
rcases EMetric.mem_nhds_iff.mp hg with ⟨r₂, h₂pos, h₂⟩
exact ⟨min r₁ r₂,
(h₁.mono (lt_min h₁.r_pos h₂pos) inf_le_left).congr
fun y hy => h₂ (EMetric.ball_subset_ball inf_le_right hy)⟩
#align has_fpower_series_at.congr HasFPowerSeriesAt.congr
protected theorem HasFPowerSeriesAt.eventually (hf : HasFPowerSeriesAt f p x) :
∀ᶠ r : ℝ≥0∞ in 𝓝[>] 0, HasFPowerSeriesOnBall f p x r :=
let ⟨_, hr⟩ := hf
mem_of_superset (Ioo_mem_nhdsWithin_Ioi (left_mem_Ico.2 hr.r_pos)) fun _ hr' =>
hr.mono hr'.1 hr'.2.le
#align has_fpower_series_at.eventually HasFPowerSeriesAt.eventually
theorem HasFPowerSeriesOnBall.eventually_hasSum (hf : HasFPowerSeriesOnBall f p x r) :
∀ᶠ y in 𝓝 0, HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y)) := by
filter_upwards [EMetric.ball_mem_nhds (0 : E) hf.r_pos] using fun _ => hf.hasSum
#align has_fpower_series_on_ball.eventually_has_sum HasFPowerSeriesOnBall.eventually_hasSum
theorem HasFPowerSeriesAt.eventually_hasSum (hf : HasFPowerSeriesAt f p x) :
∀ᶠ y in 𝓝 0, HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y)) :=
let ⟨_, hr⟩ := hf
hr.eventually_hasSum
#align has_fpower_series_at.eventually_has_sum HasFPowerSeriesAt.eventually_hasSum
theorem HasFPowerSeriesOnBall.eventually_hasSum_sub (hf : HasFPowerSeriesOnBall f p x r) :
∀ᶠ y in 𝓝 x, HasSum (fun n : ℕ => p n fun _ : Fin n => y - x) (f y) := by
filter_upwards [EMetric.ball_mem_nhds x hf.r_pos] with y using hf.hasSum_sub
#align has_fpower_series_on_ball.eventually_has_sum_sub HasFPowerSeriesOnBall.eventually_hasSum_sub
theorem HasFPowerSeriesAt.eventually_hasSum_sub (hf : HasFPowerSeriesAt f p x) :
∀ᶠ y in 𝓝 x, HasSum (fun n : ℕ => p n fun _ : Fin n => y - x) (f y) :=
let ⟨_, hr⟩ := hf
hr.eventually_hasSum_sub
#align has_fpower_series_at.eventually_has_sum_sub HasFPowerSeriesAt.eventually_hasSum_sub
theorem HasFPowerSeriesOnBall.eventually_eq_zero
(hf : HasFPowerSeriesOnBall f (0 : FormalMultilinearSeries 𝕜 E F) x r) :
∀ᶠ z in 𝓝 x, f z = 0 := by
filter_upwards [hf.eventually_hasSum_sub] with z hz using hz.unique hasSum_zero
#align has_fpower_series_on_ball.eventually_eq_zero HasFPowerSeriesOnBall.eventually_eq_zero
theorem HasFPowerSeriesAt.eventually_eq_zero
(hf : HasFPowerSeriesAt f (0 : FormalMultilinearSeries 𝕜 E F) x) : ∀ᶠ z in 𝓝 x, f z = 0 :=
let ⟨_, hr⟩ := hf
hr.eventually_eq_zero
#align has_fpower_series_at.eventually_eq_zero HasFPowerSeriesAt.eventually_eq_zero
theorem hasFPowerSeriesOnBall_const {c : F} {e : E} :
HasFPowerSeriesOnBall (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e ⊤ := by
refine' ⟨by simp, WithTop.zero_lt_top, fun _ => hasSum_single 0 fun n hn => _⟩
simp [constFormalMultilinearSeries_apply hn]
#align has_fpower_series_on_ball_const hasFPowerSeriesOnBall_const
theorem hasFPowerSeriesAt_const {c : F} {e : E} :
HasFPowerSeriesAt (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e :=
⟨⊤, hasFPowerSeriesOnBall_const⟩
#align has_fpower_series_at_const hasFPowerSeriesAt_const
theorem analyticAt_const {v : F} : AnalyticAt 𝕜 (fun _ => v) x :=
⟨constFormalMultilinearSeries 𝕜 E v, hasFPowerSeriesAt_const⟩
#align analytic_at_const analyticAt_const
theorem analyticOn_const {v : F} {s : Set E} : AnalyticOn 𝕜 (fun _ => v) s :=
fun _ _ => analyticAt_const
#align analytic_on_const analyticOn_const
theorem HasFPowerSeriesOnBall.add (hf : HasFPowerSeriesOnBall f pf x r)
(hg : HasFPowerSeriesOnBall g pg x r) : HasFPowerSeriesOnBall (f + g) (pf + pg) x r :=
{ r_le := le_trans (le_min_iff.2 ⟨hf.r_le, hg.r_le⟩) (pf.min_radius_le_radius_add pg)
r_pos := hf.r_pos
hasSum := fun hy => (hf.hasSum hy).add (hg.hasSum hy) }
#align has_fpower_series_on_ball.add HasFPowerSeriesOnBall.add
theorem HasFPowerSeriesAt.add (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x) :
HasFPowerSeriesAt (f + g) (pf + pg) x := by
rcases (hf.eventually.and hg.eventually).exists with ⟨r, hr⟩
exact ⟨r, hr.1.add hr.2⟩
#align has_fpower_series_at.add HasFPowerSeriesAt.add
theorem AnalyticAt.congr (hf : AnalyticAt 𝕜 f x) (hg : f =ᶠ[𝓝 x] g) : AnalyticAt 𝕜 g x :=
let ⟨_, hpf⟩ := hf
(hpf.congr hg).analyticAt
theorem analyticAt_congr (h : f =ᶠ[𝓝 x] g) : AnalyticAt 𝕜 f x ↔ AnalyticAt 𝕜 g x :=
⟨fun hf ↦ hf.congr h, fun hg ↦ hg.congr h.symm⟩
theorem AnalyticAt.add (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) : AnalyticAt 𝕜 (f + g) x :=
let ⟨_, hpf⟩ := hf
let ⟨_, hqf⟩ := hg
(hpf.add hqf).analyticAt
#align analytic_at.add AnalyticAt.add
theorem HasFPowerSeriesOnBall.neg (hf : HasFPowerSeriesOnBall f pf x r) :
HasFPowerSeriesOnBall (-f) (-pf) x r :=
{ r_le := by
rw [pf.radius_neg]
exact hf.r_le
r_pos := hf.r_pos
hasSum := fun hy => (hf.hasSum hy).neg }
#align has_fpower_series_on_ball.neg HasFPowerSeriesOnBall.neg
theorem HasFPowerSeriesAt.neg (hf : HasFPowerSeriesAt f pf x) : HasFPowerSeriesAt (-f) (-pf) x :=
let ⟨_, hrf⟩ := hf
hrf.neg.hasFPowerSeriesAt
#align has_fpower_series_at.neg HasFPowerSeriesAt.neg
theorem AnalyticAt.neg (hf : AnalyticAt 𝕜 f x) : AnalyticAt 𝕜 (-f) x :=
let ⟨_, hpf⟩ := hf
hpf.neg.analyticAt
#align analytic_at.neg AnalyticAt.neg
theorem HasFPowerSeriesOnBall.sub (hf : HasFPowerSeriesOnBall f pf x r)
(hg : HasFPowerSeriesOnBall g pg x r) : HasFPowerSeriesOnBall (f - g) (pf - pg) x r := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align has_fpower_series_on_ball.sub HasFPowerSeriesOnBall.sub
theorem HasFPowerSeriesAt.sub (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x) :
HasFPowerSeriesAt (f - g) (pf - pg) x := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align has_fpower_series_at.sub HasFPowerSeriesAt.sub
theorem AnalyticAt.sub (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) :
AnalyticAt 𝕜 (f - g) x := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align analytic_at.sub AnalyticAt.sub
theorem AnalyticOn.mono {s t : Set E} (hf : AnalyticOn 𝕜 f t) (hst : s ⊆ t) : AnalyticOn 𝕜 f s :=
fun z hz => hf z (hst hz)
#align analytic_on.mono AnalyticOn.mono
theorem AnalyticOn.congr' {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : f =ᶠ[𝓝ˢ s] g) :
AnalyticOn 𝕜 g s :=
fun z hz => (hf z hz).congr (mem_nhdsSet_iff_forall.mp hg z hz)
theorem analyticOn_congr' {s : Set E} (h : f =ᶠ[𝓝ˢ s] g) : AnalyticOn 𝕜 f s ↔ AnalyticOn 𝕜 g s :=
⟨fun hf => hf.congr' h, fun hg => hg.congr' h.symm⟩
theorem AnalyticOn.congr {s : Set E} (hs : IsOpen s) (hf : AnalyticOn 𝕜 f s) (hg : s.EqOn f g) :
AnalyticOn 𝕜 g s :=
hf.congr' $ mem_nhdsSet_iff_forall.mpr
(fun _ hz => eventuallyEq_iff_exists_mem.mpr ⟨s, hs.mem_nhds hz, hg⟩)
theorem analyticOn_congr {s : Set E} (hs : IsOpen s) (h : s.EqOn f g) : AnalyticOn 𝕜 f s ↔
AnalyticOn 𝕜 g s := ⟨fun hf => hf.congr hs h, fun hg => hg.congr hs h.symm⟩
theorem AnalyticOn.add {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
AnalyticOn 𝕜 (f + g) s :=
fun z hz => (hf z hz).add (hg z hz)
#align analytic_on.add AnalyticOn.add
theorem AnalyticOn.sub {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
AnalyticOn 𝕜 (f - g) s :=
fun z hz => (hf z hz).sub (hg z hz)
#align analytic_on.sub AnalyticOn.sub
theorem HasFPowerSeriesOnBall.coeff_zero (hf : HasFPowerSeriesOnBall f pf x r) (v : Fin 0 → E) :
pf 0 v = f x := by
have v_eq : v = fun i => 0 := Subsingleton.elim _ _
have zero_mem : (0 : E) ∈ EMetric.ball (0 : E) r := by simp [hf.r_pos]
have : ∀ i, i ≠ 0 → (pf i fun j => 0) = 0 := by
intro i hi
have : 0 < i := pos_iff_ne_zero.2 hi
exact ContinuousMultilinearMap.map_coord_zero _ (⟨0, this⟩ : Fin i) rfl
have A := (hf.hasSum zero_mem).unique (hasSum_single _ this)
simpa [v_eq] using A.symm
#align has_fpower_series_on_ball.coeff_zero HasFPowerSeriesOnBall.coeff_zero
theorem HasFPowerSeriesAt.coeff_zero (hf : HasFPowerSeriesAt f pf x) (v : Fin 0 → E) :
pf 0 v = f x :=
let ⟨_, hrf⟩ := hf
hrf.coeff_zero v
#align has_fpower_series_at.coeff_zero HasFPowerSeriesAt.coeff_zero
/-- If a function `f` has a power series `p` on a ball and `g` is linear, then `g ∘ f` has the
power series `g ∘ p` on the same ball. -/
theorem ContinuousLinearMap.comp_hasFPowerSeriesOnBall (g : F →L[𝕜] G)
(h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (g ∘ f) (g.compFormalMultilinearSeries p) x r :=
{ r_le := h.r_le.trans (p.radius_le_radius_continuousLinearMap_comp _)
r_pos := h.r_pos
hasSum := fun hy => by
simpa only [ContinuousLinearMap.compFormalMultilinearSeries_apply,
ContinuousLinearMap.compContinuousMultilinearMap_coe, Function.comp_apply] using
g.hasSum (h.hasSum hy) }
#align continuous_linear_map.comp_has_fpower_series_on_ball ContinuousLinearMap.comp_hasFPowerSeriesOnBall
/-- If a function `f` is analytic on a set `s` and `g` is linear, then `g ∘ f` is analytic
on `s`. -/
theorem ContinuousLinearMap.comp_analyticOn {s : Set E} (g : F →L[𝕜] G) (h : AnalyticOn 𝕜 f s) :
AnalyticOn 𝕜 (g ∘ f) s := by
rintro x hx
rcases h x hx with ⟨p, r, hp⟩
exact ⟨g.compFormalMultilinearSeries p, r, g.comp_hasFPowerSeriesOnBall hp⟩
#align continuous_linear_map.comp_analytic_on ContinuousLinearMap.comp_analyticOn
/-- If a function admits a power series expansion, then it is exponentially close to the partial
sums of this power series on strict subdisks of the disk of convergence.
This version provides an upper estimate that decreases both in `‖y‖` and `n`. See also
`HasFPowerSeriesOnBall.uniform_geometric_approx` for a weaker version. -/
theorem HasFPowerSeriesOnBall.uniform_geometric_approx' {r' : ℝ≥0}
(hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n := by
obtain ⟨a, ha, C, hC, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n, ‖p n‖ * (r' : ℝ) ^ n ≤ C * a ^ n :=
p.norm_mul_pow_le_mul_pow_of_lt_radius (h.trans_le hf.r_le)
refine' ⟨a, ha, C / (1 - a), div_pos hC (sub_pos.2 ha.2), fun y hy n => _⟩
have yr' : ‖y‖ < r' := by
rw [ball_zero_eq] at hy
exact hy
have hr'0 : 0 < (r' : ℝ) := (norm_nonneg _).trans_lt yr'
have : y ∈ EMetric.ball (0 : E) r := by
refine' mem_emetric_ball_zero_iff.2 (lt_trans _ h)
exact mod_cast yr'
rw [norm_sub_rev, ← mul_div_right_comm]
have ya : a * (‖y‖ / ↑r') ≤ a :=
mul_le_of_le_one_right ha.1.le (div_le_one_of_le yr'.le r'.coe_nonneg)
suffices ‖p.partialSum n y - f (x + y)‖ ≤ C * (a * (‖y‖ / r')) ^ n / (1 - a * (‖y‖ / r')) by
refine' this.trans _
have : 0 < a := ha.1
gcongr
apply_rules [sub_pos.2, ha.2]
apply norm_sub_le_of_geometric_bound_of_hasSum (ya.trans_lt ha.2) _ (hf.hasSum this)
intro n
calc
‖(p n) fun _ : Fin n => y‖
_ ≤ ‖p n‖ * ∏ _i : Fin n, ‖y‖ := ContinuousMultilinearMap.le_op_norm _ _
_ = ‖p n‖ * (r' : ℝ) ^ n * (‖y‖ / r') ^ n := by field_simp [mul_right_comm]
_ ≤ C * a ^ n * (‖y‖ / r') ^ n := by gcongr ?_ * _; apply hp
_ ≤ C * (a * (‖y‖ / r')) ^ n := by rw [mul_pow, mul_assoc]
#align has_fpower_series_on_ball.uniform_geometric_approx' HasFPowerSeriesOnBall.uniform_geometric_approx'
/-- If a function admits a power series expansion, then it is exponentially close to the partial
sums of this power series on strict subdisks of the disk of convergence. -/
theorem HasFPowerSeriesOnBall.uniform_geometric_approx {r' : ℝ≥0}
(hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
∃ a ∈ Ioo (0 : ℝ) 1,
∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n, ‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n := by
obtain ⟨a, ha, C, hC, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n :=
hf.uniform_geometric_approx' h
refine' ⟨a, ha, C, hC, fun y hy n => (hp y hy n).trans _⟩
have yr' : ‖y‖ < r' := by rwa [ball_zero_eq] at hy
gcongr
exacts [mul_nonneg ha.1.le (div_nonneg (norm_nonneg y) r'.coe_nonneg),
mul_le_of_le_one_right ha.1.le (div_le_one_of_le yr'.le r'.coe_nonneg)]
#align has_fpower_series_on_ball.uniform_geometric_approx HasFPowerSeriesOnBall.uniform_geometric_approx
/-- Taylor formula for an analytic function, `IsBigO` version. -/
theorem HasFPowerSeriesAt.isBigO_sub_partialSum_pow (hf : HasFPowerSeriesAt f p x) (n : ℕ) :
(fun y : E => f (x + y) - p.partialSum n y) =O[𝓝 0] fun y => ‖y‖ ^ n := by
rcases hf with ⟨r, hf⟩
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hf.r_pos with ⟨r', r'0, h⟩
obtain ⟨a, -, C, -, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n :=
hf.uniform_geometric_approx' h
refine' isBigO_iff.2 ⟨C * (a / r') ^ n, _⟩
replace r'0 : 0 < (r' : ℝ); · exact mod_cast r'0
filter_upwards [Metric.ball_mem_nhds (0 : E) r'0] with y hy
simpa [mul_pow, mul_div_assoc, mul_assoc, div_mul_eq_mul_div] using hp y hy n
set_option linter.uppercaseLean3 false in
#align has_fpower_series_at.is_O_sub_partial_sum_pow HasFPowerSeriesAt.isBigO_sub_partialSum_pow
/-- If `f` has formal power series `∑ n, pₙ` on a ball of radius `r`, then for `y, z` in any smaller
ball, the norm of the difference `f y - f z - p 1 (fun _ ↦ y - z)` is bounded above by
`C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖`. This lemma formulates this property using `IsBigO` and
`Filter.principal` on `E × E`. -/
theorem HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal
(hf : HasFPowerSeriesOnBall f p x r) (hr : r' < r) :
(fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) =O[𝓟 (EMetric.ball (x, x) r')]
fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖ := by
lift r' to ℝ≥0 using ne_top_of_lt hr
rcases (zero_le r').eq_or_lt with (rfl | hr'0)
· simp only [isBigO_bot, EMetric.ball_zero, principal_empty, ENNReal.coe_zero]
obtain ⟨a, ha, C, hC : 0 < C, hp⟩ :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n : ℕ, ‖p n‖ * (r' : ℝ) ^ n ≤ C * a ^ n
exact p.norm_mul_pow_le_mul_pow_of_lt_radius (hr.trans_le hf.r_le)
simp only [← le_div_iff (pow_pos (NNReal.coe_pos.2 hr'0) _)] at hp
set L : E × E → ℝ := fun y =>
C * (a / r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * (a / (1 - a) ^ 2 + 2 / (1 - a))
have hL : ∀ y ∈ EMetric.ball (x, x) r', ‖f y.1 - f y.2 - p 1 fun _ => y.1 - y.2‖ ≤ L y := by
intro y hy'
have hy : y ∈ EMetric.ball x r ×ˢ EMetric.ball x r := by
rw [EMetric.ball_prod_same]
exact EMetric.ball_subset_ball hr.le hy'
set A : ℕ → F := fun n => (p n fun _ => y.1 - x) - p n fun _ => y.2 - x
have hA : HasSum (fun n => A (n + 2)) (f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) := by
convert (hasSum_nat_add_iff' 2).2 ((hf.hasSum_sub hy.1).sub (hf.hasSum_sub hy.2)) using 1
rw [Finset.sum_range_succ, Finset.sum_range_one, hf.coeff_zero, hf.coeff_zero, sub_self,
zero_add, ← Subsingleton.pi_single_eq (0 : Fin 1) (y.1 - x), Pi.single,
← Subsingleton.pi_single_eq (0 : Fin 1) (y.2 - x), Pi.single, ← (p 1).map_sub, ← Pi.single,
Subsingleton.pi_single_eq, sub_sub_sub_cancel_right]
rw [EMetric.mem_ball, edist_eq_coe_nnnorm_sub, ENNReal.coe_lt_coe] at hy'
set B : ℕ → ℝ := fun n => C * (a / r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * ((n + 2) * a ^ n)
have hAB : ∀ n, ‖A (n + 2)‖ ≤ B n := fun n =>
calc
‖A (n + 2)‖ ≤ ‖p (n + 2)‖ * ↑(n + 2) * ‖y - (x, x)‖ ^ (n + 1) * ‖y.1 - y.2‖ := by
-- porting note: `pi_norm_const` was `pi_norm_const (_ : E)`
simpa only [Fintype.card_fin, pi_norm_const, Prod.norm_def, Pi.sub_def,
Prod.fst_sub, Prod.snd_sub, sub_sub_sub_cancel_right] using
(p <| n + 2).norm_image_sub_le (fun _ => y.1 - x) fun _ => y.2 - x
_ = ‖p (n + 2)‖ * ‖y - (x, x)‖ ^ n * (↑(n + 2) * ‖y - (x, x)‖ * ‖y.1 - y.2‖) := by
rw [pow_succ ‖y - (x, x)‖]
ring
-- porting note: the two `↑` in `↑r'` are new, without them, Lean fails to synthesize
-- instances `HDiv ℝ ℝ≥0 ?m` or `HMul ℝ ℝ≥0 ?m`
_ ≤ C * a ^ (n + 2) / ↑r' ^ (n + 2)
* ↑r' ^ n * (↑(n + 2) * ‖y - (x, x)‖ * ‖y.1 - y.2‖) := by
have : 0 < a := ha.1
gcongr
· apply hp
· apply hy'.le
_ = B n := by
-- porting note: in the original, `B` was in the `field_simp`, but now Lean does not
-- accept it. The current proof works in Lean 4, but does not in Lean 3.
field_simp [pow_succ]
simp only [mul_assoc, mul_comm, mul_left_comm]
have hBL : HasSum B (L y) := by
apply HasSum.mul_left
simp only [add_mul]
have : ‖a‖ < 1 := by simp only [Real.norm_eq_abs, abs_of_pos ha.1, ha.2]
rw [div_eq_mul_inv, div_eq_mul_inv]
exact (hasSum_coe_mul_geometric_of_norm_lt_1 this).add -- porting note: was `convert`!
((hasSum_geometric_of_norm_lt_1 this).mul_left 2)
exact hA.norm_le_of_bounded hBL hAB
suffices L =O[𝓟 (EMetric.ball (x, x) r')] fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖ by
refine' (IsBigO.of_bound 1 (eventually_principal.2 fun y hy => _)).trans this
rw [one_mul]
exact (hL y hy).trans (le_abs_self _)
simp_rw [mul_right_comm _ (_ * _)] -- porting note: there was an `L` inside the `simp_rw`.
exact (isBigO_refl _ _).const_mul_left _
set_option linter.uppercaseLean3 false in
#align has_fpower_series_on_ball.is_O_image_sub_image_sub_deriv_principal HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal
/-- If `f` has formal power series `∑ n, pₙ` on a ball of radius `r`, then for `y, z` in any smaller
ball, the norm of the difference `f y - f z - p 1 (fun _ ↦ y - z)` is bounded above by
`C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖`. -/
theorem HasFPowerSeriesOnBall.image_sub_sub_deriv_le (hf : HasFPowerSeriesOnBall f p x r)
(hr : r' < r) :
∃ C, ∀ᵉ (y ∈ EMetric.ball x r') (z ∈ EMetric.ball x r'),
‖f y - f z - p 1 fun _ => y - z‖ ≤ C * max ‖y - x‖ ‖z - x‖ * ‖y - z‖ := by
simpa only [isBigO_principal, mul_assoc, norm_mul, norm_norm, Prod.forall, EMetric.mem_ball,
Prod.edist_eq, max_lt_iff, and_imp, @forall_swap (_ < _) E] using
hf.isBigO_image_sub_image_sub_deriv_principal hr
#align has_fpower_series_on_ball.image_sub_sub_deriv_le HasFPowerSeriesOnBall.image_sub_sub_deriv_le
/-- If `f` has formal power series `∑ n, pₙ` at `x`, then
`f y - f z - p 1 (fun _ ↦ y - z) = O(‖(y, z) - (x, x)‖ * ‖y - z‖)` as `(y, z) → (x, x)`.
In particular, `f` is strictly differentiable at `x`. -/
theorem HasFPowerSeriesAt.isBigO_image_sub_norm_mul_norm_sub (hf : HasFPowerSeriesAt f p x) :
(fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) =O[𝓝 (x, x)] fun y =>
‖y - (x, x)‖ * ‖y.1 - y.2‖ := by
rcases hf with ⟨r, hf⟩
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hf.r_pos with ⟨r', r'0, h⟩
refine' (hf.isBigO_image_sub_image_sub_deriv_principal h).mono _
exact le_principal_iff.2 (EMetric.ball_mem_nhds _ r'0)
set_option linter.uppercaseLean3 false in
#align has_fpower_series_at.is_O_image_sub_norm_mul_norm_sub HasFPowerSeriesAt.isBigO_image_sub_norm_mul_norm_sub
/-- If a function admits a power series expansion at `x`, then it is the uniform limit of the
partial sums of this power series on strict subdisks of the disk of convergence, i.e., `f (x + y)`
is the uniform limit of `p.partialSum n y` there. -/
theorem HasFPowerSeriesOnBall.tendstoUniformlyOn {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r)
(h : (r' : ℝ≥0∞) < r) :
TendstoUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
(Metric.ball (0 : E) r') := by
obtain ⟨a, ha, C, -, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n
exact hf.uniform_geometric_approx h
refine' Metric.tendstoUniformlyOn_iff.2 fun ε εpos => _
have L : Tendsto (fun n => (C : ℝ) * a ^ n) atTop (𝓝 ((C : ℝ) * 0)) :=
tendsto_const_nhds.mul (tendsto_pow_atTop_nhds_0_of_lt_1 ha.1.le ha.2)
rw [mul_zero] at L
refine' (L.eventually (gt_mem_nhds εpos)).mono fun n hn y hy => _
rw [dist_eq_norm]
exact (hp y hy n).trans_lt hn
#align has_fpower_series_on_ball.tendsto_uniformly_on HasFPowerSeriesOnBall.tendstoUniformlyOn
/-- If a function admits a power series expansion at `x`, then it is the locally uniform limit of
the partial sums of this power series on the disk of convergence, i.e., `f (x + y)`
is the locally uniform limit of `p.partialSum n y` there. -/
theorem HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn (hf : HasFPowerSeriesOnBall f p x r) :
TendstoLocallyUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
(EMetric.ball (0 : E) r) := by
intro u hu x hx
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hx with ⟨r', xr', hr'⟩
have : EMetric.ball (0 : E) r' ∈ 𝓝 x := IsOpen.mem_nhds EMetric.isOpen_ball xr'
refine' ⟨EMetric.ball (0 : E) r', mem_nhdsWithin_of_mem_nhds this, _⟩
simpa [Metric.emetric_ball_nnreal] using hf.tendstoUniformlyOn hr' u hu
#align has_fpower_series_on_ball.tendsto_locally_uniformly_on HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn
/-- If a function admits a power series expansion at `x`, then it is the uniform limit of the
partial sums of this power series on strict subdisks of the disk of convergence, i.e., `f y`
is the uniform limit of `p.partialSum n (y - x)` there. -/
theorem HasFPowerSeriesOnBall.tendstoUniformlyOn' {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r)
(h : (r' : ℝ≥0∞) < r) :
TendstoUniformlyOn (fun n y => p.partialSum n (y - x)) f atTop (Metric.ball (x : E) r') := by
convert (hf.tendstoUniformlyOn h).comp fun y => y - x using 1
· simp [(· ∘ ·)]
· ext z
simp [dist_eq_norm]
#align has_fpower_series_on_ball.tendsto_uniformly_on' HasFPowerSeriesOnBall.tendstoUniformlyOn'
/-- If a function admits a power series expansion at `x`, then it is the locally uniform limit of
the partial sums of this power series on the disk of convergence, i.e., `f y`
is the locally uniform limit of `p.partialSum n (y - x)` there. -/
theorem HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn' (hf : HasFPowerSeriesOnBall f p x r) :
TendstoLocallyUniformlyOn (fun n y => p.partialSum n (y - x)) f atTop
(EMetric.ball (x : E) r) := by
have A : ContinuousOn (fun y : E => y - x) (EMetric.ball (x : E) r) :=
(continuous_id.sub continuous_const).continuousOn
convert hf.tendstoLocallyUniformlyOn.comp (fun y : E => y - x) _ A using 1
· ext z
simp
· intro z
simp [edist_eq_coe_nnnorm, edist_eq_coe_nnnorm_sub]
#align has_fpower_series_on_ball.tendsto_locally_uniformly_on' HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn'
/-- If a function admits a power series expansion on a disk, then it is continuous there. -/
protected theorem HasFPowerSeriesOnBall.continuousOn (hf : HasFPowerSeriesOnBall f p x r) :
ContinuousOn f (EMetric.ball x r) :=
hf.tendstoLocallyUniformlyOn'.continuousOn <|
eventually_of_forall fun n =>
((p.partialSum_continuous n).comp (continuous_id.sub continuous_const)).continuousOn
#align has_fpower_series_on_ball.continuous_on HasFPowerSeriesOnBall.continuousOn
protected theorem HasFPowerSeriesAt.continuousAt (hf : HasFPowerSeriesAt f p x) :
ContinuousAt f x :=
let ⟨_, hr⟩ := hf
hr.continuousOn.continuousAt (EMetric.ball_mem_nhds x hr.r_pos)
#align has_fpower_series_at.continuous_at HasFPowerSeriesAt.continuousAt
protected theorem AnalyticAt.continuousAt (hf : AnalyticAt 𝕜 f x) : ContinuousAt f x :=
let ⟨_, hp⟩ := hf
hp.continuousAt
#align analytic_at.continuous_at AnalyticAt.continuousAt
protected theorem AnalyticOn.continuousOn {s : Set E} (hf : AnalyticOn 𝕜 f s) : ContinuousOn f s :=
fun x hx => (hf x hx).continuousAt.continuousWithinAt
#align analytic_on.continuous_on AnalyticOn.continuousOn
/-- Analytic everywhere implies continuous -/
theorem AnalyticOn.continuous {f : E → F} (fa : AnalyticOn 𝕜 f univ) : Continuous f := by
rw [continuous_iff_continuousOn_univ]; exact fa.continuousOn
/-- In a complete space, the sum of a converging power series `p` admits `p` as a power series.
This is not totally obvious as we need to check the convergence of the series. -/
protected theorem FormalMultilinearSeries.hasFPowerSeriesOnBall [CompleteSpace F]
(p : FormalMultilinearSeries 𝕜 E F) (h : 0 < p.radius) :
HasFPowerSeriesOnBall p.sum p 0 p.radius :=
{ r_le := le_rfl
r_pos := h
hasSum := fun hy => by
rw [zero_add]
exact p.hasSum hy }
#align formal_multilinear_series.has_fpower_series_on_ball FormalMultilinearSeries.hasFPowerSeriesOnBall
theorem HasFPowerSeriesOnBall.sum (h : HasFPowerSeriesOnBall f p x r) {y : E}
(hy : y ∈ EMetric.ball (0 : E) r) : f (x + y) = p.sum y :=
(h.hasSum hy).tsum_eq.symm
#align has_fpower_series_on_ball.sum HasFPowerSeriesOnBall.sum
/-- The sum of a converging power series is continuous in its disk of convergence. -/
protected theorem FormalMultilinearSeries.continuousOn [CompleteSpace F] :
ContinuousOn p.sum (EMetric.ball 0 p.radius) := by
rcases (zero_le p.radius).eq_or_lt with h | h
· simp [← h, continuousOn_empty]
· exact (p.hasFPowerSeriesOnBall h).continuousOn
#align formal_multilinear_series.continuous_on FormalMultilinearSeries.continuousOn
end
/-!
### Uniqueness of power series
If a function `f : E → F` has two representations as power series at a point `x : E`, corresponding
to formal multilinear series `p₁` and `p₂`, then these representations agree term-by-term. That is,
for any `n : ℕ` and `y : E`, `p₁ n (fun i ↦ y) = p₂ n (fun i ↦ y)`. In the one-dimensional case,
when `f : 𝕜 → E`, the continuous multilinear maps `p₁ n` and `p₂ n` are given by
`ContinuousMultilinearMap.mkPiField`, and hence are determined completely by the value of
`p₁ n (fun i ↦ 1)`, so `p₁ = p₂`. Consequently, the radius of convergence for one series can be
transferred to the other.
-/
section Uniqueness
open ContinuousMultilinearMap
theorem Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero {n : ℕ} {p : E[×n]→L[𝕜] F}
(h : (fun y => p fun _ => y) =O[𝓝 0] fun y => ‖y‖ ^ (n + 1)) (y : E) : (p fun _ => y) = 0 := by
obtain ⟨c, c_pos, hc⟩ := h.exists_pos
obtain ⟨t, ht, t_open, z_mem⟩ := eventually_nhds_iff.mp (isBigOWith_iff.mp hc)
obtain ⟨δ, δ_pos, δε⟩ := (Metric.isOpen_iff.mp t_open) 0 z_mem
clear h hc z_mem
cases' n with n
· exact norm_eq_zero.mp (by
-- porting note: the symmetric difference of the `simpa only` sets:
-- added `Nat.zero_eq, zero_add, pow_one`
-- removed `zero_pow', Ne.def, Nat.one_ne_zero, not_false_iff`
simpa only [Nat.zero_eq, fin0_apply_norm, norm_eq_zero, norm_zero, zero_add, pow_one,
mul_zero, norm_le_zero_iff] using ht 0 (δε (Metric.mem_ball_self δ_pos)))
· refine' Or.elim (Classical.em (y = 0))
(fun hy => by simpa only [hy] using p.map_zero) fun hy => _
replace hy := norm_pos_iff.mpr hy
refine' norm_eq_zero.mp (le_antisymm (le_of_forall_pos_le_add fun ε ε_pos => _) (norm_nonneg _))
have h₀ := _root_.mul_pos c_pos (pow_pos hy (n.succ + 1))
obtain ⟨k, k_pos, k_norm⟩ := NormedField.exists_norm_lt 𝕜
(lt_min (mul_pos δ_pos (inv_pos.mpr hy)) (mul_pos ε_pos (inv_pos.mpr h₀)))
have h₁ : ‖k • y‖ < δ := by
rw [norm_smul]
exact inv_mul_cancel_right₀ hy.ne.symm δ ▸
mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_left _ _)) hy
have h₂ :=
calc
‖p fun _ => k • y‖ ≤ c * ‖k • y‖ ^ (n.succ + 1) := by
-- porting note: now Lean wants `_root_.`
simpa only [norm_pow, _root_.norm_norm] using ht (k • y) (δε (mem_ball_zero_iff.mpr h₁))
--simpa only [norm_pow, norm_norm] using ht (k • y) (δε (mem_ball_zero_iff.mpr h₁))
_ = ‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))) := by
-- porting note: added `Nat.succ_eq_add_one` since otherwise `ring` does not conclude.
simp only [norm_smul, mul_pow, Nat.succ_eq_add_one]
-- porting note: removed `rw [pow_succ]`, since it now becomes superfluous.
ring
have h₃ : ‖k‖ * (c * ‖y‖ ^ (n.succ + 1)) < ε :=
inv_mul_cancel_right₀ h₀.ne.symm ε ▸
mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_right _ _)) h₀
calc
‖p fun _ => y‖ = ‖k⁻¹ ^ n.succ‖ * ‖p fun _ => k • y‖ := by
simpa only [inv_smul_smul₀ (norm_pos_iff.mp k_pos), norm_smul, Finset.prod_const,
Finset.card_fin] using
congr_arg norm (p.map_smul_univ (fun _ : Fin n.succ => k⁻¹) fun _ : Fin n.succ => k • y)
_ ≤ ‖k⁻¹ ^ n.succ‖ * (‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1)))) := by gcongr
_ = ‖(k⁻¹ * k) ^ n.succ‖ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))) := by
rw [← mul_assoc]
simp [norm_mul, mul_pow]
_ ≤ 0 + ε := by
rw [inv_mul_cancel (norm_pos_iff.mp k_pos)]
simpa using h₃.le
set_option linter.uppercaseLean3 false in
#align asymptotics.is_O.continuous_multilinear_map_apply_eq_zero Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero
/-- If a formal multilinear series `p` represents the zero function at `x : E`, then the
terms `p n (fun i ↦ y)` appearing in the sum are zero for any `n : ℕ`, `y : E`. -/
theorem HasFPowerSeriesAt.apply_eq_zero {p : FormalMultilinearSeries 𝕜 E F} {x : E}
(h : HasFPowerSeriesAt 0 p x) (n : ℕ) : ∀ y : E, (p n fun _ => y) = 0 := by
refine' Nat.strong_induction_on n fun k hk => _
have psum_eq : p.partialSum (k + 1) = fun y => p k fun _ => y := by
funext z
refine' Finset.sum_eq_single _ (fun b hb hnb => _) fun hn => _
· have := Finset.mem_range_succ_iff.mp hb
simp only [hk b (this.lt_of_ne hnb), Pi.zero_apply]
· exact False.elim (hn (Finset.mem_range.mpr (lt_add_one k)))
replace h := h.isBigO_sub_partialSum_pow k.succ
simp only [psum_eq, zero_sub, Pi.zero_apply, Asymptotics.isBigO_neg_left] at h
exact h.continuousMultilinearMap_apply_eq_zero
#align has_fpower_series_at.apply_eq_zero HasFPowerSeriesAt.apply_eq_zero
/-- A one-dimensional formal multilinear series representing the zero function is zero. -/
theorem HasFPowerSeriesAt.eq_zero {p : FormalMultilinearSeries 𝕜 𝕜 E} {x : 𝕜}
(h : HasFPowerSeriesAt 0 p x) : p = 0 := by
-- porting note: `funext; ext` was `ext (n x)`
funext n
ext x
rw [← mkPiField_apply_one_eq_self (p n)]
-- porting note: nasty hack, was `simp [h.apply_eq_zero n 1]`
have := Or.intro_right ?_ (h.apply_eq_zero n 1)
simpa using this
#align has_fpower_series_at.eq_zero HasFPowerSeriesAt.eq_zero
/-- One-dimensional formal multilinear series representing the same function are equal. -/
theorem HasFPowerSeriesAt.eq_formalMultilinearSeries {p₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E}
{f : 𝕜 → E} {x : 𝕜} (h₁ : HasFPowerSeriesAt f p₁ x) (h₂ : HasFPowerSeriesAt f p₂ x) : p₁ = p₂ :=
sub_eq_zero.mp (HasFPowerSeriesAt.eq_zero (by simpa only [sub_self] using h₁.sub h₂))
#align has_fpower_series_at.eq_formal_multilinear_series HasFPowerSeriesAt.eq_formalMultilinearSeries
theorem HasFPowerSeriesAt.eq_formalMultilinearSeries_of_eventually
{p q : FormalMultilinearSeries 𝕜 𝕜 E} {f g : 𝕜 → E} {x : 𝕜} (hp : HasFPowerSeriesAt f p x)
(hq : HasFPowerSeriesAt g q x) (heq : ∀ᶠ z in 𝓝 x, f z = g z) : p = q :=
(hp.congr heq).eq_formalMultilinearSeries hq
#align has_fpower_series_at.eq_formal_multilinear_series_of_eventually HasFPowerSeriesAt.eq_formalMultilinearSeries_of_eventually
/-- A one-dimensional formal multilinear series representing a locally zero function is zero. -/
theorem HasFPowerSeriesAt.eq_zero_of_eventually {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E}
{x : 𝕜} (hp : HasFPowerSeriesAt f p x) (hf : f =ᶠ[𝓝 x] 0) : p = 0 :=
(hp.congr hf).eq_zero
#align has_fpower_series_at.eq_zero_of_eventually HasFPowerSeriesAt.eq_zero_of_eventually
/-- If a function `f : 𝕜 → E` has two power series representations at `x`, then the given radii in
which convergence is guaranteed may be interchanged. This can be useful when the formal multilinear
series in one representation has a particularly nice form, but the other has a larger radius. -/
theorem HasFPowerSeriesOnBall.exchange_radius {p₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E}
{r₁ r₂ : ℝ≥0∞} {x : 𝕜} (h₁ : HasFPowerSeriesOnBall f p₁ x r₁)
(h₂ : HasFPowerSeriesOnBall f p₂ x r₂) : HasFPowerSeriesOnBall f p₁ x r₂ :=
h₂.hasFPowerSeriesAt.eq_formalMultilinearSeries h₁.hasFPowerSeriesAt ▸ h₂
#align has_fpower_series_on_ball.exchange_radius HasFPowerSeriesOnBall.exchange_radius
/-- If a function `f : 𝕜 → E` has power series representation `p` on a ball of some radius and for
each positive radius it has some power series representation, then `p` converges to `f` on the whole
`𝕜`. -/
theorem HasFPowerSeriesOnBall.r_eq_top_of_exists {f : 𝕜 → E} {r : ℝ≥0∞} {x : 𝕜}
{p : FormalMultilinearSeries 𝕜 𝕜 E} (h : HasFPowerSeriesOnBall f p x r)
(h' : ∀ (r' : ℝ≥0) (_ : 0 < r'), ∃ p' : FormalMultilinearSeries 𝕜 𝕜 E,
HasFPowerSeriesOnBall f p' x r') :
HasFPowerSeriesOnBall f p x ∞ :=
{ r_le := ENNReal.le_of_forall_pos_nnreal_lt fun r hr _ =>
let ⟨_, hp'⟩ := h' r hr
(h.exchange_radius hp').r_le
r_pos := ENNReal.coe_lt_top
hasSum := fun {y} _ =>
let ⟨r', hr'⟩ := exists_gt ‖y‖₊
let ⟨_, hp'⟩ := h' r' hr'.ne_bot.bot_lt
(h.exchange_radius hp').hasSum <| mem_emetric_ball_zero_iff.mpr (ENNReal.coe_lt_coe.2 hr') }
#align has_fpower_series_on_ball.r_eq_top_of_exists HasFPowerSeriesOnBall.r_eq_top_of_exists
end Uniqueness
/-!
### Changing origin in a power series
If a function is analytic in a disk `D(x, R)`, then it is analytic in any disk contained in that
one. Indeed, one can write
$$
f (x + y + z) = \sum_{n} p_n (y + z)^n = \sum_{n, k} \binom{n}{k} p_n y^{n-k} z^k
= \sum_{k} \Bigl(\sum_{n} \binom{n}{k} p_n y^{n-k}\Bigr) z^k.
$$
The corresponding power series has thus a `k`-th coefficient equal to
$\sum_{n} \binom{n}{k} p_n y^{n-k}$. In the general case where `pₙ` is a multilinear map, this has
to be interpreted suitably: instead of having a binomial coefficient, one should sum over all
possible subsets `s` of `Fin n` of cardinal `k`, and attribute `z` to the indices in `s` and
`y` to the indices outside of `s`.
In this paragraph, we implement this. The new power series is called `p.changeOrigin y`. Then, we
check its convergence and the fact that its sum coincides with the original sum. The outcome of this
discussion is that the set of points where a function is analytic is open.
-/
namespace FormalMultilinearSeries
section
variable (p : FormalMultilinearSeries 𝕜 E F) {x y : E} {r R : ℝ≥0}
/-- A term of `FormalMultilinearSeries.changeOriginSeries`.
Given a formal multilinear series `p` and a point `x` in its ball of convergence,
`p.changeOrigin x` is a formal multilinear series such that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense. Each term of `p.changeOrigin x`
is itself an analytic function of `x` given by the series `p.changeOriginSeries`. Each term in
`changeOriginSeries` is the sum of `changeOriginSeriesTerm`'s over all `s` of cardinality `l`.
The definition is such that `p.changeOriginSeriesTerm k l s hs (fun _ ↦ x) (fun _ ↦ y) =
p (k + l) (s.piecewise (fun _ ↦ x) (fun _ ↦ y))`
-/
def changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
E[×l]→L[𝕜] E[×k]→L[𝕜] F := by
let a := ContinuousMultilinearMap.curryFinFinset 𝕜 E F hs
(by erw [Finset.card_compl, Fintype.card_fin, hs, add_tsub_cancel_right])
exact a (p (k + l))
#align formal_multilinear_series.change_origin_series_term FormalMultilinearSeries.changeOriginSeriesTerm
theorem changeOriginSeriesTerm_apply (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l)
(x y : E) :
(p.changeOriginSeriesTerm k l s hs (fun _ => x) fun _ => y) =
p (k + l) (s.piecewise (fun _ => x) fun _ => y) :=
ContinuousMultilinearMap.curryFinFinset_apply_const _ _ _ _ _
#align formal_multilinear_series.change_origin_series_term_apply FormalMultilinearSeries.changeOriginSeriesTerm_apply
@[simp]
theorem norm_changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
‖p.changeOriginSeriesTerm k l s hs‖ = ‖p (k + l)‖ := by
simp only [changeOriginSeriesTerm, LinearIsometryEquiv.norm_map]
#align formal_multilinear_series.norm_change_origin_series_term FormalMultilinearSeries.norm_changeOriginSeriesTerm
@[simp]
theorem nnnorm_changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
‖p.changeOriginSeriesTerm k l s hs‖₊ = ‖p (k + l)‖₊ := by
simp only [changeOriginSeriesTerm, LinearIsometryEquiv.nnnorm_map]
#align formal_multilinear_series.nnnorm_change_origin_series_term FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm
theorem nnnorm_changeOriginSeriesTerm_apply_le (k l : ℕ) (s : Finset (Fin (k + l)))
(hs : s.card = l) (x y : E) :
‖p.changeOriginSeriesTerm k l s hs (fun _ => x) fun _ => y‖₊ ≤
‖p (k + l)‖₊ * ‖x‖₊ ^ l * ‖y‖₊ ^ k := by
rw [← p.nnnorm_changeOriginSeriesTerm k l s hs, ← Fin.prod_const, ← Fin.prod_const]
apply ContinuousMultilinearMap.le_of_op_nnnorm_le
apply ContinuousMultilinearMap.le_op_nnnorm
#align formal_multilinear_series.nnnorm_change_origin_series_term_apply_le FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm_apply_le
/-- The power series for `f.changeOrigin k`.
Given a formal multilinear series `p` and a point `x` in its ball of convergence,
`p.changeOrigin x` is a formal multilinear series such that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense. Its `k`-th term is the sum of
the series `p.changeOriginSeries k`. -/
def changeOriginSeries (k : ℕ) : FormalMultilinearSeries 𝕜 E (E[×k]→L[𝕜] F) := fun l =>
∑ s : { s : Finset (Fin (k + l)) // Finset.card s = l }, p.changeOriginSeriesTerm k l s s.2
#align formal_multilinear_series.change_origin_series FormalMultilinearSeries.changeOriginSeries
theorem nnnorm_changeOriginSeries_le_tsum (k l : ℕ) :
‖p.changeOriginSeries k l‖₊ ≤
∑' _ : { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + l)‖₊ :=
(nnnorm_sum_le _ (fun t => changeOriginSeriesTerm p k l (Subtype.val t) t.prop)).trans_eq <| by
simp_rw [tsum_fintype, nnnorm_changeOriginSeriesTerm (p := p) (k := k) (l := l)]
#align formal_multilinear_series.nnnorm_change_origin_series_le_tsum FormalMultilinearSeries.nnnorm_changeOriginSeries_le_tsum
theorem nnnorm_changeOriginSeries_apply_le_tsum (k l : ℕ) (x : E) :
‖p.changeOriginSeries k l fun _ => x‖₊ ≤
∑' _ : { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + l)‖₊ * ‖x‖₊ ^ l := by
rw [NNReal.tsum_mul_right, ← Fin.prod_const]
exact (p.changeOriginSeries k l).le_of_op_nnnorm_le _ (p.nnnorm_changeOriginSeries_le_tsum _ _)
#align formal_multilinear_series.nnnorm_change_origin_series_apply_le_tsum FormalMultilinearSeries.nnnorm_changeOriginSeries_apply_le_tsum
/-- Changing the origin of a formal multilinear series `p`, so that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense.
-/
def changeOrigin (x : E) : FormalMultilinearSeries 𝕜 E F :=
fun k => (p.changeOriginSeries k).sum x
#align formal_multilinear_series.change_origin FormalMultilinearSeries.changeOrigin
/-- An auxiliary equivalence useful in the proofs about
`FormalMultilinearSeries.changeOriginSeries`: the set of triples `(k, l, s)`, where `s` is a
`Finset (Fin (k + l))` of cardinality `l` is equivalent to the set of pairs `(n, s)`, where `s` is a
`Finset (Fin n)`.
The forward map sends `(k, l, s)` to `(k + l, s)` and the inverse map sends `(n, s)` to
`(n - Finset.card s, Finset.card s, s)`. The actual definition is less readable because of problems
with non-definitional equalities. -/
@[simps]
def changeOriginIndexEquiv :
(Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) ≃ Σn : ℕ, Finset (Fin n) where
toFun s := ⟨s.1 + s.2.1, s.2.2⟩
invFun s :=
⟨s.1 - s.2.card, s.2.card,
⟨s.2.map
(Fin.castIso <| (tsub_add_cancel_of_le <| card_finset_fin_le s.2).symm).toEquiv.toEmbedding,
Finset.card_map _⟩⟩
left_inv := by
rintro ⟨k, l, ⟨s : Finset (Fin <| k + l), hs : s.card = l⟩⟩
dsimp only [Subtype.coe_mk]
-- Lean can't automatically generalize `k' = k + l - s.card`, `l' = s.card`, so we explicitly
-- formulate the generalized goal
suffices ∀ k' l', k' = k → l' = l → ∀ (hkl : k + l = k' + l') (hs'),
(⟨k', l', ⟨Finset.map (Fin.castIso hkl).toEquiv.toEmbedding s, hs'⟩⟩ :
Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) = ⟨k, l, ⟨s, hs⟩⟩ by
apply this <;> simp only [hs, add_tsub_cancel_right]
rintro _ _ rfl rfl hkl hs'
simp only [Equiv.refl_toEmbedding, Fin.castIso_refl, Finset.map_refl, eq_self_iff_true,
OrderIso.refl_toEquiv, and_self_iff, heq_iff_eq]
right_inv := by
rintro ⟨n, s⟩
simp [tsub_add_cancel_of_le (card_finset_fin_le s), Fin.castIso_to_equiv]
#align formal_multilinear_series.change_origin_index_equiv FormalMultilinearSeries.changeOriginIndexEquiv
theorem changeOriginSeries_summable_aux₁ {r r' : ℝ≥0} (hr : (r + r' : ℝ≥0∞) < p.radius) :
Summable fun s : Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l } =>
‖p (s.1 + s.2.1)‖₊ * r ^ s.2.1 * r' ^ s.1 := by
rw [← changeOriginIndexEquiv.symm.summable_iff]
dsimp only [Function.comp_def, changeOriginIndexEquiv_symm_apply_fst,
changeOriginIndexEquiv_symm_apply_snd_fst]
have : ∀ n : ℕ,
HasSum (fun s : Finset (Fin n) => ‖p (n - s.card + s.card)‖₊ * r ^ s.card * r' ^ (n - s.card))
(‖p n‖₊ * (r + r') ^ n) := by
intro n
-- TODO: why `simp only [tsub_add_cancel_of_le (card_finset_fin_le _)]` fails?
convert_to HasSum (fun s : Finset (Fin n) => ‖p n‖₊ * (r ^ s.card * r' ^ (n - s.card))) _
· ext1 s
rw [tsub_add_cancel_of_le (card_finset_fin_le _), mul_assoc]
rw [← Fin.sum_pow_mul_eq_add_pow]
exact (hasSum_fintype _).mul_left _
refine' NNReal.summable_sigma.2 ⟨fun n => (this n).summable, _⟩
simp only [(this _).tsum_eq]
exact p.summable_nnnorm_mul_pow hr
#align formal_multilinear_series.change_origin_series_summable_aux₁ FormalMultilinearSeries.changeOriginSeries_summable_aux₁
theorem changeOriginSeries_summable_aux₂ (hr : (r : ℝ≥0∞) < p.radius) (k : ℕ) :
Summable fun s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l } =>
‖p (k + s.1)‖₊ * r ^ s.1 := by
rcases ENNReal.lt_iff_exists_add_pos_lt.1 hr with ⟨r', h0, hr'⟩
simpa only [mul_inv_cancel_right₀ (pow_pos h0 _).ne'] using
((NNReal.summable_sigma.1 (p.changeOriginSeries_summable_aux₁ hr')).1 k).mul_right (r' ^ k)⁻¹
#align formal_multilinear_series.change_origin_series_summable_aux₂ FormalMultilinearSeries.changeOriginSeries_summable_aux₂
theorem changeOriginSeries_summable_aux₃ {r : ℝ≥0} (hr : ↑r < p.radius) (k : ℕ) :
Summable fun l : ℕ => ‖p.changeOriginSeries k l‖₊ * r ^ l := by
refine' NNReal.summable_of_le
(fun n => _) (NNReal.summable_sigma.1 <| p.changeOriginSeries_summable_aux₂ hr k).2
simp only [NNReal.tsum_mul_right]
exact mul_le_mul' (p.nnnorm_changeOriginSeries_le_tsum _ _) le_rfl
#align formal_multilinear_series.change_origin_series_summable_aux₃ FormalMultilinearSeries.changeOriginSeries_summable_aux₃
theorem le_changeOriginSeries_radius (k : ℕ) : p.radius ≤ (p.changeOriginSeries k).radius :=
ENNReal.le_of_forall_nnreal_lt fun _r hr =>
le_radius_of_summable_nnnorm _ (p.changeOriginSeries_summable_aux₃ hr k)
#align formal_multilinear_series.le_change_origin_series_radius FormalMultilinearSeries.le_changeOriginSeries_radius
theorem nnnorm_changeOrigin_le (k : ℕ) (h : (‖x‖₊ : ℝ≥0∞) < p.radius) :
‖p.changeOrigin x k‖₊ ≤
∑' s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + s.1)‖₊ * ‖x‖₊ ^ s.1 := by
refine' tsum_of_nnnorm_bounded _ fun l => p.nnnorm_changeOriginSeries_apply_le_tsum k l x
have := p.changeOriginSeries_summable_aux₂ h k
refine' HasSum.sigma this.hasSum fun l => _
exact ((NNReal.summable_sigma.1 this).1 l).hasSum
#align formal_multilinear_series.nnnorm_change_origin_le FormalMultilinearSeries.nnnorm_changeOrigin_le
/-- The radius of convergence of `p.changeOrigin x` is at least `p.radius - ‖x‖`. In other words,
`p.changeOrigin x` is well defined on the largest ball contained in the original ball of
convergence. -/
theorem changeOrigin_radius : p.radius - ‖x‖₊ ≤ (p.changeOrigin x).radius := by
refine' ENNReal.le_of_forall_pos_nnreal_lt fun r _h0 hr => _
rw [lt_tsub_iff_right, add_comm] at hr
have hr' : (‖x‖₊ : ℝ≥0∞) < p.radius := (le_add_right le_rfl).trans_lt hr
apply le_radius_of_summable_nnnorm
have : ∀ k : ℕ,
‖p.changeOrigin x k‖₊ * r ^ k ≤
(∑' s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + s.1)‖₊ * ‖x‖₊ ^ s.1) *
r ^ k :=
fun k => mul_le_mul_right' (p.nnnorm_changeOrigin_le k hr') (r ^ k)
refine' NNReal.summable_of_le this _
simpa only [← NNReal.tsum_mul_right] using
(NNReal.summable_sigma.1 (p.changeOriginSeries_summable_aux₁ hr)).2
#align formal_multilinear_series.change_origin_radius FormalMultilinearSeries.changeOrigin_radius
end
-- From this point on, assume that the space is complete, to make sure that series that converge
-- in norm also converge in `F`.
variable [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x y : E} {r R : ℝ≥0}
theorem hasFPowerSeriesOnBall_changeOrigin (k : ℕ) (hr : 0 < p.radius) :
HasFPowerSeriesOnBall (fun x => p.changeOrigin x k) (p.changeOriginSeries k) 0 p.radius :=
have := p.le_changeOriginSeries_radius k
((p.changeOriginSeries k).hasFPowerSeriesOnBall (hr.trans_le this)).mono hr this
#align formal_multilinear_series.has_fpower_series_on_ball_change_origin FormalMultilinearSeries.hasFPowerSeriesOnBall_changeOrigin
/-- Summing the series `p.changeOrigin x` at a point `y` gives back `p (x + y)`. -/
theorem changeOrigin_eval (h : (‖x‖₊ + ‖y‖₊ : ℝ≥0∞) < p.radius) :
(p.changeOrigin x).sum y = p.sum (x + y) := by
have radius_pos : 0 < p.radius := lt_of_le_of_lt (zero_le _) h
have x_mem_ball : x ∈ EMetric.ball (0 : E) p.radius :=
mem_emetric_ball_zero_iff.2 ((le_add_right le_rfl).trans_lt h)
have y_mem_ball : y ∈ EMetric.ball (0 : E) (p.changeOrigin x).radius := by
refine' mem_emetric_ball_zero_iff.2 (lt_of_lt_of_le _ p.changeOrigin_radius)
rwa [lt_tsub_iff_right, add_comm]
have x_add_y_mem_ball : x + y ∈ EMetric.ball (0 : E) p.radius := by
refine' mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt _ h)
exact mod_cast nnnorm_add_le x y
set f : (Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) → F := fun s =>
p.changeOriginSeriesTerm s.1 s.2.1 s.2.2 s.2.2.2 (fun _ => x) fun _ => y
have hsf : Summable f := by
refine' .of_nnnorm_bounded _ (p.changeOriginSeries_summable_aux₁ h) _
rintro ⟨k, l, s, hs⟩
dsimp only [Subtype.coe_mk]
exact p.nnnorm_changeOriginSeriesTerm_apply_le _ _ _ _ _ _
have hf : HasSum f ((p.changeOrigin x).sum y) := by
refine' HasSum.sigma_of_hasSum ((p.changeOrigin x).summable y_mem_ball).hasSum (fun k => _) hsf
· dsimp only
refine' ContinuousMultilinearMap.hasSum_eval _ _
have := (p.hasFPowerSeriesOnBall_changeOrigin k radius_pos).hasSum x_mem_ball
rw [zero_add] at this
refine' HasSum.sigma_of_hasSum this (fun l => _) _
· simp only [changeOriginSeries, ContinuousMultilinearMap.sum_apply]
apply hasSum_fintype
· refine' .of_nnnorm_bounded _
(p.changeOriginSeries_summable_aux₂ (mem_emetric_ball_zero_iff.1 x_mem_ball) k) fun s => _
refine' (ContinuousMultilinearMap.le_op_nnnorm _ _).trans_eq _
simp
refine' hf.unique (changeOriginIndexEquiv.symm.hasSum_iff.1 _)
refine' HasSum.sigma_of_hasSum
(p.hasSum x_add_y_mem_ball) (fun n => _) (changeOriginIndexEquiv.symm.summable_iff.2 hsf)
erw [(p n).map_add_univ (fun _ => x) fun _ => y]
-- porting note: added explicit function
convert hasSum_fintype (fun c : Finset (Fin n) => f (changeOriginIndexEquiv.symm ⟨n, c⟩))
rename_i s _
dsimp only [changeOriginSeriesTerm, (· ∘ ·), changeOriginIndexEquiv_symm_apply_fst,
changeOriginIndexEquiv_symm_apply_snd_fst, changeOriginIndexEquiv_symm_apply_snd_snd_coe]
rw [ContinuousMultilinearMap.curryFinFinset_apply_const]
have : ∀ (m) (hm : n = m), p n (s.piecewise (fun _ => x) fun _ => y) =
p m ((s.map (Fin.castIso hm).toEquiv.toEmbedding).piecewise (fun _ => x) fun _ => y) := by
rintro m rfl
simp (config := { unfoldPartialApp := true }) [Finset.piecewise]
apply this
#align formal_multilinear_series.change_origin_eval FormalMultilinearSeries.changeOrigin_eval
/-- Power series terms are analytic as we vary the origin -/
theorem analyticAt_changeOrigin (p : FormalMultilinearSeries 𝕜 E F) (rp : p.radius > 0) (n : ℕ) :
AnalyticAt 𝕜 (fun x ↦ p.changeOrigin x n) 0 :=
(FormalMultilinearSeries.hasFPowerSeriesOnBall_changeOrigin p n rp).analyticAt
end FormalMultilinearSeries
section
variable [CompleteSpace F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x y : E} {r : ℝ≥0∞}
/-- If a function admits a power series expansion `p` on a ball `B (x, r)`, then it also admits a
power series on any subball of this ball (even with a different center), given by `p.changeOrigin`.
-/
theorem HasFPowerSeriesOnBall.changeOrigin (hf : HasFPowerSeriesOnBall f p x r)
(h : (‖y‖₊ : ℝ≥0∞) < r) : HasFPowerSeriesOnBall f (p.changeOrigin y) (x + y) (r - ‖y‖₊) :=
{ r_le := by
apply le_trans _ p.changeOrigin_radius
exact tsub_le_tsub hf.r_le le_rfl
r_pos := by simp [h]
hasSum := fun {z} hz => by
have : f (x + y + z) =
FormalMultilinearSeries.sum (FormalMultilinearSeries.changeOrigin p y) z := by
rw [mem_emetric_ball_zero_iff, lt_tsub_iff_right, add_comm] at hz
rw [p.changeOrigin_eval (hz.trans_le hf.r_le), add_assoc, hf.sum]
refine' mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt _ hz)
exact mod_cast nnnorm_add_le y z
rw [this]
apply (p.changeOrigin y).hasSum
refine' EMetric.ball_subset_ball (le_trans _ p.changeOrigin_radius) hz
exact tsub_le_tsub hf.r_le le_rfl }
#align has_fpower_series_on_ball.change_origin HasFPowerSeriesOnBall.changeOrigin
/-- If a function admits a power series expansion `p` on an open ball `B (x, r)`, then
it is analytic at every point of this ball. -/
theorem HasFPowerSeriesOnBall.analyticAt_of_mem (hf : HasFPowerSeriesOnBall f p x r)
(h : y ∈ EMetric.ball x r) : AnalyticAt 𝕜 f y := by
have : (‖y - x‖₊ : ℝ≥0∞) < r := by simpa [edist_eq_coe_nnnorm_sub] using h
have := hf.changeOrigin this
rw [add_sub_cancel'_right] at this
exact this.analyticAt
#align has_fpower_series_on_ball.analytic_at_of_mem HasFPowerSeriesOnBall.analyticAt_of_mem
theorem HasFPowerSeriesOnBall.analyticOn (hf : HasFPowerSeriesOnBall f p x r) :
AnalyticOn 𝕜 f (EMetric.ball x r) :=
fun _y hy => hf.analyticAt_of_mem hy
#align has_fpower_series_on_ball.analytic_on HasFPowerSeriesOnBall.analyticOn
variable (𝕜 f)
/-- For any function `f` from a normed vector space to a Banach space, the set of points `x` such
that `f` is analytic at `x` is open. -/
theorem isOpen_analyticAt : IsOpen { x | AnalyticAt 𝕜 f x } := by
rw [isOpen_iff_mem_nhds]
rintro x ⟨p, r, hr⟩
exact mem_of_superset (EMetric.ball_mem_nhds _ hr.r_pos) fun y hy => hr.analyticAt_of_mem hy
#align is_open_analytic_at isOpen_analyticAt
variable {𝕜}
theorem AnalyticAt.eventually_analyticAt {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∀ᶠ y in 𝓝 x, AnalyticAt 𝕜 f y :=
(isOpen_analyticAt 𝕜 f).mem_nhds h
theorem AnalyticAt.exists_mem_nhds_analyticOn {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∃ s ∈ 𝓝 x, AnalyticOn 𝕜 f s :=
h.eventually_analyticAt.exists_mem
/-- If we're analytic at a point, we're analytic in a nonempty ball -/
theorem AnalyticAt.exists_ball_analyticOn {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∃ r : ℝ, 0 < r ∧ AnalyticOn 𝕜 f (Metric.ball x r) :=
Metric.isOpen_iff.mp (isOpen_analyticAt _ _) _ h
end
section
open FormalMultilinearSeries
variable {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E} {z₀ : 𝕜}
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z)) := by
refine' ⟨fun ⟨r, _, r_pos, h⟩ =>
eventually_of_mem (EMetric.ball_mem_nhds 0 r_pos) fun _ => by simpa using h, _⟩
simp only [Metric.eventually_nhds_iff]
|
rintro ⟨r, r_pos, h⟩
|
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z)) := by
refine' ⟨fun ⟨r, _, r_pos, h⟩ =>
eventually_of_mem (EMetric.ball_mem_nhds 0 r_pos) fun _ => by simpa using h, _⟩
simp only [Metric.eventually_nhds_iff]
|
Mathlib.Analysis.Analytic.Basic.1430_0.jQw1fRSE1vGpOll
|
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z))
|
Mathlib_Analysis_Analytic_Basic
|
case intro.intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
p : FormalMultilinearSeries 𝕜 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
r : ℝ
r_pos : r > 0
h : ∀ ⦃y : 𝕜⦄, dist y 0 < r → HasSum (fun n => y ^ n • coeff p n) (f (z₀ + y))
⊢ HasFPowerSeriesAt f p z₀
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Logic.Equiv.Fin
import Mathlib.Topology.Algebra.InfiniteSum.Module
#align_import analysis.analytic.basic from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
/-!
# Analytic functions
A function is analytic in one dimension around `0` if it can be written as a converging power series
`Σ pₙ zⁿ`. This definition can be extended to any dimension (even in infinite dimension) by
requiring that `pₙ` is a continuous `n`-multilinear map. In general, `pₙ` is not unique (in two
dimensions, taking `p₂ (x, y) (x', y') = x y'` or `y x'` gives the same map when applied to a
vector `(x, y) (x, y)`). A way to guarantee uniqueness is to take a symmetric `pₙ`, but this is not
always possible in nonzero characteristic (in characteristic 2, the previous example has no
symmetric representative). Therefore, we do not insist on symmetry or uniqueness in the definition,
and we only require the existence of a converging series.
The general framework is important to say that the exponential map on bounded operators on a Banach
space is analytic, as well as the inverse on invertible operators.
## Main definitions
Let `p` be a formal multilinear series from `E` to `F`, i.e., `p n` is a multilinear map on `E^n`
for `n : ℕ`.
* `p.radius`: the largest `r : ℝ≥0∞` such that `‖p n‖ * r^n` grows subexponentially.
* `p.le_radius_of_bound`, `p.le_radius_of_bound_nnreal`, `p.le_radius_of_isBigO`: if `‖p n‖ * r ^ n`
is bounded above, then `r ≤ p.radius`;
* `p.isLittleO_of_lt_radius`, `p.norm_mul_pow_le_mul_pow_of_lt_radius`,
`p.isLittleO_one_of_lt_radius`,
`p.norm_mul_pow_le_of_lt_radius`, `p.nnnorm_mul_pow_le_of_lt_radius`: if `r < p.radius`, then
`‖p n‖ * r ^ n` tends to zero exponentially;
* `p.lt_radius_of_isBigO`: if `r ≠ 0` and `‖p n‖ * r ^ n = O(a ^ n)` for some `-1 < a < 1`, then
`r < p.radius`;
* `p.partialSum n x`: the sum `∑_{i = 0}^{n-1} pᵢ xⁱ`.
* `p.sum x`: the sum `∑'_{i = 0}^{∞} pᵢ xⁱ`.
Additionally, let `f` be a function from `E` to `F`.
* `HasFPowerSeriesOnBall f p x r`: on the ball of center `x` with radius `r`,
`f (x + y) = ∑'_n pₙ yⁿ`.
* `HasFPowerSeriesAt f p x`: on some ball of center `x` with positive radius, holds
`HasFPowerSeriesOnBall f p x r`.
* `AnalyticAt 𝕜 f x`: there exists a power series `p` such that holds `HasFPowerSeriesAt f p x`.
* `AnalyticOn 𝕜 f s`: the function `f` is analytic at every point of `s`.
We develop the basic properties of these notions, notably:
* If a function admits a power series, it is continuous (see
`HasFPowerSeriesOnBall.continuousOn` and `HasFPowerSeriesAt.continuousAt` and
`AnalyticAt.continuousAt`).
* In a complete space, the sum of a formal power series with positive radius is well defined on the
disk of convergence, see `FormalMultilinearSeries.hasFPowerSeriesOnBall`.
* If a function admits a power series in a ball, then it is analytic at any point `y` of this ball,
and the power series there can be expressed in terms of the initial power series `p` as
`p.changeOrigin y`. See `HasFPowerSeriesOnBall.changeOrigin`. It follows in particular that
the set of points at which a given function is analytic is open, see `isOpen_analyticAt`.
## Implementation details
We only introduce the radius of convergence of a power series, as `p.radius`.
For a power series in finitely many dimensions, there is a finer (directional, coordinate-dependent)
notion, describing the polydisk of convergence. This notion is more specific, and not necessary to
build the general theory. We do not define it here.
-/
noncomputable section
variable {𝕜 E F G : Type*}
open Topology Classical BigOperators NNReal Filter ENNReal
open Set Filter Asymptotics
namespace FormalMultilinearSeries
variable [Ring 𝕜] [AddCommGroup E] [AddCommGroup F] [Module 𝕜 E] [Module 𝕜 F]
variable [TopologicalSpace E] [TopologicalSpace F]
variable [TopologicalAddGroup E] [TopologicalAddGroup F]
variable [ContinuousConstSMul 𝕜 E] [ContinuousConstSMul 𝕜 F]
/-- Given a formal multilinear series `p` and a vector `x`, then `p.sum x` is the sum `Σ pₙ xⁿ`. A
priori, it only behaves well when `‖x‖ < p.radius`. -/
protected def sum (p : FormalMultilinearSeries 𝕜 E F) (x : E) : F :=
∑' n : ℕ, p n fun _ => x
#align formal_multilinear_series.sum FormalMultilinearSeries.sum
/-- Given a formal multilinear series `p` and a vector `x`, then `p.partialSum n x` is the sum
`Σ pₖ xᵏ` for `k ∈ {0,..., n-1}`. -/
def partialSum (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) (x : E) : F :=
∑ k in Finset.range n, p k fun _ : Fin k => x
#align formal_multilinear_series.partial_sum FormalMultilinearSeries.partialSum
/-- The partial sums of a formal multilinear series are continuous. -/
theorem partialSum_continuous (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) :
Continuous (p.partialSum n) := by
unfold partialSum -- Porting note: added
continuity
#align formal_multilinear_series.partial_sum_continuous FormalMultilinearSeries.partialSum_continuous
end FormalMultilinearSeries
/-! ### The radius of a formal multilinear series -/
variable [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F]
[NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G]
namespace FormalMultilinearSeries
variable (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
/-- The radius of a formal multilinear series is the largest `r` such that the sum `Σ ‖pₙ‖ ‖y‖ⁿ`
converges for all `‖y‖ < r`. This implies that `Σ pₙ yⁿ` converges for all `‖y‖ < r`, but these
definitions are *not* equivalent in general. -/
def radius (p : FormalMultilinearSeries 𝕜 E F) : ℝ≥0∞ :=
⨆ (r : ℝ≥0) (C : ℝ) (_ : ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C), (r : ℝ≥0∞)
#align formal_multilinear_series.radius FormalMultilinearSeries.radius
/-- If `‖pₙ‖ rⁿ` is bounded in `n`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_bound (C : ℝ) {r : ℝ≥0} (h : ∀ n : ℕ, ‖p n‖ * (r : ℝ) ^ n ≤ C) :
(r : ℝ≥0∞) ≤ p.radius :=
le_iSup_of_le r <| le_iSup_of_le C <| le_iSup (fun _ => (r : ℝ≥0∞)) h
#align formal_multilinear_series.le_radius_of_bound FormalMultilinearSeries.le_radius_of_bound
/-- If `‖pₙ‖ rⁿ` is bounded in `n`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_bound_nnreal (C : ℝ≥0) {r : ℝ≥0} (h : ∀ n : ℕ, ‖p n‖₊ * r ^ n ≤ C) :
(r : ℝ≥0∞) ≤ p.radius :=
p.le_radius_of_bound C fun n => mod_cast h n
#align formal_multilinear_series.le_radius_of_bound_nnreal FormalMultilinearSeries.le_radius_of_bound_nnreal
/-- If `‖pₙ‖ rⁿ = O(1)`, as `n → ∞`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_isBigO (h : (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] fun _ => (1 : ℝ)) :
↑r ≤ p.radius :=
Exists.elim (isBigO_one_nat_atTop_iff.1 h) fun C hC =>
p.le_radius_of_bound C fun n => (le_abs_self _).trans (hC n)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.le_radius_of_is_O FormalMultilinearSeries.le_radius_of_isBigO
theorem le_radius_of_eventually_le (C) (h : ∀ᶠ n in atTop, ‖p n‖ * (r : ℝ) ^ n ≤ C) :
↑r ≤ p.radius :=
p.le_radius_of_isBigO <| IsBigO.of_bound C <| h.mono fun n hn => by simpa
#align formal_multilinear_series.le_radius_of_eventually_le FormalMultilinearSeries.le_radius_of_eventually_le
theorem le_radius_of_summable_nnnorm (h : Summable fun n => ‖p n‖₊ * r ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_bound_nnreal (∑' n, ‖p n‖₊ * r ^ n) fun _ => le_tsum' h _
#align formal_multilinear_series.le_radius_of_summable_nnnorm FormalMultilinearSeries.le_radius_of_summable_nnnorm
theorem le_radius_of_summable (h : Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_summable_nnnorm <| by
simp only [← coe_nnnorm] at h
exact mod_cast h
#align formal_multilinear_series.le_radius_of_summable FormalMultilinearSeries.le_radius_of_summable
theorem radius_eq_top_of_forall_nnreal_isBigO
(h : ∀ r : ℝ≥0, (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] fun _ => (1 : ℝ)) : p.radius = ∞ :=
ENNReal.eq_top_of_forall_nnreal_le fun r => p.le_radius_of_isBigO (h r)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.radius_eq_top_of_forall_nnreal_is_O FormalMultilinearSeries.radius_eq_top_of_forall_nnreal_isBigO
theorem radius_eq_top_of_eventually_eq_zero (h : ∀ᶠ n in atTop, p n = 0) : p.radius = ∞ :=
p.radius_eq_top_of_forall_nnreal_isBigO fun r =>
(isBigO_zero _ _).congr' (h.mono fun n hn => by simp [hn]) EventuallyEq.rfl
#align formal_multilinear_series.radius_eq_top_of_eventually_eq_zero FormalMultilinearSeries.radius_eq_top_of_eventually_eq_zero
theorem radius_eq_top_of_forall_image_add_eq_zero (n : ℕ) (hn : ∀ m, p (m + n) = 0) :
p.radius = ∞ :=
p.radius_eq_top_of_eventually_eq_zero <|
mem_atTop_sets.2 ⟨n, fun _ hk => tsub_add_cancel_of_le hk ▸ hn _⟩
#align formal_multilinear_series.radius_eq_top_of_forall_image_add_eq_zero FormalMultilinearSeries.radius_eq_top_of_forall_image_add_eq_zero
@[simp]
theorem constFormalMultilinearSeries_radius {v : F} :
(constFormalMultilinearSeries 𝕜 E v).radius = ⊤ :=
(constFormalMultilinearSeries 𝕜 E v).radius_eq_top_of_forall_image_add_eq_zero 1
(by simp [constFormalMultilinearSeries])
#align formal_multilinear_series.const_formal_multilinear_series_radius FormalMultilinearSeries.constFormalMultilinearSeries_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` tends to zero exponentially:
for some `0 < a < 1`, `‖p n‖ rⁿ = o(aⁿ)`. -/
theorem isLittleO_of_lt_radius (h : ↑r < p.radius) :
∃ a ∈ Ioo (0 : ℝ) 1, (fun n => ‖p n‖ * (r : ℝ) ^ n) =o[atTop] (a ^ ·) := by
have := (TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 4
rw [this]
-- Porting note: was
-- rw [(TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 4]
simp only [radius, lt_iSup_iff] at h
rcases h with ⟨t, C, hC, rt⟩
rw [ENNReal.coe_lt_coe, ← NNReal.coe_lt_coe] at rt
have : 0 < (t : ℝ) := r.coe_nonneg.trans_lt rt
rw [← div_lt_one this] at rt
refine' ⟨_, rt, C, Or.inr zero_lt_one, fun n => _⟩
calc
|‖p n‖ * (r : ℝ) ^ n| = ‖p n‖ * (t : ℝ) ^ n * (r / t : ℝ) ^ n := by
field_simp [mul_right_comm, abs_mul]
_ ≤ C * (r / t : ℝ) ^ n := by gcongr; apply hC
#align formal_multilinear_series.is_o_of_lt_radius FormalMultilinearSeries.isLittleO_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ = o(1)`. -/
theorem isLittleO_one_of_lt_radius (h : ↑r < p.radius) :
(fun n => ‖p n‖ * (r : ℝ) ^ n) =o[atTop] (fun _ => 1 : ℕ → ℝ) :=
let ⟨_, ha, hp⟩ := p.isLittleO_of_lt_radius h
hp.trans <| (isLittleO_pow_pow_of_lt_left ha.1.le ha.2).congr (fun _ => rfl) one_pow
#align formal_multilinear_series.is_o_one_of_lt_radius FormalMultilinearSeries.isLittleO_one_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` tends to zero exponentially:
for some `0 < a < 1` and `C > 0`, `‖p n‖ * r ^ n ≤ C * a ^ n`. -/
theorem norm_mul_pow_le_mul_pow_of_lt_radius (h : ↑r < p.radius) :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C * a ^ n := by
-- Porting note: moved out of `rcases`
have := ((TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 5).mp
(p.isLittleO_of_lt_radius h)
rcases this with ⟨a, ha, C, hC, H⟩
exact ⟨a, ha, C, hC, fun n => (le_abs_self _).trans (H n)⟩
#align formal_multilinear_series.norm_mul_pow_le_mul_pow_of_lt_radius FormalMultilinearSeries.norm_mul_pow_le_mul_pow_of_lt_radius
/-- If `r ≠ 0` and `‖pₙ‖ rⁿ = O(aⁿ)` for some `-1 < a < 1`, then `r < p.radius`. -/
theorem lt_radius_of_isBigO (h₀ : r ≠ 0) {a : ℝ} (ha : a ∈ Ioo (-1 : ℝ) 1)
(hp : (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] (a ^ ·)) : ↑r < p.radius := by
-- Porting note: moved out of `rcases`
have := ((TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 2 5)
rcases this.mp ⟨a, ha, hp⟩ with ⟨a, ha, C, hC, hp⟩
rw [← pos_iff_ne_zero, ← NNReal.coe_pos] at h₀
lift a to ℝ≥0 using ha.1.le
have : (r : ℝ) < r / a := by
simpa only [div_one] using (div_lt_div_left h₀ zero_lt_one ha.1).2 ha.2
norm_cast at this
rw [← ENNReal.coe_lt_coe] at this
refine' this.trans_le (p.le_radius_of_bound C fun n => _)
rw [NNReal.coe_div, div_pow, ← mul_div_assoc, div_le_iff (pow_pos ha.1 n)]
exact (le_abs_self _).trans (hp n)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.lt_radius_of_is_O FormalMultilinearSeries.lt_radius_of_isBigO
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem norm_mul_pow_le_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C :=
let ⟨_, ha, C, hC, h⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius h
⟨C, hC, fun n => (h n).trans <| mul_le_of_le_one_right hC.lt.le (pow_le_one _ ha.1.le ha.2.le)⟩
#align formal_multilinear_series.norm_mul_pow_le_of_lt_radius FormalMultilinearSeries.norm_mul_pow_le_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem norm_le_div_pow_of_pos_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h0 : 0 < r) (h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖ ≤ C / (r : ℝ) ^ n :=
let ⟨C, hC, hp⟩ := p.norm_mul_pow_le_of_lt_radius h
⟨C, hC, fun n => Iff.mpr (le_div_iff (pow_pos h0 _)) (hp n)⟩
#align formal_multilinear_series.norm_le_div_pow_of_pos_of_lt_radius FormalMultilinearSeries.norm_le_div_pow_of_pos_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem nnnorm_mul_pow_le_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖₊ * r ^ n ≤ C :=
let ⟨C, hC, hp⟩ := p.norm_mul_pow_le_of_lt_radius h
⟨⟨C, hC.lt.le⟩, hC, mod_cast hp⟩
#align formal_multilinear_series.nnnorm_mul_pow_le_of_lt_radius FormalMultilinearSeries.nnnorm_mul_pow_le_of_lt_radius
theorem le_radius_of_tendsto (p : FormalMultilinearSeries 𝕜 E F) {l : ℝ}
(h : Tendsto (fun n => ‖p n‖ * (r : ℝ) ^ n) atTop (𝓝 l)) : ↑r ≤ p.radius :=
p.le_radius_of_isBigO (h.isBigO_one _)
#align formal_multilinear_series.le_radius_of_tendsto FormalMultilinearSeries.le_radius_of_tendsto
theorem le_radius_of_summable_norm (p : FormalMultilinearSeries 𝕜 E F)
(hs : Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_tendsto hs.tendsto_atTop_zero
#align formal_multilinear_series.le_radius_of_summable_norm FormalMultilinearSeries.le_radius_of_summable_norm
theorem not_summable_norm_of_radius_lt_nnnorm (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(h : p.radius < ‖x‖₊) : ¬Summable fun n => ‖p n‖ * ‖x‖ ^ n :=
fun hs => not_le_of_lt h (p.le_radius_of_summable_norm hs)
#align formal_multilinear_series.not_summable_norm_of_radius_lt_nnnorm FormalMultilinearSeries.not_summable_norm_of_radius_lt_nnnorm
theorem summable_norm_mul_pow (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0} (h : ↑r < p.radius) :
Summable fun n : ℕ => ‖p n‖ * (r : ℝ) ^ n := by
obtain ⟨a, ha : a ∈ Ioo (0 : ℝ) 1, C, - : 0 < C, hp⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius h
exact .of_nonneg_of_le (fun n => mul_nonneg (norm_nonneg _) (pow_nonneg r.coe_nonneg _))
hp ((summable_geometric_of_lt_1 ha.1.le ha.2).mul_left _)
#align formal_multilinear_series.summable_norm_mul_pow FormalMultilinearSeries.summable_norm_mul_pow
theorem summable_norm_apply (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : Summable fun n : ℕ => ‖p n fun _ => x‖ := by
rw [mem_emetric_ball_zero_iff] at hx
refine' .of_nonneg_of_le
(fun _ => norm_nonneg _) (fun n => ((p n).le_op_norm _).trans_eq _) (p.summable_norm_mul_pow hx)
simp
#align formal_multilinear_series.summable_norm_apply FormalMultilinearSeries.summable_norm_apply
theorem summable_nnnorm_mul_pow (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0} (h : ↑r < p.radius) :
Summable fun n : ℕ => ‖p n‖₊ * r ^ n := by
rw [← NNReal.summable_coe]
push_cast
exact p.summable_norm_mul_pow h
#align formal_multilinear_series.summable_nnnorm_mul_pow FormalMultilinearSeries.summable_nnnorm_mul_pow
protected theorem summable [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : Summable fun n : ℕ => p n fun _ => x :=
(p.summable_norm_apply hx).of_norm
#align formal_multilinear_series.summable FormalMultilinearSeries.summable
theorem radius_eq_top_of_summable_norm (p : FormalMultilinearSeries 𝕜 E F)
(hs : ∀ r : ℝ≥0, Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : p.radius = ∞ :=
ENNReal.eq_top_of_forall_nnreal_le fun r => p.le_radius_of_summable_norm (hs r)
#align formal_multilinear_series.radius_eq_top_of_summable_norm FormalMultilinearSeries.radius_eq_top_of_summable_norm
theorem radius_eq_top_iff_summable_norm (p : FormalMultilinearSeries 𝕜 E F) :
p.radius = ∞ ↔ ∀ r : ℝ≥0, Summable fun n => ‖p n‖ * (r : ℝ) ^ n := by
constructor
· intro h r
obtain ⟨a, ha : a ∈ Ioo (0 : ℝ) 1, C, - : 0 < C, hp⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius
(show (r : ℝ≥0∞) < p.radius from h.symm ▸ ENNReal.coe_lt_top)
refine' .of_norm_bounded
(fun n => (C : ℝ) * a ^ n) ((summable_geometric_of_lt_1 ha.1.le ha.2).mul_left _) fun n => _
specialize hp n
rwa [Real.norm_of_nonneg (mul_nonneg (norm_nonneg _) (pow_nonneg r.coe_nonneg n))]
· exact p.radius_eq_top_of_summable_norm
#align formal_multilinear_series.radius_eq_top_iff_summable_norm FormalMultilinearSeries.radius_eq_top_iff_summable_norm
/-- If the radius of `p` is positive, then `‖pₙ‖` grows at most geometrically. -/
theorem le_mul_pow_of_radius_pos (p : FormalMultilinearSeries 𝕜 E F) (h : 0 < p.radius) :
∃ (C r : _) (hC : 0 < C) (_ : 0 < r), ∀ n, ‖p n‖ ≤ C * r ^ n := by
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 h with ⟨r, r0, rlt⟩
have rpos : 0 < (r : ℝ) := by simp [ENNReal.coe_pos.1 r0]
rcases norm_le_div_pow_of_pos_of_lt_radius p rpos rlt with ⟨C, Cpos, hCp⟩
refine' ⟨C, r⁻¹, Cpos, by simp only [inv_pos, rpos], fun n => _⟩
-- Porting note: was `convert`
rw [inv_pow, ← div_eq_mul_inv]
exact hCp n
#align formal_multilinear_series.le_mul_pow_of_radius_pos FormalMultilinearSeries.le_mul_pow_of_radius_pos
/-- The radius of the sum of two formal series is at least the minimum of their two radii. -/
theorem min_radius_le_radius_add (p q : FormalMultilinearSeries 𝕜 E F) :
min p.radius q.radius ≤ (p + q).radius := by
refine' ENNReal.le_of_forall_nnreal_lt fun r hr => _
rw [lt_min_iff] at hr
have := ((p.isLittleO_one_of_lt_radius hr.1).add (q.isLittleO_one_of_lt_radius hr.2)).isBigO
refine' (p + q).le_radius_of_isBigO ((isBigO_of_le _ fun n => _).trans this)
rw [← add_mul, norm_mul, norm_mul, norm_norm]
exact mul_le_mul_of_nonneg_right ((norm_add_le _ _).trans (le_abs_self _)) (norm_nonneg _)
#align formal_multilinear_series.min_radius_le_radius_add FormalMultilinearSeries.min_radius_le_radius_add
@[simp]
theorem radius_neg (p : FormalMultilinearSeries 𝕜 E F) : (-p).radius = p.radius := by
simp only [radius, neg_apply, norm_neg]
#align formal_multilinear_series.radius_neg FormalMultilinearSeries.radius_neg
protected theorem hasSum [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : HasSum (fun n : ℕ => p n fun _ => x) (p.sum x) :=
(p.summable hx).hasSum
#align formal_multilinear_series.has_sum FormalMultilinearSeries.hasSum
theorem radius_le_radius_continuousLinearMap_comp (p : FormalMultilinearSeries 𝕜 E F)
(f : F →L[𝕜] G) : p.radius ≤ (f.compFormalMultilinearSeries p).radius := by
refine' ENNReal.le_of_forall_nnreal_lt fun r hr => _
apply le_radius_of_isBigO
apply (IsBigO.trans_isLittleO _ (p.isLittleO_one_of_lt_radius hr)).isBigO
refine' IsBigO.mul (@IsBigOWith.isBigO _ _ _ _ _ ‖f‖ _ _ _ _) (isBigO_refl _ _)
refine IsBigOWith.of_bound (eventually_of_forall fun n => ?_)
simpa only [norm_norm] using f.norm_compContinuousMultilinearMap_le (p n)
#align formal_multilinear_series.radius_le_radius_continuous_linear_map_comp FormalMultilinearSeries.radius_le_radius_continuousLinearMap_comp
end FormalMultilinearSeries
/-! ### Expanding a function as a power series -/
section
variable {f g : E → F} {p pf pg : FormalMultilinearSeries 𝕜 E F} {x : E} {r r' : ℝ≥0∞}
/-- Given a function `f : E → F` and a formal multilinear series `p`, we say that `f` has `p` as
a power series on the ball of radius `r > 0` around `x` if `f (x + y) = ∑' pₙ yⁿ` for all `‖y‖ < r`.
-/
structure HasFPowerSeriesOnBall (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (x : E) (r : ℝ≥0∞) :
Prop where
r_le : r ≤ p.radius
r_pos : 0 < r
hasSum :
∀ {y}, y ∈ EMetric.ball (0 : E) r → HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y))
#align has_fpower_series_on_ball HasFPowerSeriesOnBall
/-- Given a function `f : E → F` and a formal multilinear series `p`, we say that `f` has `p` as
a power series around `x` if `f (x + y) = ∑' pₙ yⁿ` for all `y` in a neighborhood of `0`. -/
def HasFPowerSeriesAt (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (x : E) :=
∃ r, HasFPowerSeriesOnBall f p x r
#align has_fpower_series_at HasFPowerSeriesAt
variable (𝕜)
/-- Given a function `f : E → F`, we say that `f` is analytic at `x` if it admits a convergent power
series expansion around `x`. -/
def AnalyticAt (f : E → F) (x : E) :=
∃ p : FormalMultilinearSeries 𝕜 E F, HasFPowerSeriesAt f p x
#align analytic_at AnalyticAt
/-- Given a function `f : E → F`, we say that `f` is analytic on a set `s` if it is analytic around
every point of `s`. -/
def AnalyticOn (f : E → F) (s : Set E) :=
∀ x, x ∈ s → AnalyticAt 𝕜 f x
#align analytic_on AnalyticOn
variable {𝕜}
theorem HasFPowerSeriesOnBall.hasFPowerSeriesAt (hf : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesAt f p x :=
⟨r, hf⟩
#align has_fpower_series_on_ball.has_fpower_series_at HasFPowerSeriesOnBall.hasFPowerSeriesAt
theorem HasFPowerSeriesAt.analyticAt (hf : HasFPowerSeriesAt f p x) : AnalyticAt 𝕜 f x :=
⟨p, hf⟩
#align has_fpower_series_at.analytic_at HasFPowerSeriesAt.analyticAt
theorem HasFPowerSeriesOnBall.analyticAt (hf : HasFPowerSeriesOnBall f p x r) : AnalyticAt 𝕜 f x :=
hf.hasFPowerSeriesAt.analyticAt
#align has_fpower_series_on_ball.analytic_at HasFPowerSeriesOnBall.analyticAt
theorem HasFPowerSeriesOnBall.congr (hf : HasFPowerSeriesOnBall f p x r)
(hg : EqOn f g (EMetric.ball x r)) : HasFPowerSeriesOnBall g p x r :=
{ r_le := hf.r_le
r_pos := hf.r_pos
hasSum := fun {y} hy => by
convert hf.hasSum hy using 1
apply hg.symm
simpa [edist_eq_coe_nnnorm_sub] using hy }
#align has_fpower_series_on_ball.congr HasFPowerSeriesOnBall.congr
/-- If a function `f` has a power series `p` around `x`, then the function `z ↦ f (z - y)` has the
same power series around `x + y`. -/
theorem HasFPowerSeriesOnBall.comp_sub (hf : HasFPowerSeriesOnBall f p x r) (y : E) :
HasFPowerSeriesOnBall (fun z => f (z - y)) p (x + y) r :=
{ r_le := hf.r_le
r_pos := hf.r_pos
hasSum := fun {z} hz => by
convert hf.hasSum hz using 2
abel }
#align has_fpower_series_on_ball.comp_sub HasFPowerSeriesOnBall.comp_sub
theorem HasFPowerSeriesOnBall.hasSum_sub (hf : HasFPowerSeriesOnBall f p x r) {y : E}
(hy : y ∈ EMetric.ball x r) : HasSum (fun n : ℕ => p n fun _ => y - x) (f y) := by
have : y - x ∈ EMetric.ball (0 : E) r := by simpa [edist_eq_coe_nnnorm_sub] using hy
simpa only [add_sub_cancel'_right] using hf.hasSum this
#align has_fpower_series_on_ball.has_sum_sub HasFPowerSeriesOnBall.hasSum_sub
theorem HasFPowerSeriesOnBall.radius_pos (hf : HasFPowerSeriesOnBall f p x r) : 0 < p.radius :=
lt_of_lt_of_le hf.r_pos hf.r_le
#align has_fpower_series_on_ball.radius_pos HasFPowerSeriesOnBall.radius_pos
theorem HasFPowerSeriesAt.radius_pos (hf : HasFPowerSeriesAt f p x) : 0 < p.radius :=
let ⟨_, hr⟩ := hf
hr.radius_pos
#align has_fpower_series_at.radius_pos HasFPowerSeriesAt.radius_pos
theorem HasFPowerSeriesOnBall.mono (hf : HasFPowerSeriesOnBall f p x r) (r'_pos : 0 < r')
(hr : r' ≤ r) : HasFPowerSeriesOnBall f p x r' :=
⟨le_trans hr hf.1, r'_pos, fun hy => hf.hasSum (EMetric.ball_subset_ball hr hy)⟩
#align has_fpower_series_on_ball.mono HasFPowerSeriesOnBall.mono
theorem HasFPowerSeriesAt.congr (hf : HasFPowerSeriesAt f p x) (hg : f =ᶠ[𝓝 x] g) :
HasFPowerSeriesAt g p x := by
rcases hf with ⟨r₁, h₁⟩
rcases EMetric.mem_nhds_iff.mp hg with ⟨r₂, h₂pos, h₂⟩
exact ⟨min r₁ r₂,
(h₁.mono (lt_min h₁.r_pos h₂pos) inf_le_left).congr
fun y hy => h₂ (EMetric.ball_subset_ball inf_le_right hy)⟩
#align has_fpower_series_at.congr HasFPowerSeriesAt.congr
protected theorem HasFPowerSeriesAt.eventually (hf : HasFPowerSeriesAt f p x) :
∀ᶠ r : ℝ≥0∞ in 𝓝[>] 0, HasFPowerSeriesOnBall f p x r :=
let ⟨_, hr⟩ := hf
mem_of_superset (Ioo_mem_nhdsWithin_Ioi (left_mem_Ico.2 hr.r_pos)) fun _ hr' =>
hr.mono hr'.1 hr'.2.le
#align has_fpower_series_at.eventually HasFPowerSeriesAt.eventually
theorem HasFPowerSeriesOnBall.eventually_hasSum (hf : HasFPowerSeriesOnBall f p x r) :
∀ᶠ y in 𝓝 0, HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y)) := by
filter_upwards [EMetric.ball_mem_nhds (0 : E) hf.r_pos] using fun _ => hf.hasSum
#align has_fpower_series_on_ball.eventually_has_sum HasFPowerSeriesOnBall.eventually_hasSum
theorem HasFPowerSeriesAt.eventually_hasSum (hf : HasFPowerSeriesAt f p x) :
∀ᶠ y in 𝓝 0, HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y)) :=
let ⟨_, hr⟩ := hf
hr.eventually_hasSum
#align has_fpower_series_at.eventually_has_sum HasFPowerSeriesAt.eventually_hasSum
theorem HasFPowerSeriesOnBall.eventually_hasSum_sub (hf : HasFPowerSeriesOnBall f p x r) :
∀ᶠ y in 𝓝 x, HasSum (fun n : ℕ => p n fun _ : Fin n => y - x) (f y) := by
filter_upwards [EMetric.ball_mem_nhds x hf.r_pos] with y using hf.hasSum_sub
#align has_fpower_series_on_ball.eventually_has_sum_sub HasFPowerSeriesOnBall.eventually_hasSum_sub
theorem HasFPowerSeriesAt.eventually_hasSum_sub (hf : HasFPowerSeriesAt f p x) :
∀ᶠ y in 𝓝 x, HasSum (fun n : ℕ => p n fun _ : Fin n => y - x) (f y) :=
let ⟨_, hr⟩ := hf
hr.eventually_hasSum_sub
#align has_fpower_series_at.eventually_has_sum_sub HasFPowerSeriesAt.eventually_hasSum_sub
theorem HasFPowerSeriesOnBall.eventually_eq_zero
(hf : HasFPowerSeriesOnBall f (0 : FormalMultilinearSeries 𝕜 E F) x r) :
∀ᶠ z in 𝓝 x, f z = 0 := by
filter_upwards [hf.eventually_hasSum_sub] with z hz using hz.unique hasSum_zero
#align has_fpower_series_on_ball.eventually_eq_zero HasFPowerSeriesOnBall.eventually_eq_zero
theorem HasFPowerSeriesAt.eventually_eq_zero
(hf : HasFPowerSeriesAt f (0 : FormalMultilinearSeries 𝕜 E F) x) : ∀ᶠ z in 𝓝 x, f z = 0 :=
let ⟨_, hr⟩ := hf
hr.eventually_eq_zero
#align has_fpower_series_at.eventually_eq_zero HasFPowerSeriesAt.eventually_eq_zero
theorem hasFPowerSeriesOnBall_const {c : F} {e : E} :
HasFPowerSeriesOnBall (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e ⊤ := by
refine' ⟨by simp, WithTop.zero_lt_top, fun _ => hasSum_single 0 fun n hn => _⟩
simp [constFormalMultilinearSeries_apply hn]
#align has_fpower_series_on_ball_const hasFPowerSeriesOnBall_const
theorem hasFPowerSeriesAt_const {c : F} {e : E} :
HasFPowerSeriesAt (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e :=
⟨⊤, hasFPowerSeriesOnBall_const⟩
#align has_fpower_series_at_const hasFPowerSeriesAt_const
theorem analyticAt_const {v : F} : AnalyticAt 𝕜 (fun _ => v) x :=
⟨constFormalMultilinearSeries 𝕜 E v, hasFPowerSeriesAt_const⟩
#align analytic_at_const analyticAt_const
theorem analyticOn_const {v : F} {s : Set E} : AnalyticOn 𝕜 (fun _ => v) s :=
fun _ _ => analyticAt_const
#align analytic_on_const analyticOn_const
theorem HasFPowerSeriesOnBall.add (hf : HasFPowerSeriesOnBall f pf x r)
(hg : HasFPowerSeriesOnBall g pg x r) : HasFPowerSeriesOnBall (f + g) (pf + pg) x r :=
{ r_le := le_trans (le_min_iff.2 ⟨hf.r_le, hg.r_le⟩) (pf.min_radius_le_radius_add pg)
r_pos := hf.r_pos
hasSum := fun hy => (hf.hasSum hy).add (hg.hasSum hy) }
#align has_fpower_series_on_ball.add HasFPowerSeriesOnBall.add
theorem HasFPowerSeriesAt.add (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x) :
HasFPowerSeriesAt (f + g) (pf + pg) x := by
rcases (hf.eventually.and hg.eventually).exists with ⟨r, hr⟩
exact ⟨r, hr.1.add hr.2⟩
#align has_fpower_series_at.add HasFPowerSeriesAt.add
theorem AnalyticAt.congr (hf : AnalyticAt 𝕜 f x) (hg : f =ᶠ[𝓝 x] g) : AnalyticAt 𝕜 g x :=
let ⟨_, hpf⟩ := hf
(hpf.congr hg).analyticAt
theorem analyticAt_congr (h : f =ᶠ[𝓝 x] g) : AnalyticAt 𝕜 f x ↔ AnalyticAt 𝕜 g x :=
⟨fun hf ↦ hf.congr h, fun hg ↦ hg.congr h.symm⟩
theorem AnalyticAt.add (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) : AnalyticAt 𝕜 (f + g) x :=
let ⟨_, hpf⟩ := hf
let ⟨_, hqf⟩ := hg
(hpf.add hqf).analyticAt
#align analytic_at.add AnalyticAt.add
theorem HasFPowerSeriesOnBall.neg (hf : HasFPowerSeriesOnBall f pf x r) :
HasFPowerSeriesOnBall (-f) (-pf) x r :=
{ r_le := by
rw [pf.radius_neg]
exact hf.r_le
r_pos := hf.r_pos
hasSum := fun hy => (hf.hasSum hy).neg }
#align has_fpower_series_on_ball.neg HasFPowerSeriesOnBall.neg
theorem HasFPowerSeriesAt.neg (hf : HasFPowerSeriesAt f pf x) : HasFPowerSeriesAt (-f) (-pf) x :=
let ⟨_, hrf⟩ := hf
hrf.neg.hasFPowerSeriesAt
#align has_fpower_series_at.neg HasFPowerSeriesAt.neg
theorem AnalyticAt.neg (hf : AnalyticAt 𝕜 f x) : AnalyticAt 𝕜 (-f) x :=
let ⟨_, hpf⟩ := hf
hpf.neg.analyticAt
#align analytic_at.neg AnalyticAt.neg
theorem HasFPowerSeriesOnBall.sub (hf : HasFPowerSeriesOnBall f pf x r)
(hg : HasFPowerSeriesOnBall g pg x r) : HasFPowerSeriesOnBall (f - g) (pf - pg) x r := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align has_fpower_series_on_ball.sub HasFPowerSeriesOnBall.sub
theorem HasFPowerSeriesAt.sub (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x) :
HasFPowerSeriesAt (f - g) (pf - pg) x := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align has_fpower_series_at.sub HasFPowerSeriesAt.sub
theorem AnalyticAt.sub (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) :
AnalyticAt 𝕜 (f - g) x := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align analytic_at.sub AnalyticAt.sub
theorem AnalyticOn.mono {s t : Set E} (hf : AnalyticOn 𝕜 f t) (hst : s ⊆ t) : AnalyticOn 𝕜 f s :=
fun z hz => hf z (hst hz)
#align analytic_on.mono AnalyticOn.mono
theorem AnalyticOn.congr' {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : f =ᶠ[𝓝ˢ s] g) :
AnalyticOn 𝕜 g s :=
fun z hz => (hf z hz).congr (mem_nhdsSet_iff_forall.mp hg z hz)
theorem analyticOn_congr' {s : Set E} (h : f =ᶠ[𝓝ˢ s] g) : AnalyticOn 𝕜 f s ↔ AnalyticOn 𝕜 g s :=
⟨fun hf => hf.congr' h, fun hg => hg.congr' h.symm⟩
theorem AnalyticOn.congr {s : Set E} (hs : IsOpen s) (hf : AnalyticOn 𝕜 f s) (hg : s.EqOn f g) :
AnalyticOn 𝕜 g s :=
hf.congr' $ mem_nhdsSet_iff_forall.mpr
(fun _ hz => eventuallyEq_iff_exists_mem.mpr ⟨s, hs.mem_nhds hz, hg⟩)
theorem analyticOn_congr {s : Set E} (hs : IsOpen s) (h : s.EqOn f g) : AnalyticOn 𝕜 f s ↔
AnalyticOn 𝕜 g s := ⟨fun hf => hf.congr hs h, fun hg => hg.congr hs h.symm⟩
theorem AnalyticOn.add {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
AnalyticOn 𝕜 (f + g) s :=
fun z hz => (hf z hz).add (hg z hz)
#align analytic_on.add AnalyticOn.add
theorem AnalyticOn.sub {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
AnalyticOn 𝕜 (f - g) s :=
fun z hz => (hf z hz).sub (hg z hz)
#align analytic_on.sub AnalyticOn.sub
theorem HasFPowerSeriesOnBall.coeff_zero (hf : HasFPowerSeriesOnBall f pf x r) (v : Fin 0 → E) :
pf 0 v = f x := by
have v_eq : v = fun i => 0 := Subsingleton.elim _ _
have zero_mem : (0 : E) ∈ EMetric.ball (0 : E) r := by simp [hf.r_pos]
have : ∀ i, i ≠ 0 → (pf i fun j => 0) = 0 := by
intro i hi
have : 0 < i := pos_iff_ne_zero.2 hi
exact ContinuousMultilinearMap.map_coord_zero _ (⟨0, this⟩ : Fin i) rfl
have A := (hf.hasSum zero_mem).unique (hasSum_single _ this)
simpa [v_eq] using A.symm
#align has_fpower_series_on_ball.coeff_zero HasFPowerSeriesOnBall.coeff_zero
theorem HasFPowerSeriesAt.coeff_zero (hf : HasFPowerSeriesAt f pf x) (v : Fin 0 → E) :
pf 0 v = f x :=
let ⟨_, hrf⟩ := hf
hrf.coeff_zero v
#align has_fpower_series_at.coeff_zero HasFPowerSeriesAt.coeff_zero
/-- If a function `f` has a power series `p` on a ball and `g` is linear, then `g ∘ f` has the
power series `g ∘ p` on the same ball. -/
theorem ContinuousLinearMap.comp_hasFPowerSeriesOnBall (g : F →L[𝕜] G)
(h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (g ∘ f) (g.compFormalMultilinearSeries p) x r :=
{ r_le := h.r_le.trans (p.radius_le_radius_continuousLinearMap_comp _)
r_pos := h.r_pos
hasSum := fun hy => by
simpa only [ContinuousLinearMap.compFormalMultilinearSeries_apply,
ContinuousLinearMap.compContinuousMultilinearMap_coe, Function.comp_apply] using
g.hasSum (h.hasSum hy) }
#align continuous_linear_map.comp_has_fpower_series_on_ball ContinuousLinearMap.comp_hasFPowerSeriesOnBall
/-- If a function `f` is analytic on a set `s` and `g` is linear, then `g ∘ f` is analytic
on `s`. -/
theorem ContinuousLinearMap.comp_analyticOn {s : Set E} (g : F →L[𝕜] G) (h : AnalyticOn 𝕜 f s) :
AnalyticOn 𝕜 (g ∘ f) s := by
rintro x hx
rcases h x hx with ⟨p, r, hp⟩
exact ⟨g.compFormalMultilinearSeries p, r, g.comp_hasFPowerSeriesOnBall hp⟩
#align continuous_linear_map.comp_analytic_on ContinuousLinearMap.comp_analyticOn
/-- If a function admits a power series expansion, then it is exponentially close to the partial
sums of this power series on strict subdisks of the disk of convergence.
This version provides an upper estimate that decreases both in `‖y‖` and `n`. See also
`HasFPowerSeriesOnBall.uniform_geometric_approx` for a weaker version. -/
theorem HasFPowerSeriesOnBall.uniform_geometric_approx' {r' : ℝ≥0}
(hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n := by
obtain ⟨a, ha, C, hC, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n, ‖p n‖ * (r' : ℝ) ^ n ≤ C * a ^ n :=
p.norm_mul_pow_le_mul_pow_of_lt_radius (h.trans_le hf.r_le)
refine' ⟨a, ha, C / (1 - a), div_pos hC (sub_pos.2 ha.2), fun y hy n => _⟩
have yr' : ‖y‖ < r' := by
rw [ball_zero_eq] at hy
exact hy
have hr'0 : 0 < (r' : ℝ) := (norm_nonneg _).trans_lt yr'
have : y ∈ EMetric.ball (0 : E) r := by
refine' mem_emetric_ball_zero_iff.2 (lt_trans _ h)
exact mod_cast yr'
rw [norm_sub_rev, ← mul_div_right_comm]
have ya : a * (‖y‖ / ↑r') ≤ a :=
mul_le_of_le_one_right ha.1.le (div_le_one_of_le yr'.le r'.coe_nonneg)
suffices ‖p.partialSum n y - f (x + y)‖ ≤ C * (a * (‖y‖ / r')) ^ n / (1 - a * (‖y‖ / r')) by
refine' this.trans _
have : 0 < a := ha.1
gcongr
apply_rules [sub_pos.2, ha.2]
apply norm_sub_le_of_geometric_bound_of_hasSum (ya.trans_lt ha.2) _ (hf.hasSum this)
intro n
calc
‖(p n) fun _ : Fin n => y‖
_ ≤ ‖p n‖ * ∏ _i : Fin n, ‖y‖ := ContinuousMultilinearMap.le_op_norm _ _
_ = ‖p n‖ * (r' : ℝ) ^ n * (‖y‖ / r') ^ n := by field_simp [mul_right_comm]
_ ≤ C * a ^ n * (‖y‖ / r') ^ n := by gcongr ?_ * _; apply hp
_ ≤ C * (a * (‖y‖ / r')) ^ n := by rw [mul_pow, mul_assoc]
#align has_fpower_series_on_ball.uniform_geometric_approx' HasFPowerSeriesOnBall.uniform_geometric_approx'
/-- If a function admits a power series expansion, then it is exponentially close to the partial
sums of this power series on strict subdisks of the disk of convergence. -/
theorem HasFPowerSeriesOnBall.uniform_geometric_approx {r' : ℝ≥0}
(hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
∃ a ∈ Ioo (0 : ℝ) 1,
∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n, ‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n := by
obtain ⟨a, ha, C, hC, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n :=
hf.uniform_geometric_approx' h
refine' ⟨a, ha, C, hC, fun y hy n => (hp y hy n).trans _⟩
have yr' : ‖y‖ < r' := by rwa [ball_zero_eq] at hy
gcongr
exacts [mul_nonneg ha.1.le (div_nonneg (norm_nonneg y) r'.coe_nonneg),
mul_le_of_le_one_right ha.1.le (div_le_one_of_le yr'.le r'.coe_nonneg)]
#align has_fpower_series_on_ball.uniform_geometric_approx HasFPowerSeriesOnBall.uniform_geometric_approx
/-- Taylor formula for an analytic function, `IsBigO` version. -/
theorem HasFPowerSeriesAt.isBigO_sub_partialSum_pow (hf : HasFPowerSeriesAt f p x) (n : ℕ) :
(fun y : E => f (x + y) - p.partialSum n y) =O[𝓝 0] fun y => ‖y‖ ^ n := by
rcases hf with ⟨r, hf⟩
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hf.r_pos with ⟨r', r'0, h⟩
obtain ⟨a, -, C, -, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n :=
hf.uniform_geometric_approx' h
refine' isBigO_iff.2 ⟨C * (a / r') ^ n, _⟩
replace r'0 : 0 < (r' : ℝ); · exact mod_cast r'0
filter_upwards [Metric.ball_mem_nhds (0 : E) r'0] with y hy
simpa [mul_pow, mul_div_assoc, mul_assoc, div_mul_eq_mul_div] using hp y hy n
set_option linter.uppercaseLean3 false in
#align has_fpower_series_at.is_O_sub_partial_sum_pow HasFPowerSeriesAt.isBigO_sub_partialSum_pow
/-- If `f` has formal power series `∑ n, pₙ` on a ball of radius `r`, then for `y, z` in any smaller
ball, the norm of the difference `f y - f z - p 1 (fun _ ↦ y - z)` is bounded above by
`C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖`. This lemma formulates this property using `IsBigO` and
`Filter.principal` on `E × E`. -/
theorem HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal
(hf : HasFPowerSeriesOnBall f p x r) (hr : r' < r) :
(fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) =O[𝓟 (EMetric.ball (x, x) r')]
fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖ := by
lift r' to ℝ≥0 using ne_top_of_lt hr
rcases (zero_le r').eq_or_lt with (rfl | hr'0)
· simp only [isBigO_bot, EMetric.ball_zero, principal_empty, ENNReal.coe_zero]
obtain ⟨a, ha, C, hC : 0 < C, hp⟩ :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n : ℕ, ‖p n‖ * (r' : ℝ) ^ n ≤ C * a ^ n
exact p.norm_mul_pow_le_mul_pow_of_lt_radius (hr.trans_le hf.r_le)
simp only [← le_div_iff (pow_pos (NNReal.coe_pos.2 hr'0) _)] at hp
set L : E × E → ℝ := fun y =>
C * (a / r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * (a / (1 - a) ^ 2 + 2 / (1 - a))
have hL : ∀ y ∈ EMetric.ball (x, x) r', ‖f y.1 - f y.2 - p 1 fun _ => y.1 - y.2‖ ≤ L y := by
intro y hy'
have hy : y ∈ EMetric.ball x r ×ˢ EMetric.ball x r := by
rw [EMetric.ball_prod_same]
exact EMetric.ball_subset_ball hr.le hy'
set A : ℕ → F := fun n => (p n fun _ => y.1 - x) - p n fun _ => y.2 - x
have hA : HasSum (fun n => A (n + 2)) (f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) := by
convert (hasSum_nat_add_iff' 2).2 ((hf.hasSum_sub hy.1).sub (hf.hasSum_sub hy.2)) using 1
rw [Finset.sum_range_succ, Finset.sum_range_one, hf.coeff_zero, hf.coeff_zero, sub_self,
zero_add, ← Subsingleton.pi_single_eq (0 : Fin 1) (y.1 - x), Pi.single,
← Subsingleton.pi_single_eq (0 : Fin 1) (y.2 - x), Pi.single, ← (p 1).map_sub, ← Pi.single,
Subsingleton.pi_single_eq, sub_sub_sub_cancel_right]
rw [EMetric.mem_ball, edist_eq_coe_nnnorm_sub, ENNReal.coe_lt_coe] at hy'
set B : ℕ → ℝ := fun n => C * (a / r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * ((n + 2) * a ^ n)
have hAB : ∀ n, ‖A (n + 2)‖ ≤ B n := fun n =>
calc
‖A (n + 2)‖ ≤ ‖p (n + 2)‖ * ↑(n + 2) * ‖y - (x, x)‖ ^ (n + 1) * ‖y.1 - y.2‖ := by
-- porting note: `pi_norm_const` was `pi_norm_const (_ : E)`
simpa only [Fintype.card_fin, pi_norm_const, Prod.norm_def, Pi.sub_def,
Prod.fst_sub, Prod.snd_sub, sub_sub_sub_cancel_right] using
(p <| n + 2).norm_image_sub_le (fun _ => y.1 - x) fun _ => y.2 - x
_ = ‖p (n + 2)‖ * ‖y - (x, x)‖ ^ n * (↑(n + 2) * ‖y - (x, x)‖ * ‖y.1 - y.2‖) := by
rw [pow_succ ‖y - (x, x)‖]
ring
-- porting note: the two `↑` in `↑r'` are new, without them, Lean fails to synthesize
-- instances `HDiv ℝ ℝ≥0 ?m` or `HMul ℝ ℝ≥0 ?m`
_ ≤ C * a ^ (n + 2) / ↑r' ^ (n + 2)
* ↑r' ^ n * (↑(n + 2) * ‖y - (x, x)‖ * ‖y.1 - y.2‖) := by
have : 0 < a := ha.1
gcongr
· apply hp
· apply hy'.le
_ = B n := by
-- porting note: in the original, `B` was in the `field_simp`, but now Lean does not
-- accept it. The current proof works in Lean 4, but does not in Lean 3.
field_simp [pow_succ]
simp only [mul_assoc, mul_comm, mul_left_comm]
have hBL : HasSum B (L y) := by
apply HasSum.mul_left
simp only [add_mul]
have : ‖a‖ < 1 := by simp only [Real.norm_eq_abs, abs_of_pos ha.1, ha.2]
rw [div_eq_mul_inv, div_eq_mul_inv]
exact (hasSum_coe_mul_geometric_of_norm_lt_1 this).add -- porting note: was `convert`!
((hasSum_geometric_of_norm_lt_1 this).mul_left 2)
exact hA.norm_le_of_bounded hBL hAB
suffices L =O[𝓟 (EMetric.ball (x, x) r')] fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖ by
refine' (IsBigO.of_bound 1 (eventually_principal.2 fun y hy => _)).trans this
rw [one_mul]
exact (hL y hy).trans (le_abs_self _)
simp_rw [mul_right_comm _ (_ * _)] -- porting note: there was an `L` inside the `simp_rw`.
exact (isBigO_refl _ _).const_mul_left _
set_option linter.uppercaseLean3 false in
#align has_fpower_series_on_ball.is_O_image_sub_image_sub_deriv_principal HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal
/-- If `f` has formal power series `∑ n, pₙ` on a ball of radius `r`, then for `y, z` in any smaller
ball, the norm of the difference `f y - f z - p 1 (fun _ ↦ y - z)` is bounded above by
`C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖`. -/
theorem HasFPowerSeriesOnBall.image_sub_sub_deriv_le (hf : HasFPowerSeriesOnBall f p x r)
(hr : r' < r) :
∃ C, ∀ᵉ (y ∈ EMetric.ball x r') (z ∈ EMetric.ball x r'),
‖f y - f z - p 1 fun _ => y - z‖ ≤ C * max ‖y - x‖ ‖z - x‖ * ‖y - z‖ := by
simpa only [isBigO_principal, mul_assoc, norm_mul, norm_norm, Prod.forall, EMetric.mem_ball,
Prod.edist_eq, max_lt_iff, and_imp, @forall_swap (_ < _) E] using
hf.isBigO_image_sub_image_sub_deriv_principal hr
#align has_fpower_series_on_ball.image_sub_sub_deriv_le HasFPowerSeriesOnBall.image_sub_sub_deriv_le
/-- If `f` has formal power series `∑ n, pₙ` at `x`, then
`f y - f z - p 1 (fun _ ↦ y - z) = O(‖(y, z) - (x, x)‖ * ‖y - z‖)` as `(y, z) → (x, x)`.
In particular, `f` is strictly differentiable at `x`. -/
theorem HasFPowerSeriesAt.isBigO_image_sub_norm_mul_norm_sub (hf : HasFPowerSeriesAt f p x) :
(fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) =O[𝓝 (x, x)] fun y =>
‖y - (x, x)‖ * ‖y.1 - y.2‖ := by
rcases hf with ⟨r, hf⟩
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hf.r_pos with ⟨r', r'0, h⟩
refine' (hf.isBigO_image_sub_image_sub_deriv_principal h).mono _
exact le_principal_iff.2 (EMetric.ball_mem_nhds _ r'0)
set_option linter.uppercaseLean3 false in
#align has_fpower_series_at.is_O_image_sub_norm_mul_norm_sub HasFPowerSeriesAt.isBigO_image_sub_norm_mul_norm_sub
/-- If a function admits a power series expansion at `x`, then it is the uniform limit of the
partial sums of this power series on strict subdisks of the disk of convergence, i.e., `f (x + y)`
is the uniform limit of `p.partialSum n y` there. -/
theorem HasFPowerSeriesOnBall.tendstoUniformlyOn {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r)
(h : (r' : ℝ≥0∞) < r) :
TendstoUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
(Metric.ball (0 : E) r') := by
obtain ⟨a, ha, C, -, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n
exact hf.uniform_geometric_approx h
refine' Metric.tendstoUniformlyOn_iff.2 fun ε εpos => _
have L : Tendsto (fun n => (C : ℝ) * a ^ n) atTop (𝓝 ((C : ℝ) * 0)) :=
tendsto_const_nhds.mul (tendsto_pow_atTop_nhds_0_of_lt_1 ha.1.le ha.2)
rw [mul_zero] at L
refine' (L.eventually (gt_mem_nhds εpos)).mono fun n hn y hy => _
rw [dist_eq_norm]
exact (hp y hy n).trans_lt hn
#align has_fpower_series_on_ball.tendsto_uniformly_on HasFPowerSeriesOnBall.tendstoUniformlyOn
/-- If a function admits a power series expansion at `x`, then it is the locally uniform limit of
the partial sums of this power series on the disk of convergence, i.e., `f (x + y)`
is the locally uniform limit of `p.partialSum n y` there. -/
theorem HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn (hf : HasFPowerSeriesOnBall f p x r) :
TendstoLocallyUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
(EMetric.ball (0 : E) r) := by
intro u hu x hx
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hx with ⟨r', xr', hr'⟩
have : EMetric.ball (0 : E) r' ∈ 𝓝 x := IsOpen.mem_nhds EMetric.isOpen_ball xr'
refine' ⟨EMetric.ball (0 : E) r', mem_nhdsWithin_of_mem_nhds this, _⟩
simpa [Metric.emetric_ball_nnreal] using hf.tendstoUniformlyOn hr' u hu
#align has_fpower_series_on_ball.tendsto_locally_uniformly_on HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn
/-- If a function admits a power series expansion at `x`, then it is the uniform limit of the
partial sums of this power series on strict subdisks of the disk of convergence, i.e., `f y`
is the uniform limit of `p.partialSum n (y - x)` there. -/
theorem HasFPowerSeriesOnBall.tendstoUniformlyOn' {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r)
(h : (r' : ℝ≥0∞) < r) :
TendstoUniformlyOn (fun n y => p.partialSum n (y - x)) f atTop (Metric.ball (x : E) r') := by
convert (hf.tendstoUniformlyOn h).comp fun y => y - x using 1
· simp [(· ∘ ·)]
· ext z
simp [dist_eq_norm]
#align has_fpower_series_on_ball.tendsto_uniformly_on' HasFPowerSeriesOnBall.tendstoUniformlyOn'
/-- If a function admits a power series expansion at `x`, then it is the locally uniform limit of
the partial sums of this power series on the disk of convergence, i.e., `f y`
is the locally uniform limit of `p.partialSum n (y - x)` there. -/
theorem HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn' (hf : HasFPowerSeriesOnBall f p x r) :
TendstoLocallyUniformlyOn (fun n y => p.partialSum n (y - x)) f atTop
(EMetric.ball (x : E) r) := by
have A : ContinuousOn (fun y : E => y - x) (EMetric.ball (x : E) r) :=
(continuous_id.sub continuous_const).continuousOn
convert hf.tendstoLocallyUniformlyOn.comp (fun y : E => y - x) _ A using 1
· ext z
simp
· intro z
simp [edist_eq_coe_nnnorm, edist_eq_coe_nnnorm_sub]
#align has_fpower_series_on_ball.tendsto_locally_uniformly_on' HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn'
/-- If a function admits a power series expansion on a disk, then it is continuous there. -/
protected theorem HasFPowerSeriesOnBall.continuousOn (hf : HasFPowerSeriesOnBall f p x r) :
ContinuousOn f (EMetric.ball x r) :=
hf.tendstoLocallyUniformlyOn'.continuousOn <|
eventually_of_forall fun n =>
((p.partialSum_continuous n).comp (continuous_id.sub continuous_const)).continuousOn
#align has_fpower_series_on_ball.continuous_on HasFPowerSeriesOnBall.continuousOn
protected theorem HasFPowerSeriesAt.continuousAt (hf : HasFPowerSeriesAt f p x) :
ContinuousAt f x :=
let ⟨_, hr⟩ := hf
hr.continuousOn.continuousAt (EMetric.ball_mem_nhds x hr.r_pos)
#align has_fpower_series_at.continuous_at HasFPowerSeriesAt.continuousAt
protected theorem AnalyticAt.continuousAt (hf : AnalyticAt 𝕜 f x) : ContinuousAt f x :=
let ⟨_, hp⟩ := hf
hp.continuousAt
#align analytic_at.continuous_at AnalyticAt.continuousAt
protected theorem AnalyticOn.continuousOn {s : Set E} (hf : AnalyticOn 𝕜 f s) : ContinuousOn f s :=
fun x hx => (hf x hx).continuousAt.continuousWithinAt
#align analytic_on.continuous_on AnalyticOn.continuousOn
/-- Analytic everywhere implies continuous -/
theorem AnalyticOn.continuous {f : E → F} (fa : AnalyticOn 𝕜 f univ) : Continuous f := by
rw [continuous_iff_continuousOn_univ]; exact fa.continuousOn
/-- In a complete space, the sum of a converging power series `p` admits `p` as a power series.
This is not totally obvious as we need to check the convergence of the series. -/
protected theorem FormalMultilinearSeries.hasFPowerSeriesOnBall [CompleteSpace F]
(p : FormalMultilinearSeries 𝕜 E F) (h : 0 < p.radius) :
HasFPowerSeriesOnBall p.sum p 0 p.radius :=
{ r_le := le_rfl
r_pos := h
hasSum := fun hy => by
rw [zero_add]
exact p.hasSum hy }
#align formal_multilinear_series.has_fpower_series_on_ball FormalMultilinearSeries.hasFPowerSeriesOnBall
theorem HasFPowerSeriesOnBall.sum (h : HasFPowerSeriesOnBall f p x r) {y : E}
(hy : y ∈ EMetric.ball (0 : E) r) : f (x + y) = p.sum y :=
(h.hasSum hy).tsum_eq.symm
#align has_fpower_series_on_ball.sum HasFPowerSeriesOnBall.sum
/-- The sum of a converging power series is continuous in its disk of convergence. -/
protected theorem FormalMultilinearSeries.continuousOn [CompleteSpace F] :
ContinuousOn p.sum (EMetric.ball 0 p.radius) := by
rcases (zero_le p.radius).eq_or_lt with h | h
· simp [← h, continuousOn_empty]
· exact (p.hasFPowerSeriesOnBall h).continuousOn
#align formal_multilinear_series.continuous_on FormalMultilinearSeries.continuousOn
end
/-!
### Uniqueness of power series
If a function `f : E → F` has two representations as power series at a point `x : E`, corresponding
to formal multilinear series `p₁` and `p₂`, then these representations agree term-by-term. That is,
for any `n : ℕ` and `y : E`, `p₁ n (fun i ↦ y) = p₂ n (fun i ↦ y)`. In the one-dimensional case,
when `f : 𝕜 → E`, the continuous multilinear maps `p₁ n` and `p₂ n` are given by
`ContinuousMultilinearMap.mkPiField`, and hence are determined completely by the value of
`p₁ n (fun i ↦ 1)`, so `p₁ = p₂`. Consequently, the radius of convergence for one series can be
transferred to the other.
-/
section Uniqueness
open ContinuousMultilinearMap
theorem Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero {n : ℕ} {p : E[×n]→L[𝕜] F}
(h : (fun y => p fun _ => y) =O[𝓝 0] fun y => ‖y‖ ^ (n + 1)) (y : E) : (p fun _ => y) = 0 := by
obtain ⟨c, c_pos, hc⟩ := h.exists_pos
obtain ⟨t, ht, t_open, z_mem⟩ := eventually_nhds_iff.mp (isBigOWith_iff.mp hc)
obtain ⟨δ, δ_pos, δε⟩ := (Metric.isOpen_iff.mp t_open) 0 z_mem
clear h hc z_mem
cases' n with n
· exact norm_eq_zero.mp (by
-- porting note: the symmetric difference of the `simpa only` sets:
-- added `Nat.zero_eq, zero_add, pow_one`
-- removed `zero_pow', Ne.def, Nat.one_ne_zero, not_false_iff`
simpa only [Nat.zero_eq, fin0_apply_norm, norm_eq_zero, norm_zero, zero_add, pow_one,
mul_zero, norm_le_zero_iff] using ht 0 (δε (Metric.mem_ball_self δ_pos)))
· refine' Or.elim (Classical.em (y = 0))
(fun hy => by simpa only [hy] using p.map_zero) fun hy => _
replace hy := norm_pos_iff.mpr hy
refine' norm_eq_zero.mp (le_antisymm (le_of_forall_pos_le_add fun ε ε_pos => _) (norm_nonneg _))
have h₀ := _root_.mul_pos c_pos (pow_pos hy (n.succ + 1))
obtain ⟨k, k_pos, k_norm⟩ := NormedField.exists_norm_lt 𝕜
(lt_min (mul_pos δ_pos (inv_pos.mpr hy)) (mul_pos ε_pos (inv_pos.mpr h₀)))
have h₁ : ‖k • y‖ < δ := by
rw [norm_smul]
exact inv_mul_cancel_right₀ hy.ne.symm δ ▸
mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_left _ _)) hy
have h₂ :=
calc
‖p fun _ => k • y‖ ≤ c * ‖k • y‖ ^ (n.succ + 1) := by
-- porting note: now Lean wants `_root_.`
simpa only [norm_pow, _root_.norm_norm] using ht (k • y) (δε (mem_ball_zero_iff.mpr h₁))
--simpa only [norm_pow, norm_norm] using ht (k • y) (δε (mem_ball_zero_iff.mpr h₁))
_ = ‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))) := by
-- porting note: added `Nat.succ_eq_add_one` since otherwise `ring` does not conclude.
simp only [norm_smul, mul_pow, Nat.succ_eq_add_one]
-- porting note: removed `rw [pow_succ]`, since it now becomes superfluous.
ring
have h₃ : ‖k‖ * (c * ‖y‖ ^ (n.succ + 1)) < ε :=
inv_mul_cancel_right₀ h₀.ne.symm ε ▸
mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_right _ _)) h₀
calc
‖p fun _ => y‖ = ‖k⁻¹ ^ n.succ‖ * ‖p fun _ => k • y‖ := by
simpa only [inv_smul_smul₀ (norm_pos_iff.mp k_pos), norm_smul, Finset.prod_const,
Finset.card_fin] using
congr_arg norm (p.map_smul_univ (fun _ : Fin n.succ => k⁻¹) fun _ : Fin n.succ => k • y)
_ ≤ ‖k⁻¹ ^ n.succ‖ * (‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1)))) := by gcongr
_ = ‖(k⁻¹ * k) ^ n.succ‖ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))) := by
rw [← mul_assoc]
simp [norm_mul, mul_pow]
_ ≤ 0 + ε := by
rw [inv_mul_cancel (norm_pos_iff.mp k_pos)]
simpa using h₃.le
set_option linter.uppercaseLean3 false in
#align asymptotics.is_O.continuous_multilinear_map_apply_eq_zero Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero
/-- If a formal multilinear series `p` represents the zero function at `x : E`, then the
terms `p n (fun i ↦ y)` appearing in the sum are zero for any `n : ℕ`, `y : E`. -/
theorem HasFPowerSeriesAt.apply_eq_zero {p : FormalMultilinearSeries 𝕜 E F} {x : E}
(h : HasFPowerSeriesAt 0 p x) (n : ℕ) : ∀ y : E, (p n fun _ => y) = 0 := by
refine' Nat.strong_induction_on n fun k hk => _
have psum_eq : p.partialSum (k + 1) = fun y => p k fun _ => y := by
funext z
refine' Finset.sum_eq_single _ (fun b hb hnb => _) fun hn => _
· have := Finset.mem_range_succ_iff.mp hb
simp only [hk b (this.lt_of_ne hnb), Pi.zero_apply]
· exact False.elim (hn (Finset.mem_range.mpr (lt_add_one k)))
replace h := h.isBigO_sub_partialSum_pow k.succ
simp only [psum_eq, zero_sub, Pi.zero_apply, Asymptotics.isBigO_neg_left] at h
exact h.continuousMultilinearMap_apply_eq_zero
#align has_fpower_series_at.apply_eq_zero HasFPowerSeriesAt.apply_eq_zero
/-- A one-dimensional formal multilinear series representing the zero function is zero. -/
theorem HasFPowerSeriesAt.eq_zero {p : FormalMultilinearSeries 𝕜 𝕜 E} {x : 𝕜}
(h : HasFPowerSeriesAt 0 p x) : p = 0 := by
-- porting note: `funext; ext` was `ext (n x)`
funext n
ext x
rw [← mkPiField_apply_one_eq_self (p n)]
-- porting note: nasty hack, was `simp [h.apply_eq_zero n 1]`
have := Or.intro_right ?_ (h.apply_eq_zero n 1)
simpa using this
#align has_fpower_series_at.eq_zero HasFPowerSeriesAt.eq_zero
/-- One-dimensional formal multilinear series representing the same function are equal. -/
theorem HasFPowerSeriesAt.eq_formalMultilinearSeries {p₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E}
{f : 𝕜 → E} {x : 𝕜} (h₁ : HasFPowerSeriesAt f p₁ x) (h₂ : HasFPowerSeriesAt f p₂ x) : p₁ = p₂ :=
sub_eq_zero.mp (HasFPowerSeriesAt.eq_zero (by simpa only [sub_self] using h₁.sub h₂))
#align has_fpower_series_at.eq_formal_multilinear_series HasFPowerSeriesAt.eq_formalMultilinearSeries
theorem HasFPowerSeriesAt.eq_formalMultilinearSeries_of_eventually
{p q : FormalMultilinearSeries 𝕜 𝕜 E} {f g : 𝕜 → E} {x : 𝕜} (hp : HasFPowerSeriesAt f p x)
(hq : HasFPowerSeriesAt g q x) (heq : ∀ᶠ z in 𝓝 x, f z = g z) : p = q :=
(hp.congr heq).eq_formalMultilinearSeries hq
#align has_fpower_series_at.eq_formal_multilinear_series_of_eventually HasFPowerSeriesAt.eq_formalMultilinearSeries_of_eventually
/-- A one-dimensional formal multilinear series representing a locally zero function is zero. -/
theorem HasFPowerSeriesAt.eq_zero_of_eventually {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E}
{x : 𝕜} (hp : HasFPowerSeriesAt f p x) (hf : f =ᶠ[𝓝 x] 0) : p = 0 :=
(hp.congr hf).eq_zero
#align has_fpower_series_at.eq_zero_of_eventually HasFPowerSeriesAt.eq_zero_of_eventually
/-- If a function `f : 𝕜 → E` has two power series representations at `x`, then the given radii in
which convergence is guaranteed may be interchanged. This can be useful when the formal multilinear
series in one representation has a particularly nice form, but the other has a larger radius. -/
theorem HasFPowerSeriesOnBall.exchange_radius {p₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E}
{r₁ r₂ : ℝ≥0∞} {x : 𝕜} (h₁ : HasFPowerSeriesOnBall f p₁ x r₁)
(h₂ : HasFPowerSeriesOnBall f p₂ x r₂) : HasFPowerSeriesOnBall f p₁ x r₂ :=
h₂.hasFPowerSeriesAt.eq_formalMultilinearSeries h₁.hasFPowerSeriesAt ▸ h₂
#align has_fpower_series_on_ball.exchange_radius HasFPowerSeriesOnBall.exchange_radius
/-- If a function `f : 𝕜 → E` has power series representation `p` on a ball of some radius and for
each positive radius it has some power series representation, then `p` converges to `f` on the whole
`𝕜`. -/
theorem HasFPowerSeriesOnBall.r_eq_top_of_exists {f : 𝕜 → E} {r : ℝ≥0∞} {x : 𝕜}
{p : FormalMultilinearSeries 𝕜 𝕜 E} (h : HasFPowerSeriesOnBall f p x r)
(h' : ∀ (r' : ℝ≥0) (_ : 0 < r'), ∃ p' : FormalMultilinearSeries 𝕜 𝕜 E,
HasFPowerSeriesOnBall f p' x r') :
HasFPowerSeriesOnBall f p x ∞ :=
{ r_le := ENNReal.le_of_forall_pos_nnreal_lt fun r hr _ =>
let ⟨_, hp'⟩ := h' r hr
(h.exchange_radius hp').r_le
r_pos := ENNReal.coe_lt_top
hasSum := fun {y} _ =>
let ⟨r', hr'⟩ := exists_gt ‖y‖₊
let ⟨_, hp'⟩ := h' r' hr'.ne_bot.bot_lt
(h.exchange_radius hp').hasSum <| mem_emetric_ball_zero_iff.mpr (ENNReal.coe_lt_coe.2 hr') }
#align has_fpower_series_on_ball.r_eq_top_of_exists HasFPowerSeriesOnBall.r_eq_top_of_exists
end Uniqueness
/-!
### Changing origin in a power series
If a function is analytic in a disk `D(x, R)`, then it is analytic in any disk contained in that
one. Indeed, one can write
$$
f (x + y + z) = \sum_{n} p_n (y + z)^n = \sum_{n, k} \binom{n}{k} p_n y^{n-k} z^k
= \sum_{k} \Bigl(\sum_{n} \binom{n}{k} p_n y^{n-k}\Bigr) z^k.
$$
The corresponding power series has thus a `k`-th coefficient equal to
$\sum_{n} \binom{n}{k} p_n y^{n-k}$. In the general case where `pₙ` is a multilinear map, this has
to be interpreted suitably: instead of having a binomial coefficient, one should sum over all
possible subsets `s` of `Fin n` of cardinal `k`, and attribute `z` to the indices in `s` and
`y` to the indices outside of `s`.
In this paragraph, we implement this. The new power series is called `p.changeOrigin y`. Then, we
check its convergence and the fact that its sum coincides with the original sum. The outcome of this
discussion is that the set of points where a function is analytic is open.
-/
namespace FormalMultilinearSeries
section
variable (p : FormalMultilinearSeries 𝕜 E F) {x y : E} {r R : ℝ≥0}
/-- A term of `FormalMultilinearSeries.changeOriginSeries`.
Given a formal multilinear series `p` and a point `x` in its ball of convergence,
`p.changeOrigin x` is a formal multilinear series such that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense. Each term of `p.changeOrigin x`
is itself an analytic function of `x` given by the series `p.changeOriginSeries`. Each term in
`changeOriginSeries` is the sum of `changeOriginSeriesTerm`'s over all `s` of cardinality `l`.
The definition is such that `p.changeOriginSeriesTerm k l s hs (fun _ ↦ x) (fun _ ↦ y) =
p (k + l) (s.piecewise (fun _ ↦ x) (fun _ ↦ y))`
-/
def changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
E[×l]→L[𝕜] E[×k]→L[𝕜] F := by
let a := ContinuousMultilinearMap.curryFinFinset 𝕜 E F hs
(by erw [Finset.card_compl, Fintype.card_fin, hs, add_tsub_cancel_right])
exact a (p (k + l))
#align formal_multilinear_series.change_origin_series_term FormalMultilinearSeries.changeOriginSeriesTerm
theorem changeOriginSeriesTerm_apply (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l)
(x y : E) :
(p.changeOriginSeriesTerm k l s hs (fun _ => x) fun _ => y) =
p (k + l) (s.piecewise (fun _ => x) fun _ => y) :=
ContinuousMultilinearMap.curryFinFinset_apply_const _ _ _ _ _
#align formal_multilinear_series.change_origin_series_term_apply FormalMultilinearSeries.changeOriginSeriesTerm_apply
@[simp]
theorem norm_changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
‖p.changeOriginSeriesTerm k l s hs‖ = ‖p (k + l)‖ := by
simp only [changeOriginSeriesTerm, LinearIsometryEquiv.norm_map]
#align formal_multilinear_series.norm_change_origin_series_term FormalMultilinearSeries.norm_changeOriginSeriesTerm
@[simp]
theorem nnnorm_changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
‖p.changeOriginSeriesTerm k l s hs‖₊ = ‖p (k + l)‖₊ := by
simp only [changeOriginSeriesTerm, LinearIsometryEquiv.nnnorm_map]
#align formal_multilinear_series.nnnorm_change_origin_series_term FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm
theorem nnnorm_changeOriginSeriesTerm_apply_le (k l : ℕ) (s : Finset (Fin (k + l)))
(hs : s.card = l) (x y : E) :
‖p.changeOriginSeriesTerm k l s hs (fun _ => x) fun _ => y‖₊ ≤
‖p (k + l)‖₊ * ‖x‖₊ ^ l * ‖y‖₊ ^ k := by
rw [← p.nnnorm_changeOriginSeriesTerm k l s hs, ← Fin.prod_const, ← Fin.prod_const]
apply ContinuousMultilinearMap.le_of_op_nnnorm_le
apply ContinuousMultilinearMap.le_op_nnnorm
#align formal_multilinear_series.nnnorm_change_origin_series_term_apply_le FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm_apply_le
/-- The power series for `f.changeOrigin k`.
Given a formal multilinear series `p` and a point `x` in its ball of convergence,
`p.changeOrigin x` is a formal multilinear series such that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense. Its `k`-th term is the sum of
the series `p.changeOriginSeries k`. -/
def changeOriginSeries (k : ℕ) : FormalMultilinearSeries 𝕜 E (E[×k]→L[𝕜] F) := fun l =>
∑ s : { s : Finset (Fin (k + l)) // Finset.card s = l }, p.changeOriginSeriesTerm k l s s.2
#align formal_multilinear_series.change_origin_series FormalMultilinearSeries.changeOriginSeries
theorem nnnorm_changeOriginSeries_le_tsum (k l : ℕ) :
‖p.changeOriginSeries k l‖₊ ≤
∑' _ : { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + l)‖₊ :=
(nnnorm_sum_le _ (fun t => changeOriginSeriesTerm p k l (Subtype.val t) t.prop)).trans_eq <| by
simp_rw [tsum_fintype, nnnorm_changeOriginSeriesTerm (p := p) (k := k) (l := l)]
#align formal_multilinear_series.nnnorm_change_origin_series_le_tsum FormalMultilinearSeries.nnnorm_changeOriginSeries_le_tsum
theorem nnnorm_changeOriginSeries_apply_le_tsum (k l : ℕ) (x : E) :
‖p.changeOriginSeries k l fun _ => x‖₊ ≤
∑' _ : { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + l)‖₊ * ‖x‖₊ ^ l := by
rw [NNReal.tsum_mul_right, ← Fin.prod_const]
exact (p.changeOriginSeries k l).le_of_op_nnnorm_le _ (p.nnnorm_changeOriginSeries_le_tsum _ _)
#align formal_multilinear_series.nnnorm_change_origin_series_apply_le_tsum FormalMultilinearSeries.nnnorm_changeOriginSeries_apply_le_tsum
/-- Changing the origin of a formal multilinear series `p`, so that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense.
-/
def changeOrigin (x : E) : FormalMultilinearSeries 𝕜 E F :=
fun k => (p.changeOriginSeries k).sum x
#align formal_multilinear_series.change_origin FormalMultilinearSeries.changeOrigin
/-- An auxiliary equivalence useful in the proofs about
`FormalMultilinearSeries.changeOriginSeries`: the set of triples `(k, l, s)`, where `s` is a
`Finset (Fin (k + l))` of cardinality `l` is equivalent to the set of pairs `(n, s)`, where `s` is a
`Finset (Fin n)`.
The forward map sends `(k, l, s)` to `(k + l, s)` and the inverse map sends `(n, s)` to
`(n - Finset.card s, Finset.card s, s)`. The actual definition is less readable because of problems
with non-definitional equalities. -/
@[simps]
def changeOriginIndexEquiv :
(Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) ≃ Σn : ℕ, Finset (Fin n) where
toFun s := ⟨s.1 + s.2.1, s.2.2⟩
invFun s :=
⟨s.1 - s.2.card, s.2.card,
⟨s.2.map
(Fin.castIso <| (tsub_add_cancel_of_le <| card_finset_fin_le s.2).symm).toEquiv.toEmbedding,
Finset.card_map _⟩⟩
left_inv := by
rintro ⟨k, l, ⟨s : Finset (Fin <| k + l), hs : s.card = l⟩⟩
dsimp only [Subtype.coe_mk]
-- Lean can't automatically generalize `k' = k + l - s.card`, `l' = s.card`, so we explicitly
-- formulate the generalized goal
suffices ∀ k' l', k' = k → l' = l → ∀ (hkl : k + l = k' + l') (hs'),
(⟨k', l', ⟨Finset.map (Fin.castIso hkl).toEquiv.toEmbedding s, hs'⟩⟩ :
Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) = ⟨k, l, ⟨s, hs⟩⟩ by
apply this <;> simp only [hs, add_tsub_cancel_right]
rintro _ _ rfl rfl hkl hs'
simp only [Equiv.refl_toEmbedding, Fin.castIso_refl, Finset.map_refl, eq_self_iff_true,
OrderIso.refl_toEquiv, and_self_iff, heq_iff_eq]
right_inv := by
rintro ⟨n, s⟩
simp [tsub_add_cancel_of_le (card_finset_fin_le s), Fin.castIso_to_equiv]
#align formal_multilinear_series.change_origin_index_equiv FormalMultilinearSeries.changeOriginIndexEquiv
theorem changeOriginSeries_summable_aux₁ {r r' : ℝ≥0} (hr : (r + r' : ℝ≥0∞) < p.radius) :
Summable fun s : Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l } =>
‖p (s.1 + s.2.1)‖₊ * r ^ s.2.1 * r' ^ s.1 := by
rw [← changeOriginIndexEquiv.symm.summable_iff]
dsimp only [Function.comp_def, changeOriginIndexEquiv_symm_apply_fst,
changeOriginIndexEquiv_symm_apply_snd_fst]
have : ∀ n : ℕ,
HasSum (fun s : Finset (Fin n) => ‖p (n - s.card + s.card)‖₊ * r ^ s.card * r' ^ (n - s.card))
(‖p n‖₊ * (r + r') ^ n) := by
intro n
-- TODO: why `simp only [tsub_add_cancel_of_le (card_finset_fin_le _)]` fails?
convert_to HasSum (fun s : Finset (Fin n) => ‖p n‖₊ * (r ^ s.card * r' ^ (n - s.card))) _
· ext1 s
rw [tsub_add_cancel_of_le (card_finset_fin_le _), mul_assoc]
rw [← Fin.sum_pow_mul_eq_add_pow]
exact (hasSum_fintype _).mul_left _
refine' NNReal.summable_sigma.2 ⟨fun n => (this n).summable, _⟩
simp only [(this _).tsum_eq]
exact p.summable_nnnorm_mul_pow hr
#align formal_multilinear_series.change_origin_series_summable_aux₁ FormalMultilinearSeries.changeOriginSeries_summable_aux₁
theorem changeOriginSeries_summable_aux₂ (hr : (r : ℝ≥0∞) < p.radius) (k : ℕ) :
Summable fun s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l } =>
‖p (k + s.1)‖₊ * r ^ s.1 := by
rcases ENNReal.lt_iff_exists_add_pos_lt.1 hr with ⟨r', h0, hr'⟩
simpa only [mul_inv_cancel_right₀ (pow_pos h0 _).ne'] using
((NNReal.summable_sigma.1 (p.changeOriginSeries_summable_aux₁ hr')).1 k).mul_right (r' ^ k)⁻¹
#align formal_multilinear_series.change_origin_series_summable_aux₂ FormalMultilinearSeries.changeOriginSeries_summable_aux₂
theorem changeOriginSeries_summable_aux₃ {r : ℝ≥0} (hr : ↑r < p.radius) (k : ℕ) :
Summable fun l : ℕ => ‖p.changeOriginSeries k l‖₊ * r ^ l := by
refine' NNReal.summable_of_le
(fun n => _) (NNReal.summable_sigma.1 <| p.changeOriginSeries_summable_aux₂ hr k).2
simp only [NNReal.tsum_mul_right]
exact mul_le_mul' (p.nnnorm_changeOriginSeries_le_tsum _ _) le_rfl
#align formal_multilinear_series.change_origin_series_summable_aux₃ FormalMultilinearSeries.changeOriginSeries_summable_aux₃
theorem le_changeOriginSeries_radius (k : ℕ) : p.radius ≤ (p.changeOriginSeries k).radius :=
ENNReal.le_of_forall_nnreal_lt fun _r hr =>
le_radius_of_summable_nnnorm _ (p.changeOriginSeries_summable_aux₃ hr k)
#align formal_multilinear_series.le_change_origin_series_radius FormalMultilinearSeries.le_changeOriginSeries_radius
theorem nnnorm_changeOrigin_le (k : ℕ) (h : (‖x‖₊ : ℝ≥0∞) < p.radius) :
‖p.changeOrigin x k‖₊ ≤
∑' s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + s.1)‖₊ * ‖x‖₊ ^ s.1 := by
refine' tsum_of_nnnorm_bounded _ fun l => p.nnnorm_changeOriginSeries_apply_le_tsum k l x
have := p.changeOriginSeries_summable_aux₂ h k
refine' HasSum.sigma this.hasSum fun l => _
exact ((NNReal.summable_sigma.1 this).1 l).hasSum
#align formal_multilinear_series.nnnorm_change_origin_le FormalMultilinearSeries.nnnorm_changeOrigin_le
/-- The radius of convergence of `p.changeOrigin x` is at least `p.radius - ‖x‖`. In other words,
`p.changeOrigin x` is well defined on the largest ball contained in the original ball of
convergence. -/
theorem changeOrigin_radius : p.radius - ‖x‖₊ ≤ (p.changeOrigin x).radius := by
refine' ENNReal.le_of_forall_pos_nnreal_lt fun r _h0 hr => _
rw [lt_tsub_iff_right, add_comm] at hr
have hr' : (‖x‖₊ : ℝ≥0∞) < p.radius := (le_add_right le_rfl).trans_lt hr
apply le_radius_of_summable_nnnorm
have : ∀ k : ℕ,
‖p.changeOrigin x k‖₊ * r ^ k ≤
(∑' s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + s.1)‖₊ * ‖x‖₊ ^ s.1) *
r ^ k :=
fun k => mul_le_mul_right' (p.nnnorm_changeOrigin_le k hr') (r ^ k)
refine' NNReal.summable_of_le this _
simpa only [← NNReal.tsum_mul_right] using
(NNReal.summable_sigma.1 (p.changeOriginSeries_summable_aux₁ hr)).2
#align formal_multilinear_series.change_origin_radius FormalMultilinearSeries.changeOrigin_radius
end
-- From this point on, assume that the space is complete, to make sure that series that converge
-- in norm also converge in `F`.
variable [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x y : E} {r R : ℝ≥0}
theorem hasFPowerSeriesOnBall_changeOrigin (k : ℕ) (hr : 0 < p.radius) :
HasFPowerSeriesOnBall (fun x => p.changeOrigin x k) (p.changeOriginSeries k) 0 p.radius :=
have := p.le_changeOriginSeries_radius k
((p.changeOriginSeries k).hasFPowerSeriesOnBall (hr.trans_le this)).mono hr this
#align formal_multilinear_series.has_fpower_series_on_ball_change_origin FormalMultilinearSeries.hasFPowerSeriesOnBall_changeOrigin
/-- Summing the series `p.changeOrigin x` at a point `y` gives back `p (x + y)`. -/
theorem changeOrigin_eval (h : (‖x‖₊ + ‖y‖₊ : ℝ≥0∞) < p.radius) :
(p.changeOrigin x).sum y = p.sum (x + y) := by
have radius_pos : 0 < p.radius := lt_of_le_of_lt (zero_le _) h
have x_mem_ball : x ∈ EMetric.ball (0 : E) p.radius :=
mem_emetric_ball_zero_iff.2 ((le_add_right le_rfl).trans_lt h)
have y_mem_ball : y ∈ EMetric.ball (0 : E) (p.changeOrigin x).radius := by
refine' mem_emetric_ball_zero_iff.2 (lt_of_lt_of_le _ p.changeOrigin_radius)
rwa [lt_tsub_iff_right, add_comm]
have x_add_y_mem_ball : x + y ∈ EMetric.ball (0 : E) p.radius := by
refine' mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt _ h)
exact mod_cast nnnorm_add_le x y
set f : (Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) → F := fun s =>
p.changeOriginSeriesTerm s.1 s.2.1 s.2.2 s.2.2.2 (fun _ => x) fun _ => y
have hsf : Summable f := by
refine' .of_nnnorm_bounded _ (p.changeOriginSeries_summable_aux₁ h) _
rintro ⟨k, l, s, hs⟩
dsimp only [Subtype.coe_mk]
exact p.nnnorm_changeOriginSeriesTerm_apply_le _ _ _ _ _ _
have hf : HasSum f ((p.changeOrigin x).sum y) := by
refine' HasSum.sigma_of_hasSum ((p.changeOrigin x).summable y_mem_ball).hasSum (fun k => _) hsf
· dsimp only
refine' ContinuousMultilinearMap.hasSum_eval _ _
have := (p.hasFPowerSeriesOnBall_changeOrigin k radius_pos).hasSum x_mem_ball
rw [zero_add] at this
refine' HasSum.sigma_of_hasSum this (fun l => _) _
· simp only [changeOriginSeries, ContinuousMultilinearMap.sum_apply]
apply hasSum_fintype
· refine' .of_nnnorm_bounded _
(p.changeOriginSeries_summable_aux₂ (mem_emetric_ball_zero_iff.1 x_mem_ball) k) fun s => _
refine' (ContinuousMultilinearMap.le_op_nnnorm _ _).trans_eq _
simp
refine' hf.unique (changeOriginIndexEquiv.symm.hasSum_iff.1 _)
refine' HasSum.sigma_of_hasSum
(p.hasSum x_add_y_mem_ball) (fun n => _) (changeOriginIndexEquiv.symm.summable_iff.2 hsf)
erw [(p n).map_add_univ (fun _ => x) fun _ => y]
-- porting note: added explicit function
convert hasSum_fintype (fun c : Finset (Fin n) => f (changeOriginIndexEquiv.symm ⟨n, c⟩))
rename_i s _
dsimp only [changeOriginSeriesTerm, (· ∘ ·), changeOriginIndexEquiv_symm_apply_fst,
changeOriginIndexEquiv_symm_apply_snd_fst, changeOriginIndexEquiv_symm_apply_snd_snd_coe]
rw [ContinuousMultilinearMap.curryFinFinset_apply_const]
have : ∀ (m) (hm : n = m), p n (s.piecewise (fun _ => x) fun _ => y) =
p m ((s.map (Fin.castIso hm).toEquiv.toEmbedding).piecewise (fun _ => x) fun _ => y) := by
rintro m rfl
simp (config := { unfoldPartialApp := true }) [Finset.piecewise]
apply this
#align formal_multilinear_series.change_origin_eval FormalMultilinearSeries.changeOrigin_eval
/-- Power series terms are analytic as we vary the origin -/
theorem analyticAt_changeOrigin (p : FormalMultilinearSeries 𝕜 E F) (rp : p.radius > 0) (n : ℕ) :
AnalyticAt 𝕜 (fun x ↦ p.changeOrigin x n) 0 :=
(FormalMultilinearSeries.hasFPowerSeriesOnBall_changeOrigin p n rp).analyticAt
end FormalMultilinearSeries
section
variable [CompleteSpace F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x y : E} {r : ℝ≥0∞}
/-- If a function admits a power series expansion `p` on a ball `B (x, r)`, then it also admits a
power series on any subball of this ball (even with a different center), given by `p.changeOrigin`.
-/
theorem HasFPowerSeriesOnBall.changeOrigin (hf : HasFPowerSeriesOnBall f p x r)
(h : (‖y‖₊ : ℝ≥0∞) < r) : HasFPowerSeriesOnBall f (p.changeOrigin y) (x + y) (r - ‖y‖₊) :=
{ r_le := by
apply le_trans _ p.changeOrigin_radius
exact tsub_le_tsub hf.r_le le_rfl
r_pos := by simp [h]
hasSum := fun {z} hz => by
have : f (x + y + z) =
FormalMultilinearSeries.sum (FormalMultilinearSeries.changeOrigin p y) z := by
rw [mem_emetric_ball_zero_iff, lt_tsub_iff_right, add_comm] at hz
rw [p.changeOrigin_eval (hz.trans_le hf.r_le), add_assoc, hf.sum]
refine' mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt _ hz)
exact mod_cast nnnorm_add_le y z
rw [this]
apply (p.changeOrigin y).hasSum
refine' EMetric.ball_subset_ball (le_trans _ p.changeOrigin_radius) hz
exact tsub_le_tsub hf.r_le le_rfl }
#align has_fpower_series_on_ball.change_origin HasFPowerSeriesOnBall.changeOrigin
/-- If a function admits a power series expansion `p` on an open ball `B (x, r)`, then
it is analytic at every point of this ball. -/
theorem HasFPowerSeriesOnBall.analyticAt_of_mem (hf : HasFPowerSeriesOnBall f p x r)
(h : y ∈ EMetric.ball x r) : AnalyticAt 𝕜 f y := by
have : (‖y - x‖₊ : ℝ≥0∞) < r := by simpa [edist_eq_coe_nnnorm_sub] using h
have := hf.changeOrigin this
rw [add_sub_cancel'_right] at this
exact this.analyticAt
#align has_fpower_series_on_ball.analytic_at_of_mem HasFPowerSeriesOnBall.analyticAt_of_mem
theorem HasFPowerSeriesOnBall.analyticOn (hf : HasFPowerSeriesOnBall f p x r) :
AnalyticOn 𝕜 f (EMetric.ball x r) :=
fun _y hy => hf.analyticAt_of_mem hy
#align has_fpower_series_on_ball.analytic_on HasFPowerSeriesOnBall.analyticOn
variable (𝕜 f)
/-- For any function `f` from a normed vector space to a Banach space, the set of points `x` such
that `f` is analytic at `x` is open. -/
theorem isOpen_analyticAt : IsOpen { x | AnalyticAt 𝕜 f x } := by
rw [isOpen_iff_mem_nhds]
rintro x ⟨p, r, hr⟩
exact mem_of_superset (EMetric.ball_mem_nhds _ hr.r_pos) fun y hy => hr.analyticAt_of_mem hy
#align is_open_analytic_at isOpen_analyticAt
variable {𝕜}
theorem AnalyticAt.eventually_analyticAt {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∀ᶠ y in 𝓝 x, AnalyticAt 𝕜 f y :=
(isOpen_analyticAt 𝕜 f).mem_nhds h
theorem AnalyticAt.exists_mem_nhds_analyticOn {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∃ s ∈ 𝓝 x, AnalyticOn 𝕜 f s :=
h.eventually_analyticAt.exists_mem
/-- If we're analytic at a point, we're analytic in a nonempty ball -/
theorem AnalyticAt.exists_ball_analyticOn {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∃ r : ℝ, 0 < r ∧ AnalyticOn 𝕜 f (Metric.ball x r) :=
Metric.isOpen_iff.mp (isOpen_analyticAt _ _) _ h
end
section
open FormalMultilinearSeries
variable {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E} {z₀ : 𝕜}
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z)) := by
refine' ⟨fun ⟨r, _, r_pos, h⟩ =>
eventually_of_mem (EMetric.ball_mem_nhds 0 r_pos) fun _ => by simpa using h, _⟩
simp only [Metric.eventually_nhds_iff]
rintro ⟨r, r_pos, h⟩
|
refine' ⟨p.radius ⊓ r.toNNReal, by simp, _, _⟩
|
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z)) := by
refine' ⟨fun ⟨r, _, r_pos, h⟩ =>
eventually_of_mem (EMetric.ball_mem_nhds 0 r_pos) fun _ => by simpa using h, _⟩
simp only [Metric.eventually_nhds_iff]
rintro ⟨r, r_pos, h⟩
|
Mathlib.Analysis.Analytic.Basic.1430_0.jQw1fRSE1vGpOll
|
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z))
|
Mathlib_Analysis_Analytic_Basic
|
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
p : FormalMultilinearSeries 𝕜 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
r : ℝ
r_pos : r > 0
h : ∀ ⦃y : 𝕜⦄, dist y 0 < r → HasSum (fun n => y ^ n • coeff p n) (f (z₀ + y))
⊢ radius p ⊓ ↑(Real.toNNReal r) ≤ radius p
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Logic.Equiv.Fin
import Mathlib.Topology.Algebra.InfiniteSum.Module
#align_import analysis.analytic.basic from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
/-!
# Analytic functions
A function is analytic in one dimension around `0` if it can be written as a converging power series
`Σ pₙ zⁿ`. This definition can be extended to any dimension (even in infinite dimension) by
requiring that `pₙ` is a continuous `n`-multilinear map. In general, `pₙ` is not unique (in two
dimensions, taking `p₂ (x, y) (x', y') = x y'` or `y x'` gives the same map when applied to a
vector `(x, y) (x, y)`). A way to guarantee uniqueness is to take a symmetric `pₙ`, but this is not
always possible in nonzero characteristic (in characteristic 2, the previous example has no
symmetric representative). Therefore, we do not insist on symmetry or uniqueness in the definition,
and we only require the existence of a converging series.
The general framework is important to say that the exponential map on bounded operators on a Banach
space is analytic, as well as the inverse on invertible operators.
## Main definitions
Let `p` be a formal multilinear series from `E` to `F`, i.e., `p n` is a multilinear map on `E^n`
for `n : ℕ`.
* `p.radius`: the largest `r : ℝ≥0∞` such that `‖p n‖ * r^n` grows subexponentially.
* `p.le_radius_of_bound`, `p.le_radius_of_bound_nnreal`, `p.le_radius_of_isBigO`: if `‖p n‖ * r ^ n`
is bounded above, then `r ≤ p.radius`;
* `p.isLittleO_of_lt_radius`, `p.norm_mul_pow_le_mul_pow_of_lt_radius`,
`p.isLittleO_one_of_lt_radius`,
`p.norm_mul_pow_le_of_lt_radius`, `p.nnnorm_mul_pow_le_of_lt_radius`: if `r < p.radius`, then
`‖p n‖ * r ^ n` tends to zero exponentially;
* `p.lt_radius_of_isBigO`: if `r ≠ 0` and `‖p n‖ * r ^ n = O(a ^ n)` for some `-1 < a < 1`, then
`r < p.radius`;
* `p.partialSum n x`: the sum `∑_{i = 0}^{n-1} pᵢ xⁱ`.
* `p.sum x`: the sum `∑'_{i = 0}^{∞} pᵢ xⁱ`.
Additionally, let `f` be a function from `E` to `F`.
* `HasFPowerSeriesOnBall f p x r`: on the ball of center `x` with radius `r`,
`f (x + y) = ∑'_n pₙ yⁿ`.
* `HasFPowerSeriesAt f p x`: on some ball of center `x` with positive radius, holds
`HasFPowerSeriesOnBall f p x r`.
* `AnalyticAt 𝕜 f x`: there exists a power series `p` such that holds `HasFPowerSeriesAt f p x`.
* `AnalyticOn 𝕜 f s`: the function `f` is analytic at every point of `s`.
We develop the basic properties of these notions, notably:
* If a function admits a power series, it is continuous (see
`HasFPowerSeriesOnBall.continuousOn` and `HasFPowerSeriesAt.continuousAt` and
`AnalyticAt.continuousAt`).
* In a complete space, the sum of a formal power series with positive radius is well defined on the
disk of convergence, see `FormalMultilinearSeries.hasFPowerSeriesOnBall`.
* If a function admits a power series in a ball, then it is analytic at any point `y` of this ball,
and the power series there can be expressed in terms of the initial power series `p` as
`p.changeOrigin y`. See `HasFPowerSeriesOnBall.changeOrigin`. It follows in particular that
the set of points at which a given function is analytic is open, see `isOpen_analyticAt`.
## Implementation details
We only introduce the radius of convergence of a power series, as `p.radius`.
For a power series in finitely many dimensions, there is a finer (directional, coordinate-dependent)
notion, describing the polydisk of convergence. This notion is more specific, and not necessary to
build the general theory. We do not define it here.
-/
noncomputable section
variable {𝕜 E F G : Type*}
open Topology Classical BigOperators NNReal Filter ENNReal
open Set Filter Asymptotics
namespace FormalMultilinearSeries
variable [Ring 𝕜] [AddCommGroup E] [AddCommGroup F] [Module 𝕜 E] [Module 𝕜 F]
variable [TopologicalSpace E] [TopologicalSpace F]
variable [TopologicalAddGroup E] [TopologicalAddGroup F]
variable [ContinuousConstSMul 𝕜 E] [ContinuousConstSMul 𝕜 F]
/-- Given a formal multilinear series `p` and a vector `x`, then `p.sum x` is the sum `Σ pₙ xⁿ`. A
priori, it only behaves well when `‖x‖ < p.radius`. -/
protected def sum (p : FormalMultilinearSeries 𝕜 E F) (x : E) : F :=
∑' n : ℕ, p n fun _ => x
#align formal_multilinear_series.sum FormalMultilinearSeries.sum
/-- Given a formal multilinear series `p` and a vector `x`, then `p.partialSum n x` is the sum
`Σ pₖ xᵏ` for `k ∈ {0,..., n-1}`. -/
def partialSum (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) (x : E) : F :=
∑ k in Finset.range n, p k fun _ : Fin k => x
#align formal_multilinear_series.partial_sum FormalMultilinearSeries.partialSum
/-- The partial sums of a formal multilinear series are continuous. -/
theorem partialSum_continuous (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) :
Continuous (p.partialSum n) := by
unfold partialSum -- Porting note: added
continuity
#align formal_multilinear_series.partial_sum_continuous FormalMultilinearSeries.partialSum_continuous
end FormalMultilinearSeries
/-! ### The radius of a formal multilinear series -/
variable [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F]
[NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G]
namespace FormalMultilinearSeries
variable (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
/-- The radius of a formal multilinear series is the largest `r` such that the sum `Σ ‖pₙ‖ ‖y‖ⁿ`
converges for all `‖y‖ < r`. This implies that `Σ pₙ yⁿ` converges for all `‖y‖ < r`, but these
definitions are *not* equivalent in general. -/
def radius (p : FormalMultilinearSeries 𝕜 E F) : ℝ≥0∞ :=
⨆ (r : ℝ≥0) (C : ℝ) (_ : ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C), (r : ℝ≥0∞)
#align formal_multilinear_series.radius FormalMultilinearSeries.radius
/-- If `‖pₙ‖ rⁿ` is bounded in `n`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_bound (C : ℝ) {r : ℝ≥0} (h : ∀ n : ℕ, ‖p n‖ * (r : ℝ) ^ n ≤ C) :
(r : ℝ≥0∞) ≤ p.radius :=
le_iSup_of_le r <| le_iSup_of_le C <| le_iSup (fun _ => (r : ℝ≥0∞)) h
#align formal_multilinear_series.le_radius_of_bound FormalMultilinearSeries.le_radius_of_bound
/-- If `‖pₙ‖ rⁿ` is bounded in `n`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_bound_nnreal (C : ℝ≥0) {r : ℝ≥0} (h : ∀ n : ℕ, ‖p n‖₊ * r ^ n ≤ C) :
(r : ℝ≥0∞) ≤ p.radius :=
p.le_radius_of_bound C fun n => mod_cast h n
#align formal_multilinear_series.le_radius_of_bound_nnreal FormalMultilinearSeries.le_radius_of_bound_nnreal
/-- If `‖pₙ‖ rⁿ = O(1)`, as `n → ∞`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_isBigO (h : (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] fun _ => (1 : ℝ)) :
↑r ≤ p.radius :=
Exists.elim (isBigO_one_nat_atTop_iff.1 h) fun C hC =>
p.le_radius_of_bound C fun n => (le_abs_self _).trans (hC n)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.le_radius_of_is_O FormalMultilinearSeries.le_radius_of_isBigO
theorem le_radius_of_eventually_le (C) (h : ∀ᶠ n in atTop, ‖p n‖ * (r : ℝ) ^ n ≤ C) :
↑r ≤ p.radius :=
p.le_radius_of_isBigO <| IsBigO.of_bound C <| h.mono fun n hn => by simpa
#align formal_multilinear_series.le_radius_of_eventually_le FormalMultilinearSeries.le_radius_of_eventually_le
theorem le_radius_of_summable_nnnorm (h : Summable fun n => ‖p n‖₊ * r ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_bound_nnreal (∑' n, ‖p n‖₊ * r ^ n) fun _ => le_tsum' h _
#align formal_multilinear_series.le_radius_of_summable_nnnorm FormalMultilinearSeries.le_radius_of_summable_nnnorm
theorem le_radius_of_summable (h : Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_summable_nnnorm <| by
simp only [← coe_nnnorm] at h
exact mod_cast h
#align formal_multilinear_series.le_radius_of_summable FormalMultilinearSeries.le_radius_of_summable
theorem radius_eq_top_of_forall_nnreal_isBigO
(h : ∀ r : ℝ≥0, (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] fun _ => (1 : ℝ)) : p.radius = ∞ :=
ENNReal.eq_top_of_forall_nnreal_le fun r => p.le_radius_of_isBigO (h r)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.radius_eq_top_of_forall_nnreal_is_O FormalMultilinearSeries.radius_eq_top_of_forall_nnreal_isBigO
theorem radius_eq_top_of_eventually_eq_zero (h : ∀ᶠ n in atTop, p n = 0) : p.radius = ∞ :=
p.radius_eq_top_of_forall_nnreal_isBigO fun r =>
(isBigO_zero _ _).congr' (h.mono fun n hn => by simp [hn]) EventuallyEq.rfl
#align formal_multilinear_series.radius_eq_top_of_eventually_eq_zero FormalMultilinearSeries.radius_eq_top_of_eventually_eq_zero
theorem radius_eq_top_of_forall_image_add_eq_zero (n : ℕ) (hn : ∀ m, p (m + n) = 0) :
p.radius = ∞ :=
p.radius_eq_top_of_eventually_eq_zero <|
mem_atTop_sets.2 ⟨n, fun _ hk => tsub_add_cancel_of_le hk ▸ hn _⟩
#align formal_multilinear_series.radius_eq_top_of_forall_image_add_eq_zero FormalMultilinearSeries.radius_eq_top_of_forall_image_add_eq_zero
@[simp]
theorem constFormalMultilinearSeries_radius {v : F} :
(constFormalMultilinearSeries 𝕜 E v).radius = ⊤ :=
(constFormalMultilinearSeries 𝕜 E v).radius_eq_top_of_forall_image_add_eq_zero 1
(by simp [constFormalMultilinearSeries])
#align formal_multilinear_series.const_formal_multilinear_series_radius FormalMultilinearSeries.constFormalMultilinearSeries_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` tends to zero exponentially:
for some `0 < a < 1`, `‖p n‖ rⁿ = o(aⁿ)`. -/
theorem isLittleO_of_lt_radius (h : ↑r < p.radius) :
∃ a ∈ Ioo (0 : ℝ) 1, (fun n => ‖p n‖ * (r : ℝ) ^ n) =o[atTop] (a ^ ·) := by
have := (TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 4
rw [this]
-- Porting note: was
-- rw [(TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 4]
simp only [radius, lt_iSup_iff] at h
rcases h with ⟨t, C, hC, rt⟩
rw [ENNReal.coe_lt_coe, ← NNReal.coe_lt_coe] at rt
have : 0 < (t : ℝ) := r.coe_nonneg.trans_lt rt
rw [← div_lt_one this] at rt
refine' ⟨_, rt, C, Or.inr zero_lt_one, fun n => _⟩
calc
|‖p n‖ * (r : ℝ) ^ n| = ‖p n‖ * (t : ℝ) ^ n * (r / t : ℝ) ^ n := by
field_simp [mul_right_comm, abs_mul]
_ ≤ C * (r / t : ℝ) ^ n := by gcongr; apply hC
#align formal_multilinear_series.is_o_of_lt_radius FormalMultilinearSeries.isLittleO_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ = o(1)`. -/
theorem isLittleO_one_of_lt_radius (h : ↑r < p.radius) :
(fun n => ‖p n‖ * (r : ℝ) ^ n) =o[atTop] (fun _ => 1 : ℕ → ℝ) :=
let ⟨_, ha, hp⟩ := p.isLittleO_of_lt_radius h
hp.trans <| (isLittleO_pow_pow_of_lt_left ha.1.le ha.2).congr (fun _ => rfl) one_pow
#align formal_multilinear_series.is_o_one_of_lt_radius FormalMultilinearSeries.isLittleO_one_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` tends to zero exponentially:
for some `0 < a < 1` and `C > 0`, `‖p n‖ * r ^ n ≤ C * a ^ n`. -/
theorem norm_mul_pow_le_mul_pow_of_lt_radius (h : ↑r < p.radius) :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C * a ^ n := by
-- Porting note: moved out of `rcases`
have := ((TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 5).mp
(p.isLittleO_of_lt_radius h)
rcases this with ⟨a, ha, C, hC, H⟩
exact ⟨a, ha, C, hC, fun n => (le_abs_self _).trans (H n)⟩
#align formal_multilinear_series.norm_mul_pow_le_mul_pow_of_lt_radius FormalMultilinearSeries.norm_mul_pow_le_mul_pow_of_lt_radius
/-- If `r ≠ 0` and `‖pₙ‖ rⁿ = O(aⁿ)` for some `-1 < a < 1`, then `r < p.radius`. -/
theorem lt_radius_of_isBigO (h₀ : r ≠ 0) {a : ℝ} (ha : a ∈ Ioo (-1 : ℝ) 1)
(hp : (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] (a ^ ·)) : ↑r < p.radius := by
-- Porting note: moved out of `rcases`
have := ((TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 2 5)
rcases this.mp ⟨a, ha, hp⟩ with ⟨a, ha, C, hC, hp⟩
rw [← pos_iff_ne_zero, ← NNReal.coe_pos] at h₀
lift a to ℝ≥0 using ha.1.le
have : (r : ℝ) < r / a := by
simpa only [div_one] using (div_lt_div_left h₀ zero_lt_one ha.1).2 ha.2
norm_cast at this
rw [← ENNReal.coe_lt_coe] at this
refine' this.trans_le (p.le_radius_of_bound C fun n => _)
rw [NNReal.coe_div, div_pow, ← mul_div_assoc, div_le_iff (pow_pos ha.1 n)]
exact (le_abs_self _).trans (hp n)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.lt_radius_of_is_O FormalMultilinearSeries.lt_radius_of_isBigO
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem norm_mul_pow_le_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C :=
let ⟨_, ha, C, hC, h⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius h
⟨C, hC, fun n => (h n).trans <| mul_le_of_le_one_right hC.lt.le (pow_le_one _ ha.1.le ha.2.le)⟩
#align formal_multilinear_series.norm_mul_pow_le_of_lt_radius FormalMultilinearSeries.norm_mul_pow_le_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem norm_le_div_pow_of_pos_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h0 : 0 < r) (h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖ ≤ C / (r : ℝ) ^ n :=
let ⟨C, hC, hp⟩ := p.norm_mul_pow_le_of_lt_radius h
⟨C, hC, fun n => Iff.mpr (le_div_iff (pow_pos h0 _)) (hp n)⟩
#align formal_multilinear_series.norm_le_div_pow_of_pos_of_lt_radius FormalMultilinearSeries.norm_le_div_pow_of_pos_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem nnnorm_mul_pow_le_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖₊ * r ^ n ≤ C :=
let ⟨C, hC, hp⟩ := p.norm_mul_pow_le_of_lt_radius h
⟨⟨C, hC.lt.le⟩, hC, mod_cast hp⟩
#align formal_multilinear_series.nnnorm_mul_pow_le_of_lt_radius FormalMultilinearSeries.nnnorm_mul_pow_le_of_lt_radius
theorem le_radius_of_tendsto (p : FormalMultilinearSeries 𝕜 E F) {l : ℝ}
(h : Tendsto (fun n => ‖p n‖ * (r : ℝ) ^ n) atTop (𝓝 l)) : ↑r ≤ p.radius :=
p.le_radius_of_isBigO (h.isBigO_one _)
#align formal_multilinear_series.le_radius_of_tendsto FormalMultilinearSeries.le_radius_of_tendsto
theorem le_radius_of_summable_norm (p : FormalMultilinearSeries 𝕜 E F)
(hs : Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_tendsto hs.tendsto_atTop_zero
#align formal_multilinear_series.le_radius_of_summable_norm FormalMultilinearSeries.le_radius_of_summable_norm
theorem not_summable_norm_of_radius_lt_nnnorm (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(h : p.radius < ‖x‖₊) : ¬Summable fun n => ‖p n‖ * ‖x‖ ^ n :=
fun hs => not_le_of_lt h (p.le_radius_of_summable_norm hs)
#align formal_multilinear_series.not_summable_norm_of_radius_lt_nnnorm FormalMultilinearSeries.not_summable_norm_of_radius_lt_nnnorm
theorem summable_norm_mul_pow (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0} (h : ↑r < p.radius) :
Summable fun n : ℕ => ‖p n‖ * (r : ℝ) ^ n := by
obtain ⟨a, ha : a ∈ Ioo (0 : ℝ) 1, C, - : 0 < C, hp⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius h
exact .of_nonneg_of_le (fun n => mul_nonneg (norm_nonneg _) (pow_nonneg r.coe_nonneg _))
hp ((summable_geometric_of_lt_1 ha.1.le ha.2).mul_left _)
#align formal_multilinear_series.summable_norm_mul_pow FormalMultilinearSeries.summable_norm_mul_pow
theorem summable_norm_apply (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : Summable fun n : ℕ => ‖p n fun _ => x‖ := by
rw [mem_emetric_ball_zero_iff] at hx
refine' .of_nonneg_of_le
(fun _ => norm_nonneg _) (fun n => ((p n).le_op_norm _).trans_eq _) (p.summable_norm_mul_pow hx)
simp
#align formal_multilinear_series.summable_norm_apply FormalMultilinearSeries.summable_norm_apply
theorem summable_nnnorm_mul_pow (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0} (h : ↑r < p.radius) :
Summable fun n : ℕ => ‖p n‖₊ * r ^ n := by
rw [← NNReal.summable_coe]
push_cast
exact p.summable_norm_mul_pow h
#align formal_multilinear_series.summable_nnnorm_mul_pow FormalMultilinearSeries.summable_nnnorm_mul_pow
protected theorem summable [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : Summable fun n : ℕ => p n fun _ => x :=
(p.summable_norm_apply hx).of_norm
#align formal_multilinear_series.summable FormalMultilinearSeries.summable
theorem radius_eq_top_of_summable_norm (p : FormalMultilinearSeries 𝕜 E F)
(hs : ∀ r : ℝ≥0, Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : p.radius = ∞ :=
ENNReal.eq_top_of_forall_nnreal_le fun r => p.le_radius_of_summable_norm (hs r)
#align formal_multilinear_series.radius_eq_top_of_summable_norm FormalMultilinearSeries.radius_eq_top_of_summable_norm
theorem radius_eq_top_iff_summable_norm (p : FormalMultilinearSeries 𝕜 E F) :
p.radius = ∞ ↔ ∀ r : ℝ≥0, Summable fun n => ‖p n‖ * (r : ℝ) ^ n := by
constructor
· intro h r
obtain ⟨a, ha : a ∈ Ioo (0 : ℝ) 1, C, - : 0 < C, hp⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius
(show (r : ℝ≥0∞) < p.radius from h.symm ▸ ENNReal.coe_lt_top)
refine' .of_norm_bounded
(fun n => (C : ℝ) * a ^ n) ((summable_geometric_of_lt_1 ha.1.le ha.2).mul_left _) fun n => _
specialize hp n
rwa [Real.norm_of_nonneg (mul_nonneg (norm_nonneg _) (pow_nonneg r.coe_nonneg n))]
· exact p.radius_eq_top_of_summable_norm
#align formal_multilinear_series.radius_eq_top_iff_summable_norm FormalMultilinearSeries.radius_eq_top_iff_summable_norm
/-- If the radius of `p` is positive, then `‖pₙ‖` grows at most geometrically. -/
theorem le_mul_pow_of_radius_pos (p : FormalMultilinearSeries 𝕜 E F) (h : 0 < p.radius) :
∃ (C r : _) (hC : 0 < C) (_ : 0 < r), ∀ n, ‖p n‖ ≤ C * r ^ n := by
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 h with ⟨r, r0, rlt⟩
have rpos : 0 < (r : ℝ) := by simp [ENNReal.coe_pos.1 r0]
rcases norm_le_div_pow_of_pos_of_lt_radius p rpos rlt with ⟨C, Cpos, hCp⟩
refine' ⟨C, r⁻¹, Cpos, by simp only [inv_pos, rpos], fun n => _⟩
-- Porting note: was `convert`
rw [inv_pow, ← div_eq_mul_inv]
exact hCp n
#align formal_multilinear_series.le_mul_pow_of_radius_pos FormalMultilinearSeries.le_mul_pow_of_radius_pos
/-- The radius of the sum of two formal series is at least the minimum of their two radii. -/
theorem min_radius_le_radius_add (p q : FormalMultilinearSeries 𝕜 E F) :
min p.radius q.radius ≤ (p + q).radius := by
refine' ENNReal.le_of_forall_nnreal_lt fun r hr => _
rw [lt_min_iff] at hr
have := ((p.isLittleO_one_of_lt_radius hr.1).add (q.isLittleO_one_of_lt_radius hr.2)).isBigO
refine' (p + q).le_radius_of_isBigO ((isBigO_of_le _ fun n => _).trans this)
rw [← add_mul, norm_mul, norm_mul, norm_norm]
exact mul_le_mul_of_nonneg_right ((norm_add_le _ _).trans (le_abs_self _)) (norm_nonneg _)
#align formal_multilinear_series.min_radius_le_radius_add FormalMultilinearSeries.min_radius_le_radius_add
@[simp]
theorem radius_neg (p : FormalMultilinearSeries 𝕜 E F) : (-p).radius = p.radius := by
simp only [radius, neg_apply, norm_neg]
#align formal_multilinear_series.radius_neg FormalMultilinearSeries.radius_neg
protected theorem hasSum [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : HasSum (fun n : ℕ => p n fun _ => x) (p.sum x) :=
(p.summable hx).hasSum
#align formal_multilinear_series.has_sum FormalMultilinearSeries.hasSum
theorem radius_le_radius_continuousLinearMap_comp (p : FormalMultilinearSeries 𝕜 E F)
(f : F →L[𝕜] G) : p.radius ≤ (f.compFormalMultilinearSeries p).radius := by
refine' ENNReal.le_of_forall_nnreal_lt fun r hr => _
apply le_radius_of_isBigO
apply (IsBigO.trans_isLittleO _ (p.isLittleO_one_of_lt_radius hr)).isBigO
refine' IsBigO.mul (@IsBigOWith.isBigO _ _ _ _ _ ‖f‖ _ _ _ _) (isBigO_refl _ _)
refine IsBigOWith.of_bound (eventually_of_forall fun n => ?_)
simpa only [norm_norm] using f.norm_compContinuousMultilinearMap_le (p n)
#align formal_multilinear_series.radius_le_radius_continuous_linear_map_comp FormalMultilinearSeries.radius_le_radius_continuousLinearMap_comp
end FormalMultilinearSeries
/-! ### Expanding a function as a power series -/
section
variable {f g : E → F} {p pf pg : FormalMultilinearSeries 𝕜 E F} {x : E} {r r' : ℝ≥0∞}
/-- Given a function `f : E → F` and a formal multilinear series `p`, we say that `f` has `p` as
a power series on the ball of radius `r > 0` around `x` if `f (x + y) = ∑' pₙ yⁿ` for all `‖y‖ < r`.
-/
structure HasFPowerSeriesOnBall (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (x : E) (r : ℝ≥0∞) :
Prop where
r_le : r ≤ p.radius
r_pos : 0 < r
hasSum :
∀ {y}, y ∈ EMetric.ball (0 : E) r → HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y))
#align has_fpower_series_on_ball HasFPowerSeriesOnBall
/-- Given a function `f : E → F` and a formal multilinear series `p`, we say that `f` has `p` as
a power series around `x` if `f (x + y) = ∑' pₙ yⁿ` for all `y` in a neighborhood of `0`. -/
def HasFPowerSeriesAt (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (x : E) :=
∃ r, HasFPowerSeriesOnBall f p x r
#align has_fpower_series_at HasFPowerSeriesAt
variable (𝕜)
/-- Given a function `f : E → F`, we say that `f` is analytic at `x` if it admits a convergent power
series expansion around `x`. -/
def AnalyticAt (f : E → F) (x : E) :=
∃ p : FormalMultilinearSeries 𝕜 E F, HasFPowerSeriesAt f p x
#align analytic_at AnalyticAt
/-- Given a function `f : E → F`, we say that `f` is analytic on a set `s` if it is analytic around
every point of `s`. -/
def AnalyticOn (f : E → F) (s : Set E) :=
∀ x, x ∈ s → AnalyticAt 𝕜 f x
#align analytic_on AnalyticOn
variable {𝕜}
theorem HasFPowerSeriesOnBall.hasFPowerSeriesAt (hf : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesAt f p x :=
⟨r, hf⟩
#align has_fpower_series_on_ball.has_fpower_series_at HasFPowerSeriesOnBall.hasFPowerSeriesAt
theorem HasFPowerSeriesAt.analyticAt (hf : HasFPowerSeriesAt f p x) : AnalyticAt 𝕜 f x :=
⟨p, hf⟩
#align has_fpower_series_at.analytic_at HasFPowerSeriesAt.analyticAt
theorem HasFPowerSeriesOnBall.analyticAt (hf : HasFPowerSeriesOnBall f p x r) : AnalyticAt 𝕜 f x :=
hf.hasFPowerSeriesAt.analyticAt
#align has_fpower_series_on_ball.analytic_at HasFPowerSeriesOnBall.analyticAt
theorem HasFPowerSeriesOnBall.congr (hf : HasFPowerSeriesOnBall f p x r)
(hg : EqOn f g (EMetric.ball x r)) : HasFPowerSeriesOnBall g p x r :=
{ r_le := hf.r_le
r_pos := hf.r_pos
hasSum := fun {y} hy => by
convert hf.hasSum hy using 1
apply hg.symm
simpa [edist_eq_coe_nnnorm_sub] using hy }
#align has_fpower_series_on_ball.congr HasFPowerSeriesOnBall.congr
/-- If a function `f` has a power series `p` around `x`, then the function `z ↦ f (z - y)` has the
same power series around `x + y`. -/
theorem HasFPowerSeriesOnBall.comp_sub (hf : HasFPowerSeriesOnBall f p x r) (y : E) :
HasFPowerSeriesOnBall (fun z => f (z - y)) p (x + y) r :=
{ r_le := hf.r_le
r_pos := hf.r_pos
hasSum := fun {z} hz => by
convert hf.hasSum hz using 2
abel }
#align has_fpower_series_on_ball.comp_sub HasFPowerSeriesOnBall.comp_sub
theorem HasFPowerSeriesOnBall.hasSum_sub (hf : HasFPowerSeriesOnBall f p x r) {y : E}
(hy : y ∈ EMetric.ball x r) : HasSum (fun n : ℕ => p n fun _ => y - x) (f y) := by
have : y - x ∈ EMetric.ball (0 : E) r := by simpa [edist_eq_coe_nnnorm_sub] using hy
simpa only [add_sub_cancel'_right] using hf.hasSum this
#align has_fpower_series_on_ball.has_sum_sub HasFPowerSeriesOnBall.hasSum_sub
theorem HasFPowerSeriesOnBall.radius_pos (hf : HasFPowerSeriesOnBall f p x r) : 0 < p.radius :=
lt_of_lt_of_le hf.r_pos hf.r_le
#align has_fpower_series_on_ball.radius_pos HasFPowerSeriesOnBall.radius_pos
theorem HasFPowerSeriesAt.radius_pos (hf : HasFPowerSeriesAt f p x) : 0 < p.radius :=
let ⟨_, hr⟩ := hf
hr.radius_pos
#align has_fpower_series_at.radius_pos HasFPowerSeriesAt.radius_pos
theorem HasFPowerSeriesOnBall.mono (hf : HasFPowerSeriesOnBall f p x r) (r'_pos : 0 < r')
(hr : r' ≤ r) : HasFPowerSeriesOnBall f p x r' :=
⟨le_trans hr hf.1, r'_pos, fun hy => hf.hasSum (EMetric.ball_subset_ball hr hy)⟩
#align has_fpower_series_on_ball.mono HasFPowerSeriesOnBall.mono
theorem HasFPowerSeriesAt.congr (hf : HasFPowerSeriesAt f p x) (hg : f =ᶠ[𝓝 x] g) :
HasFPowerSeriesAt g p x := by
rcases hf with ⟨r₁, h₁⟩
rcases EMetric.mem_nhds_iff.mp hg with ⟨r₂, h₂pos, h₂⟩
exact ⟨min r₁ r₂,
(h₁.mono (lt_min h₁.r_pos h₂pos) inf_le_left).congr
fun y hy => h₂ (EMetric.ball_subset_ball inf_le_right hy)⟩
#align has_fpower_series_at.congr HasFPowerSeriesAt.congr
protected theorem HasFPowerSeriesAt.eventually (hf : HasFPowerSeriesAt f p x) :
∀ᶠ r : ℝ≥0∞ in 𝓝[>] 0, HasFPowerSeriesOnBall f p x r :=
let ⟨_, hr⟩ := hf
mem_of_superset (Ioo_mem_nhdsWithin_Ioi (left_mem_Ico.2 hr.r_pos)) fun _ hr' =>
hr.mono hr'.1 hr'.2.le
#align has_fpower_series_at.eventually HasFPowerSeriesAt.eventually
theorem HasFPowerSeriesOnBall.eventually_hasSum (hf : HasFPowerSeriesOnBall f p x r) :
∀ᶠ y in 𝓝 0, HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y)) := by
filter_upwards [EMetric.ball_mem_nhds (0 : E) hf.r_pos] using fun _ => hf.hasSum
#align has_fpower_series_on_ball.eventually_has_sum HasFPowerSeriesOnBall.eventually_hasSum
theorem HasFPowerSeriesAt.eventually_hasSum (hf : HasFPowerSeriesAt f p x) :
∀ᶠ y in 𝓝 0, HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y)) :=
let ⟨_, hr⟩ := hf
hr.eventually_hasSum
#align has_fpower_series_at.eventually_has_sum HasFPowerSeriesAt.eventually_hasSum
theorem HasFPowerSeriesOnBall.eventually_hasSum_sub (hf : HasFPowerSeriesOnBall f p x r) :
∀ᶠ y in 𝓝 x, HasSum (fun n : ℕ => p n fun _ : Fin n => y - x) (f y) := by
filter_upwards [EMetric.ball_mem_nhds x hf.r_pos] with y using hf.hasSum_sub
#align has_fpower_series_on_ball.eventually_has_sum_sub HasFPowerSeriesOnBall.eventually_hasSum_sub
theorem HasFPowerSeriesAt.eventually_hasSum_sub (hf : HasFPowerSeriesAt f p x) :
∀ᶠ y in 𝓝 x, HasSum (fun n : ℕ => p n fun _ : Fin n => y - x) (f y) :=
let ⟨_, hr⟩ := hf
hr.eventually_hasSum_sub
#align has_fpower_series_at.eventually_has_sum_sub HasFPowerSeriesAt.eventually_hasSum_sub
theorem HasFPowerSeriesOnBall.eventually_eq_zero
(hf : HasFPowerSeriesOnBall f (0 : FormalMultilinearSeries 𝕜 E F) x r) :
∀ᶠ z in 𝓝 x, f z = 0 := by
filter_upwards [hf.eventually_hasSum_sub] with z hz using hz.unique hasSum_zero
#align has_fpower_series_on_ball.eventually_eq_zero HasFPowerSeriesOnBall.eventually_eq_zero
theorem HasFPowerSeriesAt.eventually_eq_zero
(hf : HasFPowerSeriesAt f (0 : FormalMultilinearSeries 𝕜 E F) x) : ∀ᶠ z in 𝓝 x, f z = 0 :=
let ⟨_, hr⟩ := hf
hr.eventually_eq_zero
#align has_fpower_series_at.eventually_eq_zero HasFPowerSeriesAt.eventually_eq_zero
theorem hasFPowerSeriesOnBall_const {c : F} {e : E} :
HasFPowerSeriesOnBall (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e ⊤ := by
refine' ⟨by simp, WithTop.zero_lt_top, fun _ => hasSum_single 0 fun n hn => _⟩
simp [constFormalMultilinearSeries_apply hn]
#align has_fpower_series_on_ball_const hasFPowerSeriesOnBall_const
theorem hasFPowerSeriesAt_const {c : F} {e : E} :
HasFPowerSeriesAt (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e :=
⟨⊤, hasFPowerSeriesOnBall_const⟩
#align has_fpower_series_at_const hasFPowerSeriesAt_const
theorem analyticAt_const {v : F} : AnalyticAt 𝕜 (fun _ => v) x :=
⟨constFormalMultilinearSeries 𝕜 E v, hasFPowerSeriesAt_const⟩
#align analytic_at_const analyticAt_const
theorem analyticOn_const {v : F} {s : Set E} : AnalyticOn 𝕜 (fun _ => v) s :=
fun _ _ => analyticAt_const
#align analytic_on_const analyticOn_const
theorem HasFPowerSeriesOnBall.add (hf : HasFPowerSeriesOnBall f pf x r)
(hg : HasFPowerSeriesOnBall g pg x r) : HasFPowerSeriesOnBall (f + g) (pf + pg) x r :=
{ r_le := le_trans (le_min_iff.2 ⟨hf.r_le, hg.r_le⟩) (pf.min_radius_le_radius_add pg)
r_pos := hf.r_pos
hasSum := fun hy => (hf.hasSum hy).add (hg.hasSum hy) }
#align has_fpower_series_on_ball.add HasFPowerSeriesOnBall.add
theorem HasFPowerSeriesAt.add (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x) :
HasFPowerSeriesAt (f + g) (pf + pg) x := by
rcases (hf.eventually.and hg.eventually).exists with ⟨r, hr⟩
exact ⟨r, hr.1.add hr.2⟩
#align has_fpower_series_at.add HasFPowerSeriesAt.add
theorem AnalyticAt.congr (hf : AnalyticAt 𝕜 f x) (hg : f =ᶠ[𝓝 x] g) : AnalyticAt 𝕜 g x :=
let ⟨_, hpf⟩ := hf
(hpf.congr hg).analyticAt
theorem analyticAt_congr (h : f =ᶠ[𝓝 x] g) : AnalyticAt 𝕜 f x ↔ AnalyticAt 𝕜 g x :=
⟨fun hf ↦ hf.congr h, fun hg ↦ hg.congr h.symm⟩
theorem AnalyticAt.add (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) : AnalyticAt 𝕜 (f + g) x :=
let ⟨_, hpf⟩ := hf
let ⟨_, hqf⟩ := hg
(hpf.add hqf).analyticAt
#align analytic_at.add AnalyticAt.add
theorem HasFPowerSeriesOnBall.neg (hf : HasFPowerSeriesOnBall f pf x r) :
HasFPowerSeriesOnBall (-f) (-pf) x r :=
{ r_le := by
rw [pf.radius_neg]
exact hf.r_le
r_pos := hf.r_pos
hasSum := fun hy => (hf.hasSum hy).neg }
#align has_fpower_series_on_ball.neg HasFPowerSeriesOnBall.neg
theorem HasFPowerSeriesAt.neg (hf : HasFPowerSeriesAt f pf x) : HasFPowerSeriesAt (-f) (-pf) x :=
let ⟨_, hrf⟩ := hf
hrf.neg.hasFPowerSeriesAt
#align has_fpower_series_at.neg HasFPowerSeriesAt.neg
theorem AnalyticAt.neg (hf : AnalyticAt 𝕜 f x) : AnalyticAt 𝕜 (-f) x :=
let ⟨_, hpf⟩ := hf
hpf.neg.analyticAt
#align analytic_at.neg AnalyticAt.neg
theorem HasFPowerSeriesOnBall.sub (hf : HasFPowerSeriesOnBall f pf x r)
(hg : HasFPowerSeriesOnBall g pg x r) : HasFPowerSeriesOnBall (f - g) (pf - pg) x r := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align has_fpower_series_on_ball.sub HasFPowerSeriesOnBall.sub
theorem HasFPowerSeriesAt.sub (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x) :
HasFPowerSeriesAt (f - g) (pf - pg) x := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align has_fpower_series_at.sub HasFPowerSeriesAt.sub
theorem AnalyticAt.sub (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) :
AnalyticAt 𝕜 (f - g) x := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align analytic_at.sub AnalyticAt.sub
theorem AnalyticOn.mono {s t : Set E} (hf : AnalyticOn 𝕜 f t) (hst : s ⊆ t) : AnalyticOn 𝕜 f s :=
fun z hz => hf z (hst hz)
#align analytic_on.mono AnalyticOn.mono
theorem AnalyticOn.congr' {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : f =ᶠ[𝓝ˢ s] g) :
AnalyticOn 𝕜 g s :=
fun z hz => (hf z hz).congr (mem_nhdsSet_iff_forall.mp hg z hz)
theorem analyticOn_congr' {s : Set E} (h : f =ᶠ[𝓝ˢ s] g) : AnalyticOn 𝕜 f s ↔ AnalyticOn 𝕜 g s :=
⟨fun hf => hf.congr' h, fun hg => hg.congr' h.symm⟩
theorem AnalyticOn.congr {s : Set E} (hs : IsOpen s) (hf : AnalyticOn 𝕜 f s) (hg : s.EqOn f g) :
AnalyticOn 𝕜 g s :=
hf.congr' $ mem_nhdsSet_iff_forall.mpr
(fun _ hz => eventuallyEq_iff_exists_mem.mpr ⟨s, hs.mem_nhds hz, hg⟩)
theorem analyticOn_congr {s : Set E} (hs : IsOpen s) (h : s.EqOn f g) : AnalyticOn 𝕜 f s ↔
AnalyticOn 𝕜 g s := ⟨fun hf => hf.congr hs h, fun hg => hg.congr hs h.symm⟩
theorem AnalyticOn.add {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
AnalyticOn 𝕜 (f + g) s :=
fun z hz => (hf z hz).add (hg z hz)
#align analytic_on.add AnalyticOn.add
theorem AnalyticOn.sub {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
AnalyticOn 𝕜 (f - g) s :=
fun z hz => (hf z hz).sub (hg z hz)
#align analytic_on.sub AnalyticOn.sub
theorem HasFPowerSeriesOnBall.coeff_zero (hf : HasFPowerSeriesOnBall f pf x r) (v : Fin 0 → E) :
pf 0 v = f x := by
have v_eq : v = fun i => 0 := Subsingleton.elim _ _
have zero_mem : (0 : E) ∈ EMetric.ball (0 : E) r := by simp [hf.r_pos]
have : ∀ i, i ≠ 0 → (pf i fun j => 0) = 0 := by
intro i hi
have : 0 < i := pos_iff_ne_zero.2 hi
exact ContinuousMultilinearMap.map_coord_zero _ (⟨0, this⟩ : Fin i) rfl
have A := (hf.hasSum zero_mem).unique (hasSum_single _ this)
simpa [v_eq] using A.symm
#align has_fpower_series_on_ball.coeff_zero HasFPowerSeriesOnBall.coeff_zero
theorem HasFPowerSeriesAt.coeff_zero (hf : HasFPowerSeriesAt f pf x) (v : Fin 0 → E) :
pf 0 v = f x :=
let ⟨_, hrf⟩ := hf
hrf.coeff_zero v
#align has_fpower_series_at.coeff_zero HasFPowerSeriesAt.coeff_zero
/-- If a function `f` has a power series `p` on a ball and `g` is linear, then `g ∘ f` has the
power series `g ∘ p` on the same ball. -/
theorem ContinuousLinearMap.comp_hasFPowerSeriesOnBall (g : F →L[𝕜] G)
(h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (g ∘ f) (g.compFormalMultilinearSeries p) x r :=
{ r_le := h.r_le.trans (p.radius_le_radius_continuousLinearMap_comp _)
r_pos := h.r_pos
hasSum := fun hy => by
simpa only [ContinuousLinearMap.compFormalMultilinearSeries_apply,
ContinuousLinearMap.compContinuousMultilinearMap_coe, Function.comp_apply] using
g.hasSum (h.hasSum hy) }
#align continuous_linear_map.comp_has_fpower_series_on_ball ContinuousLinearMap.comp_hasFPowerSeriesOnBall
/-- If a function `f` is analytic on a set `s` and `g` is linear, then `g ∘ f` is analytic
on `s`. -/
theorem ContinuousLinearMap.comp_analyticOn {s : Set E} (g : F →L[𝕜] G) (h : AnalyticOn 𝕜 f s) :
AnalyticOn 𝕜 (g ∘ f) s := by
rintro x hx
rcases h x hx with ⟨p, r, hp⟩
exact ⟨g.compFormalMultilinearSeries p, r, g.comp_hasFPowerSeriesOnBall hp⟩
#align continuous_linear_map.comp_analytic_on ContinuousLinearMap.comp_analyticOn
/-- If a function admits a power series expansion, then it is exponentially close to the partial
sums of this power series on strict subdisks of the disk of convergence.
This version provides an upper estimate that decreases both in `‖y‖` and `n`. See also
`HasFPowerSeriesOnBall.uniform_geometric_approx` for a weaker version. -/
theorem HasFPowerSeriesOnBall.uniform_geometric_approx' {r' : ℝ≥0}
(hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n := by
obtain ⟨a, ha, C, hC, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n, ‖p n‖ * (r' : ℝ) ^ n ≤ C * a ^ n :=
p.norm_mul_pow_le_mul_pow_of_lt_radius (h.trans_le hf.r_le)
refine' ⟨a, ha, C / (1 - a), div_pos hC (sub_pos.2 ha.2), fun y hy n => _⟩
have yr' : ‖y‖ < r' := by
rw [ball_zero_eq] at hy
exact hy
have hr'0 : 0 < (r' : ℝ) := (norm_nonneg _).trans_lt yr'
have : y ∈ EMetric.ball (0 : E) r := by
refine' mem_emetric_ball_zero_iff.2 (lt_trans _ h)
exact mod_cast yr'
rw [norm_sub_rev, ← mul_div_right_comm]
have ya : a * (‖y‖ / ↑r') ≤ a :=
mul_le_of_le_one_right ha.1.le (div_le_one_of_le yr'.le r'.coe_nonneg)
suffices ‖p.partialSum n y - f (x + y)‖ ≤ C * (a * (‖y‖ / r')) ^ n / (1 - a * (‖y‖ / r')) by
refine' this.trans _
have : 0 < a := ha.1
gcongr
apply_rules [sub_pos.2, ha.2]
apply norm_sub_le_of_geometric_bound_of_hasSum (ya.trans_lt ha.2) _ (hf.hasSum this)
intro n
calc
‖(p n) fun _ : Fin n => y‖
_ ≤ ‖p n‖ * ∏ _i : Fin n, ‖y‖ := ContinuousMultilinearMap.le_op_norm _ _
_ = ‖p n‖ * (r' : ℝ) ^ n * (‖y‖ / r') ^ n := by field_simp [mul_right_comm]
_ ≤ C * a ^ n * (‖y‖ / r') ^ n := by gcongr ?_ * _; apply hp
_ ≤ C * (a * (‖y‖ / r')) ^ n := by rw [mul_pow, mul_assoc]
#align has_fpower_series_on_ball.uniform_geometric_approx' HasFPowerSeriesOnBall.uniform_geometric_approx'
/-- If a function admits a power series expansion, then it is exponentially close to the partial
sums of this power series on strict subdisks of the disk of convergence. -/
theorem HasFPowerSeriesOnBall.uniform_geometric_approx {r' : ℝ≥0}
(hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
∃ a ∈ Ioo (0 : ℝ) 1,
∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n, ‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n := by
obtain ⟨a, ha, C, hC, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n :=
hf.uniform_geometric_approx' h
refine' ⟨a, ha, C, hC, fun y hy n => (hp y hy n).trans _⟩
have yr' : ‖y‖ < r' := by rwa [ball_zero_eq] at hy
gcongr
exacts [mul_nonneg ha.1.le (div_nonneg (norm_nonneg y) r'.coe_nonneg),
mul_le_of_le_one_right ha.1.le (div_le_one_of_le yr'.le r'.coe_nonneg)]
#align has_fpower_series_on_ball.uniform_geometric_approx HasFPowerSeriesOnBall.uniform_geometric_approx
/-- Taylor formula for an analytic function, `IsBigO` version. -/
theorem HasFPowerSeriesAt.isBigO_sub_partialSum_pow (hf : HasFPowerSeriesAt f p x) (n : ℕ) :
(fun y : E => f (x + y) - p.partialSum n y) =O[𝓝 0] fun y => ‖y‖ ^ n := by
rcases hf with ⟨r, hf⟩
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hf.r_pos with ⟨r', r'0, h⟩
obtain ⟨a, -, C, -, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n :=
hf.uniform_geometric_approx' h
refine' isBigO_iff.2 ⟨C * (a / r') ^ n, _⟩
replace r'0 : 0 < (r' : ℝ); · exact mod_cast r'0
filter_upwards [Metric.ball_mem_nhds (0 : E) r'0] with y hy
simpa [mul_pow, mul_div_assoc, mul_assoc, div_mul_eq_mul_div] using hp y hy n
set_option linter.uppercaseLean3 false in
#align has_fpower_series_at.is_O_sub_partial_sum_pow HasFPowerSeriesAt.isBigO_sub_partialSum_pow
/-- If `f` has formal power series `∑ n, pₙ` on a ball of radius `r`, then for `y, z` in any smaller
ball, the norm of the difference `f y - f z - p 1 (fun _ ↦ y - z)` is bounded above by
`C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖`. This lemma formulates this property using `IsBigO` and
`Filter.principal` on `E × E`. -/
theorem HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal
(hf : HasFPowerSeriesOnBall f p x r) (hr : r' < r) :
(fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) =O[𝓟 (EMetric.ball (x, x) r')]
fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖ := by
lift r' to ℝ≥0 using ne_top_of_lt hr
rcases (zero_le r').eq_or_lt with (rfl | hr'0)
· simp only [isBigO_bot, EMetric.ball_zero, principal_empty, ENNReal.coe_zero]
obtain ⟨a, ha, C, hC : 0 < C, hp⟩ :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n : ℕ, ‖p n‖ * (r' : ℝ) ^ n ≤ C * a ^ n
exact p.norm_mul_pow_le_mul_pow_of_lt_radius (hr.trans_le hf.r_le)
simp only [← le_div_iff (pow_pos (NNReal.coe_pos.2 hr'0) _)] at hp
set L : E × E → ℝ := fun y =>
C * (a / r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * (a / (1 - a) ^ 2 + 2 / (1 - a))
have hL : ∀ y ∈ EMetric.ball (x, x) r', ‖f y.1 - f y.2 - p 1 fun _ => y.1 - y.2‖ ≤ L y := by
intro y hy'
have hy : y ∈ EMetric.ball x r ×ˢ EMetric.ball x r := by
rw [EMetric.ball_prod_same]
exact EMetric.ball_subset_ball hr.le hy'
set A : ℕ → F := fun n => (p n fun _ => y.1 - x) - p n fun _ => y.2 - x
have hA : HasSum (fun n => A (n + 2)) (f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) := by
convert (hasSum_nat_add_iff' 2).2 ((hf.hasSum_sub hy.1).sub (hf.hasSum_sub hy.2)) using 1
rw [Finset.sum_range_succ, Finset.sum_range_one, hf.coeff_zero, hf.coeff_zero, sub_self,
zero_add, ← Subsingleton.pi_single_eq (0 : Fin 1) (y.1 - x), Pi.single,
← Subsingleton.pi_single_eq (0 : Fin 1) (y.2 - x), Pi.single, ← (p 1).map_sub, ← Pi.single,
Subsingleton.pi_single_eq, sub_sub_sub_cancel_right]
rw [EMetric.mem_ball, edist_eq_coe_nnnorm_sub, ENNReal.coe_lt_coe] at hy'
set B : ℕ → ℝ := fun n => C * (a / r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * ((n + 2) * a ^ n)
have hAB : ∀ n, ‖A (n + 2)‖ ≤ B n := fun n =>
calc
‖A (n + 2)‖ ≤ ‖p (n + 2)‖ * ↑(n + 2) * ‖y - (x, x)‖ ^ (n + 1) * ‖y.1 - y.2‖ := by
-- porting note: `pi_norm_const` was `pi_norm_const (_ : E)`
simpa only [Fintype.card_fin, pi_norm_const, Prod.norm_def, Pi.sub_def,
Prod.fst_sub, Prod.snd_sub, sub_sub_sub_cancel_right] using
(p <| n + 2).norm_image_sub_le (fun _ => y.1 - x) fun _ => y.2 - x
_ = ‖p (n + 2)‖ * ‖y - (x, x)‖ ^ n * (↑(n + 2) * ‖y - (x, x)‖ * ‖y.1 - y.2‖) := by
rw [pow_succ ‖y - (x, x)‖]
ring
-- porting note: the two `↑` in `↑r'` are new, without them, Lean fails to synthesize
-- instances `HDiv ℝ ℝ≥0 ?m` or `HMul ℝ ℝ≥0 ?m`
_ ≤ C * a ^ (n + 2) / ↑r' ^ (n + 2)
* ↑r' ^ n * (↑(n + 2) * ‖y - (x, x)‖ * ‖y.1 - y.2‖) := by
have : 0 < a := ha.1
gcongr
· apply hp
· apply hy'.le
_ = B n := by
-- porting note: in the original, `B` was in the `field_simp`, but now Lean does not
-- accept it. The current proof works in Lean 4, but does not in Lean 3.
field_simp [pow_succ]
simp only [mul_assoc, mul_comm, mul_left_comm]
have hBL : HasSum B (L y) := by
apply HasSum.mul_left
simp only [add_mul]
have : ‖a‖ < 1 := by simp only [Real.norm_eq_abs, abs_of_pos ha.1, ha.2]
rw [div_eq_mul_inv, div_eq_mul_inv]
exact (hasSum_coe_mul_geometric_of_norm_lt_1 this).add -- porting note: was `convert`!
((hasSum_geometric_of_norm_lt_1 this).mul_left 2)
exact hA.norm_le_of_bounded hBL hAB
suffices L =O[𝓟 (EMetric.ball (x, x) r')] fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖ by
refine' (IsBigO.of_bound 1 (eventually_principal.2 fun y hy => _)).trans this
rw [one_mul]
exact (hL y hy).trans (le_abs_self _)
simp_rw [mul_right_comm _ (_ * _)] -- porting note: there was an `L` inside the `simp_rw`.
exact (isBigO_refl _ _).const_mul_left _
set_option linter.uppercaseLean3 false in
#align has_fpower_series_on_ball.is_O_image_sub_image_sub_deriv_principal HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal
/-- If `f` has formal power series `∑ n, pₙ` on a ball of radius `r`, then for `y, z` in any smaller
ball, the norm of the difference `f y - f z - p 1 (fun _ ↦ y - z)` is bounded above by
`C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖`. -/
theorem HasFPowerSeriesOnBall.image_sub_sub_deriv_le (hf : HasFPowerSeriesOnBall f p x r)
(hr : r' < r) :
∃ C, ∀ᵉ (y ∈ EMetric.ball x r') (z ∈ EMetric.ball x r'),
‖f y - f z - p 1 fun _ => y - z‖ ≤ C * max ‖y - x‖ ‖z - x‖ * ‖y - z‖ := by
simpa only [isBigO_principal, mul_assoc, norm_mul, norm_norm, Prod.forall, EMetric.mem_ball,
Prod.edist_eq, max_lt_iff, and_imp, @forall_swap (_ < _) E] using
hf.isBigO_image_sub_image_sub_deriv_principal hr
#align has_fpower_series_on_ball.image_sub_sub_deriv_le HasFPowerSeriesOnBall.image_sub_sub_deriv_le
/-- If `f` has formal power series `∑ n, pₙ` at `x`, then
`f y - f z - p 1 (fun _ ↦ y - z) = O(‖(y, z) - (x, x)‖ * ‖y - z‖)` as `(y, z) → (x, x)`.
In particular, `f` is strictly differentiable at `x`. -/
theorem HasFPowerSeriesAt.isBigO_image_sub_norm_mul_norm_sub (hf : HasFPowerSeriesAt f p x) :
(fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) =O[𝓝 (x, x)] fun y =>
‖y - (x, x)‖ * ‖y.1 - y.2‖ := by
rcases hf with ⟨r, hf⟩
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hf.r_pos with ⟨r', r'0, h⟩
refine' (hf.isBigO_image_sub_image_sub_deriv_principal h).mono _
exact le_principal_iff.2 (EMetric.ball_mem_nhds _ r'0)
set_option linter.uppercaseLean3 false in
#align has_fpower_series_at.is_O_image_sub_norm_mul_norm_sub HasFPowerSeriesAt.isBigO_image_sub_norm_mul_norm_sub
/-- If a function admits a power series expansion at `x`, then it is the uniform limit of the
partial sums of this power series on strict subdisks of the disk of convergence, i.e., `f (x + y)`
is the uniform limit of `p.partialSum n y` there. -/
theorem HasFPowerSeriesOnBall.tendstoUniformlyOn {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r)
(h : (r' : ℝ≥0∞) < r) :
TendstoUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
(Metric.ball (0 : E) r') := by
obtain ⟨a, ha, C, -, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n
exact hf.uniform_geometric_approx h
refine' Metric.tendstoUniformlyOn_iff.2 fun ε εpos => _
have L : Tendsto (fun n => (C : ℝ) * a ^ n) atTop (𝓝 ((C : ℝ) * 0)) :=
tendsto_const_nhds.mul (tendsto_pow_atTop_nhds_0_of_lt_1 ha.1.le ha.2)
rw [mul_zero] at L
refine' (L.eventually (gt_mem_nhds εpos)).mono fun n hn y hy => _
rw [dist_eq_norm]
exact (hp y hy n).trans_lt hn
#align has_fpower_series_on_ball.tendsto_uniformly_on HasFPowerSeriesOnBall.tendstoUniformlyOn
/-- If a function admits a power series expansion at `x`, then it is the locally uniform limit of
the partial sums of this power series on the disk of convergence, i.e., `f (x + y)`
is the locally uniform limit of `p.partialSum n y` there. -/
theorem HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn (hf : HasFPowerSeriesOnBall f p x r) :
TendstoLocallyUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
(EMetric.ball (0 : E) r) := by
intro u hu x hx
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hx with ⟨r', xr', hr'⟩
have : EMetric.ball (0 : E) r' ∈ 𝓝 x := IsOpen.mem_nhds EMetric.isOpen_ball xr'
refine' ⟨EMetric.ball (0 : E) r', mem_nhdsWithin_of_mem_nhds this, _⟩
simpa [Metric.emetric_ball_nnreal] using hf.tendstoUniformlyOn hr' u hu
#align has_fpower_series_on_ball.tendsto_locally_uniformly_on HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn
/-- If a function admits a power series expansion at `x`, then it is the uniform limit of the
partial sums of this power series on strict subdisks of the disk of convergence, i.e., `f y`
is the uniform limit of `p.partialSum n (y - x)` there. -/
theorem HasFPowerSeriesOnBall.tendstoUniformlyOn' {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r)
(h : (r' : ℝ≥0∞) < r) :
TendstoUniformlyOn (fun n y => p.partialSum n (y - x)) f atTop (Metric.ball (x : E) r') := by
convert (hf.tendstoUniformlyOn h).comp fun y => y - x using 1
· simp [(· ∘ ·)]
· ext z
simp [dist_eq_norm]
#align has_fpower_series_on_ball.tendsto_uniformly_on' HasFPowerSeriesOnBall.tendstoUniformlyOn'
/-- If a function admits a power series expansion at `x`, then it is the locally uniform limit of
the partial sums of this power series on the disk of convergence, i.e., `f y`
is the locally uniform limit of `p.partialSum n (y - x)` there. -/
theorem HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn' (hf : HasFPowerSeriesOnBall f p x r) :
TendstoLocallyUniformlyOn (fun n y => p.partialSum n (y - x)) f atTop
(EMetric.ball (x : E) r) := by
have A : ContinuousOn (fun y : E => y - x) (EMetric.ball (x : E) r) :=
(continuous_id.sub continuous_const).continuousOn
convert hf.tendstoLocallyUniformlyOn.comp (fun y : E => y - x) _ A using 1
· ext z
simp
· intro z
simp [edist_eq_coe_nnnorm, edist_eq_coe_nnnorm_sub]
#align has_fpower_series_on_ball.tendsto_locally_uniformly_on' HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn'
/-- If a function admits a power series expansion on a disk, then it is continuous there. -/
protected theorem HasFPowerSeriesOnBall.continuousOn (hf : HasFPowerSeriesOnBall f p x r) :
ContinuousOn f (EMetric.ball x r) :=
hf.tendstoLocallyUniformlyOn'.continuousOn <|
eventually_of_forall fun n =>
((p.partialSum_continuous n).comp (continuous_id.sub continuous_const)).continuousOn
#align has_fpower_series_on_ball.continuous_on HasFPowerSeriesOnBall.continuousOn
protected theorem HasFPowerSeriesAt.continuousAt (hf : HasFPowerSeriesAt f p x) :
ContinuousAt f x :=
let ⟨_, hr⟩ := hf
hr.continuousOn.continuousAt (EMetric.ball_mem_nhds x hr.r_pos)
#align has_fpower_series_at.continuous_at HasFPowerSeriesAt.continuousAt
protected theorem AnalyticAt.continuousAt (hf : AnalyticAt 𝕜 f x) : ContinuousAt f x :=
let ⟨_, hp⟩ := hf
hp.continuousAt
#align analytic_at.continuous_at AnalyticAt.continuousAt
protected theorem AnalyticOn.continuousOn {s : Set E} (hf : AnalyticOn 𝕜 f s) : ContinuousOn f s :=
fun x hx => (hf x hx).continuousAt.continuousWithinAt
#align analytic_on.continuous_on AnalyticOn.continuousOn
/-- Analytic everywhere implies continuous -/
theorem AnalyticOn.continuous {f : E → F} (fa : AnalyticOn 𝕜 f univ) : Continuous f := by
rw [continuous_iff_continuousOn_univ]; exact fa.continuousOn
/-- In a complete space, the sum of a converging power series `p` admits `p` as a power series.
This is not totally obvious as we need to check the convergence of the series. -/
protected theorem FormalMultilinearSeries.hasFPowerSeriesOnBall [CompleteSpace F]
(p : FormalMultilinearSeries 𝕜 E F) (h : 0 < p.radius) :
HasFPowerSeriesOnBall p.sum p 0 p.radius :=
{ r_le := le_rfl
r_pos := h
hasSum := fun hy => by
rw [zero_add]
exact p.hasSum hy }
#align formal_multilinear_series.has_fpower_series_on_ball FormalMultilinearSeries.hasFPowerSeriesOnBall
theorem HasFPowerSeriesOnBall.sum (h : HasFPowerSeriesOnBall f p x r) {y : E}
(hy : y ∈ EMetric.ball (0 : E) r) : f (x + y) = p.sum y :=
(h.hasSum hy).tsum_eq.symm
#align has_fpower_series_on_ball.sum HasFPowerSeriesOnBall.sum
/-- The sum of a converging power series is continuous in its disk of convergence. -/
protected theorem FormalMultilinearSeries.continuousOn [CompleteSpace F] :
ContinuousOn p.sum (EMetric.ball 0 p.radius) := by
rcases (zero_le p.radius).eq_or_lt with h | h
· simp [← h, continuousOn_empty]
· exact (p.hasFPowerSeriesOnBall h).continuousOn
#align formal_multilinear_series.continuous_on FormalMultilinearSeries.continuousOn
end
/-!
### Uniqueness of power series
If a function `f : E → F` has two representations as power series at a point `x : E`, corresponding
to formal multilinear series `p₁` and `p₂`, then these representations agree term-by-term. That is,
for any `n : ℕ` and `y : E`, `p₁ n (fun i ↦ y) = p₂ n (fun i ↦ y)`. In the one-dimensional case,
when `f : 𝕜 → E`, the continuous multilinear maps `p₁ n` and `p₂ n` are given by
`ContinuousMultilinearMap.mkPiField`, and hence are determined completely by the value of
`p₁ n (fun i ↦ 1)`, so `p₁ = p₂`. Consequently, the radius of convergence for one series can be
transferred to the other.
-/
section Uniqueness
open ContinuousMultilinearMap
theorem Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero {n : ℕ} {p : E[×n]→L[𝕜] F}
(h : (fun y => p fun _ => y) =O[𝓝 0] fun y => ‖y‖ ^ (n + 1)) (y : E) : (p fun _ => y) = 0 := by
obtain ⟨c, c_pos, hc⟩ := h.exists_pos
obtain ⟨t, ht, t_open, z_mem⟩ := eventually_nhds_iff.mp (isBigOWith_iff.mp hc)
obtain ⟨δ, δ_pos, δε⟩ := (Metric.isOpen_iff.mp t_open) 0 z_mem
clear h hc z_mem
cases' n with n
· exact norm_eq_zero.mp (by
-- porting note: the symmetric difference of the `simpa only` sets:
-- added `Nat.zero_eq, zero_add, pow_one`
-- removed `zero_pow', Ne.def, Nat.one_ne_zero, not_false_iff`
simpa only [Nat.zero_eq, fin0_apply_norm, norm_eq_zero, norm_zero, zero_add, pow_one,
mul_zero, norm_le_zero_iff] using ht 0 (δε (Metric.mem_ball_self δ_pos)))
· refine' Or.elim (Classical.em (y = 0))
(fun hy => by simpa only [hy] using p.map_zero) fun hy => _
replace hy := norm_pos_iff.mpr hy
refine' norm_eq_zero.mp (le_antisymm (le_of_forall_pos_le_add fun ε ε_pos => _) (norm_nonneg _))
have h₀ := _root_.mul_pos c_pos (pow_pos hy (n.succ + 1))
obtain ⟨k, k_pos, k_norm⟩ := NormedField.exists_norm_lt 𝕜
(lt_min (mul_pos δ_pos (inv_pos.mpr hy)) (mul_pos ε_pos (inv_pos.mpr h₀)))
have h₁ : ‖k • y‖ < δ := by
rw [norm_smul]
exact inv_mul_cancel_right₀ hy.ne.symm δ ▸
mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_left _ _)) hy
have h₂ :=
calc
‖p fun _ => k • y‖ ≤ c * ‖k • y‖ ^ (n.succ + 1) := by
-- porting note: now Lean wants `_root_.`
simpa only [norm_pow, _root_.norm_norm] using ht (k • y) (δε (mem_ball_zero_iff.mpr h₁))
--simpa only [norm_pow, norm_norm] using ht (k • y) (δε (mem_ball_zero_iff.mpr h₁))
_ = ‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))) := by
-- porting note: added `Nat.succ_eq_add_one` since otherwise `ring` does not conclude.
simp only [norm_smul, mul_pow, Nat.succ_eq_add_one]
-- porting note: removed `rw [pow_succ]`, since it now becomes superfluous.
ring
have h₃ : ‖k‖ * (c * ‖y‖ ^ (n.succ + 1)) < ε :=
inv_mul_cancel_right₀ h₀.ne.symm ε ▸
mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_right _ _)) h₀
calc
‖p fun _ => y‖ = ‖k⁻¹ ^ n.succ‖ * ‖p fun _ => k • y‖ := by
simpa only [inv_smul_smul₀ (norm_pos_iff.mp k_pos), norm_smul, Finset.prod_const,
Finset.card_fin] using
congr_arg norm (p.map_smul_univ (fun _ : Fin n.succ => k⁻¹) fun _ : Fin n.succ => k • y)
_ ≤ ‖k⁻¹ ^ n.succ‖ * (‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1)))) := by gcongr
_ = ‖(k⁻¹ * k) ^ n.succ‖ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))) := by
rw [← mul_assoc]
simp [norm_mul, mul_pow]
_ ≤ 0 + ε := by
rw [inv_mul_cancel (norm_pos_iff.mp k_pos)]
simpa using h₃.le
set_option linter.uppercaseLean3 false in
#align asymptotics.is_O.continuous_multilinear_map_apply_eq_zero Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero
/-- If a formal multilinear series `p` represents the zero function at `x : E`, then the
terms `p n (fun i ↦ y)` appearing in the sum are zero for any `n : ℕ`, `y : E`. -/
theorem HasFPowerSeriesAt.apply_eq_zero {p : FormalMultilinearSeries 𝕜 E F} {x : E}
(h : HasFPowerSeriesAt 0 p x) (n : ℕ) : ∀ y : E, (p n fun _ => y) = 0 := by
refine' Nat.strong_induction_on n fun k hk => _
have psum_eq : p.partialSum (k + 1) = fun y => p k fun _ => y := by
funext z
refine' Finset.sum_eq_single _ (fun b hb hnb => _) fun hn => _
· have := Finset.mem_range_succ_iff.mp hb
simp only [hk b (this.lt_of_ne hnb), Pi.zero_apply]
· exact False.elim (hn (Finset.mem_range.mpr (lt_add_one k)))
replace h := h.isBigO_sub_partialSum_pow k.succ
simp only [psum_eq, zero_sub, Pi.zero_apply, Asymptotics.isBigO_neg_left] at h
exact h.continuousMultilinearMap_apply_eq_zero
#align has_fpower_series_at.apply_eq_zero HasFPowerSeriesAt.apply_eq_zero
/-- A one-dimensional formal multilinear series representing the zero function is zero. -/
theorem HasFPowerSeriesAt.eq_zero {p : FormalMultilinearSeries 𝕜 𝕜 E} {x : 𝕜}
(h : HasFPowerSeriesAt 0 p x) : p = 0 := by
-- porting note: `funext; ext` was `ext (n x)`
funext n
ext x
rw [← mkPiField_apply_one_eq_self (p n)]
-- porting note: nasty hack, was `simp [h.apply_eq_zero n 1]`
have := Or.intro_right ?_ (h.apply_eq_zero n 1)
simpa using this
#align has_fpower_series_at.eq_zero HasFPowerSeriesAt.eq_zero
/-- One-dimensional formal multilinear series representing the same function are equal. -/
theorem HasFPowerSeriesAt.eq_formalMultilinearSeries {p₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E}
{f : 𝕜 → E} {x : 𝕜} (h₁ : HasFPowerSeriesAt f p₁ x) (h₂ : HasFPowerSeriesAt f p₂ x) : p₁ = p₂ :=
sub_eq_zero.mp (HasFPowerSeriesAt.eq_zero (by simpa only [sub_self] using h₁.sub h₂))
#align has_fpower_series_at.eq_formal_multilinear_series HasFPowerSeriesAt.eq_formalMultilinearSeries
theorem HasFPowerSeriesAt.eq_formalMultilinearSeries_of_eventually
{p q : FormalMultilinearSeries 𝕜 𝕜 E} {f g : 𝕜 → E} {x : 𝕜} (hp : HasFPowerSeriesAt f p x)
(hq : HasFPowerSeriesAt g q x) (heq : ∀ᶠ z in 𝓝 x, f z = g z) : p = q :=
(hp.congr heq).eq_formalMultilinearSeries hq
#align has_fpower_series_at.eq_formal_multilinear_series_of_eventually HasFPowerSeriesAt.eq_formalMultilinearSeries_of_eventually
/-- A one-dimensional formal multilinear series representing a locally zero function is zero. -/
theorem HasFPowerSeriesAt.eq_zero_of_eventually {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E}
{x : 𝕜} (hp : HasFPowerSeriesAt f p x) (hf : f =ᶠ[𝓝 x] 0) : p = 0 :=
(hp.congr hf).eq_zero
#align has_fpower_series_at.eq_zero_of_eventually HasFPowerSeriesAt.eq_zero_of_eventually
/-- If a function `f : 𝕜 → E` has two power series representations at `x`, then the given radii in
which convergence is guaranteed may be interchanged. This can be useful when the formal multilinear
series in one representation has a particularly nice form, but the other has a larger radius. -/
theorem HasFPowerSeriesOnBall.exchange_radius {p₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E}
{r₁ r₂ : ℝ≥0∞} {x : 𝕜} (h₁ : HasFPowerSeriesOnBall f p₁ x r₁)
(h₂ : HasFPowerSeriesOnBall f p₂ x r₂) : HasFPowerSeriesOnBall f p₁ x r₂ :=
h₂.hasFPowerSeriesAt.eq_formalMultilinearSeries h₁.hasFPowerSeriesAt ▸ h₂
#align has_fpower_series_on_ball.exchange_radius HasFPowerSeriesOnBall.exchange_radius
/-- If a function `f : 𝕜 → E` has power series representation `p` on a ball of some radius and for
each positive radius it has some power series representation, then `p` converges to `f` on the whole
`𝕜`. -/
theorem HasFPowerSeriesOnBall.r_eq_top_of_exists {f : 𝕜 → E} {r : ℝ≥0∞} {x : 𝕜}
{p : FormalMultilinearSeries 𝕜 𝕜 E} (h : HasFPowerSeriesOnBall f p x r)
(h' : ∀ (r' : ℝ≥0) (_ : 0 < r'), ∃ p' : FormalMultilinearSeries 𝕜 𝕜 E,
HasFPowerSeriesOnBall f p' x r') :
HasFPowerSeriesOnBall f p x ∞ :=
{ r_le := ENNReal.le_of_forall_pos_nnreal_lt fun r hr _ =>
let ⟨_, hp'⟩ := h' r hr
(h.exchange_radius hp').r_le
r_pos := ENNReal.coe_lt_top
hasSum := fun {y} _ =>
let ⟨r', hr'⟩ := exists_gt ‖y‖₊
let ⟨_, hp'⟩ := h' r' hr'.ne_bot.bot_lt
(h.exchange_radius hp').hasSum <| mem_emetric_ball_zero_iff.mpr (ENNReal.coe_lt_coe.2 hr') }
#align has_fpower_series_on_ball.r_eq_top_of_exists HasFPowerSeriesOnBall.r_eq_top_of_exists
end Uniqueness
/-!
### Changing origin in a power series
If a function is analytic in a disk `D(x, R)`, then it is analytic in any disk contained in that
one. Indeed, one can write
$$
f (x + y + z) = \sum_{n} p_n (y + z)^n = \sum_{n, k} \binom{n}{k} p_n y^{n-k} z^k
= \sum_{k} \Bigl(\sum_{n} \binom{n}{k} p_n y^{n-k}\Bigr) z^k.
$$
The corresponding power series has thus a `k`-th coefficient equal to
$\sum_{n} \binom{n}{k} p_n y^{n-k}$. In the general case where `pₙ` is a multilinear map, this has
to be interpreted suitably: instead of having a binomial coefficient, one should sum over all
possible subsets `s` of `Fin n` of cardinal `k`, and attribute `z` to the indices in `s` and
`y` to the indices outside of `s`.
In this paragraph, we implement this. The new power series is called `p.changeOrigin y`. Then, we
check its convergence and the fact that its sum coincides with the original sum. The outcome of this
discussion is that the set of points where a function is analytic is open.
-/
namespace FormalMultilinearSeries
section
variable (p : FormalMultilinearSeries 𝕜 E F) {x y : E} {r R : ℝ≥0}
/-- A term of `FormalMultilinearSeries.changeOriginSeries`.
Given a formal multilinear series `p` and a point `x` in its ball of convergence,
`p.changeOrigin x` is a formal multilinear series such that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense. Each term of `p.changeOrigin x`
is itself an analytic function of `x` given by the series `p.changeOriginSeries`. Each term in
`changeOriginSeries` is the sum of `changeOriginSeriesTerm`'s over all `s` of cardinality `l`.
The definition is such that `p.changeOriginSeriesTerm k l s hs (fun _ ↦ x) (fun _ ↦ y) =
p (k + l) (s.piecewise (fun _ ↦ x) (fun _ ↦ y))`
-/
def changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
E[×l]→L[𝕜] E[×k]→L[𝕜] F := by
let a := ContinuousMultilinearMap.curryFinFinset 𝕜 E F hs
(by erw [Finset.card_compl, Fintype.card_fin, hs, add_tsub_cancel_right])
exact a (p (k + l))
#align formal_multilinear_series.change_origin_series_term FormalMultilinearSeries.changeOriginSeriesTerm
theorem changeOriginSeriesTerm_apply (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l)
(x y : E) :
(p.changeOriginSeriesTerm k l s hs (fun _ => x) fun _ => y) =
p (k + l) (s.piecewise (fun _ => x) fun _ => y) :=
ContinuousMultilinearMap.curryFinFinset_apply_const _ _ _ _ _
#align formal_multilinear_series.change_origin_series_term_apply FormalMultilinearSeries.changeOriginSeriesTerm_apply
@[simp]
theorem norm_changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
‖p.changeOriginSeriesTerm k l s hs‖ = ‖p (k + l)‖ := by
simp only [changeOriginSeriesTerm, LinearIsometryEquiv.norm_map]
#align formal_multilinear_series.norm_change_origin_series_term FormalMultilinearSeries.norm_changeOriginSeriesTerm
@[simp]
theorem nnnorm_changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
‖p.changeOriginSeriesTerm k l s hs‖₊ = ‖p (k + l)‖₊ := by
simp only [changeOriginSeriesTerm, LinearIsometryEquiv.nnnorm_map]
#align formal_multilinear_series.nnnorm_change_origin_series_term FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm
theorem nnnorm_changeOriginSeriesTerm_apply_le (k l : ℕ) (s : Finset (Fin (k + l)))
(hs : s.card = l) (x y : E) :
‖p.changeOriginSeriesTerm k l s hs (fun _ => x) fun _ => y‖₊ ≤
‖p (k + l)‖₊ * ‖x‖₊ ^ l * ‖y‖₊ ^ k := by
rw [← p.nnnorm_changeOriginSeriesTerm k l s hs, ← Fin.prod_const, ← Fin.prod_const]
apply ContinuousMultilinearMap.le_of_op_nnnorm_le
apply ContinuousMultilinearMap.le_op_nnnorm
#align formal_multilinear_series.nnnorm_change_origin_series_term_apply_le FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm_apply_le
/-- The power series for `f.changeOrigin k`.
Given a formal multilinear series `p` and a point `x` in its ball of convergence,
`p.changeOrigin x` is a formal multilinear series such that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense. Its `k`-th term is the sum of
the series `p.changeOriginSeries k`. -/
def changeOriginSeries (k : ℕ) : FormalMultilinearSeries 𝕜 E (E[×k]→L[𝕜] F) := fun l =>
∑ s : { s : Finset (Fin (k + l)) // Finset.card s = l }, p.changeOriginSeriesTerm k l s s.2
#align formal_multilinear_series.change_origin_series FormalMultilinearSeries.changeOriginSeries
theorem nnnorm_changeOriginSeries_le_tsum (k l : ℕ) :
‖p.changeOriginSeries k l‖₊ ≤
∑' _ : { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + l)‖₊ :=
(nnnorm_sum_le _ (fun t => changeOriginSeriesTerm p k l (Subtype.val t) t.prop)).trans_eq <| by
simp_rw [tsum_fintype, nnnorm_changeOriginSeriesTerm (p := p) (k := k) (l := l)]
#align formal_multilinear_series.nnnorm_change_origin_series_le_tsum FormalMultilinearSeries.nnnorm_changeOriginSeries_le_tsum
theorem nnnorm_changeOriginSeries_apply_le_tsum (k l : ℕ) (x : E) :
‖p.changeOriginSeries k l fun _ => x‖₊ ≤
∑' _ : { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + l)‖₊ * ‖x‖₊ ^ l := by
rw [NNReal.tsum_mul_right, ← Fin.prod_const]
exact (p.changeOriginSeries k l).le_of_op_nnnorm_le _ (p.nnnorm_changeOriginSeries_le_tsum _ _)
#align formal_multilinear_series.nnnorm_change_origin_series_apply_le_tsum FormalMultilinearSeries.nnnorm_changeOriginSeries_apply_le_tsum
/-- Changing the origin of a formal multilinear series `p`, so that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense.
-/
def changeOrigin (x : E) : FormalMultilinearSeries 𝕜 E F :=
fun k => (p.changeOriginSeries k).sum x
#align formal_multilinear_series.change_origin FormalMultilinearSeries.changeOrigin
/-- An auxiliary equivalence useful in the proofs about
`FormalMultilinearSeries.changeOriginSeries`: the set of triples `(k, l, s)`, where `s` is a
`Finset (Fin (k + l))` of cardinality `l` is equivalent to the set of pairs `(n, s)`, where `s` is a
`Finset (Fin n)`.
The forward map sends `(k, l, s)` to `(k + l, s)` and the inverse map sends `(n, s)` to
`(n - Finset.card s, Finset.card s, s)`. The actual definition is less readable because of problems
with non-definitional equalities. -/
@[simps]
def changeOriginIndexEquiv :
(Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) ≃ Σn : ℕ, Finset (Fin n) where
toFun s := ⟨s.1 + s.2.1, s.2.2⟩
invFun s :=
⟨s.1 - s.2.card, s.2.card,
⟨s.2.map
(Fin.castIso <| (tsub_add_cancel_of_le <| card_finset_fin_le s.2).symm).toEquiv.toEmbedding,
Finset.card_map _⟩⟩
left_inv := by
rintro ⟨k, l, ⟨s : Finset (Fin <| k + l), hs : s.card = l⟩⟩
dsimp only [Subtype.coe_mk]
-- Lean can't automatically generalize `k' = k + l - s.card`, `l' = s.card`, so we explicitly
-- formulate the generalized goal
suffices ∀ k' l', k' = k → l' = l → ∀ (hkl : k + l = k' + l') (hs'),
(⟨k', l', ⟨Finset.map (Fin.castIso hkl).toEquiv.toEmbedding s, hs'⟩⟩ :
Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) = ⟨k, l, ⟨s, hs⟩⟩ by
apply this <;> simp only [hs, add_tsub_cancel_right]
rintro _ _ rfl rfl hkl hs'
simp only [Equiv.refl_toEmbedding, Fin.castIso_refl, Finset.map_refl, eq_self_iff_true,
OrderIso.refl_toEquiv, and_self_iff, heq_iff_eq]
right_inv := by
rintro ⟨n, s⟩
simp [tsub_add_cancel_of_le (card_finset_fin_le s), Fin.castIso_to_equiv]
#align formal_multilinear_series.change_origin_index_equiv FormalMultilinearSeries.changeOriginIndexEquiv
theorem changeOriginSeries_summable_aux₁ {r r' : ℝ≥0} (hr : (r + r' : ℝ≥0∞) < p.radius) :
Summable fun s : Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l } =>
‖p (s.1 + s.2.1)‖₊ * r ^ s.2.1 * r' ^ s.1 := by
rw [← changeOriginIndexEquiv.symm.summable_iff]
dsimp only [Function.comp_def, changeOriginIndexEquiv_symm_apply_fst,
changeOriginIndexEquiv_symm_apply_snd_fst]
have : ∀ n : ℕ,
HasSum (fun s : Finset (Fin n) => ‖p (n - s.card + s.card)‖₊ * r ^ s.card * r' ^ (n - s.card))
(‖p n‖₊ * (r + r') ^ n) := by
intro n
-- TODO: why `simp only [tsub_add_cancel_of_le (card_finset_fin_le _)]` fails?
convert_to HasSum (fun s : Finset (Fin n) => ‖p n‖₊ * (r ^ s.card * r' ^ (n - s.card))) _
· ext1 s
rw [tsub_add_cancel_of_le (card_finset_fin_le _), mul_assoc]
rw [← Fin.sum_pow_mul_eq_add_pow]
exact (hasSum_fintype _).mul_left _
refine' NNReal.summable_sigma.2 ⟨fun n => (this n).summable, _⟩
simp only [(this _).tsum_eq]
exact p.summable_nnnorm_mul_pow hr
#align formal_multilinear_series.change_origin_series_summable_aux₁ FormalMultilinearSeries.changeOriginSeries_summable_aux₁
theorem changeOriginSeries_summable_aux₂ (hr : (r : ℝ≥0∞) < p.radius) (k : ℕ) :
Summable fun s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l } =>
‖p (k + s.1)‖₊ * r ^ s.1 := by
rcases ENNReal.lt_iff_exists_add_pos_lt.1 hr with ⟨r', h0, hr'⟩
simpa only [mul_inv_cancel_right₀ (pow_pos h0 _).ne'] using
((NNReal.summable_sigma.1 (p.changeOriginSeries_summable_aux₁ hr')).1 k).mul_right (r' ^ k)⁻¹
#align formal_multilinear_series.change_origin_series_summable_aux₂ FormalMultilinearSeries.changeOriginSeries_summable_aux₂
theorem changeOriginSeries_summable_aux₃ {r : ℝ≥0} (hr : ↑r < p.radius) (k : ℕ) :
Summable fun l : ℕ => ‖p.changeOriginSeries k l‖₊ * r ^ l := by
refine' NNReal.summable_of_le
(fun n => _) (NNReal.summable_sigma.1 <| p.changeOriginSeries_summable_aux₂ hr k).2
simp only [NNReal.tsum_mul_right]
exact mul_le_mul' (p.nnnorm_changeOriginSeries_le_tsum _ _) le_rfl
#align formal_multilinear_series.change_origin_series_summable_aux₃ FormalMultilinearSeries.changeOriginSeries_summable_aux₃
theorem le_changeOriginSeries_radius (k : ℕ) : p.radius ≤ (p.changeOriginSeries k).radius :=
ENNReal.le_of_forall_nnreal_lt fun _r hr =>
le_radius_of_summable_nnnorm _ (p.changeOriginSeries_summable_aux₃ hr k)
#align formal_multilinear_series.le_change_origin_series_radius FormalMultilinearSeries.le_changeOriginSeries_radius
theorem nnnorm_changeOrigin_le (k : ℕ) (h : (‖x‖₊ : ℝ≥0∞) < p.radius) :
‖p.changeOrigin x k‖₊ ≤
∑' s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + s.1)‖₊ * ‖x‖₊ ^ s.1 := by
refine' tsum_of_nnnorm_bounded _ fun l => p.nnnorm_changeOriginSeries_apply_le_tsum k l x
have := p.changeOriginSeries_summable_aux₂ h k
refine' HasSum.sigma this.hasSum fun l => _
exact ((NNReal.summable_sigma.1 this).1 l).hasSum
#align formal_multilinear_series.nnnorm_change_origin_le FormalMultilinearSeries.nnnorm_changeOrigin_le
/-- The radius of convergence of `p.changeOrigin x` is at least `p.radius - ‖x‖`. In other words,
`p.changeOrigin x` is well defined on the largest ball contained in the original ball of
convergence. -/
theorem changeOrigin_radius : p.radius - ‖x‖₊ ≤ (p.changeOrigin x).radius := by
refine' ENNReal.le_of_forall_pos_nnreal_lt fun r _h0 hr => _
rw [lt_tsub_iff_right, add_comm] at hr
have hr' : (‖x‖₊ : ℝ≥0∞) < p.radius := (le_add_right le_rfl).trans_lt hr
apply le_radius_of_summable_nnnorm
have : ∀ k : ℕ,
‖p.changeOrigin x k‖₊ * r ^ k ≤
(∑' s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + s.1)‖₊ * ‖x‖₊ ^ s.1) *
r ^ k :=
fun k => mul_le_mul_right' (p.nnnorm_changeOrigin_le k hr') (r ^ k)
refine' NNReal.summable_of_le this _
simpa only [← NNReal.tsum_mul_right] using
(NNReal.summable_sigma.1 (p.changeOriginSeries_summable_aux₁ hr)).2
#align formal_multilinear_series.change_origin_radius FormalMultilinearSeries.changeOrigin_radius
end
-- From this point on, assume that the space is complete, to make sure that series that converge
-- in norm also converge in `F`.
variable [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x y : E} {r R : ℝ≥0}
theorem hasFPowerSeriesOnBall_changeOrigin (k : ℕ) (hr : 0 < p.radius) :
HasFPowerSeriesOnBall (fun x => p.changeOrigin x k) (p.changeOriginSeries k) 0 p.radius :=
have := p.le_changeOriginSeries_radius k
((p.changeOriginSeries k).hasFPowerSeriesOnBall (hr.trans_le this)).mono hr this
#align formal_multilinear_series.has_fpower_series_on_ball_change_origin FormalMultilinearSeries.hasFPowerSeriesOnBall_changeOrigin
/-- Summing the series `p.changeOrigin x` at a point `y` gives back `p (x + y)`. -/
theorem changeOrigin_eval (h : (‖x‖₊ + ‖y‖₊ : ℝ≥0∞) < p.radius) :
(p.changeOrigin x).sum y = p.sum (x + y) := by
have radius_pos : 0 < p.radius := lt_of_le_of_lt (zero_le _) h
have x_mem_ball : x ∈ EMetric.ball (0 : E) p.radius :=
mem_emetric_ball_zero_iff.2 ((le_add_right le_rfl).trans_lt h)
have y_mem_ball : y ∈ EMetric.ball (0 : E) (p.changeOrigin x).radius := by
refine' mem_emetric_ball_zero_iff.2 (lt_of_lt_of_le _ p.changeOrigin_radius)
rwa [lt_tsub_iff_right, add_comm]
have x_add_y_mem_ball : x + y ∈ EMetric.ball (0 : E) p.radius := by
refine' mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt _ h)
exact mod_cast nnnorm_add_le x y
set f : (Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) → F := fun s =>
p.changeOriginSeriesTerm s.1 s.2.1 s.2.2 s.2.2.2 (fun _ => x) fun _ => y
have hsf : Summable f := by
refine' .of_nnnorm_bounded _ (p.changeOriginSeries_summable_aux₁ h) _
rintro ⟨k, l, s, hs⟩
dsimp only [Subtype.coe_mk]
exact p.nnnorm_changeOriginSeriesTerm_apply_le _ _ _ _ _ _
have hf : HasSum f ((p.changeOrigin x).sum y) := by
refine' HasSum.sigma_of_hasSum ((p.changeOrigin x).summable y_mem_ball).hasSum (fun k => _) hsf
· dsimp only
refine' ContinuousMultilinearMap.hasSum_eval _ _
have := (p.hasFPowerSeriesOnBall_changeOrigin k radius_pos).hasSum x_mem_ball
rw [zero_add] at this
refine' HasSum.sigma_of_hasSum this (fun l => _) _
· simp only [changeOriginSeries, ContinuousMultilinearMap.sum_apply]
apply hasSum_fintype
· refine' .of_nnnorm_bounded _
(p.changeOriginSeries_summable_aux₂ (mem_emetric_ball_zero_iff.1 x_mem_ball) k) fun s => _
refine' (ContinuousMultilinearMap.le_op_nnnorm _ _).trans_eq _
simp
refine' hf.unique (changeOriginIndexEquiv.symm.hasSum_iff.1 _)
refine' HasSum.sigma_of_hasSum
(p.hasSum x_add_y_mem_ball) (fun n => _) (changeOriginIndexEquiv.symm.summable_iff.2 hsf)
erw [(p n).map_add_univ (fun _ => x) fun _ => y]
-- porting note: added explicit function
convert hasSum_fintype (fun c : Finset (Fin n) => f (changeOriginIndexEquiv.symm ⟨n, c⟩))
rename_i s _
dsimp only [changeOriginSeriesTerm, (· ∘ ·), changeOriginIndexEquiv_symm_apply_fst,
changeOriginIndexEquiv_symm_apply_snd_fst, changeOriginIndexEquiv_symm_apply_snd_snd_coe]
rw [ContinuousMultilinearMap.curryFinFinset_apply_const]
have : ∀ (m) (hm : n = m), p n (s.piecewise (fun _ => x) fun _ => y) =
p m ((s.map (Fin.castIso hm).toEquiv.toEmbedding).piecewise (fun _ => x) fun _ => y) := by
rintro m rfl
simp (config := { unfoldPartialApp := true }) [Finset.piecewise]
apply this
#align formal_multilinear_series.change_origin_eval FormalMultilinearSeries.changeOrigin_eval
/-- Power series terms are analytic as we vary the origin -/
theorem analyticAt_changeOrigin (p : FormalMultilinearSeries 𝕜 E F) (rp : p.radius > 0) (n : ℕ) :
AnalyticAt 𝕜 (fun x ↦ p.changeOrigin x n) 0 :=
(FormalMultilinearSeries.hasFPowerSeriesOnBall_changeOrigin p n rp).analyticAt
end FormalMultilinearSeries
section
variable [CompleteSpace F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x y : E} {r : ℝ≥0∞}
/-- If a function admits a power series expansion `p` on a ball `B (x, r)`, then it also admits a
power series on any subball of this ball (even with a different center), given by `p.changeOrigin`.
-/
theorem HasFPowerSeriesOnBall.changeOrigin (hf : HasFPowerSeriesOnBall f p x r)
(h : (‖y‖₊ : ℝ≥0∞) < r) : HasFPowerSeriesOnBall f (p.changeOrigin y) (x + y) (r - ‖y‖₊) :=
{ r_le := by
apply le_trans _ p.changeOrigin_radius
exact tsub_le_tsub hf.r_le le_rfl
r_pos := by simp [h]
hasSum := fun {z} hz => by
have : f (x + y + z) =
FormalMultilinearSeries.sum (FormalMultilinearSeries.changeOrigin p y) z := by
rw [mem_emetric_ball_zero_iff, lt_tsub_iff_right, add_comm] at hz
rw [p.changeOrigin_eval (hz.trans_le hf.r_le), add_assoc, hf.sum]
refine' mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt _ hz)
exact mod_cast nnnorm_add_le y z
rw [this]
apply (p.changeOrigin y).hasSum
refine' EMetric.ball_subset_ball (le_trans _ p.changeOrigin_radius) hz
exact tsub_le_tsub hf.r_le le_rfl }
#align has_fpower_series_on_ball.change_origin HasFPowerSeriesOnBall.changeOrigin
/-- If a function admits a power series expansion `p` on an open ball `B (x, r)`, then
it is analytic at every point of this ball. -/
theorem HasFPowerSeriesOnBall.analyticAt_of_mem (hf : HasFPowerSeriesOnBall f p x r)
(h : y ∈ EMetric.ball x r) : AnalyticAt 𝕜 f y := by
have : (‖y - x‖₊ : ℝ≥0∞) < r := by simpa [edist_eq_coe_nnnorm_sub] using h
have := hf.changeOrigin this
rw [add_sub_cancel'_right] at this
exact this.analyticAt
#align has_fpower_series_on_ball.analytic_at_of_mem HasFPowerSeriesOnBall.analyticAt_of_mem
theorem HasFPowerSeriesOnBall.analyticOn (hf : HasFPowerSeriesOnBall f p x r) :
AnalyticOn 𝕜 f (EMetric.ball x r) :=
fun _y hy => hf.analyticAt_of_mem hy
#align has_fpower_series_on_ball.analytic_on HasFPowerSeriesOnBall.analyticOn
variable (𝕜 f)
/-- For any function `f` from a normed vector space to a Banach space, the set of points `x` such
that `f` is analytic at `x` is open. -/
theorem isOpen_analyticAt : IsOpen { x | AnalyticAt 𝕜 f x } := by
rw [isOpen_iff_mem_nhds]
rintro x ⟨p, r, hr⟩
exact mem_of_superset (EMetric.ball_mem_nhds _ hr.r_pos) fun y hy => hr.analyticAt_of_mem hy
#align is_open_analytic_at isOpen_analyticAt
variable {𝕜}
theorem AnalyticAt.eventually_analyticAt {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∀ᶠ y in 𝓝 x, AnalyticAt 𝕜 f y :=
(isOpen_analyticAt 𝕜 f).mem_nhds h
theorem AnalyticAt.exists_mem_nhds_analyticOn {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∃ s ∈ 𝓝 x, AnalyticOn 𝕜 f s :=
h.eventually_analyticAt.exists_mem
/-- If we're analytic at a point, we're analytic in a nonempty ball -/
theorem AnalyticAt.exists_ball_analyticOn {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∃ r : ℝ, 0 < r ∧ AnalyticOn 𝕜 f (Metric.ball x r) :=
Metric.isOpen_iff.mp (isOpen_analyticAt _ _) _ h
end
section
open FormalMultilinearSeries
variable {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E} {z₀ : 𝕜}
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z)) := by
refine' ⟨fun ⟨r, _, r_pos, h⟩ =>
eventually_of_mem (EMetric.ball_mem_nhds 0 r_pos) fun _ => by simpa using h, _⟩
simp only [Metric.eventually_nhds_iff]
rintro ⟨r, r_pos, h⟩
refine' ⟨p.radius ⊓ r.toNNReal, by
|
simp
|
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z)) := by
refine' ⟨fun ⟨r, _, r_pos, h⟩ =>
eventually_of_mem (EMetric.ball_mem_nhds 0 r_pos) fun _ => by simpa using h, _⟩
simp only [Metric.eventually_nhds_iff]
rintro ⟨r, r_pos, h⟩
refine' ⟨p.radius ⊓ r.toNNReal, by
|
Mathlib.Analysis.Analytic.Basic.1430_0.jQw1fRSE1vGpOll
|
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z))
|
Mathlib_Analysis_Analytic_Basic
|
case intro.intro.refine'_1
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
p : FormalMultilinearSeries 𝕜 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
r : ℝ
r_pos : r > 0
h : ∀ ⦃y : 𝕜⦄, dist y 0 < r → HasSum (fun n => y ^ n • coeff p n) (f (z₀ + y))
⊢ 0 < radius p ⊓ ↑(Real.toNNReal r)
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Logic.Equiv.Fin
import Mathlib.Topology.Algebra.InfiniteSum.Module
#align_import analysis.analytic.basic from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
/-!
# Analytic functions
A function is analytic in one dimension around `0` if it can be written as a converging power series
`Σ pₙ zⁿ`. This definition can be extended to any dimension (even in infinite dimension) by
requiring that `pₙ` is a continuous `n`-multilinear map. In general, `pₙ` is not unique (in two
dimensions, taking `p₂ (x, y) (x', y') = x y'` or `y x'` gives the same map when applied to a
vector `(x, y) (x, y)`). A way to guarantee uniqueness is to take a symmetric `pₙ`, but this is not
always possible in nonzero characteristic (in characteristic 2, the previous example has no
symmetric representative). Therefore, we do not insist on symmetry or uniqueness in the definition,
and we only require the existence of a converging series.
The general framework is important to say that the exponential map on bounded operators on a Banach
space is analytic, as well as the inverse on invertible operators.
## Main definitions
Let `p` be a formal multilinear series from `E` to `F`, i.e., `p n` is a multilinear map on `E^n`
for `n : ℕ`.
* `p.radius`: the largest `r : ℝ≥0∞` such that `‖p n‖ * r^n` grows subexponentially.
* `p.le_radius_of_bound`, `p.le_radius_of_bound_nnreal`, `p.le_radius_of_isBigO`: if `‖p n‖ * r ^ n`
is bounded above, then `r ≤ p.radius`;
* `p.isLittleO_of_lt_radius`, `p.norm_mul_pow_le_mul_pow_of_lt_radius`,
`p.isLittleO_one_of_lt_radius`,
`p.norm_mul_pow_le_of_lt_radius`, `p.nnnorm_mul_pow_le_of_lt_radius`: if `r < p.radius`, then
`‖p n‖ * r ^ n` tends to zero exponentially;
* `p.lt_radius_of_isBigO`: if `r ≠ 0` and `‖p n‖ * r ^ n = O(a ^ n)` for some `-1 < a < 1`, then
`r < p.radius`;
* `p.partialSum n x`: the sum `∑_{i = 0}^{n-1} pᵢ xⁱ`.
* `p.sum x`: the sum `∑'_{i = 0}^{∞} pᵢ xⁱ`.
Additionally, let `f` be a function from `E` to `F`.
* `HasFPowerSeriesOnBall f p x r`: on the ball of center `x` with radius `r`,
`f (x + y) = ∑'_n pₙ yⁿ`.
* `HasFPowerSeriesAt f p x`: on some ball of center `x` with positive radius, holds
`HasFPowerSeriesOnBall f p x r`.
* `AnalyticAt 𝕜 f x`: there exists a power series `p` such that holds `HasFPowerSeriesAt f p x`.
* `AnalyticOn 𝕜 f s`: the function `f` is analytic at every point of `s`.
We develop the basic properties of these notions, notably:
* If a function admits a power series, it is continuous (see
`HasFPowerSeriesOnBall.continuousOn` and `HasFPowerSeriesAt.continuousAt` and
`AnalyticAt.continuousAt`).
* In a complete space, the sum of a formal power series with positive radius is well defined on the
disk of convergence, see `FormalMultilinearSeries.hasFPowerSeriesOnBall`.
* If a function admits a power series in a ball, then it is analytic at any point `y` of this ball,
and the power series there can be expressed in terms of the initial power series `p` as
`p.changeOrigin y`. See `HasFPowerSeriesOnBall.changeOrigin`. It follows in particular that
the set of points at which a given function is analytic is open, see `isOpen_analyticAt`.
## Implementation details
We only introduce the radius of convergence of a power series, as `p.radius`.
For a power series in finitely many dimensions, there is a finer (directional, coordinate-dependent)
notion, describing the polydisk of convergence. This notion is more specific, and not necessary to
build the general theory. We do not define it here.
-/
noncomputable section
variable {𝕜 E F G : Type*}
open Topology Classical BigOperators NNReal Filter ENNReal
open Set Filter Asymptotics
namespace FormalMultilinearSeries
variable [Ring 𝕜] [AddCommGroup E] [AddCommGroup F] [Module 𝕜 E] [Module 𝕜 F]
variable [TopologicalSpace E] [TopologicalSpace F]
variable [TopologicalAddGroup E] [TopologicalAddGroup F]
variable [ContinuousConstSMul 𝕜 E] [ContinuousConstSMul 𝕜 F]
/-- Given a formal multilinear series `p` and a vector `x`, then `p.sum x` is the sum `Σ pₙ xⁿ`. A
priori, it only behaves well when `‖x‖ < p.radius`. -/
protected def sum (p : FormalMultilinearSeries 𝕜 E F) (x : E) : F :=
∑' n : ℕ, p n fun _ => x
#align formal_multilinear_series.sum FormalMultilinearSeries.sum
/-- Given a formal multilinear series `p` and a vector `x`, then `p.partialSum n x` is the sum
`Σ pₖ xᵏ` for `k ∈ {0,..., n-1}`. -/
def partialSum (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) (x : E) : F :=
∑ k in Finset.range n, p k fun _ : Fin k => x
#align formal_multilinear_series.partial_sum FormalMultilinearSeries.partialSum
/-- The partial sums of a formal multilinear series are continuous. -/
theorem partialSum_continuous (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) :
Continuous (p.partialSum n) := by
unfold partialSum -- Porting note: added
continuity
#align formal_multilinear_series.partial_sum_continuous FormalMultilinearSeries.partialSum_continuous
end FormalMultilinearSeries
/-! ### The radius of a formal multilinear series -/
variable [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F]
[NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G]
namespace FormalMultilinearSeries
variable (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
/-- The radius of a formal multilinear series is the largest `r` such that the sum `Σ ‖pₙ‖ ‖y‖ⁿ`
converges for all `‖y‖ < r`. This implies that `Σ pₙ yⁿ` converges for all `‖y‖ < r`, but these
definitions are *not* equivalent in general. -/
def radius (p : FormalMultilinearSeries 𝕜 E F) : ℝ≥0∞ :=
⨆ (r : ℝ≥0) (C : ℝ) (_ : ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C), (r : ℝ≥0∞)
#align formal_multilinear_series.radius FormalMultilinearSeries.radius
/-- If `‖pₙ‖ rⁿ` is bounded in `n`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_bound (C : ℝ) {r : ℝ≥0} (h : ∀ n : ℕ, ‖p n‖ * (r : ℝ) ^ n ≤ C) :
(r : ℝ≥0∞) ≤ p.radius :=
le_iSup_of_le r <| le_iSup_of_le C <| le_iSup (fun _ => (r : ℝ≥0∞)) h
#align formal_multilinear_series.le_radius_of_bound FormalMultilinearSeries.le_radius_of_bound
/-- If `‖pₙ‖ rⁿ` is bounded in `n`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_bound_nnreal (C : ℝ≥0) {r : ℝ≥0} (h : ∀ n : ℕ, ‖p n‖₊ * r ^ n ≤ C) :
(r : ℝ≥0∞) ≤ p.radius :=
p.le_radius_of_bound C fun n => mod_cast h n
#align formal_multilinear_series.le_radius_of_bound_nnreal FormalMultilinearSeries.le_radius_of_bound_nnreal
/-- If `‖pₙ‖ rⁿ = O(1)`, as `n → ∞`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_isBigO (h : (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] fun _ => (1 : ℝ)) :
↑r ≤ p.radius :=
Exists.elim (isBigO_one_nat_atTop_iff.1 h) fun C hC =>
p.le_radius_of_bound C fun n => (le_abs_self _).trans (hC n)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.le_radius_of_is_O FormalMultilinearSeries.le_radius_of_isBigO
theorem le_radius_of_eventually_le (C) (h : ∀ᶠ n in atTop, ‖p n‖ * (r : ℝ) ^ n ≤ C) :
↑r ≤ p.radius :=
p.le_radius_of_isBigO <| IsBigO.of_bound C <| h.mono fun n hn => by simpa
#align formal_multilinear_series.le_radius_of_eventually_le FormalMultilinearSeries.le_radius_of_eventually_le
theorem le_radius_of_summable_nnnorm (h : Summable fun n => ‖p n‖₊ * r ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_bound_nnreal (∑' n, ‖p n‖₊ * r ^ n) fun _ => le_tsum' h _
#align formal_multilinear_series.le_radius_of_summable_nnnorm FormalMultilinearSeries.le_radius_of_summable_nnnorm
theorem le_radius_of_summable (h : Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_summable_nnnorm <| by
simp only [← coe_nnnorm] at h
exact mod_cast h
#align formal_multilinear_series.le_radius_of_summable FormalMultilinearSeries.le_radius_of_summable
theorem radius_eq_top_of_forall_nnreal_isBigO
(h : ∀ r : ℝ≥0, (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] fun _ => (1 : ℝ)) : p.radius = ∞ :=
ENNReal.eq_top_of_forall_nnreal_le fun r => p.le_radius_of_isBigO (h r)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.radius_eq_top_of_forall_nnreal_is_O FormalMultilinearSeries.radius_eq_top_of_forall_nnreal_isBigO
theorem radius_eq_top_of_eventually_eq_zero (h : ∀ᶠ n in atTop, p n = 0) : p.radius = ∞ :=
p.radius_eq_top_of_forall_nnreal_isBigO fun r =>
(isBigO_zero _ _).congr' (h.mono fun n hn => by simp [hn]) EventuallyEq.rfl
#align formal_multilinear_series.radius_eq_top_of_eventually_eq_zero FormalMultilinearSeries.radius_eq_top_of_eventually_eq_zero
theorem radius_eq_top_of_forall_image_add_eq_zero (n : ℕ) (hn : ∀ m, p (m + n) = 0) :
p.radius = ∞ :=
p.radius_eq_top_of_eventually_eq_zero <|
mem_atTop_sets.2 ⟨n, fun _ hk => tsub_add_cancel_of_le hk ▸ hn _⟩
#align formal_multilinear_series.radius_eq_top_of_forall_image_add_eq_zero FormalMultilinearSeries.radius_eq_top_of_forall_image_add_eq_zero
@[simp]
theorem constFormalMultilinearSeries_radius {v : F} :
(constFormalMultilinearSeries 𝕜 E v).radius = ⊤ :=
(constFormalMultilinearSeries 𝕜 E v).radius_eq_top_of_forall_image_add_eq_zero 1
(by simp [constFormalMultilinearSeries])
#align formal_multilinear_series.const_formal_multilinear_series_radius FormalMultilinearSeries.constFormalMultilinearSeries_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` tends to zero exponentially:
for some `0 < a < 1`, `‖p n‖ rⁿ = o(aⁿ)`. -/
theorem isLittleO_of_lt_radius (h : ↑r < p.radius) :
∃ a ∈ Ioo (0 : ℝ) 1, (fun n => ‖p n‖ * (r : ℝ) ^ n) =o[atTop] (a ^ ·) := by
have := (TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 4
rw [this]
-- Porting note: was
-- rw [(TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 4]
simp only [radius, lt_iSup_iff] at h
rcases h with ⟨t, C, hC, rt⟩
rw [ENNReal.coe_lt_coe, ← NNReal.coe_lt_coe] at rt
have : 0 < (t : ℝ) := r.coe_nonneg.trans_lt rt
rw [← div_lt_one this] at rt
refine' ⟨_, rt, C, Or.inr zero_lt_one, fun n => _⟩
calc
|‖p n‖ * (r : ℝ) ^ n| = ‖p n‖ * (t : ℝ) ^ n * (r / t : ℝ) ^ n := by
field_simp [mul_right_comm, abs_mul]
_ ≤ C * (r / t : ℝ) ^ n := by gcongr; apply hC
#align formal_multilinear_series.is_o_of_lt_radius FormalMultilinearSeries.isLittleO_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ = o(1)`. -/
theorem isLittleO_one_of_lt_radius (h : ↑r < p.radius) :
(fun n => ‖p n‖ * (r : ℝ) ^ n) =o[atTop] (fun _ => 1 : ℕ → ℝ) :=
let ⟨_, ha, hp⟩ := p.isLittleO_of_lt_radius h
hp.trans <| (isLittleO_pow_pow_of_lt_left ha.1.le ha.2).congr (fun _ => rfl) one_pow
#align formal_multilinear_series.is_o_one_of_lt_radius FormalMultilinearSeries.isLittleO_one_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` tends to zero exponentially:
for some `0 < a < 1` and `C > 0`, `‖p n‖ * r ^ n ≤ C * a ^ n`. -/
theorem norm_mul_pow_le_mul_pow_of_lt_radius (h : ↑r < p.radius) :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C * a ^ n := by
-- Porting note: moved out of `rcases`
have := ((TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 5).mp
(p.isLittleO_of_lt_radius h)
rcases this with ⟨a, ha, C, hC, H⟩
exact ⟨a, ha, C, hC, fun n => (le_abs_self _).trans (H n)⟩
#align formal_multilinear_series.norm_mul_pow_le_mul_pow_of_lt_radius FormalMultilinearSeries.norm_mul_pow_le_mul_pow_of_lt_radius
/-- If `r ≠ 0` and `‖pₙ‖ rⁿ = O(aⁿ)` for some `-1 < a < 1`, then `r < p.radius`. -/
theorem lt_radius_of_isBigO (h₀ : r ≠ 0) {a : ℝ} (ha : a ∈ Ioo (-1 : ℝ) 1)
(hp : (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] (a ^ ·)) : ↑r < p.radius := by
-- Porting note: moved out of `rcases`
have := ((TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 2 5)
rcases this.mp ⟨a, ha, hp⟩ with ⟨a, ha, C, hC, hp⟩
rw [← pos_iff_ne_zero, ← NNReal.coe_pos] at h₀
lift a to ℝ≥0 using ha.1.le
have : (r : ℝ) < r / a := by
simpa only [div_one] using (div_lt_div_left h₀ zero_lt_one ha.1).2 ha.2
norm_cast at this
rw [← ENNReal.coe_lt_coe] at this
refine' this.trans_le (p.le_radius_of_bound C fun n => _)
rw [NNReal.coe_div, div_pow, ← mul_div_assoc, div_le_iff (pow_pos ha.1 n)]
exact (le_abs_self _).trans (hp n)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.lt_radius_of_is_O FormalMultilinearSeries.lt_radius_of_isBigO
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem norm_mul_pow_le_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C :=
let ⟨_, ha, C, hC, h⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius h
⟨C, hC, fun n => (h n).trans <| mul_le_of_le_one_right hC.lt.le (pow_le_one _ ha.1.le ha.2.le)⟩
#align formal_multilinear_series.norm_mul_pow_le_of_lt_radius FormalMultilinearSeries.norm_mul_pow_le_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem norm_le_div_pow_of_pos_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h0 : 0 < r) (h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖ ≤ C / (r : ℝ) ^ n :=
let ⟨C, hC, hp⟩ := p.norm_mul_pow_le_of_lt_radius h
⟨C, hC, fun n => Iff.mpr (le_div_iff (pow_pos h0 _)) (hp n)⟩
#align formal_multilinear_series.norm_le_div_pow_of_pos_of_lt_radius FormalMultilinearSeries.norm_le_div_pow_of_pos_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem nnnorm_mul_pow_le_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖₊ * r ^ n ≤ C :=
let ⟨C, hC, hp⟩ := p.norm_mul_pow_le_of_lt_radius h
⟨⟨C, hC.lt.le⟩, hC, mod_cast hp⟩
#align formal_multilinear_series.nnnorm_mul_pow_le_of_lt_radius FormalMultilinearSeries.nnnorm_mul_pow_le_of_lt_radius
theorem le_radius_of_tendsto (p : FormalMultilinearSeries 𝕜 E F) {l : ℝ}
(h : Tendsto (fun n => ‖p n‖ * (r : ℝ) ^ n) atTop (𝓝 l)) : ↑r ≤ p.radius :=
p.le_radius_of_isBigO (h.isBigO_one _)
#align formal_multilinear_series.le_radius_of_tendsto FormalMultilinearSeries.le_radius_of_tendsto
theorem le_radius_of_summable_norm (p : FormalMultilinearSeries 𝕜 E F)
(hs : Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_tendsto hs.tendsto_atTop_zero
#align formal_multilinear_series.le_radius_of_summable_norm FormalMultilinearSeries.le_radius_of_summable_norm
theorem not_summable_norm_of_radius_lt_nnnorm (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(h : p.radius < ‖x‖₊) : ¬Summable fun n => ‖p n‖ * ‖x‖ ^ n :=
fun hs => not_le_of_lt h (p.le_radius_of_summable_norm hs)
#align formal_multilinear_series.not_summable_norm_of_radius_lt_nnnorm FormalMultilinearSeries.not_summable_norm_of_radius_lt_nnnorm
theorem summable_norm_mul_pow (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0} (h : ↑r < p.radius) :
Summable fun n : ℕ => ‖p n‖ * (r : ℝ) ^ n := by
obtain ⟨a, ha : a ∈ Ioo (0 : ℝ) 1, C, - : 0 < C, hp⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius h
exact .of_nonneg_of_le (fun n => mul_nonneg (norm_nonneg _) (pow_nonneg r.coe_nonneg _))
hp ((summable_geometric_of_lt_1 ha.1.le ha.2).mul_left _)
#align formal_multilinear_series.summable_norm_mul_pow FormalMultilinearSeries.summable_norm_mul_pow
theorem summable_norm_apply (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : Summable fun n : ℕ => ‖p n fun _ => x‖ := by
rw [mem_emetric_ball_zero_iff] at hx
refine' .of_nonneg_of_le
(fun _ => norm_nonneg _) (fun n => ((p n).le_op_norm _).trans_eq _) (p.summable_norm_mul_pow hx)
simp
#align formal_multilinear_series.summable_norm_apply FormalMultilinearSeries.summable_norm_apply
theorem summable_nnnorm_mul_pow (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0} (h : ↑r < p.radius) :
Summable fun n : ℕ => ‖p n‖₊ * r ^ n := by
rw [← NNReal.summable_coe]
push_cast
exact p.summable_norm_mul_pow h
#align formal_multilinear_series.summable_nnnorm_mul_pow FormalMultilinearSeries.summable_nnnorm_mul_pow
protected theorem summable [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : Summable fun n : ℕ => p n fun _ => x :=
(p.summable_norm_apply hx).of_norm
#align formal_multilinear_series.summable FormalMultilinearSeries.summable
theorem radius_eq_top_of_summable_norm (p : FormalMultilinearSeries 𝕜 E F)
(hs : ∀ r : ℝ≥0, Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : p.radius = ∞ :=
ENNReal.eq_top_of_forall_nnreal_le fun r => p.le_radius_of_summable_norm (hs r)
#align formal_multilinear_series.radius_eq_top_of_summable_norm FormalMultilinearSeries.radius_eq_top_of_summable_norm
theorem radius_eq_top_iff_summable_norm (p : FormalMultilinearSeries 𝕜 E F) :
p.radius = ∞ ↔ ∀ r : ℝ≥0, Summable fun n => ‖p n‖ * (r : ℝ) ^ n := by
constructor
· intro h r
obtain ⟨a, ha : a ∈ Ioo (0 : ℝ) 1, C, - : 0 < C, hp⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius
(show (r : ℝ≥0∞) < p.radius from h.symm ▸ ENNReal.coe_lt_top)
refine' .of_norm_bounded
(fun n => (C : ℝ) * a ^ n) ((summable_geometric_of_lt_1 ha.1.le ha.2).mul_left _) fun n => _
specialize hp n
rwa [Real.norm_of_nonneg (mul_nonneg (norm_nonneg _) (pow_nonneg r.coe_nonneg n))]
· exact p.radius_eq_top_of_summable_norm
#align formal_multilinear_series.radius_eq_top_iff_summable_norm FormalMultilinearSeries.radius_eq_top_iff_summable_norm
/-- If the radius of `p` is positive, then `‖pₙ‖` grows at most geometrically. -/
theorem le_mul_pow_of_radius_pos (p : FormalMultilinearSeries 𝕜 E F) (h : 0 < p.radius) :
∃ (C r : _) (hC : 0 < C) (_ : 0 < r), ∀ n, ‖p n‖ ≤ C * r ^ n := by
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 h with ⟨r, r0, rlt⟩
have rpos : 0 < (r : ℝ) := by simp [ENNReal.coe_pos.1 r0]
rcases norm_le_div_pow_of_pos_of_lt_radius p rpos rlt with ⟨C, Cpos, hCp⟩
refine' ⟨C, r⁻¹, Cpos, by simp only [inv_pos, rpos], fun n => _⟩
-- Porting note: was `convert`
rw [inv_pow, ← div_eq_mul_inv]
exact hCp n
#align formal_multilinear_series.le_mul_pow_of_radius_pos FormalMultilinearSeries.le_mul_pow_of_radius_pos
/-- The radius of the sum of two formal series is at least the minimum of their two radii. -/
theorem min_radius_le_radius_add (p q : FormalMultilinearSeries 𝕜 E F) :
min p.radius q.radius ≤ (p + q).radius := by
refine' ENNReal.le_of_forall_nnreal_lt fun r hr => _
rw [lt_min_iff] at hr
have := ((p.isLittleO_one_of_lt_radius hr.1).add (q.isLittleO_one_of_lt_radius hr.2)).isBigO
refine' (p + q).le_radius_of_isBigO ((isBigO_of_le _ fun n => _).trans this)
rw [← add_mul, norm_mul, norm_mul, norm_norm]
exact mul_le_mul_of_nonneg_right ((norm_add_le _ _).trans (le_abs_self _)) (norm_nonneg _)
#align formal_multilinear_series.min_radius_le_radius_add FormalMultilinearSeries.min_radius_le_radius_add
@[simp]
theorem radius_neg (p : FormalMultilinearSeries 𝕜 E F) : (-p).radius = p.radius := by
simp only [radius, neg_apply, norm_neg]
#align formal_multilinear_series.radius_neg FormalMultilinearSeries.radius_neg
protected theorem hasSum [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : HasSum (fun n : ℕ => p n fun _ => x) (p.sum x) :=
(p.summable hx).hasSum
#align formal_multilinear_series.has_sum FormalMultilinearSeries.hasSum
theorem radius_le_radius_continuousLinearMap_comp (p : FormalMultilinearSeries 𝕜 E F)
(f : F →L[𝕜] G) : p.radius ≤ (f.compFormalMultilinearSeries p).radius := by
refine' ENNReal.le_of_forall_nnreal_lt fun r hr => _
apply le_radius_of_isBigO
apply (IsBigO.trans_isLittleO _ (p.isLittleO_one_of_lt_radius hr)).isBigO
refine' IsBigO.mul (@IsBigOWith.isBigO _ _ _ _ _ ‖f‖ _ _ _ _) (isBigO_refl _ _)
refine IsBigOWith.of_bound (eventually_of_forall fun n => ?_)
simpa only [norm_norm] using f.norm_compContinuousMultilinearMap_le (p n)
#align formal_multilinear_series.radius_le_radius_continuous_linear_map_comp FormalMultilinearSeries.radius_le_radius_continuousLinearMap_comp
end FormalMultilinearSeries
/-! ### Expanding a function as a power series -/
section
variable {f g : E → F} {p pf pg : FormalMultilinearSeries 𝕜 E F} {x : E} {r r' : ℝ≥0∞}
/-- Given a function `f : E → F` and a formal multilinear series `p`, we say that `f` has `p` as
a power series on the ball of radius `r > 0` around `x` if `f (x + y) = ∑' pₙ yⁿ` for all `‖y‖ < r`.
-/
structure HasFPowerSeriesOnBall (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (x : E) (r : ℝ≥0∞) :
Prop where
r_le : r ≤ p.radius
r_pos : 0 < r
hasSum :
∀ {y}, y ∈ EMetric.ball (0 : E) r → HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y))
#align has_fpower_series_on_ball HasFPowerSeriesOnBall
/-- Given a function `f : E → F` and a formal multilinear series `p`, we say that `f` has `p` as
a power series around `x` if `f (x + y) = ∑' pₙ yⁿ` for all `y` in a neighborhood of `0`. -/
def HasFPowerSeriesAt (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (x : E) :=
∃ r, HasFPowerSeriesOnBall f p x r
#align has_fpower_series_at HasFPowerSeriesAt
variable (𝕜)
/-- Given a function `f : E → F`, we say that `f` is analytic at `x` if it admits a convergent power
series expansion around `x`. -/
def AnalyticAt (f : E → F) (x : E) :=
∃ p : FormalMultilinearSeries 𝕜 E F, HasFPowerSeriesAt f p x
#align analytic_at AnalyticAt
/-- Given a function `f : E → F`, we say that `f` is analytic on a set `s` if it is analytic around
every point of `s`. -/
def AnalyticOn (f : E → F) (s : Set E) :=
∀ x, x ∈ s → AnalyticAt 𝕜 f x
#align analytic_on AnalyticOn
variable {𝕜}
theorem HasFPowerSeriesOnBall.hasFPowerSeriesAt (hf : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesAt f p x :=
⟨r, hf⟩
#align has_fpower_series_on_ball.has_fpower_series_at HasFPowerSeriesOnBall.hasFPowerSeriesAt
theorem HasFPowerSeriesAt.analyticAt (hf : HasFPowerSeriesAt f p x) : AnalyticAt 𝕜 f x :=
⟨p, hf⟩
#align has_fpower_series_at.analytic_at HasFPowerSeriesAt.analyticAt
theorem HasFPowerSeriesOnBall.analyticAt (hf : HasFPowerSeriesOnBall f p x r) : AnalyticAt 𝕜 f x :=
hf.hasFPowerSeriesAt.analyticAt
#align has_fpower_series_on_ball.analytic_at HasFPowerSeriesOnBall.analyticAt
theorem HasFPowerSeriesOnBall.congr (hf : HasFPowerSeriesOnBall f p x r)
(hg : EqOn f g (EMetric.ball x r)) : HasFPowerSeriesOnBall g p x r :=
{ r_le := hf.r_le
r_pos := hf.r_pos
hasSum := fun {y} hy => by
convert hf.hasSum hy using 1
apply hg.symm
simpa [edist_eq_coe_nnnorm_sub] using hy }
#align has_fpower_series_on_ball.congr HasFPowerSeriesOnBall.congr
/-- If a function `f` has a power series `p` around `x`, then the function `z ↦ f (z - y)` has the
same power series around `x + y`. -/
theorem HasFPowerSeriesOnBall.comp_sub (hf : HasFPowerSeriesOnBall f p x r) (y : E) :
HasFPowerSeriesOnBall (fun z => f (z - y)) p (x + y) r :=
{ r_le := hf.r_le
r_pos := hf.r_pos
hasSum := fun {z} hz => by
convert hf.hasSum hz using 2
abel }
#align has_fpower_series_on_ball.comp_sub HasFPowerSeriesOnBall.comp_sub
theorem HasFPowerSeriesOnBall.hasSum_sub (hf : HasFPowerSeriesOnBall f p x r) {y : E}
(hy : y ∈ EMetric.ball x r) : HasSum (fun n : ℕ => p n fun _ => y - x) (f y) := by
have : y - x ∈ EMetric.ball (0 : E) r := by simpa [edist_eq_coe_nnnorm_sub] using hy
simpa only [add_sub_cancel'_right] using hf.hasSum this
#align has_fpower_series_on_ball.has_sum_sub HasFPowerSeriesOnBall.hasSum_sub
theorem HasFPowerSeriesOnBall.radius_pos (hf : HasFPowerSeriesOnBall f p x r) : 0 < p.radius :=
lt_of_lt_of_le hf.r_pos hf.r_le
#align has_fpower_series_on_ball.radius_pos HasFPowerSeriesOnBall.radius_pos
theorem HasFPowerSeriesAt.radius_pos (hf : HasFPowerSeriesAt f p x) : 0 < p.radius :=
let ⟨_, hr⟩ := hf
hr.radius_pos
#align has_fpower_series_at.radius_pos HasFPowerSeriesAt.radius_pos
theorem HasFPowerSeriesOnBall.mono (hf : HasFPowerSeriesOnBall f p x r) (r'_pos : 0 < r')
(hr : r' ≤ r) : HasFPowerSeriesOnBall f p x r' :=
⟨le_trans hr hf.1, r'_pos, fun hy => hf.hasSum (EMetric.ball_subset_ball hr hy)⟩
#align has_fpower_series_on_ball.mono HasFPowerSeriesOnBall.mono
theorem HasFPowerSeriesAt.congr (hf : HasFPowerSeriesAt f p x) (hg : f =ᶠ[𝓝 x] g) :
HasFPowerSeriesAt g p x := by
rcases hf with ⟨r₁, h₁⟩
rcases EMetric.mem_nhds_iff.mp hg with ⟨r₂, h₂pos, h₂⟩
exact ⟨min r₁ r₂,
(h₁.mono (lt_min h₁.r_pos h₂pos) inf_le_left).congr
fun y hy => h₂ (EMetric.ball_subset_ball inf_le_right hy)⟩
#align has_fpower_series_at.congr HasFPowerSeriesAt.congr
protected theorem HasFPowerSeriesAt.eventually (hf : HasFPowerSeriesAt f p x) :
∀ᶠ r : ℝ≥0∞ in 𝓝[>] 0, HasFPowerSeriesOnBall f p x r :=
let ⟨_, hr⟩ := hf
mem_of_superset (Ioo_mem_nhdsWithin_Ioi (left_mem_Ico.2 hr.r_pos)) fun _ hr' =>
hr.mono hr'.1 hr'.2.le
#align has_fpower_series_at.eventually HasFPowerSeriesAt.eventually
theorem HasFPowerSeriesOnBall.eventually_hasSum (hf : HasFPowerSeriesOnBall f p x r) :
∀ᶠ y in 𝓝 0, HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y)) := by
filter_upwards [EMetric.ball_mem_nhds (0 : E) hf.r_pos] using fun _ => hf.hasSum
#align has_fpower_series_on_ball.eventually_has_sum HasFPowerSeriesOnBall.eventually_hasSum
theorem HasFPowerSeriesAt.eventually_hasSum (hf : HasFPowerSeriesAt f p x) :
∀ᶠ y in 𝓝 0, HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y)) :=
let ⟨_, hr⟩ := hf
hr.eventually_hasSum
#align has_fpower_series_at.eventually_has_sum HasFPowerSeriesAt.eventually_hasSum
theorem HasFPowerSeriesOnBall.eventually_hasSum_sub (hf : HasFPowerSeriesOnBall f p x r) :
∀ᶠ y in 𝓝 x, HasSum (fun n : ℕ => p n fun _ : Fin n => y - x) (f y) := by
filter_upwards [EMetric.ball_mem_nhds x hf.r_pos] with y using hf.hasSum_sub
#align has_fpower_series_on_ball.eventually_has_sum_sub HasFPowerSeriesOnBall.eventually_hasSum_sub
theorem HasFPowerSeriesAt.eventually_hasSum_sub (hf : HasFPowerSeriesAt f p x) :
∀ᶠ y in 𝓝 x, HasSum (fun n : ℕ => p n fun _ : Fin n => y - x) (f y) :=
let ⟨_, hr⟩ := hf
hr.eventually_hasSum_sub
#align has_fpower_series_at.eventually_has_sum_sub HasFPowerSeriesAt.eventually_hasSum_sub
theorem HasFPowerSeriesOnBall.eventually_eq_zero
(hf : HasFPowerSeriesOnBall f (0 : FormalMultilinearSeries 𝕜 E F) x r) :
∀ᶠ z in 𝓝 x, f z = 0 := by
filter_upwards [hf.eventually_hasSum_sub] with z hz using hz.unique hasSum_zero
#align has_fpower_series_on_ball.eventually_eq_zero HasFPowerSeriesOnBall.eventually_eq_zero
theorem HasFPowerSeriesAt.eventually_eq_zero
(hf : HasFPowerSeriesAt f (0 : FormalMultilinearSeries 𝕜 E F) x) : ∀ᶠ z in 𝓝 x, f z = 0 :=
let ⟨_, hr⟩ := hf
hr.eventually_eq_zero
#align has_fpower_series_at.eventually_eq_zero HasFPowerSeriesAt.eventually_eq_zero
theorem hasFPowerSeriesOnBall_const {c : F} {e : E} :
HasFPowerSeriesOnBall (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e ⊤ := by
refine' ⟨by simp, WithTop.zero_lt_top, fun _ => hasSum_single 0 fun n hn => _⟩
simp [constFormalMultilinearSeries_apply hn]
#align has_fpower_series_on_ball_const hasFPowerSeriesOnBall_const
theorem hasFPowerSeriesAt_const {c : F} {e : E} :
HasFPowerSeriesAt (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e :=
⟨⊤, hasFPowerSeriesOnBall_const⟩
#align has_fpower_series_at_const hasFPowerSeriesAt_const
theorem analyticAt_const {v : F} : AnalyticAt 𝕜 (fun _ => v) x :=
⟨constFormalMultilinearSeries 𝕜 E v, hasFPowerSeriesAt_const⟩
#align analytic_at_const analyticAt_const
theorem analyticOn_const {v : F} {s : Set E} : AnalyticOn 𝕜 (fun _ => v) s :=
fun _ _ => analyticAt_const
#align analytic_on_const analyticOn_const
theorem HasFPowerSeriesOnBall.add (hf : HasFPowerSeriesOnBall f pf x r)
(hg : HasFPowerSeriesOnBall g pg x r) : HasFPowerSeriesOnBall (f + g) (pf + pg) x r :=
{ r_le := le_trans (le_min_iff.2 ⟨hf.r_le, hg.r_le⟩) (pf.min_radius_le_radius_add pg)
r_pos := hf.r_pos
hasSum := fun hy => (hf.hasSum hy).add (hg.hasSum hy) }
#align has_fpower_series_on_ball.add HasFPowerSeriesOnBall.add
theorem HasFPowerSeriesAt.add (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x) :
HasFPowerSeriesAt (f + g) (pf + pg) x := by
rcases (hf.eventually.and hg.eventually).exists with ⟨r, hr⟩
exact ⟨r, hr.1.add hr.2⟩
#align has_fpower_series_at.add HasFPowerSeriesAt.add
theorem AnalyticAt.congr (hf : AnalyticAt 𝕜 f x) (hg : f =ᶠ[𝓝 x] g) : AnalyticAt 𝕜 g x :=
let ⟨_, hpf⟩ := hf
(hpf.congr hg).analyticAt
theorem analyticAt_congr (h : f =ᶠ[𝓝 x] g) : AnalyticAt 𝕜 f x ↔ AnalyticAt 𝕜 g x :=
⟨fun hf ↦ hf.congr h, fun hg ↦ hg.congr h.symm⟩
theorem AnalyticAt.add (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) : AnalyticAt 𝕜 (f + g) x :=
let ⟨_, hpf⟩ := hf
let ⟨_, hqf⟩ := hg
(hpf.add hqf).analyticAt
#align analytic_at.add AnalyticAt.add
theorem HasFPowerSeriesOnBall.neg (hf : HasFPowerSeriesOnBall f pf x r) :
HasFPowerSeriesOnBall (-f) (-pf) x r :=
{ r_le := by
rw [pf.radius_neg]
exact hf.r_le
r_pos := hf.r_pos
hasSum := fun hy => (hf.hasSum hy).neg }
#align has_fpower_series_on_ball.neg HasFPowerSeriesOnBall.neg
theorem HasFPowerSeriesAt.neg (hf : HasFPowerSeriesAt f pf x) : HasFPowerSeriesAt (-f) (-pf) x :=
let ⟨_, hrf⟩ := hf
hrf.neg.hasFPowerSeriesAt
#align has_fpower_series_at.neg HasFPowerSeriesAt.neg
theorem AnalyticAt.neg (hf : AnalyticAt 𝕜 f x) : AnalyticAt 𝕜 (-f) x :=
let ⟨_, hpf⟩ := hf
hpf.neg.analyticAt
#align analytic_at.neg AnalyticAt.neg
theorem HasFPowerSeriesOnBall.sub (hf : HasFPowerSeriesOnBall f pf x r)
(hg : HasFPowerSeriesOnBall g pg x r) : HasFPowerSeriesOnBall (f - g) (pf - pg) x r := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align has_fpower_series_on_ball.sub HasFPowerSeriesOnBall.sub
theorem HasFPowerSeriesAt.sub (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x) :
HasFPowerSeriesAt (f - g) (pf - pg) x := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align has_fpower_series_at.sub HasFPowerSeriesAt.sub
theorem AnalyticAt.sub (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) :
AnalyticAt 𝕜 (f - g) x := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align analytic_at.sub AnalyticAt.sub
theorem AnalyticOn.mono {s t : Set E} (hf : AnalyticOn 𝕜 f t) (hst : s ⊆ t) : AnalyticOn 𝕜 f s :=
fun z hz => hf z (hst hz)
#align analytic_on.mono AnalyticOn.mono
theorem AnalyticOn.congr' {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : f =ᶠ[𝓝ˢ s] g) :
AnalyticOn 𝕜 g s :=
fun z hz => (hf z hz).congr (mem_nhdsSet_iff_forall.mp hg z hz)
theorem analyticOn_congr' {s : Set E} (h : f =ᶠ[𝓝ˢ s] g) : AnalyticOn 𝕜 f s ↔ AnalyticOn 𝕜 g s :=
⟨fun hf => hf.congr' h, fun hg => hg.congr' h.symm⟩
theorem AnalyticOn.congr {s : Set E} (hs : IsOpen s) (hf : AnalyticOn 𝕜 f s) (hg : s.EqOn f g) :
AnalyticOn 𝕜 g s :=
hf.congr' $ mem_nhdsSet_iff_forall.mpr
(fun _ hz => eventuallyEq_iff_exists_mem.mpr ⟨s, hs.mem_nhds hz, hg⟩)
theorem analyticOn_congr {s : Set E} (hs : IsOpen s) (h : s.EqOn f g) : AnalyticOn 𝕜 f s ↔
AnalyticOn 𝕜 g s := ⟨fun hf => hf.congr hs h, fun hg => hg.congr hs h.symm⟩
theorem AnalyticOn.add {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
AnalyticOn 𝕜 (f + g) s :=
fun z hz => (hf z hz).add (hg z hz)
#align analytic_on.add AnalyticOn.add
theorem AnalyticOn.sub {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
AnalyticOn 𝕜 (f - g) s :=
fun z hz => (hf z hz).sub (hg z hz)
#align analytic_on.sub AnalyticOn.sub
theorem HasFPowerSeriesOnBall.coeff_zero (hf : HasFPowerSeriesOnBall f pf x r) (v : Fin 0 → E) :
pf 0 v = f x := by
have v_eq : v = fun i => 0 := Subsingleton.elim _ _
have zero_mem : (0 : E) ∈ EMetric.ball (0 : E) r := by simp [hf.r_pos]
have : ∀ i, i ≠ 0 → (pf i fun j => 0) = 0 := by
intro i hi
have : 0 < i := pos_iff_ne_zero.2 hi
exact ContinuousMultilinearMap.map_coord_zero _ (⟨0, this⟩ : Fin i) rfl
have A := (hf.hasSum zero_mem).unique (hasSum_single _ this)
simpa [v_eq] using A.symm
#align has_fpower_series_on_ball.coeff_zero HasFPowerSeriesOnBall.coeff_zero
theorem HasFPowerSeriesAt.coeff_zero (hf : HasFPowerSeriesAt f pf x) (v : Fin 0 → E) :
pf 0 v = f x :=
let ⟨_, hrf⟩ := hf
hrf.coeff_zero v
#align has_fpower_series_at.coeff_zero HasFPowerSeriesAt.coeff_zero
/-- If a function `f` has a power series `p` on a ball and `g` is linear, then `g ∘ f` has the
power series `g ∘ p` on the same ball. -/
theorem ContinuousLinearMap.comp_hasFPowerSeriesOnBall (g : F →L[𝕜] G)
(h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (g ∘ f) (g.compFormalMultilinearSeries p) x r :=
{ r_le := h.r_le.trans (p.radius_le_radius_continuousLinearMap_comp _)
r_pos := h.r_pos
hasSum := fun hy => by
simpa only [ContinuousLinearMap.compFormalMultilinearSeries_apply,
ContinuousLinearMap.compContinuousMultilinearMap_coe, Function.comp_apply] using
g.hasSum (h.hasSum hy) }
#align continuous_linear_map.comp_has_fpower_series_on_ball ContinuousLinearMap.comp_hasFPowerSeriesOnBall
/-- If a function `f` is analytic on a set `s` and `g` is linear, then `g ∘ f` is analytic
on `s`. -/
theorem ContinuousLinearMap.comp_analyticOn {s : Set E} (g : F →L[𝕜] G) (h : AnalyticOn 𝕜 f s) :
AnalyticOn 𝕜 (g ∘ f) s := by
rintro x hx
rcases h x hx with ⟨p, r, hp⟩
exact ⟨g.compFormalMultilinearSeries p, r, g.comp_hasFPowerSeriesOnBall hp⟩
#align continuous_linear_map.comp_analytic_on ContinuousLinearMap.comp_analyticOn
/-- If a function admits a power series expansion, then it is exponentially close to the partial
sums of this power series on strict subdisks of the disk of convergence.
This version provides an upper estimate that decreases both in `‖y‖` and `n`. See also
`HasFPowerSeriesOnBall.uniform_geometric_approx` for a weaker version. -/
theorem HasFPowerSeriesOnBall.uniform_geometric_approx' {r' : ℝ≥0}
(hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n := by
obtain ⟨a, ha, C, hC, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n, ‖p n‖ * (r' : ℝ) ^ n ≤ C * a ^ n :=
p.norm_mul_pow_le_mul_pow_of_lt_radius (h.trans_le hf.r_le)
refine' ⟨a, ha, C / (1 - a), div_pos hC (sub_pos.2 ha.2), fun y hy n => _⟩
have yr' : ‖y‖ < r' := by
rw [ball_zero_eq] at hy
exact hy
have hr'0 : 0 < (r' : ℝ) := (norm_nonneg _).trans_lt yr'
have : y ∈ EMetric.ball (0 : E) r := by
refine' mem_emetric_ball_zero_iff.2 (lt_trans _ h)
exact mod_cast yr'
rw [norm_sub_rev, ← mul_div_right_comm]
have ya : a * (‖y‖ / ↑r') ≤ a :=
mul_le_of_le_one_right ha.1.le (div_le_one_of_le yr'.le r'.coe_nonneg)
suffices ‖p.partialSum n y - f (x + y)‖ ≤ C * (a * (‖y‖ / r')) ^ n / (1 - a * (‖y‖ / r')) by
refine' this.trans _
have : 0 < a := ha.1
gcongr
apply_rules [sub_pos.2, ha.2]
apply norm_sub_le_of_geometric_bound_of_hasSum (ya.trans_lt ha.2) _ (hf.hasSum this)
intro n
calc
‖(p n) fun _ : Fin n => y‖
_ ≤ ‖p n‖ * ∏ _i : Fin n, ‖y‖ := ContinuousMultilinearMap.le_op_norm _ _
_ = ‖p n‖ * (r' : ℝ) ^ n * (‖y‖ / r') ^ n := by field_simp [mul_right_comm]
_ ≤ C * a ^ n * (‖y‖ / r') ^ n := by gcongr ?_ * _; apply hp
_ ≤ C * (a * (‖y‖ / r')) ^ n := by rw [mul_pow, mul_assoc]
#align has_fpower_series_on_ball.uniform_geometric_approx' HasFPowerSeriesOnBall.uniform_geometric_approx'
/-- If a function admits a power series expansion, then it is exponentially close to the partial
sums of this power series on strict subdisks of the disk of convergence. -/
theorem HasFPowerSeriesOnBall.uniform_geometric_approx {r' : ℝ≥0}
(hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
∃ a ∈ Ioo (0 : ℝ) 1,
∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n, ‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n := by
obtain ⟨a, ha, C, hC, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n :=
hf.uniform_geometric_approx' h
refine' ⟨a, ha, C, hC, fun y hy n => (hp y hy n).trans _⟩
have yr' : ‖y‖ < r' := by rwa [ball_zero_eq] at hy
gcongr
exacts [mul_nonneg ha.1.le (div_nonneg (norm_nonneg y) r'.coe_nonneg),
mul_le_of_le_one_right ha.1.le (div_le_one_of_le yr'.le r'.coe_nonneg)]
#align has_fpower_series_on_ball.uniform_geometric_approx HasFPowerSeriesOnBall.uniform_geometric_approx
/-- Taylor formula for an analytic function, `IsBigO` version. -/
theorem HasFPowerSeriesAt.isBigO_sub_partialSum_pow (hf : HasFPowerSeriesAt f p x) (n : ℕ) :
(fun y : E => f (x + y) - p.partialSum n y) =O[𝓝 0] fun y => ‖y‖ ^ n := by
rcases hf with ⟨r, hf⟩
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hf.r_pos with ⟨r', r'0, h⟩
obtain ⟨a, -, C, -, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n :=
hf.uniform_geometric_approx' h
refine' isBigO_iff.2 ⟨C * (a / r') ^ n, _⟩
replace r'0 : 0 < (r' : ℝ); · exact mod_cast r'0
filter_upwards [Metric.ball_mem_nhds (0 : E) r'0] with y hy
simpa [mul_pow, mul_div_assoc, mul_assoc, div_mul_eq_mul_div] using hp y hy n
set_option linter.uppercaseLean3 false in
#align has_fpower_series_at.is_O_sub_partial_sum_pow HasFPowerSeriesAt.isBigO_sub_partialSum_pow
/-- If `f` has formal power series `∑ n, pₙ` on a ball of radius `r`, then for `y, z` in any smaller
ball, the norm of the difference `f y - f z - p 1 (fun _ ↦ y - z)` is bounded above by
`C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖`. This lemma formulates this property using `IsBigO` and
`Filter.principal` on `E × E`. -/
theorem HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal
(hf : HasFPowerSeriesOnBall f p x r) (hr : r' < r) :
(fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) =O[𝓟 (EMetric.ball (x, x) r')]
fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖ := by
lift r' to ℝ≥0 using ne_top_of_lt hr
rcases (zero_le r').eq_or_lt with (rfl | hr'0)
· simp only [isBigO_bot, EMetric.ball_zero, principal_empty, ENNReal.coe_zero]
obtain ⟨a, ha, C, hC : 0 < C, hp⟩ :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n : ℕ, ‖p n‖ * (r' : ℝ) ^ n ≤ C * a ^ n
exact p.norm_mul_pow_le_mul_pow_of_lt_radius (hr.trans_le hf.r_le)
simp only [← le_div_iff (pow_pos (NNReal.coe_pos.2 hr'0) _)] at hp
set L : E × E → ℝ := fun y =>
C * (a / r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * (a / (1 - a) ^ 2 + 2 / (1 - a))
have hL : ∀ y ∈ EMetric.ball (x, x) r', ‖f y.1 - f y.2 - p 1 fun _ => y.1 - y.2‖ ≤ L y := by
intro y hy'
have hy : y ∈ EMetric.ball x r ×ˢ EMetric.ball x r := by
rw [EMetric.ball_prod_same]
exact EMetric.ball_subset_ball hr.le hy'
set A : ℕ → F := fun n => (p n fun _ => y.1 - x) - p n fun _ => y.2 - x
have hA : HasSum (fun n => A (n + 2)) (f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) := by
convert (hasSum_nat_add_iff' 2).2 ((hf.hasSum_sub hy.1).sub (hf.hasSum_sub hy.2)) using 1
rw [Finset.sum_range_succ, Finset.sum_range_one, hf.coeff_zero, hf.coeff_zero, sub_self,
zero_add, ← Subsingleton.pi_single_eq (0 : Fin 1) (y.1 - x), Pi.single,
← Subsingleton.pi_single_eq (0 : Fin 1) (y.2 - x), Pi.single, ← (p 1).map_sub, ← Pi.single,
Subsingleton.pi_single_eq, sub_sub_sub_cancel_right]
rw [EMetric.mem_ball, edist_eq_coe_nnnorm_sub, ENNReal.coe_lt_coe] at hy'
set B : ℕ → ℝ := fun n => C * (a / r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * ((n + 2) * a ^ n)
have hAB : ∀ n, ‖A (n + 2)‖ ≤ B n := fun n =>
calc
‖A (n + 2)‖ ≤ ‖p (n + 2)‖ * ↑(n + 2) * ‖y - (x, x)‖ ^ (n + 1) * ‖y.1 - y.2‖ := by
-- porting note: `pi_norm_const` was `pi_norm_const (_ : E)`
simpa only [Fintype.card_fin, pi_norm_const, Prod.norm_def, Pi.sub_def,
Prod.fst_sub, Prod.snd_sub, sub_sub_sub_cancel_right] using
(p <| n + 2).norm_image_sub_le (fun _ => y.1 - x) fun _ => y.2 - x
_ = ‖p (n + 2)‖ * ‖y - (x, x)‖ ^ n * (↑(n + 2) * ‖y - (x, x)‖ * ‖y.1 - y.2‖) := by
rw [pow_succ ‖y - (x, x)‖]
ring
-- porting note: the two `↑` in `↑r'` are new, without them, Lean fails to synthesize
-- instances `HDiv ℝ ℝ≥0 ?m` or `HMul ℝ ℝ≥0 ?m`
_ ≤ C * a ^ (n + 2) / ↑r' ^ (n + 2)
* ↑r' ^ n * (↑(n + 2) * ‖y - (x, x)‖ * ‖y.1 - y.2‖) := by
have : 0 < a := ha.1
gcongr
· apply hp
· apply hy'.le
_ = B n := by
-- porting note: in the original, `B` was in the `field_simp`, but now Lean does not
-- accept it. The current proof works in Lean 4, but does not in Lean 3.
field_simp [pow_succ]
simp only [mul_assoc, mul_comm, mul_left_comm]
have hBL : HasSum B (L y) := by
apply HasSum.mul_left
simp only [add_mul]
have : ‖a‖ < 1 := by simp only [Real.norm_eq_abs, abs_of_pos ha.1, ha.2]
rw [div_eq_mul_inv, div_eq_mul_inv]
exact (hasSum_coe_mul_geometric_of_norm_lt_1 this).add -- porting note: was `convert`!
((hasSum_geometric_of_norm_lt_1 this).mul_left 2)
exact hA.norm_le_of_bounded hBL hAB
suffices L =O[𝓟 (EMetric.ball (x, x) r')] fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖ by
refine' (IsBigO.of_bound 1 (eventually_principal.2 fun y hy => _)).trans this
rw [one_mul]
exact (hL y hy).trans (le_abs_self _)
simp_rw [mul_right_comm _ (_ * _)] -- porting note: there was an `L` inside the `simp_rw`.
exact (isBigO_refl _ _).const_mul_left _
set_option linter.uppercaseLean3 false in
#align has_fpower_series_on_ball.is_O_image_sub_image_sub_deriv_principal HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal
/-- If `f` has formal power series `∑ n, pₙ` on a ball of radius `r`, then for `y, z` in any smaller
ball, the norm of the difference `f y - f z - p 1 (fun _ ↦ y - z)` is bounded above by
`C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖`. -/
theorem HasFPowerSeriesOnBall.image_sub_sub_deriv_le (hf : HasFPowerSeriesOnBall f p x r)
(hr : r' < r) :
∃ C, ∀ᵉ (y ∈ EMetric.ball x r') (z ∈ EMetric.ball x r'),
‖f y - f z - p 1 fun _ => y - z‖ ≤ C * max ‖y - x‖ ‖z - x‖ * ‖y - z‖ := by
simpa only [isBigO_principal, mul_assoc, norm_mul, norm_norm, Prod.forall, EMetric.mem_ball,
Prod.edist_eq, max_lt_iff, and_imp, @forall_swap (_ < _) E] using
hf.isBigO_image_sub_image_sub_deriv_principal hr
#align has_fpower_series_on_ball.image_sub_sub_deriv_le HasFPowerSeriesOnBall.image_sub_sub_deriv_le
/-- If `f` has formal power series `∑ n, pₙ` at `x`, then
`f y - f z - p 1 (fun _ ↦ y - z) = O(‖(y, z) - (x, x)‖ * ‖y - z‖)` as `(y, z) → (x, x)`.
In particular, `f` is strictly differentiable at `x`. -/
theorem HasFPowerSeriesAt.isBigO_image_sub_norm_mul_norm_sub (hf : HasFPowerSeriesAt f p x) :
(fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) =O[𝓝 (x, x)] fun y =>
‖y - (x, x)‖ * ‖y.1 - y.2‖ := by
rcases hf with ⟨r, hf⟩
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hf.r_pos with ⟨r', r'0, h⟩
refine' (hf.isBigO_image_sub_image_sub_deriv_principal h).mono _
exact le_principal_iff.2 (EMetric.ball_mem_nhds _ r'0)
set_option linter.uppercaseLean3 false in
#align has_fpower_series_at.is_O_image_sub_norm_mul_norm_sub HasFPowerSeriesAt.isBigO_image_sub_norm_mul_norm_sub
/-- If a function admits a power series expansion at `x`, then it is the uniform limit of the
partial sums of this power series on strict subdisks of the disk of convergence, i.e., `f (x + y)`
is the uniform limit of `p.partialSum n y` there. -/
theorem HasFPowerSeriesOnBall.tendstoUniformlyOn {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r)
(h : (r' : ℝ≥0∞) < r) :
TendstoUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
(Metric.ball (0 : E) r') := by
obtain ⟨a, ha, C, -, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n
exact hf.uniform_geometric_approx h
refine' Metric.tendstoUniformlyOn_iff.2 fun ε εpos => _
have L : Tendsto (fun n => (C : ℝ) * a ^ n) atTop (𝓝 ((C : ℝ) * 0)) :=
tendsto_const_nhds.mul (tendsto_pow_atTop_nhds_0_of_lt_1 ha.1.le ha.2)
rw [mul_zero] at L
refine' (L.eventually (gt_mem_nhds εpos)).mono fun n hn y hy => _
rw [dist_eq_norm]
exact (hp y hy n).trans_lt hn
#align has_fpower_series_on_ball.tendsto_uniformly_on HasFPowerSeriesOnBall.tendstoUniformlyOn
/-- If a function admits a power series expansion at `x`, then it is the locally uniform limit of
the partial sums of this power series on the disk of convergence, i.e., `f (x + y)`
is the locally uniform limit of `p.partialSum n y` there. -/
theorem HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn (hf : HasFPowerSeriesOnBall f p x r) :
TendstoLocallyUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
(EMetric.ball (0 : E) r) := by
intro u hu x hx
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hx with ⟨r', xr', hr'⟩
have : EMetric.ball (0 : E) r' ∈ 𝓝 x := IsOpen.mem_nhds EMetric.isOpen_ball xr'
refine' ⟨EMetric.ball (0 : E) r', mem_nhdsWithin_of_mem_nhds this, _⟩
simpa [Metric.emetric_ball_nnreal] using hf.tendstoUniformlyOn hr' u hu
#align has_fpower_series_on_ball.tendsto_locally_uniformly_on HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn
/-- If a function admits a power series expansion at `x`, then it is the uniform limit of the
partial sums of this power series on strict subdisks of the disk of convergence, i.e., `f y`
is the uniform limit of `p.partialSum n (y - x)` there. -/
theorem HasFPowerSeriesOnBall.tendstoUniformlyOn' {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r)
(h : (r' : ℝ≥0∞) < r) :
TendstoUniformlyOn (fun n y => p.partialSum n (y - x)) f atTop (Metric.ball (x : E) r') := by
convert (hf.tendstoUniformlyOn h).comp fun y => y - x using 1
· simp [(· ∘ ·)]
· ext z
simp [dist_eq_norm]
#align has_fpower_series_on_ball.tendsto_uniformly_on' HasFPowerSeriesOnBall.tendstoUniformlyOn'
/-- If a function admits a power series expansion at `x`, then it is the locally uniform limit of
the partial sums of this power series on the disk of convergence, i.e., `f y`
is the locally uniform limit of `p.partialSum n (y - x)` there. -/
theorem HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn' (hf : HasFPowerSeriesOnBall f p x r) :
TendstoLocallyUniformlyOn (fun n y => p.partialSum n (y - x)) f atTop
(EMetric.ball (x : E) r) := by
have A : ContinuousOn (fun y : E => y - x) (EMetric.ball (x : E) r) :=
(continuous_id.sub continuous_const).continuousOn
convert hf.tendstoLocallyUniformlyOn.comp (fun y : E => y - x) _ A using 1
· ext z
simp
· intro z
simp [edist_eq_coe_nnnorm, edist_eq_coe_nnnorm_sub]
#align has_fpower_series_on_ball.tendsto_locally_uniformly_on' HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn'
/-- If a function admits a power series expansion on a disk, then it is continuous there. -/
protected theorem HasFPowerSeriesOnBall.continuousOn (hf : HasFPowerSeriesOnBall f p x r) :
ContinuousOn f (EMetric.ball x r) :=
hf.tendstoLocallyUniformlyOn'.continuousOn <|
eventually_of_forall fun n =>
((p.partialSum_continuous n).comp (continuous_id.sub continuous_const)).continuousOn
#align has_fpower_series_on_ball.continuous_on HasFPowerSeriesOnBall.continuousOn
protected theorem HasFPowerSeriesAt.continuousAt (hf : HasFPowerSeriesAt f p x) :
ContinuousAt f x :=
let ⟨_, hr⟩ := hf
hr.continuousOn.continuousAt (EMetric.ball_mem_nhds x hr.r_pos)
#align has_fpower_series_at.continuous_at HasFPowerSeriesAt.continuousAt
protected theorem AnalyticAt.continuousAt (hf : AnalyticAt 𝕜 f x) : ContinuousAt f x :=
let ⟨_, hp⟩ := hf
hp.continuousAt
#align analytic_at.continuous_at AnalyticAt.continuousAt
protected theorem AnalyticOn.continuousOn {s : Set E} (hf : AnalyticOn 𝕜 f s) : ContinuousOn f s :=
fun x hx => (hf x hx).continuousAt.continuousWithinAt
#align analytic_on.continuous_on AnalyticOn.continuousOn
/-- Analytic everywhere implies continuous -/
theorem AnalyticOn.continuous {f : E → F} (fa : AnalyticOn 𝕜 f univ) : Continuous f := by
rw [continuous_iff_continuousOn_univ]; exact fa.continuousOn
/-- In a complete space, the sum of a converging power series `p` admits `p` as a power series.
This is not totally obvious as we need to check the convergence of the series. -/
protected theorem FormalMultilinearSeries.hasFPowerSeriesOnBall [CompleteSpace F]
(p : FormalMultilinearSeries 𝕜 E F) (h : 0 < p.radius) :
HasFPowerSeriesOnBall p.sum p 0 p.radius :=
{ r_le := le_rfl
r_pos := h
hasSum := fun hy => by
rw [zero_add]
exact p.hasSum hy }
#align formal_multilinear_series.has_fpower_series_on_ball FormalMultilinearSeries.hasFPowerSeriesOnBall
theorem HasFPowerSeriesOnBall.sum (h : HasFPowerSeriesOnBall f p x r) {y : E}
(hy : y ∈ EMetric.ball (0 : E) r) : f (x + y) = p.sum y :=
(h.hasSum hy).tsum_eq.symm
#align has_fpower_series_on_ball.sum HasFPowerSeriesOnBall.sum
/-- The sum of a converging power series is continuous in its disk of convergence. -/
protected theorem FormalMultilinearSeries.continuousOn [CompleteSpace F] :
ContinuousOn p.sum (EMetric.ball 0 p.radius) := by
rcases (zero_le p.radius).eq_or_lt with h | h
· simp [← h, continuousOn_empty]
· exact (p.hasFPowerSeriesOnBall h).continuousOn
#align formal_multilinear_series.continuous_on FormalMultilinearSeries.continuousOn
end
/-!
### Uniqueness of power series
If a function `f : E → F` has two representations as power series at a point `x : E`, corresponding
to formal multilinear series `p₁` and `p₂`, then these representations agree term-by-term. That is,
for any `n : ℕ` and `y : E`, `p₁ n (fun i ↦ y) = p₂ n (fun i ↦ y)`. In the one-dimensional case,
when `f : 𝕜 → E`, the continuous multilinear maps `p₁ n` and `p₂ n` are given by
`ContinuousMultilinearMap.mkPiField`, and hence are determined completely by the value of
`p₁ n (fun i ↦ 1)`, so `p₁ = p₂`. Consequently, the radius of convergence for one series can be
transferred to the other.
-/
section Uniqueness
open ContinuousMultilinearMap
theorem Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero {n : ℕ} {p : E[×n]→L[𝕜] F}
(h : (fun y => p fun _ => y) =O[𝓝 0] fun y => ‖y‖ ^ (n + 1)) (y : E) : (p fun _ => y) = 0 := by
obtain ⟨c, c_pos, hc⟩ := h.exists_pos
obtain ⟨t, ht, t_open, z_mem⟩ := eventually_nhds_iff.mp (isBigOWith_iff.mp hc)
obtain ⟨δ, δ_pos, δε⟩ := (Metric.isOpen_iff.mp t_open) 0 z_mem
clear h hc z_mem
cases' n with n
· exact norm_eq_zero.mp (by
-- porting note: the symmetric difference of the `simpa only` sets:
-- added `Nat.zero_eq, zero_add, pow_one`
-- removed `zero_pow', Ne.def, Nat.one_ne_zero, not_false_iff`
simpa only [Nat.zero_eq, fin0_apply_norm, norm_eq_zero, norm_zero, zero_add, pow_one,
mul_zero, norm_le_zero_iff] using ht 0 (δε (Metric.mem_ball_self δ_pos)))
· refine' Or.elim (Classical.em (y = 0))
(fun hy => by simpa only [hy] using p.map_zero) fun hy => _
replace hy := norm_pos_iff.mpr hy
refine' norm_eq_zero.mp (le_antisymm (le_of_forall_pos_le_add fun ε ε_pos => _) (norm_nonneg _))
have h₀ := _root_.mul_pos c_pos (pow_pos hy (n.succ + 1))
obtain ⟨k, k_pos, k_norm⟩ := NormedField.exists_norm_lt 𝕜
(lt_min (mul_pos δ_pos (inv_pos.mpr hy)) (mul_pos ε_pos (inv_pos.mpr h₀)))
have h₁ : ‖k • y‖ < δ := by
rw [norm_smul]
exact inv_mul_cancel_right₀ hy.ne.symm δ ▸
mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_left _ _)) hy
have h₂ :=
calc
‖p fun _ => k • y‖ ≤ c * ‖k • y‖ ^ (n.succ + 1) := by
-- porting note: now Lean wants `_root_.`
simpa only [norm_pow, _root_.norm_norm] using ht (k • y) (δε (mem_ball_zero_iff.mpr h₁))
--simpa only [norm_pow, norm_norm] using ht (k • y) (δε (mem_ball_zero_iff.mpr h₁))
_ = ‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))) := by
-- porting note: added `Nat.succ_eq_add_one` since otherwise `ring` does not conclude.
simp only [norm_smul, mul_pow, Nat.succ_eq_add_one]
-- porting note: removed `rw [pow_succ]`, since it now becomes superfluous.
ring
have h₃ : ‖k‖ * (c * ‖y‖ ^ (n.succ + 1)) < ε :=
inv_mul_cancel_right₀ h₀.ne.symm ε ▸
mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_right _ _)) h₀
calc
‖p fun _ => y‖ = ‖k⁻¹ ^ n.succ‖ * ‖p fun _ => k • y‖ := by
simpa only [inv_smul_smul₀ (norm_pos_iff.mp k_pos), norm_smul, Finset.prod_const,
Finset.card_fin] using
congr_arg norm (p.map_smul_univ (fun _ : Fin n.succ => k⁻¹) fun _ : Fin n.succ => k • y)
_ ≤ ‖k⁻¹ ^ n.succ‖ * (‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1)))) := by gcongr
_ = ‖(k⁻¹ * k) ^ n.succ‖ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))) := by
rw [← mul_assoc]
simp [norm_mul, mul_pow]
_ ≤ 0 + ε := by
rw [inv_mul_cancel (norm_pos_iff.mp k_pos)]
simpa using h₃.le
set_option linter.uppercaseLean3 false in
#align asymptotics.is_O.continuous_multilinear_map_apply_eq_zero Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero
/-- If a formal multilinear series `p` represents the zero function at `x : E`, then the
terms `p n (fun i ↦ y)` appearing in the sum are zero for any `n : ℕ`, `y : E`. -/
theorem HasFPowerSeriesAt.apply_eq_zero {p : FormalMultilinearSeries 𝕜 E F} {x : E}
(h : HasFPowerSeriesAt 0 p x) (n : ℕ) : ∀ y : E, (p n fun _ => y) = 0 := by
refine' Nat.strong_induction_on n fun k hk => _
have psum_eq : p.partialSum (k + 1) = fun y => p k fun _ => y := by
funext z
refine' Finset.sum_eq_single _ (fun b hb hnb => _) fun hn => _
· have := Finset.mem_range_succ_iff.mp hb
simp only [hk b (this.lt_of_ne hnb), Pi.zero_apply]
· exact False.elim (hn (Finset.mem_range.mpr (lt_add_one k)))
replace h := h.isBigO_sub_partialSum_pow k.succ
simp only [psum_eq, zero_sub, Pi.zero_apply, Asymptotics.isBigO_neg_left] at h
exact h.continuousMultilinearMap_apply_eq_zero
#align has_fpower_series_at.apply_eq_zero HasFPowerSeriesAt.apply_eq_zero
/-- A one-dimensional formal multilinear series representing the zero function is zero. -/
theorem HasFPowerSeriesAt.eq_zero {p : FormalMultilinearSeries 𝕜 𝕜 E} {x : 𝕜}
(h : HasFPowerSeriesAt 0 p x) : p = 0 := by
-- porting note: `funext; ext` was `ext (n x)`
funext n
ext x
rw [← mkPiField_apply_one_eq_self (p n)]
-- porting note: nasty hack, was `simp [h.apply_eq_zero n 1]`
have := Or.intro_right ?_ (h.apply_eq_zero n 1)
simpa using this
#align has_fpower_series_at.eq_zero HasFPowerSeriesAt.eq_zero
/-- One-dimensional formal multilinear series representing the same function are equal. -/
theorem HasFPowerSeriesAt.eq_formalMultilinearSeries {p₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E}
{f : 𝕜 → E} {x : 𝕜} (h₁ : HasFPowerSeriesAt f p₁ x) (h₂ : HasFPowerSeriesAt f p₂ x) : p₁ = p₂ :=
sub_eq_zero.mp (HasFPowerSeriesAt.eq_zero (by simpa only [sub_self] using h₁.sub h₂))
#align has_fpower_series_at.eq_formal_multilinear_series HasFPowerSeriesAt.eq_formalMultilinearSeries
theorem HasFPowerSeriesAt.eq_formalMultilinearSeries_of_eventually
{p q : FormalMultilinearSeries 𝕜 𝕜 E} {f g : 𝕜 → E} {x : 𝕜} (hp : HasFPowerSeriesAt f p x)
(hq : HasFPowerSeriesAt g q x) (heq : ∀ᶠ z in 𝓝 x, f z = g z) : p = q :=
(hp.congr heq).eq_formalMultilinearSeries hq
#align has_fpower_series_at.eq_formal_multilinear_series_of_eventually HasFPowerSeriesAt.eq_formalMultilinearSeries_of_eventually
/-- A one-dimensional formal multilinear series representing a locally zero function is zero. -/
theorem HasFPowerSeriesAt.eq_zero_of_eventually {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E}
{x : 𝕜} (hp : HasFPowerSeriesAt f p x) (hf : f =ᶠ[𝓝 x] 0) : p = 0 :=
(hp.congr hf).eq_zero
#align has_fpower_series_at.eq_zero_of_eventually HasFPowerSeriesAt.eq_zero_of_eventually
/-- If a function `f : 𝕜 → E` has two power series representations at `x`, then the given radii in
which convergence is guaranteed may be interchanged. This can be useful when the formal multilinear
series in one representation has a particularly nice form, but the other has a larger radius. -/
theorem HasFPowerSeriesOnBall.exchange_radius {p₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E}
{r₁ r₂ : ℝ≥0∞} {x : 𝕜} (h₁ : HasFPowerSeriesOnBall f p₁ x r₁)
(h₂ : HasFPowerSeriesOnBall f p₂ x r₂) : HasFPowerSeriesOnBall f p₁ x r₂ :=
h₂.hasFPowerSeriesAt.eq_formalMultilinearSeries h₁.hasFPowerSeriesAt ▸ h₂
#align has_fpower_series_on_ball.exchange_radius HasFPowerSeriesOnBall.exchange_radius
/-- If a function `f : 𝕜 → E` has power series representation `p` on a ball of some radius and for
each positive radius it has some power series representation, then `p` converges to `f` on the whole
`𝕜`. -/
theorem HasFPowerSeriesOnBall.r_eq_top_of_exists {f : 𝕜 → E} {r : ℝ≥0∞} {x : 𝕜}
{p : FormalMultilinearSeries 𝕜 𝕜 E} (h : HasFPowerSeriesOnBall f p x r)
(h' : ∀ (r' : ℝ≥0) (_ : 0 < r'), ∃ p' : FormalMultilinearSeries 𝕜 𝕜 E,
HasFPowerSeriesOnBall f p' x r') :
HasFPowerSeriesOnBall f p x ∞ :=
{ r_le := ENNReal.le_of_forall_pos_nnreal_lt fun r hr _ =>
let ⟨_, hp'⟩ := h' r hr
(h.exchange_radius hp').r_le
r_pos := ENNReal.coe_lt_top
hasSum := fun {y} _ =>
let ⟨r', hr'⟩ := exists_gt ‖y‖₊
let ⟨_, hp'⟩ := h' r' hr'.ne_bot.bot_lt
(h.exchange_radius hp').hasSum <| mem_emetric_ball_zero_iff.mpr (ENNReal.coe_lt_coe.2 hr') }
#align has_fpower_series_on_ball.r_eq_top_of_exists HasFPowerSeriesOnBall.r_eq_top_of_exists
end Uniqueness
/-!
### Changing origin in a power series
If a function is analytic in a disk `D(x, R)`, then it is analytic in any disk contained in that
one. Indeed, one can write
$$
f (x + y + z) = \sum_{n} p_n (y + z)^n = \sum_{n, k} \binom{n}{k} p_n y^{n-k} z^k
= \sum_{k} \Bigl(\sum_{n} \binom{n}{k} p_n y^{n-k}\Bigr) z^k.
$$
The corresponding power series has thus a `k`-th coefficient equal to
$\sum_{n} \binom{n}{k} p_n y^{n-k}$. In the general case where `pₙ` is a multilinear map, this has
to be interpreted suitably: instead of having a binomial coefficient, one should sum over all
possible subsets `s` of `Fin n` of cardinal `k`, and attribute `z` to the indices in `s` and
`y` to the indices outside of `s`.
In this paragraph, we implement this. The new power series is called `p.changeOrigin y`. Then, we
check its convergence and the fact that its sum coincides with the original sum. The outcome of this
discussion is that the set of points where a function is analytic is open.
-/
namespace FormalMultilinearSeries
section
variable (p : FormalMultilinearSeries 𝕜 E F) {x y : E} {r R : ℝ≥0}
/-- A term of `FormalMultilinearSeries.changeOriginSeries`.
Given a formal multilinear series `p` and a point `x` in its ball of convergence,
`p.changeOrigin x` is a formal multilinear series such that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense. Each term of `p.changeOrigin x`
is itself an analytic function of `x` given by the series `p.changeOriginSeries`. Each term in
`changeOriginSeries` is the sum of `changeOriginSeriesTerm`'s over all `s` of cardinality `l`.
The definition is such that `p.changeOriginSeriesTerm k l s hs (fun _ ↦ x) (fun _ ↦ y) =
p (k + l) (s.piecewise (fun _ ↦ x) (fun _ ↦ y))`
-/
def changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
E[×l]→L[𝕜] E[×k]→L[𝕜] F := by
let a := ContinuousMultilinearMap.curryFinFinset 𝕜 E F hs
(by erw [Finset.card_compl, Fintype.card_fin, hs, add_tsub_cancel_right])
exact a (p (k + l))
#align formal_multilinear_series.change_origin_series_term FormalMultilinearSeries.changeOriginSeriesTerm
theorem changeOriginSeriesTerm_apply (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l)
(x y : E) :
(p.changeOriginSeriesTerm k l s hs (fun _ => x) fun _ => y) =
p (k + l) (s.piecewise (fun _ => x) fun _ => y) :=
ContinuousMultilinearMap.curryFinFinset_apply_const _ _ _ _ _
#align formal_multilinear_series.change_origin_series_term_apply FormalMultilinearSeries.changeOriginSeriesTerm_apply
@[simp]
theorem norm_changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
‖p.changeOriginSeriesTerm k l s hs‖ = ‖p (k + l)‖ := by
simp only [changeOriginSeriesTerm, LinearIsometryEquiv.norm_map]
#align formal_multilinear_series.norm_change_origin_series_term FormalMultilinearSeries.norm_changeOriginSeriesTerm
@[simp]
theorem nnnorm_changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
‖p.changeOriginSeriesTerm k l s hs‖₊ = ‖p (k + l)‖₊ := by
simp only [changeOriginSeriesTerm, LinearIsometryEquiv.nnnorm_map]
#align formal_multilinear_series.nnnorm_change_origin_series_term FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm
theorem nnnorm_changeOriginSeriesTerm_apply_le (k l : ℕ) (s : Finset (Fin (k + l)))
(hs : s.card = l) (x y : E) :
‖p.changeOriginSeriesTerm k l s hs (fun _ => x) fun _ => y‖₊ ≤
‖p (k + l)‖₊ * ‖x‖₊ ^ l * ‖y‖₊ ^ k := by
rw [← p.nnnorm_changeOriginSeriesTerm k l s hs, ← Fin.prod_const, ← Fin.prod_const]
apply ContinuousMultilinearMap.le_of_op_nnnorm_le
apply ContinuousMultilinearMap.le_op_nnnorm
#align formal_multilinear_series.nnnorm_change_origin_series_term_apply_le FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm_apply_le
/-- The power series for `f.changeOrigin k`.
Given a formal multilinear series `p` and a point `x` in its ball of convergence,
`p.changeOrigin x` is a formal multilinear series such that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense. Its `k`-th term is the sum of
the series `p.changeOriginSeries k`. -/
def changeOriginSeries (k : ℕ) : FormalMultilinearSeries 𝕜 E (E[×k]→L[𝕜] F) := fun l =>
∑ s : { s : Finset (Fin (k + l)) // Finset.card s = l }, p.changeOriginSeriesTerm k l s s.2
#align formal_multilinear_series.change_origin_series FormalMultilinearSeries.changeOriginSeries
theorem nnnorm_changeOriginSeries_le_tsum (k l : ℕ) :
‖p.changeOriginSeries k l‖₊ ≤
∑' _ : { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + l)‖₊ :=
(nnnorm_sum_le _ (fun t => changeOriginSeriesTerm p k l (Subtype.val t) t.prop)).trans_eq <| by
simp_rw [tsum_fintype, nnnorm_changeOriginSeriesTerm (p := p) (k := k) (l := l)]
#align formal_multilinear_series.nnnorm_change_origin_series_le_tsum FormalMultilinearSeries.nnnorm_changeOriginSeries_le_tsum
theorem nnnorm_changeOriginSeries_apply_le_tsum (k l : ℕ) (x : E) :
‖p.changeOriginSeries k l fun _ => x‖₊ ≤
∑' _ : { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + l)‖₊ * ‖x‖₊ ^ l := by
rw [NNReal.tsum_mul_right, ← Fin.prod_const]
exact (p.changeOriginSeries k l).le_of_op_nnnorm_le _ (p.nnnorm_changeOriginSeries_le_tsum _ _)
#align formal_multilinear_series.nnnorm_change_origin_series_apply_le_tsum FormalMultilinearSeries.nnnorm_changeOriginSeries_apply_le_tsum
/-- Changing the origin of a formal multilinear series `p`, so that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense.
-/
def changeOrigin (x : E) : FormalMultilinearSeries 𝕜 E F :=
fun k => (p.changeOriginSeries k).sum x
#align formal_multilinear_series.change_origin FormalMultilinearSeries.changeOrigin
/-- An auxiliary equivalence useful in the proofs about
`FormalMultilinearSeries.changeOriginSeries`: the set of triples `(k, l, s)`, where `s` is a
`Finset (Fin (k + l))` of cardinality `l` is equivalent to the set of pairs `(n, s)`, where `s` is a
`Finset (Fin n)`.
The forward map sends `(k, l, s)` to `(k + l, s)` and the inverse map sends `(n, s)` to
`(n - Finset.card s, Finset.card s, s)`. The actual definition is less readable because of problems
with non-definitional equalities. -/
@[simps]
def changeOriginIndexEquiv :
(Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) ≃ Σn : ℕ, Finset (Fin n) where
toFun s := ⟨s.1 + s.2.1, s.2.2⟩
invFun s :=
⟨s.1 - s.2.card, s.2.card,
⟨s.2.map
(Fin.castIso <| (tsub_add_cancel_of_le <| card_finset_fin_le s.2).symm).toEquiv.toEmbedding,
Finset.card_map _⟩⟩
left_inv := by
rintro ⟨k, l, ⟨s : Finset (Fin <| k + l), hs : s.card = l⟩⟩
dsimp only [Subtype.coe_mk]
-- Lean can't automatically generalize `k' = k + l - s.card`, `l' = s.card`, so we explicitly
-- formulate the generalized goal
suffices ∀ k' l', k' = k → l' = l → ∀ (hkl : k + l = k' + l') (hs'),
(⟨k', l', ⟨Finset.map (Fin.castIso hkl).toEquiv.toEmbedding s, hs'⟩⟩ :
Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) = ⟨k, l, ⟨s, hs⟩⟩ by
apply this <;> simp only [hs, add_tsub_cancel_right]
rintro _ _ rfl rfl hkl hs'
simp only [Equiv.refl_toEmbedding, Fin.castIso_refl, Finset.map_refl, eq_self_iff_true,
OrderIso.refl_toEquiv, and_self_iff, heq_iff_eq]
right_inv := by
rintro ⟨n, s⟩
simp [tsub_add_cancel_of_le (card_finset_fin_le s), Fin.castIso_to_equiv]
#align formal_multilinear_series.change_origin_index_equiv FormalMultilinearSeries.changeOriginIndexEquiv
theorem changeOriginSeries_summable_aux₁ {r r' : ℝ≥0} (hr : (r + r' : ℝ≥0∞) < p.radius) :
Summable fun s : Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l } =>
‖p (s.1 + s.2.1)‖₊ * r ^ s.2.1 * r' ^ s.1 := by
rw [← changeOriginIndexEquiv.symm.summable_iff]
dsimp only [Function.comp_def, changeOriginIndexEquiv_symm_apply_fst,
changeOriginIndexEquiv_symm_apply_snd_fst]
have : ∀ n : ℕ,
HasSum (fun s : Finset (Fin n) => ‖p (n - s.card + s.card)‖₊ * r ^ s.card * r' ^ (n - s.card))
(‖p n‖₊ * (r + r') ^ n) := by
intro n
-- TODO: why `simp only [tsub_add_cancel_of_le (card_finset_fin_le _)]` fails?
convert_to HasSum (fun s : Finset (Fin n) => ‖p n‖₊ * (r ^ s.card * r' ^ (n - s.card))) _
· ext1 s
rw [tsub_add_cancel_of_le (card_finset_fin_le _), mul_assoc]
rw [← Fin.sum_pow_mul_eq_add_pow]
exact (hasSum_fintype _).mul_left _
refine' NNReal.summable_sigma.2 ⟨fun n => (this n).summable, _⟩
simp only [(this _).tsum_eq]
exact p.summable_nnnorm_mul_pow hr
#align formal_multilinear_series.change_origin_series_summable_aux₁ FormalMultilinearSeries.changeOriginSeries_summable_aux₁
theorem changeOriginSeries_summable_aux₂ (hr : (r : ℝ≥0∞) < p.radius) (k : ℕ) :
Summable fun s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l } =>
‖p (k + s.1)‖₊ * r ^ s.1 := by
rcases ENNReal.lt_iff_exists_add_pos_lt.1 hr with ⟨r', h0, hr'⟩
simpa only [mul_inv_cancel_right₀ (pow_pos h0 _).ne'] using
((NNReal.summable_sigma.1 (p.changeOriginSeries_summable_aux₁ hr')).1 k).mul_right (r' ^ k)⁻¹
#align formal_multilinear_series.change_origin_series_summable_aux₂ FormalMultilinearSeries.changeOriginSeries_summable_aux₂
theorem changeOriginSeries_summable_aux₃ {r : ℝ≥0} (hr : ↑r < p.radius) (k : ℕ) :
Summable fun l : ℕ => ‖p.changeOriginSeries k l‖₊ * r ^ l := by
refine' NNReal.summable_of_le
(fun n => _) (NNReal.summable_sigma.1 <| p.changeOriginSeries_summable_aux₂ hr k).2
simp only [NNReal.tsum_mul_right]
exact mul_le_mul' (p.nnnorm_changeOriginSeries_le_tsum _ _) le_rfl
#align formal_multilinear_series.change_origin_series_summable_aux₃ FormalMultilinearSeries.changeOriginSeries_summable_aux₃
theorem le_changeOriginSeries_radius (k : ℕ) : p.radius ≤ (p.changeOriginSeries k).radius :=
ENNReal.le_of_forall_nnreal_lt fun _r hr =>
le_radius_of_summable_nnnorm _ (p.changeOriginSeries_summable_aux₃ hr k)
#align formal_multilinear_series.le_change_origin_series_radius FormalMultilinearSeries.le_changeOriginSeries_radius
theorem nnnorm_changeOrigin_le (k : ℕ) (h : (‖x‖₊ : ℝ≥0∞) < p.radius) :
‖p.changeOrigin x k‖₊ ≤
∑' s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + s.1)‖₊ * ‖x‖₊ ^ s.1 := by
refine' tsum_of_nnnorm_bounded _ fun l => p.nnnorm_changeOriginSeries_apply_le_tsum k l x
have := p.changeOriginSeries_summable_aux₂ h k
refine' HasSum.sigma this.hasSum fun l => _
exact ((NNReal.summable_sigma.1 this).1 l).hasSum
#align formal_multilinear_series.nnnorm_change_origin_le FormalMultilinearSeries.nnnorm_changeOrigin_le
/-- The radius of convergence of `p.changeOrigin x` is at least `p.radius - ‖x‖`. In other words,
`p.changeOrigin x` is well defined on the largest ball contained in the original ball of
convergence. -/
theorem changeOrigin_radius : p.radius - ‖x‖₊ ≤ (p.changeOrigin x).radius := by
refine' ENNReal.le_of_forall_pos_nnreal_lt fun r _h0 hr => _
rw [lt_tsub_iff_right, add_comm] at hr
have hr' : (‖x‖₊ : ℝ≥0∞) < p.radius := (le_add_right le_rfl).trans_lt hr
apply le_radius_of_summable_nnnorm
have : ∀ k : ℕ,
‖p.changeOrigin x k‖₊ * r ^ k ≤
(∑' s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + s.1)‖₊ * ‖x‖₊ ^ s.1) *
r ^ k :=
fun k => mul_le_mul_right' (p.nnnorm_changeOrigin_le k hr') (r ^ k)
refine' NNReal.summable_of_le this _
simpa only [← NNReal.tsum_mul_right] using
(NNReal.summable_sigma.1 (p.changeOriginSeries_summable_aux₁ hr)).2
#align formal_multilinear_series.change_origin_radius FormalMultilinearSeries.changeOrigin_radius
end
-- From this point on, assume that the space is complete, to make sure that series that converge
-- in norm also converge in `F`.
variable [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x y : E} {r R : ℝ≥0}
theorem hasFPowerSeriesOnBall_changeOrigin (k : ℕ) (hr : 0 < p.radius) :
HasFPowerSeriesOnBall (fun x => p.changeOrigin x k) (p.changeOriginSeries k) 0 p.radius :=
have := p.le_changeOriginSeries_radius k
((p.changeOriginSeries k).hasFPowerSeriesOnBall (hr.trans_le this)).mono hr this
#align formal_multilinear_series.has_fpower_series_on_ball_change_origin FormalMultilinearSeries.hasFPowerSeriesOnBall_changeOrigin
/-- Summing the series `p.changeOrigin x` at a point `y` gives back `p (x + y)`. -/
theorem changeOrigin_eval (h : (‖x‖₊ + ‖y‖₊ : ℝ≥0∞) < p.radius) :
(p.changeOrigin x).sum y = p.sum (x + y) := by
have radius_pos : 0 < p.radius := lt_of_le_of_lt (zero_le _) h
have x_mem_ball : x ∈ EMetric.ball (0 : E) p.radius :=
mem_emetric_ball_zero_iff.2 ((le_add_right le_rfl).trans_lt h)
have y_mem_ball : y ∈ EMetric.ball (0 : E) (p.changeOrigin x).radius := by
refine' mem_emetric_ball_zero_iff.2 (lt_of_lt_of_le _ p.changeOrigin_radius)
rwa [lt_tsub_iff_right, add_comm]
have x_add_y_mem_ball : x + y ∈ EMetric.ball (0 : E) p.radius := by
refine' mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt _ h)
exact mod_cast nnnorm_add_le x y
set f : (Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) → F := fun s =>
p.changeOriginSeriesTerm s.1 s.2.1 s.2.2 s.2.2.2 (fun _ => x) fun _ => y
have hsf : Summable f := by
refine' .of_nnnorm_bounded _ (p.changeOriginSeries_summable_aux₁ h) _
rintro ⟨k, l, s, hs⟩
dsimp only [Subtype.coe_mk]
exact p.nnnorm_changeOriginSeriesTerm_apply_le _ _ _ _ _ _
have hf : HasSum f ((p.changeOrigin x).sum y) := by
refine' HasSum.sigma_of_hasSum ((p.changeOrigin x).summable y_mem_ball).hasSum (fun k => _) hsf
· dsimp only
refine' ContinuousMultilinearMap.hasSum_eval _ _
have := (p.hasFPowerSeriesOnBall_changeOrigin k radius_pos).hasSum x_mem_ball
rw [zero_add] at this
refine' HasSum.sigma_of_hasSum this (fun l => _) _
· simp only [changeOriginSeries, ContinuousMultilinearMap.sum_apply]
apply hasSum_fintype
· refine' .of_nnnorm_bounded _
(p.changeOriginSeries_summable_aux₂ (mem_emetric_ball_zero_iff.1 x_mem_ball) k) fun s => _
refine' (ContinuousMultilinearMap.le_op_nnnorm _ _).trans_eq _
simp
refine' hf.unique (changeOriginIndexEquiv.symm.hasSum_iff.1 _)
refine' HasSum.sigma_of_hasSum
(p.hasSum x_add_y_mem_ball) (fun n => _) (changeOriginIndexEquiv.symm.summable_iff.2 hsf)
erw [(p n).map_add_univ (fun _ => x) fun _ => y]
-- porting note: added explicit function
convert hasSum_fintype (fun c : Finset (Fin n) => f (changeOriginIndexEquiv.symm ⟨n, c⟩))
rename_i s _
dsimp only [changeOriginSeriesTerm, (· ∘ ·), changeOriginIndexEquiv_symm_apply_fst,
changeOriginIndexEquiv_symm_apply_snd_fst, changeOriginIndexEquiv_symm_apply_snd_snd_coe]
rw [ContinuousMultilinearMap.curryFinFinset_apply_const]
have : ∀ (m) (hm : n = m), p n (s.piecewise (fun _ => x) fun _ => y) =
p m ((s.map (Fin.castIso hm).toEquiv.toEmbedding).piecewise (fun _ => x) fun _ => y) := by
rintro m rfl
simp (config := { unfoldPartialApp := true }) [Finset.piecewise]
apply this
#align formal_multilinear_series.change_origin_eval FormalMultilinearSeries.changeOrigin_eval
/-- Power series terms are analytic as we vary the origin -/
theorem analyticAt_changeOrigin (p : FormalMultilinearSeries 𝕜 E F) (rp : p.radius > 0) (n : ℕ) :
AnalyticAt 𝕜 (fun x ↦ p.changeOrigin x n) 0 :=
(FormalMultilinearSeries.hasFPowerSeriesOnBall_changeOrigin p n rp).analyticAt
end FormalMultilinearSeries
section
variable [CompleteSpace F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x y : E} {r : ℝ≥0∞}
/-- If a function admits a power series expansion `p` on a ball `B (x, r)`, then it also admits a
power series on any subball of this ball (even with a different center), given by `p.changeOrigin`.
-/
theorem HasFPowerSeriesOnBall.changeOrigin (hf : HasFPowerSeriesOnBall f p x r)
(h : (‖y‖₊ : ℝ≥0∞) < r) : HasFPowerSeriesOnBall f (p.changeOrigin y) (x + y) (r - ‖y‖₊) :=
{ r_le := by
apply le_trans _ p.changeOrigin_radius
exact tsub_le_tsub hf.r_le le_rfl
r_pos := by simp [h]
hasSum := fun {z} hz => by
have : f (x + y + z) =
FormalMultilinearSeries.sum (FormalMultilinearSeries.changeOrigin p y) z := by
rw [mem_emetric_ball_zero_iff, lt_tsub_iff_right, add_comm] at hz
rw [p.changeOrigin_eval (hz.trans_le hf.r_le), add_assoc, hf.sum]
refine' mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt _ hz)
exact mod_cast nnnorm_add_le y z
rw [this]
apply (p.changeOrigin y).hasSum
refine' EMetric.ball_subset_ball (le_trans _ p.changeOrigin_radius) hz
exact tsub_le_tsub hf.r_le le_rfl }
#align has_fpower_series_on_ball.change_origin HasFPowerSeriesOnBall.changeOrigin
/-- If a function admits a power series expansion `p` on an open ball `B (x, r)`, then
it is analytic at every point of this ball. -/
theorem HasFPowerSeriesOnBall.analyticAt_of_mem (hf : HasFPowerSeriesOnBall f p x r)
(h : y ∈ EMetric.ball x r) : AnalyticAt 𝕜 f y := by
have : (‖y - x‖₊ : ℝ≥0∞) < r := by simpa [edist_eq_coe_nnnorm_sub] using h
have := hf.changeOrigin this
rw [add_sub_cancel'_right] at this
exact this.analyticAt
#align has_fpower_series_on_ball.analytic_at_of_mem HasFPowerSeriesOnBall.analyticAt_of_mem
theorem HasFPowerSeriesOnBall.analyticOn (hf : HasFPowerSeriesOnBall f p x r) :
AnalyticOn 𝕜 f (EMetric.ball x r) :=
fun _y hy => hf.analyticAt_of_mem hy
#align has_fpower_series_on_ball.analytic_on HasFPowerSeriesOnBall.analyticOn
variable (𝕜 f)
/-- For any function `f` from a normed vector space to a Banach space, the set of points `x` such
that `f` is analytic at `x` is open. -/
theorem isOpen_analyticAt : IsOpen { x | AnalyticAt 𝕜 f x } := by
rw [isOpen_iff_mem_nhds]
rintro x ⟨p, r, hr⟩
exact mem_of_superset (EMetric.ball_mem_nhds _ hr.r_pos) fun y hy => hr.analyticAt_of_mem hy
#align is_open_analytic_at isOpen_analyticAt
variable {𝕜}
theorem AnalyticAt.eventually_analyticAt {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∀ᶠ y in 𝓝 x, AnalyticAt 𝕜 f y :=
(isOpen_analyticAt 𝕜 f).mem_nhds h
theorem AnalyticAt.exists_mem_nhds_analyticOn {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∃ s ∈ 𝓝 x, AnalyticOn 𝕜 f s :=
h.eventually_analyticAt.exists_mem
/-- If we're analytic at a point, we're analytic in a nonempty ball -/
theorem AnalyticAt.exists_ball_analyticOn {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∃ r : ℝ, 0 < r ∧ AnalyticOn 𝕜 f (Metric.ball x r) :=
Metric.isOpen_iff.mp (isOpen_analyticAt _ _) _ h
end
section
open FormalMultilinearSeries
variable {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E} {z₀ : 𝕜}
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z)) := by
refine' ⟨fun ⟨r, _, r_pos, h⟩ =>
eventually_of_mem (EMetric.ball_mem_nhds 0 r_pos) fun _ => by simpa using h, _⟩
simp only [Metric.eventually_nhds_iff]
rintro ⟨r, r_pos, h⟩
refine' ⟨p.radius ⊓ r.toNNReal, by simp, _, _⟩
·
|
simp only [r_pos.lt, lt_inf_iff, ENNReal.coe_pos, Real.toNNReal_pos, and_true_iff]
|
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z)) := by
refine' ⟨fun ⟨r, _, r_pos, h⟩ =>
eventually_of_mem (EMetric.ball_mem_nhds 0 r_pos) fun _ => by simpa using h, _⟩
simp only [Metric.eventually_nhds_iff]
rintro ⟨r, r_pos, h⟩
refine' ⟨p.radius ⊓ r.toNNReal, by simp, _, _⟩
·
|
Mathlib.Analysis.Analytic.Basic.1430_0.jQw1fRSE1vGpOll
|
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z))
|
Mathlib_Analysis_Analytic_Basic
|
case intro.intro.refine'_1
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
p : FormalMultilinearSeries 𝕜 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
r : ℝ
r_pos : r > 0
h : ∀ ⦃y : 𝕜⦄, dist y 0 < r → HasSum (fun n => y ^ n • coeff p n) (f (z₀ + y))
⊢ 0 < radius p
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Logic.Equiv.Fin
import Mathlib.Topology.Algebra.InfiniteSum.Module
#align_import analysis.analytic.basic from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
/-!
# Analytic functions
A function is analytic in one dimension around `0` if it can be written as a converging power series
`Σ pₙ zⁿ`. This definition can be extended to any dimension (even in infinite dimension) by
requiring that `pₙ` is a continuous `n`-multilinear map. In general, `pₙ` is not unique (in two
dimensions, taking `p₂ (x, y) (x', y') = x y'` or `y x'` gives the same map when applied to a
vector `(x, y) (x, y)`). A way to guarantee uniqueness is to take a symmetric `pₙ`, but this is not
always possible in nonzero characteristic (in characteristic 2, the previous example has no
symmetric representative). Therefore, we do not insist on symmetry or uniqueness in the definition,
and we only require the existence of a converging series.
The general framework is important to say that the exponential map on bounded operators on a Banach
space is analytic, as well as the inverse on invertible operators.
## Main definitions
Let `p` be a formal multilinear series from `E` to `F`, i.e., `p n` is a multilinear map on `E^n`
for `n : ℕ`.
* `p.radius`: the largest `r : ℝ≥0∞` such that `‖p n‖ * r^n` grows subexponentially.
* `p.le_radius_of_bound`, `p.le_radius_of_bound_nnreal`, `p.le_radius_of_isBigO`: if `‖p n‖ * r ^ n`
is bounded above, then `r ≤ p.radius`;
* `p.isLittleO_of_lt_radius`, `p.norm_mul_pow_le_mul_pow_of_lt_radius`,
`p.isLittleO_one_of_lt_radius`,
`p.norm_mul_pow_le_of_lt_radius`, `p.nnnorm_mul_pow_le_of_lt_radius`: if `r < p.radius`, then
`‖p n‖ * r ^ n` tends to zero exponentially;
* `p.lt_radius_of_isBigO`: if `r ≠ 0` and `‖p n‖ * r ^ n = O(a ^ n)` for some `-1 < a < 1`, then
`r < p.radius`;
* `p.partialSum n x`: the sum `∑_{i = 0}^{n-1} pᵢ xⁱ`.
* `p.sum x`: the sum `∑'_{i = 0}^{∞} pᵢ xⁱ`.
Additionally, let `f` be a function from `E` to `F`.
* `HasFPowerSeriesOnBall f p x r`: on the ball of center `x` with radius `r`,
`f (x + y) = ∑'_n pₙ yⁿ`.
* `HasFPowerSeriesAt f p x`: on some ball of center `x` with positive radius, holds
`HasFPowerSeriesOnBall f p x r`.
* `AnalyticAt 𝕜 f x`: there exists a power series `p` such that holds `HasFPowerSeriesAt f p x`.
* `AnalyticOn 𝕜 f s`: the function `f` is analytic at every point of `s`.
We develop the basic properties of these notions, notably:
* If a function admits a power series, it is continuous (see
`HasFPowerSeriesOnBall.continuousOn` and `HasFPowerSeriesAt.continuousAt` and
`AnalyticAt.continuousAt`).
* In a complete space, the sum of a formal power series with positive radius is well defined on the
disk of convergence, see `FormalMultilinearSeries.hasFPowerSeriesOnBall`.
* If a function admits a power series in a ball, then it is analytic at any point `y` of this ball,
and the power series there can be expressed in terms of the initial power series `p` as
`p.changeOrigin y`. See `HasFPowerSeriesOnBall.changeOrigin`. It follows in particular that
the set of points at which a given function is analytic is open, see `isOpen_analyticAt`.
## Implementation details
We only introduce the radius of convergence of a power series, as `p.radius`.
For a power series in finitely many dimensions, there is a finer (directional, coordinate-dependent)
notion, describing the polydisk of convergence. This notion is more specific, and not necessary to
build the general theory. We do not define it here.
-/
noncomputable section
variable {𝕜 E F G : Type*}
open Topology Classical BigOperators NNReal Filter ENNReal
open Set Filter Asymptotics
namespace FormalMultilinearSeries
variable [Ring 𝕜] [AddCommGroup E] [AddCommGroup F] [Module 𝕜 E] [Module 𝕜 F]
variable [TopologicalSpace E] [TopologicalSpace F]
variable [TopologicalAddGroup E] [TopologicalAddGroup F]
variable [ContinuousConstSMul 𝕜 E] [ContinuousConstSMul 𝕜 F]
/-- Given a formal multilinear series `p` and a vector `x`, then `p.sum x` is the sum `Σ pₙ xⁿ`. A
priori, it only behaves well when `‖x‖ < p.radius`. -/
protected def sum (p : FormalMultilinearSeries 𝕜 E F) (x : E) : F :=
∑' n : ℕ, p n fun _ => x
#align formal_multilinear_series.sum FormalMultilinearSeries.sum
/-- Given a formal multilinear series `p` and a vector `x`, then `p.partialSum n x` is the sum
`Σ pₖ xᵏ` for `k ∈ {0,..., n-1}`. -/
def partialSum (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) (x : E) : F :=
∑ k in Finset.range n, p k fun _ : Fin k => x
#align formal_multilinear_series.partial_sum FormalMultilinearSeries.partialSum
/-- The partial sums of a formal multilinear series are continuous. -/
theorem partialSum_continuous (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) :
Continuous (p.partialSum n) := by
unfold partialSum -- Porting note: added
continuity
#align formal_multilinear_series.partial_sum_continuous FormalMultilinearSeries.partialSum_continuous
end FormalMultilinearSeries
/-! ### The radius of a formal multilinear series -/
variable [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F]
[NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G]
namespace FormalMultilinearSeries
variable (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
/-- The radius of a formal multilinear series is the largest `r` such that the sum `Σ ‖pₙ‖ ‖y‖ⁿ`
converges for all `‖y‖ < r`. This implies that `Σ pₙ yⁿ` converges for all `‖y‖ < r`, but these
definitions are *not* equivalent in general. -/
def radius (p : FormalMultilinearSeries 𝕜 E F) : ℝ≥0∞ :=
⨆ (r : ℝ≥0) (C : ℝ) (_ : ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C), (r : ℝ≥0∞)
#align formal_multilinear_series.radius FormalMultilinearSeries.radius
/-- If `‖pₙ‖ rⁿ` is bounded in `n`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_bound (C : ℝ) {r : ℝ≥0} (h : ∀ n : ℕ, ‖p n‖ * (r : ℝ) ^ n ≤ C) :
(r : ℝ≥0∞) ≤ p.radius :=
le_iSup_of_le r <| le_iSup_of_le C <| le_iSup (fun _ => (r : ℝ≥0∞)) h
#align formal_multilinear_series.le_radius_of_bound FormalMultilinearSeries.le_radius_of_bound
/-- If `‖pₙ‖ rⁿ` is bounded in `n`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_bound_nnreal (C : ℝ≥0) {r : ℝ≥0} (h : ∀ n : ℕ, ‖p n‖₊ * r ^ n ≤ C) :
(r : ℝ≥0∞) ≤ p.radius :=
p.le_radius_of_bound C fun n => mod_cast h n
#align formal_multilinear_series.le_radius_of_bound_nnreal FormalMultilinearSeries.le_radius_of_bound_nnreal
/-- If `‖pₙ‖ rⁿ = O(1)`, as `n → ∞`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_isBigO (h : (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] fun _ => (1 : ℝ)) :
↑r ≤ p.radius :=
Exists.elim (isBigO_one_nat_atTop_iff.1 h) fun C hC =>
p.le_radius_of_bound C fun n => (le_abs_self _).trans (hC n)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.le_radius_of_is_O FormalMultilinearSeries.le_radius_of_isBigO
theorem le_radius_of_eventually_le (C) (h : ∀ᶠ n in atTop, ‖p n‖ * (r : ℝ) ^ n ≤ C) :
↑r ≤ p.radius :=
p.le_radius_of_isBigO <| IsBigO.of_bound C <| h.mono fun n hn => by simpa
#align formal_multilinear_series.le_radius_of_eventually_le FormalMultilinearSeries.le_radius_of_eventually_le
theorem le_radius_of_summable_nnnorm (h : Summable fun n => ‖p n‖₊ * r ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_bound_nnreal (∑' n, ‖p n‖₊ * r ^ n) fun _ => le_tsum' h _
#align formal_multilinear_series.le_radius_of_summable_nnnorm FormalMultilinearSeries.le_radius_of_summable_nnnorm
theorem le_radius_of_summable (h : Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_summable_nnnorm <| by
simp only [← coe_nnnorm] at h
exact mod_cast h
#align formal_multilinear_series.le_radius_of_summable FormalMultilinearSeries.le_radius_of_summable
theorem radius_eq_top_of_forall_nnreal_isBigO
(h : ∀ r : ℝ≥0, (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] fun _ => (1 : ℝ)) : p.radius = ∞ :=
ENNReal.eq_top_of_forall_nnreal_le fun r => p.le_radius_of_isBigO (h r)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.radius_eq_top_of_forall_nnreal_is_O FormalMultilinearSeries.radius_eq_top_of_forall_nnreal_isBigO
theorem radius_eq_top_of_eventually_eq_zero (h : ∀ᶠ n in atTop, p n = 0) : p.radius = ∞ :=
p.radius_eq_top_of_forall_nnreal_isBigO fun r =>
(isBigO_zero _ _).congr' (h.mono fun n hn => by simp [hn]) EventuallyEq.rfl
#align formal_multilinear_series.radius_eq_top_of_eventually_eq_zero FormalMultilinearSeries.radius_eq_top_of_eventually_eq_zero
theorem radius_eq_top_of_forall_image_add_eq_zero (n : ℕ) (hn : ∀ m, p (m + n) = 0) :
p.radius = ∞ :=
p.radius_eq_top_of_eventually_eq_zero <|
mem_atTop_sets.2 ⟨n, fun _ hk => tsub_add_cancel_of_le hk ▸ hn _⟩
#align formal_multilinear_series.radius_eq_top_of_forall_image_add_eq_zero FormalMultilinearSeries.radius_eq_top_of_forall_image_add_eq_zero
@[simp]
theorem constFormalMultilinearSeries_radius {v : F} :
(constFormalMultilinearSeries 𝕜 E v).radius = ⊤ :=
(constFormalMultilinearSeries 𝕜 E v).radius_eq_top_of_forall_image_add_eq_zero 1
(by simp [constFormalMultilinearSeries])
#align formal_multilinear_series.const_formal_multilinear_series_radius FormalMultilinearSeries.constFormalMultilinearSeries_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` tends to zero exponentially:
for some `0 < a < 1`, `‖p n‖ rⁿ = o(aⁿ)`. -/
theorem isLittleO_of_lt_radius (h : ↑r < p.radius) :
∃ a ∈ Ioo (0 : ℝ) 1, (fun n => ‖p n‖ * (r : ℝ) ^ n) =o[atTop] (a ^ ·) := by
have := (TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 4
rw [this]
-- Porting note: was
-- rw [(TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 4]
simp only [radius, lt_iSup_iff] at h
rcases h with ⟨t, C, hC, rt⟩
rw [ENNReal.coe_lt_coe, ← NNReal.coe_lt_coe] at rt
have : 0 < (t : ℝ) := r.coe_nonneg.trans_lt rt
rw [← div_lt_one this] at rt
refine' ⟨_, rt, C, Or.inr zero_lt_one, fun n => _⟩
calc
|‖p n‖ * (r : ℝ) ^ n| = ‖p n‖ * (t : ℝ) ^ n * (r / t : ℝ) ^ n := by
field_simp [mul_right_comm, abs_mul]
_ ≤ C * (r / t : ℝ) ^ n := by gcongr; apply hC
#align formal_multilinear_series.is_o_of_lt_radius FormalMultilinearSeries.isLittleO_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ = o(1)`. -/
theorem isLittleO_one_of_lt_radius (h : ↑r < p.radius) :
(fun n => ‖p n‖ * (r : ℝ) ^ n) =o[atTop] (fun _ => 1 : ℕ → ℝ) :=
let ⟨_, ha, hp⟩ := p.isLittleO_of_lt_radius h
hp.trans <| (isLittleO_pow_pow_of_lt_left ha.1.le ha.2).congr (fun _ => rfl) one_pow
#align formal_multilinear_series.is_o_one_of_lt_radius FormalMultilinearSeries.isLittleO_one_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` tends to zero exponentially:
for some `0 < a < 1` and `C > 0`, `‖p n‖ * r ^ n ≤ C * a ^ n`. -/
theorem norm_mul_pow_le_mul_pow_of_lt_radius (h : ↑r < p.radius) :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C * a ^ n := by
-- Porting note: moved out of `rcases`
have := ((TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 5).mp
(p.isLittleO_of_lt_radius h)
rcases this with ⟨a, ha, C, hC, H⟩
exact ⟨a, ha, C, hC, fun n => (le_abs_self _).trans (H n)⟩
#align formal_multilinear_series.norm_mul_pow_le_mul_pow_of_lt_radius FormalMultilinearSeries.norm_mul_pow_le_mul_pow_of_lt_radius
/-- If `r ≠ 0` and `‖pₙ‖ rⁿ = O(aⁿ)` for some `-1 < a < 1`, then `r < p.radius`. -/
theorem lt_radius_of_isBigO (h₀ : r ≠ 0) {a : ℝ} (ha : a ∈ Ioo (-1 : ℝ) 1)
(hp : (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] (a ^ ·)) : ↑r < p.radius := by
-- Porting note: moved out of `rcases`
have := ((TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 2 5)
rcases this.mp ⟨a, ha, hp⟩ with ⟨a, ha, C, hC, hp⟩
rw [← pos_iff_ne_zero, ← NNReal.coe_pos] at h₀
lift a to ℝ≥0 using ha.1.le
have : (r : ℝ) < r / a := by
simpa only [div_one] using (div_lt_div_left h₀ zero_lt_one ha.1).2 ha.2
norm_cast at this
rw [← ENNReal.coe_lt_coe] at this
refine' this.trans_le (p.le_radius_of_bound C fun n => _)
rw [NNReal.coe_div, div_pow, ← mul_div_assoc, div_le_iff (pow_pos ha.1 n)]
exact (le_abs_self _).trans (hp n)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.lt_radius_of_is_O FormalMultilinearSeries.lt_radius_of_isBigO
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem norm_mul_pow_le_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C :=
let ⟨_, ha, C, hC, h⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius h
⟨C, hC, fun n => (h n).trans <| mul_le_of_le_one_right hC.lt.le (pow_le_one _ ha.1.le ha.2.le)⟩
#align formal_multilinear_series.norm_mul_pow_le_of_lt_radius FormalMultilinearSeries.norm_mul_pow_le_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem norm_le_div_pow_of_pos_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h0 : 0 < r) (h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖ ≤ C / (r : ℝ) ^ n :=
let ⟨C, hC, hp⟩ := p.norm_mul_pow_le_of_lt_radius h
⟨C, hC, fun n => Iff.mpr (le_div_iff (pow_pos h0 _)) (hp n)⟩
#align formal_multilinear_series.norm_le_div_pow_of_pos_of_lt_radius FormalMultilinearSeries.norm_le_div_pow_of_pos_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem nnnorm_mul_pow_le_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖₊ * r ^ n ≤ C :=
let ⟨C, hC, hp⟩ := p.norm_mul_pow_le_of_lt_radius h
⟨⟨C, hC.lt.le⟩, hC, mod_cast hp⟩
#align formal_multilinear_series.nnnorm_mul_pow_le_of_lt_radius FormalMultilinearSeries.nnnorm_mul_pow_le_of_lt_radius
theorem le_radius_of_tendsto (p : FormalMultilinearSeries 𝕜 E F) {l : ℝ}
(h : Tendsto (fun n => ‖p n‖ * (r : ℝ) ^ n) atTop (𝓝 l)) : ↑r ≤ p.radius :=
p.le_radius_of_isBigO (h.isBigO_one _)
#align formal_multilinear_series.le_radius_of_tendsto FormalMultilinearSeries.le_radius_of_tendsto
theorem le_radius_of_summable_norm (p : FormalMultilinearSeries 𝕜 E F)
(hs : Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_tendsto hs.tendsto_atTop_zero
#align formal_multilinear_series.le_radius_of_summable_norm FormalMultilinearSeries.le_radius_of_summable_norm
theorem not_summable_norm_of_radius_lt_nnnorm (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(h : p.radius < ‖x‖₊) : ¬Summable fun n => ‖p n‖ * ‖x‖ ^ n :=
fun hs => not_le_of_lt h (p.le_radius_of_summable_norm hs)
#align formal_multilinear_series.not_summable_norm_of_radius_lt_nnnorm FormalMultilinearSeries.not_summable_norm_of_radius_lt_nnnorm
theorem summable_norm_mul_pow (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0} (h : ↑r < p.radius) :
Summable fun n : ℕ => ‖p n‖ * (r : ℝ) ^ n := by
obtain ⟨a, ha : a ∈ Ioo (0 : ℝ) 1, C, - : 0 < C, hp⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius h
exact .of_nonneg_of_le (fun n => mul_nonneg (norm_nonneg _) (pow_nonneg r.coe_nonneg _))
hp ((summable_geometric_of_lt_1 ha.1.le ha.2).mul_left _)
#align formal_multilinear_series.summable_norm_mul_pow FormalMultilinearSeries.summable_norm_mul_pow
theorem summable_norm_apply (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : Summable fun n : ℕ => ‖p n fun _ => x‖ := by
rw [mem_emetric_ball_zero_iff] at hx
refine' .of_nonneg_of_le
(fun _ => norm_nonneg _) (fun n => ((p n).le_op_norm _).trans_eq _) (p.summable_norm_mul_pow hx)
simp
#align formal_multilinear_series.summable_norm_apply FormalMultilinearSeries.summable_norm_apply
theorem summable_nnnorm_mul_pow (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0} (h : ↑r < p.radius) :
Summable fun n : ℕ => ‖p n‖₊ * r ^ n := by
rw [← NNReal.summable_coe]
push_cast
exact p.summable_norm_mul_pow h
#align formal_multilinear_series.summable_nnnorm_mul_pow FormalMultilinearSeries.summable_nnnorm_mul_pow
protected theorem summable [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : Summable fun n : ℕ => p n fun _ => x :=
(p.summable_norm_apply hx).of_norm
#align formal_multilinear_series.summable FormalMultilinearSeries.summable
theorem radius_eq_top_of_summable_norm (p : FormalMultilinearSeries 𝕜 E F)
(hs : ∀ r : ℝ≥0, Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : p.radius = ∞ :=
ENNReal.eq_top_of_forall_nnreal_le fun r => p.le_radius_of_summable_norm (hs r)
#align formal_multilinear_series.radius_eq_top_of_summable_norm FormalMultilinearSeries.radius_eq_top_of_summable_norm
theorem radius_eq_top_iff_summable_norm (p : FormalMultilinearSeries 𝕜 E F) :
p.radius = ∞ ↔ ∀ r : ℝ≥0, Summable fun n => ‖p n‖ * (r : ℝ) ^ n := by
constructor
· intro h r
obtain ⟨a, ha : a ∈ Ioo (0 : ℝ) 1, C, - : 0 < C, hp⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius
(show (r : ℝ≥0∞) < p.radius from h.symm ▸ ENNReal.coe_lt_top)
refine' .of_norm_bounded
(fun n => (C : ℝ) * a ^ n) ((summable_geometric_of_lt_1 ha.1.le ha.2).mul_left _) fun n => _
specialize hp n
rwa [Real.norm_of_nonneg (mul_nonneg (norm_nonneg _) (pow_nonneg r.coe_nonneg n))]
· exact p.radius_eq_top_of_summable_norm
#align formal_multilinear_series.radius_eq_top_iff_summable_norm FormalMultilinearSeries.radius_eq_top_iff_summable_norm
/-- If the radius of `p` is positive, then `‖pₙ‖` grows at most geometrically. -/
theorem le_mul_pow_of_radius_pos (p : FormalMultilinearSeries 𝕜 E F) (h : 0 < p.radius) :
∃ (C r : _) (hC : 0 < C) (_ : 0 < r), ∀ n, ‖p n‖ ≤ C * r ^ n := by
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 h with ⟨r, r0, rlt⟩
have rpos : 0 < (r : ℝ) := by simp [ENNReal.coe_pos.1 r0]
rcases norm_le_div_pow_of_pos_of_lt_radius p rpos rlt with ⟨C, Cpos, hCp⟩
refine' ⟨C, r⁻¹, Cpos, by simp only [inv_pos, rpos], fun n => _⟩
-- Porting note: was `convert`
rw [inv_pow, ← div_eq_mul_inv]
exact hCp n
#align formal_multilinear_series.le_mul_pow_of_radius_pos FormalMultilinearSeries.le_mul_pow_of_radius_pos
/-- The radius of the sum of two formal series is at least the minimum of their two radii. -/
theorem min_radius_le_radius_add (p q : FormalMultilinearSeries 𝕜 E F) :
min p.radius q.radius ≤ (p + q).radius := by
refine' ENNReal.le_of_forall_nnreal_lt fun r hr => _
rw [lt_min_iff] at hr
have := ((p.isLittleO_one_of_lt_radius hr.1).add (q.isLittleO_one_of_lt_radius hr.2)).isBigO
refine' (p + q).le_radius_of_isBigO ((isBigO_of_le _ fun n => _).trans this)
rw [← add_mul, norm_mul, norm_mul, norm_norm]
exact mul_le_mul_of_nonneg_right ((norm_add_le _ _).trans (le_abs_self _)) (norm_nonneg _)
#align formal_multilinear_series.min_radius_le_radius_add FormalMultilinearSeries.min_radius_le_radius_add
@[simp]
theorem radius_neg (p : FormalMultilinearSeries 𝕜 E F) : (-p).radius = p.radius := by
simp only [radius, neg_apply, norm_neg]
#align formal_multilinear_series.radius_neg FormalMultilinearSeries.radius_neg
protected theorem hasSum [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : HasSum (fun n : ℕ => p n fun _ => x) (p.sum x) :=
(p.summable hx).hasSum
#align formal_multilinear_series.has_sum FormalMultilinearSeries.hasSum
theorem radius_le_radius_continuousLinearMap_comp (p : FormalMultilinearSeries 𝕜 E F)
(f : F →L[𝕜] G) : p.radius ≤ (f.compFormalMultilinearSeries p).radius := by
refine' ENNReal.le_of_forall_nnreal_lt fun r hr => _
apply le_radius_of_isBigO
apply (IsBigO.trans_isLittleO _ (p.isLittleO_one_of_lt_radius hr)).isBigO
refine' IsBigO.mul (@IsBigOWith.isBigO _ _ _ _ _ ‖f‖ _ _ _ _) (isBigO_refl _ _)
refine IsBigOWith.of_bound (eventually_of_forall fun n => ?_)
simpa only [norm_norm] using f.norm_compContinuousMultilinearMap_le (p n)
#align formal_multilinear_series.radius_le_radius_continuous_linear_map_comp FormalMultilinearSeries.radius_le_radius_continuousLinearMap_comp
end FormalMultilinearSeries
/-! ### Expanding a function as a power series -/
section
variable {f g : E → F} {p pf pg : FormalMultilinearSeries 𝕜 E F} {x : E} {r r' : ℝ≥0∞}
/-- Given a function `f : E → F` and a formal multilinear series `p`, we say that `f` has `p` as
a power series on the ball of radius `r > 0` around `x` if `f (x + y) = ∑' pₙ yⁿ` for all `‖y‖ < r`.
-/
structure HasFPowerSeriesOnBall (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (x : E) (r : ℝ≥0∞) :
Prop where
r_le : r ≤ p.radius
r_pos : 0 < r
hasSum :
∀ {y}, y ∈ EMetric.ball (0 : E) r → HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y))
#align has_fpower_series_on_ball HasFPowerSeriesOnBall
/-- Given a function `f : E → F` and a formal multilinear series `p`, we say that `f` has `p` as
a power series around `x` if `f (x + y) = ∑' pₙ yⁿ` for all `y` in a neighborhood of `0`. -/
def HasFPowerSeriesAt (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (x : E) :=
∃ r, HasFPowerSeriesOnBall f p x r
#align has_fpower_series_at HasFPowerSeriesAt
variable (𝕜)
/-- Given a function `f : E → F`, we say that `f` is analytic at `x` if it admits a convergent power
series expansion around `x`. -/
def AnalyticAt (f : E → F) (x : E) :=
∃ p : FormalMultilinearSeries 𝕜 E F, HasFPowerSeriesAt f p x
#align analytic_at AnalyticAt
/-- Given a function `f : E → F`, we say that `f` is analytic on a set `s` if it is analytic around
every point of `s`. -/
def AnalyticOn (f : E → F) (s : Set E) :=
∀ x, x ∈ s → AnalyticAt 𝕜 f x
#align analytic_on AnalyticOn
variable {𝕜}
theorem HasFPowerSeriesOnBall.hasFPowerSeriesAt (hf : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesAt f p x :=
⟨r, hf⟩
#align has_fpower_series_on_ball.has_fpower_series_at HasFPowerSeriesOnBall.hasFPowerSeriesAt
theorem HasFPowerSeriesAt.analyticAt (hf : HasFPowerSeriesAt f p x) : AnalyticAt 𝕜 f x :=
⟨p, hf⟩
#align has_fpower_series_at.analytic_at HasFPowerSeriesAt.analyticAt
theorem HasFPowerSeriesOnBall.analyticAt (hf : HasFPowerSeriesOnBall f p x r) : AnalyticAt 𝕜 f x :=
hf.hasFPowerSeriesAt.analyticAt
#align has_fpower_series_on_ball.analytic_at HasFPowerSeriesOnBall.analyticAt
theorem HasFPowerSeriesOnBall.congr (hf : HasFPowerSeriesOnBall f p x r)
(hg : EqOn f g (EMetric.ball x r)) : HasFPowerSeriesOnBall g p x r :=
{ r_le := hf.r_le
r_pos := hf.r_pos
hasSum := fun {y} hy => by
convert hf.hasSum hy using 1
apply hg.symm
simpa [edist_eq_coe_nnnorm_sub] using hy }
#align has_fpower_series_on_ball.congr HasFPowerSeriesOnBall.congr
/-- If a function `f` has a power series `p` around `x`, then the function `z ↦ f (z - y)` has the
same power series around `x + y`. -/
theorem HasFPowerSeriesOnBall.comp_sub (hf : HasFPowerSeriesOnBall f p x r) (y : E) :
HasFPowerSeriesOnBall (fun z => f (z - y)) p (x + y) r :=
{ r_le := hf.r_le
r_pos := hf.r_pos
hasSum := fun {z} hz => by
convert hf.hasSum hz using 2
abel }
#align has_fpower_series_on_ball.comp_sub HasFPowerSeriesOnBall.comp_sub
theorem HasFPowerSeriesOnBall.hasSum_sub (hf : HasFPowerSeriesOnBall f p x r) {y : E}
(hy : y ∈ EMetric.ball x r) : HasSum (fun n : ℕ => p n fun _ => y - x) (f y) := by
have : y - x ∈ EMetric.ball (0 : E) r := by simpa [edist_eq_coe_nnnorm_sub] using hy
simpa only [add_sub_cancel'_right] using hf.hasSum this
#align has_fpower_series_on_ball.has_sum_sub HasFPowerSeriesOnBall.hasSum_sub
theorem HasFPowerSeriesOnBall.radius_pos (hf : HasFPowerSeriesOnBall f p x r) : 0 < p.radius :=
lt_of_lt_of_le hf.r_pos hf.r_le
#align has_fpower_series_on_ball.radius_pos HasFPowerSeriesOnBall.radius_pos
theorem HasFPowerSeriesAt.radius_pos (hf : HasFPowerSeriesAt f p x) : 0 < p.radius :=
let ⟨_, hr⟩ := hf
hr.radius_pos
#align has_fpower_series_at.radius_pos HasFPowerSeriesAt.radius_pos
theorem HasFPowerSeriesOnBall.mono (hf : HasFPowerSeriesOnBall f p x r) (r'_pos : 0 < r')
(hr : r' ≤ r) : HasFPowerSeriesOnBall f p x r' :=
⟨le_trans hr hf.1, r'_pos, fun hy => hf.hasSum (EMetric.ball_subset_ball hr hy)⟩
#align has_fpower_series_on_ball.mono HasFPowerSeriesOnBall.mono
theorem HasFPowerSeriesAt.congr (hf : HasFPowerSeriesAt f p x) (hg : f =ᶠ[𝓝 x] g) :
HasFPowerSeriesAt g p x := by
rcases hf with ⟨r₁, h₁⟩
rcases EMetric.mem_nhds_iff.mp hg with ⟨r₂, h₂pos, h₂⟩
exact ⟨min r₁ r₂,
(h₁.mono (lt_min h₁.r_pos h₂pos) inf_le_left).congr
fun y hy => h₂ (EMetric.ball_subset_ball inf_le_right hy)⟩
#align has_fpower_series_at.congr HasFPowerSeriesAt.congr
protected theorem HasFPowerSeriesAt.eventually (hf : HasFPowerSeriesAt f p x) :
∀ᶠ r : ℝ≥0∞ in 𝓝[>] 0, HasFPowerSeriesOnBall f p x r :=
let ⟨_, hr⟩ := hf
mem_of_superset (Ioo_mem_nhdsWithin_Ioi (left_mem_Ico.2 hr.r_pos)) fun _ hr' =>
hr.mono hr'.1 hr'.2.le
#align has_fpower_series_at.eventually HasFPowerSeriesAt.eventually
theorem HasFPowerSeriesOnBall.eventually_hasSum (hf : HasFPowerSeriesOnBall f p x r) :
∀ᶠ y in 𝓝 0, HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y)) := by
filter_upwards [EMetric.ball_mem_nhds (0 : E) hf.r_pos] using fun _ => hf.hasSum
#align has_fpower_series_on_ball.eventually_has_sum HasFPowerSeriesOnBall.eventually_hasSum
theorem HasFPowerSeriesAt.eventually_hasSum (hf : HasFPowerSeriesAt f p x) :
∀ᶠ y in 𝓝 0, HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y)) :=
let ⟨_, hr⟩ := hf
hr.eventually_hasSum
#align has_fpower_series_at.eventually_has_sum HasFPowerSeriesAt.eventually_hasSum
theorem HasFPowerSeriesOnBall.eventually_hasSum_sub (hf : HasFPowerSeriesOnBall f p x r) :
∀ᶠ y in 𝓝 x, HasSum (fun n : ℕ => p n fun _ : Fin n => y - x) (f y) := by
filter_upwards [EMetric.ball_mem_nhds x hf.r_pos] with y using hf.hasSum_sub
#align has_fpower_series_on_ball.eventually_has_sum_sub HasFPowerSeriesOnBall.eventually_hasSum_sub
theorem HasFPowerSeriesAt.eventually_hasSum_sub (hf : HasFPowerSeriesAt f p x) :
∀ᶠ y in 𝓝 x, HasSum (fun n : ℕ => p n fun _ : Fin n => y - x) (f y) :=
let ⟨_, hr⟩ := hf
hr.eventually_hasSum_sub
#align has_fpower_series_at.eventually_has_sum_sub HasFPowerSeriesAt.eventually_hasSum_sub
theorem HasFPowerSeriesOnBall.eventually_eq_zero
(hf : HasFPowerSeriesOnBall f (0 : FormalMultilinearSeries 𝕜 E F) x r) :
∀ᶠ z in 𝓝 x, f z = 0 := by
filter_upwards [hf.eventually_hasSum_sub] with z hz using hz.unique hasSum_zero
#align has_fpower_series_on_ball.eventually_eq_zero HasFPowerSeriesOnBall.eventually_eq_zero
theorem HasFPowerSeriesAt.eventually_eq_zero
(hf : HasFPowerSeriesAt f (0 : FormalMultilinearSeries 𝕜 E F) x) : ∀ᶠ z in 𝓝 x, f z = 0 :=
let ⟨_, hr⟩ := hf
hr.eventually_eq_zero
#align has_fpower_series_at.eventually_eq_zero HasFPowerSeriesAt.eventually_eq_zero
theorem hasFPowerSeriesOnBall_const {c : F} {e : E} :
HasFPowerSeriesOnBall (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e ⊤ := by
refine' ⟨by simp, WithTop.zero_lt_top, fun _ => hasSum_single 0 fun n hn => _⟩
simp [constFormalMultilinearSeries_apply hn]
#align has_fpower_series_on_ball_const hasFPowerSeriesOnBall_const
theorem hasFPowerSeriesAt_const {c : F} {e : E} :
HasFPowerSeriesAt (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e :=
⟨⊤, hasFPowerSeriesOnBall_const⟩
#align has_fpower_series_at_const hasFPowerSeriesAt_const
theorem analyticAt_const {v : F} : AnalyticAt 𝕜 (fun _ => v) x :=
⟨constFormalMultilinearSeries 𝕜 E v, hasFPowerSeriesAt_const⟩
#align analytic_at_const analyticAt_const
theorem analyticOn_const {v : F} {s : Set E} : AnalyticOn 𝕜 (fun _ => v) s :=
fun _ _ => analyticAt_const
#align analytic_on_const analyticOn_const
theorem HasFPowerSeriesOnBall.add (hf : HasFPowerSeriesOnBall f pf x r)
(hg : HasFPowerSeriesOnBall g pg x r) : HasFPowerSeriesOnBall (f + g) (pf + pg) x r :=
{ r_le := le_trans (le_min_iff.2 ⟨hf.r_le, hg.r_le⟩) (pf.min_radius_le_radius_add pg)
r_pos := hf.r_pos
hasSum := fun hy => (hf.hasSum hy).add (hg.hasSum hy) }
#align has_fpower_series_on_ball.add HasFPowerSeriesOnBall.add
theorem HasFPowerSeriesAt.add (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x) :
HasFPowerSeriesAt (f + g) (pf + pg) x := by
rcases (hf.eventually.and hg.eventually).exists with ⟨r, hr⟩
exact ⟨r, hr.1.add hr.2⟩
#align has_fpower_series_at.add HasFPowerSeriesAt.add
theorem AnalyticAt.congr (hf : AnalyticAt 𝕜 f x) (hg : f =ᶠ[𝓝 x] g) : AnalyticAt 𝕜 g x :=
let ⟨_, hpf⟩ := hf
(hpf.congr hg).analyticAt
theorem analyticAt_congr (h : f =ᶠ[𝓝 x] g) : AnalyticAt 𝕜 f x ↔ AnalyticAt 𝕜 g x :=
⟨fun hf ↦ hf.congr h, fun hg ↦ hg.congr h.symm⟩
theorem AnalyticAt.add (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) : AnalyticAt 𝕜 (f + g) x :=
let ⟨_, hpf⟩ := hf
let ⟨_, hqf⟩ := hg
(hpf.add hqf).analyticAt
#align analytic_at.add AnalyticAt.add
theorem HasFPowerSeriesOnBall.neg (hf : HasFPowerSeriesOnBall f pf x r) :
HasFPowerSeriesOnBall (-f) (-pf) x r :=
{ r_le := by
rw [pf.radius_neg]
exact hf.r_le
r_pos := hf.r_pos
hasSum := fun hy => (hf.hasSum hy).neg }
#align has_fpower_series_on_ball.neg HasFPowerSeriesOnBall.neg
theorem HasFPowerSeriesAt.neg (hf : HasFPowerSeriesAt f pf x) : HasFPowerSeriesAt (-f) (-pf) x :=
let ⟨_, hrf⟩ := hf
hrf.neg.hasFPowerSeriesAt
#align has_fpower_series_at.neg HasFPowerSeriesAt.neg
theorem AnalyticAt.neg (hf : AnalyticAt 𝕜 f x) : AnalyticAt 𝕜 (-f) x :=
let ⟨_, hpf⟩ := hf
hpf.neg.analyticAt
#align analytic_at.neg AnalyticAt.neg
theorem HasFPowerSeriesOnBall.sub (hf : HasFPowerSeriesOnBall f pf x r)
(hg : HasFPowerSeriesOnBall g pg x r) : HasFPowerSeriesOnBall (f - g) (pf - pg) x r := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align has_fpower_series_on_ball.sub HasFPowerSeriesOnBall.sub
theorem HasFPowerSeriesAt.sub (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x) :
HasFPowerSeriesAt (f - g) (pf - pg) x := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align has_fpower_series_at.sub HasFPowerSeriesAt.sub
theorem AnalyticAt.sub (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) :
AnalyticAt 𝕜 (f - g) x := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align analytic_at.sub AnalyticAt.sub
theorem AnalyticOn.mono {s t : Set E} (hf : AnalyticOn 𝕜 f t) (hst : s ⊆ t) : AnalyticOn 𝕜 f s :=
fun z hz => hf z (hst hz)
#align analytic_on.mono AnalyticOn.mono
theorem AnalyticOn.congr' {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : f =ᶠ[𝓝ˢ s] g) :
AnalyticOn 𝕜 g s :=
fun z hz => (hf z hz).congr (mem_nhdsSet_iff_forall.mp hg z hz)
theorem analyticOn_congr' {s : Set E} (h : f =ᶠ[𝓝ˢ s] g) : AnalyticOn 𝕜 f s ↔ AnalyticOn 𝕜 g s :=
⟨fun hf => hf.congr' h, fun hg => hg.congr' h.symm⟩
theorem AnalyticOn.congr {s : Set E} (hs : IsOpen s) (hf : AnalyticOn 𝕜 f s) (hg : s.EqOn f g) :
AnalyticOn 𝕜 g s :=
hf.congr' $ mem_nhdsSet_iff_forall.mpr
(fun _ hz => eventuallyEq_iff_exists_mem.mpr ⟨s, hs.mem_nhds hz, hg⟩)
theorem analyticOn_congr {s : Set E} (hs : IsOpen s) (h : s.EqOn f g) : AnalyticOn 𝕜 f s ↔
AnalyticOn 𝕜 g s := ⟨fun hf => hf.congr hs h, fun hg => hg.congr hs h.symm⟩
theorem AnalyticOn.add {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
AnalyticOn 𝕜 (f + g) s :=
fun z hz => (hf z hz).add (hg z hz)
#align analytic_on.add AnalyticOn.add
theorem AnalyticOn.sub {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
AnalyticOn 𝕜 (f - g) s :=
fun z hz => (hf z hz).sub (hg z hz)
#align analytic_on.sub AnalyticOn.sub
theorem HasFPowerSeriesOnBall.coeff_zero (hf : HasFPowerSeriesOnBall f pf x r) (v : Fin 0 → E) :
pf 0 v = f x := by
have v_eq : v = fun i => 0 := Subsingleton.elim _ _
have zero_mem : (0 : E) ∈ EMetric.ball (0 : E) r := by simp [hf.r_pos]
have : ∀ i, i ≠ 0 → (pf i fun j => 0) = 0 := by
intro i hi
have : 0 < i := pos_iff_ne_zero.2 hi
exact ContinuousMultilinearMap.map_coord_zero _ (⟨0, this⟩ : Fin i) rfl
have A := (hf.hasSum zero_mem).unique (hasSum_single _ this)
simpa [v_eq] using A.symm
#align has_fpower_series_on_ball.coeff_zero HasFPowerSeriesOnBall.coeff_zero
theorem HasFPowerSeriesAt.coeff_zero (hf : HasFPowerSeriesAt f pf x) (v : Fin 0 → E) :
pf 0 v = f x :=
let ⟨_, hrf⟩ := hf
hrf.coeff_zero v
#align has_fpower_series_at.coeff_zero HasFPowerSeriesAt.coeff_zero
/-- If a function `f` has a power series `p` on a ball and `g` is linear, then `g ∘ f` has the
power series `g ∘ p` on the same ball. -/
theorem ContinuousLinearMap.comp_hasFPowerSeriesOnBall (g : F →L[𝕜] G)
(h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (g ∘ f) (g.compFormalMultilinearSeries p) x r :=
{ r_le := h.r_le.trans (p.radius_le_radius_continuousLinearMap_comp _)
r_pos := h.r_pos
hasSum := fun hy => by
simpa only [ContinuousLinearMap.compFormalMultilinearSeries_apply,
ContinuousLinearMap.compContinuousMultilinearMap_coe, Function.comp_apply] using
g.hasSum (h.hasSum hy) }
#align continuous_linear_map.comp_has_fpower_series_on_ball ContinuousLinearMap.comp_hasFPowerSeriesOnBall
/-- If a function `f` is analytic on a set `s` and `g` is linear, then `g ∘ f` is analytic
on `s`. -/
theorem ContinuousLinearMap.comp_analyticOn {s : Set E} (g : F →L[𝕜] G) (h : AnalyticOn 𝕜 f s) :
AnalyticOn 𝕜 (g ∘ f) s := by
rintro x hx
rcases h x hx with ⟨p, r, hp⟩
exact ⟨g.compFormalMultilinearSeries p, r, g.comp_hasFPowerSeriesOnBall hp⟩
#align continuous_linear_map.comp_analytic_on ContinuousLinearMap.comp_analyticOn
/-- If a function admits a power series expansion, then it is exponentially close to the partial
sums of this power series on strict subdisks of the disk of convergence.
This version provides an upper estimate that decreases both in `‖y‖` and `n`. See also
`HasFPowerSeriesOnBall.uniform_geometric_approx` for a weaker version. -/
theorem HasFPowerSeriesOnBall.uniform_geometric_approx' {r' : ℝ≥0}
(hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n := by
obtain ⟨a, ha, C, hC, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n, ‖p n‖ * (r' : ℝ) ^ n ≤ C * a ^ n :=
p.norm_mul_pow_le_mul_pow_of_lt_radius (h.trans_le hf.r_le)
refine' ⟨a, ha, C / (1 - a), div_pos hC (sub_pos.2 ha.2), fun y hy n => _⟩
have yr' : ‖y‖ < r' := by
rw [ball_zero_eq] at hy
exact hy
have hr'0 : 0 < (r' : ℝ) := (norm_nonneg _).trans_lt yr'
have : y ∈ EMetric.ball (0 : E) r := by
refine' mem_emetric_ball_zero_iff.2 (lt_trans _ h)
exact mod_cast yr'
rw [norm_sub_rev, ← mul_div_right_comm]
have ya : a * (‖y‖ / ↑r') ≤ a :=
mul_le_of_le_one_right ha.1.le (div_le_one_of_le yr'.le r'.coe_nonneg)
suffices ‖p.partialSum n y - f (x + y)‖ ≤ C * (a * (‖y‖ / r')) ^ n / (1 - a * (‖y‖ / r')) by
refine' this.trans _
have : 0 < a := ha.1
gcongr
apply_rules [sub_pos.2, ha.2]
apply norm_sub_le_of_geometric_bound_of_hasSum (ya.trans_lt ha.2) _ (hf.hasSum this)
intro n
calc
‖(p n) fun _ : Fin n => y‖
_ ≤ ‖p n‖ * ∏ _i : Fin n, ‖y‖ := ContinuousMultilinearMap.le_op_norm _ _
_ = ‖p n‖ * (r' : ℝ) ^ n * (‖y‖ / r') ^ n := by field_simp [mul_right_comm]
_ ≤ C * a ^ n * (‖y‖ / r') ^ n := by gcongr ?_ * _; apply hp
_ ≤ C * (a * (‖y‖ / r')) ^ n := by rw [mul_pow, mul_assoc]
#align has_fpower_series_on_ball.uniform_geometric_approx' HasFPowerSeriesOnBall.uniform_geometric_approx'
/-- If a function admits a power series expansion, then it is exponentially close to the partial
sums of this power series on strict subdisks of the disk of convergence. -/
theorem HasFPowerSeriesOnBall.uniform_geometric_approx {r' : ℝ≥0}
(hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
∃ a ∈ Ioo (0 : ℝ) 1,
∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n, ‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n := by
obtain ⟨a, ha, C, hC, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n :=
hf.uniform_geometric_approx' h
refine' ⟨a, ha, C, hC, fun y hy n => (hp y hy n).trans _⟩
have yr' : ‖y‖ < r' := by rwa [ball_zero_eq] at hy
gcongr
exacts [mul_nonneg ha.1.le (div_nonneg (norm_nonneg y) r'.coe_nonneg),
mul_le_of_le_one_right ha.1.le (div_le_one_of_le yr'.le r'.coe_nonneg)]
#align has_fpower_series_on_ball.uniform_geometric_approx HasFPowerSeriesOnBall.uniform_geometric_approx
/-- Taylor formula for an analytic function, `IsBigO` version. -/
theorem HasFPowerSeriesAt.isBigO_sub_partialSum_pow (hf : HasFPowerSeriesAt f p x) (n : ℕ) :
(fun y : E => f (x + y) - p.partialSum n y) =O[𝓝 0] fun y => ‖y‖ ^ n := by
rcases hf with ⟨r, hf⟩
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hf.r_pos with ⟨r', r'0, h⟩
obtain ⟨a, -, C, -, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n :=
hf.uniform_geometric_approx' h
refine' isBigO_iff.2 ⟨C * (a / r') ^ n, _⟩
replace r'0 : 0 < (r' : ℝ); · exact mod_cast r'0
filter_upwards [Metric.ball_mem_nhds (0 : E) r'0] with y hy
simpa [mul_pow, mul_div_assoc, mul_assoc, div_mul_eq_mul_div] using hp y hy n
set_option linter.uppercaseLean3 false in
#align has_fpower_series_at.is_O_sub_partial_sum_pow HasFPowerSeriesAt.isBigO_sub_partialSum_pow
/-- If `f` has formal power series `∑ n, pₙ` on a ball of radius `r`, then for `y, z` in any smaller
ball, the norm of the difference `f y - f z - p 1 (fun _ ↦ y - z)` is bounded above by
`C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖`. This lemma formulates this property using `IsBigO` and
`Filter.principal` on `E × E`. -/
theorem HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal
(hf : HasFPowerSeriesOnBall f p x r) (hr : r' < r) :
(fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) =O[𝓟 (EMetric.ball (x, x) r')]
fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖ := by
lift r' to ℝ≥0 using ne_top_of_lt hr
rcases (zero_le r').eq_or_lt with (rfl | hr'0)
· simp only [isBigO_bot, EMetric.ball_zero, principal_empty, ENNReal.coe_zero]
obtain ⟨a, ha, C, hC : 0 < C, hp⟩ :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n : ℕ, ‖p n‖ * (r' : ℝ) ^ n ≤ C * a ^ n
exact p.norm_mul_pow_le_mul_pow_of_lt_radius (hr.trans_le hf.r_le)
simp only [← le_div_iff (pow_pos (NNReal.coe_pos.2 hr'0) _)] at hp
set L : E × E → ℝ := fun y =>
C * (a / r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * (a / (1 - a) ^ 2 + 2 / (1 - a))
have hL : ∀ y ∈ EMetric.ball (x, x) r', ‖f y.1 - f y.2 - p 1 fun _ => y.1 - y.2‖ ≤ L y := by
intro y hy'
have hy : y ∈ EMetric.ball x r ×ˢ EMetric.ball x r := by
rw [EMetric.ball_prod_same]
exact EMetric.ball_subset_ball hr.le hy'
set A : ℕ → F := fun n => (p n fun _ => y.1 - x) - p n fun _ => y.2 - x
have hA : HasSum (fun n => A (n + 2)) (f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) := by
convert (hasSum_nat_add_iff' 2).2 ((hf.hasSum_sub hy.1).sub (hf.hasSum_sub hy.2)) using 1
rw [Finset.sum_range_succ, Finset.sum_range_one, hf.coeff_zero, hf.coeff_zero, sub_self,
zero_add, ← Subsingleton.pi_single_eq (0 : Fin 1) (y.1 - x), Pi.single,
← Subsingleton.pi_single_eq (0 : Fin 1) (y.2 - x), Pi.single, ← (p 1).map_sub, ← Pi.single,
Subsingleton.pi_single_eq, sub_sub_sub_cancel_right]
rw [EMetric.mem_ball, edist_eq_coe_nnnorm_sub, ENNReal.coe_lt_coe] at hy'
set B : ℕ → ℝ := fun n => C * (a / r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * ((n + 2) * a ^ n)
have hAB : ∀ n, ‖A (n + 2)‖ ≤ B n := fun n =>
calc
‖A (n + 2)‖ ≤ ‖p (n + 2)‖ * ↑(n + 2) * ‖y - (x, x)‖ ^ (n + 1) * ‖y.1 - y.2‖ := by
-- porting note: `pi_norm_const` was `pi_norm_const (_ : E)`
simpa only [Fintype.card_fin, pi_norm_const, Prod.norm_def, Pi.sub_def,
Prod.fst_sub, Prod.snd_sub, sub_sub_sub_cancel_right] using
(p <| n + 2).norm_image_sub_le (fun _ => y.1 - x) fun _ => y.2 - x
_ = ‖p (n + 2)‖ * ‖y - (x, x)‖ ^ n * (↑(n + 2) * ‖y - (x, x)‖ * ‖y.1 - y.2‖) := by
rw [pow_succ ‖y - (x, x)‖]
ring
-- porting note: the two `↑` in `↑r'` are new, without them, Lean fails to synthesize
-- instances `HDiv ℝ ℝ≥0 ?m` or `HMul ℝ ℝ≥0 ?m`
_ ≤ C * a ^ (n + 2) / ↑r' ^ (n + 2)
* ↑r' ^ n * (↑(n + 2) * ‖y - (x, x)‖ * ‖y.1 - y.2‖) := by
have : 0 < a := ha.1
gcongr
· apply hp
· apply hy'.le
_ = B n := by
-- porting note: in the original, `B` was in the `field_simp`, but now Lean does not
-- accept it. The current proof works in Lean 4, but does not in Lean 3.
field_simp [pow_succ]
simp only [mul_assoc, mul_comm, mul_left_comm]
have hBL : HasSum B (L y) := by
apply HasSum.mul_left
simp only [add_mul]
have : ‖a‖ < 1 := by simp only [Real.norm_eq_abs, abs_of_pos ha.1, ha.2]
rw [div_eq_mul_inv, div_eq_mul_inv]
exact (hasSum_coe_mul_geometric_of_norm_lt_1 this).add -- porting note: was `convert`!
((hasSum_geometric_of_norm_lt_1 this).mul_left 2)
exact hA.norm_le_of_bounded hBL hAB
suffices L =O[𝓟 (EMetric.ball (x, x) r')] fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖ by
refine' (IsBigO.of_bound 1 (eventually_principal.2 fun y hy => _)).trans this
rw [one_mul]
exact (hL y hy).trans (le_abs_self _)
simp_rw [mul_right_comm _ (_ * _)] -- porting note: there was an `L` inside the `simp_rw`.
exact (isBigO_refl _ _).const_mul_left _
set_option linter.uppercaseLean3 false in
#align has_fpower_series_on_ball.is_O_image_sub_image_sub_deriv_principal HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal
/-- If `f` has formal power series `∑ n, pₙ` on a ball of radius `r`, then for `y, z` in any smaller
ball, the norm of the difference `f y - f z - p 1 (fun _ ↦ y - z)` is bounded above by
`C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖`. -/
theorem HasFPowerSeriesOnBall.image_sub_sub_deriv_le (hf : HasFPowerSeriesOnBall f p x r)
(hr : r' < r) :
∃ C, ∀ᵉ (y ∈ EMetric.ball x r') (z ∈ EMetric.ball x r'),
‖f y - f z - p 1 fun _ => y - z‖ ≤ C * max ‖y - x‖ ‖z - x‖ * ‖y - z‖ := by
simpa only [isBigO_principal, mul_assoc, norm_mul, norm_norm, Prod.forall, EMetric.mem_ball,
Prod.edist_eq, max_lt_iff, and_imp, @forall_swap (_ < _) E] using
hf.isBigO_image_sub_image_sub_deriv_principal hr
#align has_fpower_series_on_ball.image_sub_sub_deriv_le HasFPowerSeriesOnBall.image_sub_sub_deriv_le
/-- If `f` has formal power series `∑ n, pₙ` at `x`, then
`f y - f z - p 1 (fun _ ↦ y - z) = O(‖(y, z) - (x, x)‖ * ‖y - z‖)` as `(y, z) → (x, x)`.
In particular, `f` is strictly differentiable at `x`. -/
theorem HasFPowerSeriesAt.isBigO_image_sub_norm_mul_norm_sub (hf : HasFPowerSeriesAt f p x) :
(fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) =O[𝓝 (x, x)] fun y =>
‖y - (x, x)‖ * ‖y.1 - y.2‖ := by
rcases hf with ⟨r, hf⟩
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hf.r_pos with ⟨r', r'0, h⟩
refine' (hf.isBigO_image_sub_image_sub_deriv_principal h).mono _
exact le_principal_iff.2 (EMetric.ball_mem_nhds _ r'0)
set_option linter.uppercaseLean3 false in
#align has_fpower_series_at.is_O_image_sub_norm_mul_norm_sub HasFPowerSeriesAt.isBigO_image_sub_norm_mul_norm_sub
/-- If a function admits a power series expansion at `x`, then it is the uniform limit of the
partial sums of this power series on strict subdisks of the disk of convergence, i.e., `f (x + y)`
is the uniform limit of `p.partialSum n y` there. -/
theorem HasFPowerSeriesOnBall.tendstoUniformlyOn {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r)
(h : (r' : ℝ≥0∞) < r) :
TendstoUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
(Metric.ball (0 : E) r') := by
obtain ⟨a, ha, C, -, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n
exact hf.uniform_geometric_approx h
refine' Metric.tendstoUniformlyOn_iff.2 fun ε εpos => _
have L : Tendsto (fun n => (C : ℝ) * a ^ n) atTop (𝓝 ((C : ℝ) * 0)) :=
tendsto_const_nhds.mul (tendsto_pow_atTop_nhds_0_of_lt_1 ha.1.le ha.2)
rw [mul_zero] at L
refine' (L.eventually (gt_mem_nhds εpos)).mono fun n hn y hy => _
rw [dist_eq_norm]
exact (hp y hy n).trans_lt hn
#align has_fpower_series_on_ball.tendsto_uniformly_on HasFPowerSeriesOnBall.tendstoUniformlyOn
/-- If a function admits a power series expansion at `x`, then it is the locally uniform limit of
the partial sums of this power series on the disk of convergence, i.e., `f (x + y)`
is the locally uniform limit of `p.partialSum n y` there. -/
theorem HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn (hf : HasFPowerSeriesOnBall f p x r) :
TendstoLocallyUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
(EMetric.ball (0 : E) r) := by
intro u hu x hx
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hx with ⟨r', xr', hr'⟩
have : EMetric.ball (0 : E) r' ∈ 𝓝 x := IsOpen.mem_nhds EMetric.isOpen_ball xr'
refine' ⟨EMetric.ball (0 : E) r', mem_nhdsWithin_of_mem_nhds this, _⟩
simpa [Metric.emetric_ball_nnreal] using hf.tendstoUniformlyOn hr' u hu
#align has_fpower_series_on_ball.tendsto_locally_uniformly_on HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn
/-- If a function admits a power series expansion at `x`, then it is the uniform limit of the
partial sums of this power series on strict subdisks of the disk of convergence, i.e., `f y`
is the uniform limit of `p.partialSum n (y - x)` there. -/
theorem HasFPowerSeriesOnBall.tendstoUniformlyOn' {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r)
(h : (r' : ℝ≥0∞) < r) :
TendstoUniformlyOn (fun n y => p.partialSum n (y - x)) f atTop (Metric.ball (x : E) r') := by
convert (hf.tendstoUniformlyOn h).comp fun y => y - x using 1
· simp [(· ∘ ·)]
· ext z
simp [dist_eq_norm]
#align has_fpower_series_on_ball.tendsto_uniformly_on' HasFPowerSeriesOnBall.tendstoUniformlyOn'
/-- If a function admits a power series expansion at `x`, then it is the locally uniform limit of
the partial sums of this power series on the disk of convergence, i.e., `f y`
is the locally uniform limit of `p.partialSum n (y - x)` there. -/
theorem HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn' (hf : HasFPowerSeriesOnBall f p x r) :
TendstoLocallyUniformlyOn (fun n y => p.partialSum n (y - x)) f atTop
(EMetric.ball (x : E) r) := by
have A : ContinuousOn (fun y : E => y - x) (EMetric.ball (x : E) r) :=
(continuous_id.sub continuous_const).continuousOn
convert hf.tendstoLocallyUniformlyOn.comp (fun y : E => y - x) _ A using 1
· ext z
simp
· intro z
simp [edist_eq_coe_nnnorm, edist_eq_coe_nnnorm_sub]
#align has_fpower_series_on_ball.tendsto_locally_uniformly_on' HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn'
/-- If a function admits a power series expansion on a disk, then it is continuous there. -/
protected theorem HasFPowerSeriesOnBall.continuousOn (hf : HasFPowerSeriesOnBall f p x r) :
ContinuousOn f (EMetric.ball x r) :=
hf.tendstoLocallyUniformlyOn'.continuousOn <|
eventually_of_forall fun n =>
((p.partialSum_continuous n).comp (continuous_id.sub continuous_const)).continuousOn
#align has_fpower_series_on_ball.continuous_on HasFPowerSeriesOnBall.continuousOn
protected theorem HasFPowerSeriesAt.continuousAt (hf : HasFPowerSeriesAt f p x) :
ContinuousAt f x :=
let ⟨_, hr⟩ := hf
hr.continuousOn.continuousAt (EMetric.ball_mem_nhds x hr.r_pos)
#align has_fpower_series_at.continuous_at HasFPowerSeriesAt.continuousAt
protected theorem AnalyticAt.continuousAt (hf : AnalyticAt 𝕜 f x) : ContinuousAt f x :=
let ⟨_, hp⟩ := hf
hp.continuousAt
#align analytic_at.continuous_at AnalyticAt.continuousAt
protected theorem AnalyticOn.continuousOn {s : Set E} (hf : AnalyticOn 𝕜 f s) : ContinuousOn f s :=
fun x hx => (hf x hx).continuousAt.continuousWithinAt
#align analytic_on.continuous_on AnalyticOn.continuousOn
/-- Analytic everywhere implies continuous -/
theorem AnalyticOn.continuous {f : E → F} (fa : AnalyticOn 𝕜 f univ) : Continuous f := by
rw [continuous_iff_continuousOn_univ]; exact fa.continuousOn
/-- In a complete space, the sum of a converging power series `p` admits `p` as a power series.
This is not totally obvious as we need to check the convergence of the series. -/
protected theorem FormalMultilinearSeries.hasFPowerSeriesOnBall [CompleteSpace F]
(p : FormalMultilinearSeries 𝕜 E F) (h : 0 < p.radius) :
HasFPowerSeriesOnBall p.sum p 0 p.radius :=
{ r_le := le_rfl
r_pos := h
hasSum := fun hy => by
rw [zero_add]
exact p.hasSum hy }
#align formal_multilinear_series.has_fpower_series_on_ball FormalMultilinearSeries.hasFPowerSeriesOnBall
theorem HasFPowerSeriesOnBall.sum (h : HasFPowerSeriesOnBall f p x r) {y : E}
(hy : y ∈ EMetric.ball (0 : E) r) : f (x + y) = p.sum y :=
(h.hasSum hy).tsum_eq.symm
#align has_fpower_series_on_ball.sum HasFPowerSeriesOnBall.sum
/-- The sum of a converging power series is continuous in its disk of convergence. -/
protected theorem FormalMultilinearSeries.continuousOn [CompleteSpace F] :
ContinuousOn p.sum (EMetric.ball 0 p.radius) := by
rcases (zero_le p.radius).eq_or_lt with h | h
· simp [← h, continuousOn_empty]
· exact (p.hasFPowerSeriesOnBall h).continuousOn
#align formal_multilinear_series.continuous_on FormalMultilinearSeries.continuousOn
end
/-!
### Uniqueness of power series
If a function `f : E → F` has two representations as power series at a point `x : E`, corresponding
to formal multilinear series `p₁` and `p₂`, then these representations agree term-by-term. That is,
for any `n : ℕ` and `y : E`, `p₁ n (fun i ↦ y) = p₂ n (fun i ↦ y)`. In the one-dimensional case,
when `f : 𝕜 → E`, the continuous multilinear maps `p₁ n` and `p₂ n` are given by
`ContinuousMultilinearMap.mkPiField`, and hence are determined completely by the value of
`p₁ n (fun i ↦ 1)`, so `p₁ = p₂`. Consequently, the radius of convergence for one series can be
transferred to the other.
-/
section Uniqueness
open ContinuousMultilinearMap
theorem Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero {n : ℕ} {p : E[×n]→L[𝕜] F}
(h : (fun y => p fun _ => y) =O[𝓝 0] fun y => ‖y‖ ^ (n + 1)) (y : E) : (p fun _ => y) = 0 := by
obtain ⟨c, c_pos, hc⟩ := h.exists_pos
obtain ⟨t, ht, t_open, z_mem⟩ := eventually_nhds_iff.mp (isBigOWith_iff.mp hc)
obtain ⟨δ, δ_pos, δε⟩ := (Metric.isOpen_iff.mp t_open) 0 z_mem
clear h hc z_mem
cases' n with n
· exact norm_eq_zero.mp (by
-- porting note: the symmetric difference of the `simpa only` sets:
-- added `Nat.zero_eq, zero_add, pow_one`
-- removed `zero_pow', Ne.def, Nat.one_ne_zero, not_false_iff`
simpa only [Nat.zero_eq, fin0_apply_norm, norm_eq_zero, norm_zero, zero_add, pow_one,
mul_zero, norm_le_zero_iff] using ht 0 (δε (Metric.mem_ball_self δ_pos)))
· refine' Or.elim (Classical.em (y = 0))
(fun hy => by simpa only [hy] using p.map_zero) fun hy => _
replace hy := norm_pos_iff.mpr hy
refine' norm_eq_zero.mp (le_antisymm (le_of_forall_pos_le_add fun ε ε_pos => _) (norm_nonneg _))
have h₀ := _root_.mul_pos c_pos (pow_pos hy (n.succ + 1))
obtain ⟨k, k_pos, k_norm⟩ := NormedField.exists_norm_lt 𝕜
(lt_min (mul_pos δ_pos (inv_pos.mpr hy)) (mul_pos ε_pos (inv_pos.mpr h₀)))
have h₁ : ‖k • y‖ < δ := by
rw [norm_smul]
exact inv_mul_cancel_right₀ hy.ne.symm δ ▸
mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_left _ _)) hy
have h₂ :=
calc
‖p fun _ => k • y‖ ≤ c * ‖k • y‖ ^ (n.succ + 1) := by
-- porting note: now Lean wants `_root_.`
simpa only [norm_pow, _root_.norm_norm] using ht (k • y) (δε (mem_ball_zero_iff.mpr h₁))
--simpa only [norm_pow, norm_norm] using ht (k • y) (δε (mem_ball_zero_iff.mpr h₁))
_ = ‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))) := by
-- porting note: added `Nat.succ_eq_add_one` since otherwise `ring` does not conclude.
simp only [norm_smul, mul_pow, Nat.succ_eq_add_one]
-- porting note: removed `rw [pow_succ]`, since it now becomes superfluous.
ring
have h₃ : ‖k‖ * (c * ‖y‖ ^ (n.succ + 1)) < ε :=
inv_mul_cancel_right₀ h₀.ne.symm ε ▸
mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_right _ _)) h₀
calc
‖p fun _ => y‖ = ‖k⁻¹ ^ n.succ‖ * ‖p fun _ => k • y‖ := by
simpa only [inv_smul_smul₀ (norm_pos_iff.mp k_pos), norm_smul, Finset.prod_const,
Finset.card_fin] using
congr_arg norm (p.map_smul_univ (fun _ : Fin n.succ => k⁻¹) fun _ : Fin n.succ => k • y)
_ ≤ ‖k⁻¹ ^ n.succ‖ * (‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1)))) := by gcongr
_ = ‖(k⁻¹ * k) ^ n.succ‖ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))) := by
rw [← mul_assoc]
simp [norm_mul, mul_pow]
_ ≤ 0 + ε := by
rw [inv_mul_cancel (norm_pos_iff.mp k_pos)]
simpa using h₃.le
set_option linter.uppercaseLean3 false in
#align asymptotics.is_O.continuous_multilinear_map_apply_eq_zero Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero
/-- If a formal multilinear series `p` represents the zero function at `x : E`, then the
terms `p n (fun i ↦ y)` appearing in the sum are zero for any `n : ℕ`, `y : E`. -/
theorem HasFPowerSeriesAt.apply_eq_zero {p : FormalMultilinearSeries 𝕜 E F} {x : E}
(h : HasFPowerSeriesAt 0 p x) (n : ℕ) : ∀ y : E, (p n fun _ => y) = 0 := by
refine' Nat.strong_induction_on n fun k hk => _
have psum_eq : p.partialSum (k + 1) = fun y => p k fun _ => y := by
funext z
refine' Finset.sum_eq_single _ (fun b hb hnb => _) fun hn => _
· have := Finset.mem_range_succ_iff.mp hb
simp only [hk b (this.lt_of_ne hnb), Pi.zero_apply]
· exact False.elim (hn (Finset.mem_range.mpr (lt_add_one k)))
replace h := h.isBigO_sub_partialSum_pow k.succ
simp only [psum_eq, zero_sub, Pi.zero_apply, Asymptotics.isBigO_neg_left] at h
exact h.continuousMultilinearMap_apply_eq_zero
#align has_fpower_series_at.apply_eq_zero HasFPowerSeriesAt.apply_eq_zero
/-- A one-dimensional formal multilinear series representing the zero function is zero. -/
theorem HasFPowerSeriesAt.eq_zero {p : FormalMultilinearSeries 𝕜 𝕜 E} {x : 𝕜}
(h : HasFPowerSeriesAt 0 p x) : p = 0 := by
-- porting note: `funext; ext` was `ext (n x)`
funext n
ext x
rw [← mkPiField_apply_one_eq_self (p n)]
-- porting note: nasty hack, was `simp [h.apply_eq_zero n 1]`
have := Or.intro_right ?_ (h.apply_eq_zero n 1)
simpa using this
#align has_fpower_series_at.eq_zero HasFPowerSeriesAt.eq_zero
/-- One-dimensional formal multilinear series representing the same function are equal. -/
theorem HasFPowerSeriesAt.eq_formalMultilinearSeries {p₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E}
{f : 𝕜 → E} {x : 𝕜} (h₁ : HasFPowerSeriesAt f p₁ x) (h₂ : HasFPowerSeriesAt f p₂ x) : p₁ = p₂ :=
sub_eq_zero.mp (HasFPowerSeriesAt.eq_zero (by simpa only [sub_self] using h₁.sub h₂))
#align has_fpower_series_at.eq_formal_multilinear_series HasFPowerSeriesAt.eq_formalMultilinearSeries
theorem HasFPowerSeriesAt.eq_formalMultilinearSeries_of_eventually
{p q : FormalMultilinearSeries 𝕜 𝕜 E} {f g : 𝕜 → E} {x : 𝕜} (hp : HasFPowerSeriesAt f p x)
(hq : HasFPowerSeriesAt g q x) (heq : ∀ᶠ z in 𝓝 x, f z = g z) : p = q :=
(hp.congr heq).eq_formalMultilinearSeries hq
#align has_fpower_series_at.eq_formal_multilinear_series_of_eventually HasFPowerSeriesAt.eq_formalMultilinearSeries_of_eventually
/-- A one-dimensional formal multilinear series representing a locally zero function is zero. -/
theorem HasFPowerSeriesAt.eq_zero_of_eventually {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E}
{x : 𝕜} (hp : HasFPowerSeriesAt f p x) (hf : f =ᶠ[𝓝 x] 0) : p = 0 :=
(hp.congr hf).eq_zero
#align has_fpower_series_at.eq_zero_of_eventually HasFPowerSeriesAt.eq_zero_of_eventually
/-- If a function `f : 𝕜 → E` has two power series representations at `x`, then the given radii in
which convergence is guaranteed may be interchanged. This can be useful when the formal multilinear
series in one representation has a particularly nice form, but the other has a larger radius. -/
theorem HasFPowerSeriesOnBall.exchange_radius {p₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E}
{r₁ r₂ : ℝ≥0∞} {x : 𝕜} (h₁ : HasFPowerSeriesOnBall f p₁ x r₁)
(h₂ : HasFPowerSeriesOnBall f p₂ x r₂) : HasFPowerSeriesOnBall f p₁ x r₂ :=
h₂.hasFPowerSeriesAt.eq_formalMultilinearSeries h₁.hasFPowerSeriesAt ▸ h₂
#align has_fpower_series_on_ball.exchange_radius HasFPowerSeriesOnBall.exchange_radius
/-- If a function `f : 𝕜 → E` has power series representation `p` on a ball of some radius and for
each positive radius it has some power series representation, then `p` converges to `f` on the whole
`𝕜`. -/
theorem HasFPowerSeriesOnBall.r_eq_top_of_exists {f : 𝕜 → E} {r : ℝ≥0∞} {x : 𝕜}
{p : FormalMultilinearSeries 𝕜 𝕜 E} (h : HasFPowerSeriesOnBall f p x r)
(h' : ∀ (r' : ℝ≥0) (_ : 0 < r'), ∃ p' : FormalMultilinearSeries 𝕜 𝕜 E,
HasFPowerSeriesOnBall f p' x r') :
HasFPowerSeriesOnBall f p x ∞ :=
{ r_le := ENNReal.le_of_forall_pos_nnreal_lt fun r hr _ =>
let ⟨_, hp'⟩ := h' r hr
(h.exchange_radius hp').r_le
r_pos := ENNReal.coe_lt_top
hasSum := fun {y} _ =>
let ⟨r', hr'⟩ := exists_gt ‖y‖₊
let ⟨_, hp'⟩ := h' r' hr'.ne_bot.bot_lt
(h.exchange_radius hp').hasSum <| mem_emetric_ball_zero_iff.mpr (ENNReal.coe_lt_coe.2 hr') }
#align has_fpower_series_on_ball.r_eq_top_of_exists HasFPowerSeriesOnBall.r_eq_top_of_exists
end Uniqueness
/-!
### Changing origin in a power series
If a function is analytic in a disk `D(x, R)`, then it is analytic in any disk contained in that
one. Indeed, one can write
$$
f (x + y + z) = \sum_{n} p_n (y + z)^n = \sum_{n, k} \binom{n}{k} p_n y^{n-k} z^k
= \sum_{k} \Bigl(\sum_{n} \binom{n}{k} p_n y^{n-k}\Bigr) z^k.
$$
The corresponding power series has thus a `k`-th coefficient equal to
$\sum_{n} \binom{n}{k} p_n y^{n-k}$. In the general case where `pₙ` is a multilinear map, this has
to be interpreted suitably: instead of having a binomial coefficient, one should sum over all
possible subsets `s` of `Fin n` of cardinal `k`, and attribute `z` to the indices in `s` and
`y` to the indices outside of `s`.
In this paragraph, we implement this. The new power series is called `p.changeOrigin y`. Then, we
check its convergence and the fact that its sum coincides with the original sum. The outcome of this
discussion is that the set of points where a function is analytic is open.
-/
namespace FormalMultilinearSeries
section
variable (p : FormalMultilinearSeries 𝕜 E F) {x y : E} {r R : ℝ≥0}
/-- A term of `FormalMultilinearSeries.changeOriginSeries`.
Given a formal multilinear series `p` and a point `x` in its ball of convergence,
`p.changeOrigin x` is a formal multilinear series such that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense. Each term of `p.changeOrigin x`
is itself an analytic function of `x` given by the series `p.changeOriginSeries`. Each term in
`changeOriginSeries` is the sum of `changeOriginSeriesTerm`'s over all `s` of cardinality `l`.
The definition is such that `p.changeOriginSeriesTerm k l s hs (fun _ ↦ x) (fun _ ↦ y) =
p (k + l) (s.piecewise (fun _ ↦ x) (fun _ ↦ y))`
-/
def changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
E[×l]→L[𝕜] E[×k]→L[𝕜] F := by
let a := ContinuousMultilinearMap.curryFinFinset 𝕜 E F hs
(by erw [Finset.card_compl, Fintype.card_fin, hs, add_tsub_cancel_right])
exact a (p (k + l))
#align formal_multilinear_series.change_origin_series_term FormalMultilinearSeries.changeOriginSeriesTerm
theorem changeOriginSeriesTerm_apply (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l)
(x y : E) :
(p.changeOriginSeriesTerm k l s hs (fun _ => x) fun _ => y) =
p (k + l) (s.piecewise (fun _ => x) fun _ => y) :=
ContinuousMultilinearMap.curryFinFinset_apply_const _ _ _ _ _
#align formal_multilinear_series.change_origin_series_term_apply FormalMultilinearSeries.changeOriginSeriesTerm_apply
@[simp]
theorem norm_changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
‖p.changeOriginSeriesTerm k l s hs‖ = ‖p (k + l)‖ := by
simp only [changeOriginSeriesTerm, LinearIsometryEquiv.norm_map]
#align formal_multilinear_series.norm_change_origin_series_term FormalMultilinearSeries.norm_changeOriginSeriesTerm
@[simp]
theorem nnnorm_changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
‖p.changeOriginSeriesTerm k l s hs‖₊ = ‖p (k + l)‖₊ := by
simp only [changeOriginSeriesTerm, LinearIsometryEquiv.nnnorm_map]
#align formal_multilinear_series.nnnorm_change_origin_series_term FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm
theorem nnnorm_changeOriginSeriesTerm_apply_le (k l : ℕ) (s : Finset (Fin (k + l)))
(hs : s.card = l) (x y : E) :
‖p.changeOriginSeriesTerm k l s hs (fun _ => x) fun _ => y‖₊ ≤
‖p (k + l)‖₊ * ‖x‖₊ ^ l * ‖y‖₊ ^ k := by
rw [← p.nnnorm_changeOriginSeriesTerm k l s hs, ← Fin.prod_const, ← Fin.prod_const]
apply ContinuousMultilinearMap.le_of_op_nnnorm_le
apply ContinuousMultilinearMap.le_op_nnnorm
#align formal_multilinear_series.nnnorm_change_origin_series_term_apply_le FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm_apply_le
/-- The power series for `f.changeOrigin k`.
Given a formal multilinear series `p` and a point `x` in its ball of convergence,
`p.changeOrigin x` is a formal multilinear series such that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense. Its `k`-th term is the sum of
the series `p.changeOriginSeries k`. -/
def changeOriginSeries (k : ℕ) : FormalMultilinearSeries 𝕜 E (E[×k]→L[𝕜] F) := fun l =>
∑ s : { s : Finset (Fin (k + l)) // Finset.card s = l }, p.changeOriginSeriesTerm k l s s.2
#align formal_multilinear_series.change_origin_series FormalMultilinearSeries.changeOriginSeries
theorem nnnorm_changeOriginSeries_le_tsum (k l : ℕ) :
‖p.changeOriginSeries k l‖₊ ≤
∑' _ : { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + l)‖₊ :=
(nnnorm_sum_le _ (fun t => changeOriginSeriesTerm p k l (Subtype.val t) t.prop)).trans_eq <| by
simp_rw [tsum_fintype, nnnorm_changeOriginSeriesTerm (p := p) (k := k) (l := l)]
#align formal_multilinear_series.nnnorm_change_origin_series_le_tsum FormalMultilinearSeries.nnnorm_changeOriginSeries_le_tsum
theorem nnnorm_changeOriginSeries_apply_le_tsum (k l : ℕ) (x : E) :
‖p.changeOriginSeries k l fun _ => x‖₊ ≤
∑' _ : { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + l)‖₊ * ‖x‖₊ ^ l := by
rw [NNReal.tsum_mul_right, ← Fin.prod_const]
exact (p.changeOriginSeries k l).le_of_op_nnnorm_le _ (p.nnnorm_changeOriginSeries_le_tsum _ _)
#align formal_multilinear_series.nnnorm_change_origin_series_apply_le_tsum FormalMultilinearSeries.nnnorm_changeOriginSeries_apply_le_tsum
/-- Changing the origin of a formal multilinear series `p`, so that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense.
-/
def changeOrigin (x : E) : FormalMultilinearSeries 𝕜 E F :=
fun k => (p.changeOriginSeries k).sum x
#align formal_multilinear_series.change_origin FormalMultilinearSeries.changeOrigin
/-- An auxiliary equivalence useful in the proofs about
`FormalMultilinearSeries.changeOriginSeries`: the set of triples `(k, l, s)`, where `s` is a
`Finset (Fin (k + l))` of cardinality `l` is equivalent to the set of pairs `(n, s)`, where `s` is a
`Finset (Fin n)`.
The forward map sends `(k, l, s)` to `(k + l, s)` and the inverse map sends `(n, s)` to
`(n - Finset.card s, Finset.card s, s)`. The actual definition is less readable because of problems
with non-definitional equalities. -/
@[simps]
def changeOriginIndexEquiv :
(Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) ≃ Σn : ℕ, Finset (Fin n) where
toFun s := ⟨s.1 + s.2.1, s.2.2⟩
invFun s :=
⟨s.1 - s.2.card, s.2.card,
⟨s.2.map
(Fin.castIso <| (tsub_add_cancel_of_le <| card_finset_fin_le s.2).symm).toEquiv.toEmbedding,
Finset.card_map _⟩⟩
left_inv := by
rintro ⟨k, l, ⟨s : Finset (Fin <| k + l), hs : s.card = l⟩⟩
dsimp only [Subtype.coe_mk]
-- Lean can't automatically generalize `k' = k + l - s.card`, `l' = s.card`, so we explicitly
-- formulate the generalized goal
suffices ∀ k' l', k' = k → l' = l → ∀ (hkl : k + l = k' + l') (hs'),
(⟨k', l', ⟨Finset.map (Fin.castIso hkl).toEquiv.toEmbedding s, hs'⟩⟩ :
Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) = ⟨k, l, ⟨s, hs⟩⟩ by
apply this <;> simp only [hs, add_tsub_cancel_right]
rintro _ _ rfl rfl hkl hs'
simp only [Equiv.refl_toEmbedding, Fin.castIso_refl, Finset.map_refl, eq_self_iff_true,
OrderIso.refl_toEquiv, and_self_iff, heq_iff_eq]
right_inv := by
rintro ⟨n, s⟩
simp [tsub_add_cancel_of_le (card_finset_fin_le s), Fin.castIso_to_equiv]
#align formal_multilinear_series.change_origin_index_equiv FormalMultilinearSeries.changeOriginIndexEquiv
theorem changeOriginSeries_summable_aux₁ {r r' : ℝ≥0} (hr : (r + r' : ℝ≥0∞) < p.radius) :
Summable fun s : Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l } =>
‖p (s.1 + s.2.1)‖₊ * r ^ s.2.1 * r' ^ s.1 := by
rw [← changeOriginIndexEquiv.symm.summable_iff]
dsimp only [Function.comp_def, changeOriginIndexEquiv_symm_apply_fst,
changeOriginIndexEquiv_symm_apply_snd_fst]
have : ∀ n : ℕ,
HasSum (fun s : Finset (Fin n) => ‖p (n - s.card + s.card)‖₊ * r ^ s.card * r' ^ (n - s.card))
(‖p n‖₊ * (r + r') ^ n) := by
intro n
-- TODO: why `simp only [tsub_add_cancel_of_le (card_finset_fin_le _)]` fails?
convert_to HasSum (fun s : Finset (Fin n) => ‖p n‖₊ * (r ^ s.card * r' ^ (n - s.card))) _
· ext1 s
rw [tsub_add_cancel_of_le (card_finset_fin_le _), mul_assoc]
rw [← Fin.sum_pow_mul_eq_add_pow]
exact (hasSum_fintype _).mul_left _
refine' NNReal.summable_sigma.2 ⟨fun n => (this n).summable, _⟩
simp only [(this _).tsum_eq]
exact p.summable_nnnorm_mul_pow hr
#align formal_multilinear_series.change_origin_series_summable_aux₁ FormalMultilinearSeries.changeOriginSeries_summable_aux₁
theorem changeOriginSeries_summable_aux₂ (hr : (r : ℝ≥0∞) < p.radius) (k : ℕ) :
Summable fun s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l } =>
‖p (k + s.1)‖₊ * r ^ s.1 := by
rcases ENNReal.lt_iff_exists_add_pos_lt.1 hr with ⟨r', h0, hr'⟩
simpa only [mul_inv_cancel_right₀ (pow_pos h0 _).ne'] using
((NNReal.summable_sigma.1 (p.changeOriginSeries_summable_aux₁ hr')).1 k).mul_right (r' ^ k)⁻¹
#align formal_multilinear_series.change_origin_series_summable_aux₂ FormalMultilinearSeries.changeOriginSeries_summable_aux₂
theorem changeOriginSeries_summable_aux₃ {r : ℝ≥0} (hr : ↑r < p.radius) (k : ℕ) :
Summable fun l : ℕ => ‖p.changeOriginSeries k l‖₊ * r ^ l := by
refine' NNReal.summable_of_le
(fun n => _) (NNReal.summable_sigma.1 <| p.changeOriginSeries_summable_aux₂ hr k).2
simp only [NNReal.tsum_mul_right]
exact mul_le_mul' (p.nnnorm_changeOriginSeries_le_tsum _ _) le_rfl
#align formal_multilinear_series.change_origin_series_summable_aux₃ FormalMultilinearSeries.changeOriginSeries_summable_aux₃
theorem le_changeOriginSeries_radius (k : ℕ) : p.radius ≤ (p.changeOriginSeries k).radius :=
ENNReal.le_of_forall_nnreal_lt fun _r hr =>
le_radius_of_summable_nnnorm _ (p.changeOriginSeries_summable_aux₃ hr k)
#align formal_multilinear_series.le_change_origin_series_radius FormalMultilinearSeries.le_changeOriginSeries_radius
theorem nnnorm_changeOrigin_le (k : ℕ) (h : (‖x‖₊ : ℝ≥0∞) < p.radius) :
‖p.changeOrigin x k‖₊ ≤
∑' s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + s.1)‖₊ * ‖x‖₊ ^ s.1 := by
refine' tsum_of_nnnorm_bounded _ fun l => p.nnnorm_changeOriginSeries_apply_le_tsum k l x
have := p.changeOriginSeries_summable_aux₂ h k
refine' HasSum.sigma this.hasSum fun l => _
exact ((NNReal.summable_sigma.1 this).1 l).hasSum
#align formal_multilinear_series.nnnorm_change_origin_le FormalMultilinearSeries.nnnorm_changeOrigin_le
/-- The radius of convergence of `p.changeOrigin x` is at least `p.radius - ‖x‖`. In other words,
`p.changeOrigin x` is well defined on the largest ball contained in the original ball of
convergence. -/
theorem changeOrigin_radius : p.radius - ‖x‖₊ ≤ (p.changeOrigin x).radius := by
refine' ENNReal.le_of_forall_pos_nnreal_lt fun r _h0 hr => _
rw [lt_tsub_iff_right, add_comm] at hr
have hr' : (‖x‖₊ : ℝ≥0∞) < p.radius := (le_add_right le_rfl).trans_lt hr
apply le_radius_of_summable_nnnorm
have : ∀ k : ℕ,
‖p.changeOrigin x k‖₊ * r ^ k ≤
(∑' s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + s.1)‖₊ * ‖x‖₊ ^ s.1) *
r ^ k :=
fun k => mul_le_mul_right' (p.nnnorm_changeOrigin_le k hr') (r ^ k)
refine' NNReal.summable_of_le this _
simpa only [← NNReal.tsum_mul_right] using
(NNReal.summable_sigma.1 (p.changeOriginSeries_summable_aux₁ hr)).2
#align formal_multilinear_series.change_origin_radius FormalMultilinearSeries.changeOrigin_radius
end
-- From this point on, assume that the space is complete, to make sure that series that converge
-- in norm also converge in `F`.
variable [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x y : E} {r R : ℝ≥0}
theorem hasFPowerSeriesOnBall_changeOrigin (k : ℕ) (hr : 0 < p.radius) :
HasFPowerSeriesOnBall (fun x => p.changeOrigin x k) (p.changeOriginSeries k) 0 p.radius :=
have := p.le_changeOriginSeries_radius k
((p.changeOriginSeries k).hasFPowerSeriesOnBall (hr.trans_le this)).mono hr this
#align formal_multilinear_series.has_fpower_series_on_ball_change_origin FormalMultilinearSeries.hasFPowerSeriesOnBall_changeOrigin
/-- Summing the series `p.changeOrigin x` at a point `y` gives back `p (x + y)`. -/
theorem changeOrigin_eval (h : (‖x‖₊ + ‖y‖₊ : ℝ≥0∞) < p.radius) :
(p.changeOrigin x).sum y = p.sum (x + y) := by
have radius_pos : 0 < p.radius := lt_of_le_of_lt (zero_le _) h
have x_mem_ball : x ∈ EMetric.ball (0 : E) p.radius :=
mem_emetric_ball_zero_iff.2 ((le_add_right le_rfl).trans_lt h)
have y_mem_ball : y ∈ EMetric.ball (0 : E) (p.changeOrigin x).radius := by
refine' mem_emetric_ball_zero_iff.2 (lt_of_lt_of_le _ p.changeOrigin_radius)
rwa [lt_tsub_iff_right, add_comm]
have x_add_y_mem_ball : x + y ∈ EMetric.ball (0 : E) p.radius := by
refine' mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt _ h)
exact mod_cast nnnorm_add_le x y
set f : (Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) → F := fun s =>
p.changeOriginSeriesTerm s.1 s.2.1 s.2.2 s.2.2.2 (fun _ => x) fun _ => y
have hsf : Summable f := by
refine' .of_nnnorm_bounded _ (p.changeOriginSeries_summable_aux₁ h) _
rintro ⟨k, l, s, hs⟩
dsimp only [Subtype.coe_mk]
exact p.nnnorm_changeOriginSeriesTerm_apply_le _ _ _ _ _ _
have hf : HasSum f ((p.changeOrigin x).sum y) := by
refine' HasSum.sigma_of_hasSum ((p.changeOrigin x).summable y_mem_ball).hasSum (fun k => _) hsf
· dsimp only
refine' ContinuousMultilinearMap.hasSum_eval _ _
have := (p.hasFPowerSeriesOnBall_changeOrigin k radius_pos).hasSum x_mem_ball
rw [zero_add] at this
refine' HasSum.sigma_of_hasSum this (fun l => _) _
· simp only [changeOriginSeries, ContinuousMultilinearMap.sum_apply]
apply hasSum_fintype
· refine' .of_nnnorm_bounded _
(p.changeOriginSeries_summable_aux₂ (mem_emetric_ball_zero_iff.1 x_mem_ball) k) fun s => _
refine' (ContinuousMultilinearMap.le_op_nnnorm _ _).trans_eq _
simp
refine' hf.unique (changeOriginIndexEquiv.symm.hasSum_iff.1 _)
refine' HasSum.sigma_of_hasSum
(p.hasSum x_add_y_mem_ball) (fun n => _) (changeOriginIndexEquiv.symm.summable_iff.2 hsf)
erw [(p n).map_add_univ (fun _ => x) fun _ => y]
-- porting note: added explicit function
convert hasSum_fintype (fun c : Finset (Fin n) => f (changeOriginIndexEquiv.symm ⟨n, c⟩))
rename_i s _
dsimp only [changeOriginSeriesTerm, (· ∘ ·), changeOriginIndexEquiv_symm_apply_fst,
changeOriginIndexEquiv_symm_apply_snd_fst, changeOriginIndexEquiv_symm_apply_snd_snd_coe]
rw [ContinuousMultilinearMap.curryFinFinset_apply_const]
have : ∀ (m) (hm : n = m), p n (s.piecewise (fun _ => x) fun _ => y) =
p m ((s.map (Fin.castIso hm).toEquiv.toEmbedding).piecewise (fun _ => x) fun _ => y) := by
rintro m rfl
simp (config := { unfoldPartialApp := true }) [Finset.piecewise]
apply this
#align formal_multilinear_series.change_origin_eval FormalMultilinearSeries.changeOrigin_eval
/-- Power series terms are analytic as we vary the origin -/
theorem analyticAt_changeOrigin (p : FormalMultilinearSeries 𝕜 E F) (rp : p.radius > 0) (n : ℕ) :
AnalyticAt 𝕜 (fun x ↦ p.changeOrigin x n) 0 :=
(FormalMultilinearSeries.hasFPowerSeriesOnBall_changeOrigin p n rp).analyticAt
end FormalMultilinearSeries
section
variable [CompleteSpace F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x y : E} {r : ℝ≥0∞}
/-- If a function admits a power series expansion `p` on a ball `B (x, r)`, then it also admits a
power series on any subball of this ball (even with a different center), given by `p.changeOrigin`.
-/
theorem HasFPowerSeriesOnBall.changeOrigin (hf : HasFPowerSeriesOnBall f p x r)
(h : (‖y‖₊ : ℝ≥0∞) < r) : HasFPowerSeriesOnBall f (p.changeOrigin y) (x + y) (r - ‖y‖₊) :=
{ r_le := by
apply le_trans _ p.changeOrigin_radius
exact tsub_le_tsub hf.r_le le_rfl
r_pos := by simp [h]
hasSum := fun {z} hz => by
have : f (x + y + z) =
FormalMultilinearSeries.sum (FormalMultilinearSeries.changeOrigin p y) z := by
rw [mem_emetric_ball_zero_iff, lt_tsub_iff_right, add_comm] at hz
rw [p.changeOrigin_eval (hz.trans_le hf.r_le), add_assoc, hf.sum]
refine' mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt _ hz)
exact mod_cast nnnorm_add_le y z
rw [this]
apply (p.changeOrigin y).hasSum
refine' EMetric.ball_subset_ball (le_trans _ p.changeOrigin_radius) hz
exact tsub_le_tsub hf.r_le le_rfl }
#align has_fpower_series_on_ball.change_origin HasFPowerSeriesOnBall.changeOrigin
/-- If a function admits a power series expansion `p` on an open ball `B (x, r)`, then
it is analytic at every point of this ball. -/
theorem HasFPowerSeriesOnBall.analyticAt_of_mem (hf : HasFPowerSeriesOnBall f p x r)
(h : y ∈ EMetric.ball x r) : AnalyticAt 𝕜 f y := by
have : (‖y - x‖₊ : ℝ≥0∞) < r := by simpa [edist_eq_coe_nnnorm_sub] using h
have := hf.changeOrigin this
rw [add_sub_cancel'_right] at this
exact this.analyticAt
#align has_fpower_series_on_ball.analytic_at_of_mem HasFPowerSeriesOnBall.analyticAt_of_mem
theorem HasFPowerSeriesOnBall.analyticOn (hf : HasFPowerSeriesOnBall f p x r) :
AnalyticOn 𝕜 f (EMetric.ball x r) :=
fun _y hy => hf.analyticAt_of_mem hy
#align has_fpower_series_on_ball.analytic_on HasFPowerSeriesOnBall.analyticOn
variable (𝕜 f)
/-- For any function `f` from a normed vector space to a Banach space, the set of points `x` such
that `f` is analytic at `x` is open. -/
theorem isOpen_analyticAt : IsOpen { x | AnalyticAt 𝕜 f x } := by
rw [isOpen_iff_mem_nhds]
rintro x ⟨p, r, hr⟩
exact mem_of_superset (EMetric.ball_mem_nhds _ hr.r_pos) fun y hy => hr.analyticAt_of_mem hy
#align is_open_analytic_at isOpen_analyticAt
variable {𝕜}
theorem AnalyticAt.eventually_analyticAt {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∀ᶠ y in 𝓝 x, AnalyticAt 𝕜 f y :=
(isOpen_analyticAt 𝕜 f).mem_nhds h
theorem AnalyticAt.exists_mem_nhds_analyticOn {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∃ s ∈ 𝓝 x, AnalyticOn 𝕜 f s :=
h.eventually_analyticAt.exists_mem
/-- If we're analytic at a point, we're analytic in a nonempty ball -/
theorem AnalyticAt.exists_ball_analyticOn {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∃ r : ℝ, 0 < r ∧ AnalyticOn 𝕜 f (Metric.ball x r) :=
Metric.isOpen_iff.mp (isOpen_analyticAt _ _) _ h
end
section
open FormalMultilinearSeries
variable {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E} {z₀ : 𝕜}
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z)) := by
refine' ⟨fun ⟨r, _, r_pos, h⟩ =>
eventually_of_mem (EMetric.ball_mem_nhds 0 r_pos) fun _ => by simpa using h, _⟩
simp only [Metric.eventually_nhds_iff]
rintro ⟨r, r_pos, h⟩
refine' ⟨p.radius ⊓ r.toNNReal, by simp, _, _⟩
· simp only [r_pos.lt, lt_inf_iff, ENNReal.coe_pos, Real.toNNReal_pos, and_true_iff]
|
obtain ⟨z, z_pos, le_z⟩ := NormedField.exists_norm_lt 𝕜 r_pos.lt
|
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z)) := by
refine' ⟨fun ⟨r, _, r_pos, h⟩ =>
eventually_of_mem (EMetric.ball_mem_nhds 0 r_pos) fun _ => by simpa using h, _⟩
simp only [Metric.eventually_nhds_iff]
rintro ⟨r, r_pos, h⟩
refine' ⟨p.radius ⊓ r.toNNReal, by simp, _, _⟩
· simp only [r_pos.lt, lt_inf_iff, ENNReal.coe_pos, Real.toNNReal_pos, and_true_iff]
|
Mathlib.Analysis.Analytic.Basic.1430_0.jQw1fRSE1vGpOll
|
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z))
|
Mathlib_Analysis_Analytic_Basic
|
case intro.intro.refine'_1.intro.intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
p : FormalMultilinearSeries 𝕜 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
r : ℝ
r_pos : r > 0
h : ∀ ⦃y : 𝕜⦄, dist y 0 < r → HasSum (fun n => y ^ n • coeff p n) (f (z₀ + y))
z : 𝕜
z_pos : 0 < ‖z‖
le_z : ‖z‖ < r
⊢ 0 < radius p
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Logic.Equiv.Fin
import Mathlib.Topology.Algebra.InfiniteSum.Module
#align_import analysis.analytic.basic from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
/-!
# Analytic functions
A function is analytic in one dimension around `0` if it can be written as a converging power series
`Σ pₙ zⁿ`. This definition can be extended to any dimension (even in infinite dimension) by
requiring that `pₙ` is a continuous `n`-multilinear map. In general, `pₙ` is not unique (in two
dimensions, taking `p₂ (x, y) (x', y') = x y'` or `y x'` gives the same map when applied to a
vector `(x, y) (x, y)`). A way to guarantee uniqueness is to take a symmetric `pₙ`, but this is not
always possible in nonzero characteristic (in characteristic 2, the previous example has no
symmetric representative). Therefore, we do not insist on symmetry or uniqueness in the definition,
and we only require the existence of a converging series.
The general framework is important to say that the exponential map on bounded operators on a Banach
space is analytic, as well as the inverse on invertible operators.
## Main definitions
Let `p` be a formal multilinear series from `E` to `F`, i.e., `p n` is a multilinear map on `E^n`
for `n : ℕ`.
* `p.radius`: the largest `r : ℝ≥0∞` such that `‖p n‖ * r^n` grows subexponentially.
* `p.le_radius_of_bound`, `p.le_radius_of_bound_nnreal`, `p.le_radius_of_isBigO`: if `‖p n‖ * r ^ n`
is bounded above, then `r ≤ p.radius`;
* `p.isLittleO_of_lt_radius`, `p.norm_mul_pow_le_mul_pow_of_lt_radius`,
`p.isLittleO_one_of_lt_radius`,
`p.norm_mul_pow_le_of_lt_radius`, `p.nnnorm_mul_pow_le_of_lt_radius`: if `r < p.radius`, then
`‖p n‖ * r ^ n` tends to zero exponentially;
* `p.lt_radius_of_isBigO`: if `r ≠ 0` and `‖p n‖ * r ^ n = O(a ^ n)` for some `-1 < a < 1`, then
`r < p.radius`;
* `p.partialSum n x`: the sum `∑_{i = 0}^{n-1} pᵢ xⁱ`.
* `p.sum x`: the sum `∑'_{i = 0}^{∞} pᵢ xⁱ`.
Additionally, let `f` be a function from `E` to `F`.
* `HasFPowerSeriesOnBall f p x r`: on the ball of center `x` with radius `r`,
`f (x + y) = ∑'_n pₙ yⁿ`.
* `HasFPowerSeriesAt f p x`: on some ball of center `x` with positive radius, holds
`HasFPowerSeriesOnBall f p x r`.
* `AnalyticAt 𝕜 f x`: there exists a power series `p` such that holds `HasFPowerSeriesAt f p x`.
* `AnalyticOn 𝕜 f s`: the function `f` is analytic at every point of `s`.
We develop the basic properties of these notions, notably:
* If a function admits a power series, it is continuous (see
`HasFPowerSeriesOnBall.continuousOn` and `HasFPowerSeriesAt.continuousAt` and
`AnalyticAt.continuousAt`).
* In a complete space, the sum of a formal power series with positive radius is well defined on the
disk of convergence, see `FormalMultilinearSeries.hasFPowerSeriesOnBall`.
* If a function admits a power series in a ball, then it is analytic at any point `y` of this ball,
and the power series there can be expressed in terms of the initial power series `p` as
`p.changeOrigin y`. See `HasFPowerSeriesOnBall.changeOrigin`. It follows in particular that
the set of points at which a given function is analytic is open, see `isOpen_analyticAt`.
## Implementation details
We only introduce the radius of convergence of a power series, as `p.radius`.
For a power series in finitely many dimensions, there is a finer (directional, coordinate-dependent)
notion, describing the polydisk of convergence. This notion is more specific, and not necessary to
build the general theory. We do not define it here.
-/
noncomputable section
variable {𝕜 E F G : Type*}
open Topology Classical BigOperators NNReal Filter ENNReal
open Set Filter Asymptotics
namespace FormalMultilinearSeries
variable [Ring 𝕜] [AddCommGroup E] [AddCommGroup F] [Module 𝕜 E] [Module 𝕜 F]
variable [TopologicalSpace E] [TopologicalSpace F]
variable [TopologicalAddGroup E] [TopologicalAddGroup F]
variable [ContinuousConstSMul 𝕜 E] [ContinuousConstSMul 𝕜 F]
/-- Given a formal multilinear series `p` and a vector `x`, then `p.sum x` is the sum `Σ pₙ xⁿ`. A
priori, it only behaves well when `‖x‖ < p.radius`. -/
protected def sum (p : FormalMultilinearSeries 𝕜 E F) (x : E) : F :=
∑' n : ℕ, p n fun _ => x
#align formal_multilinear_series.sum FormalMultilinearSeries.sum
/-- Given a formal multilinear series `p` and a vector `x`, then `p.partialSum n x` is the sum
`Σ pₖ xᵏ` for `k ∈ {0,..., n-1}`. -/
def partialSum (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) (x : E) : F :=
∑ k in Finset.range n, p k fun _ : Fin k => x
#align formal_multilinear_series.partial_sum FormalMultilinearSeries.partialSum
/-- The partial sums of a formal multilinear series are continuous. -/
theorem partialSum_continuous (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) :
Continuous (p.partialSum n) := by
unfold partialSum -- Porting note: added
continuity
#align formal_multilinear_series.partial_sum_continuous FormalMultilinearSeries.partialSum_continuous
end FormalMultilinearSeries
/-! ### The radius of a formal multilinear series -/
variable [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F]
[NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G]
namespace FormalMultilinearSeries
variable (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
/-- The radius of a formal multilinear series is the largest `r` such that the sum `Σ ‖pₙ‖ ‖y‖ⁿ`
converges for all `‖y‖ < r`. This implies that `Σ pₙ yⁿ` converges for all `‖y‖ < r`, but these
definitions are *not* equivalent in general. -/
def radius (p : FormalMultilinearSeries 𝕜 E F) : ℝ≥0∞ :=
⨆ (r : ℝ≥0) (C : ℝ) (_ : ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C), (r : ℝ≥0∞)
#align formal_multilinear_series.radius FormalMultilinearSeries.radius
/-- If `‖pₙ‖ rⁿ` is bounded in `n`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_bound (C : ℝ) {r : ℝ≥0} (h : ∀ n : ℕ, ‖p n‖ * (r : ℝ) ^ n ≤ C) :
(r : ℝ≥0∞) ≤ p.radius :=
le_iSup_of_le r <| le_iSup_of_le C <| le_iSup (fun _ => (r : ℝ≥0∞)) h
#align formal_multilinear_series.le_radius_of_bound FormalMultilinearSeries.le_radius_of_bound
/-- If `‖pₙ‖ rⁿ` is bounded in `n`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_bound_nnreal (C : ℝ≥0) {r : ℝ≥0} (h : ∀ n : ℕ, ‖p n‖₊ * r ^ n ≤ C) :
(r : ℝ≥0∞) ≤ p.radius :=
p.le_radius_of_bound C fun n => mod_cast h n
#align formal_multilinear_series.le_radius_of_bound_nnreal FormalMultilinearSeries.le_radius_of_bound_nnreal
/-- If `‖pₙ‖ rⁿ = O(1)`, as `n → ∞`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_isBigO (h : (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] fun _ => (1 : ℝ)) :
↑r ≤ p.radius :=
Exists.elim (isBigO_one_nat_atTop_iff.1 h) fun C hC =>
p.le_radius_of_bound C fun n => (le_abs_self _).trans (hC n)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.le_radius_of_is_O FormalMultilinearSeries.le_radius_of_isBigO
theorem le_radius_of_eventually_le (C) (h : ∀ᶠ n in atTop, ‖p n‖ * (r : ℝ) ^ n ≤ C) :
↑r ≤ p.radius :=
p.le_radius_of_isBigO <| IsBigO.of_bound C <| h.mono fun n hn => by simpa
#align formal_multilinear_series.le_radius_of_eventually_le FormalMultilinearSeries.le_radius_of_eventually_le
theorem le_radius_of_summable_nnnorm (h : Summable fun n => ‖p n‖₊ * r ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_bound_nnreal (∑' n, ‖p n‖₊ * r ^ n) fun _ => le_tsum' h _
#align formal_multilinear_series.le_radius_of_summable_nnnorm FormalMultilinearSeries.le_radius_of_summable_nnnorm
theorem le_radius_of_summable (h : Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_summable_nnnorm <| by
simp only [← coe_nnnorm] at h
exact mod_cast h
#align formal_multilinear_series.le_radius_of_summable FormalMultilinearSeries.le_radius_of_summable
theorem radius_eq_top_of_forall_nnreal_isBigO
(h : ∀ r : ℝ≥0, (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] fun _ => (1 : ℝ)) : p.radius = ∞ :=
ENNReal.eq_top_of_forall_nnreal_le fun r => p.le_radius_of_isBigO (h r)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.radius_eq_top_of_forall_nnreal_is_O FormalMultilinearSeries.radius_eq_top_of_forall_nnreal_isBigO
theorem radius_eq_top_of_eventually_eq_zero (h : ∀ᶠ n in atTop, p n = 0) : p.radius = ∞ :=
p.radius_eq_top_of_forall_nnreal_isBigO fun r =>
(isBigO_zero _ _).congr' (h.mono fun n hn => by simp [hn]) EventuallyEq.rfl
#align formal_multilinear_series.radius_eq_top_of_eventually_eq_zero FormalMultilinearSeries.radius_eq_top_of_eventually_eq_zero
theorem radius_eq_top_of_forall_image_add_eq_zero (n : ℕ) (hn : ∀ m, p (m + n) = 0) :
p.radius = ∞ :=
p.radius_eq_top_of_eventually_eq_zero <|
mem_atTop_sets.2 ⟨n, fun _ hk => tsub_add_cancel_of_le hk ▸ hn _⟩
#align formal_multilinear_series.radius_eq_top_of_forall_image_add_eq_zero FormalMultilinearSeries.radius_eq_top_of_forall_image_add_eq_zero
@[simp]
theorem constFormalMultilinearSeries_radius {v : F} :
(constFormalMultilinearSeries 𝕜 E v).radius = ⊤ :=
(constFormalMultilinearSeries 𝕜 E v).radius_eq_top_of_forall_image_add_eq_zero 1
(by simp [constFormalMultilinearSeries])
#align formal_multilinear_series.const_formal_multilinear_series_radius FormalMultilinearSeries.constFormalMultilinearSeries_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` tends to zero exponentially:
for some `0 < a < 1`, `‖p n‖ rⁿ = o(aⁿ)`. -/
theorem isLittleO_of_lt_radius (h : ↑r < p.radius) :
∃ a ∈ Ioo (0 : ℝ) 1, (fun n => ‖p n‖ * (r : ℝ) ^ n) =o[atTop] (a ^ ·) := by
have := (TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 4
rw [this]
-- Porting note: was
-- rw [(TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 4]
simp only [radius, lt_iSup_iff] at h
rcases h with ⟨t, C, hC, rt⟩
rw [ENNReal.coe_lt_coe, ← NNReal.coe_lt_coe] at rt
have : 0 < (t : ℝ) := r.coe_nonneg.trans_lt rt
rw [← div_lt_one this] at rt
refine' ⟨_, rt, C, Or.inr zero_lt_one, fun n => _⟩
calc
|‖p n‖ * (r : ℝ) ^ n| = ‖p n‖ * (t : ℝ) ^ n * (r / t : ℝ) ^ n := by
field_simp [mul_right_comm, abs_mul]
_ ≤ C * (r / t : ℝ) ^ n := by gcongr; apply hC
#align formal_multilinear_series.is_o_of_lt_radius FormalMultilinearSeries.isLittleO_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ = o(1)`. -/
theorem isLittleO_one_of_lt_radius (h : ↑r < p.radius) :
(fun n => ‖p n‖ * (r : ℝ) ^ n) =o[atTop] (fun _ => 1 : ℕ → ℝ) :=
let ⟨_, ha, hp⟩ := p.isLittleO_of_lt_radius h
hp.trans <| (isLittleO_pow_pow_of_lt_left ha.1.le ha.2).congr (fun _ => rfl) one_pow
#align formal_multilinear_series.is_o_one_of_lt_radius FormalMultilinearSeries.isLittleO_one_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` tends to zero exponentially:
for some `0 < a < 1` and `C > 0`, `‖p n‖ * r ^ n ≤ C * a ^ n`. -/
theorem norm_mul_pow_le_mul_pow_of_lt_radius (h : ↑r < p.radius) :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C * a ^ n := by
-- Porting note: moved out of `rcases`
have := ((TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 5).mp
(p.isLittleO_of_lt_radius h)
rcases this with ⟨a, ha, C, hC, H⟩
exact ⟨a, ha, C, hC, fun n => (le_abs_self _).trans (H n)⟩
#align formal_multilinear_series.norm_mul_pow_le_mul_pow_of_lt_radius FormalMultilinearSeries.norm_mul_pow_le_mul_pow_of_lt_radius
/-- If `r ≠ 0` and `‖pₙ‖ rⁿ = O(aⁿ)` for some `-1 < a < 1`, then `r < p.radius`. -/
theorem lt_radius_of_isBigO (h₀ : r ≠ 0) {a : ℝ} (ha : a ∈ Ioo (-1 : ℝ) 1)
(hp : (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] (a ^ ·)) : ↑r < p.radius := by
-- Porting note: moved out of `rcases`
have := ((TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 2 5)
rcases this.mp ⟨a, ha, hp⟩ with ⟨a, ha, C, hC, hp⟩
rw [← pos_iff_ne_zero, ← NNReal.coe_pos] at h₀
lift a to ℝ≥0 using ha.1.le
have : (r : ℝ) < r / a := by
simpa only [div_one] using (div_lt_div_left h₀ zero_lt_one ha.1).2 ha.2
norm_cast at this
rw [← ENNReal.coe_lt_coe] at this
refine' this.trans_le (p.le_radius_of_bound C fun n => _)
rw [NNReal.coe_div, div_pow, ← mul_div_assoc, div_le_iff (pow_pos ha.1 n)]
exact (le_abs_self _).trans (hp n)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.lt_radius_of_is_O FormalMultilinearSeries.lt_radius_of_isBigO
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem norm_mul_pow_le_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C :=
let ⟨_, ha, C, hC, h⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius h
⟨C, hC, fun n => (h n).trans <| mul_le_of_le_one_right hC.lt.le (pow_le_one _ ha.1.le ha.2.le)⟩
#align formal_multilinear_series.norm_mul_pow_le_of_lt_radius FormalMultilinearSeries.norm_mul_pow_le_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem norm_le_div_pow_of_pos_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h0 : 0 < r) (h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖ ≤ C / (r : ℝ) ^ n :=
let ⟨C, hC, hp⟩ := p.norm_mul_pow_le_of_lt_radius h
⟨C, hC, fun n => Iff.mpr (le_div_iff (pow_pos h0 _)) (hp n)⟩
#align formal_multilinear_series.norm_le_div_pow_of_pos_of_lt_radius FormalMultilinearSeries.norm_le_div_pow_of_pos_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem nnnorm_mul_pow_le_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖₊ * r ^ n ≤ C :=
let ⟨C, hC, hp⟩ := p.norm_mul_pow_le_of_lt_radius h
⟨⟨C, hC.lt.le⟩, hC, mod_cast hp⟩
#align formal_multilinear_series.nnnorm_mul_pow_le_of_lt_radius FormalMultilinearSeries.nnnorm_mul_pow_le_of_lt_radius
theorem le_radius_of_tendsto (p : FormalMultilinearSeries 𝕜 E F) {l : ℝ}
(h : Tendsto (fun n => ‖p n‖ * (r : ℝ) ^ n) atTop (𝓝 l)) : ↑r ≤ p.radius :=
p.le_radius_of_isBigO (h.isBigO_one _)
#align formal_multilinear_series.le_radius_of_tendsto FormalMultilinearSeries.le_radius_of_tendsto
theorem le_radius_of_summable_norm (p : FormalMultilinearSeries 𝕜 E F)
(hs : Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_tendsto hs.tendsto_atTop_zero
#align formal_multilinear_series.le_radius_of_summable_norm FormalMultilinearSeries.le_radius_of_summable_norm
theorem not_summable_norm_of_radius_lt_nnnorm (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(h : p.radius < ‖x‖₊) : ¬Summable fun n => ‖p n‖ * ‖x‖ ^ n :=
fun hs => not_le_of_lt h (p.le_radius_of_summable_norm hs)
#align formal_multilinear_series.not_summable_norm_of_radius_lt_nnnorm FormalMultilinearSeries.not_summable_norm_of_radius_lt_nnnorm
theorem summable_norm_mul_pow (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0} (h : ↑r < p.radius) :
Summable fun n : ℕ => ‖p n‖ * (r : ℝ) ^ n := by
obtain ⟨a, ha : a ∈ Ioo (0 : ℝ) 1, C, - : 0 < C, hp⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius h
exact .of_nonneg_of_le (fun n => mul_nonneg (norm_nonneg _) (pow_nonneg r.coe_nonneg _))
hp ((summable_geometric_of_lt_1 ha.1.le ha.2).mul_left _)
#align formal_multilinear_series.summable_norm_mul_pow FormalMultilinearSeries.summable_norm_mul_pow
theorem summable_norm_apply (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : Summable fun n : ℕ => ‖p n fun _ => x‖ := by
rw [mem_emetric_ball_zero_iff] at hx
refine' .of_nonneg_of_le
(fun _ => norm_nonneg _) (fun n => ((p n).le_op_norm _).trans_eq _) (p.summable_norm_mul_pow hx)
simp
#align formal_multilinear_series.summable_norm_apply FormalMultilinearSeries.summable_norm_apply
theorem summable_nnnorm_mul_pow (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0} (h : ↑r < p.radius) :
Summable fun n : ℕ => ‖p n‖₊ * r ^ n := by
rw [← NNReal.summable_coe]
push_cast
exact p.summable_norm_mul_pow h
#align formal_multilinear_series.summable_nnnorm_mul_pow FormalMultilinearSeries.summable_nnnorm_mul_pow
protected theorem summable [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : Summable fun n : ℕ => p n fun _ => x :=
(p.summable_norm_apply hx).of_norm
#align formal_multilinear_series.summable FormalMultilinearSeries.summable
theorem radius_eq_top_of_summable_norm (p : FormalMultilinearSeries 𝕜 E F)
(hs : ∀ r : ℝ≥0, Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : p.radius = ∞ :=
ENNReal.eq_top_of_forall_nnreal_le fun r => p.le_radius_of_summable_norm (hs r)
#align formal_multilinear_series.radius_eq_top_of_summable_norm FormalMultilinearSeries.radius_eq_top_of_summable_norm
theorem radius_eq_top_iff_summable_norm (p : FormalMultilinearSeries 𝕜 E F) :
p.radius = ∞ ↔ ∀ r : ℝ≥0, Summable fun n => ‖p n‖ * (r : ℝ) ^ n := by
constructor
· intro h r
obtain ⟨a, ha : a ∈ Ioo (0 : ℝ) 1, C, - : 0 < C, hp⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius
(show (r : ℝ≥0∞) < p.radius from h.symm ▸ ENNReal.coe_lt_top)
refine' .of_norm_bounded
(fun n => (C : ℝ) * a ^ n) ((summable_geometric_of_lt_1 ha.1.le ha.2).mul_left _) fun n => _
specialize hp n
rwa [Real.norm_of_nonneg (mul_nonneg (norm_nonneg _) (pow_nonneg r.coe_nonneg n))]
· exact p.radius_eq_top_of_summable_norm
#align formal_multilinear_series.radius_eq_top_iff_summable_norm FormalMultilinearSeries.radius_eq_top_iff_summable_norm
/-- If the radius of `p` is positive, then `‖pₙ‖` grows at most geometrically. -/
theorem le_mul_pow_of_radius_pos (p : FormalMultilinearSeries 𝕜 E F) (h : 0 < p.radius) :
∃ (C r : _) (hC : 0 < C) (_ : 0 < r), ∀ n, ‖p n‖ ≤ C * r ^ n := by
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 h with ⟨r, r0, rlt⟩
have rpos : 0 < (r : ℝ) := by simp [ENNReal.coe_pos.1 r0]
rcases norm_le_div_pow_of_pos_of_lt_radius p rpos rlt with ⟨C, Cpos, hCp⟩
refine' ⟨C, r⁻¹, Cpos, by simp only [inv_pos, rpos], fun n => _⟩
-- Porting note: was `convert`
rw [inv_pow, ← div_eq_mul_inv]
exact hCp n
#align formal_multilinear_series.le_mul_pow_of_radius_pos FormalMultilinearSeries.le_mul_pow_of_radius_pos
/-- The radius of the sum of two formal series is at least the minimum of their two radii. -/
theorem min_radius_le_radius_add (p q : FormalMultilinearSeries 𝕜 E F) :
min p.radius q.radius ≤ (p + q).radius := by
refine' ENNReal.le_of_forall_nnreal_lt fun r hr => _
rw [lt_min_iff] at hr
have := ((p.isLittleO_one_of_lt_radius hr.1).add (q.isLittleO_one_of_lt_radius hr.2)).isBigO
refine' (p + q).le_radius_of_isBigO ((isBigO_of_le _ fun n => _).trans this)
rw [← add_mul, norm_mul, norm_mul, norm_norm]
exact mul_le_mul_of_nonneg_right ((norm_add_le _ _).trans (le_abs_self _)) (norm_nonneg _)
#align formal_multilinear_series.min_radius_le_radius_add FormalMultilinearSeries.min_radius_le_radius_add
@[simp]
theorem radius_neg (p : FormalMultilinearSeries 𝕜 E F) : (-p).radius = p.radius := by
simp only [radius, neg_apply, norm_neg]
#align formal_multilinear_series.radius_neg FormalMultilinearSeries.radius_neg
protected theorem hasSum [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : HasSum (fun n : ℕ => p n fun _ => x) (p.sum x) :=
(p.summable hx).hasSum
#align formal_multilinear_series.has_sum FormalMultilinearSeries.hasSum
theorem radius_le_radius_continuousLinearMap_comp (p : FormalMultilinearSeries 𝕜 E F)
(f : F →L[𝕜] G) : p.radius ≤ (f.compFormalMultilinearSeries p).radius := by
refine' ENNReal.le_of_forall_nnreal_lt fun r hr => _
apply le_radius_of_isBigO
apply (IsBigO.trans_isLittleO _ (p.isLittleO_one_of_lt_radius hr)).isBigO
refine' IsBigO.mul (@IsBigOWith.isBigO _ _ _ _ _ ‖f‖ _ _ _ _) (isBigO_refl _ _)
refine IsBigOWith.of_bound (eventually_of_forall fun n => ?_)
simpa only [norm_norm] using f.norm_compContinuousMultilinearMap_le (p n)
#align formal_multilinear_series.radius_le_radius_continuous_linear_map_comp FormalMultilinearSeries.radius_le_radius_continuousLinearMap_comp
end FormalMultilinearSeries
/-! ### Expanding a function as a power series -/
section
variable {f g : E → F} {p pf pg : FormalMultilinearSeries 𝕜 E F} {x : E} {r r' : ℝ≥0∞}
/-- Given a function `f : E → F` and a formal multilinear series `p`, we say that `f` has `p` as
a power series on the ball of radius `r > 0` around `x` if `f (x + y) = ∑' pₙ yⁿ` for all `‖y‖ < r`.
-/
structure HasFPowerSeriesOnBall (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (x : E) (r : ℝ≥0∞) :
Prop where
r_le : r ≤ p.radius
r_pos : 0 < r
hasSum :
∀ {y}, y ∈ EMetric.ball (0 : E) r → HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y))
#align has_fpower_series_on_ball HasFPowerSeriesOnBall
/-- Given a function `f : E → F` and a formal multilinear series `p`, we say that `f` has `p` as
a power series around `x` if `f (x + y) = ∑' pₙ yⁿ` for all `y` in a neighborhood of `0`. -/
def HasFPowerSeriesAt (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (x : E) :=
∃ r, HasFPowerSeriesOnBall f p x r
#align has_fpower_series_at HasFPowerSeriesAt
variable (𝕜)
/-- Given a function `f : E → F`, we say that `f` is analytic at `x` if it admits a convergent power
series expansion around `x`. -/
def AnalyticAt (f : E → F) (x : E) :=
∃ p : FormalMultilinearSeries 𝕜 E F, HasFPowerSeriesAt f p x
#align analytic_at AnalyticAt
/-- Given a function `f : E → F`, we say that `f` is analytic on a set `s` if it is analytic around
every point of `s`. -/
def AnalyticOn (f : E → F) (s : Set E) :=
∀ x, x ∈ s → AnalyticAt 𝕜 f x
#align analytic_on AnalyticOn
variable {𝕜}
theorem HasFPowerSeriesOnBall.hasFPowerSeriesAt (hf : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesAt f p x :=
⟨r, hf⟩
#align has_fpower_series_on_ball.has_fpower_series_at HasFPowerSeriesOnBall.hasFPowerSeriesAt
theorem HasFPowerSeriesAt.analyticAt (hf : HasFPowerSeriesAt f p x) : AnalyticAt 𝕜 f x :=
⟨p, hf⟩
#align has_fpower_series_at.analytic_at HasFPowerSeriesAt.analyticAt
theorem HasFPowerSeriesOnBall.analyticAt (hf : HasFPowerSeriesOnBall f p x r) : AnalyticAt 𝕜 f x :=
hf.hasFPowerSeriesAt.analyticAt
#align has_fpower_series_on_ball.analytic_at HasFPowerSeriesOnBall.analyticAt
theorem HasFPowerSeriesOnBall.congr (hf : HasFPowerSeriesOnBall f p x r)
(hg : EqOn f g (EMetric.ball x r)) : HasFPowerSeriesOnBall g p x r :=
{ r_le := hf.r_le
r_pos := hf.r_pos
hasSum := fun {y} hy => by
convert hf.hasSum hy using 1
apply hg.symm
simpa [edist_eq_coe_nnnorm_sub] using hy }
#align has_fpower_series_on_ball.congr HasFPowerSeriesOnBall.congr
/-- If a function `f` has a power series `p` around `x`, then the function `z ↦ f (z - y)` has the
same power series around `x + y`. -/
theorem HasFPowerSeriesOnBall.comp_sub (hf : HasFPowerSeriesOnBall f p x r) (y : E) :
HasFPowerSeriesOnBall (fun z => f (z - y)) p (x + y) r :=
{ r_le := hf.r_le
r_pos := hf.r_pos
hasSum := fun {z} hz => by
convert hf.hasSum hz using 2
abel }
#align has_fpower_series_on_ball.comp_sub HasFPowerSeriesOnBall.comp_sub
theorem HasFPowerSeriesOnBall.hasSum_sub (hf : HasFPowerSeriesOnBall f p x r) {y : E}
(hy : y ∈ EMetric.ball x r) : HasSum (fun n : ℕ => p n fun _ => y - x) (f y) := by
have : y - x ∈ EMetric.ball (0 : E) r := by simpa [edist_eq_coe_nnnorm_sub] using hy
simpa only [add_sub_cancel'_right] using hf.hasSum this
#align has_fpower_series_on_ball.has_sum_sub HasFPowerSeriesOnBall.hasSum_sub
theorem HasFPowerSeriesOnBall.radius_pos (hf : HasFPowerSeriesOnBall f p x r) : 0 < p.radius :=
lt_of_lt_of_le hf.r_pos hf.r_le
#align has_fpower_series_on_ball.radius_pos HasFPowerSeriesOnBall.radius_pos
theorem HasFPowerSeriesAt.radius_pos (hf : HasFPowerSeriesAt f p x) : 0 < p.radius :=
let ⟨_, hr⟩ := hf
hr.radius_pos
#align has_fpower_series_at.radius_pos HasFPowerSeriesAt.radius_pos
theorem HasFPowerSeriesOnBall.mono (hf : HasFPowerSeriesOnBall f p x r) (r'_pos : 0 < r')
(hr : r' ≤ r) : HasFPowerSeriesOnBall f p x r' :=
⟨le_trans hr hf.1, r'_pos, fun hy => hf.hasSum (EMetric.ball_subset_ball hr hy)⟩
#align has_fpower_series_on_ball.mono HasFPowerSeriesOnBall.mono
theorem HasFPowerSeriesAt.congr (hf : HasFPowerSeriesAt f p x) (hg : f =ᶠ[𝓝 x] g) :
HasFPowerSeriesAt g p x := by
rcases hf with ⟨r₁, h₁⟩
rcases EMetric.mem_nhds_iff.mp hg with ⟨r₂, h₂pos, h₂⟩
exact ⟨min r₁ r₂,
(h₁.mono (lt_min h₁.r_pos h₂pos) inf_le_left).congr
fun y hy => h₂ (EMetric.ball_subset_ball inf_le_right hy)⟩
#align has_fpower_series_at.congr HasFPowerSeriesAt.congr
protected theorem HasFPowerSeriesAt.eventually (hf : HasFPowerSeriesAt f p x) :
∀ᶠ r : ℝ≥0∞ in 𝓝[>] 0, HasFPowerSeriesOnBall f p x r :=
let ⟨_, hr⟩ := hf
mem_of_superset (Ioo_mem_nhdsWithin_Ioi (left_mem_Ico.2 hr.r_pos)) fun _ hr' =>
hr.mono hr'.1 hr'.2.le
#align has_fpower_series_at.eventually HasFPowerSeriesAt.eventually
theorem HasFPowerSeriesOnBall.eventually_hasSum (hf : HasFPowerSeriesOnBall f p x r) :
∀ᶠ y in 𝓝 0, HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y)) := by
filter_upwards [EMetric.ball_mem_nhds (0 : E) hf.r_pos] using fun _ => hf.hasSum
#align has_fpower_series_on_ball.eventually_has_sum HasFPowerSeriesOnBall.eventually_hasSum
theorem HasFPowerSeriesAt.eventually_hasSum (hf : HasFPowerSeriesAt f p x) :
∀ᶠ y in 𝓝 0, HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y)) :=
let ⟨_, hr⟩ := hf
hr.eventually_hasSum
#align has_fpower_series_at.eventually_has_sum HasFPowerSeriesAt.eventually_hasSum
theorem HasFPowerSeriesOnBall.eventually_hasSum_sub (hf : HasFPowerSeriesOnBall f p x r) :
∀ᶠ y in 𝓝 x, HasSum (fun n : ℕ => p n fun _ : Fin n => y - x) (f y) := by
filter_upwards [EMetric.ball_mem_nhds x hf.r_pos] with y using hf.hasSum_sub
#align has_fpower_series_on_ball.eventually_has_sum_sub HasFPowerSeriesOnBall.eventually_hasSum_sub
theorem HasFPowerSeriesAt.eventually_hasSum_sub (hf : HasFPowerSeriesAt f p x) :
∀ᶠ y in 𝓝 x, HasSum (fun n : ℕ => p n fun _ : Fin n => y - x) (f y) :=
let ⟨_, hr⟩ := hf
hr.eventually_hasSum_sub
#align has_fpower_series_at.eventually_has_sum_sub HasFPowerSeriesAt.eventually_hasSum_sub
theorem HasFPowerSeriesOnBall.eventually_eq_zero
(hf : HasFPowerSeriesOnBall f (0 : FormalMultilinearSeries 𝕜 E F) x r) :
∀ᶠ z in 𝓝 x, f z = 0 := by
filter_upwards [hf.eventually_hasSum_sub] with z hz using hz.unique hasSum_zero
#align has_fpower_series_on_ball.eventually_eq_zero HasFPowerSeriesOnBall.eventually_eq_zero
theorem HasFPowerSeriesAt.eventually_eq_zero
(hf : HasFPowerSeriesAt f (0 : FormalMultilinearSeries 𝕜 E F) x) : ∀ᶠ z in 𝓝 x, f z = 0 :=
let ⟨_, hr⟩ := hf
hr.eventually_eq_zero
#align has_fpower_series_at.eventually_eq_zero HasFPowerSeriesAt.eventually_eq_zero
theorem hasFPowerSeriesOnBall_const {c : F} {e : E} :
HasFPowerSeriesOnBall (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e ⊤ := by
refine' ⟨by simp, WithTop.zero_lt_top, fun _ => hasSum_single 0 fun n hn => _⟩
simp [constFormalMultilinearSeries_apply hn]
#align has_fpower_series_on_ball_const hasFPowerSeriesOnBall_const
theorem hasFPowerSeriesAt_const {c : F} {e : E} :
HasFPowerSeriesAt (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e :=
⟨⊤, hasFPowerSeriesOnBall_const⟩
#align has_fpower_series_at_const hasFPowerSeriesAt_const
theorem analyticAt_const {v : F} : AnalyticAt 𝕜 (fun _ => v) x :=
⟨constFormalMultilinearSeries 𝕜 E v, hasFPowerSeriesAt_const⟩
#align analytic_at_const analyticAt_const
theorem analyticOn_const {v : F} {s : Set E} : AnalyticOn 𝕜 (fun _ => v) s :=
fun _ _ => analyticAt_const
#align analytic_on_const analyticOn_const
theorem HasFPowerSeriesOnBall.add (hf : HasFPowerSeriesOnBall f pf x r)
(hg : HasFPowerSeriesOnBall g pg x r) : HasFPowerSeriesOnBall (f + g) (pf + pg) x r :=
{ r_le := le_trans (le_min_iff.2 ⟨hf.r_le, hg.r_le⟩) (pf.min_radius_le_radius_add pg)
r_pos := hf.r_pos
hasSum := fun hy => (hf.hasSum hy).add (hg.hasSum hy) }
#align has_fpower_series_on_ball.add HasFPowerSeriesOnBall.add
theorem HasFPowerSeriesAt.add (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x) :
HasFPowerSeriesAt (f + g) (pf + pg) x := by
rcases (hf.eventually.and hg.eventually).exists with ⟨r, hr⟩
exact ⟨r, hr.1.add hr.2⟩
#align has_fpower_series_at.add HasFPowerSeriesAt.add
theorem AnalyticAt.congr (hf : AnalyticAt 𝕜 f x) (hg : f =ᶠ[𝓝 x] g) : AnalyticAt 𝕜 g x :=
let ⟨_, hpf⟩ := hf
(hpf.congr hg).analyticAt
theorem analyticAt_congr (h : f =ᶠ[𝓝 x] g) : AnalyticAt 𝕜 f x ↔ AnalyticAt 𝕜 g x :=
⟨fun hf ↦ hf.congr h, fun hg ↦ hg.congr h.symm⟩
theorem AnalyticAt.add (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) : AnalyticAt 𝕜 (f + g) x :=
let ⟨_, hpf⟩ := hf
let ⟨_, hqf⟩ := hg
(hpf.add hqf).analyticAt
#align analytic_at.add AnalyticAt.add
theorem HasFPowerSeriesOnBall.neg (hf : HasFPowerSeriesOnBall f pf x r) :
HasFPowerSeriesOnBall (-f) (-pf) x r :=
{ r_le := by
rw [pf.radius_neg]
exact hf.r_le
r_pos := hf.r_pos
hasSum := fun hy => (hf.hasSum hy).neg }
#align has_fpower_series_on_ball.neg HasFPowerSeriesOnBall.neg
theorem HasFPowerSeriesAt.neg (hf : HasFPowerSeriesAt f pf x) : HasFPowerSeriesAt (-f) (-pf) x :=
let ⟨_, hrf⟩ := hf
hrf.neg.hasFPowerSeriesAt
#align has_fpower_series_at.neg HasFPowerSeriesAt.neg
theorem AnalyticAt.neg (hf : AnalyticAt 𝕜 f x) : AnalyticAt 𝕜 (-f) x :=
let ⟨_, hpf⟩ := hf
hpf.neg.analyticAt
#align analytic_at.neg AnalyticAt.neg
theorem HasFPowerSeriesOnBall.sub (hf : HasFPowerSeriesOnBall f pf x r)
(hg : HasFPowerSeriesOnBall g pg x r) : HasFPowerSeriesOnBall (f - g) (pf - pg) x r := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align has_fpower_series_on_ball.sub HasFPowerSeriesOnBall.sub
theorem HasFPowerSeriesAt.sub (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x) :
HasFPowerSeriesAt (f - g) (pf - pg) x := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align has_fpower_series_at.sub HasFPowerSeriesAt.sub
theorem AnalyticAt.sub (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) :
AnalyticAt 𝕜 (f - g) x := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align analytic_at.sub AnalyticAt.sub
theorem AnalyticOn.mono {s t : Set E} (hf : AnalyticOn 𝕜 f t) (hst : s ⊆ t) : AnalyticOn 𝕜 f s :=
fun z hz => hf z (hst hz)
#align analytic_on.mono AnalyticOn.mono
theorem AnalyticOn.congr' {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : f =ᶠ[𝓝ˢ s] g) :
AnalyticOn 𝕜 g s :=
fun z hz => (hf z hz).congr (mem_nhdsSet_iff_forall.mp hg z hz)
theorem analyticOn_congr' {s : Set E} (h : f =ᶠ[𝓝ˢ s] g) : AnalyticOn 𝕜 f s ↔ AnalyticOn 𝕜 g s :=
⟨fun hf => hf.congr' h, fun hg => hg.congr' h.symm⟩
theorem AnalyticOn.congr {s : Set E} (hs : IsOpen s) (hf : AnalyticOn 𝕜 f s) (hg : s.EqOn f g) :
AnalyticOn 𝕜 g s :=
hf.congr' $ mem_nhdsSet_iff_forall.mpr
(fun _ hz => eventuallyEq_iff_exists_mem.mpr ⟨s, hs.mem_nhds hz, hg⟩)
theorem analyticOn_congr {s : Set E} (hs : IsOpen s) (h : s.EqOn f g) : AnalyticOn 𝕜 f s ↔
AnalyticOn 𝕜 g s := ⟨fun hf => hf.congr hs h, fun hg => hg.congr hs h.symm⟩
theorem AnalyticOn.add {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
AnalyticOn 𝕜 (f + g) s :=
fun z hz => (hf z hz).add (hg z hz)
#align analytic_on.add AnalyticOn.add
theorem AnalyticOn.sub {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
AnalyticOn 𝕜 (f - g) s :=
fun z hz => (hf z hz).sub (hg z hz)
#align analytic_on.sub AnalyticOn.sub
theorem HasFPowerSeriesOnBall.coeff_zero (hf : HasFPowerSeriesOnBall f pf x r) (v : Fin 0 → E) :
pf 0 v = f x := by
have v_eq : v = fun i => 0 := Subsingleton.elim _ _
have zero_mem : (0 : E) ∈ EMetric.ball (0 : E) r := by simp [hf.r_pos]
have : ∀ i, i ≠ 0 → (pf i fun j => 0) = 0 := by
intro i hi
have : 0 < i := pos_iff_ne_zero.2 hi
exact ContinuousMultilinearMap.map_coord_zero _ (⟨0, this⟩ : Fin i) rfl
have A := (hf.hasSum zero_mem).unique (hasSum_single _ this)
simpa [v_eq] using A.symm
#align has_fpower_series_on_ball.coeff_zero HasFPowerSeriesOnBall.coeff_zero
theorem HasFPowerSeriesAt.coeff_zero (hf : HasFPowerSeriesAt f pf x) (v : Fin 0 → E) :
pf 0 v = f x :=
let ⟨_, hrf⟩ := hf
hrf.coeff_zero v
#align has_fpower_series_at.coeff_zero HasFPowerSeriesAt.coeff_zero
/-- If a function `f` has a power series `p` on a ball and `g` is linear, then `g ∘ f` has the
power series `g ∘ p` on the same ball. -/
theorem ContinuousLinearMap.comp_hasFPowerSeriesOnBall (g : F →L[𝕜] G)
(h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (g ∘ f) (g.compFormalMultilinearSeries p) x r :=
{ r_le := h.r_le.trans (p.radius_le_radius_continuousLinearMap_comp _)
r_pos := h.r_pos
hasSum := fun hy => by
simpa only [ContinuousLinearMap.compFormalMultilinearSeries_apply,
ContinuousLinearMap.compContinuousMultilinearMap_coe, Function.comp_apply] using
g.hasSum (h.hasSum hy) }
#align continuous_linear_map.comp_has_fpower_series_on_ball ContinuousLinearMap.comp_hasFPowerSeriesOnBall
/-- If a function `f` is analytic on a set `s` and `g` is linear, then `g ∘ f` is analytic
on `s`. -/
theorem ContinuousLinearMap.comp_analyticOn {s : Set E} (g : F →L[𝕜] G) (h : AnalyticOn 𝕜 f s) :
AnalyticOn 𝕜 (g ∘ f) s := by
rintro x hx
rcases h x hx with ⟨p, r, hp⟩
exact ⟨g.compFormalMultilinearSeries p, r, g.comp_hasFPowerSeriesOnBall hp⟩
#align continuous_linear_map.comp_analytic_on ContinuousLinearMap.comp_analyticOn
/-- If a function admits a power series expansion, then it is exponentially close to the partial
sums of this power series on strict subdisks of the disk of convergence.
This version provides an upper estimate that decreases both in `‖y‖` and `n`. See also
`HasFPowerSeriesOnBall.uniform_geometric_approx` for a weaker version. -/
theorem HasFPowerSeriesOnBall.uniform_geometric_approx' {r' : ℝ≥0}
(hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n := by
obtain ⟨a, ha, C, hC, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n, ‖p n‖ * (r' : ℝ) ^ n ≤ C * a ^ n :=
p.norm_mul_pow_le_mul_pow_of_lt_radius (h.trans_le hf.r_le)
refine' ⟨a, ha, C / (1 - a), div_pos hC (sub_pos.2 ha.2), fun y hy n => _⟩
have yr' : ‖y‖ < r' := by
rw [ball_zero_eq] at hy
exact hy
have hr'0 : 0 < (r' : ℝ) := (norm_nonneg _).trans_lt yr'
have : y ∈ EMetric.ball (0 : E) r := by
refine' mem_emetric_ball_zero_iff.2 (lt_trans _ h)
exact mod_cast yr'
rw [norm_sub_rev, ← mul_div_right_comm]
have ya : a * (‖y‖ / ↑r') ≤ a :=
mul_le_of_le_one_right ha.1.le (div_le_one_of_le yr'.le r'.coe_nonneg)
suffices ‖p.partialSum n y - f (x + y)‖ ≤ C * (a * (‖y‖ / r')) ^ n / (1 - a * (‖y‖ / r')) by
refine' this.trans _
have : 0 < a := ha.1
gcongr
apply_rules [sub_pos.2, ha.2]
apply norm_sub_le_of_geometric_bound_of_hasSum (ya.trans_lt ha.2) _ (hf.hasSum this)
intro n
calc
‖(p n) fun _ : Fin n => y‖
_ ≤ ‖p n‖ * ∏ _i : Fin n, ‖y‖ := ContinuousMultilinearMap.le_op_norm _ _
_ = ‖p n‖ * (r' : ℝ) ^ n * (‖y‖ / r') ^ n := by field_simp [mul_right_comm]
_ ≤ C * a ^ n * (‖y‖ / r') ^ n := by gcongr ?_ * _; apply hp
_ ≤ C * (a * (‖y‖ / r')) ^ n := by rw [mul_pow, mul_assoc]
#align has_fpower_series_on_ball.uniform_geometric_approx' HasFPowerSeriesOnBall.uniform_geometric_approx'
/-- If a function admits a power series expansion, then it is exponentially close to the partial
sums of this power series on strict subdisks of the disk of convergence. -/
theorem HasFPowerSeriesOnBall.uniform_geometric_approx {r' : ℝ≥0}
(hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
∃ a ∈ Ioo (0 : ℝ) 1,
∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n, ‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n := by
obtain ⟨a, ha, C, hC, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n :=
hf.uniform_geometric_approx' h
refine' ⟨a, ha, C, hC, fun y hy n => (hp y hy n).trans _⟩
have yr' : ‖y‖ < r' := by rwa [ball_zero_eq] at hy
gcongr
exacts [mul_nonneg ha.1.le (div_nonneg (norm_nonneg y) r'.coe_nonneg),
mul_le_of_le_one_right ha.1.le (div_le_one_of_le yr'.le r'.coe_nonneg)]
#align has_fpower_series_on_ball.uniform_geometric_approx HasFPowerSeriesOnBall.uniform_geometric_approx
/-- Taylor formula for an analytic function, `IsBigO` version. -/
theorem HasFPowerSeriesAt.isBigO_sub_partialSum_pow (hf : HasFPowerSeriesAt f p x) (n : ℕ) :
(fun y : E => f (x + y) - p.partialSum n y) =O[𝓝 0] fun y => ‖y‖ ^ n := by
rcases hf with ⟨r, hf⟩
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hf.r_pos with ⟨r', r'0, h⟩
obtain ⟨a, -, C, -, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n :=
hf.uniform_geometric_approx' h
refine' isBigO_iff.2 ⟨C * (a / r') ^ n, _⟩
replace r'0 : 0 < (r' : ℝ); · exact mod_cast r'0
filter_upwards [Metric.ball_mem_nhds (0 : E) r'0] with y hy
simpa [mul_pow, mul_div_assoc, mul_assoc, div_mul_eq_mul_div] using hp y hy n
set_option linter.uppercaseLean3 false in
#align has_fpower_series_at.is_O_sub_partial_sum_pow HasFPowerSeriesAt.isBigO_sub_partialSum_pow
/-- If `f` has formal power series `∑ n, pₙ` on a ball of radius `r`, then for `y, z` in any smaller
ball, the norm of the difference `f y - f z - p 1 (fun _ ↦ y - z)` is bounded above by
`C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖`. This lemma formulates this property using `IsBigO` and
`Filter.principal` on `E × E`. -/
theorem HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal
(hf : HasFPowerSeriesOnBall f p x r) (hr : r' < r) :
(fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) =O[𝓟 (EMetric.ball (x, x) r')]
fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖ := by
lift r' to ℝ≥0 using ne_top_of_lt hr
rcases (zero_le r').eq_or_lt with (rfl | hr'0)
· simp only [isBigO_bot, EMetric.ball_zero, principal_empty, ENNReal.coe_zero]
obtain ⟨a, ha, C, hC : 0 < C, hp⟩ :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n : ℕ, ‖p n‖ * (r' : ℝ) ^ n ≤ C * a ^ n
exact p.norm_mul_pow_le_mul_pow_of_lt_radius (hr.trans_le hf.r_le)
simp only [← le_div_iff (pow_pos (NNReal.coe_pos.2 hr'0) _)] at hp
set L : E × E → ℝ := fun y =>
C * (a / r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * (a / (1 - a) ^ 2 + 2 / (1 - a))
have hL : ∀ y ∈ EMetric.ball (x, x) r', ‖f y.1 - f y.2 - p 1 fun _ => y.1 - y.2‖ ≤ L y := by
intro y hy'
have hy : y ∈ EMetric.ball x r ×ˢ EMetric.ball x r := by
rw [EMetric.ball_prod_same]
exact EMetric.ball_subset_ball hr.le hy'
set A : ℕ → F := fun n => (p n fun _ => y.1 - x) - p n fun _ => y.2 - x
have hA : HasSum (fun n => A (n + 2)) (f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) := by
convert (hasSum_nat_add_iff' 2).2 ((hf.hasSum_sub hy.1).sub (hf.hasSum_sub hy.2)) using 1
rw [Finset.sum_range_succ, Finset.sum_range_one, hf.coeff_zero, hf.coeff_zero, sub_self,
zero_add, ← Subsingleton.pi_single_eq (0 : Fin 1) (y.1 - x), Pi.single,
← Subsingleton.pi_single_eq (0 : Fin 1) (y.2 - x), Pi.single, ← (p 1).map_sub, ← Pi.single,
Subsingleton.pi_single_eq, sub_sub_sub_cancel_right]
rw [EMetric.mem_ball, edist_eq_coe_nnnorm_sub, ENNReal.coe_lt_coe] at hy'
set B : ℕ → ℝ := fun n => C * (a / r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * ((n + 2) * a ^ n)
have hAB : ∀ n, ‖A (n + 2)‖ ≤ B n := fun n =>
calc
‖A (n + 2)‖ ≤ ‖p (n + 2)‖ * ↑(n + 2) * ‖y - (x, x)‖ ^ (n + 1) * ‖y.1 - y.2‖ := by
-- porting note: `pi_norm_const` was `pi_norm_const (_ : E)`
simpa only [Fintype.card_fin, pi_norm_const, Prod.norm_def, Pi.sub_def,
Prod.fst_sub, Prod.snd_sub, sub_sub_sub_cancel_right] using
(p <| n + 2).norm_image_sub_le (fun _ => y.1 - x) fun _ => y.2 - x
_ = ‖p (n + 2)‖ * ‖y - (x, x)‖ ^ n * (↑(n + 2) * ‖y - (x, x)‖ * ‖y.1 - y.2‖) := by
rw [pow_succ ‖y - (x, x)‖]
ring
-- porting note: the two `↑` in `↑r'` are new, without them, Lean fails to synthesize
-- instances `HDiv ℝ ℝ≥0 ?m` or `HMul ℝ ℝ≥0 ?m`
_ ≤ C * a ^ (n + 2) / ↑r' ^ (n + 2)
* ↑r' ^ n * (↑(n + 2) * ‖y - (x, x)‖ * ‖y.1 - y.2‖) := by
have : 0 < a := ha.1
gcongr
· apply hp
· apply hy'.le
_ = B n := by
-- porting note: in the original, `B` was in the `field_simp`, but now Lean does not
-- accept it. The current proof works in Lean 4, but does not in Lean 3.
field_simp [pow_succ]
simp only [mul_assoc, mul_comm, mul_left_comm]
have hBL : HasSum B (L y) := by
apply HasSum.mul_left
simp only [add_mul]
have : ‖a‖ < 1 := by simp only [Real.norm_eq_abs, abs_of_pos ha.1, ha.2]
rw [div_eq_mul_inv, div_eq_mul_inv]
exact (hasSum_coe_mul_geometric_of_norm_lt_1 this).add -- porting note: was `convert`!
((hasSum_geometric_of_norm_lt_1 this).mul_left 2)
exact hA.norm_le_of_bounded hBL hAB
suffices L =O[𝓟 (EMetric.ball (x, x) r')] fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖ by
refine' (IsBigO.of_bound 1 (eventually_principal.2 fun y hy => _)).trans this
rw [one_mul]
exact (hL y hy).trans (le_abs_self _)
simp_rw [mul_right_comm _ (_ * _)] -- porting note: there was an `L` inside the `simp_rw`.
exact (isBigO_refl _ _).const_mul_left _
set_option linter.uppercaseLean3 false in
#align has_fpower_series_on_ball.is_O_image_sub_image_sub_deriv_principal HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal
/-- If `f` has formal power series `∑ n, pₙ` on a ball of radius `r`, then for `y, z` in any smaller
ball, the norm of the difference `f y - f z - p 1 (fun _ ↦ y - z)` is bounded above by
`C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖`. -/
theorem HasFPowerSeriesOnBall.image_sub_sub_deriv_le (hf : HasFPowerSeriesOnBall f p x r)
(hr : r' < r) :
∃ C, ∀ᵉ (y ∈ EMetric.ball x r') (z ∈ EMetric.ball x r'),
‖f y - f z - p 1 fun _ => y - z‖ ≤ C * max ‖y - x‖ ‖z - x‖ * ‖y - z‖ := by
simpa only [isBigO_principal, mul_assoc, norm_mul, norm_norm, Prod.forall, EMetric.mem_ball,
Prod.edist_eq, max_lt_iff, and_imp, @forall_swap (_ < _) E] using
hf.isBigO_image_sub_image_sub_deriv_principal hr
#align has_fpower_series_on_ball.image_sub_sub_deriv_le HasFPowerSeriesOnBall.image_sub_sub_deriv_le
/-- If `f` has formal power series `∑ n, pₙ` at `x`, then
`f y - f z - p 1 (fun _ ↦ y - z) = O(‖(y, z) - (x, x)‖ * ‖y - z‖)` as `(y, z) → (x, x)`.
In particular, `f` is strictly differentiable at `x`. -/
theorem HasFPowerSeriesAt.isBigO_image_sub_norm_mul_norm_sub (hf : HasFPowerSeriesAt f p x) :
(fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) =O[𝓝 (x, x)] fun y =>
‖y - (x, x)‖ * ‖y.1 - y.2‖ := by
rcases hf with ⟨r, hf⟩
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hf.r_pos with ⟨r', r'0, h⟩
refine' (hf.isBigO_image_sub_image_sub_deriv_principal h).mono _
exact le_principal_iff.2 (EMetric.ball_mem_nhds _ r'0)
set_option linter.uppercaseLean3 false in
#align has_fpower_series_at.is_O_image_sub_norm_mul_norm_sub HasFPowerSeriesAt.isBigO_image_sub_norm_mul_norm_sub
/-- If a function admits a power series expansion at `x`, then it is the uniform limit of the
partial sums of this power series on strict subdisks of the disk of convergence, i.e., `f (x + y)`
is the uniform limit of `p.partialSum n y` there. -/
theorem HasFPowerSeriesOnBall.tendstoUniformlyOn {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r)
(h : (r' : ℝ≥0∞) < r) :
TendstoUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
(Metric.ball (0 : E) r') := by
obtain ⟨a, ha, C, -, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n
exact hf.uniform_geometric_approx h
refine' Metric.tendstoUniformlyOn_iff.2 fun ε εpos => _
have L : Tendsto (fun n => (C : ℝ) * a ^ n) atTop (𝓝 ((C : ℝ) * 0)) :=
tendsto_const_nhds.mul (tendsto_pow_atTop_nhds_0_of_lt_1 ha.1.le ha.2)
rw [mul_zero] at L
refine' (L.eventually (gt_mem_nhds εpos)).mono fun n hn y hy => _
rw [dist_eq_norm]
exact (hp y hy n).trans_lt hn
#align has_fpower_series_on_ball.tendsto_uniformly_on HasFPowerSeriesOnBall.tendstoUniformlyOn
/-- If a function admits a power series expansion at `x`, then it is the locally uniform limit of
the partial sums of this power series on the disk of convergence, i.e., `f (x + y)`
is the locally uniform limit of `p.partialSum n y` there. -/
theorem HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn (hf : HasFPowerSeriesOnBall f p x r) :
TendstoLocallyUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
(EMetric.ball (0 : E) r) := by
intro u hu x hx
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hx with ⟨r', xr', hr'⟩
have : EMetric.ball (0 : E) r' ∈ 𝓝 x := IsOpen.mem_nhds EMetric.isOpen_ball xr'
refine' ⟨EMetric.ball (0 : E) r', mem_nhdsWithin_of_mem_nhds this, _⟩
simpa [Metric.emetric_ball_nnreal] using hf.tendstoUniformlyOn hr' u hu
#align has_fpower_series_on_ball.tendsto_locally_uniformly_on HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn
/-- If a function admits a power series expansion at `x`, then it is the uniform limit of the
partial sums of this power series on strict subdisks of the disk of convergence, i.e., `f y`
is the uniform limit of `p.partialSum n (y - x)` there. -/
theorem HasFPowerSeriesOnBall.tendstoUniformlyOn' {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r)
(h : (r' : ℝ≥0∞) < r) :
TendstoUniformlyOn (fun n y => p.partialSum n (y - x)) f atTop (Metric.ball (x : E) r') := by
convert (hf.tendstoUniformlyOn h).comp fun y => y - x using 1
· simp [(· ∘ ·)]
· ext z
simp [dist_eq_norm]
#align has_fpower_series_on_ball.tendsto_uniformly_on' HasFPowerSeriesOnBall.tendstoUniformlyOn'
/-- If a function admits a power series expansion at `x`, then it is the locally uniform limit of
the partial sums of this power series on the disk of convergence, i.e., `f y`
is the locally uniform limit of `p.partialSum n (y - x)` there. -/
theorem HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn' (hf : HasFPowerSeriesOnBall f p x r) :
TendstoLocallyUniformlyOn (fun n y => p.partialSum n (y - x)) f atTop
(EMetric.ball (x : E) r) := by
have A : ContinuousOn (fun y : E => y - x) (EMetric.ball (x : E) r) :=
(continuous_id.sub continuous_const).continuousOn
convert hf.tendstoLocallyUniformlyOn.comp (fun y : E => y - x) _ A using 1
· ext z
simp
· intro z
simp [edist_eq_coe_nnnorm, edist_eq_coe_nnnorm_sub]
#align has_fpower_series_on_ball.tendsto_locally_uniformly_on' HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn'
/-- If a function admits a power series expansion on a disk, then it is continuous there. -/
protected theorem HasFPowerSeriesOnBall.continuousOn (hf : HasFPowerSeriesOnBall f p x r) :
ContinuousOn f (EMetric.ball x r) :=
hf.tendstoLocallyUniformlyOn'.continuousOn <|
eventually_of_forall fun n =>
((p.partialSum_continuous n).comp (continuous_id.sub continuous_const)).continuousOn
#align has_fpower_series_on_ball.continuous_on HasFPowerSeriesOnBall.continuousOn
protected theorem HasFPowerSeriesAt.continuousAt (hf : HasFPowerSeriesAt f p x) :
ContinuousAt f x :=
let ⟨_, hr⟩ := hf
hr.continuousOn.continuousAt (EMetric.ball_mem_nhds x hr.r_pos)
#align has_fpower_series_at.continuous_at HasFPowerSeriesAt.continuousAt
protected theorem AnalyticAt.continuousAt (hf : AnalyticAt 𝕜 f x) : ContinuousAt f x :=
let ⟨_, hp⟩ := hf
hp.continuousAt
#align analytic_at.continuous_at AnalyticAt.continuousAt
protected theorem AnalyticOn.continuousOn {s : Set E} (hf : AnalyticOn 𝕜 f s) : ContinuousOn f s :=
fun x hx => (hf x hx).continuousAt.continuousWithinAt
#align analytic_on.continuous_on AnalyticOn.continuousOn
/-- Analytic everywhere implies continuous -/
theorem AnalyticOn.continuous {f : E → F} (fa : AnalyticOn 𝕜 f univ) : Continuous f := by
rw [continuous_iff_continuousOn_univ]; exact fa.continuousOn
/-- In a complete space, the sum of a converging power series `p` admits `p` as a power series.
This is not totally obvious as we need to check the convergence of the series. -/
protected theorem FormalMultilinearSeries.hasFPowerSeriesOnBall [CompleteSpace F]
(p : FormalMultilinearSeries 𝕜 E F) (h : 0 < p.radius) :
HasFPowerSeriesOnBall p.sum p 0 p.radius :=
{ r_le := le_rfl
r_pos := h
hasSum := fun hy => by
rw [zero_add]
exact p.hasSum hy }
#align formal_multilinear_series.has_fpower_series_on_ball FormalMultilinearSeries.hasFPowerSeriesOnBall
theorem HasFPowerSeriesOnBall.sum (h : HasFPowerSeriesOnBall f p x r) {y : E}
(hy : y ∈ EMetric.ball (0 : E) r) : f (x + y) = p.sum y :=
(h.hasSum hy).tsum_eq.symm
#align has_fpower_series_on_ball.sum HasFPowerSeriesOnBall.sum
/-- The sum of a converging power series is continuous in its disk of convergence. -/
protected theorem FormalMultilinearSeries.continuousOn [CompleteSpace F] :
ContinuousOn p.sum (EMetric.ball 0 p.radius) := by
rcases (zero_le p.radius).eq_or_lt with h | h
· simp [← h, continuousOn_empty]
· exact (p.hasFPowerSeriesOnBall h).continuousOn
#align formal_multilinear_series.continuous_on FormalMultilinearSeries.continuousOn
end
/-!
### Uniqueness of power series
If a function `f : E → F` has two representations as power series at a point `x : E`, corresponding
to formal multilinear series `p₁` and `p₂`, then these representations agree term-by-term. That is,
for any `n : ℕ` and `y : E`, `p₁ n (fun i ↦ y) = p₂ n (fun i ↦ y)`. In the one-dimensional case,
when `f : 𝕜 → E`, the continuous multilinear maps `p₁ n` and `p₂ n` are given by
`ContinuousMultilinearMap.mkPiField`, and hence are determined completely by the value of
`p₁ n (fun i ↦ 1)`, so `p₁ = p₂`. Consequently, the radius of convergence for one series can be
transferred to the other.
-/
section Uniqueness
open ContinuousMultilinearMap
theorem Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero {n : ℕ} {p : E[×n]→L[𝕜] F}
(h : (fun y => p fun _ => y) =O[𝓝 0] fun y => ‖y‖ ^ (n + 1)) (y : E) : (p fun _ => y) = 0 := by
obtain ⟨c, c_pos, hc⟩ := h.exists_pos
obtain ⟨t, ht, t_open, z_mem⟩ := eventually_nhds_iff.mp (isBigOWith_iff.mp hc)
obtain ⟨δ, δ_pos, δε⟩ := (Metric.isOpen_iff.mp t_open) 0 z_mem
clear h hc z_mem
cases' n with n
· exact norm_eq_zero.mp (by
-- porting note: the symmetric difference of the `simpa only` sets:
-- added `Nat.zero_eq, zero_add, pow_one`
-- removed `zero_pow', Ne.def, Nat.one_ne_zero, not_false_iff`
simpa only [Nat.zero_eq, fin0_apply_norm, norm_eq_zero, norm_zero, zero_add, pow_one,
mul_zero, norm_le_zero_iff] using ht 0 (δε (Metric.mem_ball_self δ_pos)))
· refine' Or.elim (Classical.em (y = 0))
(fun hy => by simpa only [hy] using p.map_zero) fun hy => _
replace hy := norm_pos_iff.mpr hy
refine' norm_eq_zero.mp (le_antisymm (le_of_forall_pos_le_add fun ε ε_pos => _) (norm_nonneg _))
have h₀ := _root_.mul_pos c_pos (pow_pos hy (n.succ + 1))
obtain ⟨k, k_pos, k_norm⟩ := NormedField.exists_norm_lt 𝕜
(lt_min (mul_pos δ_pos (inv_pos.mpr hy)) (mul_pos ε_pos (inv_pos.mpr h₀)))
have h₁ : ‖k • y‖ < δ := by
rw [norm_smul]
exact inv_mul_cancel_right₀ hy.ne.symm δ ▸
mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_left _ _)) hy
have h₂ :=
calc
‖p fun _ => k • y‖ ≤ c * ‖k • y‖ ^ (n.succ + 1) := by
-- porting note: now Lean wants `_root_.`
simpa only [norm_pow, _root_.norm_norm] using ht (k • y) (δε (mem_ball_zero_iff.mpr h₁))
--simpa only [norm_pow, norm_norm] using ht (k • y) (δε (mem_ball_zero_iff.mpr h₁))
_ = ‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))) := by
-- porting note: added `Nat.succ_eq_add_one` since otherwise `ring` does not conclude.
simp only [norm_smul, mul_pow, Nat.succ_eq_add_one]
-- porting note: removed `rw [pow_succ]`, since it now becomes superfluous.
ring
have h₃ : ‖k‖ * (c * ‖y‖ ^ (n.succ + 1)) < ε :=
inv_mul_cancel_right₀ h₀.ne.symm ε ▸
mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_right _ _)) h₀
calc
‖p fun _ => y‖ = ‖k⁻¹ ^ n.succ‖ * ‖p fun _ => k • y‖ := by
simpa only [inv_smul_smul₀ (norm_pos_iff.mp k_pos), norm_smul, Finset.prod_const,
Finset.card_fin] using
congr_arg norm (p.map_smul_univ (fun _ : Fin n.succ => k⁻¹) fun _ : Fin n.succ => k • y)
_ ≤ ‖k⁻¹ ^ n.succ‖ * (‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1)))) := by gcongr
_ = ‖(k⁻¹ * k) ^ n.succ‖ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))) := by
rw [← mul_assoc]
simp [norm_mul, mul_pow]
_ ≤ 0 + ε := by
rw [inv_mul_cancel (norm_pos_iff.mp k_pos)]
simpa using h₃.le
set_option linter.uppercaseLean3 false in
#align asymptotics.is_O.continuous_multilinear_map_apply_eq_zero Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero
/-- If a formal multilinear series `p` represents the zero function at `x : E`, then the
terms `p n (fun i ↦ y)` appearing in the sum are zero for any `n : ℕ`, `y : E`. -/
theorem HasFPowerSeriesAt.apply_eq_zero {p : FormalMultilinearSeries 𝕜 E F} {x : E}
(h : HasFPowerSeriesAt 0 p x) (n : ℕ) : ∀ y : E, (p n fun _ => y) = 0 := by
refine' Nat.strong_induction_on n fun k hk => _
have psum_eq : p.partialSum (k + 1) = fun y => p k fun _ => y := by
funext z
refine' Finset.sum_eq_single _ (fun b hb hnb => _) fun hn => _
· have := Finset.mem_range_succ_iff.mp hb
simp only [hk b (this.lt_of_ne hnb), Pi.zero_apply]
· exact False.elim (hn (Finset.mem_range.mpr (lt_add_one k)))
replace h := h.isBigO_sub_partialSum_pow k.succ
simp only [psum_eq, zero_sub, Pi.zero_apply, Asymptotics.isBigO_neg_left] at h
exact h.continuousMultilinearMap_apply_eq_zero
#align has_fpower_series_at.apply_eq_zero HasFPowerSeriesAt.apply_eq_zero
/-- A one-dimensional formal multilinear series representing the zero function is zero. -/
theorem HasFPowerSeriesAt.eq_zero {p : FormalMultilinearSeries 𝕜 𝕜 E} {x : 𝕜}
(h : HasFPowerSeriesAt 0 p x) : p = 0 := by
-- porting note: `funext; ext` was `ext (n x)`
funext n
ext x
rw [← mkPiField_apply_one_eq_self (p n)]
-- porting note: nasty hack, was `simp [h.apply_eq_zero n 1]`
have := Or.intro_right ?_ (h.apply_eq_zero n 1)
simpa using this
#align has_fpower_series_at.eq_zero HasFPowerSeriesAt.eq_zero
/-- One-dimensional formal multilinear series representing the same function are equal. -/
theorem HasFPowerSeriesAt.eq_formalMultilinearSeries {p₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E}
{f : 𝕜 → E} {x : 𝕜} (h₁ : HasFPowerSeriesAt f p₁ x) (h₂ : HasFPowerSeriesAt f p₂ x) : p₁ = p₂ :=
sub_eq_zero.mp (HasFPowerSeriesAt.eq_zero (by simpa only [sub_self] using h₁.sub h₂))
#align has_fpower_series_at.eq_formal_multilinear_series HasFPowerSeriesAt.eq_formalMultilinearSeries
theorem HasFPowerSeriesAt.eq_formalMultilinearSeries_of_eventually
{p q : FormalMultilinearSeries 𝕜 𝕜 E} {f g : 𝕜 → E} {x : 𝕜} (hp : HasFPowerSeriesAt f p x)
(hq : HasFPowerSeriesAt g q x) (heq : ∀ᶠ z in 𝓝 x, f z = g z) : p = q :=
(hp.congr heq).eq_formalMultilinearSeries hq
#align has_fpower_series_at.eq_formal_multilinear_series_of_eventually HasFPowerSeriesAt.eq_formalMultilinearSeries_of_eventually
/-- A one-dimensional formal multilinear series representing a locally zero function is zero. -/
theorem HasFPowerSeriesAt.eq_zero_of_eventually {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E}
{x : 𝕜} (hp : HasFPowerSeriesAt f p x) (hf : f =ᶠ[𝓝 x] 0) : p = 0 :=
(hp.congr hf).eq_zero
#align has_fpower_series_at.eq_zero_of_eventually HasFPowerSeriesAt.eq_zero_of_eventually
/-- If a function `f : 𝕜 → E` has two power series representations at `x`, then the given radii in
which convergence is guaranteed may be interchanged. This can be useful when the formal multilinear
series in one representation has a particularly nice form, but the other has a larger radius. -/
theorem HasFPowerSeriesOnBall.exchange_radius {p₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E}
{r₁ r₂ : ℝ≥0∞} {x : 𝕜} (h₁ : HasFPowerSeriesOnBall f p₁ x r₁)
(h₂ : HasFPowerSeriesOnBall f p₂ x r₂) : HasFPowerSeriesOnBall f p₁ x r₂ :=
h₂.hasFPowerSeriesAt.eq_formalMultilinearSeries h₁.hasFPowerSeriesAt ▸ h₂
#align has_fpower_series_on_ball.exchange_radius HasFPowerSeriesOnBall.exchange_radius
/-- If a function `f : 𝕜 → E` has power series representation `p` on a ball of some radius and for
each positive radius it has some power series representation, then `p` converges to `f` on the whole
`𝕜`. -/
theorem HasFPowerSeriesOnBall.r_eq_top_of_exists {f : 𝕜 → E} {r : ℝ≥0∞} {x : 𝕜}
{p : FormalMultilinearSeries 𝕜 𝕜 E} (h : HasFPowerSeriesOnBall f p x r)
(h' : ∀ (r' : ℝ≥0) (_ : 0 < r'), ∃ p' : FormalMultilinearSeries 𝕜 𝕜 E,
HasFPowerSeriesOnBall f p' x r') :
HasFPowerSeriesOnBall f p x ∞ :=
{ r_le := ENNReal.le_of_forall_pos_nnreal_lt fun r hr _ =>
let ⟨_, hp'⟩ := h' r hr
(h.exchange_radius hp').r_le
r_pos := ENNReal.coe_lt_top
hasSum := fun {y} _ =>
let ⟨r', hr'⟩ := exists_gt ‖y‖₊
let ⟨_, hp'⟩ := h' r' hr'.ne_bot.bot_lt
(h.exchange_radius hp').hasSum <| mem_emetric_ball_zero_iff.mpr (ENNReal.coe_lt_coe.2 hr') }
#align has_fpower_series_on_ball.r_eq_top_of_exists HasFPowerSeriesOnBall.r_eq_top_of_exists
end Uniqueness
/-!
### Changing origin in a power series
If a function is analytic in a disk `D(x, R)`, then it is analytic in any disk contained in that
one. Indeed, one can write
$$
f (x + y + z) = \sum_{n} p_n (y + z)^n = \sum_{n, k} \binom{n}{k} p_n y^{n-k} z^k
= \sum_{k} \Bigl(\sum_{n} \binom{n}{k} p_n y^{n-k}\Bigr) z^k.
$$
The corresponding power series has thus a `k`-th coefficient equal to
$\sum_{n} \binom{n}{k} p_n y^{n-k}$. In the general case where `pₙ` is a multilinear map, this has
to be interpreted suitably: instead of having a binomial coefficient, one should sum over all
possible subsets `s` of `Fin n` of cardinal `k`, and attribute `z` to the indices in `s` and
`y` to the indices outside of `s`.
In this paragraph, we implement this. The new power series is called `p.changeOrigin y`. Then, we
check its convergence and the fact that its sum coincides with the original sum. The outcome of this
discussion is that the set of points where a function is analytic is open.
-/
namespace FormalMultilinearSeries
section
variable (p : FormalMultilinearSeries 𝕜 E F) {x y : E} {r R : ℝ≥0}
/-- A term of `FormalMultilinearSeries.changeOriginSeries`.
Given a formal multilinear series `p` and a point `x` in its ball of convergence,
`p.changeOrigin x` is a formal multilinear series such that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense. Each term of `p.changeOrigin x`
is itself an analytic function of `x` given by the series `p.changeOriginSeries`. Each term in
`changeOriginSeries` is the sum of `changeOriginSeriesTerm`'s over all `s` of cardinality `l`.
The definition is such that `p.changeOriginSeriesTerm k l s hs (fun _ ↦ x) (fun _ ↦ y) =
p (k + l) (s.piecewise (fun _ ↦ x) (fun _ ↦ y))`
-/
def changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
E[×l]→L[𝕜] E[×k]→L[𝕜] F := by
let a := ContinuousMultilinearMap.curryFinFinset 𝕜 E F hs
(by erw [Finset.card_compl, Fintype.card_fin, hs, add_tsub_cancel_right])
exact a (p (k + l))
#align formal_multilinear_series.change_origin_series_term FormalMultilinearSeries.changeOriginSeriesTerm
theorem changeOriginSeriesTerm_apply (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l)
(x y : E) :
(p.changeOriginSeriesTerm k l s hs (fun _ => x) fun _ => y) =
p (k + l) (s.piecewise (fun _ => x) fun _ => y) :=
ContinuousMultilinearMap.curryFinFinset_apply_const _ _ _ _ _
#align formal_multilinear_series.change_origin_series_term_apply FormalMultilinearSeries.changeOriginSeriesTerm_apply
@[simp]
theorem norm_changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
‖p.changeOriginSeriesTerm k l s hs‖ = ‖p (k + l)‖ := by
simp only [changeOriginSeriesTerm, LinearIsometryEquiv.norm_map]
#align formal_multilinear_series.norm_change_origin_series_term FormalMultilinearSeries.norm_changeOriginSeriesTerm
@[simp]
theorem nnnorm_changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
‖p.changeOriginSeriesTerm k l s hs‖₊ = ‖p (k + l)‖₊ := by
simp only [changeOriginSeriesTerm, LinearIsometryEquiv.nnnorm_map]
#align formal_multilinear_series.nnnorm_change_origin_series_term FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm
theorem nnnorm_changeOriginSeriesTerm_apply_le (k l : ℕ) (s : Finset (Fin (k + l)))
(hs : s.card = l) (x y : E) :
‖p.changeOriginSeriesTerm k l s hs (fun _ => x) fun _ => y‖₊ ≤
‖p (k + l)‖₊ * ‖x‖₊ ^ l * ‖y‖₊ ^ k := by
rw [← p.nnnorm_changeOriginSeriesTerm k l s hs, ← Fin.prod_const, ← Fin.prod_const]
apply ContinuousMultilinearMap.le_of_op_nnnorm_le
apply ContinuousMultilinearMap.le_op_nnnorm
#align formal_multilinear_series.nnnorm_change_origin_series_term_apply_le FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm_apply_le
/-- The power series for `f.changeOrigin k`.
Given a formal multilinear series `p` and a point `x` in its ball of convergence,
`p.changeOrigin x` is a formal multilinear series such that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense. Its `k`-th term is the sum of
the series `p.changeOriginSeries k`. -/
def changeOriginSeries (k : ℕ) : FormalMultilinearSeries 𝕜 E (E[×k]→L[𝕜] F) := fun l =>
∑ s : { s : Finset (Fin (k + l)) // Finset.card s = l }, p.changeOriginSeriesTerm k l s s.2
#align formal_multilinear_series.change_origin_series FormalMultilinearSeries.changeOriginSeries
theorem nnnorm_changeOriginSeries_le_tsum (k l : ℕ) :
‖p.changeOriginSeries k l‖₊ ≤
∑' _ : { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + l)‖₊ :=
(nnnorm_sum_le _ (fun t => changeOriginSeriesTerm p k l (Subtype.val t) t.prop)).trans_eq <| by
simp_rw [tsum_fintype, nnnorm_changeOriginSeriesTerm (p := p) (k := k) (l := l)]
#align formal_multilinear_series.nnnorm_change_origin_series_le_tsum FormalMultilinearSeries.nnnorm_changeOriginSeries_le_tsum
theorem nnnorm_changeOriginSeries_apply_le_tsum (k l : ℕ) (x : E) :
‖p.changeOriginSeries k l fun _ => x‖₊ ≤
∑' _ : { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + l)‖₊ * ‖x‖₊ ^ l := by
rw [NNReal.tsum_mul_right, ← Fin.prod_const]
exact (p.changeOriginSeries k l).le_of_op_nnnorm_le _ (p.nnnorm_changeOriginSeries_le_tsum _ _)
#align formal_multilinear_series.nnnorm_change_origin_series_apply_le_tsum FormalMultilinearSeries.nnnorm_changeOriginSeries_apply_le_tsum
/-- Changing the origin of a formal multilinear series `p`, so that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense.
-/
def changeOrigin (x : E) : FormalMultilinearSeries 𝕜 E F :=
fun k => (p.changeOriginSeries k).sum x
#align formal_multilinear_series.change_origin FormalMultilinearSeries.changeOrigin
/-- An auxiliary equivalence useful in the proofs about
`FormalMultilinearSeries.changeOriginSeries`: the set of triples `(k, l, s)`, where `s` is a
`Finset (Fin (k + l))` of cardinality `l` is equivalent to the set of pairs `(n, s)`, where `s` is a
`Finset (Fin n)`.
The forward map sends `(k, l, s)` to `(k + l, s)` and the inverse map sends `(n, s)` to
`(n - Finset.card s, Finset.card s, s)`. The actual definition is less readable because of problems
with non-definitional equalities. -/
@[simps]
def changeOriginIndexEquiv :
(Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) ≃ Σn : ℕ, Finset (Fin n) where
toFun s := ⟨s.1 + s.2.1, s.2.2⟩
invFun s :=
⟨s.1 - s.2.card, s.2.card,
⟨s.2.map
(Fin.castIso <| (tsub_add_cancel_of_le <| card_finset_fin_le s.2).symm).toEquiv.toEmbedding,
Finset.card_map _⟩⟩
left_inv := by
rintro ⟨k, l, ⟨s : Finset (Fin <| k + l), hs : s.card = l⟩⟩
dsimp only [Subtype.coe_mk]
-- Lean can't automatically generalize `k' = k + l - s.card`, `l' = s.card`, so we explicitly
-- formulate the generalized goal
suffices ∀ k' l', k' = k → l' = l → ∀ (hkl : k + l = k' + l') (hs'),
(⟨k', l', ⟨Finset.map (Fin.castIso hkl).toEquiv.toEmbedding s, hs'⟩⟩ :
Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) = ⟨k, l, ⟨s, hs⟩⟩ by
apply this <;> simp only [hs, add_tsub_cancel_right]
rintro _ _ rfl rfl hkl hs'
simp only [Equiv.refl_toEmbedding, Fin.castIso_refl, Finset.map_refl, eq_self_iff_true,
OrderIso.refl_toEquiv, and_self_iff, heq_iff_eq]
right_inv := by
rintro ⟨n, s⟩
simp [tsub_add_cancel_of_le (card_finset_fin_le s), Fin.castIso_to_equiv]
#align formal_multilinear_series.change_origin_index_equiv FormalMultilinearSeries.changeOriginIndexEquiv
theorem changeOriginSeries_summable_aux₁ {r r' : ℝ≥0} (hr : (r + r' : ℝ≥0∞) < p.radius) :
Summable fun s : Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l } =>
‖p (s.1 + s.2.1)‖₊ * r ^ s.2.1 * r' ^ s.1 := by
rw [← changeOriginIndexEquiv.symm.summable_iff]
dsimp only [Function.comp_def, changeOriginIndexEquiv_symm_apply_fst,
changeOriginIndexEquiv_symm_apply_snd_fst]
have : ∀ n : ℕ,
HasSum (fun s : Finset (Fin n) => ‖p (n - s.card + s.card)‖₊ * r ^ s.card * r' ^ (n - s.card))
(‖p n‖₊ * (r + r') ^ n) := by
intro n
-- TODO: why `simp only [tsub_add_cancel_of_le (card_finset_fin_le _)]` fails?
convert_to HasSum (fun s : Finset (Fin n) => ‖p n‖₊ * (r ^ s.card * r' ^ (n - s.card))) _
· ext1 s
rw [tsub_add_cancel_of_le (card_finset_fin_le _), mul_assoc]
rw [← Fin.sum_pow_mul_eq_add_pow]
exact (hasSum_fintype _).mul_left _
refine' NNReal.summable_sigma.2 ⟨fun n => (this n).summable, _⟩
simp only [(this _).tsum_eq]
exact p.summable_nnnorm_mul_pow hr
#align formal_multilinear_series.change_origin_series_summable_aux₁ FormalMultilinearSeries.changeOriginSeries_summable_aux₁
theorem changeOriginSeries_summable_aux₂ (hr : (r : ℝ≥0∞) < p.radius) (k : ℕ) :
Summable fun s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l } =>
‖p (k + s.1)‖₊ * r ^ s.1 := by
rcases ENNReal.lt_iff_exists_add_pos_lt.1 hr with ⟨r', h0, hr'⟩
simpa only [mul_inv_cancel_right₀ (pow_pos h0 _).ne'] using
((NNReal.summable_sigma.1 (p.changeOriginSeries_summable_aux₁ hr')).1 k).mul_right (r' ^ k)⁻¹
#align formal_multilinear_series.change_origin_series_summable_aux₂ FormalMultilinearSeries.changeOriginSeries_summable_aux₂
theorem changeOriginSeries_summable_aux₃ {r : ℝ≥0} (hr : ↑r < p.radius) (k : ℕ) :
Summable fun l : ℕ => ‖p.changeOriginSeries k l‖₊ * r ^ l := by
refine' NNReal.summable_of_le
(fun n => _) (NNReal.summable_sigma.1 <| p.changeOriginSeries_summable_aux₂ hr k).2
simp only [NNReal.tsum_mul_right]
exact mul_le_mul' (p.nnnorm_changeOriginSeries_le_tsum _ _) le_rfl
#align formal_multilinear_series.change_origin_series_summable_aux₃ FormalMultilinearSeries.changeOriginSeries_summable_aux₃
theorem le_changeOriginSeries_radius (k : ℕ) : p.radius ≤ (p.changeOriginSeries k).radius :=
ENNReal.le_of_forall_nnreal_lt fun _r hr =>
le_radius_of_summable_nnnorm _ (p.changeOriginSeries_summable_aux₃ hr k)
#align formal_multilinear_series.le_change_origin_series_radius FormalMultilinearSeries.le_changeOriginSeries_radius
theorem nnnorm_changeOrigin_le (k : ℕ) (h : (‖x‖₊ : ℝ≥0∞) < p.radius) :
‖p.changeOrigin x k‖₊ ≤
∑' s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + s.1)‖₊ * ‖x‖₊ ^ s.1 := by
refine' tsum_of_nnnorm_bounded _ fun l => p.nnnorm_changeOriginSeries_apply_le_tsum k l x
have := p.changeOriginSeries_summable_aux₂ h k
refine' HasSum.sigma this.hasSum fun l => _
exact ((NNReal.summable_sigma.1 this).1 l).hasSum
#align formal_multilinear_series.nnnorm_change_origin_le FormalMultilinearSeries.nnnorm_changeOrigin_le
/-- The radius of convergence of `p.changeOrigin x` is at least `p.radius - ‖x‖`. In other words,
`p.changeOrigin x` is well defined on the largest ball contained in the original ball of
convergence. -/
theorem changeOrigin_radius : p.radius - ‖x‖₊ ≤ (p.changeOrigin x).radius := by
refine' ENNReal.le_of_forall_pos_nnreal_lt fun r _h0 hr => _
rw [lt_tsub_iff_right, add_comm] at hr
have hr' : (‖x‖₊ : ℝ≥0∞) < p.radius := (le_add_right le_rfl).trans_lt hr
apply le_radius_of_summable_nnnorm
have : ∀ k : ℕ,
‖p.changeOrigin x k‖₊ * r ^ k ≤
(∑' s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + s.1)‖₊ * ‖x‖₊ ^ s.1) *
r ^ k :=
fun k => mul_le_mul_right' (p.nnnorm_changeOrigin_le k hr') (r ^ k)
refine' NNReal.summable_of_le this _
simpa only [← NNReal.tsum_mul_right] using
(NNReal.summable_sigma.1 (p.changeOriginSeries_summable_aux₁ hr)).2
#align formal_multilinear_series.change_origin_radius FormalMultilinearSeries.changeOrigin_radius
end
-- From this point on, assume that the space is complete, to make sure that series that converge
-- in norm also converge in `F`.
variable [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x y : E} {r R : ℝ≥0}
theorem hasFPowerSeriesOnBall_changeOrigin (k : ℕ) (hr : 0 < p.radius) :
HasFPowerSeriesOnBall (fun x => p.changeOrigin x k) (p.changeOriginSeries k) 0 p.radius :=
have := p.le_changeOriginSeries_radius k
((p.changeOriginSeries k).hasFPowerSeriesOnBall (hr.trans_le this)).mono hr this
#align formal_multilinear_series.has_fpower_series_on_ball_change_origin FormalMultilinearSeries.hasFPowerSeriesOnBall_changeOrigin
/-- Summing the series `p.changeOrigin x` at a point `y` gives back `p (x + y)`. -/
theorem changeOrigin_eval (h : (‖x‖₊ + ‖y‖₊ : ℝ≥0∞) < p.radius) :
(p.changeOrigin x).sum y = p.sum (x + y) := by
have radius_pos : 0 < p.radius := lt_of_le_of_lt (zero_le _) h
have x_mem_ball : x ∈ EMetric.ball (0 : E) p.radius :=
mem_emetric_ball_zero_iff.2 ((le_add_right le_rfl).trans_lt h)
have y_mem_ball : y ∈ EMetric.ball (0 : E) (p.changeOrigin x).radius := by
refine' mem_emetric_ball_zero_iff.2 (lt_of_lt_of_le _ p.changeOrigin_radius)
rwa [lt_tsub_iff_right, add_comm]
have x_add_y_mem_ball : x + y ∈ EMetric.ball (0 : E) p.radius := by
refine' mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt _ h)
exact mod_cast nnnorm_add_le x y
set f : (Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) → F := fun s =>
p.changeOriginSeriesTerm s.1 s.2.1 s.2.2 s.2.2.2 (fun _ => x) fun _ => y
have hsf : Summable f := by
refine' .of_nnnorm_bounded _ (p.changeOriginSeries_summable_aux₁ h) _
rintro ⟨k, l, s, hs⟩
dsimp only [Subtype.coe_mk]
exact p.nnnorm_changeOriginSeriesTerm_apply_le _ _ _ _ _ _
have hf : HasSum f ((p.changeOrigin x).sum y) := by
refine' HasSum.sigma_of_hasSum ((p.changeOrigin x).summable y_mem_ball).hasSum (fun k => _) hsf
· dsimp only
refine' ContinuousMultilinearMap.hasSum_eval _ _
have := (p.hasFPowerSeriesOnBall_changeOrigin k radius_pos).hasSum x_mem_ball
rw [zero_add] at this
refine' HasSum.sigma_of_hasSum this (fun l => _) _
· simp only [changeOriginSeries, ContinuousMultilinearMap.sum_apply]
apply hasSum_fintype
· refine' .of_nnnorm_bounded _
(p.changeOriginSeries_summable_aux₂ (mem_emetric_ball_zero_iff.1 x_mem_ball) k) fun s => _
refine' (ContinuousMultilinearMap.le_op_nnnorm _ _).trans_eq _
simp
refine' hf.unique (changeOriginIndexEquiv.symm.hasSum_iff.1 _)
refine' HasSum.sigma_of_hasSum
(p.hasSum x_add_y_mem_ball) (fun n => _) (changeOriginIndexEquiv.symm.summable_iff.2 hsf)
erw [(p n).map_add_univ (fun _ => x) fun _ => y]
-- porting note: added explicit function
convert hasSum_fintype (fun c : Finset (Fin n) => f (changeOriginIndexEquiv.symm ⟨n, c⟩))
rename_i s _
dsimp only [changeOriginSeriesTerm, (· ∘ ·), changeOriginIndexEquiv_symm_apply_fst,
changeOriginIndexEquiv_symm_apply_snd_fst, changeOriginIndexEquiv_symm_apply_snd_snd_coe]
rw [ContinuousMultilinearMap.curryFinFinset_apply_const]
have : ∀ (m) (hm : n = m), p n (s.piecewise (fun _ => x) fun _ => y) =
p m ((s.map (Fin.castIso hm).toEquiv.toEmbedding).piecewise (fun _ => x) fun _ => y) := by
rintro m rfl
simp (config := { unfoldPartialApp := true }) [Finset.piecewise]
apply this
#align formal_multilinear_series.change_origin_eval FormalMultilinearSeries.changeOrigin_eval
/-- Power series terms are analytic as we vary the origin -/
theorem analyticAt_changeOrigin (p : FormalMultilinearSeries 𝕜 E F) (rp : p.radius > 0) (n : ℕ) :
AnalyticAt 𝕜 (fun x ↦ p.changeOrigin x n) 0 :=
(FormalMultilinearSeries.hasFPowerSeriesOnBall_changeOrigin p n rp).analyticAt
end FormalMultilinearSeries
section
variable [CompleteSpace F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x y : E} {r : ℝ≥0∞}
/-- If a function admits a power series expansion `p` on a ball `B (x, r)`, then it also admits a
power series on any subball of this ball (even with a different center), given by `p.changeOrigin`.
-/
theorem HasFPowerSeriesOnBall.changeOrigin (hf : HasFPowerSeriesOnBall f p x r)
(h : (‖y‖₊ : ℝ≥0∞) < r) : HasFPowerSeriesOnBall f (p.changeOrigin y) (x + y) (r - ‖y‖₊) :=
{ r_le := by
apply le_trans _ p.changeOrigin_radius
exact tsub_le_tsub hf.r_le le_rfl
r_pos := by simp [h]
hasSum := fun {z} hz => by
have : f (x + y + z) =
FormalMultilinearSeries.sum (FormalMultilinearSeries.changeOrigin p y) z := by
rw [mem_emetric_ball_zero_iff, lt_tsub_iff_right, add_comm] at hz
rw [p.changeOrigin_eval (hz.trans_le hf.r_le), add_assoc, hf.sum]
refine' mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt _ hz)
exact mod_cast nnnorm_add_le y z
rw [this]
apply (p.changeOrigin y).hasSum
refine' EMetric.ball_subset_ball (le_trans _ p.changeOrigin_radius) hz
exact tsub_le_tsub hf.r_le le_rfl }
#align has_fpower_series_on_ball.change_origin HasFPowerSeriesOnBall.changeOrigin
/-- If a function admits a power series expansion `p` on an open ball `B (x, r)`, then
it is analytic at every point of this ball. -/
theorem HasFPowerSeriesOnBall.analyticAt_of_mem (hf : HasFPowerSeriesOnBall f p x r)
(h : y ∈ EMetric.ball x r) : AnalyticAt 𝕜 f y := by
have : (‖y - x‖₊ : ℝ≥0∞) < r := by simpa [edist_eq_coe_nnnorm_sub] using h
have := hf.changeOrigin this
rw [add_sub_cancel'_right] at this
exact this.analyticAt
#align has_fpower_series_on_ball.analytic_at_of_mem HasFPowerSeriesOnBall.analyticAt_of_mem
theorem HasFPowerSeriesOnBall.analyticOn (hf : HasFPowerSeriesOnBall f p x r) :
AnalyticOn 𝕜 f (EMetric.ball x r) :=
fun _y hy => hf.analyticAt_of_mem hy
#align has_fpower_series_on_ball.analytic_on HasFPowerSeriesOnBall.analyticOn
variable (𝕜 f)
/-- For any function `f` from a normed vector space to a Banach space, the set of points `x` such
that `f` is analytic at `x` is open. -/
theorem isOpen_analyticAt : IsOpen { x | AnalyticAt 𝕜 f x } := by
rw [isOpen_iff_mem_nhds]
rintro x ⟨p, r, hr⟩
exact mem_of_superset (EMetric.ball_mem_nhds _ hr.r_pos) fun y hy => hr.analyticAt_of_mem hy
#align is_open_analytic_at isOpen_analyticAt
variable {𝕜}
theorem AnalyticAt.eventually_analyticAt {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∀ᶠ y in 𝓝 x, AnalyticAt 𝕜 f y :=
(isOpen_analyticAt 𝕜 f).mem_nhds h
theorem AnalyticAt.exists_mem_nhds_analyticOn {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∃ s ∈ 𝓝 x, AnalyticOn 𝕜 f s :=
h.eventually_analyticAt.exists_mem
/-- If we're analytic at a point, we're analytic in a nonempty ball -/
theorem AnalyticAt.exists_ball_analyticOn {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∃ r : ℝ, 0 < r ∧ AnalyticOn 𝕜 f (Metric.ball x r) :=
Metric.isOpen_iff.mp (isOpen_analyticAt _ _) _ h
end
section
open FormalMultilinearSeries
variable {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E} {z₀ : 𝕜}
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z)) := by
refine' ⟨fun ⟨r, _, r_pos, h⟩ =>
eventually_of_mem (EMetric.ball_mem_nhds 0 r_pos) fun _ => by simpa using h, _⟩
simp only [Metric.eventually_nhds_iff]
rintro ⟨r, r_pos, h⟩
refine' ⟨p.radius ⊓ r.toNNReal, by simp, _, _⟩
· simp only [r_pos.lt, lt_inf_iff, ENNReal.coe_pos, Real.toNNReal_pos, and_true_iff]
obtain ⟨z, z_pos, le_z⟩ := NormedField.exists_norm_lt 𝕜 r_pos.lt
|
have : (‖z‖₊ : ENNReal) ≤ p.radius := by
simp only [dist_zero_right] at h
apply FormalMultilinearSeries.le_radius_of_tendsto
convert tendsto_norm.comp (h le_z).summable.tendsto_atTop_zero
funext
simp [norm_smul, mul_comm]
|
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z)) := by
refine' ⟨fun ⟨r, _, r_pos, h⟩ =>
eventually_of_mem (EMetric.ball_mem_nhds 0 r_pos) fun _ => by simpa using h, _⟩
simp only [Metric.eventually_nhds_iff]
rintro ⟨r, r_pos, h⟩
refine' ⟨p.radius ⊓ r.toNNReal, by simp, _, _⟩
· simp only [r_pos.lt, lt_inf_iff, ENNReal.coe_pos, Real.toNNReal_pos, and_true_iff]
obtain ⟨z, z_pos, le_z⟩ := NormedField.exists_norm_lt 𝕜 r_pos.lt
|
Mathlib.Analysis.Analytic.Basic.1430_0.jQw1fRSE1vGpOll
|
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z))
|
Mathlib_Analysis_Analytic_Basic
|
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
p : FormalMultilinearSeries 𝕜 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
r : ℝ
r_pos : r > 0
h : ∀ ⦃y : 𝕜⦄, dist y 0 < r → HasSum (fun n => y ^ n • coeff p n) (f (z₀ + y))
z : 𝕜
z_pos : 0 < ‖z‖
le_z : ‖z‖ < r
⊢ ↑‖z‖₊ ≤ radius p
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Logic.Equiv.Fin
import Mathlib.Topology.Algebra.InfiniteSum.Module
#align_import analysis.analytic.basic from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
/-!
# Analytic functions
A function is analytic in one dimension around `0` if it can be written as a converging power series
`Σ pₙ zⁿ`. This definition can be extended to any dimension (even in infinite dimension) by
requiring that `pₙ` is a continuous `n`-multilinear map. In general, `pₙ` is not unique (in two
dimensions, taking `p₂ (x, y) (x', y') = x y'` or `y x'` gives the same map when applied to a
vector `(x, y) (x, y)`). A way to guarantee uniqueness is to take a symmetric `pₙ`, but this is not
always possible in nonzero characteristic (in characteristic 2, the previous example has no
symmetric representative). Therefore, we do not insist on symmetry or uniqueness in the definition,
and we only require the existence of a converging series.
The general framework is important to say that the exponential map on bounded operators on a Banach
space is analytic, as well as the inverse on invertible operators.
## Main definitions
Let `p` be a formal multilinear series from `E` to `F`, i.e., `p n` is a multilinear map on `E^n`
for `n : ℕ`.
* `p.radius`: the largest `r : ℝ≥0∞` such that `‖p n‖ * r^n` grows subexponentially.
* `p.le_radius_of_bound`, `p.le_radius_of_bound_nnreal`, `p.le_radius_of_isBigO`: if `‖p n‖ * r ^ n`
is bounded above, then `r ≤ p.radius`;
* `p.isLittleO_of_lt_radius`, `p.norm_mul_pow_le_mul_pow_of_lt_radius`,
`p.isLittleO_one_of_lt_radius`,
`p.norm_mul_pow_le_of_lt_radius`, `p.nnnorm_mul_pow_le_of_lt_radius`: if `r < p.radius`, then
`‖p n‖ * r ^ n` tends to zero exponentially;
* `p.lt_radius_of_isBigO`: if `r ≠ 0` and `‖p n‖ * r ^ n = O(a ^ n)` for some `-1 < a < 1`, then
`r < p.radius`;
* `p.partialSum n x`: the sum `∑_{i = 0}^{n-1} pᵢ xⁱ`.
* `p.sum x`: the sum `∑'_{i = 0}^{∞} pᵢ xⁱ`.
Additionally, let `f` be a function from `E` to `F`.
* `HasFPowerSeriesOnBall f p x r`: on the ball of center `x` with radius `r`,
`f (x + y) = ∑'_n pₙ yⁿ`.
* `HasFPowerSeriesAt f p x`: on some ball of center `x` with positive radius, holds
`HasFPowerSeriesOnBall f p x r`.
* `AnalyticAt 𝕜 f x`: there exists a power series `p` such that holds `HasFPowerSeriesAt f p x`.
* `AnalyticOn 𝕜 f s`: the function `f` is analytic at every point of `s`.
We develop the basic properties of these notions, notably:
* If a function admits a power series, it is continuous (see
`HasFPowerSeriesOnBall.continuousOn` and `HasFPowerSeriesAt.continuousAt` and
`AnalyticAt.continuousAt`).
* In a complete space, the sum of a formal power series with positive radius is well defined on the
disk of convergence, see `FormalMultilinearSeries.hasFPowerSeriesOnBall`.
* If a function admits a power series in a ball, then it is analytic at any point `y` of this ball,
and the power series there can be expressed in terms of the initial power series `p` as
`p.changeOrigin y`. See `HasFPowerSeriesOnBall.changeOrigin`. It follows in particular that
the set of points at which a given function is analytic is open, see `isOpen_analyticAt`.
## Implementation details
We only introduce the radius of convergence of a power series, as `p.radius`.
For a power series in finitely many dimensions, there is a finer (directional, coordinate-dependent)
notion, describing the polydisk of convergence. This notion is more specific, and not necessary to
build the general theory. We do not define it here.
-/
noncomputable section
variable {𝕜 E F G : Type*}
open Topology Classical BigOperators NNReal Filter ENNReal
open Set Filter Asymptotics
namespace FormalMultilinearSeries
variable [Ring 𝕜] [AddCommGroup E] [AddCommGroup F] [Module 𝕜 E] [Module 𝕜 F]
variable [TopologicalSpace E] [TopologicalSpace F]
variable [TopologicalAddGroup E] [TopologicalAddGroup F]
variable [ContinuousConstSMul 𝕜 E] [ContinuousConstSMul 𝕜 F]
/-- Given a formal multilinear series `p` and a vector `x`, then `p.sum x` is the sum `Σ pₙ xⁿ`. A
priori, it only behaves well when `‖x‖ < p.radius`. -/
protected def sum (p : FormalMultilinearSeries 𝕜 E F) (x : E) : F :=
∑' n : ℕ, p n fun _ => x
#align formal_multilinear_series.sum FormalMultilinearSeries.sum
/-- Given a formal multilinear series `p` and a vector `x`, then `p.partialSum n x` is the sum
`Σ pₖ xᵏ` for `k ∈ {0,..., n-1}`. -/
def partialSum (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) (x : E) : F :=
∑ k in Finset.range n, p k fun _ : Fin k => x
#align formal_multilinear_series.partial_sum FormalMultilinearSeries.partialSum
/-- The partial sums of a formal multilinear series are continuous. -/
theorem partialSum_continuous (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) :
Continuous (p.partialSum n) := by
unfold partialSum -- Porting note: added
continuity
#align formal_multilinear_series.partial_sum_continuous FormalMultilinearSeries.partialSum_continuous
end FormalMultilinearSeries
/-! ### The radius of a formal multilinear series -/
variable [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F]
[NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G]
namespace FormalMultilinearSeries
variable (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
/-- The radius of a formal multilinear series is the largest `r` such that the sum `Σ ‖pₙ‖ ‖y‖ⁿ`
converges for all `‖y‖ < r`. This implies that `Σ pₙ yⁿ` converges for all `‖y‖ < r`, but these
definitions are *not* equivalent in general. -/
def radius (p : FormalMultilinearSeries 𝕜 E F) : ℝ≥0∞ :=
⨆ (r : ℝ≥0) (C : ℝ) (_ : ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C), (r : ℝ≥0∞)
#align formal_multilinear_series.radius FormalMultilinearSeries.radius
/-- If `‖pₙ‖ rⁿ` is bounded in `n`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_bound (C : ℝ) {r : ℝ≥0} (h : ∀ n : ℕ, ‖p n‖ * (r : ℝ) ^ n ≤ C) :
(r : ℝ≥0∞) ≤ p.radius :=
le_iSup_of_le r <| le_iSup_of_le C <| le_iSup (fun _ => (r : ℝ≥0∞)) h
#align formal_multilinear_series.le_radius_of_bound FormalMultilinearSeries.le_radius_of_bound
/-- If `‖pₙ‖ rⁿ` is bounded in `n`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_bound_nnreal (C : ℝ≥0) {r : ℝ≥0} (h : ∀ n : ℕ, ‖p n‖₊ * r ^ n ≤ C) :
(r : ℝ≥0∞) ≤ p.radius :=
p.le_radius_of_bound C fun n => mod_cast h n
#align formal_multilinear_series.le_radius_of_bound_nnreal FormalMultilinearSeries.le_radius_of_bound_nnreal
/-- If `‖pₙ‖ rⁿ = O(1)`, as `n → ∞`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_isBigO (h : (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] fun _ => (1 : ℝ)) :
↑r ≤ p.radius :=
Exists.elim (isBigO_one_nat_atTop_iff.1 h) fun C hC =>
p.le_radius_of_bound C fun n => (le_abs_self _).trans (hC n)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.le_radius_of_is_O FormalMultilinearSeries.le_radius_of_isBigO
theorem le_radius_of_eventually_le (C) (h : ∀ᶠ n in atTop, ‖p n‖ * (r : ℝ) ^ n ≤ C) :
↑r ≤ p.radius :=
p.le_radius_of_isBigO <| IsBigO.of_bound C <| h.mono fun n hn => by simpa
#align formal_multilinear_series.le_radius_of_eventually_le FormalMultilinearSeries.le_radius_of_eventually_le
theorem le_radius_of_summable_nnnorm (h : Summable fun n => ‖p n‖₊ * r ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_bound_nnreal (∑' n, ‖p n‖₊ * r ^ n) fun _ => le_tsum' h _
#align formal_multilinear_series.le_radius_of_summable_nnnorm FormalMultilinearSeries.le_radius_of_summable_nnnorm
theorem le_radius_of_summable (h : Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_summable_nnnorm <| by
simp only [← coe_nnnorm] at h
exact mod_cast h
#align formal_multilinear_series.le_radius_of_summable FormalMultilinearSeries.le_radius_of_summable
theorem radius_eq_top_of_forall_nnreal_isBigO
(h : ∀ r : ℝ≥0, (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] fun _ => (1 : ℝ)) : p.radius = ∞ :=
ENNReal.eq_top_of_forall_nnreal_le fun r => p.le_radius_of_isBigO (h r)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.radius_eq_top_of_forall_nnreal_is_O FormalMultilinearSeries.radius_eq_top_of_forall_nnreal_isBigO
theorem radius_eq_top_of_eventually_eq_zero (h : ∀ᶠ n in atTop, p n = 0) : p.radius = ∞ :=
p.radius_eq_top_of_forall_nnreal_isBigO fun r =>
(isBigO_zero _ _).congr' (h.mono fun n hn => by simp [hn]) EventuallyEq.rfl
#align formal_multilinear_series.radius_eq_top_of_eventually_eq_zero FormalMultilinearSeries.radius_eq_top_of_eventually_eq_zero
theorem radius_eq_top_of_forall_image_add_eq_zero (n : ℕ) (hn : ∀ m, p (m + n) = 0) :
p.radius = ∞ :=
p.radius_eq_top_of_eventually_eq_zero <|
mem_atTop_sets.2 ⟨n, fun _ hk => tsub_add_cancel_of_le hk ▸ hn _⟩
#align formal_multilinear_series.radius_eq_top_of_forall_image_add_eq_zero FormalMultilinearSeries.radius_eq_top_of_forall_image_add_eq_zero
@[simp]
theorem constFormalMultilinearSeries_radius {v : F} :
(constFormalMultilinearSeries 𝕜 E v).radius = ⊤ :=
(constFormalMultilinearSeries 𝕜 E v).radius_eq_top_of_forall_image_add_eq_zero 1
(by simp [constFormalMultilinearSeries])
#align formal_multilinear_series.const_formal_multilinear_series_radius FormalMultilinearSeries.constFormalMultilinearSeries_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` tends to zero exponentially:
for some `0 < a < 1`, `‖p n‖ rⁿ = o(aⁿ)`. -/
theorem isLittleO_of_lt_radius (h : ↑r < p.radius) :
∃ a ∈ Ioo (0 : ℝ) 1, (fun n => ‖p n‖ * (r : ℝ) ^ n) =o[atTop] (a ^ ·) := by
have := (TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 4
rw [this]
-- Porting note: was
-- rw [(TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 4]
simp only [radius, lt_iSup_iff] at h
rcases h with ⟨t, C, hC, rt⟩
rw [ENNReal.coe_lt_coe, ← NNReal.coe_lt_coe] at rt
have : 0 < (t : ℝ) := r.coe_nonneg.trans_lt rt
rw [← div_lt_one this] at rt
refine' ⟨_, rt, C, Or.inr zero_lt_one, fun n => _⟩
calc
|‖p n‖ * (r : ℝ) ^ n| = ‖p n‖ * (t : ℝ) ^ n * (r / t : ℝ) ^ n := by
field_simp [mul_right_comm, abs_mul]
_ ≤ C * (r / t : ℝ) ^ n := by gcongr; apply hC
#align formal_multilinear_series.is_o_of_lt_radius FormalMultilinearSeries.isLittleO_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ = o(1)`. -/
theorem isLittleO_one_of_lt_radius (h : ↑r < p.radius) :
(fun n => ‖p n‖ * (r : ℝ) ^ n) =o[atTop] (fun _ => 1 : ℕ → ℝ) :=
let ⟨_, ha, hp⟩ := p.isLittleO_of_lt_radius h
hp.trans <| (isLittleO_pow_pow_of_lt_left ha.1.le ha.2).congr (fun _ => rfl) one_pow
#align formal_multilinear_series.is_o_one_of_lt_radius FormalMultilinearSeries.isLittleO_one_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` tends to zero exponentially:
for some `0 < a < 1` and `C > 0`, `‖p n‖ * r ^ n ≤ C * a ^ n`. -/
theorem norm_mul_pow_le_mul_pow_of_lt_radius (h : ↑r < p.radius) :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C * a ^ n := by
-- Porting note: moved out of `rcases`
have := ((TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 5).mp
(p.isLittleO_of_lt_radius h)
rcases this with ⟨a, ha, C, hC, H⟩
exact ⟨a, ha, C, hC, fun n => (le_abs_self _).trans (H n)⟩
#align formal_multilinear_series.norm_mul_pow_le_mul_pow_of_lt_radius FormalMultilinearSeries.norm_mul_pow_le_mul_pow_of_lt_radius
/-- If `r ≠ 0` and `‖pₙ‖ rⁿ = O(aⁿ)` for some `-1 < a < 1`, then `r < p.radius`. -/
theorem lt_radius_of_isBigO (h₀ : r ≠ 0) {a : ℝ} (ha : a ∈ Ioo (-1 : ℝ) 1)
(hp : (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] (a ^ ·)) : ↑r < p.radius := by
-- Porting note: moved out of `rcases`
have := ((TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 2 5)
rcases this.mp ⟨a, ha, hp⟩ with ⟨a, ha, C, hC, hp⟩
rw [← pos_iff_ne_zero, ← NNReal.coe_pos] at h₀
lift a to ℝ≥0 using ha.1.le
have : (r : ℝ) < r / a := by
simpa only [div_one] using (div_lt_div_left h₀ zero_lt_one ha.1).2 ha.2
norm_cast at this
rw [← ENNReal.coe_lt_coe] at this
refine' this.trans_le (p.le_radius_of_bound C fun n => _)
rw [NNReal.coe_div, div_pow, ← mul_div_assoc, div_le_iff (pow_pos ha.1 n)]
exact (le_abs_self _).trans (hp n)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.lt_radius_of_is_O FormalMultilinearSeries.lt_radius_of_isBigO
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem norm_mul_pow_le_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C :=
let ⟨_, ha, C, hC, h⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius h
⟨C, hC, fun n => (h n).trans <| mul_le_of_le_one_right hC.lt.le (pow_le_one _ ha.1.le ha.2.le)⟩
#align formal_multilinear_series.norm_mul_pow_le_of_lt_radius FormalMultilinearSeries.norm_mul_pow_le_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem norm_le_div_pow_of_pos_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h0 : 0 < r) (h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖ ≤ C / (r : ℝ) ^ n :=
let ⟨C, hC, hp⟩ := p.norm_mul_pow_le_of_lt_radius h
⟨C, hC, fun n => Iff.mpr (le_div_iff (pow_pos h0 _)) (hp n)⟩
#align formal_multilinear_series.norm_le_div_pow_of_pos_of_lt_radius FormalMultilinearSeries.norm_le_div_pow_of_pos_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem nnnorm_mul_pow_le_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖₊ * r ^ n ≤ C :=
let ⟨C, hC, hp⟩ := p.norm_mul_pow_le_of_lt_radius h
⟨⟨C, hC.lt.le⟩, hC, mod_cast hp⟩
#align formal_multilinear_series.nnnorm_mul_pow_le_of_lt_radius FormalMultilinearSeries.nnnorm_mul_pow_le_of_lt_radius
theorem le_radius_of_tendsto (p : FormalMultilinearSeries 𝕜 E F) {l : ℝ}
(h : Tendsto (fun n => ‖p n‖ * (r : ℝ) ^ n) atTop (𝓝 l)) : ↑r ≤ p.radius :=
p.le_radius_of_isBigO (h.isBigO_one _)
#align formal_multilinear_series.le_radius_of_tendsto FormalMultilinearSeries.le_radius_of_tendsto
theorem le_radius_of_summable_norm (p : FormalMultilinearSeries 𝕜 E F)
(hs : Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_tendsto hs.tendsto_atTop_zero
#align formal_multilinear_series.le_radius_of_summable_norm FormalMultilinearSeries.le_radius_of_summable_norm
theorem not_summable_norm_of_radius_lt_nnnorm (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(h : p.radius < ‖x‖₊) : ¬Summable fun n => ‖p n‖ * ‖x‖ ^ n :=
fun hs => not_le_of_lt h (p.le_radius_of_summable_norm hs)
#align formal_multilinear_series.not_summable_norm_of_radius_lt_nnnorm FormalMultilinearSeries.not_summable_norm_of_radius_lt_nnnorm
theorem summable_norm_mul_pow (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0} (h : ↑r < p.radius) :
Summable fun n : ℕ => ‖p n‖ * (r : ℝ) ^ n := by
obtain ⟨a, ha : a ∈ Ioo (0 : ℝ) 1, C, - : 0 < C, hp⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius h
exact .of_nonneg_of_le (fun n => mul_nonneg (norm_nonneg _) (pow_nonneg r.coe_nonneg _))
hp ((summable_geometric_of_lt_1 ha.1.le ha.2).mul_left _)
#align formal_multilinear_series.summable_norm_mul_pow FormalMultilinearSeries.summable_norm_mul_pow
theorem summable_norm_apply (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : Summable fun n : ℕ => ‖p n fun _ => x‖ := by
rw [mem_emetric_ball_zero_iff] at hx
refine' .of_nonneg_of_le
(fun _ => norm_nonneg _) (fun n => ((p n).le_op_norm _).trans_eq _) (p.summable_norm_mul_pow hx)
simp
#align formal_multilinear_series.summable_norm_apply FormalMultilinearSeries.summable_norm_apply
theorem summable_nnnorm_mul_pow (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0} (h : ↑r < p.radius) :
Summable fun n : ℕ => ‖p n‖₊ * r ^ n := by
rw [← NNReal.summable_coe]
push_cast
exact p.summable_norm_mul_pow h
#align formal_multilinear_series.summable_nnnorm_mul_pow FormalMultilinearSeries.summable_nnnorm_mul_pow
protected theorem summable [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : Summable fun n : ℕ => p n fun _ => x :=
(p.summable_norm_apply hx).of_norm
#align formal_multilinear_series.summable FormalMultilinearSeries.summable
theorem radius_eq_top_of_summable_norm (p : FormalMultilinearSeries 𝕜 E F)
(hs : ∀ r : ℝ≥0, Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : p.radius = ∞ :=
ENNReal.eq_top_of_forall_nnreal_le fun r => p.le_radius_of_summable_norm (hs r)
#align formal_multilinear_series.radius_eq_top_of_summable_norm FormalMultilinearSeries.radius_eq_top_of_summable_norm
theorem radius_eq_top_iff_summable_norm (p : FormalMultilinearSeries 𝕜 E F) :
p.radius = ∞ ↔ ∀ r : ℝ≥0, Summable fun n => ‖p n‖ * (r : ℝ) ^ n := by
constructor
· intro h r
obtain ⟨a, ha : a ∈ Ioo (0 : ℝ) 1, C, - : 0 < C, hp⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius
(show (r : ℝ≥0∞) < p.radius from h.symm ▸ ENNReal.coe_lt_top)
refine' .of_norm_bounded
(fun n => (C : ℝ) * a ^ n) ((summable_geometric_of_lt_1 ha.1.le ha.2).mul_left _) fun n => _
specialize hp n
rwa [Real.norm_of_nonneg (mul_nonneg (norm_nonneg _) (pow_nonneg r.coe_nonneg n))]
· exact p.radius_eq_top_of_summable_norm
#align formal_multilinear_series.radius_eq_top_iff_summable_norm FormalMultilinearSeries.radius_eq_top_iff_summable_norm
/-- If the radius of `p` is positive, then `‖pₙ‖` grows at most geometrically. -/
theorem le_mul_pow_of_radius_pos (p : FormalMultilinearSeries 𝕜 E F) (h : 0 < p.radius) :
∃ (C r : _) (hC : 0 < C) (_ : 0 < r), ∀ n, ‖p n‖ ≤ C * r ^ n := by
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 h with ⟨r, r0, rlt⟩
have rpos : 0 < (r : ℝ) := by simp [ENNReal.coe_pos.1 r0]
rcases norm_le_div_pow_of_pos_of_lt_radius p rpos rlt with ⟨C, Cpos, hCp⟩
refine' ⟨C, r⁻¹, Cpos, by simp only [inv_pos, rpos], fun n => _⟩
-- Porting note: was `convert`
rw [inv_pow, ← div_eq_mul_inv]
exact hCp n
#align formal_multilinear_series.le_mul_pow_of_radius_pos FormalMultilinearSeries.le_mul_pow_of_radius_pos
/-- The radius of the sum of two formal series is at least the minimum of their two radii. -/
theorem min_radius_le_radius_add (p q : FormalMultilinearSeries 𝕜 E F) :
min p.radius q.radius ≤ (p + q).radius := by
refine' ENNReal.le_of_forall_nnreal_lt fun r hr => _
rw [lt_min_iff] at hr
have := ((p.isLittleO_one_of_lt_radius hr.1).add (q.isLittleO_one_of_lt_radius hr.2)).isBigO
refine' (p + q).le_radius_of_isBigO ((isBigO_of_le _ fun n => _).trans this)
rw [← add_mul, norm_mul, norm_mul, norm_norm]
exact mul_le_mul_of_nonneg_right ((norm_add_le _ _).trans (le_abs_self _)) (norm_nonneg _)
#align formal_multilinear_series.min_radius_le_radius_add FormalMultilinearSeries.min_radius_le_radius_add
@[simp]
theorem radius_neg (p : FormalMultilinearSeries 𝕜 E F) : (-p).radius = p.radius := by
simp only [radius, neg_apply, norm_neg]
#align formal_multilinear_series.radius_neg FormalMultilinearSeries.radius_neg
protected theorem hasSum [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : HasSum (fun n : ℕ => p n fun _ => x) (p.sum x) :=
(p.summable hx).hasSum
#align formal_multilinear_series.has_sum FormalMultilinearSeries.hasSum
theorem radius_le_radius_continuousLinearMap_comp (p : FormalMultilinearSeries 𝕜 E F)
(f : F →L[𝕜] G) : p.radius ≤ (f.compFormalMultilinearSeries p).radius := by
refine' ENNReal.le_of_forall_nnreal_lt fun r hr => _
apply le_radius_of_isBigO
apply (IsBigO.trans_isLittleO _ (p.isLittleO_one_of_lt_radius hr)).isBigO
refine' IsBigO.mul (@IsBigOWith.isBigO _ _ _ _ _ ‖f‖ _ _ _ _) (isBigO_refl _ _)
refine IsBigOWith.of_bound (eventually_of_forall fun n => ?_)
simpa only [norm_norm] using f.norm_compContinuousMultilinearMap_le (p n)
#align formal_multilinear_series.radius_le_radius_continuous_linear_map_comp FormalMultilinearSeries.radius_le_radius_continuousLinearMap_comp
end FormalMultilinearSeries
/-! ### Expanding a function as a power series -/
section
variable {f g : E → F} {p pf pg : FormalMultilinearSeries 𝕜 E F} {x : E} {r r' : ℝ≥0∞}
/-- Given a function `f : E → F` and a formal multilinear series `p`, we say that `f` has `p` as
a power series on the ball of radius `r > 0` around `x` if `f (x + y) = ∑' pₙ yⁿ` for all `‖y‖ < r`.
-/
structure HasFPowerSeriesOnBall (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (x : E) (r : ℝ≥0∞) :
Prop where
r_le : r ≤ p.radius
r_pos : 0 < r
hasSum :
∀ {y}, y ∈ EMetric.ball (0 : E) r → HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y))
#align has_fpower_series_on_ball HasFPowerSeriesOnBall
/-- Given a function `f : E → F` and a formal multilinear series `p`, we say that `f` has `p` as
a power series around `x` if `f (x + y) = ∑' pₙ yⁿ` for all `y` in a neighborhood of `0`. -/
def HasFPowerSeriesAt (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (x : E) :=
∃ r, HasFPowerSeriesOnBall f p x r
#align has_fpower_series_at HasFPowerSeriesAt
variable (𝕜)
/-- Given a function `f : E → F`, we say that `f` is analytic at `x` if it admits a convergent power
series expansion around `x`. -/
def AnalyticAt (f : E → F) (x : E) :=
∃ p : FormalMultilinearSeries 𝕜 E F, HasFPowerSeriesAt f p x
#align analytic_at AnalyticAt
/-- Given a function `f : E → F`, we say that `f` is analytic on a set `s` if it is analytic around
every point of `s`. -/
def AnalyticOn (f : E → F) (s : Set E) :=
∀ x, x ∈ s → AnalyticAt 𝕜 f x
#align analytic_on AnalyticOn
variable {𝕜}
theorem HasFPowerSeriesOnBall.hasFPowerSeriesAt (hf : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesAt f p x :=
⟨r, hf⟩
#align has_fpower_series_on_ball.has_fpower_series_at HasFPowerSeriesOnBall.hasFPowerSeriesAt
theorem HasFPowerSeriesAt.analyticAt (hf : HasFPowerSeriesAt f p x) : AnalyticAt 𝕜 f x :=
⟨p, hf⟩
#align has_fpower_series_at.analytic_at HasFPowerSeriesAt.analyticAt
theorem HasFPowerSeriesOnBall.analyticAt (hf : HasFPowerSeriesOnBall f p x r) : AnalyticAt 𝕜 f x :=
hf.hasFPowerSeriesAt.analyticAt
#align has_fpower_series_on_ball.analytic_at HasFPowerSeriesOnBall.analyticAt
theorem HasFPowerSeriesOnBall.congr (hf : HasFPowerSeriesOnBall f p x r)
(hg : EqOn f g (EMetric.ball x r)) : HasFPowerSeriesOnBall g p x r :=
{ r_le := hf.r_le
r_pos := hf.r_pos
hasSum := fun {y} hy => by
convert hf.hasSum hy using 1
apply hg.symm
simpa [edist_eq_coe_nnnorm_sub] using hy }
#align has_fpower_series_on_ball.congr HasFPowerSeriesOnBall.congr
/-- If a function `f` has a power series `p` around `x`, then the function `z ↦ f (z - y)` has the
same power series around `x + y`. -/
theorem HasFPowerSeriesOnBall.comp_sub (hf : HasFPowerSeriesOnBall f p x r) (y : E) :
HasFPowerSeriesOnBall (fun z => f (z - y)) p (x + y) r :=
{ r_le := hf.r_le
r_pos := hf.r_pos
hasSum := fun {z} hz => by
convert hf.hasSum hz using 2
abel }
#align has_fpower_series_on_ball.comp_sub HasFPowerSeriesOnBall.comp_sub
theorem HasFPowerSeriesOnBall.hasSum_sub (hf : HasFPowerSeriesOnBall f p x r) {y : E}
(hy : y ∈ EMetric.ball x r) : HasSum (fun n : ℕ => p n fun _ => y - x) (f y) := by
have : y - x ∈ EMetric.ball (0 : E) r := by simpa [edist_eq_coe_nnnorm_sub] using hy
simpa only [add_sub_cancel'_right] using hf.hasSum this
#align has_fpower_series_on_ball.has_sum_sub HasFPowerSeriesOnBall.hasSum_sub
theorem HasFPowerSeriesOnBall.radius_pos (hf : HasFPowerSeriesOnBall f p x r) : 0 < p.radius :=
lt_of_lt_of_le hf.r_pos hf.r_le
#align has_fpower_series_on_ball.radius_pos HasFPowerSeriesOnBall.radius_pos
theorem HasFPowerSeriesAt.radius_pos (hf : HasFPowerSeriesAt f p x) : 0 < p.radius :=
let ⟨_, hr⟩ := hf
hr.radius_pos
#align has_fpower_series_at.radius_pos HasFPowerSeriesAt.radius_pos
theorem HasFPowerSeriesOnBall.mono (hf : HasFPowerSeriesOnBall f p x r) (r'_pos : 0 < r')
(hr : r' ≤ r) : HasFPowerSeriesOnBall f p x r' :=
⟨le_trans hr hf.1, r'_pos, fun hy => hf.hasSum (EMetric.ball_subset_ball hr hy)⟩
#align has_fpower_series_on_ball.mono HasFPowerSeriesOnBall.mono
theorem HasFPowerSeriesAt.congr (hf : HasFPowerSeriesAt f p x) (hg : f =ᶠ[𝓝 x] g) :
HasFPowerSeriesAt g p x := by
rcases hf with ⟨r₁, h₁⟩
rcases EMetric.mem_nhds_iff.mp hg with ⟨r₂, h₂pos, h₂⟩
exact ⟨min r₁ r₂,
(h₁.mono (lt_min h₁.r_pos h₂pos) inf_le_left).congr
fun y hy => h₂ (EMetric.ball_subset_ball inf_le_right hy)⟩
#align has_fpower_series_at.congr HasFPowerSeriesAt.congr
protected theorem HasFPowerSeriesAt.eventually (hf : HasFPowerSeriesAt f p x) :
∀ᶠ r : ℝ≥0∞ in 𝓝[>] 0, HasFPowerSeriesOnBall f p x r :=
let ⟨_, hr⟩ := hf
mem_of_superset (Ioo_mem_nhdsWithin_Ioi (left_mem_Ico.2 hr.r_pos)) fun _ hr' =>
hr.mono hr'.1 hr'.2.le
#align has_fpower_series_at.eventually HasFPowerSeriesAt.eventually
theorem HasFPowerSeriesOnBall.eventually_hasSum (hf : HasFPowerSeriesOnBall f p x r) :
∀ᶠ y in 𝓝 0, HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y)) := by
filter_upwards [EMetric.ball_mem_nhds (0 : E) hf.r_pos] using fun _ => hf.hasSum
#align has_fpower_series_on_ball.eventually_has_sum HasFPowerSeriesOnBall.eventually_hasSum
theorem HasFPowerSeriesAt.eventually_hasSum (hf : HasFPowerSeriesAt f p x) :
∀ᶠ y in 𝓝 0, HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y)) :=
let ⟨_, hr⟩ := hf
hr.eventually_hasSum
#align has_fpower_series_at.eventually_has_sum HasFPowerSeriesAt.eventually_hasSum
theorem HasFPowerSeriesOnBall.eventually_hasSum_sub (hf : HasFPowerSeriesOnBall f p x r) :
∀ᶠ y in 𝓝 x, HasSum (fun n : ℕ => p n fun _ : Fin n => y - x) (f y) := by
filter_upwards [EMetric.ball_mem_nhds x hf.r_pos] with y using hf.hasSum_sub
#align has_fpower_series_on_ball.eventually_has_sum_sub HasFPowerSeriesOnBall.eventually_hasSum_sub
theorem HasFPowerSeriesAt.eventually_hasSum_sub (hf : HasFPowerSeriesAt f p x) :
∀ᶠ y in 𝓝 x, HasSum (fun n : ℕ => p n fun _ : Fin n => y - x) (f y) :=
let ⟨_, hr⟩ := hf
hr.eventually_hasSum_sub
#align has_fpower_series_at.eventually_has_sum_sub HasFPowerSeriesAt.eventually_hasSum_sub
theorem HasFPowerSeriesOnBall.eventually_eq_zero
(hf : HasFPowerSeriesOnBall f (0 : FormalMultilinearSeries 𝕜 E F) x r) :
∀ᶠ z in 𝓝 x, f z = 0 := by
filter_upwards [hf.eventually_hasSum_sub] with z hz using hz.unique hasSum_zero
#align has_fpower_series_on_ball.eventually_eq_zero HasFPowerSeriesOnBall.eventually_eq_zero
theorem HasFPowerSeriesAt.eventually_eq_zero
(hf : HasFPowerSeriesAt f (0 : FormalMultilinearSeries 𝕜 E F) x) : ∀ᶠ z in 𝓝 x, f z = 0 :=
let ⟨_, hr⟩ := hf
hr.eventually_eq_zero
#align has_fpower_series_at.eventually_eq_zero HasFPowerSeriesAt.eventually_eq_zero
theorem hasFPowerSeriesOnBall_const {c : F} {e : E} :
HasFPowerSeriesOnBall (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e ⊤ := by
refine' ⟨by simp, WithTop.zero_lt_top, fun _ => hasSum_single 0 fun n hn => _⟩
simp [constFormalMultilinearSeries_apply hn]
#align has_fpower_series_on_ball_const hasFPowerSeriesOnBall_const
theorem hasFPowerSeriesAt_const {c : F} {e : E} :
HasFPowerSeriesAt (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e :=
⟨⊤, hasFPowerSeriesOnBall_const⟩
#align has_fpower_series_at_const hasFPowerSeriesAt_const
theorem analyticAt_const {v : F} : AnalyticAt 𝕜 (fun _ => v) x :=
⟨constFormalMultilinearSeries 𝕜 E v, hasFPowerSeriesAt_const⟩
#align analytic_at_const analyticAt_const
theorem analyticOn_const {v : F} {s : Set E} : AnalyticOn 𝕜 (fun _ => v) s :=
fun _ _ => analyticAt_const
#align analytic_on_const analyticOn_const
theorem HasFPowerSeriesOnBall.add (hf : HasFPowerSeriesOnBall f pf x r)
(hg : HasFPowerSeriesOnBall g pg x r) : HasFPowerSeriesOnBall (f + g) (pf + pg) x r :=
{ r_le := le_trans (le_min_iff.2 ⟨hf.r_le, hg.r_le⟩) (pf.min_radius_le_radius_add pg)
r_pos := hf.r_pos
hasSum := fun hy => (hf.hasSum hy).add (hg.hasSum hy) }
#align has_fpower_series_on_ball.add HasFPowerSeriesOnBall.add
theorem HasFPowerSeriesAt.add (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x) :
HasFPowerSeriesAt (f + g) (pf + pg) x := by
rcases (hf.eventually.and hg.eventually).exists with ⟨r, hr⟩
exact ⟨r, hr.1.add hr.2⟩
#align has_fpower_series_at.add HasFPowerSeriesAt.add
theorem AnalyticAt.congr (hf : AnalyticAt 𝕜 f x) (hg : f =ᶠ[𝓝 x] g) : AnalyticAt 𝕜 g x :=
let ⟨_, hpf⟩ := hf
(hpf.congr hg).analyticAt
theorem analyticAt_congr (h : f =ᶠ[𝓝 x] g) : AnalyticAt 𝕜 f x ↔ AnalyticAt 𝕜 g x :=
⟨fun hf ↦ hf.congr h, fun hg ↦ hg.congr h.symm⟩
theorem AnalyticAt.add (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) : AnalyticAt 𝕜 (f + g) x :=
let ⟨_, hpf⟩ := hf
let ⟨_, hqf⟩ := hg
(hpf.add hqf).analyticAt
#align analytic_at.add AnalyticAt.add
theorem HasFPowerSeriesOnBall.neg (hf : HasFPowerSeriesOnBall f pf x r) :
HasFPowerSeriesOnBall (-f) (-pf) x r :=
{ r_le := by
rw [pf.radius_neg]
exact hf.r_le
r_pos := hf.r_pos
hasSum := fun hy => (hf.hasSum hy).neg }
#align has_fpower_series_on_ball.neg HasFPowerSeriesOnBall.neg
theorem HasFPowerSeriesAt.neg (hf : HasFPowerSeriesAt f pf x) : HasFPowerSeriesAt (-f) (-pf) x :=
let ⟨_, hrf⟩ := hf
hrf.neg.hasFPowerSeriesAt
#align has_fpower_series_at.neg HasFPowerSeriesAt.neg
theorem AnalyticAt.neg (hf : AnalyticAt 𝕜 f x) : AnalyticAt 𝕜 (-f) x :=
let ⟨_, hpf⟩ := hf
hpf.neg.analyticAt
#align analytic_at.neg AnalyticAt.neg
theorem HasFPowerSeriesOnBall.sub (hf : HasFPowerSeriesOnBall f pf x r)
(hg : HasFPowerSeriesOnBall g pg x r) : HasFPowerSeriesOnBall (f - g) (pf - pg) x r := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align has_fpower_series_on_ball.sub HasFPowerSeriesOnBall.sub
theorem HasFPowerSeriesAt.sub (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x) :
HasFPowerSeriesAt (f - g) (pf - pg) x := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align has_fpower_series_at.sub HasFPowerSeriesAt.sub
theorem AnalyticAt.sub (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) :
AnalyticAt 𝕜 (f - g) x := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align analytic_at.sub AnalyticAt.sub
theorem AnalyticOn.mono {s t : Set E} (hf : AnalyticOn 𝕜 f t) (hst : s ⊆ t) : AnalyticOn 𝕜 f s :=
fun z hz => hf z (hst hz)
#align analytic_on.mono AnalyticOn.mono
theorem AnalyticOn.congr' {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : f =ᶠ[𝓝ˢ s] g) :
AnalyticOn 𝕜 g s :=
fun z hz => (hf z hz).congr (mem_nhdsSet_iff_forall.mp hg z hz)
theorem analyticOn_congr' {s : Set E} (h : f =ᶠ[𝓝ˢ s] g) : AnalyticOn 𝕜 f s ↔ AnalyticOn 𝕜 g s :=
⟨fun hf => hf.congr' h, fun hg => hg.congr' h.symm⟩
theorem AnalyticOn.congr {s : Set E} (hs : IsOpen s) (hf : AnalyticOn 𝕜 f s) (hg : s.EqOn f g) :
AnalyticOn 𝕜 g s :=
hf.congr' $ mem_nhdsSet_iff_forall.mpr
(fun _ hz => eventuallyEq_iff_exists_mem.mpr ⟨s, hs.mem_nhds hz, hg⟩)
theorem analyticOn_congr {s : Set E} (hs : IsOpen s) (h : s.EqOn f g) : AnalyticOn 𝕜 f s ↔
AnalyticOn 𝕜 g s := ⟨fun hf => hf.congr hs h, fun hg => hg.congr hs h.symm⟩
theorem AnalyticOn.add {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
AnalyticOn 𝕜 (f + g) s :=
fun z hz => (hf z hz).add (hg z hz)
#align analytic_on.add AnalyticOn.add
theorem AnalyticOn.sub {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
AnalyticOn 𝕜 (f - g) s :=
fun z hz => (hf z hz).sub (hg z hz)
#align analytic_on.sub AnalyticOn.sub
theorem HasFPowerSeriesOnBall.coeff_zero (hf : HasFPowerSeriesOnBall f pf x r) (v : Fin 0 → E) :
pf 0 v = f x := by
have v_eq : v = fun i => 0 := Subsingleton.elim _ _
have zero_mem : (0 : E) ∈ EMetric.ball (0 : E) r := by simp [hf.r_pos]
have : ∀ i, i ≠ 0 → (pf i fun j => 0) = 0 := by
intro i hi
have : 0 < i := pos_iff_ne_zero.2 hi
exact ContinuousMultilinearMap.map_coord_zero _ (⟨0, this⟩ : Fin i) rfl
have A := (hf.hasSum zero_mem).unique (hasSum_single _ this)
simpa [v_eq] using A.symm
#align has_fpower_series_on_ball.coeff_zero HasFPowerSeriesOnBall.coeff_zero
theorem HasFPowerSeriesAt.coeff_zero (hf : HasFPowerSeriesAt f pf x) (v : Fin 0 → E) :
pf 0 v = f x :=
let ⟨_, hrf⟩ := hf
hrf.coeff_zero v
#align has_fpower_series_at.coeff_zero HasFPowerSeriesAt.coeff_zero
/-- If a function `f` has a power series `p` on a ball and `g` is linear, then `g ∘ f` has the
power series `g ∘ p` on the same ball. -/
theorem ContinuousLinearMap.comp_hasFPowerSeriesOnBall (g : F →L[𝕜] G)
(h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (g ∘ f) (g.compFormalMultilinearSeries p) x r :=
{ r_le := h.r_le.trans (p.radius_le_radius_continuousLinearMap_comp _)
r_pos := h.r_pos
hasSum := fun hy => by
simpa only [ContinuousLinearMap.compFormalMultilinearSeries_apply,
ContinuousLinearMap.compContinuousMultilinearMap_coe, Function.comp_apply] using
g.hasSum (h.hasSum hy) }
#align continuous_linear_map.comp_has_fpower_series_on_ball ContinuousLinearMap.comp_hasFPowerSeriesOnBall
/-- If a function `f` is analytic on a set `s` and `g` is linear, then `g ∘ f` is analytic
on `s`. -/
theorem ContinuousLinearMap.comp_analyticOn {s : Set E} (g : F →L[𝕜] G) (h : AnalyticOn 𝕜 f s) :
AnalyticOn 𝕜 (g ∘ f) s := by
rintro x hx
rcases h x hx with ⟨p, r, hp⟩
exact ⟨g.compFormalMultilinearSeries p, r, g.comp_hasFPowerSeriesOnBall hp⟩
#align continuous_linear_map.comp_analytic_on ContinuousLinearMap.comp_analyticOn
/-- If a function admits a power series expansion, then it is exponentially close to the partial
sums of this power series on strict subdisks of the disk of convergence.
This version provides an upper estimate that decreases both in `‖y‖` and `n`. See also
`HasFPowerSeriesOnBall.uniform_geometric_approx` for a weaker version. -/
theorem HasFPowerSeriesOnBall.uniform_geometric_approx' {r' : ℝ≥0}
(hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n := by
obtain ⟨a, ha, C, hC, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n, ‖p n‖ * (r' : ℝ) ^ n ≤ C * a ^ n :=
p.norm_mul_pow_le_mul_pow_of_lt_radius (h.trans_le hf.r_le)
refine' ⟨a, ha, C / (1 - a), div_pos hC (sub_pos.2 ha.2), fun y hy n => _⟩
have yr' : ‖y‖ < r' := by
rw [ball_zero_eq] at hy
exact hy
have hr'0 : 0 < (r' : ℝ) := (norm_nonneg _).trans_lt yr'
have : y ∈ EMetric.ball (0 : E) r := by
refine' mem_emetric_ball_zero_iff.2 (lt_trans _ h)
exact mod_cast yr'
rw [norm_sub_rev, ← mul_div_right_comm]
have ya : a * (‖y‖ / ↑r') ≤ a :=
mul_le_of_le_one_right ha.1.le (div_le_one_of_le yr'.le r'.coe_nonneg)
suffices ‖p.partialSum n y - f (x + y)‖ ≤ C * (a * (‖y‖ / r')) ^ n / (1 - a * (‖y‖ / r')) by
refine' this.trans _
have : 0 < a := ha.1
gcongr
apply_rules [sub_pos.2, ha.2]
apply norm_sub_le_of_geometric_bound_of_hasSum (ya.trans_lt ha.2) _ (hf.hasSum this)
intro n
calc
‖(p n) fun _ : Fin n => y‖
_ ≤ ‖p n‖ * ∏ _i : Fin n, ‖y‖ := ContinuousMultilinearMap.le_op_norm _ _
_ = ‖p n‖ * (r' : ℝ) ^ n * (‖y‖ / r') ^ n := by field_simp [mul_right_comm]
_ ≤ C * a ^ n * (‖y‖ / r') ^ n := by gcongr ?_ * _; apply hp
_ ≤ C * (a * (‖y‖ / r')) ^ n := by rw [mul_pow, mul_assoc]
#align has_fpower_series_on_ball.uniform_geometric_approx' HasFPowerSeriesOnBall.uniform_geometric_approx'
/-- If a function admits a power series expansion, then it is exponentially close to the partial
sums of this power series on strict subdisks of the disk of convergence. -/
theorem HasFPowerSeriesOnBall.uniform_geometric_approx {r' : ℝ≥0}
(hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
∃ a ∈ Ioo (0 : ℝ) 1,
∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n, ‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n := by
obtain ⟨a, ha, C, hC, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n :=
hf.uniform_geometric_approx' h
refine' ⟨a, ha, C, hC, fun y hy n => (hp y hy n).trans _⟩
have yr' : ‖y‖ < r' := by rwa [ball_zero_eq] at hy
gcongr
exacts [mul_nonneg ha.1.le (div_nonneg (norm_nonneg y) r'.coe_nonneg),
mul_le_of_le_one_right ha.1.le (div_le_one_of_le yr'.le r'.coe_nonneg)]
#align has_fpower_series_on_ball.uniform_geometric_approx HasFPowerSeriesOnBall.uniform_geometric_approx
/-- Taylor formula for an analytic function, `IsBigO` version. -/
theorem HasFPowerSeriesAt.isBigO_sub_partialSum_pow (hf : HasFPowerSeriesAt f p x) (n : ℕ) :
(fun y : E => f (x + y) - p.partialSum n y) =O[𝓝 0] fun y => ‖y‖ ^ n := by
rcases hf with ⟨r, hf⟩
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hf.r_pos with ⟨r', r'0, h⟩
obtain ⟨a, -, C, -, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n :=
hf.uniform_geometric_approx' h
refine' isBigO_iff.2 ⟨C * (a / r') ^ n, _⟩
replace r'0 : 0 < (r' : ℝ); · exact mod_cast r'0
filter_upwards [Metric.ball_mem_nhds (0 : E) r'0] with y hy
simpa [mul_pow, mul_div_assoc, mul_assoc, div_mul_eq_mul_div] using hp y hy n
set_option linter.uppercaseLean3 false in
#align has_fpower_series_at.is_O_sub_partial_sum_pow HasFPowerSeriesAt.isBigO_sub_partialSum_pow
/-- If `f` has formal power series `∑ n, pₙ` on a ball of radius `r`, then for `y, z` in any smaller
ball, the norm of the difference `f y - f z - p 1 (fun _ ↦ y - z)` is bounded above by
`C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖`. This lemma formulates this property using `IsBigO` and
`Filter.principal` on `E × E`. -/
theorem HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal
(hf : HasFPowerSeriesOnBall f p x r) (hr : r' < r) :
(fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) =O[𝓟 (EMetric.ball (x, x) r')]
fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖ := by
lift r' to ℝ≥0 using ne_top_of_lt hr
rcases (zero_le r').eq_or_lt with (rfl | hr'0)
· simp only [isBigO_bot, EMetric.ball_zero, principal_empty, ENNReal.coe_zero]
obtain ⟨a, ha, C, hC : 0 < C, hp⟩ :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n : ℕ, ‖p n‖ * (r' : ℝ) ^ n ≤ C * a ^ n
exact p.norm_mul_pow_le_mul_pow_of_lt_radius (hr.trans_le hf.r_le)
simp only [← le_div_iff (pow_pos (NNReal.coe_pos.2 hr'0) _)] at hp
set L : E × E → ℝ := fun y =>
C * (a / r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * (a / (1 - a) ^ 2 + 2 / (1 - a))
have hL : ∀ y ∈ EMetric.ball (x, x) r', ‖f y.1 - f y.2 - p 1 fun _ => y.1 - y.2‖ ≤ L y := by
intro y hy'
have hy : y ∈ EMetric.ball x r ×ˢ EMetric.ball x r := by
rw [EMetric.ball_prod_same]
exact EMetric.ball_subset_ball hr.le hy'
set A : ℕ → F := fun n => (p n fun _ => y.1 - x) - p n fun _ => y.2 - x
have hA : HasSum (fun n => A (n + 2)) (f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) := by
convert (hasSum_nat_add_iff' 2).2 ((hf.hasSum_sub hy.1).sub (hf.hasSum_sub hy.2)) using 1
rw [Finset.sum_range_succ, Finset.sum_range_one, hf.coeff_zero, hf.coeff_zero, sub_self,
zero_add, ← Subsingleton.pi_single_eq (0 : Fin 1) (y.1 - x), Pi.single,
← Subsingleton.pi_single_eq (0 : Fin 1) (y.2 - x), Pi.single, ← (p 1).map_sub, ← Pi.single,
Subsingleton.pi_single_eq, sub_sub_sub_cancel_right]
rw [EMetric.mem_ball, edist_eq_coe_nnnorm_sub, ENNReal.coe_lt_coe] at hy'
set B : ℕ → ℝ := fun n => C * (a / r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * ((n + 2) * a ^ n)
have hAB : ∀ n, ‖A (n + 2)‖ ≤ B n := fun n =>
calc
‖A (n + 2)‖ ≤ ‖p (n + 2)‖ * ↑(n + 2) * ‖y - (x, x)‖ ^ (n + 1) * ‖y.1 - y.2‖ := by
-- porting note: `pi_norm_const` was `pi_norm_const (_ : E)`
simpa only [Fintype.card_fin, pi_norm_const, Prod.norm_def, Pi.sub_def,
Prod.fst_sub, Prod.snd_sub, sub_sub_sub_cancel_right] using
(p <| n + 2).norm_image_sub_le (fun _ => y.1 - x) fun _ => y.2 - x
_ = ‖p (n + 2)‖ * ‖y - (x, x)‖ ^ n * (↑(n + 2) * ‖y - (x, x)‖ * ‖y.1 - y.2‖) := by
rw [pow_succ ‖y - (x, x)‖]
ring
-- porting note: the two `↑` in `↑r'` are new, without them, Lean fails to synthesize
-- instances `HDiv ℝ ℝ≥0 ?m` or `HMul ℝ ℝ≥0 ?m`
_ ≤ C * a ^ (n + 2) / ↑r' ^ (n + 2)
* ↑r' ^ n * (↑(n + 2) * ‖y - (x, x)‖ * ‖y.1 - y.2‖) := by
have : 0 < a := ha.1
gcongr
· apply hp
· apply hy'.le
_ = B n := by
-- porting note: in the original, `B` was in the `field_simp`, but now Lean does not
-- accept it. The current proof works in Lean 4, but does not in Lean 3.
field_simp [pow_succ]
simp only [mul_assoc, mul_comm, mul_left_comm]
have hBL : HasSum B (L y) := by
apply HasSum.mul_left
simp only [add_mul]
have : ‖a‖ < 1 := by simp only [Real.norm_eq_abs, abs_of_pos ha.1, ha.2]
rw [div_eq_mul_inv, div_eq_mul_inv]
exact (hasSum_coe_mul_geometric_of_norm_lt_1 this).add -- porting note: was `convert`!
((hasSum_geometric_of_norm_lt_1 this).mul_left 2)
exact hA.norm_le_of_bounded hBL hAB
suffices L =O[𝓟 (EMetric.ball (x, x) r')] fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖ by
refine' (IsBigO.of_bound 1 (eventually_principal.2 fun y hy => _)).trans this
rw [one_mul]
exact (hL y hy).trans (le_abs_self _)
simp_rw [mul_right_comm _ (_ * _)] -- porting note: there was an `L` inside the `simp_rw`.
exact (isBigO_refl _ _).const_mul_left _
set_option linter.uppercaseLean3 false in
#align has_fpower_series_on_ball.is_O_image_sub_image_sub_deriv_principal HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal
/-- If `f` has formal power series `∑ n, pₙ` on a ball of radius `r`, then for `y, z` in any smaller
ball, the norm of the difference `f y - f z - p 1 (fun _ ↦ y - z)` is bounded above by
`C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖`. -/
theorem HasFPowerSeriesOnBall.image_sub_sub_deriv_le (hf : HasFPowerSeriesOnBall f p x r)
(hr : r' < r) :
∃ C, ∀ᵉ (y ∈ EMetric.ball x r') (z ∈ EMetric.ball x r'),
‖f y - f z - p 1 fun _ => y - z‖ ≤ C * max ‖y - x‖ ‖z - x‖ * ‖y - z‖ := by
simpa only [isBigO_principal, mul_assoc, norm_mul, norm_norm, Prod.forall, EMetric.mem_ball,
Prod.edist_eq, max_lt_iff, and_imp, @forall_swap (_ < _) E] using
hf.isBigO_image_sub_image_sub_deriv_principal hr
#align has_fpower_series_on_ball.image_sub_sub_deriv_le HasFPowerSeriesOnBall.image_sub_sub_deriv_le
/-- If `f` has formal power series `∑ n, pₙ` at `x`, then
`f y - f z - p 1 (fun _ ↦ y - z) = O(‖(y, z) - (x, x)‖ * ‖y - z‖)` as `(y, z) → (x, x)`.
In particular, `f` is strictly differentiable at `x`. -/
theorem HasFPowerSeriesAt.isBigO_image_sub_norm_mul_norm_sub (hf : HasFPowerSeriesAt f p x) :
(fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) =O[𝓝 (x, x)] fun y =>
‖y - (x, x)‖ * ‖y.1 - y.2‖ := by
rcases hf with ⟨r, hf⟩
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hf.r_pos with ⟨r', r'0, h⟩
refine' (hf.isBigO_image_sub_image_sub_deriv_principal h).mono _
exact le_principal_iff.2 (EMetric.ball_mem_nhds _ r'0)
set_option linter.uppercaseLean3 false in
#align has_fpower_series_at.is_O_image_sub_norm_mul_norm_sub HasFPowerSeriesAt.isBigO_image_sub_norm_mul_norm_sub
/-- If a function admits a power series expansion at `x`, then it is the uniform limit of the
partial sums of this power series on strict subdisks of the disk of convergence, i.e., `f (x + y)`
is the uniform limit of `p.partialSum n y` there. -/
theorem HasFPowerSeriesOnBall.tendstoUniformlyOn {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r)
(h : (r' : ℝ≥0∞) < r) :
TendstoUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
(Metric.ball (0 : E) r') := by
obtain ⟨a, ha, C, -, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n
exact hf.uniform_geometric_approx h
refine' Metric.tendstoUniformlyOn_iff.2 fun ε εpos => _
have L : Tendsto (fun n => (C : ℝ) * a ^ n) atTop (𝓝 ((C : ℝ) * 0)) :=
tendsto_const_nhds.mul (tendsto_pow_atTop_nhds_0_of_lt_1 ha.1.le ha.2)
rw [mul_zero] at L
refine' (L.eventually (gt_mem_nhds εpos)).mono fun n hn y hy => _
rw [dist_eq_norm]
exact (hp y hy n).trans_lt hn
#align has_fpower_series_on_ball.tendsto_uniformly_on HasFPowerSeriesOnBall.tendstoUniformlyOn
/-- If a function admits a power series expansion at `x`, then it is the locally uniform limit of
the partial sums of this power series on the disk of convergence, i.e., `f (x + y)`
is the locally uniform limit of `p.partialSum n y` there. -/
theorem HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn (hf : HasFPowerSeriesOnBall f p x r) :
TendstoLocallyUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
(EMetric.ball (0 : E) r) := by
intro u hu x hx
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hx with ⟨r', xr', hr'⟩
have : EMetric.ball (0 : E) r' ∈ 𝓝 x := IsOpen.mem_nhds EMetric.isOpen_ball xr'
refine' ⟨EMetric.ball (0 : E) r', mem_nhdsWithin_of_mem_nhds this, _⟩
simpa [Metric.emetric_ball_nnreal] using hf.tendstoUniformlyOn hr' u hu
#align has_fpower_series_on_ball.tendsto_locally_uniformly_on HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn
/-- If a function admits a power series expansion at `x`, then it is the uniform limit of the
partial sums of this power series on strict subdisks of the disk of convergence, i.e., `f y`
is the uniform limit of `p.partialSum n (y - x)` there. -/
theorem HasFPowerSeriesOnBall.tendstoUniformlyOn' {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r)
(h : (r' : ℝ≥0∞) < r) :
TendstoUniformlyOn (fun n y => p.partialSum n (y - x)) f atTop (Metric.ball (x : E) r') := by
convert (hf.tendstoUniformlyOn h).comp fun y => y - x using 1
· simp [(· ∘ ·)]
· ext z
simp [dist_eq_norm]
#align has_fpower_series_on_ball.tendsto_uniformly_on' HasFPowerSeriesOnBall.tendstoUniformlyOn'
/-- If a function admits a power series expansion at `x`, then it is the locally uniform limit of
the partial sums of this power series on the disk of convergence, i.e., `f y`
is the locally uniform limit of `p.partialSum n (y - x)` there. -/
theorem HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn' (hf : HasFPowerSeriesOnBall f p x r) :
TendstoLocallyUniformlyOn (fun n y => p.partialSum n (y - x)) f atTop
(EMetric.ball (x : E) r) := by
have A : ContinuousOn (fun y : E => y - x) (EMetric.ball (x : E) r) :=
(continuous_id.sub continuous_const).continuousOn
convert hf.tendstoLocallyUniformlyOn.comp (fun y : E => y - x) _ A using 1
· ext z
simp
· intro z
simp [edist_eq_coe_nnnorm, edist_eq_coe_nnnorm_sub]
#align has_fpower_series_on_ball.tendsto_locally_uniformly_on' HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn'
/-- If a function admits a power series expansion on a disk, then it is continuous there. -/
protected theorem HasFPowerSeriesOnBall.continuousOn (hf : HasFPowerSeriesOnBall f p x r) :
ContinuousOn f (EMetric.ball x r) :=
hf.tendstoLocallyUniformlyOn'.continuousOn <|
eventually_of_forall fun n =>
((p.partialSum_continuous n).comp (continuous_id.sub continuous_const)).continuousOn
#align has_fpower_series_on_ball.continuous_on HasFPowerSeriesOnBall.continuousOn
protected theorem HasFPowerSeriesAt.continuousAt (hf : HasFPowerSeriesAt f p x) :
ContinuousAt f x :=
let ⟨_, hr⟩ := hf
hr.continuousOn.continuousAt (EMetric.ball_mem_nhds x hr.r_pos)
#align has_fpower_series_at.continuous_at HasFPowerSeriesAt.continuousAt
protected theorem AnalyticAt.continuousAt (hf : AnalyticAt 𝕜 f x) : ContinuousAt f x :=
let ⟨_, hp⟩ := hf
hp.continuousAt
#align analytic_at.continuous_at AnalyticAt.continuousAt
protected theorem AnalyticOn.continuousOn {s : Set E} (hf : AnalyticOn 𝕜 f s) : ContinuousOn f s :=
fun x hx => (hf x hx).continuousAt.continuousWithinAt
#align analytic_on.continuous_on AnalyticOn.continuousOn
/-- Analytic everywhere implies continuous -/
theorem AnalyticOn.continuous {f : E → F} (fa : AnalyticOn 𝕜 f univ) : Continuous f := by
rw [continuous_iff_continuousOn_univ]; exact fa.continuousOn
/-- In a complete space, the sum of a converging power series `p` admits `p` as a power series.
This is not totally obvious as we need to check the convergence of the series. -/
protected theorem FormalMultilinearSeries.hasFPowerSeriesOnBall [CompleteSpace F]
(p : FormalMultilinearSeries 𝕜 E F) (h : 0 < p.radius) :
HasFPowerSeriesOnBall p.sum p 0 p.radius :=
{ r_le := le_rfl
r_pos := h
hasSum := fun hy => by
rw [zero_add]
exact p.hasSum hy }
#align formal_multilinear_series.has_fpower_series_on_ball FormalMultilinearSeries.hasFPowerSeriesOnBall
theorem HasFPowerSeriesOnBall.sum (h : HasFPowerSeriesOnBall f p x r) {y : E}
(hy : y ∈ EMetric.ball (0 : E) r) : f (x + y) = p.sum y :=
(h.hasSum hy).tsum_eq.symm
#align has_fpower_series_on_ball.sum HasFPowerSeriesOnBall.sum
/-- The sum of a converging power series is continuous in its disk of convergence. -/
protected theorem FormalMultilinearSeries.continuousOn [CompleteSpace F] :
ContinuousOn p.sum (EMetric.ball 0 p.radius) := by
rcases (zero_le p.radius).eq_or_lt with h | h
· simp [← h, continuousOn_empty]
· exact (p.hasFPowerSeriesOnBall h).continuousOn
#align formal_multilinear_series.continuous_on FormalMultilinearSeries.continuousOn
end
/-!
### Uniqueness of power series
If a function `f : E → F` has two representations as power series at a point `x : E`, corresponding
to formal multilinear series `p₁` and `p₂`, then these representations agree term-by-term. That is,
for any `n : ℕ` and `y : E`, `p₁ n (fun i ↦ y) = p₂ n (fun i ↦ y)`. In the one-dimensional case,
when `f : 𝕜 → E`, the continuous multilinear maps `p₁ n` and `p₂ n` are given by
`ContinuousMultilinearMap.mkPiField`, and hence are determined completely by the value of
`p₁ n (fun i ↦ 1)`, so `p₁ = p₂`. Consequently, the radius of convergence for one series can be
transferred to the other.
-/
section Uniqueness
open ContinuousMultilinearMap
theorem Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero {n : ℕ} {p : E[×n]→L[𝕜] F}
(h : (fun y => p fun _ => y) =O[𝓝 0] fun y => ‖y‖ ^ (n + 1)) (y : E) : (p fun _ => y) = 0 := by
obtain ⟨c, c_pos, hc⟩ := h.exists_pos
obtain ⟨t, ht, t_open, z_mem⟩ := eventually_nhds_iff.mp (isBigOWith_iff.mp hc)
obtain ⟨δ, δ_pos, δε⟩ := (Metric.isOpen_iff.mp t_open) 0 z_mem
clear h hc z_mem
cases' n with n
· exact norm_eq_zero.mp (by
-- porting note: the symmetric difference of the `simpa only` sets:
-- added `Nat.zero_eq, zero_add, pow_one`
-- removed `zero_pow', Ne.def, Nat.one_ne_zero, not_false_iff`
simpa only [Nat.zero_eq, fin0_apply_norm, norm_eq_zero, norm_zero, zero_add, pow_one,
mul_zero, norm_le_zero_iff] using ht 0 (δε (Metric.mem_ball_self δ_pos)))
· refine' Or.elim (Classical.em (y = 0))
(fun hy => by simpa only [hy] using p.map_zero) fun hy => _
replace hy := norm_pos_iff.mpr hy
refine' norm_eq_zero.mp (le_antisymm (le_of_forall_pos_le_add fun ε ε_pos => _) (norm_nonneg _))
have h₀ := _root_.mul_pos c_pos (pow_pos hy (n.succ + 1))
obtain ⟨k, k_pos, k_norm⟩ := NormedField.exists_norm_lt 𝕜
(lt_min (mul_pos δ_pos (inv_pos.mpr hy)) (mul_pos ε_pos (inv_pos.mpr h₀)))
have h₁ : ‖k • y‖ < δ := by
rw [norm_smul]
exact inv_mul_cancel_right₀ hy.ne.symm δ ▸
mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_left _ _)) hy
have h₂ :=
calc
‖p fun _ => k • y‖ ≤ c * ‖k • y‖ ^ (n.succ + 1) := by
-- porting note: now Lean wants `_root_.`
simpa only [norm_pow, _root_.norm_norm] using ht (k • y) (δε (mem_ball_zero_iff.mpr h₁))
--simpa only [norm_pow, norm_norm] using ht (k • y) (δε (mem_ball_zero_iff.mpr h₁))
_ = ‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))) := by
-- porting note: added `Nat.succ_eq_add_one` since otherwise `ring` does not conclude.
simp only [norm_smul, mul_pow, Nat.succ_eq_add_one]
-- porting note: removed `rw [pow_succ]`, since it now becomes superfluous.
ring
have h₃ : ‖k‖ * (c * ‖y‖ ^ (n.succ + 1)) < ε :=
inv_mul_cancel_right₀ h₀.ne.symm ε ▸
mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_right _ _)) h₀
calc
‖p fun _ => y‖ = ‖k⁻¹ ^ n.succ‖ * ‖p fun _ => k • y‖ := by
simpa only [inv_smul_smul₀ (norm_pos_iff.mp k_pos), norm_smul, Finset.prod_const,
Finset.card_fin] using
congr_arg norm (p.map_smul_univ (fun _ : Fin n.succ => k⁻¹) fun _ : Fin n.succ => k • y)
_ ≤ ‖k⁻¹ ^ n.succ‖ * (‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1)))) := by gcongr
_ = ‖(k⁻¹ * k) ^ n.succ‖ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))) := by
rw [← mul_assoc]
simp [norm_mul, mul_pow]
_ ≤ 0 + ε := by
rw [inv_mul_cancel (norm_pos_iff.mp k_pos)]
simpa using h₃.le
set_option linter.uppercaseLean3 false in
#align asymptotics.is_O.continuous_multilinear_map_apply_eq_zero Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero
/-- If a formal multilinear series `p` represents the zero function at `x : E`, then the
terms `p n (fun i ↦ y)` appearing in the sum are zero for any `n : ℕ`, `y : E`. -/
theorem HasFPowerSeriesAt.apply_eq_zero {p : FormalMultilinearSeries 𝕜 E F} {x : E}
(h : HasFPowerSeriesAt 0 p x) (n : ℕ) : ∀ y : E, (p n fun _ => y) = 0 := by
refine' Nat.strong_induction_on n fun k hk => _
have psum_eq : p.partialSum (k + 1) = fun y => p k fun _ => y := by
funext z
refine' Finset.sum_eq_single _ (fun b hb hnb => _) fun hn => _
· have := Finset.mem_range_succ_iff.mp hb
simp only [hk b (this.lt_of_ne hnb), Pi.zero_apply]
· exact False.elim (hn (Finset.mem_range.mpr (lt_add_one k)))
replace h := h.isBigO_sub_partialSum_pow k.succ
simp only [psum_eq, zero_sub, Pi.zero_apply, Asymptotics.isBigO_neg_left] at h
exact h.continuousMultilinearMap_apply_eq_zero
#align has_fpower_series_at.apply_eq_zero HasFPowerSeriesAt.apply_eq_zero
/-- A one-dimensional formal multilinear series representing the zero function is zero. -/
theorem HasFPowerSeriesAt.eq_zero {p : FormalMultilinearSeries 𝕜 𝕜 E} {x : 𝕜}
(h : HasFPowerSeriesAt 0 p x) : p = 0 := by
-- porting note: `funext; ext` was `ext (n x)`
funext n
ext x
rw [← mkPiField_apply_one_eq_self (p n)]
-- porting note: nasty hack, was `simp [h.apply_eq_zero n 1]`
have := Or.intro_right ?_ (h.apply_eq_zero n 1)
simpa using this
#align has_fpower_series_at.eq_zero HasFPowerSeriesAt.eq_zero
/-- One-dimensional formal multilinear series representing the same function are equal. -/
theorem HasFPowerSeriesAt.eq_formalMultilinearSeries {p₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E}
{f : 𝕜 → E} {x : 𝕜} (h₁ : HasFPowerSeriesAt f p₁ x) (h₂ : HasFPowerSeriesAt f p₂ x) : p₁ = p₂ :=
sub_eq_zero.mp (HasFPowerSeriesAt.eq_zero (by simpa only [sub_self] using h₁.sub h₂))
#align has_fpower_series_at.eq_formal_multilinear_series HasFPowerSeriesAt.eq_formalMultilinearSeries
theorem HasFPowerSeriesAt.eq_formalMultilinearSeries_of_eventually
{p q : FormalMultilinearSeries 𝕜 𝕜 E} {f g : 𝕜 → E} {x : 𝕜} (hp : HasFPowerSeriesAt f p x)
(hq : HasFPowerSeriesAt g q x) (heq : ∀ᶠ z in 𝓝 x, f z = g z) : p = q :=
(hp.congr heq).eq_formalMultilinearSeries hq
#align has_fpower_series_at.eq_formal_multilinear_series_of_eventually HasFPowerSeriesAt.eq_formalMultilinearSeries_of_eventually
/-- A one-dimensional formal multilinear series representing a locally zero function is zero. -/
theorem HasFPowerSeriesAt.eq_zero_of_eventually {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E}
{x : 𝕜} (hp : HasFPowerSeriesAt f p x) (hf : f =ᶠ[𝓝 x] 0) : p = 0 :=
(hp.congr hf).eq_zero
#align has_fpower_series_at.eq_zero_of_eventually HasFPowerSeriesAt.eq_zero_of_eventually
/-- If a function `f : 𝕜 → E` has two power series representations at `x`, then the given radii in
which convergence is guaranteed may be interchanged. This can be useful when the formal multilinear
series in one representation has a particularly nice form, but the other has a larger radius. -/
theorem HasFPowerSeriesOnBall.exchange_radius {p₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E}
{r₁ r₂ : ℝ≥0∞} {x : 𝕜} (h₁ : HasFPowerSeriesOnBall f p₁ x r₁)
(h₂ : HasFPowerSeriesOnBall f p₂ x r₂) : HasFPowerSeriesOnBall f p₁ x r₂ :=
h₂.hasFPowerSeriesAt.eq_formalMultilinearSeries h₁.hasFPowerSeriesAt ▸ h₂
#align has_fpower_series_on_ball.exchange_radius HasFPowerSeriesOnBall.exchange_radius
/-- If a function `f : 𝕜 → E` has power series representation `p` on a ball of some radius and for
each positive radius it has some power series representation, then `p` converges to `f` on the whole
`𝕜`. -/
theorem HasFPowerSeriesOnBall.r_eq_top_of_exists {f : 𝕜 → E} {r : ℝ≥0∞} {x : 𝕜}
{p : FormalMultilinearSeries 𝕜 𝕜 E} (h : HasFPowerSeriesOnBall f p x r)
(h' : ∀ (r' : ℝ≥0) (_ : 0 < r'), ∃ p' : FormalMultilinearSeries 𝕜 𝕜 E,
HasFPowerSeriesOnBall f p' x r') :
HasFPowerSeriesOnBall f p x ∞ :=
{ r_le := ENNReal.le_of_forall_pos_nnreal_lt fun r hr _ =>
let ⟨_, hp'⟩ := h' r hr
(h.exchange_radius hp').r_le
r_pos := ENNReal.coe_lt_top
hasSum := fun {y} _ =>
let ⟨r', hr'⟩ := exists_gt ‖y‖₊
let ⟨_, hp'⟩ := h' r' hr'.ne_bot.bot_lt
(h.exchange_radius hp').hasSum <| mem_emetric_ball_zero_iff.mpr (ENNReal.coe_lt_coe.2 hr') }
#align has_fpower_series_on_ball.r_eq_top_of_exists HasFPowerSeriesOnBall.r_eq_top_of_exists
end Uniqueness
/-!
### Changing origin in a power series
If a function is analytic in a disk `D(x, R)`, then it is analytic in any disk contained in that
one. Indeed, one can write
$$
f (x + y + z) = \sum_{n} p_n (y + z)^n = \sum_{n, k} \binom{n}{k} p_n y^{n-k} z^k
= \sum_{k} \Bigl(\sum_{n} \binom{n}{k} p_n y^{n-k}\Bigr) z^k.
$$
The corresponding power series has thus a `k`-th coefficient equal to
$\sum_{n} \binom{n}{k} p_n y^{n-k}$. In the general case where `pₙ` is a multilinear map, this has
to be interpreted suitably: instead of having a binomial coefficient, one should sum over all
possible subsets `s` of `Fin n` of cardinal `k`, and attribute `z` to the indices in `s` and
`y` to the indices outside of `s`.
In this paragraph, we implement this. The new power series is called `p.changeOrigin y`. Then, we
check its convergence and the fact that its sum coincides with the original sum. The outcome of this
discussion is that the set of points where a function is analytic is open.
-/
namespace FormalMultilinearSeries
section
variable (p : FormalMultilinearSeries 𝕜 E F) {x y : E} {r R : ℝ≥0}
/-- A term of `FormalMultilinearSeries.changeOriginSeries`.
Given a formal multilinear series `p` and a point `x` in its ball of convergence,
`p.changeOrigin x` is a formal multilinear series such that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense. Each term of `p.changeOrigin x`
is itself an analytic function of `x` given by the series `p.changeOriginSeries`. Each term in
`changeOriginSeries` is the sum of `changeOriginSeriesTerm`'s over all `s` of cardinality `l`.
The definition is such that `p.changeOriginSeriesTerm k l s hs (fun _ ↦ x) (fun _ ↦ y) =
p (k + l) (s.piecewise (fun _ ↦ x) (fun _ ↦ y))`
-/
def changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
E[×l]→L[𝕜] E[×k]→L[𝕜] F := by
let a := ContinuousMultilinearMap.curryFinFinset 𝕜 E F hs
(by erw [Finset.card_compl, Fintype.card_fin, hs, add_tsub_cancel_right])
exact a (p (k + l))
#align formal_multilinear_series.change_origin_series_term FormalMultilinearSeries.changeOriginSeriesTerm
theorem changeOriginSeriesTerm_apply (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l)
(x y : E) :
(p.changeOriginSeriesTerm k l s hs (fun _ => x) fun _ => y) =
p (k + l) (s.piecewise (fun _ => x) fun _ => y) :=
ContinuousMultilinearMap.curryFinFinset_apply_const _ _ _ _ _
#align formal_multilinear_series.change_origin_series_term_apply FormalMultilinearSeries.changeOriginSeriesTerm_apply
@[simp]
theorem norm_changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
‖p.changeOriginSeriesTerm k l s hs‖ = ‖p (k + l)‖ := by
simp only [changeOriginSeriesTerm, LinearIsometryEquiv.norm_map]
#align formal_multilinear_series.norm_change_origin_series_term FormalMultilinearSeries.norm_changeOriginSeriesTerm
@[simp]
theorem nnnorm_changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
‖p.changeOriginSeriesTerm k l s hs‖₊ = ‖p (k + l)‖₊ := by
simp only [changeOriginSeriesTerm, LinearIsometryEquiv.nnnorm_map]
#align formal_multilinear_series.nnnorm_change_origin_series_term FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm
theorem nnnorm_changeOriginSeriesTerm_apply_le (k l : ℕ) (s : Finset (Fin (k + l)))
(hs : s.card = l) (x y : E) :
‖p.changeOriginSeriesTerm k l s hs (fun _ => x) fun _ => y‖₊ ≤
‖p (k + l)‖₊ * ‖x‖₊ ^ l * ‖y‖₊ ^ k := by
rw [← p.nnnorm_changeOriginSeriesTerm k l s hs, ← Fin.prod_const, ← Fin.prod_const]
apply ContinuousMultilinearMap.le_of_op_nnnorm_le
apply ContinuousMultilinearMap.le_op_nnnorm
#align formal_multilinear_series.nnnorm_change_origin_series_term_apply_le FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm_apply_le
/-- The power series for `f.changeOrigin k`.
Given a formal multilinear series `p` and a point `x` in its ball of convergence,
`p.changeOrigin x` is a formal multilinear series such that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense. Its `k`-th term is the sum of
the series `p.changeOriginSeries k`. -/
def changeOriginSeries (k : ℕ) : FormalMultilinearSeries 𝕜 E (E[×k]→L[𝕜] F) := fun l =>
∑ s : { s : Finset (Fin (k + l)) // Finset.card s = l }, p.changeOriginSeriesTerm k l s s.2
#align formal_multilinear_series.change_origin_series FormalMultilinearSeries.changeOriginSeries
theorem nnnorm_changeOriginSeries_le_tsum (k l : ℕ) :
‖p.changeOriginSeries k l‖₊ ≤
∑' _ : { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + l)‖₊ :=
(nnnorm_sum_le _ (fun t => changeOriginSeriesTerm p k l (Subtype.val t) t.prop)).trans_eq <| by
simp_rw [tsum_fintype, nnnorm_changeOriginSeriesTerm (p := p) (k := k) (l := l)]
#align formal_multilinear_series.nnnorm_change_origin_series_le_tsum FormalMultilinearSeries.nnnorm_changeOriginSeries_le_tsum
theorem nnnorm_changeOriginSeries_apply_le_tsum (k l : ℕ) (x : E) :
‖p.changeOriginSeries k l fun _ => x‖₊ ≤
∑' _ : { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + l)‖₊ * ‖x‖₊ ^ l := by
rw [NNReal.tsum_mul_right, ← Fin.prod_const]
exact (p.changeOriginSeries k l).le_of_op_nnnorm_le _ (p.nnnorm_changeOriginSeries_le_tsum _ _)
#align formal_multilinear_series.nnnorm_change_origin_series_apply_le_tsum FormalMultilinearSeries.nnnorm_changeOriginSeries_apply_le_tsum
/-- Changing the origin of a formal multilinear series `p`, so that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense.
-/
def changeOrigin (x : E) : FormalMultilinearSeries 𝕜 E F :=
fun k => (p.changeOriginSeries k).sum x
#align formal_multilinear_series.change_origin FormalMultilinearSeries.changeOrigin
/-- An auxiliary equivalence useful in the proofs about
`FormalMultilinearSeries.changeOriginSeries`: the set of triples `(k, l, s)`, where `s` is a
`Finset (Fin (k + l))` of cardinality `l` is equivalent to the set of pairs `(n, s)`, where `s` is a
`Finset (Fin n)`.
The forward map sends `(k, l, s)` to `(k + l, s)` and the inverse map sends `(n, s)` to
`(n - Finset.card s, Finset.card s, s)`. The actual definition is less readable because of problems
with non-definitional equalities. -/
@[simps]
def changeOriginIndexEquiv :
(Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) ≃ Σn : ℕ, Finset (Fin n) where
toFun s := ⟨s.1 + s.2.1, s.2.2⟩
invFun s :=
⟨s.1 - s.2.card, s.2.card,
⟨s.2.map
(Fin.castIso <| (tsub_add_cancel_of_le <| card_finset_fin_le s.2).symm).toEquiv.toEmbedding,
Finset.card_map _⟩⟩
left_inv := by
rintro ⟨k, l, ⟨s : Finset (Fin <| k + l), hs : s.card = l⟩⟩
dsimp only [Subtype.coe_mk]
-- Lean can't automatically generalize `k' = k + l - s.card`, `l' = s.card`, so we explicitly
-- formulate the generalized goal
suffices ∀ k' l', k' = k → l' = l → ∀ (hkl : k + l = k' + l') (hs'),
(⟨k', l', ⟨Finset.map (Fin.castIso hkl).toEquiv.toEmbedding s, hs'⟩⟩ :
Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) = ⟨k, l, ⟨s, hs⟩⟩ by
apply this <;> simp only [hs, add_tsub_cancel_right]
rintro _ _ rfl rfl hkl hs'
simp only [Equiv.refl_toEmbedding, Fin.castIso_refl, Finset.map_refl, eq_self_iff_true,
OrderIso.refl_toEquiv, and_self_iff, heq_iff_eq]
right_inv := by
rintro ⟨n, s⟩
simp [tsub_add_cancel_of_le (card_finset_fin_le s), Fin.castIso_to_equiv]
#align formal_multilinear_series.change_origin_index_equiv FormalMultilinearSeries.changeOriginIndexEquiv
theorem changeOriginSeries_summable_aux₁ {r r' : ℝ≥0} (hr : (r + r' : ℝ≥0∞) < p.radius) :
Summable fun s : Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l } =>
‖p (s.1 + s.2.1)‖₊ * r ^ s.2.1 * r' ^ s.1 := by
rw [← changeOriginIndexEquiv.symm.summable_iff]
dsimp only [Function.comp_def, changeOriginIndexEquiv_symm_apply_fst,
changeOriginIndexEquiv_symm_apply_snd_fst]
have : ∀ n : ℕ,
HasSum (fun s : Finset (Fin n) => ‖p (n - s.card + s.card)‖₊ * r ^ s.card * r' ^ (n - s.card))
(‖p n‖₊ * (r + r') ^ n) := by
intro n
-- TODO: why `simp only [tsub_add_cancel_of_le (card_finset_fin_le _)]` fails?
convert_to HasSum (fun s : Finset (Fin n) => ‖p n‖₊ * (r ^ s.card * r' ^ (n - s.card))) _
· ext1 s
rw [tsub_add_cancel_of_le (card_finset_fin_le _), mul_assoc]
rw [← Fin.sum_pow_mul_eq_add_pow]
exact (hasSum_fintype _).mul_left _
refine' NNReal.summable_sigma.2 ⟨fun n => (this n).summable, _⟩
simp only [(this _).tsum_eq]
exact p.summable_nnnorm_mul_pow hr
#align formal_multilinear_series.change_origin_series_summable_aux₁ FormalMultilinearSeries.changeOriginSeries_summable_aux₁
theorem changeOriginSeries_summable_aux₂ (hr : (r : ℝ≥0∞) < p.radius) (k : ℕ) :
Summable fun s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l } =>
‖p (k + s.1)‖₊ * r ^ s.1 := by
rcases ENNReal.lt_iff_exists_add_pos_lt.1 hr with ⟨r', h0, hr'⟩
simpa only [mul_inv_cancel_right₀ (pow_pos h0 _).ne'] using
((NNReal.summable_sigma.1 (p.changeOriginSeries_summable_aux₁ hr')).1 k).mul_right (r' ^ k)⁻¹
#align formal_multilinear_series.change_origin_series_summable_aux₂ FormalMultilinearSeries.changeOriginSeries_summable_aux₂
theorem changeOriginSeries_summable_aux₃ {r : ℝ≥0} (hr : ↑r < p.radius) (k : ℕ) :
Summable fun l : ℕ => ‖p.changeOriginSeries k l‖₊ * r ^ l := by
refine' NNReal.summable_of_le
(fun n => _) (NNReal.summable_sigma.1 <| p.changeOriginSeries_summable_aux₂ hr k).2
simp only [NNReal.tsum_mul_right]
exact mul_le_mul' (p.nnnorm_changeOriginSeries_le_tsum _ _) le_rfl
#align formal_multilinear_series.change_origin_series_summable_aux₃ FormalMultilinearSeries.changeOriginSeries_summable_aux₃
theorem le_changeOriginSeries_radius (k : ℕ) : p.radius ≤ (p.changeOriginSeries k).radius :=
ENNReal.le_of_forall_nnreal_lt fun _r hr =>
le_radius_of_summable_nnnorm _ (p.changeOriginSeries_summable_aux₃ hr k)
#align formal_multilinear_series.le_change_origin_series_radius FormalMultilinearSeries.le_changeOriginSeries_radius
theorem nnnorm_changeOrigin_le (k : ℕ) (h : (‖x‖₊ : ℝ≥0∞) < p.radius) :
‖p.changeOrigin x k‖₊ ≤
∑' s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + s.1)‖₊ * ‖x‖₊ ^ s.1 := by
refine' tsum_of_nnnorm_bounded _ fun l => p.nnnorm_changeOriginSeries_apply_le_tsum k l x
have := p.changeOriginSeries_summable_aux₂ h k
refine' HasSum.sigma this.hasSum fun l => _
exact ((NNReal.summable_sigma.1 this).1 l).hasSum
#align formal_multilinear_series.nnnorm_change_origin_le FormalMultilinearSeries.nnnorm_changeOrigin_le
/-- The radius of convergence of `p.changeOrigin x` is at least `p.radius - ‖x‖`. In other words,
`p.changeOrigin x` is well defined on the largest ball contained in the original ball of
convergence. -/
theorem changeOrigin_radius : p.radius - ‖x‖₊ ≤ (p.changeOrigin x).radius := by
refine' ENNReal.le_of_forall_pos_nnreal_lt fun r _h0 hr => _
rw [lt_tsub_iff_right, add_comm] at hr
have hr' : (‖x‖₊ : ℝ≥0∞) < p.radius := (le_add_right le_rfl).trans_lt hr
apply le_radius_of_summable_nnnorm
have : ∀ k : ℕ,
‖p.changeOrigin x k‖₊ * r ^ k ≤
(∑' s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + s.1)‖₊ * ‖x‖₊ ^ s.1) *
r ^ k :=
fun k => mul_le_mul_right' (p.nnnorm_changeOrigin_le k hr') (r ^ k)
refine' NNReal.summable_of_le this _
simpa only [← NNReal.tsum_mul_right] using
(NNReal.summable_sigma.1 (p.changeOriginSeries_summable_aux₁ hr)).2
#align formal_multilinear_series.change_origin_radius FormalMultilinearSeries.changeOrigin_radius
end
-- From this point on, assume that the space is complete, to make sure that series that converge
-- in norm also converge in `F`.
variable [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x y : E} {r R : ℝ≥0}
theorem hasFPowerSeriesOnBall_changeOrigin (k : ℕ) (hr : 0 < p.radius) :
HasFPowerSeriesOnBall (fun x => p.changeOrigin x k) (p.changeOriginSeries k) 0 p.radius :=
have := p.le_changeOriginSeries_radius k
((p.changeOriginSeries k).hasFPowerSeriesOnBall (hr.trans_le this)).mono hr this
#align formal_multilinear_series.has_fpower_series_on_ball_change_origin FormalMultilinearSeries.hasFPowerSeriesOnBall_changeOrigin
/-- Summing the series `p.changeOrigin x` at a point `y` gives back `p (x + y)`. -/
theorem changeOrigin_eval (h : (‖x‖₊ + ‖y‖₊ : ℝ≥0∞) < p.radius) :
(p.changeOrigin x).sum y = p.sum (x + y) := by
have radius_pos : 0 < p.radius := lt_of_le_of_lt (zero_le _) h
have x_mem_ball : x ∈ EMetric.ball (0 : E) p.radius :=
mem_emetric_ball_zero_iff.2 ((le_add_right le_rfl).trans_lt h)
have y_mem_ball : y ∈ EMetric.ball (0 : E) (p.changeOrigin x).radius := by
refine' mem_emetric_ball_zero_iff.2 (lt_of_lt_of_le _ p.changeOrigin_radius)
rwa [lt_tsub_iff_right, add_comm]
have x_add_y_mem_ball : x + y ∈ EMetric.ball (0 : E) p.radius := by
refine' mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt _ h)
exact mod_cast nnnorm_add_le x y
set f : (Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) → F := fun s =>
p.changeOriginSeriesTerm s.1 s.2.1 s.2.2 s.2.2.2 (fun _ => x) fun _ => y
have hsf : Summable f := by
refine' .of_nnnorm_bounded _ (p.changeOriginSeries_summable_aux₁ h) _
rintro ⟨k, l, s, hs⟩
dsimp only [Subtype.coe_mk]
exact p.nnnorm_changeOriginSeriesTerm_apply_le _ _ _ _ _ _
have hf : HasSum f ((p.changeOrigin x).sum y) := by
refine' HasSum.sigma_of_hasSum ((p.changeOrigin x).summable y_mem_ball).hasSum (fun k => _) hsf
· dsimp only
refine' ContinuousMultilinearMap.hasSum_eval _ _
have := (p.hasFPowerSeriesOnBall_changeOrigin k radius_pos).hasSum x_mem_ball
rw [zero_add] at this
refine' HasSum.sigma_of_hasSum this (fun l => _) _
· simp only [changeOriginSeries, ContinuousMultilinearMap.sum_apply]
apply hasSum_fintype
· refine' .of_nnnorm_bounded _
(p.changeOriginSeries_summable_aux₂ (mem_emetric_ball_zero_iff.1 x_mem_ball) k) fun s => _
refine' (ContinuousMultilinearMap.le_op_nnnorm _ _).trans_eq _
simp
refine' hf.unique (changeOriginIndexEquiv.symm.hasSum_iff.1 _)
refine' HasSum.sigma_of_hasSum
(p.hasSum x_add_y_mem_ball) (fun n => _) (changeOriginIndexEquiv.symm.summable_iff.2 hsf)
erw [(p n).map_add_univ (fun _ => x) fun _ => y]
-- porting note: added explicit function
convert hasSum_fintype (fun c : Finset (Fin n) => f (changeOriginIndexEquiv.symm ⟨n, c⟩))
rename_i s _
dsimp only [changeOriginSeriesTerm, (· ∘ ·), changeOriginIndexEquiv_symm_apply_fst,
changeOriginIndexEquiv_symm_apply_snd_fst, changeOriginIndexEquiv_symm_apply_snd_snd_coe]
rw [ContinuousMultilinearMap.curryFinFinset_apply_const]
have : ∀ (m) (hm : n = m), p n (s.piecewise (fun _ => x) fun _ => y) =
p m ((s.map (Fin.castIso hm).toEquiv.toEmbedding).piecewise (fun _ => x) fun _ => y) := by
rintro m rfl
simp (config := { unfoldPartialApp := true }) [Finset.piecewise]
apply this
#align formal_multilinear_series.change_origin_eval FormalMultilinearSeries.changeOrigin_eval
/-- Power series terms are analytic as we vary the origin -/
theorem analyticAt_changeOrigin (p : FormalMultilinearSeries 𝕜 E F) (rp : p.radius > 0) (n : ℕ) :
AnalyticAt 𝕜 (fun x ↦ p.changeOrigin x n) 0 :=
(FormalMultilinearSeries.hasFPowerSeriesOnBall_changeOrigin p n rp).analyticAt
end FormalMultilinearSeries
section
variable [CompleteSpace F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x y : E} {r : ℝ≥0∞}
/-- If a function admits a power series expansion `p` on a ball `B (x, r)`, then it also admits a
power series on any subball of this ball (even with a different center), given by `p.changeOrigin`.
-/
theorem HasFPowerSeriesOnBall.changeOrigin (hf : HasFPowerSeriesOnBall f p x r)
(h : (‖y‖₊ : ℝ≥0∞) < r) : HasFPowerSeriesOnBall f (p.changeOrigin y) (x + y) (r - ‖y‖₊) :=
{ r_le := by
apply le_trans _ p.changeOrigin_radius
exact tsub_le_tsub hf.r_le le_rfl
r_pos := by simp [h]
hasSum := fun {z} hz => by
have : f (x + y + z) =
FormalMultilinearSeries.sum (FormalMultilinearSeries.changeOrigin p y) z := by
rw [mem_emetric_ball_zero_iff, lt_tsub_iff_right, add_comm] at hz
rw [p.changeOrigin_eval (hz.trans_le hf.r_le), add_assoc, hf.sum]
refine' mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt _ hz)
exact mod_cast nnnorm_add_le y z
rw [this]
apply (p.changeOrigin y).hasSum
refine' EMetric.ball_subset_ball (le_trans _ p.changeOrigin_radius) hz
exact tsub_le_tsub hf.r_le le_rfl }
#align has_fpower_series_on_ball.change_origin HasFPowerSeriesOnBall.changeOrigin
/-- If a function admits a power series expansion `p` on an open ball `B (x, r)`, then
it is analytic at every point of this ball. -/
theorem HasFPowerSeriesOnBall.analyticAt_of_mem (hf : HasFPowerSeriesOnBall f p x r)
(h : y ∈ EMetric.ball x r) : AnalyticAt 𝕜 f y := by
have : (‖y - x‖₊ : ℝ≥0∞) < r := by simpa [edist_eq_coe_nnnorm_sub] using h
have := hf.changeOrigin this
rw [add_sub_cancel'_right] at this
exact this.analyticAt
#align has_fpower_series_on_ball.analytic_at_of_mem HasFPowerSeriesOnBall.analyticAt_of_mem
theorem HasFPowerSeriesOnBall.analyticOn (hf : HasFPowerSeriesOnBall f p x r) :
AnalyticOn 𝕜 f (EMetric.ball x r) :=
fun _y hy => hf.analyticAt_of_mem hy
#align has_fpower_series_on_ball.analytic_on HasFPowerSeriesOnBall.analyticOn
variable (𝕜 f)
/-- For any function `f` from a normed vector space to a Banach space, the set of points `x` such
that `f` is analytic at `x` is open. -/
theorem isOpen_analyticAt : IsOpen { x | AnalyticAt 𝕜 f x } := by
rw [isOpen_iff_mem_nhds]
rintro x ⟨p, r, hr⟩
exact mem_of_superset (EMetric.ball_mem_nhds _ hr.r_pos) fun y hy => hr.analyticAt_of_mem hy
#align is_open_analytic_at isOpen_analyticAt
variable {𝕜}
theorem AnalyticAt.eventually_analyticAt {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∀ᶠ y in 𝓝 x, AnalyticAt 𝕜 f y :=
(isOpen_analyticAt 𝕜 f).mem_nhds h
theorem AnalyticAt.exists_mem_nhds_analyticOn {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∃ s ∈ 𝓝 x, AnalyticOn 𝕜 f s :=
h.eventually_analyticAt.exists_mem
/-- If we're analytic at a point, we're analytic in a nonempty ball -/
theorem AnalyticAt.exists_ball_analyticOn {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∃ r : ℝ, 0 < r ∧ AnalyticOn 𝕜 f (Metric.ball x r) :=
Metric.isOpen_iff.mp (isOpen_analyticAt _ _) _ h
end
section
open FormalMultilinearSeries
variable {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E} {z₀ : 𝕜}
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z)) := by
refine' ⟨fun ⟨r, _, r_pos, h⟩ =>
eventually_of_mem (EMetric.ball_mem_nhds 0 r_pos) fun _ => by simpa using h, _⟩
simp only [Metric.eventually_nhds_iff]
rintro ⟨r, r_pos, h⟩
refine' ⟨p.radius ⊓ r.toNNReal, by simp, _, _⟩
· simp only [r_pos.lt, lt_inf_iff, ENNReal.coe_pos, Real.toNNReal_pos, and_true_iff]
obtain ⟨z, z_pos, le_z⟩ := NormedField.exists_norm_lt 𝕜 r_pos.lt
have : (‖z‖₊ : ENNReal) ≤ p.radius := by
|
simp only [dist_zero_right] at h
|
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z)) := by
refine' ⟨fun ⟨r, _, r_pos, h⟩ =>
eventually_of_mem (EMetric.ball_mem_nhds 0 r_pos) fun _ => by simpa using h, _⟩
simp only [Metric.eventually_nhds_iff]
rintro ⟨r, r_pos, h⟩
refine' ⟨p.radius ⊓ r.toNNReal, by simp, _, _⟩
· simp only [r_pos.lt, lt_inf_iff, ENNReal.coe_pos, Real.toNNReal_pos, and_true_iff]
obtain ⟨z, z_pos, le_z⟩ := NormedField.exists_norm_lt 𝕜 r_pos.lt
have : (‖z‖₊ : ENNReal) ≤ p.radius := by
|
Mathlib.Analysis.Analytic.Basic.1430_0.jQw1fRSE1vGpOll
|
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z))
|
Mathlib_Analysis_Analytic_Basic
|
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
p : FormalMultilinearSeries 𝕜 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
r : ℝ
r_pos : r > 0
z : 𝕜
z_pos : 0 < ‖z‖
le_z : ‖z‖ < r
h : ∀ ⦃y : 𝕜⦄, ‖y‖ < r → HasSum (fun n => y ^ n • coeff p n) (f (z₀ + y))
⊢ ↑‖z‖₊ ≤ radius p
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Logic.Equiv.Fin
import Mathlib.Topology.Algebra.InfiniteSum.Module
#align_import analysis.analytic.basic from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
/-!
# Analytic functions
A function is analytic in one dimension around `0` if it can be written as a converging power series
`Σ pₙ zⁿ`. This definition can be extended to any dimension (even in infinite dimension) by
requiring that `pₙ` is a continuous `n`-multilinear map. In general, `pₙ` is not unique (in two
dimensions, taking `p₂ (x, y) (x', y') = x y'` or `y x'` gives the same map when applied to a
vector `(x, y) (x, y)`). A way to guarantee uniqueness is to take a symmetric `pₙ`, but this is not
always possible in nonzero characteristic (in characteristic 2, the previous example has no
symmetric representative). Therefore, we do not insist on symmetry or uniqueness in the definition,
and we only require the existence of a converging series.
The general framework is important to say that the exponential map on bounded operators on a Banach
space is analytic, as well as the inverse on invertible operators.
## Main definitions
Let `p` be a formal multilinear series from `E` to `F`, i.e., `p n` is a multilinear map on `E^n`
for `n : ℕ`.
* `p.radius`: the largest `r : ℝ≥0∞` such that `‖p n‖ * r^n` grows subexponentially.
* `p.le_radius_of_bound`, `p.le_radius_of_bound_nnreal`, `p.le_radius_of_isBigO`: if `‖p n‖ * r ^ n`
is bounded above, then `r ≤ p.radius`;
* `p.isLittleO_of_lt_radius`, `p.norm_mul_pow_le_mul_pow_of_lt_radius`,
`p.isLittleO_one_of_lt_radius`,
`p.norm_mul_pow_le_of_lt_radius`, `p.nnnorm_mul_pow_le_of_lt_radius`: if `r < p.radius`, then
`‖p n‖ * r ^ n` tends to zero exponentially;
* `p.lt_radius_of_isBigO`: if `r ≠ 0` and `‖p n‖ * r ^ n = O(a ^ n)` for some `-1 < a < 1`, then
`r < p.radius`;
* `p.partialSum n x`: the sum `∑_{i = 0}^{n-1} pᵢ xⁱ`.
* `p.sum x`: the sum `∑'_{i = 0}^{∞} pᵢ xⁱ`.
Additionally, let `f` be a function from `E` to `F`.
* `HasFPowerSeriesOnBall f p x r`: on the ball of center `x` with radius `r`,
`f (x + y) = ∑'_n pₙ yⁿ`.
* `HasFPowerSeriesAt f p x`: on some ball of center `x` with positive radius, holds
`HasFPowerSeriesOnBall f p x r`.
* `AnalyticAt 𝕜 f x`: there exists a power series `p` such that holds `HasFPowerSeriesAt f p x`.
* `AnalyticOn 𝕜 f s`: the function `f` is analytic at every point of `s`.
We develop the basic properties of these notions, notably:
* If a function admits a power series, it is continuous (see
`HasFPowerSeriesOnBall.continuousOn` and `HasFPowerSeriesAt.continuousAt` and
`AnalyticAt.continuousAt`).
* In a complete space, the sum of a formal power series with positive radius is well defined on the
disk of convergence, see `FormalMultilinearSeries.hasFPowerSeriesOnBall`.
* If a function admits a power series in a ball, then it is analytic at any point `y` of this ball,
and the power series there can be expressed in terms of the initial power series `p` as
`p.changeOrigin y`. See `HasFPowerSeriesOnBall.changeOrigin`. It follows in particular that
the set of points at which a given function is analytic is open, see `isOpen_analyticAt`.
## Implementation details
We only introduce the radius of convergence of a power series, as `p.radius`.
For a power series in finitely many dimensions, there is a finer (directional, coordinate-dependent)
notion, describing the polydisk of convergence. This notion is more specific, and not necessary to
build the general theory. We do not define it here.
-/
noncomputable section
variable {𝕜 E F G : Type*}
open Topology Classical BigOperators NNReal Filter ENNReal
open Set Filter Asymptotics
namespace FormalMultilinearSeries
variable [Ring 𝕜] [AddCommGroup E] [AddCommGroup F] [Module 𝕜 E] [Module 𝕜 F]
variable [TopologicalSpace E] [TopologicalSpace F]
variable [TopologicalAddGroup E] [TopologicalAddGroup F]
variable [ContinuousConstSMul 𝕜 E] [ContinuousConstSMul 𝕜 F]
/-- Given a formal multilinear series `p` and a vector `x`, then `p.sum x` is the sum `Σ pₙ xⁿ`. A
priori, it only behaves well when `‖x‖ < p.radius`. -/
protected def sum (p : FormalMultilinearSeries 𝕜 E F) (x : E) : F :=
∑' n : ℕ, p n fun _ => x
#align formal_multilinear_series.sum FormalMultilinearSeries.sum
/-- Given a formal multilinear series `p` and a vector `x`, then `p.partialSum n x` is the sum
`Σ pₖ xᵏ` for `k ∈ {0,..., n-1}`. -/
def partialSum (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) (x : E) : F :=
∑ k in Finset.range n, p k fun _ : Fin k => x
#align formal_multilinear_series.partial_sum FormalMultilinearSeries.partialSum
/-- The partial sums of a formal multilinear series are continuous. -/
theorem partialSum_continuous (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) :
Continuous (p.partialSum n) := by
unfold partialSum -- Porting note: added
continuity
#align formal_multilinear_series.partial_sum_continuous FormalMultilinearSeries.partialSum_continuous
end FormalMultilinearSeries
/-! ### The radius of a formal multilinear series -/
variable [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F]
[NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G]
namespace FormalMultilinearSeries
variable (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
/-- The radius of a formal multilinear series is the largest `r` such that the sum `Σ ‖pₙ‖ ‖y‖ⁿ`
converges for all `‖y‖ < r`. This implies that `Σ pₙ yⁿ` converges for all `‖y‖ < r`, but these
definitions are *not* equivalent in general. -/
def radius (p : FormalMultilinearSeries 𝕜 E F) : ℝ≥0∞ :=
⨆ (r : ℝ≥0) (C : ℝ) (_ : ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C), (r : ℝ≥0∞)
#align formal_multilinear_series.radius FormalMultilinearSeries.radius
/-- If `‖pₙ‖ rⁿ` is bounded in `n`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_bound (C : ℝ) {r : ℝ≥0} (h : ∀ n : ℕ, ‖p n‖ * (r : ℝ) ^ n ≤ C) :
(r : ℝ≥0∞) ≤ p.radius :=
le_iSup_of_le r <| le_iSup_of_le C <| le_iSup (fun _ => (r : ℝ≥0∞)) h
#align formal_multilinear_series.le_radius_of_bound FormalMultilinearSeries.le_radius_of_bound
/-- If `‖pₙ‖ rⁿ` is bounded in `n`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_bound_nnreal (C : ℝ≥0) {r : ℝ≥0} (h : ∀ n : ℕ, ‖p n‖₊ * r ^ n ≤ C) :
(r : ℝ≥0∞) ≤ p.radius :=
p.le_radius_of_bound C fun n => mod_cast h n
#align formal_multilinear_series.le_radius_of_bound_nnreal FormalMultilinearSeries.le_radius_of_bound_nnreal
/-- If `‖pₙ‖ rⁿ = O(1)`, as `n → ∞`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_isBigO (h : (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] fun _ => (1 : ℝ)) :
↑r ≤ p.radius :=
Exists.elim (isBigO_one_nat_atTop_iff.1 h) fun C hC =>
p.le_radius_of_bound C fun n => (le_abs_self _).trans (hC n)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.le_radius_of_is_O FormalMultilinearSeries.le_radius_of_isBigO
theorem le_radius_of_eventually_le (C) (h : ∀ᶠ n in atTop, ‖p n‖ * (r : ℝ) ^ n ≤ C) :
↑r ≤ p.radius :=
p.le_radius_of_isBigO <| IsBigO.of_bound C <| h.mono fun n hn => by simpa
#align formal_multilinear_series.le_radius_of_eventually_le FormalMultilinearSeries.le_radius_of_eventually_le
theorem le_radius_of_summable_nnnorm (h : Summable fun n => ‖p n‖₊ * r ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_bound_nnreal (∑' n, ‖p n‖₊ * r ^ n) fun _ => le_tsum' h _
#align formal_multilinear_series.le_radius_of_summable_nnnorm FormalMultilinearSeries.le_radius_of_summable_nnnorm
theorem le_radius_of_summable (h : Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_summable_nnnorm <| by
simp only [← coe_nnnorm] at h
exact mod_cast h
#align formal_multilinear_series.le_radius_of_summable FormalMultilinearSeries.le_radius_of_summable
theorem radius_eq_top_of_forall_nnreal_isBigO
(h : ∀ r : ℝ≥0, (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] fun _ => (1 : ℝ)) : p.radius = ∞ :=
ENNReal.eq_top_of_forall_nnreal_le fun r => p.le_radius_of_isBigO (h r)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.radius_eq_top_of_forall_nnreal_is_O FormalMultilinearSeries.radius_eq_top_of_forall_nnreal_isBigO
theorem radius_eq_top_of_eventually_eq_zero (h : ∀ᶠ n in atTop, p n = 0) : p.radius = ∞ :=
p.radius_eq_top_of_forall_nnreal_isBigO fun r =>
(isBigO_zero _ _).congr' (h.mono fun n hn => by simp [hn]) EventuallyEq.rfl
#align formal_multilinear_series.radius_eq_top_of_eventually_eq_zero FormalMultilinearSeries.radius_eq_top_of_eventually_eq_zero
theorem radius_eq_top_of_forall_image_add_eq_zero (n : ℕ) (hn : ∀ m, p (m + n) = 0) :
p.radius = ∞ :=
p.radius_eq_top_of_eventually_eq_zero <|
mem_atTop_sets.2 ⟨n, fun _ hk => tsub_add_cancel_of_le hk ▸ hn _⟩
#align formal_multilinear_series.radius_eq_top_of_forall_image_add_eq_zero FormalMultilinearSeries.radius_eq_top_of_forall_image_add_eq_zero
@[simp]
theorem constFormalMultilinearSeries_radius {v : F} :
(constFormalMultilinearSeries 𝕜 E v).radius = ⊤ :=
(constFormalMultilinearSeries 𝕜 E v).radius_eq_top_of_forall_image_add_eq_zero 1
(by simp [constFormalMultilinearSeries])
#align formal_multilinear_series.const_formal_multilinear_series_radius FormalMultilinearSeries.constFormalMultilinearSeries_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` tends to zero exponentially:
for some `0 < a < 1`, `‖p n‖ rⁿ = o(aⁿ)`. -/
theorem isLittleO_of_lt_radius (h : ↑r < p.radius) :
∃ a ∈ Ioo (0 : ℝ) 1, (fun n => ‖p n‖ * (r : ℝ) ^ n) =o[atTop] (a ^ ·) := by
have := (TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 4
rw [this]
-- Porting note: was
-- rw [(TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 4]
simp only [radius, lt_iSup_iff] at h
rcases h with ⟨t, C, hC, rt⟩
rw [ENNReal.coe_lt_coe, ← NNReal.coe_lt_coe] at rt
have : 0 < (t : ℝ) := r.coe_nonneg.trans_lt rt
rw [← div_lt_one this] at rt
refine' ⟨_, rt, C, Or.inr zero_lt_one, fun n => _⟩
calc
|‖p n‖ * (r : ℝ) ^ n| = ‖p n‖ * (t : ℝ) ^ n * (r / t : ℝ) ^ n := by
field_simp [mul_right_comm, abs_mul]
_ ≤ C * (r / t : ℝ) ^ n := by gcongr; apply hC
#align formal_multilinear_series.is_o_of_lt_radius FormalMultilinearSeries.isLittleO_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ = o(1)`. -/
theorem isLittleO_one_of_lt_radius (h : ↑r < p.radius) :
(fun n => ‖p n‖ * (r : ℝ) ^ n) =o[atTop] (fun _ => 1 : ℕ → ℝ) :=
let ⟨_, ha, hp⟩ := p.isLittleO_of_lt_radius h
hp.trans <| (isLittleO_pow_pow_of_lt_left ha.1.le ha.2).congr (fun _ => rfl) one_pow
#align formal_multilinear_series.is_o_one_of_lt_radius FormalMultilinearSeries.isLittleO_one_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` tends to zero exponentially:
for some `0 < a < 1` and `C > 0`, `‖p n‖ * r ^ n ≤ C * a ^ n`. -/
theorem norm_mul_pow_le_mul_pow_of_lt_radius (h : ↑r < p.radius) :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C * a ^ n := by
-- Porting note: moved out of `rcases`
have := ((TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 5).mp
(p.isLittleO_of_lt_radius h)
rcases this with ⟨a, ha, C, hC, H⟩
exact ⟨a, ha, C, hC, fun n => (le_abs_self _).trans (H n)⟩
#align formal_multilinear_series.norm_mul_pow_le_mul_pow_of_lt_radius FormalMultilinearSeries.norm_mul_pow_le_mul_pow_of_lt_radius
/-- If `r ≠ 0` and `‖pₙ‖ rⁿ = O(aⁿ)` for some `-1 < a < 1`, then `r < p.radius`. -/
theorem lt_radius_of_isBigO (h₀ : r ≠ 0) {a : ℝ} (ha : a ∈ Ioo (-1 : ℝ) 1)
(hp : (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] (a ^ ·)) : ↑r < p.radius := by
-- Porting note: moved out of `rcases`
have := ((TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 2 5)
rcases this.mp ⟨a, ha, hp⟩ with ⟨a, ha, C, hC, hp⟩
rw [← pos_iff_ne_zero, ← NNReal.coe_pos] at h₀
lift a to ℝ≥0 using ha.1.le
have : (r : ℝ) < r / a := by
simpa only [div_one] using (div_lt_div_left h₀ zero_lt_one ha.1).2 ha.2
norm_cast at this
rw [← ENNReal.coe_lt_coe] at this
refine' this.trans_le (p.le_radius_of_bound C fun n => _)
rw [NNReal.coe_div, div_pow, ← mul_div_assoc, div_le_iff (pow_pos ha.1 n)]
exact (le_abs_self _).trans (hp n)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.lt_radius_of_is_O FormalMultilinearSeries.lt_radius_of_isBigO
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem norm_mul_pow_le_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C :=
let ⟨_, ha, C, hC, h⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius h
⟨C, hC, fun n => (h n).trans <| mul_le_of_le_one_right hC.lt.le (pow_le_one _ ha.1.le ha.2.le)⟩
#align formal_multilinear_series.norm_mul_pow_le_of_lt_radius FormalMultilinearSeries.norm_mul_pow_le_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem norm_le_div_pow_of_pos_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h0 : 0 < r) (h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖ ≤ C / (r : ℝ) ^ n :=
let ⟨C, hC, hp⟩ := p.norm_mul_pow_le_of_lt_radius h
⟨C, hC, fun n => Iff.mpr (le_div_iff (pow_pos h0 _)) (hp n)⟩
#align formal_multilinear_series.norm_le_div_pow_of_pos_of_lt_radius FormalMultilinearSeries.norm_le_div_pow_of_pos_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem nnnorm_mul_pow_le_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖₊ * r ^ n ≤ C :=
let ⟨C, hC, hp⟩ := p.norm_mul_pow_le_of_lt_radius h
⟨⟨C, hC.lt.le⟩, hC, mod_cast hp⟩
#align formal_multilinear_series.nnnorm_mul_pow_le_of_lt_radius FormalMultilinearSeries.nnnorm_mul_pow_le_of_lt_radius
theorem le_radius_of_tendsto (p : FormalMultilinearSeries 𝕜 E F) {l : ℝ}
(h : Tendsto (fun n => ‖p n‖ * (r : ℝ) ^ n) atTop (𝓝 l)) : ↑r ≤ p.radius :=
p.le_radius_of_isBigO (h.isBigO_one _)
#align formal_multilinear_series.le_radius_of_tendsto FormalMultilinearSeries.le_radius_of_tendsto
theorem le_radius_of_summable_norm (p : FormalMultilinearSeries 𝕜 E F)
(hs : Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_tendsto hs.tendsto_atTop_zero
#align formal_multilinear_series.le_radius_of_summable_norm FormalMultilinearSeries.le_radius_of_summable_norm
theorem not_summable_norm_of_radius_lt_nnnorm (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(h : p.radius < ‖x‖₊) : ¬Summable fun n => ‖p n‖ * ‖x‖ ^ n :=
fun hs => not_le_of_lt h (p.le_radius_of_summable_norm hs)
#align formal_multilinear_series.not_summable_norm_of_radius_lt_nnnorm FormalMultilinearSeries.not_summable_norm_of_radius_lt_nnnorm
theorem summable_norm_mul_pow (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0} (h : ↑r < p.radius) :
Summable fun n : ℕ => ‖p n‖ * (r : ℝ) ^ n := by
obtain ⟨a, ha : a ∈ Ioo (0 : ℝ) 1, C, - : 0 < C, hp⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius h
exact .of_nonneg_of_le (fun n => mul_nonneg (norm_nonneg _) (pow_nonneg r.coe_nonneg _))
hp ((summable_geometric_of_lt_1 ha.1.le ha.2).mul_left _)
#align formal_multilinear_series.summable_norm_mul_pow FormalMultilinearSeries.summable_norm_mul_pow
theorem summable_norm_apply (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : Summable fun n : ℕ => ‖p n fun _ => x‖ := by
rw [mem_emetric_ball_zero_iff] at hx
refine' .of_nonneg_of_le
(fun _ => norm_nonneg _) (fun n => ((p n).le_op_norm _).trans_eq _) (p.summable_norm_mul_pow hx)
simp
#align formal_multilinear_series.summable_norm_apply FormalMultilinearSeries.summable_norm_apply
theorem summable_nnnorm_mul_pow (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0} (h : ↑r < p.radius) :
Summable fun n : ℕ => ‖p n‖₊ * r ^ n := by
rw [← NNReal.summable_coe]
push_cast
exact p.summable_norm_mul_pow h
#align formal_multilinear_series.summable_nnnorm_mul_pow FormalMultilinearSeries.summable_nnnorm_mul_pow
protected theorem summable [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : Summable fun n : ℕ => p n fun _ => x :=
(p.summable_norm_apply hx).of_norm
#align formal_multilinear_series.summable FormalMultilinearSeries.summable
theorem radius_eq_top_of_summable_norm (p : FormalMultilinearSeries 𝕜 E F)
(hs : ∀ r : ℝ≥0, Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : p.radius = ∞ :=
ENNReal.eq_top_of_forall_nnreal_le fun r => p.le_radius_of_summable_norm (hs r)
#align formal_multilinear_series.radius_eq_top_of_summable_norm FormalMultilinearSeries.radius_eq_top_of_summable_norm
theorem radius_eq_top_iff_summable_norm (p : FormalMultilinearSeries 𝕜 E F) :
p.radius = ∞ ↔ ∀ r : ℝ≥0, Summable fun n => ‖p n‖ * (r : ℝ) ^ n := by
constructor
· intro h r
obtain ⟨a, ha : a ∈ Ioo (0 : ℝ) 1, C, - : 0 < C, hp⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius
(show (r : ℝ≥0∞) < p.radius from h.symm ▸ ENNReal.coe_lt_top)
refine' .of_norm_bounded
(fun n => (C : ℝ) * a ^ n) ((summable_geometric_of_lt_1 ha.1.le ha.2).mul_left _) fun n => _
specialize hp n
rwa [Real.norm_of_nonneg (mul_nonneg (norm_nonneg _) (pow_nonneg r.coe_nonneg n))]
· exact p.radius_eq_top_of_summable_norm
#align formal_multilinear_series.radius_eq_top_iff_summable_norm FormalMultilinearSeries.radius_eq_top_iff_summable_norm
/-- If the radius of `p` is positive, then `‖pₙ‖` grows at most geometrically. -/
theorem le_mul_pow_of_radius_pos (p : FormalMultilinearSeries 𝕜 E F) (h : 0 < p.radius) :
∃ (C r : _) (hC : 0 < C) (_ : 0 < r), ∀ n, ‖p n‖ ≤ C * r ^ n := by
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 h with ⟨r, r0, rlt⟩
have rpos : 0 < (r : ℝ) := by simp [ENNReal.coe_pos.1 r0]
rcases norm_le_div_pow_of_pos_of_lt_radius p rpos rlt with ⟨C, Cpos, hCp⟩
refine' ⟨C, r⁻¹, Cpos, by simp only [inv_pos, rpos], fun n => _⟩
-- Porting note: was `convert`
rw [inv_pow, ← div_eq_mul_inv]
exact hCp n
#align formal_multilinear_series.le_mul_pow_of_radius_pos FormalMultilinearSeries.le_mul_pow_of_radius_pos
/-- The radius of the sum of two formal series is at least the minimum of their two radii. -/
theorem min_radius_le_radius_add (p q : FormalMultilinearSeries 𝕜 E F) :
min p.radius q.radius ≤ (p + q).radius := by
refine' ENNReal.le_of_forall_nnreal_lt fun r hr => _
rw [lt_min_iff] at hr
have := ((p.isLittleO_one_of_lt_radius hr.1).add (q.isLittleO_one_of_lt_radius hr.2)).isBigO
refine' (p + q).le_radius_of_isBigO ((isBigO_of_le _ fun n => _).trans this)
rw [← add_mul, norm_mul, norm_mul, norm_norm]
exact mul_le_mul_of_nonneg_right ((norm_add_le _ _).trans (le_abs_self _)) (norm_nonneg _)
#align formal_multilinear_series.min_radius_le_radius_add FormalMultilinearSeries.min_radius_le_radius_add
@[simp]
theorem radius_neg (p : FormalMultilinearSeries 𝕜 E F) : (-p).radius = p.radius := by
simp only [radius, neg_apply, norm_neg]
#align formal_multilinear_series.radius_neg FormalMultilinearSeries.radius_neg
protected theorem hasSum [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : HasSum (fun n : ℕ => p n fun _ => x) (p.sum x) :=
(p.summable hx).hasSum
#align formal_multilinear_series.has_sum FormalMultilinearSeries.hasSum
theorem radius_le_radius_continuousLinearMap_comp (p : FormalMultilinearSeries 𝕜 E F)
(f : F →L[𝕜] G) : p.radius ≤ (f.compFormalMultilinearSeries p).radius := by
refine' ENNReal.le_of_forall_nnreal_lt fun r hr => _
apply le_radius_of_isBigO
apply (IsBigO.trans_isLittleO _ (p.isLittleO_one_of_lt_radius hr)).isBigO
refine' IsBigO.mul (@IsBigOWith.isBigO _ _ _ _ _ ‖f‖ _ _ _ _) (isBigO_refl _ _)
refine IsBigOWith.of_bound (eventually_of_forall fun n => ?_)
simpa only [norm_norm] using f.norm_compContinuousMultilinearMap_le (p n)
#align formal_multilinear_series.radius_le_radius_continuous_linear_map_comp FormalMultilinearSeries.radius_le_radius_continuousLinearMap_comp
end FormalMultilinearSeries
/-! ### Expanding a function as a power series -/
section
variable {f g : E → F} {p pf pg : FormalMultilinearSeries 𝕜 E F} {x : E} {r r' : ℝ≥0∞}
/-- Given a function `f : E → F` and a formal multilinear series `p`, we say that `f` has `p` as
a power series on the ball of radius `r > 0` around `x` if `f (x + y) = ∑' pₙ yⁿ` for all `‖y‖ < r`.
-/
structure HasFPowerSeriesOnBall (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (x : E) (r : ℝ≥0∞) :
Prop where
r_le : r ≤ p.radius
r_pos : 0 < r
hasSum :
∀ {y}, y ∈ EMetric.ball (0 : E) r → HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y))
#align has_fpower_series_on_ball HasFPowerSeriesOnBall
/-- Given a function `f : E → F` and a formal multilinear series `p`, we say that `f` has `p` as
a power series around `x` if `f (x + y) = ∑' pₙ yⁿ` for all `y` in a neighborhood of `0`. -/
def HasFPowerSeriesAt (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (x : E) :=
∃ r, HasFPowerSeriesOnBall f p x r
#align has_fpower_series_at HasFPowerSeriesAt
variable (𝕜)
/-- Given a function `f : E → F`, we say that `f` is analytic at `x` if it admits a convergent power
series expansion around `x`. -/
def AnalyticAt (f : E → F) (x : E) :=
∃ p : FormalMultilinearSeries 𝕜 E F, HasFPowerSeriesAt f p x
#align analytic_at AnalyticAt
/-- Given a function `f : E → F`, we say that `f` is analytic on a set `s` if it is analytic around
every point of `s`. -/
def AnalyticOn (f : E → F) (s : Set E) :=
∀ x, x ∈ s → AnalyticAt 𝕜 f x
#align analytic_on AnalyticOn
variable {𝕜}
theorem HasFPowerSeriesOnBall.hasFPowerSeriesAt (hf : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesAt f p x :=
⟨r, hf⟩
#align has_fpower_series_on_ball.has_fpower_series_at HasFPowerSeriesOnBall.hasFPowerSeriesAt
theorem HasFPowerSeriesAt.analyticAt (hf : HasFPowerSeriesAt f p x) : AnalyticAt 𝕜 f x :=
⟨p, hf⟩
#align has_fpower_series_at.analytic_at HasFPowerSeriesAt.analyticAt
theorem HasFPowerSeriesOnBall.analyticAt (hf : HasFPowerSeriesOnBall f p x r) : AnalyticAt 𝕜 f x :=
hf.hasFPowerSeriesAt.analyticAt
#align has_fpower_series_on_ball.analytic_at HasFPowerSeriesOnBall.analyticAt
theorem HasFPowerSeriesOnBall.congr (hf : HasFPowerSeriesOnBall f p x r)
(hg : EqOn f g (EMetric.ball x r)) : HasFPowerSeriesOnBall g p x r :=
{ r_le := hf.r_le
r_pos := hf.r_pos
hasSum := fun {y} hy => by
convert hf.hasSum hy using 1
apply hg.symm
simpa [edist_eq_coe_nnnorm_sub] using hy }
#align has_fpower_series_on_ball.congr HasFPowerSeriesOnBall.congr
/-- If a function `f` has a power series `p` around `x`, then the function `z ↦ f (z - y)` has the
same power series around `x + y`. -/
theorem HasFPowerSeriesOnBall.comp_sub (hf : HasFPowerSeriesOnBall f p x r) (y : E) :
HasFPowerSeriesOnBall (fun z => f (z - y)) p (x + y) r :=
{ r_le := hf.r_le
r_pos := hf.r_pos
hasSum := fun {z} hz => by
convert hf.hasSum hz using 2
abel }
#align has_fpower_series_on_ball.comp_sub HasFPowerSeriesOnBall.comp_sub
theorem HasFPowerSeriesOnBall.hasSum_sub (hf : HasFPowerSeriesOnBall f p x r) {y : E}
(hy : y ∈ EMetric.ball x r) : HasSum (fun n : ℕ => p n fun _ => y - x) (f y) := by
have : y - x ∈ EMetric.ball (0 : E) r := by simpa [edist_eq_coe_nnnorm_sub] using hy
simpa only [add_sub_cancel'_right] using hf.hasSum this
#align has_fpower_series_on_ball.has_sum_sub HasFPowerSeriesOnBall.hasSum_sub
theorem HasFPowerSeriesOnBall.radius_pos (hf : HasFPowerSeriesOnBall f p x r) : 0 < p.radius :=
lt_of_lt_of_le hf.r_pos hf.r_le
#align has_fpower_series_on_ball.radius_pos HasFPowerSeriesOnBall.radius_pos
theorem HasFPowerSeriesAt.radius_pos (hf : HasFPowerSeriesAt f p x) : 0 < p.radius :=
let ⟨_, hr⟩ := hf
hr.radius_pos
#align has_fpower_series_at.radius_pos HasFPowerSeriesAt.radius_pos
theorem HasFPowerSeriesOnBall.mono (hf : HasFPowerSeriesOnBall f p x r) (r'_pos : 0 < r')
(hr : r' ≤ r) : HasFPowerSeriesOnBall f p x r' :=
⟨le_trans hr hf.1, r'_pos, fun hy => hf.hasSum (EMetric.ball_subset_ball hr hy)⟩
#align has_fpower_series_on_ball.mono HasFPowerSeriesOnBall.mono
theorem HasFPowerSeriesAt.congr (hf : HasFPowerSeriesAt f p x) (hg : f =ᶠ[𝓝 x] g) :
HasFPowerSeriesAt g p x := by
rcases hf with ⟨r₁, h₁⟩
rcases EMetric.mem_nhds_iff.mp hg with ⟨r₂, h₂pos, h₂⟩
exact ⟨min r₁ r₂,
(h₁.mono (lt_min h₁.r_pos h₂pos) inf_le_left).congr
fun y hy => h₂ (EMetric.ball_subset_ball inf_le_right hy)⟩
#align has_fpower_series_at.congr HasFPowerSeriesAt.congr
protected theorem HasFPowerSeriesAt.eventually (hf : HasFPowerSeriesAt f p x) :
∀ᶠ r : ℝ≥0∞ in 𝓝[>] 0, HasFPowerSeriesOnBall f p x r :=
let ⟨_, hr⟩ := hf
mem_of_superset (Ioo_mem_nhdsWithin_Ioi (left_mem_Ico.2 hr.r_pos)) fun _ hr' =>
hr.mono hr'.1 hr'.2.le
#align has_fpower_series_at.eventually HasFPowerSeriesAt.eventually
theorem HasFPowerSeriesOnBall.eventually_hasSum (hf : HasFPowerSeriesOnBall f p x r) :
∀ᶠ y in 𝓝 0, HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y)) := by
filter_upwards [EMetric.ball_mem_nhds (0 : E) hf.r_pos] using fun _ => hf.hasSum
#align has_fpower_series_on_ball.eventually_has_sum HasFPowerSeriesOnBall.eventually_hasSum
theorem HasFPowerSeriesAt.eventually_hasSum (hf : HasFPowerSeriesAt f p x) :
∀ᶠ y in 𝓝 0, HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y)) :=
let ⟨_, hr⟩ := hf
hr.eventually_hasSum
#align has_fpower_series_at.eventually_has_sum HasFPowerSeriesAt.eventually_hasSum
theorem HasFPowerSeriesOnBall.eventually_hasSum_sub (hf : HasFPowerSeriesOnBall f p x r) :
∀ᶠ y in 𝓝 x, HasSum (fun n : ℕ => p n fun _ : Fin n => y - x) (f y) := by
filter_upwards [EMetric.ball_mem_nhds x hf.r_pos] with y using hf.hasSum_sub
#align has_fpower_series_on_ball.eventually_has_sum_sub HasFPowerSeriesOnBall.eventually_hasSum_sub
theorem HasFPowerSeriesAt.eventually_hasSum_sub (hf : HasFPowerSeriesAt f p x) :
∀ᶠ y in 𝓝 x, HasSum (fun n : ℕ => p n fun _ : Fin n => y - x) (f y) :=
let ⟨_, hr⟩ := hf
hr.eventually_hasSum_sub
#align has_fpower_series_at.eventually_has_sum_sub HasFPowerSeriesAt.eventually_hasSum_sub
theorem HasFPowerSeriesOnBall.eventually_eq_zero
(hf : HasFPowerSeriesOnBall f (0 : FormalMultilinearSeries 𝕜 E F) x r) :
∀ᶠ z in 𝓝 x, f z = 0 := by
filter_upwards [hf.eventually_hasSum_sub] with z hz using hz.unique hasSum_zero
#align has_fpower_series_on_ball.eventually_eq_zero HasFPowerSeriesOnBall.eventually_eq_zero
theorem HasFPowerSeriesAt.eventually_eq_zero
(hf : HasFPowerSeriesAt f (0 : FormalMultilinearSeries 𝕜 E F) x) : ∀ᶠ z in 𝓝 x, f z = 0 :=
let ⟨_, hr⟩ := hf
hr.eventually_eq_zero
#align has_fpower_series_at.eventually_eq_zero HasFPowerSeriesAt.eventually_eq_zero
theorem hasFPowerSeriesOnBall_const {c : F} {e : E} :
HasFPowerSeriesOnBall (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e ⊤ := by
refine' ⟨by simp, WithTop.zero_lt_top, fun _ => hasSum_single 0 fun n hn => _⟩
simp [constFormalMultilinearSeries_apply hn]
#align has_fpower_series_on_ball_const hasFPowerSeriesOnBall_const
theorem hasFPowerSeriesAt_const {c : F} {e : E} :
HasFPowerSeriesAt (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e :=
⟨⊤, hasFPowerSeriesOnBall_const⟩
#align has_fpower_series_at_const hasFPowerSeriesAt_const
theorem analyticAt_const {v : F} : AnalyticAt 𝕜 (fun _ => v) x :=
⟨constFormalMultilinearSeries 𝕜 E v, hasFPowerSeriesAt_const⟩
#align analytic_at_const analyticAt_const
theorem analyticOn_const {v : F} {s : Set E} : AnalyticOn 𝕜 (fun _ => v) s :=
fun _ _ => analyticAt_const
#align analytic_on_const analyticOn_const
theorem HasFPowerSeriesOnBall.add (hf : HasFPowerSeriesOnBall f pf x r)
(hg : HasFPowerSeriesOnBall g pg x r) : HasFPowerSeriesOnBall (f + g) (pf + pg) x r :=
{ r_le := le_trans (le_min_iff.2 ⟨hf.r_le, hg.r_le⟩) (pf.min_radius_le_radius_add pg)
r_pos := hf.r_pos
hasSum := fun hy => (hf.hasSum hy).add (hg.hasSum hy) }
#align has_fpower_series_on_ball.add HasFPowerSeriesOnBall.add
theorem HasFPowerSeriesAt.add (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x) :
HasFPowerSeriesAt (f + g) (pf + pg) x := by
rcases (hf.eventually.and hg.eventually).exists with ⟨r, hr⟩
exact ⟨r, hr.1.add hr.2⟩
#align has_fpower_series_at.add HasFPowerSeriesAt.add
theorem AnalyticAt.congr (hf : AnalyticAt 𝕜 f x) (hg : f =ᶠ[𝓝 x] g) : AnalyticAt 𝕜 g x :=
let ⟨_, hpf⟩ := hf
(hpf.congr hg).analyticAt
theorem analyticAt_congr (h : f =ᶠ[𝓝 x] g) : AnalyticAt 𝕜 f x ↔ AnalyticAt 𝕜 g x :=
⟨fun hf ↦ hf.congr h, fun hg ↦ hg.congr h.symm⟩
theorem AnalyticAt.add (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) : AnalyticAt 𝕜 (f + g) x :=
let ⟨_, hpf⟩ := hf
let ⟨_, hqf⟩ := hg
(hpf.add hqf).analyticAt
#align analytic_at.add AnalyticAt.add
theorem HasFPowerSeriesOnBall.neg (hf : HasFPowerSeriesOnBall f pf x r) :
HasFPowerSeriesOnBall (-f) (-pf) x r :=
{ r_le := by
rw [pf.radius_neg]
exact hf.r_le
r_pos := hf.r_pos
hasSum := fun hy => (hf.hasSum hy).neg }
#align has_fpower_series_on_ball.neg HasFPowerSeriesOnBall.neg
theorem HasFPowerSeriesAt.neg (hf : HasFPowerSeriesAt f pf x) : HasFPowerSeriesAt (-f) (-pf) x :=
let ⟨_, hrf⟩ := hf
hrf.neg.hasFPowerSeriesAt
#align has_fpower_series_at.neg HasFPowerSeriesAt.neg
theorem AnalyticAt.neg (hf : AnalyticAt 𝕜 f x) : AnalyticAt 𝕜 (-f) x :=
let ⟨_, hpf⟩ := hf
hpf.neg.analyticAt
#align analytic_at.neg AnalyticAt.neg
theorem HasFPowerSeriesOnBall.sub (hf : HasFPowerSeriesOnBall f pf x r)
(hg : HasFPowerSeriesOnBall g pg x r) : HasFPowerSeriesOnBall (f - g) (pf - pg) x r := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align has_fpower_series_on_ball.sub HasFPowerSeriesOnBall.sub
theorem HasFPowerSeriesAt.sub (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x) :
HasFPowerSeriesAt (f - g) (pf - pg) x := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align has_fpower_series_at.sub HasFPowerSeriesAt.sub
theorem AnalyticAt.sub (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) :
AnalyticAt 𝕜 (f - g) x := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align analytic_at.sub AnalyticAt.sub
theorem AnalyticOn.mono {s t : Set E} (hf : AnalyticOn 𝕜 f t) (hst : s ⊆ t) : AnalyticOn 𝕜 f s :=
fun z hz => hf z (hst hz)
#align analytic_on.mono AnalyticOn.mono
theorem AnalyticOn.congr' {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : f =ᶠ[𝓝ˢ s] g) :
AnalyticOn 𝕜 g s :=
fun z hz => (hf z hz).congr (mem_nhdsSet_iff_forall.mp hg z hz)
theorem analyticOn_congr' {s : Set E} (h : f =ᶠ[𝓝ˢ s] g) : AnalyticOn 𝕜 f s ↔ AnalyticOn 𝕜 g s :=
⟨fun hf => hf.congr' h, fun hg => hg.congr' h.symm⟩
theorem AnalyticOn.congr {s : Set E} (hs : IsOpen s) (hf : AnalyticOn 𝕜 f s) (hg : s.EqOn f g) :
AnalyticOn 𝕜 g s :=
hf.congr' $ mem_nhdsSet_iff_forall.mpr
(fun _ hz => eventuallyEq_iff_exists_mem.mpr ⟨s, hs.mem_nhds hz, hg⟩)
theorem analyticOn_congr {s : Set E} (hs : IsOpen s) (h : s.EqOn f g) : AnalyticOn 𝕜 f s ↔
AnalyticOn 𝕜 g s := ⟨fun hf => hf.congr hs h, fun hg => hg.congr hs h.symm⟩
theorem AnalyticOn.add {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
AnalyticOn 𝕜 (f + g) s :=
fun z hz => (hf z hz).add (hg z hz)
#align analytic_on.add AnalyticOn.add
theorem AnalyticOn.sub {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
AnalyticOn 𝕜 (f - g) s :=
fun z hz => (hf z hz).sub (hg z hz)
#align analytic_on.sub AnalyticOn.sub
theorem HasFPowerSeriesOnBall.coeff_zero (hf : HasFPowerSeriesOnBall f pf x r) (v : Fin 0 → E) :
pf 0 v = f x := by
have v_eq : v = fun i => 0 := Subsingleton.elim _ _
have zero_mem : (0 : E) ∈ EMetric.ball (0 : E) r := by simp [hf.r_pos]
have : ∀ i, i ≠ 0 → (pf i fun j => 0) = 0 := by
intro i hi
have : 0 < i := pos_iff_ne_zero.2 hi
exact ContinuousMultilinearMap.map_coord_zero _ (⟨0, this⟩ : Fin i) rfl
have A := (hf.hasSum zero_mem).unique (hasSum_single _ this)
simpa [v_eq] using A.symm
#align has_fpower_series_on_ball.coeff_zero HasFPowerSeriesOnBall.coeff_zero
theorem HasFPowerSeriesAt.coeff_zero (hf : HasFPowerSeriesAt f pf x) (v : Fin 0 → E) :
pf 0 v = f x :=
let ⟨_, hrf⟩ := hf
hrf.coeff_zero v
#align has_fpower_series_at.coeff_zero HasFPowerSeriesAt.coeff_zero
/-- If a function `f` has a power series `p` on a ball and `g` is linear, then `g ∘ f` has the
power series `g ∘ p` on the same ball. -/
theorem ContinuousLinearMap.comp_hasFPowerSeriesOnBall (g : F →L[𝕜] G)
(h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (g ∘ f) (g.compFormalMultilinearSeries p) x r :=
{ r_le := h.r_le.trans (p.radius_le_radius_continuousLinearMap_comp _)
r_pos := h.r_pos
hasSum := fun hy => by
simpa only [ContinuousLinearMap.compFormalMultilinearSeries_apply,
ContinuousLinearMap.compContinuousMultilinearMap_coe, Function.comp_apply] using
g.hasSum (h.hasSum hy) }
#align continuous_linear_map.comp_has_fpower_series_on_ball ContinuousLinearMap.comp_hasFPowerSeriesOnBall
/-- If a function `f` is analytic on a set `s` and `g` is linear, then `g ∘ f` is analytic
on `s`. -/
theorem ContinuousLinearMap.comp_analyticOn {s : Set E} (g : F →L[𝕜] G) (h : AnalyticOn 𝕜 f s) :
AnalyticOn 𝕜 (g ∘ f) s := by
rintro x hx
rcases h x hx with ⟨p, r, hp⟩
exact ⟨g.compFormalMultilinearSeries p, r, g.comp_hasFPowerSeriesOnBall hp⟩
#align continuous_linear_map.comp_analytic_on ContinuousLinearMap.comp_analyticOn
/-- If a function admits a power series expansion, then it is exponentially close to the partial
sums of this power series on strict subdisks of the disk of convergence.
This version provides an upper estimate that decreases both in `‖y‖` and `n`. See also
`HasFPowerSeriesOnBall.uniform_geometric_approx` for a weaker version. -/
theorem HasFPowerSeriesOnBall.uniform_geometric_approx' {r' : ℝ≥0}
(hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n := by
obtain ⟨a, ha, C, hC, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n, ‖p n‖ * (r' : ℝ) ^ n ≤ C * a ^ n :=
p.norm_mul_pow_le_mul_pow_of_lt_radius (h.trans_le hf.r_le)
refine' ⟨a, ha, C / (1 - a), div_pos hC (sub_pos.2 ha.2), fun y hy n => _⟩
have yr' : ‖y‖ < r' := by
rw [ball_zero_eq] at hy
exact hy
have hr'0 : 0 < (r' : ℝ) := (norm_nonneg _).trans_lt yr'
have : y ∈ EMetric.ball (0 : E) r := by
refine' mem_emetric_ball_zero_iff.2 (lt_trans _ h)
exact mod_cast yr'
rw [norm_sub_rev, ← mul_div_right_comm]
have ya : a * (‖y‖ / ↑r') ≤ a :=
mul_le_of_le_one_right ha.1.le (div_le_one_of_le yr'.le r'.coe_nonneg)
suffices ‖p.partialSum n y - f (x + y)‖ ≤ C * (a * (‖y‖ / r')) ^ n / (1 - a * (‖y‖ / r')) by
refine' this.trans _
have : 0 < a := ha.1
gcongr
apply_rules [sub_pos.2, ha.2]
apply norm_sub_le_of_geometric_bound_of_hasSum (ya.trans_lt ha.2) _ (hf.hasSum this)
intro n
calc
‖(p n) fun _ : Fin n => y‖
_ ≤ ‖p n‖ * ∏ _i : Fin n, ‖y‖ := ContinuousMultilinearMap.le_op_norm _ _
_ = ‖p n‖ * (r' : ℝ) ^ n * (‖y‖ / r') ^ n := by field_simp [mul_right_comm]
_ ≤ C * a ^ n * (‖y‖ / r') ^ n := by gcongr ?_ * _; apply hp
_ ≤ C * (a * (‖y‖ / r')) ^ n := by rw [mul_pow, mul_assoc]
#align has_fpower_series_on_ball.uniform_geometric_approx' HasFPowerSeriesOnBall.uniform_geometric_approx'
/-- If a function admits a power series expansion, then it is exponentially close to the partial
sums of this power series on strict subdisks of the disk of convergence. -/
theorem HasFPowerSeriesOnBall.uniform_geometric_approx {r' : ℝ≥0}
(hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
∃ a ∈ Ioo (0 : ℝ) 1,
∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n, ‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n := by
obtain ⟨a, ha, C, hC, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n :=
hf.uniform_geometric_approx' h
refine' ⟨a, ha, C, hC, fun y hy n => (hp y hy n).trans _⟩
have yr' : ‖y‖ < r' := by rwa [ball_zero_eq] at hy
gcongr
exacts [mul_nonneg ha.1.le (div_nonneg (norm_nonneg y) r'.coe_nonneg),
mul_le_of_le_one_right ha.1.le (div_le_one_of_le yr'.le r'.coe_nonneg)]
#align has_fpower_series_on_ball.uniform_geometric_approx HasFPowerSeriesOnBall.uniform_geometric_approx
/-- Taylor formula for an analytic function, `IsBigO` version. -/
theorem HasFPowerSeriesAt.isBigO_sub_partialSum_pow (hf : HasFPowerSeriesAt f p x) (n : ℕ) :
(fun y : E => f (x + y) - p.partialSum n y) =O[𝓝 0] fun y => ‖y‖ ^ n := by
rcases hf with ⟨r, hf⟩
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hf.r_pos with ⟨r', r'0, h⟩
obtain ⟨a, -, C, -, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n :=
hf.uniform_geometric_approx' h
refine' isBigO_iff.2 ⟨C * (a / r') ^ n, _⟩
replace r'0 : 0 < (r' : ℝ); · exact mod_cast r'0
filter_upwards [Metric.ball_mem_nhds (0 : E) r'0] with y hy
simpa [mul_pow, mul_div_assoc, mul_assoc, div_mul_eq_mul_div] using hp y hy n
set_option linter.uppercaseLean3 false in
#align has_fpower_series_at.is_O_sub_partial_sum_pow HasFPowerSeriesAt.isBigO_sub_partialSum_pow
/-- If `f` has formal power series `∑ n, pₙ` on a ball of radius `r`, then for `y, z` in any smaller
ball, the norm of the difference `f y - f z - p 1 (fun _ ↦ y - z)` is bounded above by
`C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖`. This lemma formulates this property using `IsBigO` and
`Filter.principal` on `E × E`. -/
theorem HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal
(hf : HasFPowerSeriesOnBall f p x r) (hr : r' < r) :
(fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) =O[𝓟 (EMetric.ball (x, x) r')]
fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖ := by
lift r' to ℝ≥0 using ne_top_of_lt hr
rcases (zero_le r').eq_or_lt with (rfl | hr'0)
· simp only [isBigO_bot, EMetric.ball_zero, principal_empty, ENNReal.coe_zero]
obtain ⟨a, ha, C, hC : 0 < C, hp⟩ :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n : ℕ, ‖p n‖ * (r' : ℝ) ^ n ≤ C * a ^ n
exact p.norm_mul_pow_le_mul_pow_of_lt_radius (hr.trans_le hf.r_le)
simp only [← le_div_iff (pow_pos (NNReal.coe_pos.2 hr'0) _)] at hp
set L : E × E → ℝ := fun y =>
C * (a / r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * (a / (1 - a) ^ 2 + 2 / (1 - a))
have hL : ∀ y ∈ EMetric.ball (x, x) r', ‖f y.1 - f y.2 - p 1 fun _ => y.1 - y.2‖ ≤ L y := by
intro y hy'
have hy : y ∈ EMetric.ball x r ×ˢ EMetric.ball x r := by
rw [EMetric.ball_prod_same]
exact EMetric.ball_subset_ball hr.le hy'
set A : ℕ → F := fun n => (p n fun _ => y.1 - x) - p n fun _ => y.2 - x
have hA : HasSum (fun n => A (n + 2)) (f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) := by
convert (hasSum_nat_add_iff' 2).2 ((hf.hasSum_sub hy.1).sub (hf.hasSum_sub hy.2)) using 1
rw [Finset.sum_range_succ, Finset.sum_range_one, hf.coeff_zero, hf.coeff_zero, sub_self,
zero_add, ← Subsingleton.pi_single_eq (0 : Fin 1) (y.1 - x), Pi.single,
← Subsingleton.pi_single_eq (0 : Fin 1) (y.2 - x), Pi.single, ← (p 1).map_sub, ← Pi.single,
Subsingleton.pi_single_eq, sub_sub_sub_cancel_right]
rw [EMetric.mem_ball, edist_eq_coe_nnnorm_sub, ENNReal.coe_lt_coe] at hy'
set B : ℕ → ℝ := fun n => C * (a / r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * ((n + 2) * a ^ n)
have hAB : ∀ n, ‖A (n + 2)‖ ≤ B n := fun n =>
calc
‖A (n + 2)‖ ≤ ‖p (n + 2)‖ * ↑(n + 2) * ‖y - (x, x)‖ ^ (n + 1) * ‖y.1 - y.2‖ := by
-- porting note: `pi_norm_const` was `pi_norm_const (_ : E)`
simpa only [Fintype.card_fin, pi_norm_const, Prod.norm_def, Pi.sub_def,
Prod.fst_sub, Prod.snd_sub, sub_sub_sub_cancel_right] using
(p <| n + 2).norm_image_sub_le (fun _ => y.1 - x) fun _ => y.2 - x
_ = ‖p (n + 2)‖ * ‖y - (x, x)‖ ^ n * (↑(n + 2) * ‖y - (x, x)‖ * ‖y.1 - y.2‖) := by
rw [pow_succ ‖y - (x, x)‖]
ring
-- porting note: the two `↑` in `↑r'` are new, without them, Lean fails to synthesize
-- instances `HDiv ℝ ℝ≥0 ?m` or `HMul ℝ ℝ≥0 ?m`
_ ≤ C * a ^ (n + 2) / ↑r' ^ (n + 2)
* ↑r' ^ n * (↑(n + 2) * ‖y - (x, x)‖ * ‖y.1 - y.2‖) := by
have : 0 < a := ha.1
gcongr
· apply hp
· apply hy'.le
_ = B n := by
-- porting note: in the original, `B` was in the `field_simp`, but now Lean does not
-- accept it. The current proof works in Lean 4, but does not in Lean 3.
field_simp [pow_succ]
simp only [mul_assoc, mul_comm, mul_left_comm]
have hBL : HasSum B (L y) := by
apply HasSum.mul_left
simp only [add_mul]
have : ‖a‖ < 1 := by simp only [Real.norm_eq_abs, abs_of_pos ha.1, ha.2]
rw [div_eq_mul_inv, div_eq_mul_inv]
exact (hasSum_coe_mul_geometric_of_norm_lt_1 this).add -- porting note: was `convert`!
((hasSum_geometric_of_norm_lt_1 this).mul_left 2)
exact hA.norm_le_of_bounded hBL hAB
suffices L =O[𝓟 (EMetric.ball (x, x) r')] fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖ by
refine' (IsBigO.of_bound 1 (eventually_principal.2 fun y hy => _)).trans this
rw [one_mul]
exact (hL y hy).trans (le_abs_self _)
simp_rw [mul_right_comm _ (_ * _)] -- porting note: there was an `L` inside the `simp_rw`.
exact (isBigO_refl _ _).const_mul_left _
set_option linter.uppercaseLean3 false in
#align has_fpower_series_on_ball.is_O_image_sub_image_sub_deriv_principal HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal
/-- If `f` has formal power series `∑ n, pₙ` on a ball of radius `r`, then for `y, z` in any smaller
ball, the norm of the difference `f y - f z - p 1 (fun _ ↦ y - z)` is bounded above by
`C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖`. -/
theorem HasFPowerSeriesOnBall.image_sub_sub_deriv_le (hf : HasFPowerSeriesOnBall f p x r)
(hr : r' < r) :
∃ C, ∀ᵉ (y ∈ EMetric.ball x r') (z ∈ EMetric.ball x r'),
‖f y - f z - p 1 fun _ => y - z‖ ≤ C * max ‖y - x‖ ‖z - x‖ * ‖y - z‖ := by
simpa only [isBigO_principal, mul_assoc, norm_mul, norm_norm, Prod.forall, EMetric.mem_ball,
Prod.edist_eq, max_lt_iff, and_imp, @forall_swap (_ < _) E] using
hf.isBigO_image_sub_image_sub_deriv_principal hr
#align has_fpower_series_on_ball.image_sub_sub_deriv_le HasFPowerSeriesOnBall.image_sub_sub_deriv_le
/-- If `f` has formal power series `∑ n, pₙ` at `x`, then
`f y - f z - p 1 (fun _ ↦ y - z) = O(‖(y, z) - (x, x)‖ * ‖y - z‖)` as `(y, z) → (x, x)`.
In particular, `f` is strictly differentiable at `x`. -/
theorem HasFPowerSeriesAt.isBigO_image_sub_norm_mul_norm_sub (hf : HasFPowerSeriesAt f p x) :
(fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) =O[𝓝 (x, x)] fun y =>
‖y - (x, x)‖ * ‖y.1 - y.2‖ := by
rcases hf with ⟨r, hf⟩
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hf.r_pos with ⟨r', r'0, h⟩
refine' (hf.isBigO_image_sub_image_sub_deriv_principal h).mono _
exact le_principal_iff.2 (EMetric.ball_mem_nhds _ r'0)
set_option linter.uppercaseLean3 false in
#align has_fpower_series_at.is_O_image_sub_norm_mul_norm_sub HasFPowerSeriesAt.isBigO_image_sub_norm_mul_norm_sub
/-- If a function admits a power series expansion at `x`, then it is the uniform limit of the
partial sums of this power series on strict subdisks of the disk of convergence, i.e., `f (x + y)`
is the uniform limit of `p.partialSum n y` there. -/
theorem HasFPowerSeriesOnBall.tendstoUniformlyOn {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r)
(h : (r' : ℝ≥0∞) < r) :
TendstoUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
(Metric.ball (0 : E) r') := by
obtain ⟨a, ha, C, -, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n
exact hf.uniform_geometric_approx h
refine' Metric.tendstoUniformlyOn_iff.2 fun ε εpos => _
have L : Tendsto (fun n => (C : ℝ) * a ^ n) atTop (𝓝 ((C : ℝ) * 0)) :=
tendsto_const_nhds.mul (tendsto_pow_atTop_nhds_0_of_lt_1 ha.1.le ha.2)
rw [mul_zero] at L
refine' (L.eventually (gt_mem_nhds εpos)).mono fun n hn y hy => _
rw [dist_eq_norm]
exact (hp y hy n).trans_lt hn
#align has_fpower_series_on_ball.tendsto_uniformly_on HasFPowerSeriesOnBall.tendstoUniformlyOn
/-- If a function admits a power series expansion at `x`, then it is the locally uniform limit of
the partial sums of this power series on the disk of convergence, i.e., `f (x + y)`
is the locally uniform limit of `p.partialSum n y` there. -/
theorem HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn (hf : HasFPowerSeriesOnBall f p x r) :
TendstoLocallyUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
(EMetric.ball (0 : E) r) := by
intro u hu x hx
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hx with ⟨r', xr', hr'⟩
have : EMetric.ball (0 : E) r' ∈ 𝓝 x := IsOpen.mem_nhds EMetric.isOpen_ball xr'
refine' ⟨EMetric.ball (0 : E) r', mem_nhdsWithin_of_mem_nhds this, _⟩
simpa [Metric.emetric_ball_nnreal] using hf.tendstoUniformlyOn hr' u hu
#align has_fpower_series_on_ball.tendsto_locally_uniformly_on HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn
/-- If a function admits a power series expansion at `x`, then it is the uniform limit of the
partial sums of this power series on strict subdisks of the disk of convergence, i.e., `f y`
is the uniform limit of `p.partialSum n (y - x)` there. -/
theorem HasFPowerSeriesOnBall.tendstoUniformlyOn' {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r)
(h : (r' : ℝ≥0∞) < r) :
TendstoUniformlyOn (fun n y => p.partialSum n (y - x)) f atTop (Metric.ball (x : E) r') := by
convert (hf.tendstoUniformlyOn h).comp fun y => y - x using 1
· simp [(· ∘ ·)]
· ext z
simp [dist_eq_norm]
#align has_fpower_series_on_ball.tendsto_uniformly_on' HasFPowerSeriesOnBall.tendstoUniformlyOn'
/-- If a function admits a power series expansion at `x`, then it is the locally uniform limit of
the partial sums of this power series on the disk of convergence, i.e., `f y`
is the locally uniform limit of `p.partialSum n (y - x)` there. -/
theorem HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn' (hf : HasFPowerSeriesOnBall f p x r) :
TendstoLocallyUniformlyOn (fun n y => p.partialSum n (y - x)) f atTop
(EMetric.ball (x : E) r) := by
have A : ContinuousOn (fun y : E => y - x) (EMetric.ball (x : E) r) :=
(continuous_id.sub continuous_const).continuousOn
convert hf.tendstoLocallyUniformlyOn.comp (fun y : E => y - x) _ A using 1
· ext z
simp
· intro z
simp [edist_eq_coe_nnnorm, edist_eq_coe_nnnorm_sub]
#align has_fpower_series_on_ball.tendsto_locally_uniformly_on' HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn'
/-- If a function admits a power series expansion on a disk, then it is continuous there. -/
protected theorem HasFPowerSeriesOnBall.continuousOn (hf : HasFPowerSeriesOnBall f p x r) :
ContinuousOn f (EMetric.ball x r) :=
hf.tendstoLocallyUniformlyOn'.continuousOn <|
eventually_of_forall fun n =>
((p.partialSum_continuous n).comp (continuous_id.sub continuous_const)).continuousOn
#align has_fpower_series_on_ball.continuous_on HasFPowerSeriesOnBall.continuousOn
protected theorem HasFPowerSeriesAt.continuousAt (hf : HasFPowerSeriesAt f p x) :
ContinuousAt f x :=
let ⟨_, hr⟩ := hf
hr.continuousOn.continuousAt (EMetric.ball_mem_nhds x hr.r_pos)
#align has_fpower_series_at.continuous_at HasFPowerSeriesAt.continuousAt
protected theorem AnalyticAt.continuousAt (hf : AnalyticAt 𝕜 f x) : ContinuousAt f x :=
let ⟨_, hp⟩ := hf
hp.continuousAt
#align analytic_at.continuous_at AnalyticAt.continuousAt
protected theorem AnalyticOn.continuousOn {s : Set E} (hf : AnalyticOn 𝕜 f s) : ContinuousOn f s :=
fun x hx => (hf x hx).continuousAt.continuousWithinAt
#align analytic_on.continuous_on AnalyticOn.continuousOn
/-- Analytic everywhere implies continuous -/
theorem AnalyticOn.continuous {f : E → F} (fa : AnalyticOn 𝕜 f univ) : Continuous f := by
rw [continuous_iff_continuousOn_univ]; exact fa.continuousOn
/-- In a complete space, the sum of a converging power series `p` admits `p` as a power series.
This is not totally obvious as we need to check the convergence of the series. -/
protected theorem FormalMultilinearSeries.hasFPowerSeriesOnBall [CompleteSpace F]
(p : FormalMultilinearSeries 𝕜 E F) (h : 0 < p.radius) :
HasFPowerSeriesOnBall p.sum p 0 p.radius :=
{ r_le := le_rfl
r_pos := h
hasSum := fun hy => by
rw [zero_add]
exact p.hasSum hy }
#align formal_multilinear_series.has_fpower_series_on_ball FormalMultilinearSeries.hasFPowerSeriesOnBall
theorem HasFPowerSeriesOnBall.sum (h : HasFPowerSeriesOnBall f p x r) {y : E}
(hy : y ∈ EMetric.ball (0 : E) r) : f (x + y) = p.sum y :=
(h.hasSum hy).tsum_eq.symm
#align has_fpower_series_on_ball.sum HasFPowerSeriesOnBall.sum
/-- The sum of a converging power series is continuous in its disk of convergence. -/
protected theorem FormalMultilinearSeries.continuousOn [CompleteSpace F] :
ContinuousOn p.sum (EMetric.ball 0 p.radius) := by
rcases (zero_le p.radius).eq_or_lt with h | h
· simp [← h, continuousOn_empty]
· exact (p.hasFPowerSeriesOnBall h).continuousOn
#align formal_multilinear_series.continuous_on FormalMultilinearSeries.continuousOn
end
/-!
### Uniqueness of power series
If a function `f : E → F` has two representations as power series at a point `x : E`, corresponding
to formal multilinear series `p₁` and `p₂`, then these representations agree term-by-term. That is,
for any `n : ℕ` and `y : E`, `p₁ n (fun i ↦ y) = p₂ n (fun i ↦ y)`. In the one-dimensional case,
when `f : 𝕜 → E`, the continuous multilinear maps `p₁ n` and `p₂ n` are given by
`ContinuousMultilinearMap.mkPiField`, and hence are determined completely by the value of
`p₁ n (fun i ↦ 1)`, so `p₁ = p₂`. Consequently, the radius of convergence for one series can be
transferred to the other.
-/
section Uniqueness
open ContinuousMultilinearMap
theorem Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero {n : ℕ} {p : E[×n]→L[𝕜] F}
(h : (fun y => p fun _ => y) =O[𝓝 0] fun y => ‖y‖ ^ (n + 1)) (y : E) : (p fun _ => y) = 0 := by
obtain ⟨c, c_pos, hc⟩ := h.exists_pos
obtain ⟨t, ht, t_open, z_mem⟩ := eventually_nhds_iff.mp (isBigOWith_iff.mp hc)
obtain ⟨δ, δ_pos, δε⟩ := (Metric.isOpen_iff.mp t_open) 0 z_mem
clear h hc z_mem
cases' n with n
· exact norm_eq_zero.mp (by
-- porting note: the symmetric difference of the `simpa only` sets:
-- added `Nat.zero_eq, zero_add, pow_one`
-- removed `zero_pow', Ne.def, Nat.one_ne_zero, not_false_iff`
simpa only [Nat.zero_eq, fin0_apply_norm, norm_eq_zero, norm_zero, zero_add, pow_one,
mul_zero, norm_le_zero_iff] using ht 0 (δε (Metric.mem_ball_self δ_pos)))
· refine' Or.elim (Classical.em (y = 0))
(fun hy => by simpa only [hy] using p.map_zero) fun hy => _
replace hy := norm_pos_iff.mpr hy
refine' norm_eq_zero.mp (le_antisymm (le_of_forall_pos_le_add fun ε ε_pos => _) (norm_nonneg _))
have h₀ := _root_.mul_pos c_pos (pow_pos hy (n.succ + 1))
obtain ⟨k, k_pos, k_norm⟩ := NormedField.exists_norm_lt 𝕜
(lt_min (mul_pos δ_pos (inv_pos.mpr hy)) (mul_pos ε_pos (inv_pos.mpr h₀)))
have h₁ : ‖k • y‖ < δ := by
rw [norm_smul]
exact inv_mul_cancel_right₀ hy.ne.symm δ ▸
mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_left _ _)) hy
have h₂ :=
calc
‖p fun _ => k • y‖ ≤ c * ‖k • y‖ ^ (n.succ + 1) := by
-- porting note: now Lean wants `_root_.`
simpa only [norm_pow, _root_.norm_norm] using ht (k • y) (δε (mem_ball_zero_iff.mpr h₁))
--simpa only [norm_pow, norm_norm] using ht (k • y) (δε (mem_ball_zero_iff.mpr h₁))
_ = ‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))) := by
-- porting note: added `Nat.succ_eq_add_one` since otherwise `ring` does not conclude.
simp only [norm_smul, mul_pow, Nat.succ_eq_add_one]
-- porting note: removed `rw [pow_succ]`, since it now becomes superfluous.
ring
have h₃ : ‖k‖ * (c * ‖y‖ ^ (n.succ + 1)) < ε :=
inv_mul_cancel_right₀ h₀.ne.symm ε ▸
mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_right _ _)) h₀
calc
‖p fun _ => y‖ = ‖k⁻¹ ^ n.succ‖ * ‖p fun _ => k • y‖ := by
simpa only [inv_smul_smul₀ (norm_pos_iff.mp k_pos), norm_smul, Finset.prod_const,
Finset.card_fin] using
congr_arg norm (p.map_smul_univ (fun _ : Fin n.succ => k⁻¹) fun _ : Fin n.succ => k • y)
_ ≤ ‖k⁻¹ ^ n.succ‖ * (‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1)))) := by gcongr
_ = ‖(k⁻¹ * k) ^ n.succ‖ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))) := by
rw [← mul_assoc]
simp [norm_mul, mul_pow]
_ ≤ 0 + ε := by
rw [inv_mul_cancel (norm_pos_iff.mp k_pos)]
simpa using h₃.le
set_option linter.uppercaseLean3 false in
#align asymptotics.is_O.continuous_multilinear_map_apply_eq_zero Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero
/-- If a formal multilinear series `p` represents the zero function at `x : E`, then the
terms `p n (fun i ↦ y)` appearing in the sum are zero for any `n : ℕ`, `y : E`. -/
theorem HasFPowerSeriesAt.apply_eq_zero {p : FormalMultilinearSeries 𝕜 E F} {x : E}
(h : HasFPowerSeriesAt 0 p x) (n : ℕ) : ∀ y : E, (p n fun _ => y) = 0 := by
refine' Nat.strong_induction_on n fun k hk => _
have psum_eq : p.partialSum (k + 1) = fun y => p k fun _ => y := by
funext z
refine' Finset.sum_eq_single _ (fun b hb hnb => _) fun hn => _
· have := Finset.mem_range_succ_iff.mp hb
simp only [hk b (this.lt_of_ne hnb), Pi.zero_apply]
· exact False.elim (hn (Finset.mem_range.mpr (lt_add_one k)))
replace h := h.isBigO_sub_partialSum_pow k.succ
simp only [psum_eq, zero_sub, Pi.zero_apply, Asymptotics.isBigO_neg_left] at h
exact h.continuousMultilinearMap_apply_eq_zero
#align has_fpower_series_at.apply_eq_zero HasFPowerSeriesAt.apply_eq_zero
/-- A one-dimensional formal multilinear series representing the zero function is zero. -/
theorem HasFPowerSeriesAt.eq_zero {p : FormalMultilinearSeries 𝕜 𝕜 E} {x : 𝕜}
(h : HasFPowerSeriesAt 0 p x) : p = 0 := by
-- porting note: `funext; ext` was `ext (n x)`
funext n
ext x
rw [← mkPiField_apply_one_eq_self (p n)]
-- porting note: nasty hack, was `simp [h.apply_eq_zero n 1]`
have := Or.intro_right ?_ (h.apply_eq_zero n 1)
simpa using this
#align has_fpower_series_at.eq_zero HasFPowerSeriesAt.eq_zero
/-- One-dimensional formal multilinear series representing the same function are equal. -/
theorem HasFPowerSeriesAt.eq_formalMultilinearSeries {p₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E}
{f : 𝕜 → E} {x : 𝕜} (h₁ : HasFPowerSeriesAt f p₁ x) (h₂ : HasFPowerSeriesAt f p₂ x) : p₁ = p₂ :=
sub_eq_zero.mp (HasFPowerSeriesAt.eq_zero (by simpa only [sub_self] using h₁.sub h₂))
#align has_fpower_series_at.eq_formal_multilinear_series HasFPowerSeriesAt.eq_formalMultilinearSeries
theorem HasFPowerSeriesAt.eq_formalMultilinearSeries_of_eventually
{p q : FormalMultilinearSeries 𝕜 𝕜 E} {f g : 𝕜 → E} {x : 𝕜} (hp : HasFPowerSeriesAt f p x)
(hq : HasFPowerSeriesAt g q x) (heq : ∀ᶠ z in 𝓝 x, f z = g z) : p = q :=
(hp.congr heq).eq_formalMultilinearSeries hq
#align has_fpower_series_at.eq_formal_multilinear_series_of_eventually HasFPowerSeriesAt.eq_formalMultilinearSeries_of_eventually
/-- A one-dimensional formal multilinear series representing a locally zero function is zero. -/
theorem HasFPowerSeriesAt.eq_zero_of_eventually {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E}
{x : 𝕜} (hp : HasFPowerSeriesAt f p x) (hf : f =ᶠ[𝓝 x] 0) : p = 0 :=
(hp.congr hf).eq_zero
#align has_fpower_series_at.eq_zero_of_eventually HasFPowerSeriesAt.eq_zero_of_eventually
/-- If a function `f : 𝕜 → E` has two power series representations at `x`, then the given radii in
which convergence is guaranteed may be interchanged. This can be useful when the formal multilinear
series in one representation has a particularly nice form, but the other has a larger radius. -/
theorem HasFPowerSeriesOnBall.exchange_radius {p₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E}
{r₁ r₂ : ℝ≥0∞} {x : 𝕜} (h₁ : HasFPowerSeriesOnBall f p₁ x r₁)
(h₂ : HasFPowerSeriesOnBall f p₂ x r₂) : HasFPowerSeriesOnBall f p₁ x r₂ :=
h₂.hasFPowerSeriesAt.eq_formalMultilinearSeries h₁.hasFPowerSeriesAt ▸ h₂
#align has_fpower_series_on_ball.exchange_radius HasFPowerSeriesOnBall.exchange_radius
/-- If a function `f : 𝕜 → E` has power series representation `p` on a ball of some radius and for
each positive radius it has some power series representation, then `p` converges to `f` on the whole
`𝕜`. -/
theorem HasFPowerSeriesOnBall.r_eq_top_of_exists {f : 𝕜 → E} {r : ℝ≥0∞} {x : 𝕜}
{p : FormalMultilinearSeries 𝕜 𝕜 E} (h : HasFPowerSeriesOnBall f p x r)
(h' : ∀ (r' : ℝ≥0) (_ : 0 < r'), ∃ p' : FormalMultilinearSeries 𝕜 𝕜 E,
HasFPowerSeriesOnBall f p' x r') :
HasFPowerSeriesOnBall f p x ∞ :=
{ r_le := ENNReal.le_of_forall_pos_nnreal_lt fun r hr _ =>
let ⟨_, hp'⟩ := h' r hr
(h.exchange_radius hp').r_le
r_pos := ENNReal.coe_lt_top
hasSum := fun {y} _ =>
let ⟨r', hr'⟩ := exists_gt ‖y‖₊
let ⟨_, hp'⟩ := h' r' hr'.ne_bot.bot_lt
(h.exchange_radius hp').hasSum <| mem_emetric_ball_zero_iff.mpr (ENNReal.coe_lt_coe.2 hr') }
#align has_fpower_series_on_ball.r_eq_top_of_exists HasFPowerSeriesOnBall.r_eq_top_of_exists
end Uniqueness
/-!
### Changing origin in a power series
If a function is analytic in a disk `D(x, R)`, then it is analytic in any disk contained in that
one. Indeed, one can write
$$
f (x + y + z) = \sum_{n} p_n (y + z)^n = \sum_{n, k} \binom{n}{k} p_n y^{n-k} z^k
= \sum_{k} \Bigl(\sum_{n} \binom{n}{k} p_n y^{n-k}\Bigr) z^k.
$$
The corresponding power series has thus a `k`-th coefficient equal to
$\sum_{n} \binom{n}{k} p_n y^{n-k}$. In the general case where `pₙ` is a multilinear map, this has
to be interpreted suitably: instead of having a binomial coefficient, one should sum over all
possible subsets `s` of `Fin n` of cardinal `k`, and attribute `z` to the indices in `s` and
`y` to the indices outside of `s`.
In this paragraph, we implement this. The new power series is called `p.changeOrigin y`. Then, we
check its convergence and the fact that its sum coincides with the original sum. The outcome of this
discussion is that the set of points where a function is analytic is open.
-/
namespace FormalMultilinearSeries
section
variable (p : FormalMultilinearSeries 𝕜 E F) {x y : E} {r R : ℝ≥0}
/-- A term of `FormalMultilinearSeries.changeOriginSeries`.
Given a formal multilinear series `p` and a point `x` in its ball of convergence,
`p.changeOrigin x` is a formal multilinear series such that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense. Each term of `p.changeOrigin x`
is itself an analytic function of `x` given by the series `p.changeOriginSeries`. Each term in
`changeOriginSeries` is the sum of `changeOriginSeriesTerm`'s over all `s` of cardinality `l`.
The definition is such that `p.changeOriginSeriesTerm k l s hs (fun _ ↦ x) (fun _ ↦ y) =
p (k + l) (s.piecewise (fun _ ↦ x) (fun _ ↦ y))`
-/
def changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
E[×l]→L[𝕜] E[×k]→L[𝕜] F := by
let a := ContinuousMultilinearMap.curryFinFinset 𝕜 E F hs
(by erw [Finset.card_compl, Fintype.card_fin, hs, add_tsub_cancel_right])
exact a (p (k + l))
#align formal_multilinear_series.change_origin_series_term FormalMultilinearSeries.changeOriginSeriesTerm
theorem changeOriginSeriesTerm_apply (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l)
(x y : E) :
(p.changeOriginSeriesTerm k l s hs (fun _ => x) fun _ => y) =
p (k + l) (s.piecewise (fun _ => x) fun _ => y) :=
ContinuousMultilinearMap.curryFinFinset_apply_const _ _ _ _ _
#align formal_multilinear_series.change_origin_series_term_apply FormalMultilinearSeries.changeOriginSeriesTerm_apply
@[simp]
theorem norm_changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
‖p.changeOriginSeriesTerm k l s hs‖ = ‖p (k + l)‖ := by
simp only [changeOriginSeriesTerm, LinearIsometryEquiv.norm_map]
#align formal_multilinear_series.norm_change_origin_series_term FormalMultilinearSeries.norm_changeOriginSeriesTerm
@[simp]
theorem nnnorm_changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
‖p.changeOriginSeriesTerm k l s hs‖₊ = ‖p (k + l)‖₊ := by
simp only [changeOriginSeriesTerm, LinearIsometryEquiv.nnnorm_map]
#align formal_multilinear_series.nnnorm_change_origin_series_term FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm
theorem nnnorm_changeOriginSeriesTerm_apply_le (k l : ℕ) (s : Finset (Fin (k + l)))
(hs : s.card = l) (x y : E) :
‖p.changeOriginSeriesTerm k l s hs (fun _ => x) fun _ => y‖₊ ≤
‖p (k + l)‖₊ * ‖x‖₊ ^ l * ‖y‖₊ ^ k := by
rw [← p.nnnorm_changeOriginSeriesTerm k l s hs, ← Fin.prod_const, ← Fin.prod_const]
apply ContinuousMultilinearMap.le_of_op_nnnorm_le
apply ContinuousMultilinearMap.le_op_nnnorm
#align formal_multilinear_series.nnnorm_change_origin_series_term_apply_le FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm_apply_le
/-- The power series for `f.changeOrigin k`.
Given a formal multilinear series `p` and a point `x` in its ball of convergence,
`p.changeOrigin x` is a formal multilinear series such that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense. Its `k`-th term is the sum of
the series `p.changeOriginSeries k`. -/
def changeOriginSeries (k : ℕ) : FormalMultilinearSeries 𝕜 E (E[×k]→L[𝕜] F) := fun l =>
∑ s : { s : Finset (Fin (k + l)) // Finset.card s = l }, p.changeOriginSeriesTerm k l s s.2
#align formal_multilinear_series.change_origin_series FormalMultilinearSeries.changeOriginSeries
theorem nnnorm_changeOriginSeries_le_tsum (k l : ℕ) :
‖p.changeOriginSeries k l‖₊ ≤
∑' _ : { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + l)‖₊ :=
(nnnorm_sum_le _ (fun t => changeOriginSeriesTerm p k l (Subtype.val t) t.prop)).trans_eq <| by
simp_rw [tsum_fintype, nnnorm_changeOriginSeriesTerm (p := p) (k := k) (l := l)]
#align formal_multilinear_series.nnnorm_change_origin_series_le_tsum FormalMultilinearSeries.nnnorm_changeOriginSeries_le_tsum
theorem nnnorm_changeOriginSeries_apply_le_tsum (k l : ℕ) (x : E) :
‖p.changeOriginSeries k l fun _ => x‖₊ ≤
∑' _ : { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + l)‖₊ * ‖x‖₊ ^ l := by
rw [NNReal.tsum_mul_right, ← Fin.prod_const]
exact (p.changeOriginSeries k l).le_of_op_nnnorm_le _ (p.nnnorm_changeOriginSeries_le_tsum _ _)
#align formal_multilinear_series.nnnorm_change_origin_series_apply_le_tsum FormalMultilinearSeries.nnnorm_changeOriginSeries_apply_le_tsum
/-- Changing the origin of a formal multilinear series `p`, so that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense.
-/
def changeOrigin (x : E) : FormalMultilinearSeries 𝕜 E F :=
fun k => (p.changeOriginSeries k).sum x
#align formal_multilinear_series.change_origin FormalMultilinearSeries.changeOrigin
/-- An auxiliary equivalence useful in the proofs about
`FormalMultilinearSeries.changeOriginSeries`: the set of triples `(k, l, s)`, where `s` is a
`Finset (Fin (k + l))` of cardinality `l` is equivalent to the set of pairs `(n, s)`, where `s` is a
`Finset (Fin n)`.
The forward map sends `(k, l, s)` to `(k + l, s)` and the inverse map sends `(n, s)` to
`(n - Finset.card s, Finset.card s, s)`. The actual definition is less readable because of problems
with non-definitional equalities. -/
@[simps]
def changeOriginIndexEquiv :
(Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) ≃ Σn : ℕ, Finset (Fin n) where
toFun s := ⟨s.1 + s.2.1, s.2.2⟩
invFun s :=
⟨s.1 - s.2.card, s.2.card,
⟨s.2.map
(Fin.castIso <| (tsub_add_cancel_of_le <| card_finset_fin_le s.2).symm).toEquiv.toEmbedding,
Finset.card_map _⟩⟩
left_inv := by
rintro ⟨k, l, ⟨s : Finset (Fin <| k + l), hs : s.card = l⟩⟩
dsimp only [Subtype.coe_mk]
-- Lean can't automatically generalize `k' = k + l - s.card`, `l' = s.card`, so we explicitly
-- formulate the generalized goal
suffices ∀ k' l', k' = k → l' = l → ∀ (hkl : k + l = k' + l') (hs'),
(⟨k', l', ⟨Finset.map (Fin.castIso hkl).toEquiv.toEmbedding s, hs'⟩⟩ :
Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) = ⟨k, l, ⟨s, hs⟩⟩ by
apply this <;> simp only [hs, add_tsub_cancel_right]
rintro _ _ rfl rfl hkl hs'
simp only [Equiv.refl_toEmbedding, Fin.castIso_refl, Finset.map_refl, eq_self_iff_true,
OrderIso.refl_toEquiv, and_self_iff, heq_iff_eq]
right_inv := by
rintro ⟨n, s⟩
simp [tsub_add_cancel_of_le (card_finset_fin_le s), Fin.castIso_to_equiv]
#align formal_multilinear_series.change_origin_index_equiv FormalMultilinearSeries.changeOriginIndexEquiv
theorem changeOriginSeries_summable_aux₁ {r r' : ℝ≥0} (hr : (r + r' : ℝ≥0∞) < p.radius) :
Summable fun s : Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l } =>
‖p (s.1 + s.2.1)‖₊ * r ^ s.2.1 * r' ^ s.1 := by
rw [← changeOriginIndexEquiv.symm.summable_iff]
dsimp only [Function.comp_def, changeOriginIndexEquiv_symm_apply_fst,
changeOriginIndexEquiv_symm_apply_snd_fst]
have : ∀ n : ℕ,
HasSum (fun s : Finset (Fin n) => ‖p (n - s.card + s.card)‖₊ * r ^ s.card * r' ^ (n - s.card))
(‖p n‖₊ * (r + r') ^ n) := by
intro n
-- TODO: why `simp only [tsub_add_cancel_of_le (card_finset_fin_le _)]` fails?
convert_to HasSum (fun s : Finset (Fin n) => ‖p n‖₊ * (r ^ s.card * r' ^ (n - s.card))) _
· ext1 s
rw [tsub_add_cancel_of_le (card_finset_fin_le _), mul_assoc]
rw [← Fin.sum_pow_mul_eq_add_pow]
exact (hasSum_fintype _).mul_left _
refine' NNReal.summable_sigma.2 ⟨fun n => (this n).summable, _⟩
simp only [(this _).tsum_eq]
exact p.summable_nnnorm_mul_pow hr
#align formal_multilinear_series.change_origin_series_summable_aux₁ FormalMultilinearSeries.changeOriginSeries_summable_aux₁
theorem changeOriginSeries_summable_aux₂ (hr : (r : ℝ≥0∞) < p.radius) (k : ℕ) :
Summable fun s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l } =>
‖p (k + s.1)‖₊ * r ^ s.1 := by
rcases ENNReal.lt_iff_exists_add_pos_lt.1 hr with ⟨r', h0, hr'⟩
simpa only [mul_inv_cancel_right₀ (pow_pos h0 _).ne'] using
((NNReal.summable_sigma.1 (p.changeOriginSeries_summable_aux₁ hr')).1 k).mul_right (r' ^ k)⁻¹
#align formal_multilinear_series.change_origin_series_summable_aux₂ FormalMultilinearSeries.changeOriginSeries_summable_aux₂
theorem changeOriginSeries_summable_aux₃ {r : ℝ≥0} (hr : ↑r < p.radius) (k : ℕ) :
Summable fun l : ℕ => ‖p.changeOriginSeries k l‖₊ * r ^ l := by
refine' NNReal.summable_of_le
(fun n => _) (NNReal.summable_sigma.1 <| p.changeOriginSeries_summable_aux₂ hr k).2
simp only [NNReal.tsum_mul_right]
exact mul_le_mul' (p.nnnorm_changeOriginSeries_le_tsum _ _) le_rfl
#align formal_multilinear_series.change_origin_series_summable_aux₃ FormalMultilinearSeries.changeOriginSeries_summable_aux₃
theorem le_changeOriginSeries_radius (k : ℕ) : p.radius ≤ (p.changeOriginSeries k).radius :=
ENNReal.le_of_forall_nnreal_lt fun _r hr =>
le_radius_of_summable_nnnorm _ (p.changeOriginSeries_summable_aux₃ hr k)
#align formal_multilinear_series.le_change_origin_series_radius FormalMultilinearSeries.le_changeOriginSeries_radius
theorem nnnorm_changeOrigin_le (k : ℕ) (h : (‖x‖₊ : ℝ≥0∞) < p.radius) :
‖p.changeOrigin x k‖₊ ≤
∑' s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + s.1)‖₊ * ‖x‖₊ ^ s.1 := by
refine' tsum_of_nnnorm_bounded _ fun l => p.nnnorm_changeOriginSeries_apply_le_tsum k l x
have := p.changeOriginSeries_summable_aux₂ h k
refine' HasSum.sigma this.hasSum fun l => _
exact ((NNReal.summable_sigma.1 this).1 l).hasSum
#align formal_multilinear_series.nnnorm_change_origin_le FormalMultilinearSeries.nnnorm_changeOrigin_le
/-- The radius of convergence of `p.changeOrigin x` is at least `p.radius - ‖x‖`. In other words,
`p.changeOrigin x` is well defined on the largest ball contained in the original ball of
convergence. -/
theorem changeOrigin_radius : p.radius - ‖x‖₊ ≤ (p.changeOrigin x).radius := by
refine' ENNReal.le_of_forall_pos_nnreal_lt fun r _h0 hr => _
rw [lt_tsub_iff_right, add_comm] at hr
have hr' : (‖x‖₊ : ℝ≥0∞) < p.radius := (le_add_right le_rfl).trans_lt hr
apply le_radius_of_summable_nnnorm
have : ∀ k : ℕ,
‖p.changeOrigin x k‖₊ * r ^ k ≤
(∑' s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + s.1)‖₊ * ‖x‖₊ ^ s.1) *
r ^ k :=
fun k => mul_le_mul_right' (p.nnnorm_changeOrigin_le k hr') (r ^ k)
refine' NNReal.summable_of_le this _
simpa only [← NNReal.tsum_mul_right] using
(NNReal.summable_sigma.1 (p.changeOriginSeries_summable_aux₁ hr)).2
#align formal_multilinear_series.change_origin_radius FormalMultilinearSeries.changeOrigin_radius
end
-- From this point on, assume that the space is complete, to make sure that series that converge
-- in norm also converge in `F`.
variable [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x y : E} {r R : ℝ≥0}
theorem hasFPowerSeriesOnBall_changeOrigin (k : ℕ) (hr : 0 < p.radius) :
HasFPowerSeriesOnBall (fun x => p.changeOrigin x k) (p.changeOriginSeries k) 0 p.radius :=
have := p.le_changeOriginSeries_radius k
((p.changeOriginSeries k).hasFPowerSeriesOnBall (hr.trans_le this)).mono hr this
#align formal_multilinear_series.has_fpower_series_on_ball_change_origin FormalMultilinearSeries.hasFPowerSeriesOnBall_changeOrigin
/-- Summing the series `p.changeOrigin x` at a point `y` gives back `p (x + y)`. -/
theorem changeOrigin_eval (h : (‖x‖₊ + ‖y‖₊ : ℝ≥0∞) < p.radius) :
(p.changeOrigin x).sum y = p.sum (x + y) := by
have radius_pos : 0 < p.radius := lt_of_le_of_lt (zero_le _) h
have x_mem_ball : x ∈ EMetric.ball (0 : E) p.radius :=
mem_emetric_ball_zero_iff.2 ((le_add_right le_rfl).trans_lt h)
have y_mem_ball : y ∈ EMetric.ball (0 : E) (p.changeOrigin x).radius := by
refine' mem_emetric_ball_zero_iff.2 (lt_of_lt_of_le _ p.changeOrigin_radius)
rwa [lt_tsub_iff_right, add_comm]
have x_add_y_mem_ball : x + y ∈ EMetric.ball (0 : E) p.radius := by
refine' mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt _ h)
exact mod_cast nnnorm_add_le x y
set f : (Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) → F := fun s =>
p.changeOriginSeriesTerm s.1 s.2.1 s.2.2 s.2.2.2 (fun _ => x) fun _ => y
have hsf : Summable f := by
refine' .of_nnnorm_bounded _ (p.changeOriginSeries_summable_aux₁ h) _
rintro ⟨k, l, s, hs⟩
dsimp only [Subtype.coe_mk]
exact p.nnnorm_changeOriginSeriesTerm_apply_le _ _ _ _ _ _
have hf : HasSum f ((p.changeOrigin x).sum y) := by
refine' HasSum.sigma_of_hasSum ((p.changeOrigin x).summable y_mem_ball).hasSum (fun k => _) hsf
· dsimp only
refine' ContinuousMultilinearMap.hasSum_eval _ _
have := (p.hasFPowerSeriesOnBall_changeOrigin k radius_pos).hasSum x_mem_ball
rw [zero_add] at this
refine' HasSum.sigma_of_hasSum this (fun l => _) _
· simp only [changeOriginSeries, ContinuousMultilinearMap.sum_apply]
apply hasSum_fintype
· refine' .of_nnnorm_bounded _
(p.changeOriginSeries_summable_aux₂ (mem_emetric_ball_zero_iff.1 x_mem_ball) k) fun s => _
refine' (ContinuousMultilinearMap.le_op_nnnorm _ _).trans_eq _
simp
refine' hf.unique (changeOriginIndexEquiv.symm.hasSum_iff.1 _)
refine' HasSum.sigma_of_hasSum
(p.hasSum x_add_y_mem_ball) (fun n => _) (changeOriginIndexEquiv.symm.summable_iff.2 hsf)
erw [(p n).map_add_univ (fun _ => x) fun _ => y]
-- porting note: added explicit function
convert hasSum_fintype (fun c : Finset (Fin n) => f (changeOriginIndexEquiv.symm ⟨n, c⟩))
rename_i s _
dsimp only [changeOriginSeriesTerm, (· ∘ ·), changeOriginIndexEquiv_symm_apply_fst,
changeOriginIndexEquiv_symm_apply_snd_fst, changeOriginIndexEquiv_symm_apply_snd_snd_coe]
rw [ContinuousMultilinearMap.curryFinFinset_apply_const]
have : ∀ (m) (hm : n = m), p n (s.piecewise (fun _ => x) fun _ => y) =
p m ((s.map (Fin.castIso hm).toEquiv.toEmbedding).piecewise (fun _ => x) fun _ => y) := by
rintro m rfl
simp (config := { unfoldPartialApp := true }) [Finset.piecewise]
apply this
#align formal_multilinear_series.change_origin_eval FormalMultilinearSeries.changeOrigin_eval
/-- Power series terms are analytic as we vary the origin -/
theorem analyticAt_changeOrigin (p : FormalMultilinearSeries 𝕜 E F) (rp : p.radius > 0) (n : ℕ) :
AnalyticAt 𝕜 (fun x ↦ p.changeOrigin x n) 0 :=
(FormalMultilinearSeries.hasFPowerSeriesOnBall_changeOrigin p n rp).analyticAt
end FormalMultilinearSeries
section
variable [CompleteSpace F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x y : E} {r : ℝ≥0∞}
/-- If a function admits a power series expansion `p` on a ball `B (x, r)`, then it also admits a
power series on any subball of this ball (even with a different center), given by `p.changeOrigin`.
-/
theorem HasFPowerSeriesOnBall.changeOrigin (hf : HasFPowerSeriesOnBall f p x r)
(h : (‖y‖₊ : ℝ≥0∞) < r) : HasFPowerSeriesOnBall f (p.changeOrigin y) (x + y) (r - ‖y‖₊) :=
{ r_le := by
apply le_trans _ p.changeOrigin_radius
exact tsub_le_tsub hf.r_le le_rfl
r_pos := by simp [h]
hasSum := fun {z} hz => by
have : f (x + y + z) =
FormalMultilinearSeries.sum (FormalMultilinearSeries.changeOrigin p y) z := by
rw [mem_emetric_ball_zero_iff, lt_tsub_iff_right, add_comm] at hz
rw [p.changeOrigin_eval (hz.trans_le hf.r_le), add_assoc, hf.sum]
refine' mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt _ hz)
exact mod_cast nnnorm_add_le y z
rw [this]
apply (p.changeOrigin y).hasSum
refine' EMetric.ball_subset_ball (le_trans _ p.changeOrigin_radius) hz
exact tsub_le_tsub hf.r_le le_rfl }
#align has_fpower_series_on_ball.change_origin HasFPowerSeriesOnBall.changeOrigin
/-- If a function admits a power series expansion `p` on an open ball `B (x, r)`, then
it is analytic at every point of this ball. -/
theorem HasFPowerSeriesOnBall.analyticAt_of_mem (hf : HasFPowerSeriesOnBall f p x r)
(h : y ∈ EMetric.ball x r) : AnalyticAt 𝕜 f y := by
have : (‖y - x‖₊ : ℝ≥0∞) < r := by simpa [edist_eq_coe_nnnorm_sub] using h
have := hf.changeOrigin this
rw [add_sub_cancel'_right] at this
exact this.analyticAt
#align has_fpower_series_on_ball.analytic_at_of_mem HasFPowerSeriesOnBall.analyticAt_of_mem
theorem HasFPowerSeriesOnBall.analyticOn (hf : HasFPowerSeriesOnBall f p x r) :
AnalyticOn 𝕜 f (EMetric.ball x r) :=
fun _y hy => hf.analyticAt_of_mem hy
#align has_fpower_series_on_ball.analytic_on HasFPowerSeriesOnBall.analyticOn
variable (𝕜 f)
/-- For any function `f` from a normed vector space to a Banach space, the set of points `x` such
that `f` is analytic at `x` is open. -/
theorem isOpen_analyticAt : IsOpen { x | AnalyticAt 𝕜 f x } := by
rw [isOpen_iff_mem_nhds]
rintro x ⟨p, r, hr⟩
exact mem_of_superset (EMetric.ball_mem_nhds _ hr.r_pos) fun y hy => hr.analyticAt_of_mem hy
#align is_open_analytic_at isOpen_analyticAt
variable {𝕜}
theorem AnalyticAt.eventually_analyticAt {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∀ᶠ y in 𝓝 x, AnalyticAt 𝕜 f y :=
(isOpen_analyticAt 𝕜 f).mem_nhds h
theorem AnalyticAt.exists_mem_nhds_analyticOn {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∃ s ∈ 𝓝 x, AnalyticOn 𝕜 f s :=
h.eventually_analyticAt.exists_mem
/-- If we're analytic at a point, we're analytic in a nonempty ball -/
theorem AnalyticAt.exists_ball_analyticOn {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∃ r : ℝ, 0 < r ∧ AnalyticOn 𝕜 f (Metric.ball x r) :=
Metric.isOpen_iff.mp (isOpen_analyticAt _ _) _ h
end
section
open FormalMultilinearSeries
variable {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E} {z₀ : 𝕜}
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z)) := by
refine' ⟨fun ⟨r, _, r_pos, h⟩ =>
eventually_of_mem (EMetric.ball_mem_nhds 0 r_pos) fun _ => by simpa using h, _⟩
simp only [Metric.eventually_nhds_iff]
rintro ⟨r, r_pos, h⟩
refine' ⟨p.radius ⊓ r.toNNReal, by simp, _, _⟩
· simp only [r_pos.lt, lt_inf_iff, ENNReal.coe_pos, Real.toNNReal_pos, and_true_iff]
obtain ⟨z, z_pos, le_z⟩ := NormedField.exists_norm_lt 𝕜 r_pos.lt
have : (‖z‖₊ : ENNReal) ≤ p.radius := by
simp only [dist_zero_right] at h
|
apply FormalMultilinearSeries.le_radius_of_tendsto
|
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z)) := by
refine' ⟨fun ⟨r, _, r_pos, h⟩ =>
eventually_of_mem (EMetric.ball_mem_nhds 0 r_pos) fun _ => by simpa using h, _⟩
simp only [Metric.eventually_nhds_iff]
rintro ⟨r, r_pos, h⟩
refine' ⟨p.radius ⊓ r.toNNReal, by simp, _, _⟩
· simp only [r_pos.lt, lt_inf_iff, ENNReal.coe_pos, Real.toNNReal_pos, and_true_iff]
obtain ⟨z, z_pos, le_z⟩ := NormedField.exists_norm_lt 𝕜 r_pos.lt
have : (‖z‖₊ : ENNReal) ≤ p.radius := by
simp only [dist_zero_right] at h
|
Mathlib.Analysis.Analytic.Basic.1430_0.jQw1fRSE1vGpOll
|
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z))
|
Mathlib_Analysis_Analytic_Basic
|
case h
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
p : FormalMultilinearSeries 𝕜 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
r : ℝ
r_pos : r > 0
z : 𝕜
z_pos : 0 < ‖z‖
le_z : ‖z‖ < r
h : ∀ ⦃y : 𝕜⦄, ‖y‖ < r → HasSum (fun n => y ^ n • coeff p n) (f (z₀ + y))
⊢ Tendsto (fun n => ‖p n‖ * ↑‖z‖₊ ^ n) atTop (𝓝 ?l)
case l
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
p : FormalMultilinearSeries 𝕜 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
r : ℝ
r_pos : r > 0
z : 𝕜
z_pos : 0 < ‖z‖
le_z : ‖z‖ < r
h : ∀ ⦃y : 𝕜⦄, ‖y‖ < r → HasSum (fun n => y ^ n • coeff p n) (f (z₀ + y))
⊢ ℝ
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Logic.Equiv.Fin
import Mathlib.Topology.Algebra.InfiniteSum.Module
#align_import analysis.analytic.basic from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
/-!
# Analytic functions
A function is analytic in one dimension around `0` if it can be written as a converging power series
`Σ pₙ zⁿ`. This definition can be extended to any dimension (even in infinite dimension) by
requiring that `pₙ` is a continuous `n`-multilinear map. In general, `pₙ` is not unique (in two
dimensions, taking `p₂ (x, y) (x', y') = x y'` or `y x'` gives the same map when applied to a
vector `(x, y) (x, y)`). A way to guarantee uniqueness is to take a symmetric `pₙ`, but this is not
always possible in nonzero characteristic (in characteristic 2, the previous example has no
symmetric representative). Therefore, we do not insist on symmetry or uniqueness in the definition,
and we only require the existence of a converging series.
The general framework is important to say that the exponential map on bounded operators on a Banach
space is analytic, as well as the inverse on invertible operators.
## Main definitions
Let `p` be a formal multilinear series from `E` to `F`, i.e., `p n` is a multilinear map on `E^n`
for `n : ℕ`.
* `p.radius`: the largest `r : ℝ≥0∞` such that `‖p n‖ * r^n` grows subexponentially.
* `p.le_radius_of_bound`, `p.le_radius_of_bound_nnreal`, `p.le_radius_of_isBigO`: if `‖p n‖ * r ^ n`
is bounded above, then `r ≤ p.radius`;
* `p.isLittleO_of_lt_radius`, `p.norm_mul_pow_le_mul_pow_of_lt_radius`,
`p.isLittleO_one_of_lt_radius`,
`p.norm_mul_pow_le_of_lt_radius`, `p.nnnorm_mul_pow_le_of_lt_radius`: if `r < p.radius`, then
`‖p n‖ * r ^ n` tends to zero exponentially;
* `p.lt_radius_of_isBigO`: if `r ≠ 0` and `‖p n‖ * r ^ n = O(a ^ n)` for some `-1 < a < 1`, then
`r < p.radius`;
* `p.partialSum n x`: the sum `∑_{i = 0}^{n-1} pᵢ xⁱ`.
* `p.sum x`: the sum `∑'_{i = 0}^{∞} pᵢ xⁱ`.
Additionally, let `f` be a function from `E` to `F`.
* `HasFPowerSeriesOnBall f p x r`: on the ball of center `x` with radius `r`,
`f (x + y) = ∑'_n pₙ yⁿ`.
* `HasFPowerSeriesAt f p x`: on some ball of center `x` with positive radius, holds
`HasFPowerSeriesOnBall f p x r`.
* `AnalyticAt 𝕜 f x`: there exists a power series `p` such that holds `HasFPowerSeriesAt f p x`.
* `AnalyticOn 𝕜 f s`: the function `f` is analytic at every point of `s`.
We develop the basic properties of these notions, notably:
* If a function admits a power series, it is continuous (see
`HasFPowerSeriesOnBall.continuousOn` and `HasFPowerSeriesAt.continuousAt` and
`AnalyticAt.continuousAt`).
* In a complete space, the sum of a formal power series with positive radius is well defined on the
disk of convergence, see `FormalMultilinearSeries.hasFPowerSeriesOnBall`.
* If a function admits a power series in a ball, then it is analytic at any point `y` of this ball,
and the power series there can be expressed in terms of the initial power series `p` as
`p.changeOrigin y`. See `HasFPowerSeriesOnBall.changeOrigin`. It follows in particular that
the set of points at which a given function is analytic is open, see `isOpen_analyticAt`.
## Implementation details
We only introduce the radius of convergence of a power series, as `p.radius`.
For a power series in finitely many dimensions, there is a finer (directional, coordinate-dependent)
notion, describing the polydisk of convergence. This notion is more specific, and not necessary to
build the general theory. We do not define it here.
-/
noncomputable section
variable {𝕜 E F G : Type*}
open Topology Classical BigOperators NNReal Filter ENNReal
open Set Filter Asymptotics
namespace FormalMultilinearSeries
variable [Ring 𝕜] [AddCommGroup E] [AddCommGroup F] [Module 𝕜 E] [Module 𝕜 F]
variable [TopologicalSpace E] [TopologicalSpace F]
variable [TopologicalAddGroup E] [TopologicalAddGroup F]
variable [ContinuousConstSMul 𝕜 E] [ContinuousConstSMul 𝕜 F]
/-- Given a formal multilinear series `p` and a vector `x`, then `p.sum x` is the sum `Σ pₙ xⁿ`. A
priori, it only behaves well when `‖x‖ < p.radius`. -/
protected def sum (p : FormalMultilinearSeries 𝕜 E F) (x : E) : F :=
∑' n : ℕ, p n fun _ => x
#align formal_multilinear_series.sum FormalMultilinearSeries.sum
/-- Given a formal multilinear series `p` and a vector `x`, then `p.partialSum n x` is the sum
`Σ pₖ xᵏ` for `k ∈ {0,..., n-1}`. -/
def partialSum (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) (x : E) : F :=
∑ k in Finset.range n, p k fun _ : Fin k => x
#align formal_multilinear_series.partial_sum FormalMultilinearSeries.partialSum
/-- The partial sums of a formal multilinear series are continuous. -/
theorem partialSum_continuous (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) :
Continuous (p.partialSum n) := by
unfold partialSum -- Porting note: added
continuity
#align formal_multilinear_series.partial_sum_continuous FormalMultilinearSeries.partialSum_continuous
end FormalMultilinearSeries
/-! ### The radius of a formal multilinear series -/
variable [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F]
[NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G]
namespace FormalMultilinearSeries
variable (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
/-- The radius of a formal multilinear series is the largest `r` such that the sum `Σ ‖pₙ‖ ‖y‖ⁿ`
converges for all `‖y‖ < r`. This implies that `Σ pₙ yⁿ` converges for all `‖y‖ < r`, but these
definitions are *not* equivalent in general. -/
def radius (p : FormalMultilinearSeries 𝕜 E F) : ℝ≥0∞ :=
⨆ (r : ℝ≥0) (C : ℝ) (_ : ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C), (r : ℝ≥0∞)
#align formal_multilinear_series.radius FormalMultilinearSeries.radius
/-- If `‖pₙ‖ rⁿ` is bounded in `n`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_bound (C : ℝ) {r : ℝ≥0} (h : ∀ n : ℕ, ‖p n‖ * (r : ℝ) ^ n ≤ C) :
(r : ℝ≥0∞) ≤ p.radius :=
le_iSup_of_le r <| le_iSup_of_le C <| le_iSup (fun _ => (r : ℝ≥0∞)) h
#align formal_multilinear_series.le_radius_of_bound FormalMultilinearSeries.le_radius_of_bound
/-- If `‖pₙ‖ rⁿ` is bounded in `n`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_bound_nnreal (C : ℝ≥0) {r : ℝ≥0} (h : ∀ n : ℕ, ‖p n‖₊ * r ^ n ≤ C) :
(r : ℝ≥0∞) ≤ p.radius :=
p.le_radius_of_bound C fun n => mod_cast h n
#align formal_multilinear_series.le_radius_of_bound_nnreal FormalMultilinearSeries.le_radius_of_bound_nnreal
/-- If `‖pₙ‖ rⁿ = O(1)`, as `n → ∞`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_isBigO (h : (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] fun _ => (1 : ℝ)) :
↑r ≤ p.radius :=
Exists.elim (isBigO_one_nat_atTop_iff.1 h) fun C hC =>
p.le_radius_of_bound C fun n => (le_abs_self _).trans (hC n)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.le_radius_of_is_O FormalMultilinearSeries.le_radius_of_isBigO
theorem le_radius_of_eventually_le (C) (h : ∀ᶠ n in atTop, ‖p n‖ * (r : ℝ) ^ n ≤ C) :
↑r ≤ p.radius :=
p.le_radius_of_isBigO <| IsBigO.of_bound C <| h.mono fun n hn => by simpa
#align formal_multilinear_series.le_radius_of_eventually_le FormalMultilinearSeries.le_radius_of_eventually_le
theorem le_radius_of_summable_nnnorm (h : Summable fun n => ‖p n‖₊ * r ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_bound_nnreal (∑' n, ‖p n‖₊ * r ^ n) fun _ => le_tsum' h _
#align formal_multilinear_series.le_radius_of_summable_nnnorm FormalMultilinearSeries.le_radius_of_summable_nnnorm
theorem le_radius_of_summable (h : Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_summable_nnnorm <| by
simp only [← coe_nnnorm] at h
exact mod_cast h
#align formal_multilinear_series.le_radius_of_summable FormalMultilinearSeries.le_radius_of_summable
theorem radius_eq_top_of_forall_nnreal_isBigO
(h : ∀ r : ℝ≥0, (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] fun _ => (1 : ℝ)) : p.radius = ∞ :=
ENNReal.eq_top_of_forall_nnreal_le fun r => p.le_radius_of_isBigO (h r)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.radius_eq_top_of_forall_nnreal_is_O FormalMultilinearSeries.radius_eq_top_of_forall_nnreal_isBigO
theorem radius_eq_top_of_eventually_eq_zero (h : ∀ᶠ n in atTop, p n = 0) : p.radius = ∞ :=
p.radius_eq_top_of_forall_nnreal_isBigO fun r =>
(isBigO_zero _ _).congr' (h.mono fun n hn => by simp [hn]) EventuallyEq.rfl
#align formal_multilinear_series.radius_eq_top_of_eventually_eq_zero FormalMultilinearSeries.radius_eq_top_of_eventually_eq_zero
theorem radius_eq_top_of_forall_image_add_eq_zero (n : ℕ) (hn : ∀ m, p (m + n) = 0) :
p.radius = ∞ :=
p.radius_eq_top_of_eventually_eq_zero <|
mem_atTop_sets.2 ⟨n, fun _ hk => tsub_add_cancel_of_le hk ▸ hn _⟩
#align formal_multilinear_series.radius_eq_top_of_forall_image_add_eq_zero FormalMultilinearSeries.radius_eq_top_of_forall_image_add_eq_zero
@[simp]
theorem constFormalMultilinearSeries_radius {v : F} :
(constFormalMultilinearSeries 𝕜 E v).radius = ⊤ :=
(constFormalMultilinearSeries 𝕜 E v).radius_eq_top_of_forall_image_add_eq_zero 1
(by simp [constFormalMultilinearSeries])
#align formal_multilinear_series.const_formal_multilinear_series_radius FormalMultilinearSeries.constFormalMultilinearSeries_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` tends to zero exponentially:
for some `0 < a < 1`, `‖p n‖ rⁿ = o(aⁿ)`. -/
theorem isLittleO_of_lt_radius (h : ↑r < p.radius) :
∃ a ∈ Ioo (0 : ℝ) 1, (fun n => ‖p n‖ * (r : ℝ) ^ n) =o[atTop] (a ^ ·) := by
have := (TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 4
rw [this]
-- Porting note: was
-- rw [(TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 4]
simp only [radius, lt_iSup_iff] at h
rcases h with ⟨t, C, hC, rt⟩
rw [ENNReal.coe_lt_coe, ← NNReal.coe_lt_coe] at rt
have : 0 < (t : ℝ) := r.coe_nonneg.trans_lt rt
rw [← div_lt_one this] at rt
refine' ⟨_, rt, C, Or.inr zero_lt_one, fun n => _⟩
calc
|‖p n‖ * (r : ℝ) ^ n| = ‖p n‖ * (t : ℝ) ^ n * (r / t : ℝ) ^ n := by
field_simp [mul_right_comm, abs_mul]
_ ≤ C * (r / t : ℝ) ^ n := by gcongr; apply hC
#align formal_multilinear_series.is_o_of_lt_radius FormalMultilinearSeries.isLittleO_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ = o(1)`. -/
theorem isLittleO_one_of_lt_radius (h : ↑r < p.radius) :
(fun n => ‖p n‖ * (r : ℝ) ^ n) =o[atTop] (fun _ => 1 : ℕ → ℝ) :=
let ⟨_, ha, hp⟩ := p.isLittleO_of_lt_radius h
hp.trans <| (isLittleO_pow_pow_of_lt_left ha.1.le ha.2).congr (fun _ => rfl) one_pow
#align formal_multilinear_series.is_o_one_of_lt_radius FormalMultilinearSeries.isLittleO_one_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` tends to zero exponentially:
for some `0 < a < 1` and `C > 0`, `‖p n‖ * r ^ n ≤ C * a ^ n`. -/
theorem norm_mul_pow_le_mul_pow_of_lt_radius (h : ↑r < p.radius) :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C * a ^ n := by
-- Porting note: moved out of `rcases`
have := ((TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 5).mp
(p.isLittleO_of_lt_radius h)
rcases this with ⟨a, ha, C, hC, H⟩
exact ⟨a, ha, C, hC, fun n => (le_abs_self _).trans (H n)⟩
#align formal_multilinear_series.norm_mul_pow_le_mul_pow_of_lt_radius FormalMultilinearSeries.norm_mul_pow_le_mul_pow_of_lt_radius
/-- If `r ≠ 0` and `‖pₙ‖ rⁿ = O(aⁿ)` for some `-1 < a < 1`, then `r < p.radius`. -/
theorem lt_radius_of_isBigO (h₀ : r ≠ 0) {a : ℝ} (ha : a ∈ Ioo (-1 : ℝ) 1)
(hp : (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] (a ^ ·)) : ↑r < p.radius := by
-- Porting note: moved out of `rcases`
have := ((TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 2 5)
rcases this.mp ⟨a, ha, hp⟩ with ⟨a, ha, C, hC, hp⟩
rw [← pos_iff_ne_zero, ← NNReal.coe_pos] at h₀
lift a to ℝ≥0 using ha.1.le
have : (r : ℝ) < r / a := by
simpa only [div_one] using (div_lt_div_left h₀ zero_lt_one ha.1).2 ha.2
norm_cast at this
rw [← ENNReal.coe_lt_coe] at this
refine' this.trans_le (p.le_radius_of_bound C fun n => _)
rw [NNReal.coe_div, div_pow, ← mul_div_assoc, div_le_iff (pow_pos ha.1 n)]
exact (le_abs_self _).trans (hp n)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.lt_radius_of_is_O FormalMultilinearSeries.lt_radius_of_isBigO
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem norm_mul_pow_le_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C :=
let ⟨_, ha, C, hC, h⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius h
⟨C, hC, fun n => (h n).trans <| mul_le_of_le_one_right hC.lt.le (pow_le_one _ ha.1.le ha.2.le)⟩
#align formal_multilinear_series.norm_mul_pow_le_of_lt_radius FormalMultilinearSeries.norm_mul_pow_le_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem norm_le_div_pow_of_pos_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h0 : 0 < r) (h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖ ≤ C / (r : ℝ) ^ n :=
let ⟨C, hC, hp⟩ := p.norm_mul_pow_le_of_lt_radius h
⟨C, hC, fun n => Iff.mpr (le_div_iff (pow_pos h0 _)) (hp n)⟩
#align formal_multilinear_series.norm_le_div_pow_of_pos_of_lt_radius FormalMultilinearSeries.norm_le_div_pow_of_pos_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem nnnorm_mul_pow_le_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖₊ * r ^ n ≤ C :=
let ⟨C, hC, hp⟩ := p.norm_mul_pow_le_of_lt_radius h
⟨⟨C, hC.lt.le⟩, hC, mod_cast hp⟩
#align formal_multilinear_series.nnnorm_mul_pow_le_of_lt_radius FormalMultilinearSeries.nnnorm_mul_pow_le_of_lt_radius
theorem le_radius_of_tendsto (p : FormalMultilinearSeries 𝕜 E F) {l : ℝ}
(h : Tendsto (fun n => ‖p n‖ * (r : ℝ) ^ n) atTop (𝓝 l)) : ↑r ≤ p.radius :=
p.le_radius_of_isBigO (h.isBigO_one _)
#align formal_multilinear_series.le_radius_of_tendsto FormalMultilinearSeries.le_radius_of_tendsto
theorem le_radius_of_summable_norm (p : FormalMultilinearSeries 𝕜 E F)
(hs : Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_tendsto hs.tendsto_atTop_zero
#align formal_multilinear_series.le_radius_of_summable_norm FormalMultilinearSeries.le_radius_of_summable_norm
theorem not_summable_norm_of_radius_lt_nnnorm (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(h : p.radius < ‖x‖₊) : ¬Summable fun n => ‖p n‖ * ‖x‖ ^ n :=
fun hs => not_le_of_lt h (p.le_radius_of_summable_norm hs)
#align formal_multilinear_series.not_summable_norm_of_radius_lt_nnnorm FormalMultilinearSeries.not_summable_norm_of_radius_lt_nnnorm
theorem summable_norm_mul_pow (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0} (h : ↑r < p.radius) :
Summable fun n : ℕ => ‖p n‖ * (r : ℝ) ^ n := by
obtain ⟨a, ha : a ∈ Ioo (0 : ℝ) 1, C, - : 0 < C, hp⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius h
exact .of_nonneg_of_le (fun n => mul_nonneg (norm_nonneg _) (pow_nonneg r.coe_nonneg _))
hp ((summable_geometric_of_lt_1 ha.1.le ha.2).mul_left _)
#align formal_multilinear_series.summable_norm_mul_pow FormalMultilinearSeries.summable_norm_mul_pow
theorem summable_norm_apply (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : Summable fun n : ℕ => ‖p n fun _ => x‖ := by
rw [mem_emetric_ball_zero_iff] at hx
refine' .of_nonneg_of_le
(fun _ => norm_nonneg _) (fun n => ((p n).le_op_norm _).trans_eq _) (p.summable_norm_mul_pow hx)
simp
#align formal_multilinear_series.summable_norm_apply FormalMultilinearSeries.summable_norm_apply
theorem summable_nnnorm_mul_pow (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0} (h : ↑r < p.radius) :
Summable fun n : ℕ => ‖p n‖₊ * r ^ n := by
rw [← NNReal.summable_coe]
push_cast
exact p.summable_norm_mul_pow h
#align formal_multilinear_series.summable_nnnorm_mul_pow FormalMultilinearSeries.summable_nnnorm_mul_pow
protected theorem summable [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : Summable fun n : ℕ => p n fun _ => x :=
(p.summable_norm_apply hx).of_norm
#align formal_multilinear_series.summable FormalMultilinearSeries.summable
theorem radius_eq_top_of_summable_norm (p : FormalMultilinearSeries 𝕜 E F)
(hs : ∀ r : ℝ≥0, Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : p.radius = ∞ :=
ENNReal.eq_top_of_forall_nnreal_le fun r => p.le_radius_of_summable_norm (hs r)
#align formal_multilinear_series.radius_eq_top_of_summable_norm FormalMultilinearSeries.radius_eq_top_of_summable_norm
theorem radius_eq_top_iff_summable_norm (p : FormalMultilinearSeries 𝕜 E F) :
p.radius = ∞ ↔ ∀ r : ℝ≥0, Summable fun n => ‖p n‖ * (r : ℝ) ^ n := by
constructor
· intro h r
obtain ⟨a, ha : a ∈ Ioo (0 : ℝ) 1, C, - : 0 < C, hp⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius
(show (r : ℝ≥0∞) < p.radius from h.symm ▸ ENNReal.coe_lt_top)
refine' .of_norm_bounded
(fun n => (C : ℝ) * a ^ n) ((summable_geometric_of_lt_1 ha.1.le ha.2).mul_left _) fun n => _
specialize hp n
rwa [Real.norm_of_nonneg (mul_nonneg (norm_nonneg _) (pow_nonneg r.coe_nonneg n))]
· exact p.radius_eq_top_of_summable_norm
#align formal_multilinear_series.radius_eq_top_iff_summable_norm FormalMultilinearSeries.radius_eq_top_iff_summable_norm
/-- If the radius of `p` is positive, then `‖pₙ‖` grows at most geometrically. -/
theorem le_mul_pow_of_radius_pos (p : FormalMultilinearSeries 𝕜 E F) (h : 0 < p.radius) :
∃ (C r : _) (hC : 0 < C) (_ : 0 < r), ∀ n, ‖p n‖ ≤ C * r ^ n := by
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 h with ⟨r, r0, rlt⟩
have rpos : 0 < (r : ℝ) := by simp [ENNReal.coe_pos.1 r0]
rcases norm_le_div_pow_of_pos_of_lt_radius p rpos rlt with ⟨C, Cpos, hCp⟩
refine' ⟨C, r⁻¹, Cpos, by simp only [inv_pos, rpos], fun n => _⟩
-- Porting note: was `convert`
rw [inv_pow, ← div_eq_mul_inv]
exact hCp n
#align formal_multilinear_series.le_mul_pow_of_radius_pos FormalMultilinearSeries.le_mul_pow_of_radius_pos
/-- The radius of the sum of two formal series is at least the minimum of their two radii. -/
theorem min_radius_le_radius_add (p q : FormalMultilinearSeries 𝕜 E F) :
min p.radius q.radius ≤ (p + q).radius := by
refine' ENNReal.le_of_forall_nnreal_lt fun r hr => _
rw [lt_min_iff] at hr
have := ((p.isLittleO_one_of_lt_radius hr.1).add (q.isLittleO_one_of_lt_radius hr.2)).isBigO
refine' (p + q).le_radius_of_isBigO ((isBigO_of_le _ fun n => _).trans this)
rw [← add_mul, norm_mul, norm_mul, norm_norm]
exact mul_le_mul_of_nonneg_right ((norm_add_le _ _).trans (le_abs_self _)) (norm_nonneg _)
#align formal_multilinear_series.min_radius_le_radius_add FormalMultilinearSeries.min_radius_le_radius_add
@[simp]
theorem radius_neg (p : FormalMultilinearSeries 𝕜 E F) : (-p).radius = p.radius := by
simp only [radius, neg_apply, norm_neg]
#align formal_multilinear_series.radius_neg FormalMultilinearSeries.radius_neg
protected theorem hasSum [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : HasSum (fun n : ℕ => p n fun _ => x) (p.sum x) :=
(p.summable hx).hasSum
#align formal_multilinear_series.has_sum FormalMultilinearSeries.hasSum
theorem radius_le_radius_continuousLinearMap_comp (p : FormalMultilinearSeries 𝕜 E F)
(f : F →L[𝕜] G) : p.radius ≤ (f.compFormalMultilinearSeries p).radius := by
refine' ENNReal.le_of_forall_nnreal_lt fun r hr => _
apply le_radius_of_isBigO
apply (IsBigO.trans_isLittleO _ (p.isLittleO_one_of_lt_radius hr)).isBigO
refine' IsBigO.mul (@IsBigOWith.isBigO _ _ _ _ _ ‖f‖ _ _ _ _) (isBigO_refl _ _)
refine IsBigOWith.of_bound (eventually_of_forall fun n => ?_)
simpa only [norm_norm] using f.norm_compContinuousMultilinearMap_le (p n)
#align formal_multilinear_series.radius_le_radius_continuous_linear_map_comp FormalMultilinearSeries.radius_le_radius_continuousLinearMap_comp
end FormalMultilinearSeries
/-! ### Expanding a function as a power series -/
section
variable {f g : E → F} {p pf pg : FormalMultilinearSeries 𝕜 E F} {x : E} {r r' : ℝ≥0∞}
/-- Given a function `f : E → F` and a formal multilinear series `p`, we say that `f` has `p` as
a power series on the ball of radius `r > 0` around `x` if `f (x + y) = ∑' pₙ yⁿ` for all `‖y‖ < r`.
-/
structure HasFPowerSeriesOnBall (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (x : E) (r : ℝ≥0∞) :
Prop where
r_le : r ≤ p.radius
r_pos : 0 < r
hasSum :
∀ {y}, y ∈ EMetric.ball (0 : E) r → HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y))
#align has_fpower_series_on_ball HasFPowerSeriesOnBall
/-- Given a function `f : E → F` and a formal multilinear series `p`, we say that `f` has `p` as
a power series around `x` if `f (x + y) = ∑' pₙ yⁿ` for all `y` in a neighborhood of `0`. -/
def HasFPowerSeriesAt (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (x : E) :=
∃ r, HasFPowerSeriesOnBall f p x r
#align has_fpower_series_at HasFPowerSeriesAt
variable (𝕜)
/-- Given a function `f : E → F`, we say that `f` is analytic at `x` if it admits a convergent power
series expansion around `x`. -/
def AnalyticAt (f : E → F) (x : E) :=
∃ p : FormalMultilinearSeries 𝕜 E F, HasFPowerSeriesAt f p x
#align analytic_at AnalyticAt
/-- Given a function `f : E → F`, we say that `f` is analytic on a set `s` if it is analytic around
every point of `s`. -/
def AnalyticOn (f : E → F) (s : Set E) :=
∀ x, x ∈ s → AnalyticAt 𝕜 f x
#align analytic_on AnalyticOn
variable {𝕜}
theorem HasFPowerSeriesOnBall.hasFPowerSeriesAt (hf : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesAt f p x :=
⟨r, hf⟩
#align has_fpower_series_on_ball.has_fpower_series_at HasFPowerSeriesOnBall.hasFPowerSeriesAt
theorem HasFPowerSeriesAt.analyticAt (hf : HasFPowerSeriesAt f p x) : AnalyticAt 𝕜 f x :=
⟨p, hf⟩
#align has_fpower_series_at.analytic_at HasFPowerSeriesAt.analyticAt
theorem HasFPowerSeriesOnBall.analyticAt (hf : HasFPowerSeriesOnBall f p x r) : AnalyticAt 𝕜 f x :=
hf.hasFPowerSeriesAt.analyticAt
#align has_fpower_series_on_ball.analytic_at HasFPowerSeriesOnBall.analyticAt
theorem HasFPowerSeriesOnBall.congr (hf : HasFPowerSeriesOnBall f p x r)
(hg : EqOn f g (EMetric.ball x r)) : HasFPowerSeriesOnBall g p x r :=
{ r_le := hf.r_le
r_pos := hf.r_pos
hasSum := fun {y} hy => by
convert hf.hasSum hy using 1
apply hg.symm
simpa [edist_eq_coe_nnnorm_sub] using hy }
#align has_fpower_series_on_ball.congr HasFPowerSeriesOnBall.congr
/-- If a function `f` has a power series `p` around `x`, then the function `z ↦ f (z - y)` has the
same power series around `x + y`. -/
theorem HasFPowerSeriesOnBall.comp_sub (hf : HasFPowerSeriesOnBall f p x r) (y : E) :
HasFPowerSeriesOnBall (fun z => f (z - y)) p (x + y) r :=
{ r_le := hf.r_le
r_pos := hf.r_pos
hasSum := fun {z} hz => by
convert hf.hasSum hz using 2
abel }
#align has_fpower_series_on_ball.comp_sub HasFPowerSeriesOnBall.comp_sub
theorem HasFPowerSeriesOnBall.hasSum_sub (hf : HasFPowerSeriesOnBall f p x r) {y : E}
(hy : y ∈ EMetric.ball x r) : HasSum (fun n : ℕ => p n fun _ => y - x) (f y) := by
have : y - x ∈ EMetric.ball (0 : E) r := by simpa [edist_eq_coe_nnnorm_sub] using hy
simpa only [add_sub_cancel'_right] using hf.hasSum this
#align has_fpower_series_on_ball.has_sum_sub HasFPowerSeriesOnBall.hasSum_sub
theorem HasFPowerSeriesOnBall.radius_pos (hf : HasFPowerSeriesOnBall f p x r) : 0 < p.radius :=
lt_of_lt_of_le hf.r_pos hf.r_le
#align has_fpower_series_on_ball.radius_pos HasFPowerSeriesOnBall.radius_pos
theorem HasFPowerSeriesAt.radius_pos (hf : HasFPowerSeriesAt f p x) : 0 < p.radius :=
let ⟨_, hr⟩ := hf
hr.radius_pos
#align has_fpower_series_at.radius_pos HasFPowerSeriesAt.radius_pos
theorem HasFPowerSeriesOnBall.mono (hf : HasFPowerSeriesOnBall f p x r) (r'_pos : 0 < r')
(hr : r' ≤ r) : HasFPowerSeriesOnBall f p x r' :=
⟨le_trans hr hf.1, r'_pos, fun hy => hf.hasSum (EMetric.ball_subset_ball hr hy)⟩
#align has_fpower_series_on_ball.mono HasFPowerSeriesOnBall.mono
theorem HasFPowerSeriesAt.congr (hf : HasFPowerSeriesAt f p x) (hg : f =ᶠ[𝓝 x] g) :
HasFPowerSeriesAt g p x := by
rcases hf with ⟨r₁, h₁⟩
rcases EMetric.mem_nhds_iff.mp hg with ⟨r₂, h₂pos, h₂⟩
exact ⟨min r₁ r₂,
(h₁.mono (lt_min h₁.r_pos h₂pos) inf_le_left).congr
fun y hy => h₂ (EMetric.ball_subset_ball inf_le_right hy)⟩
#align has_fpower_series_at.congr HasFPowerSeriesAt.congr
protected theorem HasFPowerSeriesAt.eventually (hf : HasFPowerSeriesAt f p x) :
∀ᶠ r : ℝ≥0∞ in 𝓝[>] 0, HasFPowerSeriesOnBall f p x r :=
let ⟨_, hr⟩ := hf
mem_of_superset (Ioo_mem_nhdsWithin_Ioi (left_mem_Ico.2 hr.r_pos)) fun _ hr' =>
hr.mono hr'.1 hr'.2.le
#align has_fpower_series_at.eventually HasFPowerSeriesAt.eventually
theorem HasFPowerSeriesOnBall.eventually_hasSum (hf : HasFPowerSeriesOnBall f p x r) :
∀ᶠ y in 𝓝 0, HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y)) := by
filter_upwards [EMetric.ball_mem_nhds (0 : E) hf.r_pos] using fun _ => hf.hasSum
#align has_fpower_series_on_ball.eventually_has_sum HasFPowerSeriesOnBall.eventually_hasSum
theorem HasFPowerSeriesAt.eventually_hasSum (hf : HasFPowerSeriesAt f p x) :
∀ᶠ y in 𝓝 0, HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y)) :=
let ⟨_, hr⟩ := hf
hr.eventually_hasSum
#align has_fpower_series_at.eventually_has_sum HasFPowerSeriesAt.eventually_hasSum
theorem HasFPowerSeriesOnBall.eventually_hasSum_sub (hf : HasFPowerSeriesOnBall f p x r) :
∀ᶠ y in 𝓝 x, HasSum (fun n : ℕ => p n fun _ : Fin n => y - x) (f y) := by
filter_upwards [EMetric.ball_mem_nhds x hf.r_pos] with y using hf.hasSum_sub
#align has_fpower_series_on_ball.eventually_has_sum_sub HasFPowerSeriesOnBall.eventually_hasSum_sub
theorem HasFPowerSeriesAt.eventually_hasSum_sub (hf : HasFPowerSeriesAt f p x) :
∀ᶠ y in 𝓝 x, HasSum (fun n : ℕ => p n fun _ : Fin n => y - x) (f y) :=
let ⟨_, hr⟩ := hf
hr.eventually_hasSum_sub
#align has_fpower_series_at.eventually_has_sum_sub HasFPowerSeriesAt.eventually_hasSum_sub
theorem HasFPowerSeriesOnBall.eventually_eq_zero
(hf : HasFPowerSeriesOnBall f (0 : FormalMultilinearSeries 𝕜 E F) x r) :
∀ᶠ z in 𝓝 x, f z = 0 := by
filter_upwards [hf.eventually_hasSum_sub] with z hz using hz.unique hasSum_zero
#align has_fpower_series_on_ball.eventually_eq_zero HasFPowerSeriesOnBall.eventually_eq_zero
theorem HasFPowerSeriesAt.eventually_eq_zero
(hf : HasFPowerSeriesAt f (0 : FormalMultilinearSeries 𝕜 E F) x) : ∀ᶠ z in 𝓝 x, f z = 0 :=
let ⟨_, hr⟩ := hf
hr.eventually_eq_zero
#align has_fpower_series_at.eventually_eq_zero HasFPowerSeriesAt.eventually_eq_zero
theorem hasFPowerSeriesOnBall_const {c : F} {e : E} :
HasFPowerSeriesOnBall (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e ⊤ := by
refine' ⟨by simp, WithTop.zero_lt_top, fun _ => hasSum_single 0 fun n hn => _⟩
simp [constFormalMultilinearSeries_apply hn]
#align has_fpower_series_on_ball_const hasFPowerSeriesOnBall_const
theorem hasFPowerSeriesAt_const {c : F} {e : E} :
HasFPowerSeriesAt (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e :=
⟨⊤, hasFPowerSeriesOnBall_const⟩
#align has_fpower_series_at_const hasFPowerSeriesAt_const
theorem analyticAt_const {v : F} : AnalyticAt 𝕜 (fun _ => v) x :=
⟨constFormalMultilinearSeries 𝕜 E v, hasFPowerSeriesAt_const⟩
#align analytic_at_const analyticAt_const
theorem analyticOn_const {v : F} {s : Set E} : AnalyticOn 𝕜 (fun _ => v) s :=
fun _ _ => analyticAt_const
#align analytic_on_const analyticOn_const
theorem HasFPowerSeriesOnBall.add (hf : HasFPowerSeriesOnBall f pf x r)
(hg : HasFPowerSeriesOnBall g pg x r) : HasFPowerSeriesOnBall (f + g) (pf + pg) x r :=
{ r_le := le_trans (le_min_iff.2 ⟨hf.r_le, hg.r_le⟩) (pf.min_radius_le_radius_add pg)
r_pos := hf.r_pos
hasSum := fun hy => (hf.hasSum hy).add (hg.hasSum hy) }
#align has_fpower_series_on_ball.add HasFPowerSeriesOnBall.add
theorem HasFPowerSeriesAt.add (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x) :
HasFPowerSeriesAt (f + g) (pf + pg) x := by
rcases (hf.eventually.and hg.eventually).exists with ⟨r, hr⟩
exact ⟨r, hr.1.add hr.2⟩
#align has_fpower_series_at.add HasFPowerSeriesAt.add
theorem AnalyticAt.congr (hf : AnalyticAt 𝕜 f x) (hg : f =ᶠ[𝓝 x] g) : AnalyticAt 𝕜 g x :=
let ⟨_, hpf⟩ := hf
(hpf.congr hg).analyticAt
theorem analyticAt_congr (h : f =ᶠ[𝓝 x] g) : AnalyticAt 𝕜 f x ↔ AnalyticAt 𝕜 g x :=
⟨fun hf ↦ hf.congr h, fun hg ↦ hg.congr h.symm⟩
theorem AnalyticAt.add (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) : AnalyticAt 𝕜 (f + g) x :=
let ⟨_, hpf⟩ := hf
let ⟨_, hqf⟩ := hg
(hpf.add hqf).analyticAt
#align analytic_at.add AnalyticAt.add
theorem HasFPowerSeriesOnBall.neg (hf : HasFPowerSeriesOnBall f pf x r) :
HasFPowerSeriesOnBall (-f) (-pf) x r :=
{ r_le := by
rw [pf.radius_neg]
exact hf.r_le
r_pos := hf.r_pos
hasSum := fun hy => (hf.hasSum hy).neg }
#align has_fpower_series_on_ball.neg HasFPowerSeriesOnBall.neg
theorem HasFPowerSeriesAt.neg (hf : HasFPowerSeriesAt f pf x) : HasFPowerSeriesAt (-f) (-pf) x :=
let ⟨_, hrf⟩ := hf
hrf.neg.hasFPowerSeriesAt
#align has_fpower_series_at.neg HasFPowerSeriesAt.neg
theorem AnalyticAt.neg (hf : AnalyticAt 𝕜 f x) : AnalyticAt 𝕜 (-f) x :=
let ⟨_, hpf⟩ := hf
hpf.neg.analyticAt
#align analytic_at.neg AnalyticAt.neg
theorem HasFPowerSeriesOnBall.sub (hf : HasFPowerSeriesOnBall f pf x r)
(hg : HasFPowerSeriesOnBall g pg x r) : HasFPowerSeriesOnBall (f - g) (pf - pg) x r := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align has_fpower_series_on_ball.sub HasFPowerSeriesOnBall.sub
theorem HasFPowerSeriesAt.sub (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x) :
HasFPowerSeriesAt (f - g) (pf - pg) x := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align has_fpower_series_at.sub HasFPowerSeriesAt.sub
theorem AnalyticAt.sub (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) :
AnalyticAt 𝕜 (f - g) x := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align analytic_at.sub AnalyticAt.sub
theorem AnalyticOn.mono {s t : Set E} (hf : AnalyticOn 𝕜 f t) (hst : s ⊆ t) : AnalyticOn 𝕜 f s :=
fun z hz => hf z (hst hz)
#align analytic_on.mono AnalyticOn.mono
theorem AnalyticOn.congr' {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : f =ᶠ[𝓝ˢ s] g) :
AnalyticOn 𝕜 g s :=
fun z hz => (hf z hz).congr (mem_nhdsSet_iff_forall.mp hg z hz)
theorem analyticOn_congr' {s : Set E} (h : f =ᶠ[𝓝ˢ s] g) : AnalyticOn 𝕜 f s ↔ AnalyticOn 𝕜 g s :=
⟨fun hf => hf.congr' h, fun hg => hg.congr' h.symm⟩
theorem AnalyticOn.congr {s : Set E} (hs : IsOpen s) (hf : AnalyticOn 𝕜 f s) (hg : s.EqOn f g) :
AnalyticOn 𝕜 g s :=
hf.congr' $ mem_nhdsSet_iff_forall.mpr
(fun _ hz => eventuallyEq_iff_exists_mem.mpr ⟨s, hs.mem_nhds hz, hg⟩)
theorem analyticOn_congr {s : Set E} (hs : IsOpen s) (h : s.EqOn f g) : AnalyticOn 𝕜 f s ↔
AnalyticOn 𝕜 g s := ⟨fun hf => hf.congr hs h, fun hg => hg.congr hs h.symm⟩
theorem AnalyticOn.add {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
AnalyticOn 𝕜 (f + g) s :=
fun z hz => (hf z hz).add (hg z hz)
#align analytic_on.add AnalyticOn.add
theorem AnalyticOn.sub {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
AnalyticOn 𝕜 (f - g) s :=
fun z hz => (hf z hz).sub (hg z hz)
#align analytic_on.sub AnalyticOn.sub
theorem HasFPowerSeriesOnBall.coeff_zero (hf : HasFPowerSeriesOnBall f pf x r) (v : Fin 0 → E) :
pf 0 v = f x := by
have v_eq : v = fun i => 0 := Subsingleton.elim _ _
have zero_mem : (0 : E) ∈ EMetric.ball (0 : E) r := by simp [hf.r_pos]
have : ∀ i, i ≠ 0 → (pf i fun j => 0) = 0 := by
intro i hi
have : 0 < i := pos_iff_ne_zero.2 hi
exact ContinuousMultilinearMap.map_coord_zero _ (⟨0, this⟩ : Fin i) rfl
have A := (hf.hasSum zero_mem).unique (hasSum_single _ this)
simpa [v_eq] using A.symm
#align has_fpower_series_on_ball.coeff_zero HasFPowerSeriesOnBall.coeff_zero
theorem HasFPowerSeriesAt.coeff_zero (hf : HasFPowerSeriesAt f pf x) (v : Fin 0 → E) :
pf 0 v = f x :=
let ⟨_, hrf⟩ := hf
hrf.coeff_zero v
#align has_fpower_series_at.coeff_zero HasFPowerSeriesAt.coeff_zero
/-- If a function `f` has a power series `p` on a ball and `g` is linear, then `g ∘ f` has the
power series `g ∘ p` on the same ball. -/
theorem ContinuousLinearMap.comp_hasFPowerSeriesOnBall (g : F →L[𝕜] G)
(h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (g ∘ f) (g.compFormalMultilinearSeries p) x r :=
{ r_le := h.r_le.trans (p.radius_le_radius_continuousLinearMap_comp _)
r_pos := h.r_pos
hasSum := fun hy => by
simpa only [ContinuousLinearMap.compFormalMultilinearSeries_apply,
ContinuousLinearMap.compContinuousMultilinearMap_coe, Function.comp_apply] using
g.hasSum (h.hasSum hy) }
#align continuous_linear_map.comp_has_fpower_series_on_ball ContinuousLinearMap.comp_hasFPowerSeriesOnBall
/-- If a function `f` is analytic on a set `s` and `g` is linear, then `g ∘ f` is analytic
on `s`. -/
theorem ContinuousLinearMap.comp_analyticOn {s : Set E} (g : F →L[𝕜] G) (h : AnalyticOn 𝕜 f s) :
AnalyticOn 𝕜 (g ∘ f) s := by
rintro x hx
rcases h x hx with ⟨p, r, hp⟩
exact ⟨g.compFormalMultilinearSeries p, r, g.comp_hasFPowerSeriesOnBall hp⟩
#align continuous_linear_map.comp_analytic_on ContinuousLinearMap.comp_analyticOn
/-- If a function admits a power series expansion, then it is exponentially close to the partial
sums of this power series on strict subdisks of the disk of convergence.
This version provides an upper estimate that decreases both in `‖y‖` and `n`. See also
`HasFPowerSeriesOnBall.uniform_geometric_approx` for a weaker version. -/
theorem HasFPowerSeriesOnBall.uniform_geometric_approx' {r' : ℝ≥0}
(hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n := by
obtain ⟨a, ha, C, hC, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n, ‖p n‖ * (r' : ℝ) ^ n ≤ C * a ^ n :=
p.norm_mul_pow_le_mul_pow_of_lt_radius (h.trans_le hf.r_le)
refine' ⟨a, ha, C / (1 - a), div_pos hC (sub_pos.2 ha.2), fun y hy n => _⟩
have yr' : ‖y‖ < r' := by
rw [ball_zero_eq] at hy
exact hy
have hr'0 : 0 < (r' : ℝ) := (norm_nonneg _).trans_lt yr'
have : y ∈ EMetric.ball (0 : E) r := by
refine' mem_emetric_ball_zero_iff.2 (lt_trans _ h)
exact mod_cast yr'
rw [norm_sub_rev, ← mul_div_right_comm]
have ya : a * (‖y‖ / ↑r') ≤ a :=
mul_le_of_le_one_right ha.1.le (div_le_one_of_le yr'.le r'.coe_nonneg)
suffices ‖p.partialSum n y - f (x + y)‖ ≤ C * (a * (‖y‖ / r')) ^ n / (1 - a * (‖y‖ / r')) by
refine' this.trans _
have : 0 < a := ha.1
gcongr
apply_rules [sub_pos.2, ha.2]
apply norm_sub_le_of_geometric_bound_of_hasSum (ya.trans_lt ha.2) _ (hf.hasSum this)
intro n
calc
‖(p n) fun _ : Fin n => y‖
_ ≤ ‖p n‖ * ∏ _i : Fin n, ‖y‖ := ContinuousMultilinearMap.le_op_norm _ _
_ = ‖p n‖ * (r' : ℝ) ^ n * (‖y‖ / r') ^ n := by field_simp [mul_right_comm]
_ ≤ C * a ^ n * (‖y‖ / r') ^ n := by gcongr ?_ * _; apply hp
_ ≤ C * (a * (‖y‖ / r')) ^ n := by rw [mul_pow, mul_assoc]
#align has_fpower_series_on_ball.uniform_geometric_approx' HasFPowerSeriesOnBall.uniform_geometric_approx'
/-- If a function admits a power series expansion, then it is exponentially close to the partial
sums of this power series on strict subdisks of the disk of convergence. -/
theorem HasFPowerSeriesOnBall.uniform_geometric_approx {r' : ℝ≥0}
(hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
∃ a ∈ Ioo (0 : ℝ) 1,
∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n, ‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n := by
obtain ⟨a, ha, C, hC, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n :=
hf.uniform_geometric_approx' h
refine' ⟨a, ha, C, hC, fun y hy n => (hp y hy n).trans _⟩
have yr' : ‖y‖ < r' := by rwa [ball_zero_eq] at hy
gcongr
exacts [mul_nonneg ha.1.le (div_nonneg (norm_nonneg y) r'.coe_nonneg),
mul_le_of_le_one_right ha.1.le (div_le_one_of_le yr'.le r'.coe_nonneg)]
#align has_fpower_series_on_ball.uniform_geometric_approx HasFPowerSeriesOnBall.uniform_geometric_approx
/-- Taylor formula for an analytic function, `IsBigO` version. -/
theorem HasFPowerSeriesAt.isBigO_sub_partialSum_pow (hf : HasFPowerSeriesAt f p x) (n : ℕ) :
(fun y : E => f (x + y) - p.partialSum n y) =O[𝓝 0] fun y => ‖y‖ ^ n := by
rcases hf with ⟨r, hf⟩
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hf.r_pos with ⟨r', r'0, h⟩
obtain ⟨a, -, C, -, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n :=
hf.uniform_geometric_approx' h
refine' isBigO_iff.2 ⟨C * (a / r') ^ n, _⟩
replace r'0 : 0 < (r' : ℝ); · exact mod_cast r'0
filter_upwards [Metric.ball_mem_nhds (0 : E) r'0] with y hy
simpa [mul_pow, mul_div_assoc, mul_assoc, div_mul_eq_mul_div] using hp y hy n
set_option linter.uppercaseLean3 false in
#align has_fpower_series_at.is_O_sub_partial_sum_pow HasFPowerSeriesAt.isBigO_sub_partialSum_pow
/-- If `f` has formal power series `∑ n, pₙ` on a ball of radius `r`, then for `y, z` in any smaller
ball, the norm of the difference `f y - f z - p 1 (fun _ ↦ y - z)` is bounded above by
`C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖`. This lemma formulates this property using `IsBigO` and
`Filter.principal` on `E × E`. -/
theorem HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal
(hf : HasFPowerSeriesOnBall f p x r) (hr : r' < r) :
(fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) =O[𝓟 (EMetric.ball (x, x) r')]
fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖ := by
lift r' to ℝ≥0 using ne_top_of_lt hr
rcases (zero_le r').eq_or_lt with (rfl | hr'0)
· simp only [isBigO_bot, EMetric.ball_zero, principal_empty, ENNReal.coe_zero]
obtain ⟨a, ha, C, hC : 0 < C, hp⟩ :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n : ℕ, ‖p n‖ * (r' : ℝ) ^ n ≤ C * a ^ n
exact p.norm_mul_pow_le_mul_pow_of_lt_radius (hr.trans_le hf.r_le)
simp only [← le_div_iff (pow_pos (NNReal.coe_pos.2 hr'0) _)] at hp
set L : E × E → ℝ := fun y =>
C * (a / r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * (a / (1 - a) ^ 2 + 2 / (1 - a))
have hL : ∀ y ∈ EMetric.ball (x, x) r', ‖f y.1 - f y.2 - p 1 fun _ => y.1 - y.2‖ ≤ L y := by
intro y hy'
have hy : y ∈ EMetric.ball x r ×ˢ EMetric.ball x r := by
rw [EMetric.ball_prod_same]
exact EMetric.ball_subset_ball hr.le hy'
set A : ℕ → F := fun n => (p n fun _ => y.1 - x) - p n fun _ => y.2 - x
have hA : HasSum (fun n => A (n + 2)) (f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) := by
convert (hasSum_nat_add_iff' 2).2 ((hf.hasSum_sub hy.1).sub (hf.hasSum_sub hy.2)) using 1
rw [Finset.sum_range_succ, Finset.sum_range_one, hf.coeff_zero, hf.coeff_zero, sub_self,
zero_add, ← Subsingleton.pi_single_eq (0 : Fin 1) (y.1 - x), Pi.single,
← Subsingleton.pi_single_eq (0 : Fin 1) (y.2 - x), Pi.single, ← (p 1).map_sub, ← Pi.single,
Subsingleton.pi_single_eq, sub_sub_sub_cancel_right]
rw [EMetric.mem_ball, edist_eq_coe_nnnorm_sub, ENNReal.coe_lt_coe] at hy'
set B : ℕ → ℝ := fun n => C * (a / r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * ((n + 2) * a ^ n)
have hAB : ∀ n, ‖A (n + 2)‖ ≤ B n := fun n =>
calc
‖A (n + 2)‖ ≤ ‖p (n + 2)‖ * ↑(n + 2) * ‖y - (x, x)‖ ^ (n + 1) * ‖y.1 - y.2‖ := by
-- porting note: `pi_norm_const` was `pi_norm_const (_ : E)`
simpa only [Fintype.card_fin, pi_norm_const, Prod.norm_def, Pi.sub_def,
Prod.fst_sub, Prod.snd_sub, sub_sub_sub_cancel_right] using
(p <| n + 2).norm_image_sub_le (fun _ => y.1 - x) fun _ => y.2 - x
_ = ‖p (n + 2)‖ * ‖y - (x, x)‖ ^ n * (↑(n + 2) * ‖y - (x, x)‖ * ‖y.1 - y.2‖) := by
rw [pow_succ ‖y - (x, x)‖]
ring
-- porting note: the two `↑` in `↑r'` are new, without them, Lean fails to synthesize
-- instances `HDiv ℝ ℝ≥0 ?m` or `HMul ℝ ℝ≥0 ?m`
_ ≤ C * a ^ (n + 2) / ↑r' ^ (n + 2)
* ↑r' ^ n * (↑(n + 2) * ‖y - (x, x)‖ * ‖y.1 - y.2‖) := by
have : 0 < a := ha.1
gcongr
· apply hp
· apply hy'.le
_ = B n := by
-- porting note: in the original, `B` was in the `field_simp`, but now Lean does not
-- accept it. The current proof works in Lean 4, but does not in Lean 3.
field_simp [pow_succ]
simp only [mul_assoc, mul_comm, mul_left_comm]
have hBL : HasSum B (L y) := by
apply HasSum.mul_left
simp only [add_mul]
have : ‖a‖ < 1 := by simp only [Real.norm_eq_abs, abs_of_pos ha.1, ha.2]
rw [div_eq_mul_inv, div_eq_mul_inv]
exact (hasSum_coe_mul_geometric_of_norm_lt_1 this).add -- porting note: was `convert`!
((hasSum_geometric_of_norm_lt_1 this).mul_left 2)
exact hA.norm_le_of_bounded hBL hAB
suffices L =O[𝓟 (EMetric.ball (x, x) r')] fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖ by
refine' (IsBigO.of_bound 1 (eventually_principal.2 fun y hy => _)).trans this
rw [one_mul]
exact (hL y hy).trans (le_abs_self _)
simp_rw [mul_right_comm _ (_ * _)] -- porting note: there was an `L` inside the `simp_rw`.
exact (isBigO_refl _ _).const_mul_left _
set_option linter.uppercaseLean3 false in
#align has_fpower_series_on_ball.is_O_image_sub_image_sub_deriv_principal HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal
/-- If `f` has formal power series `∑ n, pₙ` on a ball of radius `r`, then for `y, z` in any smaller
ball, the norm of the difference `f y - f z - p 1 (fun _ ↦ y - z)` is bounded above by
`C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖`. -/
theorem HasFPowerSeriesOnBall.image_sub_sub_deriv_le (hf : HasFPowerSeriesOnBall f p x r)
(hr : r' < r) :
∃ C, ∀ᵉ (y ∈ EMetric.ball x r') (z ∈ EMetric.ball x r'),
‖f y - f z - p 1 fun _ => y - z‖ ≤ C * max ‖y - x‖ ‖z - x‖ * ‖y - z‖ := by
simpa only [isBigO_principal, mul_assoc, norm_mul, norm_norm, Prod.forall, EMetric.mem_ball,
Prod.edist_eq, max_lt_iff, and_imp, @forall_swap (_ < _) E] using
hf.isBigO_image_sub_image_sub_deriv_principal hr
#align has_fpower_series_on_ball.image_sub_sub_deriv_le HasFPowerSeriesOnBall.image_sub_sub_deriv_le
/-- If `f` has formal power series `∑ n, pₙ` at `x`, then
`f y - f z - p 1 (fun _ ↦ y - z) = O(‖(y, z) - (x, x)‖ * ‖y - z‖)` as `(y, z) → (x, x)`.
In particular, `f` is strictly differentiable at `x`. -/
theorem HasFPowerSeriesAt.isBigO_image_sub_norm_mul_norm_sub (hf : HasFPowerSeriesAt f p x) :
(fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) =O[𝓝 (x, x)] fun y =>
‖y - (x, x)‖ * ‖y.1 - y.2‖ := by
rcases hf with ⟨r, hf⟩
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hf.r_pos with ⟨r', r'0, h⟩
refine' (hf.isBigO_image_sub_image_sub_deriv_principal h).mono _
exact le_principal_iff.2 (EMetric.ball_mem_nhds _ r'0)
set_option linter.uppercaseLean3 false in
#align has_fpower_series_at.is_O_image_sub_norm_mul_norm_sub HasFPowerSeriesAt.isBigO_image_sub_norm_mul_norm_sub
/-- If a function admits a power series expansion at `x`, then it is the uniform limit of the
partial sums of this power series on strict subdisks of the disk of convergence, i.e., `f (x + y)`
is the uniform limit of `p.partialSum n y` there. -/
theorem HasFPowerSeriesOnBall.tendstoUniformlyOn {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r)
(h : (r' : ℝ≥0∞) < r) :
TendstoUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
(Metric.ball (0 : E) r') := by
obtain ⟨a, ha, C, -, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n
exact hf.uniform_geometric_approx h
refine' Metric.tendstoUniformlyOn_iff.2 fun ε εpos => _
have L : Tendsto (fun n => (C : ℝ) * a ^ n) atTop (𝓝 ((C : ℝ) * 0)) :=
tendsto_const_nhds.mul (tendsto_pow_atTop_nhds_0_of_lt_1 ha.1.le ha.2)
rw [mul_zero] at L
refine' (L.eventually (gt_mem_nhds εpos)).mono fun n hn y hy => _
rw [dist_eq_norm]
exact (hp y hy n).trans_lt hn
#align has_fpower_series_on_ball.tendsto_uniformly_on HasFPowerSeriesOnBall.tendstoUniformlyOn
/-- If a function admits a power series expansion at `x`, then it is the locally uniform limit of
the partial sums of this power series on the disk of convergence, i.e., `f (x + y)`
is the locally uniform limit of `p.partialSum n y` there. -/
theorem HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn (hf : HasFPowerSeriesOnBall f p x r) :
TendstoLocallyUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
(EMetric.ball (0 : E) r) := by
intro u hu x hx
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hx with ⟨r', xr', hr'⟩
have : EMetric.ball (0 : E) r' ∈ 𝓝 x := IsOpen.mem_nhds EMetric.isOpen_ball xr'
refine' ⟨EMetric.ball (0 : E) r', mem_nhdsWithin_of_mem_nhds this, _⟩
simpa [Metric.emetric_ball_nnreal] using hf.tendstoUniformlyOn hr' u hu
#align has_fpower_series_on_ball.tendsto_locally_uniformly_on HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn
/-- If a function admits a power series expansion at `x`, then it is the uniform limit of the
partial sums of this power series on strict subdisks of the disk of convergence, i.e., `f y`
is the uniform limit of `p.partialSum n (y - x)` there. -/
theorem HasFPowerSeriesOnBall.tendstoUniformlyOn' {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r)
(h : (r' : ℝ≥0∞) < r) :
TendstoUniformlyOn (fun n y => p.partialSum n (y - x)) f atTop (Metric.ball (x : E) r') := by
convert (hf.tendstoUniformlyOn h).comp fun y => y - x using 1
· simp [(· ∘ ·)]
· ext z
simp [dist_eq_norm]
#align has_fpower_series_on_ball.tendsto_uniformly_on' HasFPowerSeriesOnBall.tendstoUniformlyOn'
/-- If a function admits a power series expansion at `x`, then it is the locally uniform limit of
the partial sums of this power series on the disk of convergence, i.e., `f y`
is the locally uniform limit of `p.partialSum n (y - x)` there. -/
theorem HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn' (hf : HasFPowerSeriesOnBall f p x r) :
TendstoLocallyUniformlyOn (fun n y => p.partialSum n (y - x)) f atTop
(EMetric.ball (x : E) r) := by
have A : ContinuousOn (fun y : E => y - x) (EMetric.ball (x : E) r) :=
(continuous_id.sub continuous_const).continuousOn
convert hf.tendstoLocallyUniformlyOn.comp (fun y : E => y - x) _ A using 1
· ext z
simp
· intro z
simp [edist_eq_coe_nnnorm, edist_eq_coe_nnnorm_sub]
#align has_fpower_series_on_ball.tendsto_locally_uniformly_on' HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn'
/-- If a function admits a power series expansion on a disk, then it is continuous there. -/
protected theorem HasFPowerSeriesOnBall.continuousOn (hf : HasFPowerSeriesOnBall f p x r) :
ContinuousOn f (EMetric.ball x r) :=
hf.tendstoLocallyUniformlyOn'.continuousOn <|
eventually_of_forall fun n =>
((p.partialSum_continuous n).comp (continuous_id.sub continuous_const)).continuousOn
#align has_fpower_series_on_ball.continuous_on HasFPowerSeriesOnBall.continuousOn
protected theorem HasFPowerSeriesAt.continuousAt (hf : HasFPowerSeriesAt f p x) :
ContinuousAt f x :=
let ⟨_, hr⟩ := hf
hr.continuousOn.continuousAt (EMetric.ball_mem_nhds x hr.r_pos)
#align has_fpower_series_at.continuous_at HasFPowerSeriesAt.continuousAt
protected theorem AnalyticAt.continuousAt (hf : AnalyticAt 𝕜 f x) : ContinuousAt f x :=
let ⟨_, hp⟩ := hf
hp.continuousAt
#align analytic_at.continuous_at AnalyticAt.continuousAt
protected theorem AnalyticOn.continuousOn {s : Set E} (hf : AnalyticOn 𝕜 f s) : ContinuousOn f s :=
fun x hx => (hf x hx).continuousAt.continuousWithinAt
#align analytic_on.continuous_on AnalyticOn.continuousOn
/-- Analytic everywhere implies continuous -/
theorem AnalyticOn.continuous {f : E → F} (fa : AnalyticOn 𝕜 f univ) : Continuous f := by
rw [continuous_iff_continuousOn_univ]; exact fa.continuousOn
/-- In a complete space, the sum of a converging power series `p` admits `p` as a power series.
This is not totally obvious as we need to check the convergence of the series. -/
protected theorem FormalMultilinearSeries.hasFPowerSeriesOnBall [CompleteSpace F]
(p : FormalMultilinearSeries 𝕜 E F) (h : 0 < p.radius) :
HasFPowerSeriesOnBall p.sum p 0 p.radius :=
{ r_le := le_rfl
r_pos := h
hasSum := fun hy => by
rw [zero_add]
exact p.hasSum hy }
#align formal_multilinear_series.has_fpower_series_on_ball FormalMultilinearSeries.hasFPowerSeriesOnBall
theorem HasFPowerSeriesOnBall.sum (h : HasFPowerSeriesOnBall f p x r) {y : E}
(hy : y ∈ EMetric.ball (0 : E) r) : f (x + y) = p.sum y :=
(h.hasSum hy).tsum_eq.symm
#align has_fpower_series_on_ball.sum HasFPowerSeriesOnBall.sum
/-- The sum of a converging power series is continuous in its disk of convergence. -/
protected theorem FormalMultilinearSeries.continuousOn [CompleteSpace F] :
ContinuousOn p.sum (EMetric.ball 0 p.radius) := by
rcases (zero_le p.radius).eq_or_lt with h | h
· simp [← h, continuousOn_empty]
· exact (p.hasFPowerSeriesOnBall h).continuousOn
#align formal_multilinear_series.continuous_on FormalMultilinearSeries.continuousOn
end
/-!
### Uniqueness of power series
If a function `f : E → F` has two representations as power series at a point `x : E`, corresponding
to formal multilinear series `p₁` and `p₂`, then these representations agree term-by-term. That is,
for any `n : ℕ` and `y : E`, `p₁ n (fun i ↦ y) = p₂ n (fun i ↦ y)`. In the one-dimensional case,
when `f : 𝕜 → E`, the continuous multilinear maps `p₁ n` and `p₂ n` are given by
`ContinuousMultilinearMap.mkPiField`, and hence are determined completely by the value of
`p₁ n (fun i ↦ 1)`, so `p₁ = p₂`. Consequently, the radius of convergence for one series can be
transferred to the other.
-/
section Uniqueness
open ContinuousMultilinearMap
theorem Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero {n : ℕ} {p : E[×n]→L[𝕜] F}
(h : (fun y => p fun _ => y) =O[𝓝 0] fun y => ‖y‖ ^ (n + 1)) (y : E) : (p fun _ => y) = 0 := by
obtain ⟨c, c_pos, hc⟩ := h.exists_pos
obtain ⟨t, ht, t_open, z_mem⟩ := eventually_nhds_iff.mp (isBigOWith_iff.mp hc)
obtain ⟨δ, δ_pos, δε⟩ := (Metric.isOpen_iff.mp t_open) 0 z_mem
clear h hc z_mem
cases' n with n
· exact norm_eq_zero.mp (by
-- porting note: the symmetric difference of the `simpa only` sets:
-- added `Nat.zero_eq, zero_add, pow_one`
-- removed `zero_pow', Ne.def, Nat.one_ne_zero, not_false_iff`
simpa only [Nat.zero_eq, fin0_apply_norm, norm_eq_zero, norm_zero, zero_add, pow_one,
mul_zero, norm_le_zero_iff] using ht 0 (δε (Metric.mem_ball_self δ_pos)))
· refine' Or.elim (Classical.em (y = 0))
(fun hy => by simpa only [hy] using p.map_zero) fun hy => _
replace hy := norm_pos_iff.mpr hy
refine' norm_eq_zero.mp (le_antisymm (le_of_forall_pos_le_add fun ε ε_pos => _) (norm_nonneg _))
have h₀ := _root_.mul_pos c_pos (pow_pos hy (n.succ + 1))
obtain ⟨k, k_pos, k_norm⟩ := NormedField.exists_norm_lt 𝕜
(lt_min (mul_pos δ_pos (inv_pos.mpr hy)) (mul_pos ε_pos (inv_pos.mpr h₀)))
have h₁ : ‖k • y‖ < δ := by
rw [norm_smul]
exact inv_mul_cancel_right₀ hy.ne.symm δ ▸
mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_left _ _)) hy
have h₂ :=
calc
‖p fun _ => k • y‖ ≤ c * ‖k • y‖ ^ (n.succ + 1) := by
-- porting note: now Lean wants `_root_.`
simpa only [norm_pow, _root_.norm_norm] using ht (k • y) (δε (mem_ball_zero_iff.mpr h₁))
--simpa only [norm_pow, norm_norm] using ht (k • y) (δε (mem_ball_zero_iff.mpr h₁))
_ = ‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))) := by
-- porting note: added `Nat.succ_eq_add_one` since otherwise `ring` does not conclude.
simp only [norm_smul, mul_pow, Nat.succ_eq_add_one]
-- porting note: removed `rw [pow_succ]`, since it now becomes superfluous.
ring
have h₃ : ‖k‖ * (c * ‖y‖ ^ (n.succ + 1)) < ε :=
inv_mul_cancel_right₀ h₀.ne.symm ε ▸
mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_right _ _)) h₀
calc
‖p fun _ => y‖ = ‖k⁻¹ ^ n.succ‖ * ‖p fun _ => k • y‖ := by
simpa only [inv_smul_smul₀ (norm_pos_iff.mp k_pos), norm_smul, Finset.prod_const,
Finset.card_fin] using
congr_arg norm (p.map_smul_univ (fun _ : Fin n.succ => k⁻¹) fun _ : Fin n.succ => k • y)
_ ≤ ‖k⁻¹ ^ n.succ‖ * (‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1)))) := by gcongr
_ = ‖(k⁻¹ * k) ^ n.succ‖ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))) := by
rw [← mul_assoc]
simp [norm_mul, mul_pow]
_ ≤ 0 + ε := by
rw [inv_mul_cancel (norm_pos_iff.mp k_pos)]
simpa using h₃.le
set_option linter.uppercaseLean3 false in
#align asymptotics.is_O.continuous_multilinear_map_apply_eq_zero Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero
/-- If a formal multilinear series `p` represents the zero function at `x : E`, then the
terms `p n (fun i ↦ y)` appearing in the sum are zero for any `n : ℕ`, `y : E`. -/
theorem HasFPowerSeriesAt.apply_eq_zero {p : FormalMultilinearSeries 𝕜 E F} {x : E}
(h : HasFPowerSeriesAt 0 p x) (n : ℕ) : ∀ y : E, (p n fun _ => y) = 0 := by
refine' Nat.strong_induction_on n fun k hk => _
have psum_eq : p.partialSum (k + 1) = fun y => p k fun _ => y := by
funext z
refine' Finset.sum_eq_single _ (fun b hb hnb => _) fun hn => _
· have := Finset.mem_range_succ_iff.mp hb
simp only [hk b (this.lt_of_ne hnb), Pi.zero_apply]
· exact False.elim (hn (Finset.mem_range.mpr (lt_add_one k)))
replace h := h.isBigO_sub_partialSum_pow k.succ
simp only [psum_eq, zero_sub, Pi.zero_apply, Asymptotics.isBigO_neg_left] at h
exact h.continuousMultilinearMap_apply_eq_zero
#align has_fpower_series_at.apply_eq_zero HasFPowerSeriesAt.apply_eq_zero
/-- A one-dimensional formal multilinear series representing the zero function is zero. -/
theorem HasFPowerSeriesAt.eq_zero {p : FormalMultilinearSeries 𝕜 𝕜 E} {x : 𝕜}
(h : HasFPowerSeriesAt 0 p x) : p = 0 := by
-- porting note: `funext; ext` was `ext (n x)`
funext n
ext x
rw [← mkPiField_apply_one_eq_self (p n)]
-- porting note: nasty hack, was `simp [h.apply_eq_zero n 1]`
have := Or.intro_right ?_ (h.apply_eq_zero n 1)
simpa using this
#align has_fpower_series_at.eq_zero HasFPowerSeriesAt.eq_zero
/-- One-dimensional formal multilinear series representing the same function are equal. -/
theorem HasFPowerSeriesAt.eq_formalMultilinearSeries {p₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E}
{f : 𝕜 → E} {x : 𝕜} (h₁ : HasFPowerSeriesAt f p₁ x) (h₂ : HasFPowerSeriesAt f p₂ x) : p₁ = p₂ :=
sub_eq_zero.mp (HasFPowerSeriesAt.eq_zero (by simpa only [sub_self] using h₁.sub h₂))
#align has_fpower_series_at.eq_formal_multilinear_series HasFPowerSeriesAt.eq_formalMultilinearSeries
theorem HasFPowerSeriesAt.eq_formalMultilinearSeries_of_eventually
{p q : FormalMultilinearSeries 𝕜 𝕜 E} {f g : 𝕜 → E} {x : 𝕜} (hp : HasFPowerSeriesAt f p x)
(hq : HasFPowerSeriesAt g q x) (heq : ∀ᶠ z in 𝓝 x, f z = g z) : p = q :=
(hp.congr heq).eq_formalMultilinearSeries hq
#align has_fpower_series_at.eq_formal_multilinear_series_of_eventually HasFPowerSeriesAt.eq_formalMultilinearSeries_of_eventually
/-- A one-dimensional formal multilinear series representing a locally zero function is zero. -/
theorem HasFPowerSeriesAt.eq_zero_of_eventually {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E}
{x : 𝕜} (hp : HasFPowerSeriesAt f p x) (hf : f =ᶠ[𝓝 x] 0) : p = 0 :=
(hp.congr hf).eq_zero
#align has_fpower_series_at.eq_zero_of_eventually HasFPowerSeriesAt.eq_zero_of_eventually
/-- If a function `f : 𝕜 → E` has two power series representations at `x`, then the given radii in
which convergence is guaranteed may be interchanged. This can be useful when the formal multilinear
series in one representation has a particularly nice form, but the other has a larger radius. -/
theorem HasFPowerSeriesOnBall.exchange_radius {p₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E}
{r₁ r₂ : ℝ≥0∞} {x : 𝕜} (h₁ : HasFPowerSeriesOnBall f p₁ x r₁)
(h₂ : HasFPowerSeriesOnBall f p₂ x r₂) : HasFPowerSeriesOnBall f p₁ x r₂ :=
h₂.hasFPowerSeriesAt.eq_formalMultilinearSeries h₁.hasFPowerSeriesAt ▸ h₂
#align has_fpower_series_on_ball.exchange_radius HasFPowerSeriesOnBall.exchange_radius
/-- If a function `f : 𝕜 → E` has power series representation `p` on a ball of some radius and for
each positive radius it has some power series representation, then `p` converges to `f` on the whole
`𝕜`. -/
theorem HasFPowerSeriesOnBall.r_eq_top_of_exists {f : 𝕜 → E} {r : ℝ≥0∞} {x : 𝕜}
{p : FormalMultilinearSeries 𝕜 𝕜 E} (h : HasFPowerSeriesOnBall f p x r)
(h' : ∀ (r' : ℝ≥0) (_ : 0 < r'), ∃ p' : FormalMultilinearSeries 𝕜 𝕜 E,
HasFPowerSeriesOnBall f p' x r') :
HasFPowerSeriesOnBall f p x ∞ :=
{ r_le := ENNReal.le_of_forall_pos_nnreal_lt fun r hr _ =>
let ⟨_, hp'⟩ := h' r hr
(h.exchange_radius hp').r_le
r_pos := ENNReal.coe_lt_top
hasSum := fun {y} _ =>
let ⟨r', hr'⟩ := exists_gt ‖y‖₊
let ⟨_, hp'⟩ := h' r' hr'.ne_bot.bot_lt
(h.exchange_radius hp').hasSum <| mem_emetric_ball_zero_iff.mpr (ENNReal.coe_lt_coe.2 hr') }
#align has_fpower_series_on_ball.r_eq_top_of_exists HasFPowerSeriesOnBall.r_eq_top_of_exists
end Uniqueness
/-!
### Changing origin in a power series
If a function is analytic in a disk `D(x, R)`, then it is analytic in any disk contained in that
one. Indeed, one can write
$$
f (x + y + z) = \sum_{n} p_n (y + z)^n = \sum_{n, k} \binom{n}{k} p_n y^{n-k} z^k
= \sum_{k} \Bigl(\sum_{n} \binom{n}{k} p_n y^{n-k}\Bigr) z^k.
$$
The corresponding power series has thus a `k`-th coefficient equal to
$\sum_{n} \binom{n}{k} p_n y^{n-k}$. In the general case where `pₙ` is a multilinear map, this has
to be interpreted suitably: instead of having a binomial coefficient, one should sum over all
possible subsets `s` of `Fin n` of cardinal `k`, and attribute `z` to the indices in `s` and
`y` to the indices outside of `s`.
In this paragraph, we implement this. The new power series is called `p.changeOrigin y`. Then, we
check its convergence and the fact that its sum coincides with the original sum. The outcome of this
discussion is that the set of points where a function is analytic is open.
-/
namespace FormalMultilinearSeries
section
variable (p : FormalMultilinearSeries 𝕜 E F) {x y : E} {r R : ℝ≥0}
/-- A term of `FormalMultilinearSeries.changeOriginSeries`.
Given a formal multilinear series `p` and a point `x` in its ball of convergence,
`p.changeOrigin x` is a formal multilinear series such that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense. Each term of `p.changeOrigin x`
is itself an analytic function of `x` given by the series `p.changeOriginSeries`. Each term in
`changeOriginSeries` is the sum of `changeOriginSeriesTerm`'s over all `s` of cardinality `l`.
The definition is such that `p.changeOriginSeriesTerm k l s hs (fun _ ↦ x) (fun _ ↦ y) =
p (k + l) (s.piecewise (fun _ ↦ x) (fun _ ↦ y))`
-/
def changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
E[×l]→L[𝕜] E[×k]→L[𝕜] F := by
let a := ContinuousMultilinearMap.curryFinFinset 𝕜 E F hs
(by erw [Finset.card_compl, Fintype.card_fin, hs, add_tsub_cancel_right])
exact a (p (k + l))
#align formal_multilinear_series.change_origin_series_term FormalMultilinearSeries.changeOriginSeriesTerm
theorem changeOriginSeriesTerm_apply (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l)
(x y : E) :
(p.changeOriginSeriesTerm k l s hs (fun _ => x) fun _ => y) =
p (k + l) (s.piecewise (fun _ => x) fun _ => y) :=
ContinuousMultilinearMap.curryFinFinset_apply_const _ _ _ _ _
#align formal_multilinear_series.change_origin_series_term_apply FormalMultilinearSeries.changeOriginSeriesTerm_apply
@[simp]
theorem norm_changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
‖p.changeOriginSeriesTerm k l s hs‖ = ‖p (k + l)‖ := by
simp only [changeOriginSeriesTerm, LinearIsometryEquiv.norm_map]
#align formal_multilinear_series.norm_change_origin_series_term FormalMultilinearSeries.norm_changeOriginSeriesTerm
@[simp]
theorem nnnorm_changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
‖p.changeOriginSeriesTerm k l s hs‖₊ = ‖p (k + l)‖₊ := by
simp only [changeOriginSeriesTerm, LinearIsometryEquiv.nnnorm_map]
#align formal_multilinear_series.nnnorm_change_origin_series_term FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm
theorem nnnorm_changeOriginSeriesTerm_apply_le (k l : ℕ) (s : Finset (Fin (k + l)))
(hs : s.card = l) (x y : E) :
‖p.changeOriginSeriesTerm k l s hs (fun _ => x) fun _ => y‖₊ ≤
‖p (k + l)‖₊ * ‖x‖₊ ^ l * ‖y‖₊ ^ k := by
rw [← p.nnnorm_changeOriginSeriesTerm k l s hs, ← Fin.prod_const, ← Fin.prod_const]
apply ContinuousMultilinearMap.le_of_op_nnnorm_le
apply ContinuousMultilinearMap.le_op_nnnorm
#align formal_multilinear_series.nnnorm_change_origin_series_term_apply_le FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm_apply_le
/-- The power series for `f.changeOrigin k`.
Given a formal multilinear series `p` and a point `x` in its ball of convergence,
`p.changeOrigin x` is a formal multilinear series such that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense. Its `k`-th term is the sum of
the series `p.changeOriginSeries k`. -/
def changeOriginSeries (k : ℕ) : FormalMultilinearSeries 𝕜 E (E[×k]→L[𝕜] F) := fun l =>
∑ s : { s : Finset (Fin (k + l)) // Finset.card s = l }, p.changeOriginSeriesTerm k l s s.2
#align formal_multilinear_series.change_origin_series FormalMultilinearSeries.changeOriginSeries
theorem nnnorm_changeOriginSeries_le_tsum (k l : ℕ) :
‖p.changeOriginSeries k l‖₊ ≤
∑' _ : { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + l)‖₊ :=
(nnnorm_sum_le _ (fun t => changeOriginSeriesTerm p k l (Subtype.val t) t.prop)).trans_eq <| by
simp_rw [tsum_fintype, nnnorm_changeOriginSeriesTerm (p := p) (k := k) (l := l)]
#align formal_multilinear_series.nnnorm_change_origin_series_le_tsum FormalMultilinearSeries.nnnorm_changeOriginSeries_le_tsum
theorem nnnorm_changeOriginSeries_apply_le_tsum (k l : ℕ) (x : E) :
‖p.changeOriginSeries k l fun _ => x‖₊ ≤
∑' _ : { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + l)‖₊ * ‖x‖₊ ^ l := by
rw [NNReal.tsum_mul_right, ← Fin.prod_const]
exact (p.changeOriginSeries k l).le_of_op_nnnorm_le _ (p.nnnorm_changeOriginSeries_le_tsum _ _)
#align formal_multilinear_series.nnnorm_change_origin_series_apply_le_tsum FormalMultilinearSeries.nnnorm_changeOriginSeries_apply_le_tsum
/-- Changing the origin of a formal multilinear series `p`, so that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense.
-/
def changeOrigin (x : E) : FormalMultilinearSeries 𝕜 E F :=
fun k => (p.changeOriginSeries k).sum x
#align formal_multilinear_series.change_origin FormalMultilinearSeries.changeOrigin
/-- An auxiliary equivalence useful in the proofs about
`FormalMultilinearSeries.changeOriginSeries`: the set of triples `(k, l, s)`, where `s` is a
`Finset (Fin (k + l))` of cardinality `l` is equivalent to the set of pairs `(n, s)`, where `s` is a
`Finset (Fin n)`.
The forward map sends `(k, l, s)` to `(k + l, s)` and the inverse map sends `(n, s)` to
`(n - Finset.card s, Finset.card s, s)`. The actual definition is less readable because of problems
with non-definitional equalities. -/
@[simps]
def changeOriginIndexEquiv :
(Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) ≃ Σn : ℕ, Finset (Fin n) where
toFun s := ⟨s.1 + s.2.1, s.2.2⟩
invFun s :=
⟨s.1 - s.2.card, s.2.card,
⟨s.2.map
(Fin.castIso <| (tsub_add_cancel_of_le <| card_finset_fin_le s.2).symm).toEquiv.toEmbedding,
Finset.card_map _⟩⟩
left_inv := by
rintro ⟨k, l, ⟨s : Finset (Fin <| k + l), hs : s.card = l⟩⟩
dsimp only [Subtype.coe_mk]
-- Lean can't automatically generalize `k' = k + l - s.card`, `l' = s.card`, so we explicitly
-- formulate the generalized goal
suffices ∀ k' l', k' = k → l' = l → ∀ (hkl : k + l = k' + l') (hs'),
(⟨k', l', ⟨Finset.map (Fin.castIso hkl).toEquiv.toEmbedding s, hs'⟩⟩ :
Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) = ⟨k, l, ⟨s, hs⟩⟩ by
apply this <;> simp only [hs, add_tsub_cancel_right]
rintro _ _ rfl rfl hkl hs'
simp only [Equiv.refl_toEmbedding, Fin.castIso_refl, Finset.map_refl, eq_self_iff_true,
OrderIso.refl_toEquiv, and_self_iff, heq_iff_eq]
right_inv := by
rintro ⟨n, s⟩
simp [tsub_add_cancel_of_le (card_finset_fin_le s), Fin.castIso_to_equiv]
#align formal_multilinear_series.change_origin_index_equiv FormalMultilinearSeries.changeOriginIndexEquiv
theorem changeOriginSeries_summable_aux₁ {r r' : ℝ≥0} (hr : (r + r' : ℝ≥0∞) < p.radius) :
Summable fun s : Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l } =>
‖p (s.1 + s.2.1)‖₊ * r ^ s.2.1 * r' ^ s.1 := by
rw [← changeOriginIndexEquiv.symm.summable_iff]
dsimp only [Function.comp_def, changeOriginIndexEquiv_symm_apply_fst,
changeOriginIndexEquiv_symm_apply_snd_fst]
have : ∀ n : ℕ,
HasSum (fun s : Finset (Fin n) => ‖p (n - s.card + s.card)‖₊ * r ^ s.card * r' ^ (n - s.card))
(‖p n‖₊ * (r + r') ^ n) := by
intro n
-- TODO: why `simp only [tsub_add_cancel_of_le (card_finset_fin_le _)]` fails?
convert_to HasSum (fun s : Finset (Fin n) => ‖p n‖₊ * (r ^ s.card * r' ^ (n - s.card))) _
· ext1 s
rw [tsub_add_cancel_of_le (card_finset_fin_le _), mul_assoc]
rw [← Fin.sum_pow_mul_eq_add_pow]
exact (hasSum_fintype _).mul_left _
refine' NNReal.summable_sigma.2 ⟨fun n => (this n).summable, _⟩
simp only [(this _).tsum_eq]
exact p.summable_nnnorm_mul_pow hr
#align formal_multilinear_series.change_origin_series_summable_aux₁ FormalMultilinearSeries.changeOriginSeries_summable_aux₁
theorem changeOriginSeries_summable_aux₂ (hr : (r : ℝ≥0∞) < p.radius) (k : ℕ) :
Summable fun s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l } =>
‖p (k + s.1)‖₊ * r ^ s.1 := by
rcases ENNReal.lt_iff_exists_add_pos_lt.1 hr with ⟨r', h0, hr'⟩
simpa only [mul_inv_cancel_right₀ (pow_pos h0 _).ne'] using
((NNReal.summable_sigma.1 (p.changeOriginSeries_summable_aux₁ hr')).1 k).mul_right (r' ^ k)⁻¹
#align formal_multilinear_series.change_origin_series_summable_aux₂ FormalMultilinearSeries.changeOriginSeries_summable_aux₂
theorem changeOriginSeries_summable_aux₃ {r : ℝ≥0} (hr : ↑r < p.radius) (k : ℕ) :
Summable fun l : ℕ => ‖p.changeOriginSeries k l‖₊ * r ^ l := by
refine' NNReal.summable_of_le
(fun n => _) (NNReal.summable_sigma.1 <| p.changeOriginSeries_summable_aux₂ hr k).2
simp only [NNReal.tsum_mul_right]
exact mul_le_mul' (p.nnnorm_changeOriginSeries_le_tsum _ _) le_rfl
#align formal_multilinear_series.change_origin_series_summable_aux₃ FormalMultilinearSeries.changeOriginSeries_summable_aux₃
theorem le_changeOriginSeries_radius (k : ℕ) : p.radius ≤ (p.changeOriginSeries k).radius :=
ENNReal.le_of_forall_nnreal_lt fun _r hr =>
le_radius_of_summable_nnnorm _ (p.changeOriginSeries_summable_aux₃ hr k)
#align formal_multilinear_series.le_change_origin_series_radius FormalMultilinearSeries.le_changeOriginSeries_radius
theorem nnnorm_changeOrigin_le (k : ℕ) (h : (‖x‖₊ : ℝ≥0∞) < p.radius) :
‖p.changeOrigin x k‖₊ ≤
∑' s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + s.1)‖₊ * ‖x‖₊ ^ s.1 := by
refine' tsum_of_nnnorm_bounded _ fun l => p.nnnorm_changeOriginSeries_apply_le_tsum k l x
have := p.changeOriginSeries_summable_aux₂ h k
refine' HasSum.sigma this.hasSum fun l => _
exact ((NNReal.summable_sigma.1 this).1 l).hasSum
#align formal_multilinear_series.nnnorm_change_origin_le FormalMultilinearSeries.nnnorm_changeOrigin_le
/-- The radius of convergence of `p.changeOrigin x` is at least `p.radius - ‖x‖`. In other words,
`p.changeOrigin x` is well defined on the largest ball contained in the original ball of
convergence. -/
theorem changeOrigin_radius : p.radius - ‖x‖₊ ≤ (p.changeOrigin x).radius := by
refine' ENNReal.le_of_forall_pos_nnreal_lt fun r _h0 hr => _
rw [lt_tsub_iff_right, add_comm] at hr
have hr' : (‖x‖₊ : ℝ≥0∞) < p.radius := (le_add_right le_rfl).trans_lt hr
apply le_radius_of_summable_nnnorm
have : ∀ k : ℕ,
‖p.changeOrigin x k‖₊ * r ^ k ≤
(∑' s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + s.1)‖₊ * ‖x‖₊ ^ s.1) *
r ^ k :=
fun k => mul_le_mul_right' (p.nnnorm_changeOrigin_le k hr') (r ^ k)
refine' NNReal.summable_of_le this _
simpa only [← NNReal.tsum_mul_right] using
(NNReal.summable_sigma.1 (p.changeOriginSeries_summable_aux₁ hr)).2
#align formal_multilinear_series.change_origin_radius FormalMultilinearSeries.changeOrigin_radius
end
-- From this point on, assume that the space is complete, to make sure that series that converge
-- in norm also converge in `F`.
variable [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x y : E} {r R : ℝ≥0}
theorem hasFPowerSeriesOnBall_changeOrigin (k : ℕ) (hr : 0 < p.radius) :
HasFPowerSeriesOnBall (fun x => p.changeOrigin x k) (p.changeOriginSeries k) 0 p.radius :=
have := p.le_changeOriginSeries_radius k
((p.changeOriginSeries k).hasFPowerSeriesOnBall (hr.trans_le this)).mono hr this
#align formal_multilinear_series.has_fpower_series_on_ball_change_origin FormalMultilinearSeries.hasFPowerSeriesOnBall_changeOrigin
/-- Summing the series `p.changeOrigin x` at a point `y` gives back `p (x + y)`. -/
theorem changeOrigin_eval (h : (‖x‖₊ + ‖y‖₊ : ℝ≥0∞) < p.radius) :
(p.changeOrigin x).sum y = p.sum (x + y) := by
have radius_pos : 0 < p.radius := lt_of_le_of_lt (zero_le _) h
have x_mem_ball : x ∈ EMetric.ball (0 : E) p.radius :=
mem_emetric_ball_zero_iff.2 ((le_add_right le_rfl).trans_lt h)
have y_mem_ball : y ∈ EMetric.ball (0 : E) (p.changeOrigin x).radius := by
refine' mem_emetric_ball_zero_iff.2 (lt_of_lt_of_le _ p.changeOrigin_radius)
rwa [lt_tsub_iff_right, add_comm]
have x_add_y_mem_ball : x + y ∈ EMetric.ball (0 : E) p.radius := by
refine' mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt _ h)
exact mod_cast nnnorm_add_le x y
set f : (Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) → F := fun s =>
p.changeOriginSeriesTerm s.1 s.2.1 s.2.2 s.2.2.2 (fun _ => x) fun _ => y
have hsf : Summable f := by
refine' .of_nnnorm_bounded _ (p.changeOriginSeries_summable_aux₁ h) _
rintro ⟨k, l, s, hs⟩
dsimp only [Subtype.coe_mk]
exact p.nnnorm_changeOriginSeriesTerm_apply_le _ _ _ _ _ _
have hf : HasSum f ((p.changeOrigin x).sum y) := by
refine' HasSum.sigma_of_hasSum ((p.changeOrigin x).summable y_mem_ball).hasSum (fun k => _) hsf
· dsimp only
refine' ContinuousMultilinearMap.hasSum_eval _ _
have := (p.hasFPowerSeriesOnBall_changeOrigin k radius_pos).hasSum x_mem_ball
rw [zero_add] at this
refine' HasSum.sigma_of_hasSum this (fun l => _) _
· simp only [changeOriginSeries, ContinuousMultilinearMap.sum_apply]
apply hasSum_fintype
· refine' .of_nnnorm_bounded _
(p.changeOriginSeries_summable_aux₂ (mem_emetric_ball_zero_iff.1 x_mem_ball) k) fun s => _
refine' (ContinuousMultilinearMap.le_op_nnnorm _ _).trans_eq _
simp
refine' hf.unique (changeOriginIndexEquiv.symm.hasSum_iff.1 _)
refine' HasSum.sigma_of_hasSum
(p.hasSum x_add_y_mem_ball) (fun n => _) (changeOriginIndexEquiv.symm.summable_iff.2 hsf)
erw [(p n).map_add_univ (fun _ => x) fun _ => y]
-- porting note: added explicit function
convert hasSum_fintype (fun c : Finset (Fin n) => f (changeOriginIndexEquiv.symm ⟨n, c⟩))
rename_i s _
dsimp only [changeOriginSeriesTerm, (· ∘ ·), changeOriginIndexEquiv_symm_apply_fst,
changeOriginIndexEquiv_symm_apply_snd_fst, changeOriginIndexEquiv_symm_apply_snd_snd_coe]
rw [ContinuousMultilinearMap.curryFinFinset_apply_const]
have : ∀ (m) (hm : n = m), p n (s.piecewise (fun _ => x) fun _ => y) =
p m ((s.map (Fin.castIso hm).toEquiv.toEmbedding).piecewise (fun _ => x) fun _ => y) := by
rintro m rfl
simp (config := { unfoldPartialApp := true }) [Finset.piecewise]
apply this
#align formal_multilinear_series.change_origin_eval FormalMultilinearSeries.changeOrigin_eval
/-- Power series terms are analytic as we vary the origin -/
theorem analyticAt_changeOrigin (p : FormalMultilinearSeries 𝕜 E F) (rp : p.radius > 0) (n : ℕ) :
AnalyticAt 𝕜 (fun x ↦ p.changeOrigin x n) 0 :=
(FormalMultilinearSeries.hasFPowerSeriesOnBall_changeOrigin p n rp).analyticAt
end FormalMultilinearSeries
section
variable [CompleteSpace F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x y : E} {r : ℝ≥0∞}
/-- If a function admits a power series expansion `p` on a ball `B (x, r)`, then it also admits a
power series on any subball of this ball (even with a different center), given by `p.changeOrigin`.
-/
theorem HasFPowerSeriesOnBall.changeOrigin (hf : HasFPowerSeriesOnBall f p x r)
(h : (‖y‖₊ : ℝ≥0∞) < r) : HasFPowerSeriesOnBall f (p.changeOrigin y) (x + y) (r - ‖y‖₊) :=
{ r_le := by
apply le_trans _ p.changeOrigin_radius
exact tsub_le_tsub hf.r_le le_rfl
r_pos := by simp [h]
hasSum := fun {z} hz => by
have : f (x + y + z) =
FormalMultilinearSeries.sum (FormalMultilinearSeries.changeOrigin p y) z := by
rw [mem_emetric_ball_zero_iff, lt_tsub_iff_right, add_comm] at hz
rw [p.changeOrigin_eval (hz.trans_le hf.r_le), add_assoc, hf.sum]
refine' mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt _ hz)
exact mod_cast nnnorm_add_le y z
rw [this]
apply (p.changeOrigin y).hasSum
refine' EMetric.ball_subset_ball (le_trans _ p.changeOrigin_radius) hz
exact tsub_le_tsub hf.r_le le_rfl }
#align has_fpower_series_on_ball.change_origin HasFPowerSeriesOnBall.changeOrigin
/-- If a function admits a power series expansion `p` on an open ball `B (x, r)`, then
it is analytic at every point of this ball. -/
theorem HasFPowerSeriesOnBall.analyticAt_of_mem (hf : HasFPowerSeriesOnBall f p x r)
(h : y ∈ EMetric.ball x r) : AnalyticAt 𝕜 f y := by
have : (‖y - x‖₊ : ℝ≥0∞) < r := by simpa [edist_eq_coe_nnnorm_sub] using h
have := hf.changeOrigin this
rw [add_sub_cancel'_right] at this
exact this.analyticAt
#align has_fpower_series_on_ball.analytic_at_of_mem HasFPowerSeriesOnBall.analyticAt_of_mem
theorem HasFPowerSeriesOnBall.analyticOn (hf : HasFPowerSeriesOnBall f p x r) :
AnalyticOn 𝕜 f (EMetric.ball x r) :=
fun _y hy => hf.analyticAt_of_mem hy
#align has_fpower_series_on_ball.analytic_on HasFPowerSeriesOnBall.analyticOn
variable (𝕜 f)
/-- For any function `f` from a normed vector space to a Banach space, the set of points `x` such
that `f` is analytic at `x` is open. -/
theorem isOpen_analyticAt : IsOpen { x | AnalyticAt 𝕜 f x } := by
rw [isOpen_iff_mem_nhds]
rintro x ⟨p, r, hr⟩
exact mem_of_superset (EMetric.ball_mem_nhds _ hr.r_pos) fun y hy => hr.analyticAt_of_mem hy
#align is_open_analytic_at isOpen_analyticAt
variable {𝕜}
theorem AnalyticAt.eventually_analyticAt {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∀ᶠ y in 𝓝 x, AnalyticAt 𝕜 f y :=
(isOpen_analyticAt 𝕜 f).mem_nhds h
theorem AnalyticAt.exists_mem_nhds_analyticOn {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∃ s ∈ 𝓝 x, AnalyticOn 𝕜 f s :=
h.eventually_analyticAt.exists_mem
/-- If we're analytic at a point, we're analytic in a nonempty ball -/
theorem AnalyticAt.exists_ball_analyticOn {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∃ r : ℝ, 0 < r ∧ AnalyticOn 𝕜 f (Metric.ball x r) :=
Metric.isOpen_iff.mp (isOpen_analyticAt _ _) _ h
end
section
open FormalMultilinearSeries
variable {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E} {z₀ : 𝕜}
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z)) := by
refine' ⟨fun ⟨r, _, r_pos, h⟩ =>
eventually_of_mem (EMetric.ball_mem_nhds 0 r_pos) fun _ => by simpa using h, _⟩
simp only [Metric.eventually_nhds_iff]
rintro ⟨r, r_pos, h⟩
refine' ⟨p.radius ⊓ r.toNNReal, by simp, _, _⟩
· simp only [r_pos.lt, lt_inf_iff, ENNReal.coe_pos, Real.toNNReal_pos, and_true_iff]
obtain ⟨z, z_pos, le_z⟩ := NormedField.exists_norm_lt 𝕜 r_pos.lt
have : (‖z‖₊ : ENNReal) ≤ p.radius := by
simp only [dist_zero_right] at h
apply FormalMultilinearSeries.le_radius_of_tendsto
|
convert tendsto_norm.comp (h le_z).summable.tendsto_atTop_zero
|
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z)) := by
refine' ⟨fun ⟨r, _, r_pos, h⟩ =>
eventually_of_mem (EMetric.ball_mem_nhds 0 r_pos) fun _ => by simpa using h, _⟩
simp only [Metric.eventually_nhds_iff]
rintro ⟨r, r_pos, h⟩
refine' ⟨p.radius ⊓ r.toNNReal, by simp, _, _⟩
· simp only [r_pos.lt, lt_inf_iff, ENNReal.coe_pos, Real.toNNReal_pos, and_true_iff]
obtain ⟨z, z_pos, le_z⟩ := NormedField.exists_norm_lt 𝕜 r_pos.lt
have : (‖z‖₊ : ENNReal) ≤ p.radius := by
simp only [dist_zero_right] at h
apply FormalMultilinearSeries.le_radius_of_tendsto
|
Mathlib.Analysis.Analytic.Basic.1430_0.jQw1fRSE1vGpOll
|
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z))
|
Mathlib_Analysis_Analytic_Basic
|
case h.e'_3.h
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
p : FormalMultilinearSeries 𝕜 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
r : ℝ
r_pos : r > 0
z : 𝕜
z_pos : 0 < ‖z‖
le_z : ‖z‖ < r
h : ∀ ⦃y : 𝕜⦄, ‖y‖ < r → HasSum (fun n => y ^ n • coeff p n) (f (z₀ + y))
x✝ : ℕ
⊢ ‖p x✝‖ * ↑‖z‖₊ ^ x✝ = ((fun a => ‖a‖) ∘ fun n => z ^ n • coeff p n) x✝
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Logic.Equiv.Fin
import Mathlib.Topology.Algebra.InfiniteSum.Module
#align_import analysis.analytic.basic from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
/-!
# Analytic functions
A function is analytic in one dimension around `0` if it can be written as a converging power series
`Σ pₙ zⁿ`. This definition can be extended to any dimension (even in infinite dimension) by
requiring that `pₙ` is a continuous `n`-multilinear map. In general, `pₙ` is not unique (in two
dimensions, taking `p₂ (x, y) (x', y') = x y'` or `y x'` gives the same map when applied to a
vector `(x, y) (x, y)`). A way to guarantee uniqueness is to take a symmetric `pₙ`, but this is not
always possible in nonzero characteristic (in characteristic 2, the previous example has no
symmetric representative). Therefore, we do not insist on symmetry or uniqueness in the definition,
and we only require the existence of a converging series.
The general framework is important to say that the exponential map on bounded operators on a Banach
space is analytic, as well as the inverse on invertible operators.
## Main definitions
Let `p` be a formal multilinear series from `E` to `F`, i.e., `p n` is a multilinear map on `E^n`
for `n : ℕ`.
* `p.radius`: the largest `r : ℝ≥0∞` such that `‖p n‖ * r^n` grows subexponentially.
* `p.le_radius_of_bound`, `p.le_radius_of_bound_nnreal`, `p.le_radius_of_isBigO`: if `‖p n‖ * r ^ n`
is bounded above, then `r ≤ p.radius`;
* `p.isLittleO_of_lt_radius`, `p.norm_mul_pow_le_mul_pow_of_lt_radius`,
`p.isLittleO_one_of_lt_radius`,
`p.norm_mul_pow_le_of_lt_radius`, `p.nnnorm_mul_pow_le_of_lt_radius`: if `r < p.radius`, then
`‖p n‖ * r ^ n` tends to zero exponentially;
* `p.lt_radius_of_isBigO`: if `r ≠ 0` and `‖p n‖ * r ^ n = O(a ^ n)` for some `-1 < a < 1`, then
`r < p.radius`;
* `p.partialSum n x`: the sum `∑_{i = 0}^{n-1} pᵢ xⁱ`.
* `p.sum x`: the sum `∑'_{i = 0}^{∞} pᵢ xⁱ`.
Additionally, let `f` be a function from `E` to `F`.
* `HasFPowerSeriesOnBall f p x r`: on the ball of center `x` with radius `r`,
`f (x + y) = ∑'_n pₙ yⁿ`.
* `HasFPowerSeriesAt f p x`: on some ball of center `x` with positive radius, holds
`HasFPowerSeriesOnBall f p x r`.
* `AnalyticAt 𝕜 f x`: there exists a power series `p` such that holds `HasFPowerSeriesAt f p x`.
* `AnalyticOn 𝕜 f s`: the function `f` is analytic at every point of `s`.
We develop the basic properties of these notions, notably:
* If a function admits a power series, it is continuous (see
`HasFPowerSeriesOnBall.continuousOn` and `HasFPowerSeriesAt.continuousAt` and
`AnalyticAt.continuousAt`).
* In a complete space, the sum of a formal power series with positive radius is well defined on the
disk of convergence, see `FormalMultilinearSeries.hasFPowerSeriesOnBall`.
* If a function admits a power series in a ball, then it is analytic at any point `y` of this ball,
and the power series there can be expressed in terms of the initial power series `p` as
`p.changeOrigin y`. See `HasFPowerSeriesOnBall.changeOrigin`. It follows in particular that
the set of points at which a given function is analytic is open, see `isOpen_analyticAt`.
## Implementation details
We only introduce the radius of convergence of a power series, as `p.radius`.
For a power series in finitely many dimensions, there is a finer (directional, coordinate-dependent)
notion, describing the polydisk of convergence. This notion is more specific, and not necessary to
build the general theory. We do not define it here.
-/
noncomputable section
variable {𝕜 E F G : Type*}
open Topology Classical BigOperators NNReal Filter ENNReal
open Set Filter Asymptotics
namespace FormalMultilinearSeries
variable [Ring 𝕜] [AddCommGroup E] [AddCommGroup F] [Module 𝕜 E] [Module 𝕜 F]
variable [TopologicalSpace E] [TopologicalSpace F]
variable [TopologicalAddGroup E] [TopologicalAddGroup F]
variable [ContinuousConstSMul 𝕜 E] [ContinuousConstSMul 𝕜 F]
/-- Given a formal multilinear series `p` and a vector `x`, then `p.sum x` is the sum `Σ pₙ xⁿ`. A
priori, it only behaves well when `‖x‖ < p.radius`. -/
protected def sum (p : FormalMultilinearSeries 𝕜 E F) (x : E) : F :=
∑' n : ℕ, p n fun _ => x
#align formal_multilinear_series.sum FormalMultilinearSeries.sum
/-- Given a formal multilinear series `p` and a vector `x`, then `p.partialSum n x` is the sum
`Σ pₖ xᵏ` for `k ∈ {0,..., n-1}`. -/
def partialSum (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) (x : E) : F :=
∑ k in Finset.range n, p k fun _ : Fin k => x
#align formal_multilinear_series.partial_sum FormalMultilinearSeries.partialSum
/-- The partial sums of a formal multilinear series are continuous. -/
theorem partialSum_continuous (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) :
Continuous (p.partialSum n) := by
unfold partialSum -- Porting note: added
continuity
#align formal_multilinear_series.partial_sum_continuous FormalMultilinearSeries.partialSum_continuous
end FormalMultilinearSeries
/-! ### The radius of a formal multilinear series -/
variable [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F]
[NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G]
namespace FormalMultilinearSeries
variable (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
/-- The radius of a formal multilinear series is the largest `r` such that the sum `Σ ‖pₙ‖ ‖y‖ⁿ`
converges for all `‖y‖ < r`. This implies that `Σ pₙ yⁿ` converges for all `‖y‖ < r`, but these
definitions are *not* equivalent in general. -/
def radius (p : FormalMultilinearSeries 𝕜 E F) : ℝ≥0∞ :=
⨆ (r : ℝ≥0) (C : ℝ) (_ : ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C), (r : ℝ≥0∞)
#align formal_multilinear_series.radius FormalMultilinearSeries.radius
/-- If `‖pₙ‖ rⁿ` is bounded in `n`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_bound (C : ℝ) {r : ℝ≥0} (h : ∀ n : ℕ, ‖p n‖ * (r : ℝ) ^ n ≤ C) :
(r : ℝ≥0∞) ≤ p.radius :=
le_iSup_of_le r <| le_iSup_of_le C <| le_iSup (fun _ => (r : ℝ≥0∞)) h
#align formal_multilinear_series.le_radius_of_bound FormalMultilinearSeries.le_radius_of_bound
/-- If `‖pₙ‖ rⁿ` is bounded in `n`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_bound_nnreal (C : ℝ≥0) {r : ℝ≥0} (h : ∀ n : ℕ, ‖p n‖₊ * r ^ n ≤ C) :
(r : ℝ≥0∞) ≤ p.radius :=
p.le_radius_of_bound C fun n => mod_cast h n
#align formal_multilinear_series.le_radius_of_bound_nnreal FormalMultilinearSeries.le_radius_of_bound_nnreal
/-- If `‖pₙ‖ rⁿ = O(1)`, as `n → ∞`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_isBigO (h : (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] fun _ => (1 : ℝ)) :
↑r ≤ p.radius :=
Exists.elim (isBigO_one_nat_atTop_iff.1 h) fun C hC =>
p.le_radius_of_bound C fun n => (le_abs_self _).trans (hC n)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.le_radius_of_is_O FormalMultilinearSeries.le_radius_of_isBigO
theorem le_radius_of_eventually_le (C) (h : ∀ᶠ n in atTop, ‖p n‖ * (r : ℝ) ^ n ≤ C) :
↑r ≤ p.radius :=
p.le_radius_of_isBigO <| IsBigO.of_bound C <| h.mono fun n hn => by simpa
#align formal_multilinear_series.le_radius_of_eventually_le FormalMultilinearSeries.le_radius_of_eventually_le
theorem le_radius_of_summable_nnnorm (h : Summable fun n => ‖p n‖₊ * r ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_bound_nnreal (∑' n, ‖p n‖₊ * r ^ n) fun _ => le_tsum' h _
#align formal_multilinear_series.le_radius_of_summable_nnnorm FormalMultilinearSeries.le_radius_of_summable_nnnorm
theorem le_radius_of_summable (h : Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_summable_nnnorm <| by
simp only [← coe_nnnorm] at h
exact mod_cast h
#align formal_multilinear_series.le_radius_of_summable FormalMultilinearSeries.le_radius_of_summable
theorem radius_eq_top_of_forall_nnreal_isBigO
(h : ∀ r : ℝ≥0, (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] fun _ => (1 : ℝ)) : p.radius = ∞ :=
ENNReal.eq_top_of_forall_nnreal_le fun r => p.le_radius_of_isBigO (h r)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.radius_eq_top_of_forall_nnreal_is_O FormalMultilinearSeries.radius_eq_top_of_forall_nnreal_isBigO
theorem radius_eq_top_of_eventually_eq_zero (h : ∀ᶠ n in atTop, p n = 0) : p.radius = ∞ :=
p.radius_eq_top_of_forall_nnreal_isBigO fun r =>
(isBigO_zero _ _).congr' (h.mono fun n hn => by simp [hn]) EventuallyEq.rfl
#align formal_multilinear_series.radius_eq_top_of_eventually_eq_zero FormalMultilinearSeries.radius_eq_top_of_eventually_eq_zero
theorem radius_eq_top_of_forall_image_add_eq_zero (n : ℕ) (hn : ∀ m, p (m + n) = 0) :
p.radius = ∞ :=
p.radius_eq_top_of_eventually_eq_zero <|
mem_atTop_sets.2 ⟨n, fun _ hk => tsub_add_cancel_of_le hk ▸ hn _⟩
#align formal_multilinear_series.radius_eq_top_of_forall_image_add_eq_zero FormalMultilinearSeries.radius_eq_top_of_forall_image_add_eq_zero
@[simp]
theorem constFormalMultilinearSeries_radius {v : F} :
(constFormalMultilinearSeries 𝕜 E v).radius = ⊤ :=
(constFormalMultilinearSeries 𝕜 E v).radius_eq_top_of_forall_image_add_eq_zero 1
(by simp [constFormalMultilinearSeries])
#align formal_multilinear_series.const_formal_multilinear_series_radius FormalMultilinearSeries.constFormalMultilinearSeries_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` tends to zero exponentially:
for some `0 < a < 1`, `‖p n‖ rⁿ = o(aⁿ)`. -/
theorem isLittleO_of_lt_radius (h : ↑r < p.radius) :
∃ a ∈ Ioo (0 : ℝ) 1, (fun n => ‖p n‖ * (r : ℝ) ^ n) =o[atTop] (a ^ ·) := by
have := (TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 4
rw [this]
-- Porting note: was
-- rw [(TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 4]
simp only [radius, lt_iSup_iff] at h
rcases h with ⟨t, C, hC, rt⟩
rw [ENNReal.coe_lt_coe, ← NNReal.coe_lt_coe] at rt
have : 0 < (t : ℝ) := r.coe_nonneg.trans_lt rt
rw [← div_lt_one this] at rt
refine' ⟨_, rt, C, Or.inr zero_lt_one, fun n => _⟩
calc
|‖p n‖ * (r : ℝ) ^ n| = ‖p n‖ * (t : ℝ) ^ n * (r / t : ℝ) ^ n := by
field_simp [mul_right_comm, abs_mul]
_ ≤ C * (r / t : ℝ) ^ n := by gcongr; apply hC
#align formal_multilinear_series.is_o_of_lt_radius FormalMultilinearSeries.isLittleO_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ = o(1)`. -/
theorem isLittleO_one_of_lt_radius (h : ↑r < p.radius) :
(fun n => ‖p n‖ * (r : ℝ) ^ n) =o[atTop] (fun _ => 1 : ℕ → ℝ) :=
let ⟨_, ha, hp⟩ := p.isLittleO_of_lt_radius h
hp.trans <| (isLittleO_pow_pow_of_lt_left ha.1.le ha.2).congr (fun _ => rfl) one_pow
#align formal_multilinear_series.is_o_one_of_lt_radius FormalMultilinearSeries.isLittleO_one_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` tends to zero exponentially:
for some `0 < a < 1` and `C > 0`, `‖p n‖ * r ^ n ≤ C * a ^ n`. -/
theorem norm_mul_pow_le_mul_pow_of_lt_radius (h : ↑r < p.radius) :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C * a ^ n := by
-- Porting note: moved out of `rcases`
have := ((TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 5).mp
(p.isLittleO_of_lt_radius h)
rcases this with ⟨a, ha, C, hC, H⟩
exact ⟨a, ha, C, hC, fun n => (le_abs_self _).trans (H n)⟩
#align formal_multilinear_series.norm_mul_pow_le_mul_pow_of_lt_radius FormalMultilinearSeries.norm_mul_pow_le_mul_pow_of_lt_radius
/-- If `r ≠ 0` and `‖pₙ‖ rⁿ = O(aⁿ)` for some `-1 < a < 1`, then `r < p.radius`. -/
theorem lt_radius_of_isBigO (h₀ : r ≠ 0) {a : ℝ} (ha : a ∈ Ioo (-1 : ℝ) 1)
(hp : (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] (a ^ ·)) : ↑r < p.radius := by
-- Porting note: moved out of `rcases`
have := ((TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 2 5)
rcases this.mp ⟨a, ha, hp⟩ with ⟨a, ha, C, hC, hp⟩
rw [← pos_iff_ne_zero, ← NNReal.coe_pos] at h₀
lift a to ℝ≥0 using ha.1.le
have : (r : ℝ) < r / a := by
simpa only [div_one] using (div_lt_div_left h₀ zero_lt_one ha.1).2 ha.2
norm_cast at this
rw [← ENNReal.coe_lt_coe] at this
refine' this.trans_le (p.le_radius_of_bound C fun n => _)
rw [NNReal.coe_div, div_pow, ← mul_div_assoc, div_le_iff (pow_pos ha.1 n)]
exact (le_abs_self _).trans (hp n)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.lt_radius_of_is_O FormalMultilinearSeries.lt_radius_of_isBigO
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem norm_mul_pow_le_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C :=
let ⟨_, ha, C, hC, h⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius h
⟨C, hC, fun n => (h n).trans <| mul_le_of_le_one_right hC.lt.le (pow_le_one _ ha.1.le ha.2.le)⟩
#align formal_multilinear_series.norm_mul_pow_le_of_lt_radius FormalMultilinearSeries.norm_mul_pow_le_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem norm_le_div_pow_of_pos_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h0 : 0 < r) (h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖ ≤ C / (r : ℝ) ^ n :=
let ⟨C, hC, hp⟩ := p.norm_mul_pow_le_of_lt_radius h
⟨C, hC, fun n => Iff.mpr (le_div_iff (pow_pos h0 _)) (hp n)⟩
#align formal_multilinear_series.norm_le_div_pow_of_pos_of_lt_radius FormalMultilinearSeries.norm_le_div_pow_of_pos_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem nnnorm_mul_pow_le_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖₊ * r ^ n ≤ C :=
let ⟨C, hC, hp⟩ := p.norm_mul_pow_le_of_lt_radius h
⟨⟨C, hC.lt.le⟩, hC, mod_cast hp⟩
#align formal_multilinear_series.nnnorm_mul_pow_le_of_lt_radius FormalMultilinearSeries.nnnorm_mul_pow_le_of_lt_radius
theorem le_radius_of_tendsto (p : FormalMultilinearSeries 𝕜 E F) {l : ℝ}
(h : Tendsto (fun n => ‖p n‖ * (r : ℝ) ^ n) atTop (𝓝 l)) : ↑r ≤ p.radius :=
p.le_radius_of_isBigO (h.isBigO_one _)
#align formal_multilinear_series.le_radius_of_tendsto FormalMultilinearSeries.le_radius_of_tendsto
theorem le_radius_of_summable_norm (p : FormalMultilinearSeries 𝕜 E F)
(hs : Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_tendsto hs.tendsto_atTop_zero
#align formal_multilinear_series.le_radius_of_summable_norm FormalMultilinearSeries.le_radius_of_summable_norm
theorem not_summable_norm_of_radius_lt_nnnorm (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(h : p.radius < ‖x‖₊) : ¬Summable fun n => ‖p n‖ * ‖x‖ ^ n :=
fun hs => not_le_of_lt h (p.le_radius_of_summable_norm hs)
#align formal_multilinear_series.not_summable_norm_of_radius_lt_nnnorm FormalMultilinearSeries.not_summable_norm_of_radius_lt_nnnorm
theorem summable_norm_mul_pow (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0} (h : ↑r < p.radius) :
Summable fun n : ℕ => ‖p n‖ * (r : ℝ) ^ n := by
obtain ⟨a, ha : a ∈ Ioo (0 : ℝ) 1, C, - : 0 < C, hp⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius h
exact .of_nonneg_of_le (fun n => mul_nonneg (norm_nonneg _) (pow_nonneg r.coe_nonneg _))
hp ((summable_geometric_of_lt_1 ha.1.le ha.2).mul_left _)
#align formal_multilinear_series.summable_norm_mul_pow FormalMultilinearSeries.summable_norm_mul_pow
theorem summable_norm_apply (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : Summable fun n : ℕ => ‖p n fun _ => x‖ := by
rw [mem_emetric_ball_zero_iff] at hx
refine' .of_nonneg_of_le
(fun _ => norm_nonneg _) (fun n => ((p n).le_op_norm _).trans_eq _) (p.summable_norm_mul_pow hx)
simp
#align formal_multilinear_series.summable_norm_apply FormalMultilinearSeries.summable_norm_apply
theorem summable_nnnorm_mul_pow (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0} (h : ↑r < p.radius) :
Summable fun n : ℕ => ‖p n‖₊ * r ^ n := by
rw [← NNReal.summable_coe]
push_cast
exact p.summable_norm_mul_pow h
#align formal_multilinear_series.summable_nnnorm_mul_pow FormalMultilinearSeries.summable_nnnorm_mul_pow
protected theorem summable [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : Summable fun n : ℕ => p n fun _ => x :=
(p.summable_norm_apply hx).of_norm
#align formal_multilinear_series.summable FormalMultilinearSeries.summable
theorem radius_eq_top_of_summable_norm (p : FormalMultilinearSeries 𝕜 E F)
(hs : ∀ r : ℝ≥0, Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : p.radius = ∞ :=
ENNReal.eq_top_of_forall_nnreal_le fun r => p.le_radius_of_summable_norm (hs r)
#align formal_multilinear_series.radius_eq_top_of_summable_norm FormalMultilinearSeries.radius_eq_top_of_summable_norm
theorem radius_eq_top_iff_summable_norm (p : FormalMultilinearSeries 𝕜 E F) :
p.radius = ∞ ↔ ∀ r : ℝ≥0, Summable fun n => ‖p n‖ * (r : ℝ) ^ n := by
constructor
· intro h r
obtain ⟨a, ha : a ∈ Ioo (0 : ℝ) 1, C, - : 0 < C, hp⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius
(show (r : ℝ≥0∞) < p.radius from h.symm ▸ ENNReal.coe_lt_top)
refine' .of_norm_bounded
(fun n => (C : ℝ) * a ^ n) ((summable_geometric_of_lt_1 ha.1.le ha.2).mul_left _) fun n => _
specialize hp n
rwa [Real.norm_of_nonneg (mul_nonneg (norm_nonneg _) (pow_nonneg r.coe_nonneg n))]
· exact p.radius_eq_top_of_summable_norm
#align formal_multilinear_series.radius_eq_top_iff_summable_norm FormalMultilinearSeries.radius_eq_top_iff_summable_norm
/-- If the radius of `p` is positive, then `‖pₙ‖` grows at most geometrically. -/
theorem le_mul_pow_of_radius_pos (p : FormalMultilinearSeries 𝕜 E F) (h : 0 < p.radius) :
∃ (C r : _) (hC : 0 < C) (_ : 0 < r), ∀ n, ‖p n‖ ≤ C * r ^ n := by
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 h with ⟨r, r0, rlt⟩
have rpos : 0 < (r : ℝ) := by simp [ENNReal.coe_pos.1 r0]
rcases norm_le_div_pow_of_pos_of_lt_radius p rpos rlt with ⟨C, Cpos, hCp⟩
refine' ⟨C, r⁻¹, Cpos, by simp only [inv_pos, rpos], fun n => _⟩
-- Porting note: was `convert`
rw [inv_pow, ← div_eq_mul_inv]
exact hCp n
#align formal_multilinear_series.le_mul_pow_of_radius_pos FormalMultilinearSeries.le_mul_pow_of_radius_pos
/-- The radius of the sum of two formal series is at least the minimum of their two radii. -/
theorem min_radius_le_radius_add (p q : FormalMultilinearSeries 𝕜 E F) :
min p.radius q.radius ≤ (p + q).radius := by
refine' ENNReal.le_of_forall_nnreal_lt fun r hr => _
rw [lt_min_iff] at hr
have := ((p.isLittleO_one_of_lt_radius hr.1).add (q.isLittleO_one_of_lt_radius hr.2)).isBigO
refine' (p + q).le_radius_of_isBigO ((isBigO_of_le _ fun n => _).trans this)
rw [← add_mul, norm_mul, norm_mul, norm_norm]
exact mul_le_mul_of_nonneg_right ((norm_add_le _ _).trans (le_abs_self _)) (norm_nonneg _)
#align formal_multilinear_series.min_radius_le_radius_add FormalMultilinearSeries.min_radius_le_radius_add
@[simp]
theorem radius_neg (p : FormalMultilinearSeries 𝕜 E F) : (-p).radius = p.radius := by
simp only [radius, neg_apply, norm_neg]
#align formal_multilinear_series.radius_neg FormalMultilinearSeries.radius_neg
protected theorem hasSum [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : HasSum (fun n : ℕ => p n fun _ => x) (p.sum x) :=
(p.summable hx).hasSum
#align formal_multilinear_series.has_sum FormalMultilinearSeries.hasSum
theorem radius_le_radius_continuousLinearMap_comp (p : FormalMultilinearSeries 𝕜 E F)
(f : F →L[𝕜] G) : p.radius ≤ (f.compFormalMultilinearSeries p).radius := by
refine' ENNReal.le_of_forall_nnreal_lt fun r hr => _
apply le_radius_of_isBigO
apply (IsBigO.trans_isLittleO _ (p.isLittleO_one_of_lt_radius hr)).isBigO
refine' IsBigO.mul (@IsBigOWith.isBigO _ _ _ _ _ ‖f‖ _ _ _ _) (isBigO_refl _ _)
refine IsBigOWith.of_bound (eventually_of_forall fun n => ?_)
simpa only [norm_norm] using f.norm_compContinuousMultilinearMap_le (p n)
#align formal_multilinear_series.radius_le_radius_continuous_linear_map_comp FormalMultilinearSeries.radius_le_radius_continuousLinearMap_comp
end FormalMultilinearSeries
/-! ### Expanding a function as a power series -/
section
variable {f g : E → F} {p pf pg : FormalMultilinearSeries 𝕜 E F} {x : E} {r r' : ℝ≥0∞}
/-- Given a function `f : E → F` and a formal multilinear series `p`, we say that `f` has `p` as
a power series on the ball of radius `r > 0` around `x` if `f (x + y) = ∑' pₙ yⁿ` for all `‖y‖ < r`.
-/
structure HasFPowerSeriesOnBall (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (x : E) (r : ℝ≥0∞) :
Prop where
r_le : r ≤ p.radius
r_pos : 0 < r
hasSum :
∀ {y}, y ∈ EMetric.ball (0 : E) r → HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y))
#align has_fpower_series_on_ball HasFPowerSeriesOnBall
/-- Given a function `f : E → F` and a formal multilinear series `p`, we say that `f` has `p` as
a power series around `x` if `f (x + y) = ∑' pₙ yⁿ` for all `y` in a neighborhood of `0`. -/
def HasFPowerSeriesAt (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (x : E) :=
∃ r, HasFPowerSeriesOnBall f p x r
#align has_fpower_series_at HasFPowerSeriesAt
variable (𝕜)
/-- Given a function `f : E → F`, we say that `f` is analytic at `x` if it admits a convergent power
series expansion around `x`. -/
def AnalyticAt (f : E → F) (x : E) :=
∃ p : FormalMultilinearSeries 𝕜 E F, HasFPowerSeriesAt f p x
#align analytic_at AnalyticAt
/-- Given a function `f : E → F`, we say that `f` is analytic on a set `s` if it is analytic around
every point of `s`. -/
def AnalyticOn (f : E → F) (s : Set E) :=
∀ x, x ∈ s → AnalyticAt 𝕜 f x
#align analytic_on AnalyticOn
variable {𝕜}
theorem HasFPowerSeriesOnBall.hasFPowerSeriesAt (hf : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesAt f p x :=
⟨r, hf⟩
#align has_fpower_series_on_ball.has_fpower_series_at HasFPowerSeriesOnBall.hasFPowerSeriesAt
theorem HasFPowerSeriesAt.analyticAt (hf : HasFPowerSeriesAt f p x) : AnalyticAt 𝕜 f x :=
⟨p, hf⟩
#align has_fpower_series_at.analytic_at HasFPowerSeriesAt.analyticAt
theorem HasFPowerSeriesOnBall.analyticAt (hf : HasFPowerSeriesOnBall f p x r) : AnalyticAt 𝕜 f x :=
hf.hasFPowerSeriesAt.analyticAt
#align has_fpower_series_on_ball.analytic_at HasFPowerSeriesOnBall.analyticAt
theorem HasFPowerSeriesOnBall.congr (hf : HasFPowerSeriesOnBall f p x r)
(hg : EqOn f g (EMetric.ball x r)) : HasFPowerSeriesOnBall g p x r :=
{ r_le := hf.r_le
r_pos := hf.r_pos
hasSum := fun {y} hy => by
convert hf.hasSum hy using 1
apply hg.symm
simpa [edist_eq_coe_nnnorm_sub] using hy }
#align has_fpower_series_on_ball.congr HasFPowerSeriesOnBall.congr
/-- If a function `f` has a power series `p` around `x`, then the function `z ↦ f (z - y)` has the
same power series around `x + y`. -/
theorem HasFPowerSeriesOnBall.comp_sub (hf : HasFPowerSeriesOnBall f p x r) (y : E) :
HasFPowerSeriesOnBall (fun z => f (z - y)) p (x + y) r :=
{ r_le := hf.r_le
r_pos := hf.r_pos
hasSum := fun {z} hz => by
convert hf.hasSum hz using 2
abel }
#align has_fpower_series_on_ball.comp_sub HasFPowerSeriesOnBall.comp_sub
theorem HasFPowerSeriesOnBall.hasSum_sub (hf : HasFPowerSeriesOnBall f p x r) {y : E}
(hy : y ∈ EMetric.ball x r) : HasSum (fun n : ℕ => p n fun _ => y - x) (f y) := by
have : y - x ∈ EMetric.ball (0 : E) r := by simpa [edist_eq_coe_nnnorm_sub] using hy
simpa only [add_sub_cancel'_right] using hf.hasSum this
#align has_fpower_series_on_ball.has_sum_sub HasFPowerSeriesOnBall.hasSum_sub
theorem HasFPowerSeriesOnBall.radius_pos (hf : HasFPowerSeriesOnBall f p x r) : 0 < p.radius :=
lt_of_lt_of_le hf.r_pos hf.r_le
#align has_fpower_series_on_ball.radius_pos HasFPowerSeriesOnBall.radius_pos
theorem HasFPowerSeriesAt.radius_pos (hf : HasFPowerSeriesAt f p x) : 0 < p.radius :=
let ⟨_, hr⟩ := hf
hr.radius_pos
#align has_fpower_series_at.radius_pos HasFPowerSeriesAt.radius_pos
theorem HasFPowerSeriesOnBall.mono (hf : HasFPowerSeriesOnBall f p x r) (r'_pos : 0 < r')
(hr : r' ≤ r) : HasFPowerSeriesOnBall f p x r' :=
⟨le_trans hr hf.1, r'_pos, fun hy => hf.hasSum (EMetric.ball_subset_ball hr hy)⟩
#align has_fpower_series_on_ball.mono HasFPowerSeriesOnBall.mono
theorem HasFPowerSeriesAt.congr (hf : HasFPowerSeriesAt f p x) (hg : f =ᶠ[𝓝 x] g) :
HasFPowerSeriesAt g p x := by
rcases hf with ⟨r₁, h₁⟩
rcases EMetric.mem_nhds_iff.mp hg with ⟨r₂, h₂pos, h₂⟩
exact ⟨min r₁ r₂,
(h₁.mono (lt_min h₁.r_pos h₂pos) inf_le_left).congr
fun y hy => h₂ (EMetric.ball_subset_ball inf_le_right hy)⟩
#align has_fpower_series_at.congr HasFPowerSeriesAt.congr
protected theorem HasFPowerSeriesAt.eventually (hf : HasFPowerSeriesAt f p x) :
∀ᶠ r : ℝ≥0∞ in 𝓝[>] 0, HasFPowerSeriesOnBall f p x r :=
let ⟨_, hr⟩ := hf
mem_of_superset (Ioo_mem_nhdsWithin_Ioi (left_mem_Ico.2 hr.r_pos)) fun _ hr' =>
hr.mono hr'.1 hr'.2.le
#align has_fpower_series_at.eventually HasFPowerSeriesAt.eventually
theorem HasFPowerSeriesOnBall.eventually_hasSum (hf : HasFPowerSeriesOnBall f p x r) :
∀ᶠ y in 𝓝 0, HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y)) := by
filter_upwards [EMetric.ball_mem_nhds (0 : E) hf.r_pos] using fun _ => hf.hasSum
#align has_fpower_series_on_ball.eventually_has_sum HasFPowerSeriesOnBall.eventually_hasSum
theorem HasFPowerSeriesAt.eventually_hasSum (hf : HasFPowerSeriesAt f p x) :
∀ᶠ y in 𝓝 0, HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y)) :=
let ⟨_, hr⟩ := hf
hr.eventually_hasSum
#align has_fpower_series_at.eventually_has_sum HasFPowerSeriesAt.eventually_hasSum
theorem HasFPowerSeriesOnBall.eventually_hasSum_sub (hf : HasFPowerSeriesOnBall f p x r) :
∀ᶠ y in 𝓝 x, HasSum (fun n : ℕ => p n fun _ : Fin n => y - x) (f y) := by
filter_upwards [EMetric.ball_mem_nhds x hf.r_pos] with y using hf.hasSum_sub
#align has_fpower_series_on_ball.eventually_has_sum_sub HasFPowerSeriesOnBall.eventually_hasSum_sub
theorem HasFPowerSeriesAt.eventually_hasSum_sub (hf : HasFPowerSeriesAt f p x) :
∀ᶠ y in 𝓝 x, HasSum (fun n : ℕ => p n fun _ : Fin n => y - x) (f y) :=
let ⟨_, hr⟩ := hf
hr.eventually_hasSum_sub
#align has_fpower_series_at.eventually_has_sum_sub HasFPowerSeriesAt.eventually_hasSum_sub
theorem HasFPowerSeriesOnBall.eventually_eq_zero
(hf : HasFPowerSeriesOnBall f (0 : FormalMultilinearSeries 𝕜 E F) x r) :
∀ᶠ z in 𝓝 x, f z = 0 := by
filter_upwards [hf.eventually_hasSum_sub] with z hz using hz.unique hasSum_zero
#align has_fpower_series_on_ball.eventually_eq_zero HasFPowerSeriesOnBall.eventually_eq_zero
theorem HasFPowerSeriesAt.eventually_eq_zero
(hf : HasFPowerSeriesAt f (0 : FormalMultilinearSeries 𝕜 E F) x) : ∀ᶠ z in 𝓝 x, f z = 0 :=
let ⟨_, hr⟩ := hf
hr.eventually_eq_zero
#align has_fpower_series_at.eventually_eq_zero HasFPowerSeriesAt.eventually_eq_zero
theorem hasFPowerSeriesOnBall_const {c : F} {e : E} :
HasFPowerSeriesOnBall (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e ⊤ := by
refine' ⟨by simp, WithTop.zero_lt_top, fun _ => hasSum_single 0 fun n hn => _⟩
simp [constFormalMultilinearSeries_apply hn]
#align has_fpower_series_on_ball_const hasFPowerSeriesOnBall_const
theorem hasFPowerSeriesAt_const {c : F} {e : E} :
HasFPowerSeriesAt (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e :=
⟨⊤, hasFPowerSeriesOnBall_const⟩
#align has_fpower_series_at_const hasFPowerSeriesAt_const
theorem analyticAt_const {v : F} : AnalyticAt 𝕜 (fun _ => v) x :=
⟨constFormalMultilinearSeries 𝕜 E v, hasFPowerSeriesAt_const⟩
#align analytic_at_const analyticAt_const
theorem analyticOn_const {v : F} {s : Set E} : AnalyticOn 𝕜 (fun _ => v) s :=
fun _ _ => analyticAt_const
#align analytic_on_const analyticOn_const
theorem HasFPowerSeriesOnBall.add (hf : HasFPowerSeriesOnBall f pf x r)
(hg : HasFPowerSeriesOnBall g pg x r) : HasFPowerSeriesOnBall (f + g) (pf + pg) x r :=
{ r_le := le_trans (le_min_iff.2 ⟨hf.r_le, hg.r_le⟩) (pf.min_radius_le_radius_add pg)
r_pos := hf.r_pos
hasSum := fun hy => (hf.hasSum hy).add (hg.hasSum hy) }
#align has_fpower_series_on_ball.add HasFPowerSeriesOnBall.add
theorem HasFPowerSeriesAt.add (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x) :
HasFPowerSeriesAt (f + g) (pf + pg) x := by
rcases (hf.eventually.and hg.eventually).exists with ⟨r, hr⟩
exact ⟨r, hr.1.add hr.2⟩
#align has_fpower_series_at.add HasFPowerSeriesAt.add
theorem AnalyticAt.congr (hf : AnalyticAt 𝕜 f x) (hg : f =ᶠ[𝓝 x] g) : AnalyticAt 𝕜 g x :=
let ⟨_, hpf⟩ := hf
(hpf.congr hg).analyticAt
theorem analyticAt_congr (h : f =ᶠ[𝓝 x] g) : AnalyticAt 𝕜 f x ↔ AnalyticAt 𝕜 g x :=
⟨fun hf ↦ hf.congr h, fun hg ↦ hg.congr h.symm⟩
theorem AnalyticAt.add (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) : AnalyticAt 𝕜 (f + g) x :=
let ⟨_, hpf⟩ := hf
let ⟨_, hqf⟩ := hg
(hpf.add hqf).analyticAt
#align analytic_at.add AnalyticAt.add
theorem HasFPowerSeriesOnBall.neg (hf : HasFPowerSeriesOnBall f pf x r) :
HasFPowerSeriesOnBall (-f) (-pf) x r :=
{ r_le := by
rw [pf.radius_neg]
exact hf.r_le
r_pos := hf.r_pos
hasSum := fun hy => (hf.hasSum hy).neg }
#align has_fpower_series_on_ball.neg HasFPowerSeriesOnBall.neg
theorem HasFPowerSeriesAt.neg (hf : HasFPowerSeriesAt f pf x) : HasFPowerSeriesAt (-f) (-pf) x :=
let ⟨_, hrf⟩ := hf
hrf.neg.hasFPowerSeriesAt
#align has_fpower_series_at.neg HasFPowerSeriesAt.neg
theorem AnalyticAt.neg (hf : AnalyticAt 𝕜 f x) : AnalyticAt 𝕜 (-f) x :=
let ⟨_, hpf⟩ := hf
hpf.neg.analyticAt
#align analytic_at.neg AnalyticAt.neg
theorem HasFPowerSeriesOnBall.sub (hf : HasFPowerSeriesOnBall f pf x r)
(hg : HasFPowerSeriesOnBall g pg x r) : HasFPowerSeriesOnBall (f - g) (pf - pg) x r := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align has_fpower_series_on_ball.sub HasFPowerSeriesOnBall.sub
theorem HasFPowerSeriesAt.sub (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x) :
HasFPowerSeriesAt (f - g) (pf - pg) x := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align has_fpower_series_at.sub HasFPowerSeriesAt.sub
theorem AnalyticAt.sub (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) :
AnalyticAt 𝕜 (f - g) x := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align analytic_at.sub AnalyticAt.sub
theorem AnalyticOn.mono {s t : Set E} (hf : AnalyticOn 𝕜 f t) (hst : s ⊆ t) : AnalyticOn 𝕜 f s :=
fun z hz => hf z (hst hz)
#align analytic_on.mono AnalyticOn.mono
theorem AnalyticOn.congr' {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : f =ᶠ[𝓝ˢ s] g) :
AnalyticOn 𝕜 g s :=
fun z hz => (hf z hz).congr (mem_nhdsSet_iff_forall.mp hg z hz)
theorem analyticOn_congr' {s : Set E} (h : f =ᶠ[𝓝ˢ s] g) : AnalyticOn 𝕜 f s ↔ AnalyticOn 𝕜 g s :=
⟨fun hf => hf.congr' h, fun hg => hg.congr' h.symm⟩
theorem AnalyticOn.congr {s : Set E} (hs : IsOpen s) (hf : AnalyticOn 𝕜 f s) (hg : s.EqOn f g) :
AnalyticOn 𝕜 g s :=
hf.congr' $ mem_nhdsSet_iff_forall.mpr
(fun _ hz => eventuallyEq_iff_exists_mem.mpr ⟨s, hs.mem_nhds hz, hg⟩)
theorem analyticOn_congr {s : Set E} (hs : IsOpen s) (h : s.EqOn f g) : AnalyticOn 𝕜 f s ↔
AnalyticOn 𝕜 g s := ⟨fun hf => hf.congr hs h, fun hg => hg.congr hs h.symm⟩
theorem AnalyticOn.add {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
AnalyticOn 𝕜 (f + g) s :=
fun z hz => (hf z hz).add (hg z hz)
#align analytic_on.add AnalyticOn.add
theorem AnalyticOn.sub {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
AnalyticOn 𝕜 (f - g) s :=
fun z hz => (hf z hz).sub (hg z hz)
#align analytic_on.sub AnalyticOn.sub
theorem HasFPowerSeriesOnBall.coeff_zero (hf : HasFPowerSeriesOnBall f pf x r) (v : Fin 0 → E) :
pf 0 v = f x := by
have v_eq : v = fun i => 0 := Subsingleton.elim _ _
have zero_mem : (0 : E) ∈ EMetric.ball (0 : E) r := by simp [hf.r_pos]
have : ∀ i, i ≠ 0 → (pf i fun j => 0) = 0 := by
intro i hi
have : 0 < i := pos_iff_ne_zero.2 hi
exact ContinuousMultilinearMap.map_coord_zero _ (⟨0, this⟩ : Fin i) rfl
have A := (hf.hasSum zero_mem).unique (hasSum_single _ this)
simpa [v_eq] using A.symm
#align has_fpower_series_on_ball.coeff_zero HasFPowerSeriesOnBall.coeff_zero
theorem HasFPowerSeriesAt.coeff_zero (hf : HasFPowerSeriesAt f pf x) (v : Fin 0 → E) :
pf 0 v = f x :=
let ⟨_, hrf⟩ := hf
hrf.coeff_zero v
#align has_fpower_series_at.coeff_zero HasFPowerSeriesAt.coeff_zero
/-- If a function `f` has a power series `p` on a ball and `g` is linear, then `g ∘ f` has the
power series `g ∘ p` on the same ball. -/
theorem ContinuousLinearMap.comp_hasFPowerSeriesOnBall (g : F →L[𝕜] G)
(h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (g ∘ f) (g.compFormalMultilinearSeries p) x r :=
{ r_le := h.r_le.trans (p.radius_le_radius_continuousLinearMap_comp _)
r_pos := h.r_pos
hasSum := fun hy => by
simpa only [ContinuousLinearMap.compFormalMultilinearSeries_apply,
ContinuousLinearMap.compContinuousMultilinearMap_coe, Function.comp_apply] using
g.hasSum (h.hasSum hy) }
#align continuous_linear_map.comp_has_fpower_series_on_ball ContinuousLinearMap.comp_hasFPowerSeriesOnBall
/-- If a function `f` is analytic on a set `s` and `g` is linear, then `g ∘ f` is analytic
on `s`. -/
theorem ContinuousLinearMap.comp_analyticOn {s : Set E} (g : F →L[𝕜] G) (h : AnalyticOn 𝕜 f s) :
AnalyticOn 𝕜 (g ∘ f) s := by
rintro x hx
rcases h x hx with ⟨p, r, hp⟩
exact ⟨g.compFormalMultilinearSeries p, r, g.comp_hasFPowerSeriesOnBall hp⟩
#align continuous_linear_map.comp_analytic_on ContinuousLinearMap.comp_analyticOn
/-- If a function admits a power series expansion, then it is exponentially close to the partial
sums of this power series on strict subdisks of the disk of convergence.
This version provides an upper estimate that decreases both in `‖y‖` and `n`. See also
`HasFPowerSeriesOnBall.uniform_geometric_approx` for a weaker version. -/
theorem HasFPowerSeriesOnBall.uniform_geometric_approx' {r' : ℝ≥0}
(hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n := by
obtain ⟨a, ha, C, hC, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n, ‖p n‖ * (r' : ℝ) ^ n ≤ C * a ^ n :=
p.norm_mul_pow_le_mul_pow_of_lt_radius (h.trans_le hf.r_le)
refine' ⟨a, ha, C / (1 - a), div_pos hC (sub_pos.2 ha.2), fun y hy n => _⟩
have yr' : ‖y‖ < r' := by
rw [ball_zero_eq] at hy
exact hy
have hr'0 : 0 < (r' : ℝ) := (norm_nonneg _).trans_lt yr'
have : y ∈ EMetric.ball (0 : E) r := by
refine' mem_emetric_ball_zero_iff.2 (lt_trans _ h)
exact mod_cast yr'
rw [norm_sub_rev, ← mul_div_right_comm]
have ya : a * (‖y‖ / ↑r') ≤ a :=
mul_le_of_le_one_right ha.1.le (div_le_one_of_le yr'.le r'.coe_nonneg)
suffices ‖p.partialSum n y - f (x + y)‖ ≤ C * (a * (‖y‖ / r')) ^ n / (1 - a * (‖y‖ / r')) by
refine' this.trans _
have : 0 < a := ha.1
gcongr
apply_rules [sub_pos.2, ha.2]
apply norm_sub_le_of_geometric_bound_of_hasSum (ya.trans_lt ha.2) _ (hf.hasSum this)
intro n
calc
‖(p n) fun _ : Fin n => y‖
_ ≤ ‖p n‖ * ∏ _i : Fin n, ‖y‖ := ContinuousMultilinearMap.le_op_norm _ _
_ = ‖p n‖ * (r' : ℝ) ^ n * (‖y‖ / r') ^ n := by field_simp [mul_right_comm]
_ ≤ C * a ^ n * (‖y‖ / r') ^ n := by gcongr ?_ * _; apply hp
_ ≤ C * (a * (‖y‖ / r')) ^ n := by rw [mul_pow, mul_assoc]
#align has_fpower_series_on_ball.uniform_geometric_approx' HasFPowerSeriesOnBall.uniform_geometric_approx'
/-- If a function admits a power series expansion, then it is exponentially close to the partial
sums of this power series on strict subdisks of the disk of convergence. -/
theorem HasFPowerSeriesOnBall.uniform_geometric_approx {r' : ℝ≥0}
(hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
∃ a ∈ Ioo (0 : ℝ) 1,
∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n, ‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n := by
obtain ⟨a, ha, C, hC, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n :=
hf.uniform_geometric_approx' h
refine' ⟨a, ha, C, hC, fun y hy n => (hp y hy n).trans _⟩
have yr' : ‖y‖ < r' := by rwa [ball_zero_eq] at hy
gcongr
exacts [mul_nonneg ha.1.le (div_nonneg (norm_nonneg y) r'.coe_nonneg),
mul_le_of_le_one_right ha.1.le (div_le_one_of_le yr'.le r'.coe_nonneg)]
#align has_fpower_series_on_ball.uniform_geometric_approx HasFPowerSeriesOnBall.uniform_geometric_approx
/-- Taylor formula for an analytic function, `IsBigO` version. -/
theorem HasFPowerSeriesAt.isBigO_sub_partialSum_pow (hf : HasFPowerSeriesAt f p x) (n : ℕ) :
(fun y : E => f (x + y) - p.partialSum n y) =O[𝓝 0] fun y => ‖y‖ ^ n := by
rcases hf with ⟨r, hf⟩
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hf.r_pos with ⟨r', r'0, h⟩
obtain ⟨a, -, C, -, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n :=
hf.uniform_geometric_approx' h
refine' isBigO_iff.2 ⟨C * (a / r') ^ n, _⟩
replace r'0 : 0 < (r' : ℝ); · exact mod_cast r'0
filter_upwards [Metric.ball_mem_nhds (0 : E) r'0] with y hy
simpa [mul_pow, mul_div_assoc, mul_assoc, div_mul_eq_mul_div] using hp y hy n
set_option linter.uppercaseLean3 false in
#align has_fpower_series_at.is_O_sub_partial_sum_pow HasFPowerSeriesAt.isBigO_sub_partialSum_pow
/-- If `f` has formal power series `∑ n, pₙ` on a ball of radius `r`, then for `y, z` in any smaller
ball, the norm of the difference `f y - f z - p 1 (fun _ ↦ y - z)` is bounded above by
`C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖`. This lemma formulates this property using `IsBigO` and
`Filter.principal` on `E × E`. -/
theorem HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal
(hf : HasFPowerSeriesOnBall f p x r) (hr : r' < r) :
(fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) =O[𝓟 (EMetric.ball (x, x) r')]
fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖ := by
lift r' to ℝ≥0 using ne_top_of_lt hr
rcases (zero_le r').eq_or_lt with (rfl | hr'0)
· simp only [isBigO_bot, EMetric.ball_zero, principal_empty, ENNReal.coe_zero]
obtain ⟨a, ha, C, hC : 0 < C, hp⟩ :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n : ℕ, ‖p n‖ * (r' : ℝ) ^ n ≤ C * a ^ n
exact p.norm_mul_pow_le_mul_pow_of_lt_radius (hr.trans_le hf.r_le)
simp only [← le_div_iff (pow_pos (NNReal.coe_pos.2 hr'0) _)] at hp
set L : E × E → ℝ := fun y =>
C * (a / r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * (a / (1 - a) ^ 2 + 2 / (1 - a))
have hL : ∀ y ∈ EMetric.ball (x, x) r', ‖f y.1 - f y.2 - p 1 fun _ => y.1 - y.2‖ ≤ L y := by
intro y hy'
have hy : y ∈ EMetric.ball x r ×ˢ EMetric.ball x r := by
rw [EMetric.ball_prod_same]
exact EMetric.ball_subset_ball hr.le hy'
set A : ℕ → F := fun n => (p n fun _ => y.1 - x) - p n fun _ => y.2 - x
have hA : HasSum (fun n => A (n + 2)) (f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) := by
convert (hasSum_nat_add_iff' 2).2 ((hf.hasSum_sub hy.1).sub (hf.hasSum_sub hy.2)) using 1
rw [Finset.sum_range_succ, Finset.sum_range_one, hf.coeff_zero, hf.coeff_zero, sub_self,
zero_add, ← Subsingleton.pi_single_eq (0 : Fin 1) (y.1 - x), Pi.single,
← Subsingleton.pi_single_eq (0 : Fin 1) (y.2 - x), Pi.single, ← (p 1).map_sub, ← Pi.single,
Subsingleton.pi_single_eq, sub_sub_sub_cancel_right]
rw [EMetric.mem_ball, edist_eq_coe_nnnorm_sub, ENNReal.coe_lt_coe] at hy'
set B : ℕ → ℝ := fun n => C * (a / r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * ((n + 2) * a ^ n)
have hAB : ∀ n, ‖A (n + 2)‖ ≤ B n := fun n =>
calc
‖A (n + 2)‖ ≤ ‖p (n + 2)‖ * ↑(n + 2) * ‖y - (x, x)‖ ^ (n + 1) * ‖y.1 - y.2‖ := by
-- porting note: `pi_norm_const` was `pi_norm_const (_ : E)`
simpa only [Fintype.card_fin, pi_norm_const, Prod.norm_def, Pi.sub_def,
Prod.fst_sub, Prod.snd_sub, sub_sub_sub_cancel_right] using
(p <| n + 2).norm_image_sub_le (fun _ => y.1 - x) fun _ => y.2 - x
_ = ‖p (n + 2)‖ * ‖y - (x, x)‖ ^ n * (↑(n + 2) * ‖y - (x, x)‖ * ‖y.1 - y.2‖) := by
rw [pow_succ ‖y - (x, x)‖]
ring
-- porting note: the two `↑` in `↑r'` are new, without them, Lean fails to synthesize
-- instances `HDiv ℝ ℝ≥0 ?m` or `HMul ℝ ℝ≥0 ?m`
_ ≤ C * a ^ (n + 2) / ↑r' ^ (n + 2)
* ↑r' ^ n * (↑(n + 2) * ‖y - (x, x)‖ * ‖y.1 - y.2‖) := by
have : 0 < a := ha.1
gcongr
· apply hp
· apply hy'.le
_ = B n := by
-- porting note: in the original, `B` was in the `field_simp`, but now Lean does not
-- accept it. The current proof works in Lean 4, but does not in Lean 3.
field_simp [pow_succ]
simp only [mul_assoc, mul_comm, mul_left_comm]
have hBL : HasSum B (L y) := by
apply HasSum.mul_left
simp only [add_mul]
have : ‖a‖ < 1 := by simp only [Real.norm_eq_abs, abs_of_pos ha.1, ha.2]
rw [div_eq_mul_inv, div_eq_mul_inv]
exact (hasSum_coe_mul_geometric_of_norm_lt_1 this).add -- porting note: was `convert`!
((hasSum_geometric_of_norm_lt_1 this).mul_left 2)
exact hA.norm_le_of_bounded hBL hAB
suffices L =O[𝓟 (EMetric.ball (x, x) r')] fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖ by
refine' (IsBigO.of_bound 1 (eventually_principal.2 fun y hy => _)).trans this
rw [one_mul]
exact (hL y hy).trans (le_abs_self _)
simp_rw [mul_right_comm _ (_ * _)] -- porting note: there was an `L` inside the `simp_rw`.
exact (isBigO_refl _ _).const_mul_left _
set_option linter.uppercaseLean3 false in
#align has_fpower_series_on_ball.is_O_image_sub_image_sub_deriv_principal HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal
/-- If `f` has formal power series `∑ n, pₙ` on a ball of radius `r`, then for `y, z` in any smaller
ball, the norm of the difference `f y - f z - p 1 (fun _ ↦ y - z)` is bounded above by
`C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖`. -/
theorem HasFPowerSeriesOnBall.image_sub_sub_deriv_le (hf : HasFPowerSeriesOnBall f p x r)
(hr : r' < r) :
∃ C, ∀ᵉ (y ∈ EMetric.ball x r') (z ∈ EMetric.ball x r'),
‖f y - f z - p 1 fun _ => y - z‖ ≤ C * max ‖y - x‖ ‖z - x‖ * ‖y - z‖ := by
simpa only [isBigO_principal, mul_assoc, norm_mul, norm_norm, Prod.forall, EMetric.mem_ball,
Prod.edist_eq, max_lt_iff, and_imp, @forall_swap (_ < _) E] using
hf.isBigO_image_sub_image_sub_deriv_principal hr
#align has_fpower_series_on_ball.image_sub_sub_deriv_le HasFPowerSeriesOnBall.image_sub_sub_deriv_le
/-- If `f` has formal power series `∑ n, pₙ` at `x`, then
`f y - f z - p 1 (fun _ ↦ y - z) = O(‖(y, z) - (x, x)‖ * ‖y - z‖)` as `(y, z) → (x, x)`.
In particular, `f` is strictly differentiable at `x`. -/
theorem HasFPowerSeriesAt.isBigO_image_sub_norm_mul_norm_sub (hf : HasFPowerSeriesAt f p x) :
(fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) =O[𝓝 (x, x)] fun y =>
‖y - (x, x)‖ * ‖y.1 - y.2‖ := by
rcases hf with ⟨r, hf⟩
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hf.r_pos with ⟨r', r'0, h⟩
refine' (hf.isBigO_image_sub_image_sub_deriv_principal h).mono _
exact le_principal_iff.2 (EMetric.ball_mem_nhds _ r'0)
set_option linter.uppercaseLean3 false in
#align has_fpower_series_at.is_O_image_sub_norm_mul_norm_sub HasFPowerSeriesAt.isBigO_image_sub_norm_mul_norm_sub
/-- If a function admits a power series expansion at `x`, then it is the uniform limit of the
partial sums of this power series on strict subdisks of the disk of convergence, i.e., `f (x + y)`
is the uniform limit of `p.partialSum n y` there. -/
theorem HasFPowerSeriesOnBall.tendstoUniformlyOn {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r)
(h : (r' : ℝ≥0∞) < r) :
TendstoUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
(Metric.ball (0 : E) r') := by
obtain ⟨a, ha, C, -, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n
exact hf.uniform_geometric_approx h
refine' Metric.tendstoUniformlyOn_iff.2 fun ε εpos => _
have L : Tendsto (fun n => (C : ℝ) * a ^ n) atTop (𝓝 ((C : ℝ) * 0)) :=
tendsto_const_nhds.mul (tendsto_pow_atTop_nhds_0_of_lt_1 ha.1.le ha.2)
rw [mul_zero] at L
refine' (L.eventually (gt_mem_nhds εpos)).mono fun n hn y hy => _
rw [dist_eq_norm]
exact (hp y hy n).trans_lt hn
#align has_fpower_series_on_ball.tendsto_uniformly_on HasFPowerSeriesOnBall.tendstoUniformlyOn
/-- If a function admits a power series expansion at `x`, then it is the locally uniform limit of
the partial sums of this power series on the disk of convergence, i.e., `f (x + y)`
is the locally uniform limit of `p.partialSum n y` there. -/
theorem HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn (hf : HasFPowerSeriesOnBall f p x r) :
TendstoLocallyUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
(EMetric.ball (0 : E) r) := by
intro u hu x hx
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hx with ⟨r', xr', hr'⟩
have : EMetric.ball (0 : E) r' ∈ 𝓝 x := IsOpen.mem_nhds EMetric.isOpen_ball xr'
refine' ⟨EMetric.ball (0 : E) r', mem_nhdsWithin_of_mem_nhds this, _⟩
simpa [Metric.emetric_ball_nnreal] using hf.tendstoUniformlyOn hr' u hu
#align has_fpower_series_on_ball.tendsto_locally_uniformly_on HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn
/-- If a function admits a power series expansion at `x`, then it is the uniform limit of the
partial sums of this power series on strict subdisks of the disk of convergence, i.e., `f y`
is the uniform limit of `p.partialSum n (y - x)` there. -/
theorem HasFPowerSeriesOnBall.tendstoUniformlyOn' {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r)
(h : (r' : ℝ≥0∞) < r) :
TendstoUniformlyOn (fun n y => p.partialSum n (y - x)) f atTop (Metric.ball (x : E) r') := by
convert (hf.tendstoUniformlyOn h).comp fun y => y - x using 1
· simp [(· ∘ ·)]
· ext z
simp [dist_eq_norm]
#align has_fpower_series_on_ball.tendsto_uniformly_on' HasFPowerSeriesOnBall.tendstoUniformlyOn'
/-- If a function admits a power series expansion at `x`, then it is the locally uniform limit of
the partial sums of this power series on the disk of convergence, i.e., `f y`
is the locally uniform limit of `p.partialSum n (y - x)` there. -/
theorem HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn' (hf : HasFPowerSeriesOnBall f p x r) :
TendstoLocallyUniformlyOn (fun n y => p.partialSum n (y - x)) f atTop
(EMetric.ball (x : E) r) := by
have A : ContinuousOn (fun y : E => y - x) (EMetric.ball (x : E) r) :=
(continuous_id.sub continuous_const).continuousOn
convert hf.tendstoLocallyUniformlyOn.comp (fun y : E => y - x) _ A using 1
· ext z
simp
· intro z
simp [edist_eq_coe_nnnorm, edist_eq_coe_nnnorm_sub]
#align has_fpower_series_on_ball.tendsto_locally_uniformly_on' HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn'
/-- If a function admits a power series expansion on a disk, then it is continuous there. -/
protected theorem HasFPowerSeriesOnBall.continuousOn (hf : HasFPowerSeriesOnBall f p x r) :
ContinuousOn f (EMetric.ball x r) :=
hf.tendstoLocallyUniformlyOn'.continuousOn <|
eventually_of_forall fun n =>
((p.partialSum_continuous n).comp (continuous_id.sub continuous_const)).continuousOn
#align has_fpower_series_on_ball.continuous_on HasFPowerSeriesOnBall.continuousOn
protected theorem HasFPowerSeriesAt.continuousAt (hf : HasFPowerSeriesAt f p x) :
ContinuousAt f x :=
let ⟨_, hr⟩ := hf
hr.continuousOn.continuousAt (EMetric.ball_mem_nhds x hr.r_pos)
#align has_fpower_series_at.continuous_at HasFPowerSeriesAt.continuousAt
protected theorem AnalyticAt.continuousAt (hf : AnalyticAt 𝕜 f x) : ContinuousAt f x :=
let ⟨_, hp⟩ := hf
hp.continuousAt
#align analytic_at.continuous_at AnalyticAt.continuousAt
protected theorem AnalyticOn.continuousOn {s : Set E} (hf : AnalyticOn 𝕜 f s) : ContinuousOn f s :=
fun x hx => (hf x hx).continuousAt.continuousWithinAt
#align analytic_on.continuous_on AnalyticOn.continuousOn
/-- Analytic everywhere implies continuous -/
theorem AnalyticOn.continuous {f : E → F} (fa : AnalyticOn 𝕜 f univ) : Continuous f := by
rw [continuous_iff_continuousOn_univ]; exact fa.continuousOn
/-- In a complete space, the sum of a converging power series `p` admits `p` as a power series.
This is not totally obvious as we need to check the convergence of the series. -/
protected theorem FormalMultilinearSeries.hasFPowerSeriesOnBall [CompleteSpace F]
(p : FormalMultilinearSeries 𝕜 E F) (h : 0 < p.radius) :
HasFPowerSeriesOnBall p.sum p 0 p.radius :=
{ r_le := le_rfl
r_pos := h
hasSum := fun hy => by
rw [zero_add]
exact p.hasSum hy }
#align formal_multilinear_series.has_fpower_series_on_ball FormalMultilinearSeries.hasFPowerSeriesOnBall
theorem HasFPowerSeriesOnBall.sum (h : HasFPowerSeriesOnBall f p x r) {y : E}
(hy : y ∈ EMetric.ball (0 : E) r) : f (x + y) = p.sum y :=
(h.hasSum hy).tsum_eq.symm
#align has_fpower_series_on_ball.sum HasFPowerSeriesOnBall.sum
/-- The sum of a converging power series is continuous in its disk of convergence. -/
protected theorem FormalMultilinearSeries.continuousOn [CompleteSpace F] :
ContinuousOn p.sum (EMetric.ball 0 p.radius) := by
rcases (zero_le p.radius).eq_or_lt with h | h
· simp [← h, continuousOn_empty]
· exact (p.hasFPowerSeriesOnBall h).continuousOn
#align formal_multilinear_series.continuous_on FormalMultilinearSeries.continuousOn
end
/-!
### Uniqueness of power series
If a function `f : E → F` has two representations as power series at a point `x : E`, corresponding
to formal multilinear series `p₁` and `p₂`, then these representations agree term-by-term. That is,
for any `n : ℕ` and `y : E`, `p₁ n (fun i ↦ y) = p₂ n (fun i ↦ y)`. In the one-dimensional case,
when `f : 𝕜 → E`, the continuous multilinear maps `p₁ n` and `p₂ n` are given by
`ContinuousMultilinearMap.mkPiField`, and hence are determined completely by the value of
`p₁ n (fun i ↦ 1)`, so `p₁ = p₂`. Consequently, the radius of convergence for one series can be
transferred to the other.
-/
section Uniqueness
open ContinuousMultilinearMap
theorem Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero {n : ℕ} {p : E[×n]→L[𝕜] F}
(h : (fun y => p fun _ => y) =O[𝓝 0] fun y => ‖y‖ ^ (n + 1)) (y : E) : (p fun _ => y) = 0 := by
obtain ⟨c, c_pos, hc⟩ := h.exists_pos
obtain ⟨t, ht, t_open, z_mem⟩ := eventually_nhds_iff.mp (isBigOWith_iff.mp hc)
obtain ⟨δ, δ_pos, δε⟩ := (Metric.isOpen_iff.mp t_open) 0 z_mem
clear h hc z_mem
cases' n with n
· exact norm_eq_zero.mp (by
-- porting note: the symmetric difference of the `simpa only` sets:
-- added `Nat.zero_eq, zero_add, pow_one`
-- removed `zero_pow', Ne.def, Nat.one_ne_zero, not_false_iff`
simpa only [Nat.zero_eq, fin0_apply_norm, norm_eq_zero, norm_zero, zero_add, pow_one,
mul_zero, norm_le_zero_iff] using ht 0 (δε (Metric.mem_ball_self δ_pos)))
· refine' Or.elim (Classical.em (y = 0))
(fun hy => by simpa only [hy] using p.map_zero) fun hy => _
replace hy := norm_pos_iff.mpr hy
refine' norm_eq_zero.mp (le_antisymm (le_of_forall_pos_le_add fun ε ε_pos => _) (norm_nonneg _))
have h₀ := _root_.mul_pos c_pos (pow_pos hy (n.succ + 1))
obtain ⟨k, k_pos, k_norm⟩ := NormedField.exists_norm_lt 𝕜
(lt_min (mul_pos δ_pos (inv_pos.mpr hy)) (mul_pos ε_pos (inv_pos.mpr h₀)))
have h₁ : ‖k • y‖ < δ := by
rw [norm_smul]
exact inv_mul_cancel_right₀ hy.ne.symm δ ▸
mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_left _ _)) hy
have h₂ :=
calc
‖p fun _ => k • y‖ ≤ c * ‖k • y‖ ^ (n.succ + 1) := by
-- porting note: now Lean wants `_root_.`
simpa only [norm_pow, _root_.norm_norm] using ht (k • y) (δε (mem_ball_zero_iff.mpr h₁))
--simpa only [norm_pow, norm_norm] using ht (k • y) (δε (mem_ball_zero_iff.mpr h₁))
_ = ‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))) := by
-- porting note: added `Nat.succ_eq_add_one` since otherwise `ring` does not conclude.
simp only [norm_smul, mul_pow, Nat.succ_eq_add_one]
-- porting note: removed `rw [pow_succ]`, since it now becomes superfluous.
ring
have h₃ : ‖k‖ * (c * ‖y‖ ^ (n.succ + 1)) < ε :=
inv_mul_cancel_right₀ h₀.ne.symm ε ▸
mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_right _ _)) h₀
calc
‖p fun _ => y‖ = ‖k⁻¹ ^ n.succ‖ * ‖p fun _ => k • y‖ := by
simpa only [inv_smul_smul₀ (norm_pos_iff.mp k_pos), norm_smul, Finset.prod_const,
Finset.card_fin] using
congr_arg norm (p.map_smul_univ (fun _ : Fin n.succ => k⁻¹) fun _ : Fin n.succ => k • y)
_ ≤ ‖k⁻¹ ^ n.succ‖ * (‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1)))) := by gcongr
_ = ‖(k⁻¹ * k) ^ n.succ‖ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))) := by
rw [← mul_assoc]
simp [norm_mul, mul_pow]
_ ≤ 0 + ε := by
rw [inv_mul_cancel (norm_pos_iff.mp k_pos)]
simpa using h₃.le
set_option linter.uppercaseLean3 false in
#align asymptotics.is_O.continuous_multilinear_map_apply_eq_zero Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero
/-- If a formal multilinear series `p` represents the zero function at `x : E`, then the
terms `p n (fun i ↦ y)` appearing in the sum are zero for any `n : ℕ`, `y : E`. -/
theorem HasFPowerSeriesAt.apply_eq_zero {p : FormalMultilinearSeries 𝕜 E F} {x : E}
(h : HasFPowerSeriesAt 0 p x) (n : ℕ) : ∀ y : E, (p n fun _ => y) = 0 := by
refine' Nat.strong_induction_on n fun k hk => _
have psum_eq : p.partialSum (k + 1) = fun y => p k fun _ => y := by
funext z
refine' Finset.sum_eq_single _ (fun b hb hnb => _) fun hn => _
· have := Finset.mem_range_succ_iff.mp hb
simp only [hk b (this.lt_of_ne hnb), Pi.zero_apply]
· exact False.elim (hn (Finset.mem_range.mpr (lt_add_one k)))
replace h := h.isBigO_sub_partialSum_pow k.succ
simp only [psum_eq, zero_sub, Pi.zero_apply, Asymptotics.isBigO_neg_left] at h
exact h.continuousMultilinearMap_apply_eq_zero
#align has_fpower_series_at.apply_eq_zero HasFPowerSeriesAt.apply_eq_zero
/-- A one-dimensional formal multilinear series representing the zero function is zero. -/
theorem HasFPowerSeriesAt.eq_zero {p : FormalMultilinearSeries 𝕜 𝕜 E} {x : 𝕜}
(h : HasFPowerSeriesAt 0 p x) : p = 0 := by
-- porting note: `funext; ext` was `ext (n x)`
funext n
ext x
rw [← mkPiField_apply_one_eq_self (p n)]
-- porting note: nasty hack, was `simp [h.apply_eq_zero n 1]`
have := Or.intro_right ?_ (h.apply_eq_zero n 1)
simpa using this
#align has_fpower_series_at.eq_zero HasFPowerSeriesAt.eq_zero
/-- One-dimensional formal multilinear series representing the same function are equal. -/
theorem HasFPowerSeriesAt.eq_formalMultilinearSeries {p₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E}
{f : 𝕜 → E} {x : 𝕜} (h₁ : HasFPowerSeriesAt f p₁ x) (h₂ : HasFPowerSeriesAt f p₂ x) : p₁ = p₂ :=
sub_eq_zero.mp (HasFPowerSeriesAt.eq_zero (by simpa only [sub_self] using h₁.sub h₂))
#align has_fpower_series_at.eq_formal_multilinear_series HasFPowerSeriesAt.eq_formalMultilinearSeries
theorem HasFPowerSeriesAt.eq_formalMultilinearSeries_of_eventually
{p q : FormalMultilinearSeries 𝕜 𝕜 E} {f g : 𝕜 → E} {x : 𝕜} (hp : HasFPowerSeriesAt f p x)
(hq : HasFPowerSeriesAt g q x) (heq : ∀ᶠ z in 𝓝 x, f z = g z) : p = q :=
(hp.congr heq).eq_formalMultilinearSeries hq
#align has_fpower_series_at.eq_formal_multilinear_series_of_eventually HasFPowerSeriesAt.eq_formalMultilinearSeries_of_eventually
/-- A one-dimensional formal multilinear series representing a locally zero function is zero. -/
theorem HasFPowerSeriesAt.eq_zero_of_eventually {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E}
{x : 𝕜} (hp : HasFPowerSeriesAt f p x) (hf : f =ᶠ[𝓝 x] 0) : p = 0 :=
(hp.congr hf).eq_zero
#align has_fpower_series_at.eq_zero_of_eventually HasFPowerSeriesAt.eq_zero_of_eventually
/-- If a function `f : 𝕜 → E` has two power series representations at `x`, then the given radii in
which convergence is guaranteed may be interchanged. This can be useful when the formal multilinear
series in one representation has a particularly nice form, but the other has a larger radius. -/
theorem HasFPowerSeriesOnBall.exchange_radius {p₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E}
{r₁ r₂ : ℝ≥0∞} {x : 𝕜} (h₁ : HasFPowerSeriesOnBall f p₁ x r₁)
(h₂ : HasFPowerSeriesOnBall f p₂ x r₂) : HasFPowerSeriesOnBall f p₁ x r₂ :=
h₂.hasFPowerSeriesAt.eq_formalMultilinearSeries h₁.hasFPowerSeriesAt ▸ h₂
#align has_fpower_series_on_ball.exchange_radius HasFPowerSeriesOnBall.exchange_radius
/-- If a function `f : 𝕜 → E` has power series representation `p` on a ball of some radius and for
each positive radius it has some power series representation, then `p` converges to `f` on the whole
`𝕜`. -/
theorem HasFPowerSeriesOnBall.r_eq_top_of_exists {f : 𝕜 → E} {r : ℝ≥0∞} {x : 𝕜}
{p : FormalMultilinearSeries 𝕜 𝕜 E} (h : HasFPowerSeriesOnBall f p x r)
(h' : ∀ (r' : ℝ≥0) (_ : 0 < r'), ∃ p' : FormalMultilinearSeries 𝕜 𝕜 E,
HasFPowerSeriesOnBall f p' x r') :
HasFPowerSeriesOnBall f p x ∞ :=
{ r_le := ENNReal.le_of_forall_pos_nnreal_lt fun r hr _ =>
let ⟨_, hp'⟩ := h' r hr
(h.exchange_radius hp').r_le
r_pos := ENNReal.coe_lt_top
hasSum := fun {y} _ =>
let ⟨r', hr'⟩ := exists_gt ‖y‖₊
let ⟨_, hp'⟩ := h' r' hr'.ne_bot.bot_lt
(h.exchange_radius hp').hasSum <| mem_emetric_ball_zero_iff.mpr (ENNReal.coe_lt_coe.2 hr') }
#align has_fpower_series_on_ball.r_eq_top_of_exists HasFPowerSeriesOnBall.r_eq_top_of_exists
end Uniqueness
/-!
### Changing origin in a power series
If a function is analytic in a disk `D(x, R)`, then it is analytic in any disk contained in that
one. Indeed, one can write
$$
f (x + y + z) = \sum_{n} p_n (y + z)^n = \sum_{n, k} \binom{n}{k} p_n y^{n-k} z^k
= \sum_{k} \Bigl(\sum_{n} \binom{n}{k} p_n y^{n-k}\Bigr) z^k.
$$
The corresponding power series has thus a `k`-th coefficient equal to
$\sum_{n} \binom{n}{k} p_n y^{n-k}$. In the general case where `pₙ` is a multilinear map, this has
to be interpreted suitably: instead of having a binomial coefficient, one should sum over all
possible subsets `s` of `Fin n` of cardinal `k`, and attribute `z` to the indices in `s` and
`y` to the indices outside of `s`.
In this paragraph, we implement this. The new power series is called `p.changeOrigin y`. Then, we
check its convergence and the fact that its sum coincides with the original sum. The outcome of this
discussion is that the set of points where a function is analytic is open.
-/
namespace FormalMultilinearSeries
section
variable (p : FormalMultilinearSeries 𝕜 E F) {x y : E} {r R : ℝ≥0}
/-- A term of `FormalMultilinearSeries.changeOriginSeries`.
Given a formal multilinear series `p` and a point `x` in its ball of convergence,
`p.changeOrigin x` is a formal multilinear series such that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense. Each term of `p.changeOrigin x`
is itself an analytic function of `x` given by the series `p.changeOriginSeries`. Each term in
`changeOriginSeries` is the sum of `changeOriginSeriesTerm`'s over all `s` of cardinality `l`.
The definition is such that `p.changeOriginSeriesTerm k l s hs (fun _ ↦ x) (fun _ ↦ y) =
p (k + l) (s.piecewise (fun _ ↦ x) (fun _ ↦ y))`
-/
def changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
E[×l]→L[𝕜] E[×k]→L[𝕜] F := by
let a := ContinuousMultilinearMap.curryFinFinset 𝕜 E F hs
(by erw [Finset.card_compl, Fintype.card_fin, hs, add_tsub_cancel_right])
exact a (p (k + l))
#align formal_multilinear_series.change_origin_series_term FormalMultilinearSeries.changeOriginSeriesTerm
theorem changeOriginSeriesTerm_apply (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l)
(x y : E) :
(p.changeOriginSeriesTerm k l s hs (fun _ => x) fun _ => y) =
p (k + l) (s.piecewise (fun _ => x) fun _ => y) :=
ContinuousMultilinearMap.curryFinFinset_apply_const _ _ _ _ _
#align formal_multilinear_series.change_origin_series_term_apply FormalMultilinearSeries.changeOriginSeriesTerm_apply
@[simp]
theorem norm_changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
‖p.changeOriginSeriesTerm k l s hs‖ = ‖p (k + l)‖ := by
simp only [changeOriginSeriesTerm, LinearIsometryEquiv.norm_map]
#align formal_multilinear_series.norm_change_origin_series_term FormalMultilinearSeries.norm_changeOriginSeriesTerm
@[simp]
theorem nnnorm_changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
‖p.changeOriginSeriesTerm k l s hs‖₊ = ‖p (k + l)‖₊ := by
simp only [changeOriginSeriesTerm, LinearIsometryEquiv.nnnorm_map]
#align formal_multilinear_series.nnnorm_change_origin_series_term FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm
theorem nnnorm_changeOriginSeriesTerm_apply_le (k l : ℕ) (s : Finset (Fin (k + l)))
(hs : s.card = l) (x y : E) :
‖p.changeOriginSeriesTerm k l s hs (fun _ => x) fun _ => y‖₊ ≤
‖p (k + l)‖₊ * ‖x‖₊ ^ l * ‖y‖₊ ^ k := by
rw [← p.nnnorm_changeOriginSeriesTerm k l s hs, ← Fin.prod_const, ← Fin.prod_const]
apply ContinuousMultilinearMap.le_of_op_nnnorm_le
apply ContinuousMultilinearMap.le_op_nnnorm
#align formal_multilinear_series.nnnorm_change_origin_series_term_apply_le FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm_apply_le
/-- The power series for `f.changeOrigin k`.
Given a formal multilinear series `p` and a point `x` in its ball of convergence,
`p.changeOrigin x` is a formal multilinear series such that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense. Its `k`-th term is the sum of
the series `p.changeOriginSeries k`. -/
def changeOriginSeries (k : ℕ) : FormalMultilinearSeries 𝕜 E (E[×k]→L[𝕜] F) := fun l =>
∑ s : { s : Finset (Fin (k + l)) // Finset.card s = l }, p.changeOriginSeriesTerm k l s s.2
#align formal_multilinear_series.change_origin_series FormalMultilinearSeries.changeOriginSeries
theorem nnnorm_changeOriginSeries_le_tsum (k l : ℕ) :
‖p.changeOriginSeries k l‖₊ ≤
∑' _ : { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + l)‖₊ :=
(nnnorm_sum_le _ (fun t => changeOriginSeriesTerm p k l (Subtype.val t) t.prop)).trans_eq <| by
simp_rw [tsum_fintype, nnnorm_changeOriginSeriesTerm (p := p) (k := k) (l := l)]
#align formal_multilinear_series.nnnorm_change_origin_series_le_tsum FormalMultilinearSeries.nnnorm_changeOriginSeries_le_tsum
theorem nnnorm_changeOriginSeries_apply_le_tsum (k l : ℕ) (x : E) :
‖p.changeOriginSeries k l fun _ => x‖₊ ≤
∑' _ : { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + l)‖₊ * ‖x‖₊ ^ l := by
rw [NNReal.tsum_mul_right, ← Fin.prod_const]
exact (p.changeOriginSeries k l).le_of_op_nnnorm_le _ (p.nnnorm_changeOriginSeries_le_tsum _ _)
#align formal_multilinear_series.nnnorm_change_origin_series_apply_le_tsum FormalMultilinearSeries.nnnorm_changeOriginSeries_apply_le_tsum
/-- Changing the origin of a formal multilinear series `p`, so that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense.
-/
def changeOrigin (x : E) : FormalMultilinearSeries 𝕜 E F :=
fun k => (p.changeOriginSeries k).sum x
#align formal_multilinear_series.change_origin FormalMultilinearSeries.changeOrigin
/-- An auxiliary equivalence useful in the proofs about
`FormalMultilinearSeries.changeOriginSeries`: the set of triples `(k, l, s)`, where `s` is a
`Finset (Fin (k + l))` of cardinality `l` is equivalent to the set of pairs `(n, s)`, where `s` is a
`Finset (Fin n)`.
The forward map sends `(k, l, s)` to `(k + l, s)` and the inverse map sends `(n, s)` to
`(n - Finset.card s, Finset.card s, s)`. The actual definition is less readable because of problems
with non-definitional equalities. -/
@[simps]
def changeOriginIndexEquiv :
(Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) ≃ Σn : ℕ, Finset (Fin n) where
toFun s := ⟨s.1 + s.2.1, s.2.2⟩
invFun s :=
⟨s.1 - s.2.card, s.2.card,
⟨s.2.map
(Fin.castIso <| (tsub_add_cancel_of_le <| card_finset_fin_le s.2).symm).toEquiv.toEmbedding,
Finset.card_map _⟩⟩
left_inv := by
rintro ⟨k, l, ⟨s : Finset (Fin <| k + l), hs : s.card = l⟩⟩
dsimp only [Subtype.coe_mk]
-- Lean can't automatically generalize `k' = k + l - s.card`, `l' = s.card`, so we explicitly
-- formulate the generalized goal
suffices ∀ k' l', k' = k → l' = l → ∀ (hkl : k + l = k' + l') (hs'),
(⟨k', l', ⟨Finset.map (Fin.castIso hkl).toEquiv.toEmbedding s, hs'⟩⟩ :
Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) = ⟨k, l, ⟨s, hs⟩⟩ by
apply this <;> simp only [hs, add_tsub_cancel_right]
rintro _ _ rfl rfl hkl hs'
simp only [Equiv.refl_toEmbedding, Fin.castIso_refl, Finset.map_refl, eq_self_iff_true,
OrderIso.refl_toEquiv, and_self_iff, heq_iff_eq]
right_inv := by
rintro ⟨n, s⟩
simp [tsub_add_cancel_of_le (card_finset_fin_le s), Fin.castIso_to_equiv]
#align formal_multilinear_series.change_origin_index_equiv FormalMultilinearSeries.changeOriginIndexEquiv
theorem changeOriginSeries_summable_aux₁ {r r' : ℝ≥0} (hr : (r + r' : ℝ≥0∞) < p.radius) :
Summable fun s : Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l } =>
‖p (s.1 + s.2.1)‖₊ * r ^ s.2.1 * r' ^ s.1 := by
rw [← changeOriginIndexEquiv.symm.summable_iff]
dsimp only [Function.comp_def, changeOriginIndexEquiv_symm_apply_fst,
changeOriginIndexEquiv_symm_apply_snd_fst]
have : ∀ n : ℕ,
HasSum (fun s : Finset (Fin n) => ‖p (n - s.card + s.card)‖₊ * r ^ s.card * r' ^ (n - s.card))
(‖p n‖₊ * (r + r') ^ n) := by
intro n
-- TODO: why `simp only [tsub_add_cancel_of_le (card_finset_fin_le _)]` fails?
convert_to HasSum (fun s : Finset (Fin n) => ‖p n‖₊ * (r ^ s.card * r' ^ (n - s.card))) _
· ext1 s
rw [tsub_add_cancel_of_le (card_finset_fin_le _), mul_assoc]
rw [← Fin.sum_pow_mul_eq_add_pow]
exact (hasSum_fintype _).mul_left _
refine' NNReal.summable_sigma.2 ⟨fun n => (this n).summable, _⟩
simp only [(this _).tsum_eq]
exact p.summable_nnnorm_mul_pow hr
#align formal_multilinear_series.change_origin_series_summable_aux₁ FormalMultilinearSeries.changeOriginSeries_summable_aux₁
theorem changeOriginSeries_summable_aux₂ (hr : (r : ℝ≥0∞) < p.radius) (k : ℕ) :
Summable fun s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l } =>
‖p (k + s.1)‖₊ * r ^ s.1 := by
rcases ENNReal.lt_iff_exists_add_pos_lt.1 hr with ⟨r', h0, hr'⟩
simpa only [mul_inv_cancel_right₀ (pow_pos h0 _).ne'] using
((NNReal.summable_sigma.1 (p.changeOriginSeries_summable_aux₁ hr')).1 k).mul_right (r' ^ k)⁻¹
#align formal_multilinear_series.change_origin_series_summable_aux₂ FormalMultilinearSeries.changeOriginSeries_summable_aux₂
theorem changeOriginSeries_summable_aux₃ {r : ℝ≥0} (hr : ↑r < p.radius) (k : ℕ) :
Summable fun l : ℕ => ‖p.changeOriginSeries k l‖₊ * r ^ l := by
refine' NNReal.summable_of_le
(fun n => _) (NNReal.summable_sigma.1 <| p.changeOriginSeries_summable_aux₂ hr k).2
simp only [NNReal.tsum_mul_right]
exact mul_le_mul' (p.nnnorm_changeOriginSeries_le_tsum _ _) le_rfl
#align formal_multilinear_series.change_origin_series_summable_aux₃ FormalMultilinearSeries.changeOriginSeries_summable_aux₃
theorem le_changeOriginSeries_radius (k : ℕ) : p.radius ≤ (p.changeOriginSeries k).radius :=
ENNReal.le_of_forall_nnreal_lt fun _r hr =>
le_radius_of_summable_nnnorm _ (p.changeOriginSeries_summable_aux₃ hr k)
#align formal_multilinear_series.le_change_origin_series_radius FormalMultilinearSeries.le_changeOriginSeries_radius
theorem nnnorm_changeOrigin_le (k : ℕ) (h : (‖x‖₊ : ℝ≥0∞) < p.radius) :
‖p.changeOrigin x k‖₊ ≤
∑' s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + s.1)‖₊ * ‖x‖₊ ^ s.1 := by
refine' tsum_of_nnnorm_bounded _ fun l => p.nnnorm_changeOriginSeries_apply_le_tsum k l x
have := p.changeOriginSeries_summable_aux₂ h k
refine' HasSum.sigma this.hasSum fun l => _
exact ((NNReal.summable_sigma.1 this).1 l).hasSum
#align formal_multilinear_series.nnnorm_change_origin_le FormalMultilinearSeries.nnnorm_changeOrigin_le
/-- The radius of convergence of `p.changeOrigin x` is at least `p.radius - ‖x‖`. In other words,
`p.changeOrigin x` is well defined on the largest ball contained in the original ball of
convergence. -/
theorem changeOrigin_radius : p.radius - ‖x‖₊ ≤ (p.changeOrigin x).radius := by
refine' ENNReal.le_of_forall_pos_nnreal_lt fun r _h0 hr => _
rw [lt_tsub_iff_right, add_comm] at hr
have hr' : (‖x‖₊ : ℝ≥0∞) < p.radius := (le_add_right le_rfl).trans_lt hr
apply le_radius_of_summable_nnnorm
have : ∀ k : ℕ,
‖p.changeOrigin x k‖₊ * r ^ k ≤
(∑' s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + s.1)‖₊ * ‖x‖₊ ^ s.1) *
r ^ k :=
fun k => mul_le_mul_right' (p.nnnorm_changeOrigin_le k hr') (r ^ k)
refine' NNReal.summable_of_le this _
simpa only [← NNReal.tsum_mul_right] using
(NNReal.summable_sigma.1 (p.changeOriginSeries_summable_aux₁ hr)).2
#align formal_multilinear_series.change_origin_radius FormalMultilinearSeries.changeOrigin_radius
end
-- From this point on, assume that the space is complete, to make sure that series that converge
-- in norm also converge in `F`.
variable [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x y : E} {r R : ℝ≥0}
theorem hasFPowerSeriesOnBall_changeOrigin (k : ℕ) (hr : 0 < p.radius) :
HasFPowerSeriesOnBall (fun x => p.changeOrigin x k) (p.changeOriginSeries k) 0 p.radius :=
have := p.le_changeOriginSeries_radius k
((p.changeOriginSeries k).hasFPowerSeriesOnBall (hr.trans_le this)).mono hr this
#align formal_multilinear_series.has_fpower_series_on_ball_change_origin FormalMultilinearSeries.hasFPowerSeriesOnBall_changeOrigin
/-- Summing the series `p.changeOrigin x` at a point `y` gives back `p (x + y)`. -/
theorem changeOrigin_eval (h : (‖x‖₊ + ‖y‖₊ : ℝ≥0∞) < p.radius) :
(p.changeOrigin x).sum y = p.sum (x + y) := by
have radius_pos : 0 < p.radius := lt_of_le_of_lt (zero_le _) h
have x_mem_ball : x ∈ EMetric.ball (0 : E) p.radius :=
mem_emetric_ball_zero_iff.2 ((le_add_right le_rfl).trans_lt h)
have y_mem_ball : y ∈ EMetric.ball (0 : E) (p.changeOrigin x).radius := by
refine' mem_emetric_ball_zero_iff.2 (lt_of_lt_of_le _ p.changeOrigin_radius)
rwa [lt_tsub_iff_right, add_comm]
have x_add_y_mem_ball : x + y ∈ EMetric.ball (0 : E) p.radius := by
refine' mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt _ h)
exact mod_cast nnnorm_add_le x y
set f : (Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) → F := fun s =>
p.changeOriginSeriesTerm s.1 s.2.1 s.2.2 s.2.2.2 (fun _ => x) fun _ => y
have hsf : Summable f := by
refine' .of_nnnorm_bounded _ (p.changeOriginSeries_summable_aux₁ h) _
rintro ⟨k, l, s, hs⟩
dsimp only [Subtype.coe_mk]
exact p.nnnorm_changeOriginSeriesTerm_apply_le _ _ _ _ _ _
have hf : HasSum f ((p.changeOrigin x).sum y) := by
refine' HasSum.sigma_of_hasSum ((p.changeOrigin x).summable y_mem_ball).hasSum (fun k => _) hsf
· dsimp only
refine' ContinuousMultilinearMap.hasSum_eval _ _
have := (p.hasFPowerSeriesOnBall_changeOrigin k radius_pos).hasSum x_mem_ball
rw [zero_add] at this
refine' HasSum.sigma_of_hasSum this (fun l => _) _
· simp only [changeOriginSeries, ContinuousMultilinearMap.sum_apply]
apply hasSum_fintype
· refine' .of_nnnorm_bounded _
(p.changeOriginSeries_summable_aux₂ (mem_emetric_ball_zero_iff.1 x_mem_ball) k) fun s => _
refine' (ContinuousMultilinearMap.le_op_nnnorm _ _).trans_eq _
simp
refine' hf.unique (changeOriginIndexEquiv.symm.hasSum_iff.1 _)
refine' HasSum.sigma_of_hasSum
(p.hasSum x_add_y_mem_ball) (fun n => _) (changeOriginIndexEquiv.symm.summable_iff.2 hsf)
erw [(p n).map_add_univ (fun _ => x) fun _ => y]
-- porting note: added explicit function
convert hasSum_fintype (fun c : Finset (Fin n) => f (changeOriginIndexEquiv.symm ⟨n, c⟩))
rename_i s _
dsimp only [changeOriginSeriesTerm, (· ∘ ·), changeOriginIndexEquiv_symm_apply_fst,
changeOriginIndexEquiv_symm_apply_snd_fst, changeOriginIndexEquiv_symm_apply_snd_snd_coe]
rw [ContinuousMultilinearMap.curryFinFinset_apply_const]
have : ∀ (m) (hm : n = m), p n (s.piecewise (fun _ => x) fun _ => y) =
p m ((s.map (Fin.castIso hm).toEquiv.toEmbedding).piecewise (fun _ => x) fun _ => y) := by
rintro m rfl
simp (config := { unfoldPartialApp := true }) [Finset.piecewise]
apply this
#align formal_multilinear_series.change_origin_eval FormalMultilinearSeries.changeOrigin_eval
/-- Power series terms are analytic as we vary the origin -/
theorem analyticAt_changeOrigin (p : FormalMultilinearSeries 𝕜 E F) (rp : p.radius > 0) (n : ℕ) :
AnalyticAt 𝕜 (fun x ↦ p.changeOrigin x n) 0 :=
(FormalMultilinearSeries.hasFPowerSeriesOnBall_changeOrigin p n rp).analyticAt
end FormalMultilinearSeries
section
variable [CompleteSpace F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x y : E} {r : ℝ≥0∞}
/-- If a function admits a power series expansion `p` on a ball `B (x, r)`, then it also admits a
power series on any subball of this ball (even with a different center), given by `p.changeOrigin`.
-/
theorem HasFPowerSeriesOnBall.changeOrigin (hf : HasFPowerSeriesOnBall f p x r)
(h : (‖y‖₊ : ℝ≥0∞) < r) : HasFPowerSeriesOnBall f (p.changeOrigin y) (x + y) (r - ‖y‖₊) :=
{ r_le := by
apply le_trans _ p.changeOrigin_radius
exact tsub_le_tsub hf.r_le le_rfl
r_pos := by simp [h]
hasSum := fun {z} hz => by
have : f (x + y + z) =
FormalMultilinearSeries.sum (FormalMultilinearSeries.changeOrigin p y) z := by
rw [mem_emetric_ball_zero_iff, lt_tsub_iff_right, add_comm] at hz
rw [p.changeOrigin_eval (hz.trans_le hf.r_le), add_assoc, hf.sum]
refine' mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt _ hz)
exact mod_cast nnnorm_add_le y z
rw [this]
apply (p.changeOrigin y).hasSum
refine' EMetric.ball_subset_ball (le_trans _ p.changeOrigin_radius) hz
exact tsub_le_tsub hf.r_le le_rfl }
#align has_fpower_series_on_ball.change_origin HasFPowerSeriesOnBall.changeOrigin
/-- If a function admits a power series expansion `p` on an open ball `B (x, r)`, then
it is analytic at every point of this ball. -/
theorem HasFPowerSeriesOnBall.analyticAt_of_mem (hf : HasFPowerSeriesOnBall f p x r)
(h : y ∈ EMetric.ball x r) : AnalyticAt 𝕜 f y := by
have : (‖y - x‖₊ : ℝ≥0∞) < r := by simpa [edist_eq_coe_nnnorm_sub] using h
have := hf.changeOrigin this
rw [add_sub_cancel'_right] at this
exact this.analyticAt
#align has_fpower_series_on_ball.analytic_at_of_mem HasFPowerSeriesOnBall.analyticAt_of_mem
theorem HasFPowerSeriesOnBall.analyticOn (hf : HasFPowerSeriesOnBall f p x r) :
AnalyticOn 𝕜 f (EMetric.ball x r) :=
fun _y hy => hf.analyticAt_of_mem hy
#align has_fpower_series_on_ball.analytic_on HasFPowerSeriesOnBall.analyticOn
variable (𝕜 f)
/-- For any function `f` from a normed vector space to a Banach space, the set of points `x` such
that `f` is analytic at `x` is open. -/
theorem isOpen_analyticAt : IsOpen { x | AnalyticAt 𝕜 f x } := by
rw [isOpen_iff_mem_nhds]
rintro x ⟨p, r, hr⟩
exact mem_of_superset (EMetric.ball_mem_nhds _ hr.r_pos) fun y hy => hr.analyticAt_of_mem hy
#align is_open_analytic_at isOpen_analyticAt
variable {𝕜}
theorem AnalyticAt.eventually_analyticAt {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∀ᶠ y in 𝓝 x, AnalyticAt 𝕜 f y :=
(isOpen_analyticAt 𝕜 f).mem_nhds h
theorem AnalyticAt.exists_mem_nhds_analyticOn {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∃ s ∈ 𝓝 x, AnalyticOn 𝕜 f s :=
h.eventually_analyticAt.exists_mem
/-- If we're analytic at a point, we're analytic in a nonempty ball -/
theorem AnalyticAt.exists_ball_analyticOn {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∃ r : ℝ, 0 < r ∧ AnalyticOn 𝕜 f (Metric.ball x r) :=
Metric.isOpen_iff.mp (isOpen_analyticAt _ _) _ h
end
section
open FormalMultilinearSeries
variable {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E} {z₀ : 𝕜}
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z)) := by
refine' ⟨fun ⟨r, _, r_pos, h⟩ =>
eventually_of_mem (EMetric.ball_mem_nhds 0 r_pos) fun _ => by simpa using h, _⟩
simp only [Metric.eventually_nhds_iff]
rintro ⟨r, r_pos, h⟩
refine' ⟨p.radius ⊓ r.toNNReal, by simp, _, _⟩
· simp only [r_pos.lt, lt_inf_iff, ENNReal.coe_pos, Real.toNNReal_pos, and_true_iff]
obtain ⟨z, z_pos, le_z⟩ := NormedField.exists_norm_lt 𝕜 r_pos.lt
have : (‖z‖₊ : ENNReal) ≤ p.radius := by
simp only [dist_zero_right] at h
apply FormalMultilinearSeries.le_radius_of_tendsto
convert tendsto_norm.comp (h le_z).summable.tendsto_atTop_zero
|
funext
|
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z)) := by
refine' ⟨fun ⟨r, _, r_pos, h⟩ =>
eventually_of_mem (EMetric.ball_mem_nhds 0 r_pos) fun _ => by simpa using h, _⟩
simp only [Metric.eventually_nhds_iff]
rintro ⟨r, r_pos, h⟩
refine' ⟨p.radius ⊓ r.toNNReal, by simp, _, _⟩
· simp only [r_pos.lt, lt_inf_iff, ENNReal.coe_pos, Real.toNNReal_pos, and_true_iff]
obtain ⟨z, z_pos, le_z⟩ := NormedField.exists_norm_lt 𝕜 r_pos.lt
have : (‖z‖₊ : ENNReal) ≤ p.radius := by
simp only [dist_zero_right] at h
apply FormalMultilinearSeries.le_radius_of_tendsto
convert tendsto_norm.comp (h le_z).summable.tendsto_atTop_zero
|
Mathlib.Analysis.Analytic.Basic.1430_0.jQw1fRSE1vGpOll
|
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z))
|
Mathlib_Analysis_Analytic_Basic
|
case h.e'_3.h
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
p : FormalMultilinearSeries 𝕜 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
r : ℝ
r_pos : r > 0
z : 𝕜
z_pos : 0 < ‖z‖
le_z : ‖z‖ < r
h : ∀ ⦃y : 𝕜⦄, ‖y‖ < r → HasSum (fun n => y ^ n • coeff p n) (f (z₀ + y))
x✝ : ℕ
⊢ ‖p x✝‖ * ↑‖z‖₊ ^ x✝ = ((fun a => ‖a‖) ∘ fun n => z ^ n • coeff p n) x✝
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Logic.Equiv.Fin
import Mathlib.Topology.Algebra.InfiniteSum.Module
#align_import analysis.analytic.basic from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
/-!
# Analytic functions
A function is analytic in one dimension around `0` if it can be written as a converging power series
`Σ pₙ zⁿ`. This definition can be extended to any dimension (even in infinite dimension) by
requiring that `pₙ` is a continuous `n`-multilinear map. In general, `pₙ` is not unique (in two
dimensions, taking `p₂ (x, y) (x', y') = x y'` or `y x'` gives the same map when applied to a
vector `(x, y) (x, y)`). A way to guarantee uniqueness is to take a symmetric `pₙ`, but this is not
always possible in nonzero characteristic (in characteristic 2, the previous example has no
symmetric representative). Therefore, we do not insist on symmetry or uniqueness in the definition,
and we only require the existence of a converging series.
The general framework is important to say that the exponential map on bounded operators on a Banach
space is analytic, as well as the inverse on invertible operators.
## Main definitions
Let `p` be a formal multilinear series from `E` to `F`, i.e., `p n` is a multilinear map on `E^n`
for `n : ℕ`.
* `p.radius`: the largest `r : ℝ≥0∞` such that `‖p n‖ * r^n` grows subexponentially.
* `p.le_radius_of_bound`, `p.le_radius_of_bound_nnreal`, `p.le_radius_of_isBigO`: if `‖p n‖ * r ^ n`
is bounded above, then `r ≤ p.radius`;
* `p.isLittleO_of_lt_radius`, `p.norm_mul_pow_le_mul_pow_of_lt_radius`,
`p.isLittleO_one_of_lt_radius`,
`p.norm_mul_pow_le_of_lt_radius`, `p.nnnorm_mul_pow_le_of_lt_radius`: if `r < p.radius`, then
`‖p n‖ * r ^ n` tends to zero exponentially;
* `p.lt_radius_of_isBigO`: if `r ≠ 0` and `‖p n‖ * r ^ n = O(a ^ n)` for some `-1 < a < 1`, then
`r < p.radius`;
* `p.partialSum n x`: the sum `∑_{i = 0}^{n-1} pᵢ xⁱ`.
* `p.sum x`: the sum `∑'_{i = 0}^{∞} pᵢ xⁱ`.
Additionally, let `f` be a function from `E` to `F`.
* `HasFPowerSeriesOnBall f p x r`: on the ball of center `x` with radius `r`,
`f (x + y) = ∑'_n pₙ yⁿ`.
* `HasFPowerSeriesAt f p x`: on some ball of center `x` with positive radius, holds
`HasFPowerSeriesOnBall f p x r`.
* `AnalyticAt 𝕜 f x`: there exists a power series `p` such that holds `HasFPowerSeriesAt f p x`.
* `AnalyticOn 𝕜 f s`: the function `f` is analytic at every point of `s`.
We develop the basic properties of these notions, notably:
* If a function admits a power series, it is continuous (see
`HasFPowerSeriesOnBall.continuousOn` and `HasFPowerSeriesAt.continuousAt` and
`AnalyticAt.continuousAt`).
* In a complete space, the sum of a formal power series with positive radius is well defined on the
disk of convergence, see `FormalMultilinearSeries.hasFPowerSeriesOnBall`.
* If a function admits a power series in a ball, then it is analytic at any point `y` of this ball,
and the power series there can be expressed in terms of the initial power series `p` as
`p.changeOrigin y`. See `HasFPowerSeriesOnBall.changeOrigin`. It follows in particular that
the set of points at which a given function is analytic is open, see `isOpen_analyticAt`.
## Implementation details
We only introduce the radius of convergence of a power series, as `p.radius`.
For a power series in finitely many dimensions, there is a finer (directional, coordinate-dependent)
notion, describing the polydisk of convergence. This notion is more specific, and not necessary to
build the general theory. We do not define it here.
-/
noncomputable section
variable {𝕜 E F G : Type*}
open Topology Classical BigOperators NNReal Filter ENNReal
open Set Filter Asymptotics
namespace FormalMultilinearSeries
variable [Ring 𝕜] [AddCommGroup E] [AddCommGroup F] [Module 𝕜 E] [Module 𝕜 F]
variable [TopologicalSpace E] [TopologicalSpace F]
variable [TopologicalAddGroup E] [TopologicalAddGroup F]
variable [ContinuousConstSMul 𝕜 E] [ContinuousConstSMul 𝕜 F]
/-- Given a formal multilinear series `p` and a vector `x`, then `p.sum x` is the sum `Σ pₙ xⁿ`. A
priori, it only behaves well when `‖x‖ < p.radius`. -/
protected def sum (p : FormalMultilinearSeries 𝕜 E F) (x : E) : F :=
∑' n : ℕ, p n fun _ => x
#align formal_multilinear_series.sum FormalMultilinearSeries.sum
/-- Given a formal multilinear series `p` and a vector `x`, then `p.partialSum n x` is the sum
`Σ pₖ xᵏ` for `k ∈ {0,..., n-1}`. -/
def partialSum (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) (x : E) : F :=
∑ k in Finset.range n, p k fun _ : Fin k => x
#align formal_multilinear_series.partial_sum FormalMultilinearSeries.partialSum
/-- The partial sums of a formal multilinear series are continuous. -/
theorem partialSum_continuous (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) :
Continuous (p.partialSum n) := by
unfold partialSum -- Porting note: added
continuity
#align formal_multilinear_series.partial_sum_continuous FormalMultilinearSeries.partialSum_continuous
end FormalMultilinearSeries
/-! ### The radius of a formal multilinear series -/
variable [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F]
[NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G]
namespace FormalMultilinearSeries
variable (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
/-- The radius of a formal multilinear series is the largest `r` such that the sum `Σ ‖pₙ‖ ‖y‖ⁿ`
converges for all `‖y‖ < r`. This implies that `Σ pₙ yⁿ` converges for all `‖y‖ < r`, but these
definitions are *not* equivalent in general. -/
def radius (p : FormalMultilinearSeries 𝕜 E F) : ℝ≥0∞ :=
⨆ (r : ℝ≥0) (C : ℝ) (_ : ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C), (r : ℝ≥0∞)
#align formal_multilinear_series.radius FormalMultilinearSeries.radius
/-- If `‖pₙ‖ rⁿ` is bounded in `n`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_bound (C : ℝ) {r : ℝ≥0} (h : ∀ n : ℕ, ‖p n‖ * (r : ℝ) ^ n ≤ C) :
(r : ℝ≥0∞) ≤ p.radius :=
le_iSup_of_le r <| le_iSup_of_le C <| le_iSup (fun _ => (r : ℝ≥0∞)) h
#align formal_multilinear_series.le_radius_of_bound FormalMultilinearSeries.le_radius_of_bound
/-- If `‖pₙ‖ rⁿ` is bounded in `n`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_bound_nnreal (C : ℝ≥0) {r : ℝ≥0} (h : ∀ n : ℕ, ‖p n‖₊ * r ^ n ≤ C) :
(r : ℝ≥0∞) ≤ p.radius :=
p.le_radius_of_bound C fun n => mod_cast h n
#align formal_multilinear_series.le_radius_of_bound_nnreal FormalMultilinearSeries.le_radius_of_bound_nnreal
/-- If `‖pₙ‖ rⁿ = O(1)`, as `n → ∞`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_isBigO (h : (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] fun _ => (1 : ℝ)) :
↑r ≤ p.radius :=
Exists.elim (isBigO_one_nat_atTop_iff.1 h) fun C hC =>
p.le_radius_of_bound C fun n => (le_abs_self _).trans (hC n)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.le_radius_of_is_O FormalMultilinearSeries.le_radius_of_isBigO
theorem le_radius_of_eventually_le (C) (h : ∀ᶠ n in atTop, ‖p n‖ * (r : ℝ) ^ n ≤ C) :
↑r ≤ p.radius :=
p.le_radius_of_isBigO <| IsBigO.of_bound C <| h.mono fun n hn => by simpa
#align formal_multilinear_series.le_radius_of_eventually_le FormalMultilinearSeries.le_radius_of_eventually_le
theorem le_radius_of_summable_nnnorm (h : Summable fun n => ‖p n‖₊ * r ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_bound_nnreal (∑' n, ‖p n‖₊ * r ^ n) fun _ => le_tsum' h _
#align formal_multilinear_series.le_radius_of_summable_nnnorm FormalMultilinearSeries.le_radius_of_summable_nnnorm
theorem le_radius_of_summable (h : Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_summable_nnnorm <| by
simp only [← coe_nnnorm] at h
exact mod_cast h
#align formal_multilinear_series.le_radius_of_summable FormalMultilinearSeries.le_radius_of_summable
theorem radius_eq_top_of_forall_nnreal_isBigO
(h : ∀ r : ℝ≥0, (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] fun _ => (1 : ℝ)) : p.radius = ∞ :=
ENNReal.eq_top_of_forall_nnreal_le fun r => p.le_radius_of_isBigO (h r)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.radius_eq_top_of_forall_nnreal_is_O FormalMultilinearSeries.radius_eq_top_of_forall_nnreal_isBigO
theorem radius_eq_top_of_eventually_eq_zero (h : ∀ᶠ n in atTop, p n = 0) : p.radius = ∞ :=
p.radius_eq_top_of_forall_nnreal_isBigO fun r =>
(isBigO_zero _ _).congr' (h.mono fun n hn => by simp [hn]) EventuallyEq.rfl
#align formal_multilinear_series.radius_eq_top_of_eventually_eq_zero FormalMultilinearSeries.radius_eq_top_of_eventually_eq_zero
theorem radius_eq_top_of_forall_image_add_eq_zero (n : ℕ) (hn : ∀ m, p (m + n) = 0) :
p.radius = ∞ :=
p.radius_eq_top_of_eventually_eq_zero <|
mem_atTop_sets.2 ⟨n, fun _ hk => tsub_add_cancel_of_le hk ▸ hn _⟩
#align formal_multilinear_series.radius_eq_top_of_forall_image_add_eq_zero FormalMultilinearSeries.radius_eq_top_of_forall_image_add_eq_zero
@[simp]
theorem constFormalMultilinearSeries_radius {v : F} :
(constFormalMultilinearSeries 𝕜 E v).radius = ⊤ :=
(constFormalMultilinearSeries 𝕜 E v).radius_eq_top_of_forall_image_add_eq_zero 1
(by simp [constFormalMultilinearSeries])
#align formal_multilinear_series.const_formal_multilinear_series_radius FormalMultilinearSeries.constFormalMultilinearSeries_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` tends to zero exponentially:
for some `0 < a < 1`, `‖p n‖ rⁿ = o(aⁿ)`. -/
theorem isLittleO_of_lt_radius (h : ↑r < p.radius) :
∃ a ∈ Ioo (0 : ℝ) 1, (fun n => ‖p n‖ * (r : ℝ) ^ n) =o[atTop] (a ^ ·) := by
have := (TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 4
rw [this]
-- Porting note: was
-- rw [(TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 4]
simp only [radius, lt_iSup_iff] at h
rcases h with ⟨t, C, hC, rt⟩
rw [ENNReal.coe_lt_coe, ← NNReal.coe_lt_coe] at rt
have : 0 < (t : ℝ) := r.coe_nonneg.trans_lt rt
rw [← div_lt_one this] at rt
refine' ⟨_, rt, C, Or.inr zero_lt_one, fun n => _⟩
calc
|‖p n‖ * (r : ℝ) ^ n| = ‖p n‖ * (t : ℝ) ^ n * (r / t : ℝ) ^ n := by
field_simp [mul_right_comm, abs_mul]
_ ≤ C * (r / t : ℝ) ^ n := by gcongr; apply hC
#align formal_multilinear_series.is_o_of_lt_radius FormalMultilinearSeries.isLittleO_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ = o(1)`. -/
theorem isLittleO_one_of_lt_radius (h : ↑r < p.radius) :
(fun n => ‖p n‖ * (r : ℝ) ^ n) =o[atTop] (fun _ => 1 : ℕ → ℝ) :=
let ⟨_, ha, hp⟩ := p.isLittleO_of_lt_radius h
hp.trans <| (isLittleO_pow_pow_of_lt_left ha.1.le ha.2).congr (fun _ => rfl) one_pow
#align formal_multilinear_series.is_o_one_of_lt_radius FormalMultilinearSeries.isLittleO_one_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` tends to zero exponentially:
for some `0 < a < 1` and `C > 0`, `‖p n‖ * r ^ n ≤ C * a ^ n`. -/
theorem norm_mul_pow_le_mul_pow_of_lt_radius (h : ↑r < p.radius) :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C * a ^ n := by
-- Porting note: moved out of `rcases`
have := ((TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 5).mp
(p.isLittleO_of_lt_radius h)
rcases this with ⟨a, ha, C, hC, H⟩
exact ⟨a, ha, C, hC, fun n => (le_abs_self _).trans (H n)⟩
#align formal_multilinear_series.norm_mul_pow_le_mul_pow_of_lt_radius FormalMultilinearSeries.norm_mul_pow_le_mul_pow_of_lt_radius
/-- If `r ≠ 0` and `‖pₙ‖ rⁿ = O(aⁿ)` for some `-1 < a < 1`, then `r < p.radius`. -/
theorem lt_radius_of_isBigO (h₀ : r ≠ 0) {a : ℝ} (ha : a ∈ Ioo (-1 : ℝ) 1)
(hp : (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] (a ^ ·)) : ↑r < p.radius := by
-- Porting note: moved out of `rcases`
have := ((TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 2 5)
rcases this.mp ⟨a, ha, hp⟩ with ⟨a, ha, C, hC, hp⟩
rw [← pos_iff_ne_zero, ← NNReal.coe_pos] at h₀
lift a to ℝ≥0 using ha.1.le
have : (r : ℝ) < r / a := by
simpa only [div_one] using (div_lt_div_left h₀ zero_lt_one ha.1).2 ha.2
norm_cast at this
rw [← ENNReal.coe_lt_coe] at this
refine' this.trans_le (p.le_radius_of_bound C fun n => _)
rw [NNReal.coe_div, div_pow, ← mul_div_assoc, div_le_iff (pow_pos ha.1 n)]
exact (le_abs_self _).trans (hp n)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.lt_radius_of_is_O FormalMultilinearSeries.lt_radius_of_isBigO
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem norm_mul_pow_le_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C :=
let ⟨_, ha, C, hC, h⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius h
⟨C, hC, fun n => (h n).trans <| mul_le_of_le_one_right hC.lt.le (pow_le_one _ ha.1.le ha.2.le)⟩
#align formal_multilinear_series.norm_mul_pow_le_of_lt_radius FormalMultilinearSeries.norm_mul_pow_le_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem norm_le_div_pow_of_pos_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h0 : 0 < r) (h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖ ≤ C / (r : ℝ) ^ n :=
let ⟨C, hC, hp⟩ := p.norm_mul_pow_le_of_lt_radius h
⟨C, hC, fun n => Iff.mpr (le_div_iff (pow_pos h0 _)) (hp n)⟩
#align formal_multilinear_series.norm_le_div_pow_of_pos_of_lt_radius FormalMultilinearSeries.norm_le_div_pow_of_pos_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem nnnorm_mul_pow_le_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖₊ * r ^ n ≤ C :=
let ⟨C, hC, hp⟩ := p.norm_mul_pow_le_of_lt_radius h
⟨⟨C, hC.lt.le⟩, hC, mod_cast hp⟩
#align formal_multilinear_series.nnnorm_mul_pow_le_of_lt_radius FormalMultilinearSeries.nnnorm_mul_pow_le_of_lt_radius
theorem le_radius_of_tendsto (p : FormalMultilinearSeries 𝕜 E F) {l : ℝ}
(h : Tendsto (fun n => ‖p n‖ * (r : ℝ) ^ n) atTop (𝓝 l)) : ↑r ≤ p.radius :=
p.le_radius_of_isBigO (h.isBigO_one _)
#align formal_multilinear_series.le_radius_of_tendsto FormalMultilinearSeries.le_radius_of_tendsto
theorem le_radius_of_summable_norm (p : FormalMultilinearSeries 𝕜 E F)
(hs : Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_tendsto hs.tendsto_atTop_zero
#align formal_multilinear_series.le_radius_of_summable_norm FormalMultilinearSeries.le_radius_of_summable_norm
theorem not_summable_norm_of_radius_lt_nnnorm (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(h : p.radius < ‖x‖₊) : ¬Summable fun n => ‖p n‖ * ‖x‖ ^ n :=
fun hs => not_le_of_lt h (p.le_radius_of_summable_norm hs)
#align formal_multilinear_series.not_summable_norm_of_radius_lt_nnnorm FormalMultilinearSeries.not_summable_norm_of_radius_lt_nnnorm
theorem summable_norm_mul_pow (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0} (h : ↑r < p.radius) :
Summable fun n : ℕ => ‖p n‖ * (r : ℝ) ^ n := by
obtain ⟨a, ha : a ∈ Ioo (0 : ℝ) 1, C, - : 0 < C, hp⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius h
exact .of_nonneg_of_le (fun n => mul_nonneg (norm_nonneg _) (pow_nonneg r.coe_nonneg _))
hp ((summable_geometric_of_lt_1 ha.1.le ha.2).mul_left _)
#align formal_multilinear_series.summable_norm_mul_pow FormalMultilinearSeries.summable_norm_mul_pow
theorem summable_norm_apply (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : Summable fun n : ℕ => ‖p n fun _ => x‖ := by
rw [mem_emetric_ball_zero_iff] at hx
refine' .of_nonneg_of_le
(fun _ => norm_nonneg _) (fun n => ((p n).le_op_norm _).trans_eq _) (p.summable_norm_mul_pow hx)
simp
#align formal_multilinear_series.summable_norm_apply FormalMultilinearSeries.summable_norm_apply
theorem summable_nnnorm_mul_pow (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0} (h : ↑r < p.radius) :
Summable fun n : ℕ => ‖p n‖₊ * r ^ n := by
rw [← NNReal.summable_coe]
push_cast
exact p.summable_norm_mul_pow h
#align formal_multilinear_series.summable_nnnorm_mul_pow FormalMultilinearSeries.summable_nnnorm_mul_pow
protected theorem summable [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : Summable fun n : ℕ => p n fun _ => x :=
(p.summable_norm_apply hx).of_norm
#align formal_multilinear_series.summable FormalMultilinearSeries.summable
theorem radius_eq_top_of_summable_norm (p : FormalMultilinearSeries 𝕜 E F)
(hs : ∀ r : ℝ≥0, Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : p.radius = ∞ :=
ENNReal.eq_top_of_forall_nnreal_le fun r => p.le_radius_of_summable_norm (hs r)
#align formal_multilinear_series.radius_eq_top_of_summable_norm FormalMultilinearSeries.radius_eq_top_of_summable_norm
theorem radius_eq_top_iff_summable_norm (p : FormalMultilinearSeries 𝕜 E F) :
p.radius = ∞ ↔ ∀ r : ℝ≥0, Summable fun n => ‖p n‖ * (r : ℝ) ^ n := by
constructor
· intro h r
obtain ⟨a, ha : a ∈ Ioo (0 : ℝ) 1, C, - : 0 < C, hp⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius
(show (r : ℝ≥0∞) < p.radius from h.symm ▸ ENNReal.coe_lt_top)
refine' .of_norm_bounded
(fun n => (C : ℝ) * a ^ n) ((summable_geometric_of_lt_1 ha.1.le ha.2).mul_left _) fun n => _
specialize hp n
rwa [Real.norm_of_nonneg (mul_nonneg (norm_nonneg _) (pow_nonneg r.coe_nonneg n))]
· exact p.radius_eq_top_of_summable_norm
#align formal_multilinear_series.radius_eq_top_iff_summable_norm FormalMultilinearSeries.radius_eq_top_iff_summable_norm
/-- If the radius of `p` is positive, then `‖pₙ‖` grows at most geometrically. -/
theorem le_mul_pow_of_radius_pos (p : FormalMultilinearSeries 𝕜 E F) (h : 0 < p.radius) :
∃ (C r : _) (hC : 0 < C) (_ : 0 < r), ∀ n, ‖p n‖ ≤ C * r ^ n := by
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 h with ⟨r, r0, rlt⟩
have rpos : 0 < (r : ℝ) := by simp [ENNReal.coe_pos.1 r0]
rcases norm_le_div_pow_of_pos_of_lt_radius p rpos rlt with ⟨C, Cpos, hCp⟩
refine' ⟨C, r⁻¹, Cpos, by simp only [inv_pos, rpos], fun n => _⟩
-- Porting note: was `convert`
rw [inv_pow, ← div_eq_mul_inv]
exact hCp n
#align formal_multilinear_series.le_mul_pow_of_radius_pos FormalMultilinearSeries.le_mul_pow_of_radius_pos
/-- The radius of the sum of two formal series is at least the minimum of their two radii. -/
theorem min_radius_le_radius_add (p q : FormalMultilinearSeries 𝕜 E F) :
min p.radius q.radius ≤ (p + q).radius := by
refine' ENNReal.le_of_forall_nnreal_lt fun r hr => _
rw [lt_min_iff] at hr
have := ((p.isLittleO_one_of_lt_radius hr.1).add (q.isLittleO_one_of_lt_radius hr.2)).isBigO
refine' (p + q).le_radius_of_isBigO ((isBigO_of_le _ fun n => _).trans this)
rw [← add_mul, norm_mul, norm_mul, norm_norm]
exact mul_le_mul_of_nonneg_right ((norm_add_le _ _).trans (le_abs_self _)) (norm_nonneg _)
#align formal_multilinear_series.min_radius_le_radius_add FormalMultilinearSeries.min_radius_le_radius_add
@[simp]
theorem radius_neg (p : FormalMultilinearSeries 𝕜 E F) : (-p).radius = p.radius := by
simp only [radius, neg_apply, norm_neg]
#align formal_multilinear_series.radius_neg FormalMultilinearSeries.radius_neg
protected theorem hasSum [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : HasSum (fun n : ℕ => p n fun _ => x) (p.sum x) :=
(p.summable hx).hasSum
#align formal_multilinear_series.has_sum FormalMultilinearSeries.hasSum
theorem radius_le_radius_continuousLinearMap_comp (p : FormalMultilinearSeries 𝕜 E F)
(f : F →L[𝕜] G) : p.radius ≤ (f.compFormalMultilinearSeries p).radius := by
refine' ENNReal.le_of_forall_nnreal_lt fun r hr => _
apply le_radius_of_isBigO
apply (IsBigO.trans_isLittleO _ (p.isLittleO_one_of_lt_radius hr)).isBigO
refine' IsBigO.mul (@IsBigOWith.isBigO _ _ _ _ _ ‖f‖ _ _ _ _) (isBigO_refl _ _)
refine IsBigOWith.of_bound (eventually_of_forall fun n => ?_)
simpa only [norm_norm] using f.norm_compContinuousMultilinearMap_le (p n)
#align formal_multilinear_series.radius_le_radius_continuous_linear_map_comp FormalMultilinearSeries.radius_le_radius_continuousLinearMap_comp
end FormalMultilinearSeries
/-! ### Expanding a function as a power series -/
section
variable {f g : E → F} {p pf pg : FormalMultilinearSeries 𝕜 E F} {x : E} {r r' : ℝ≥0∞}
/-- Given a function `f : E → F` and a formal multilinear series `p`, we say that `f` has `p` as
a power series on the ball of radius `r > 0` around `x` if `f (x + y) = ∑' pₙ yⁿ` for all `‖y‖ < r`.
-/
structure HasFPowerSeriesOnBall (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (x : E) (r : ℝ≥0∞) :
Prop where
r_le : r ≤ p.radius
r_pos : 0 < r
hasSum :
∀ {y}, y ∈ EMetric.ball (0 : E) r → HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y))
#align has_fpower_series_on_ball HasFPowerSeriesOnBall
/-- Given a function `f : E → F` and a formal multilinear series `p`, we say that `f` has `p` as
a power series around `x` if `f (x + y) = ∑' pₙ yⁿ` for all `y` in a neighborhood of `0`. -/
def HasFPowerSeriesAt (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (x : E) :=
∃ r, HasFPowerSeriesOnBall f p x r
#align has_fpower_series_at HasFPowerSeriesAt
variable (𝕜)
/-- Given a function `f : E → F`, we say that `f` is analytic at `x` if it admits a convergent power
series expansion around `x`. -/
def AnalyticAt (f : E → F) (x : E) :=
∃ p : FormalMultilinearSeries 𝕜 E F, HasFPowerSeriesAt f p x
#align analytic_at AnalyticAt
/-- Given a function `f : E → F`, we say that `f` is analytic on a set `s` if it is analytic around
every point of `s`. -/
def AnalyticOn (f : E → F) (s : Set E) :=
∀ x, x ∈ s → AnalyticAt 𝕜 f x
#align analytic_on AnalyticOn
variable {𝕜}
theorem HasFPowerSeriesOnBall.hasFPowerSeriesAt (hf : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesAt f p x :=
⟨r, hf⟩
#align has_fpower_series_on_ball.has_fpower_series_at HasFPowerSeriesOnBall.hasFPowerSeriesAt
theorem HasFPowerSeriesAt.analyticAt (hf : HasFPowerSeriesAt f p x) : AnalyticAt 𝕜 f x :=
⟨p, hf⟩
#align has_fpower_series_at.analytic_at HasFPowerSeriesAt.analyticAt
theorem HasFPowerSeriesOnBall.analyticAt (hf : HasFPowerSeriesOnBall f p x r) : AnalyticAt 𝕜 f x :=
hf.hasFPowerSeriesAt.analyticAt
#align has_fpower_series_on_ball.analytic_at HasFPowerSeriesOnBall.analyticAt
theorem HasFPowerSeriesOnBall.congr (hf : HasFPowerSeriesOnBall f p x r)
(hg : EqOn f g (EMetric.ball x r)) : HasFPowerSeriesOnBall g p x r :=
{ r_le := hf.r_le
r_pos := hf.r_pos
hasSum := fun {y} hy => by
convert hf.hasSum hy using 1
apply hg.symm
simpa [edist_eq_coe_nnnorm_sub] using hy }
#align has_fpower_series_on_ball.congr HasFPowerSeriesOnBall.congr
/-- If a function `f` has a power series `p` around `x`, then the function `z ↦ f (z - y)` has the
same power series around `x + y`. -/
theorem HasFPowerSeriesOnBall.comp_sub (hf : HasFPowerSeriesOnBall f p x r) (y : E) :
HasFPowerSeriesOnBall (fun z => f (z - y)) p (x + y) r :=
{ r_le := hf.r_le
r_pos := hf.r_pos
hasSum := fun {z} hz => by
convert hf.hasSum hz using 2
abel }
#align has_fpower_series_on_ball.comp_sub HasFPowerSeriesOnBall.comp_sub
theorem HasFPowerSeriesOnBall.hasSum_sub (hf : HasFPowerSeriesOnBall f p x r) {y : E}
(hy : y ∈ EMetric.ball x r) : HasSum (fun n : ℕ => p n fun _ => y - x) (f y) := by
have : y - x ∈ EMetric.ball (0 : E) r := by simpa [edist_eq_coe_nnnorm_sub] using hy
simpa only [add_sub_cancel'_right] using hf.hasSum this
#align has_fpower_series_on_ball.has_sum_sub HasFPowerSeriesOnBall.hasSum_sub
theorem HasFPowerSeriesOnBall.radius_pos (hf : HasFPowerSeriesOnBall f p x r) : 0 < p.radius :=
lt_of_lt_of_le hf.r_pos hf.r_le
#align has_fpower_series_on_ball.radius_pos HasFPowerSeriesOnBall.radius_pos
theorem HasFPowerSeriesAt.radius_pos (hf : HasFPowerSeriesAt f p x) : 0 < p.radius :=
let ⟨_, hr⟩ := hf
hr.radius_pos
#align has_fpower_series_at.radius_pos HasFPowerSeriesAt.radius_pos
theorem HasFPowerSeriesOnBall.mono (hf : HasFPowerSeriesOnBall f p x r) (r'_pos : 0 < r')
(hr : r' ≤ r) : HasFPowerSeriesOnBall f p x r' :=
⟨le_trans hr hf.1, r'_pos, fun hy => hf.hasSum (EMetric.ball_subset_ball hr hy)⟩
#align has_fpower_series_on_ball.mono HasFPowerSeriesOnBall.mono
theorem HasFPowerSeriesAt.congr (hf : HasFPowerSeriesAt f p x) (hg : f =ᶠ[𝓝 x] g) :
HasFPowerSeriesAt g p x := by
rcases hf with ⟨r₁, h₁⟩
rcases EMetric.mem_nhds_iff.mp hg with ⟨r₂, h₂pos, h₂⟩
exact ⟨min r₁ r₂,
(h₁.mono (lt_min h₁.r_pos h₂pos) inf_le_left).congr
fun y hy => h₂ (EMetric.ball_subset_ball inf_le_right hy)⟩
#align has_fpower_series_at.congr HasFPowerSeriesAt.congr
protected theorem HasFPowerSeriesAt.eventually (hf : HasFPowerSeriesAt f p x) :
∀ᶠ r : ℝ≥0∞ in 𝓝[>] 0, HasFPowerSeriesOnBall f p x r :=
let ⟨_, hr⟩ := hf
mem_of_superset (Ioo_mem_nhdsWithin_Ioi (left_mem_Ico.2 hr.r_pos)) fun _ hr' =>
hr.mono hr'.1 hr'.2.le
#align has_fpower_series_at.eventually HasFPowerSeriesAt.eventually
theorem HasFPowerSeriesOnBall.eventually_hasSum (hf : HasFPowerSeriesOnBall f p x r) :
∀ᶠ y in 𝓝 0, HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y)) := by
filter_upwards [EMetric.ball_mem_nhds (0 : E) hf.r_pos] using fun _ => hf.hasSum
#align has_fpower_series_on_ball.eventually_has_sum HasFPowerSeriesOnBall.eventually_hasSum
theorem HasFPowerSeriesAt.eventually_hasSum (hf : HasFPowerSeriesAt f p x) :
∀ᶠ y in 𝓝 0, HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y)) :=
let ⟨_, hr⟩ := hf
hr.eventually_hasSum
#align has_fpower_series_at.eventually_has_sum HasFPowerSeriesAt.eventually_hasSum
theorem HasFPowerSeriesOnBall.eventually_hasSum_sub (hf : HasFPowerSeriesOnBall f p x r) :
∀ᶠ y in 𝓝 x, HasSum (fun n : ℕ => p n fun _ : Fin n => y - x) (f y) := by
filter_upwards [EMetric.ball_mem_nhds x hf.r_pos] with y using hf.hasSum_sub
#align has_fpower_series_on_ball.eventually_has_sum_sub HasFPowerSeriesOnBall.eventually_hasSum_sub
theorem HasFPowerSeriesAt.eventually_hasSum_sub (hf : HasFPowerSeriesAt f p x) :
∀ᶠ y in 𝓝 x, HasSum (fun n : ℕ => p n fun _ : Fin n => y - x) (f y) :=
let ⟨_, hr⟩ := hf
hr.eventually_hasSum_sub
#align has_fpower_series_at.eventually_has_sum_sub HasFPowerSeriesAt.eventually_hasSum_sub
theorem HasFPowerSeriesOnBall.eventually_eq_zero
(hf : HasFPowerSeriesOnBall f (0 : FormalMultilinearSeries 𝕜 E F) x r) :
∀ᶠ z in 𝓝 x, f z = 0 := by
filter_upwards [hf.eventually_hasSum_sub] with z hz using hz.unique hasSum_zero
#align has_fpower_series_on_ball.eventually_eq_zero HasFPowerSeriesOnBall.eventually_eq_zero
theorem HasFPowerSeriesAt.eventually_eq_zero
(hf : HasFPowerSeriesAt f (0 : FormalMultilinearSeries 𝕜 E F) x) : ∀ᶠ z in 𝓝 x, f z = 0 :=
let ⟨_, hr⟩ := hf
hr.eventually_eq_zero
#align has_fpower_series_at.eventually_eq_zero HasFPowerSeriesAt.eventually_eq_zero
theorem hasFPowerSeriesOnBall_const {c : F} {e : E} :
HasFPowerSeriesOnBall (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e ⊤ := by
refine' ⟨by simp, WithTop.zero_lt_top, fun _ => hasSum_single 0 fun n hn => _⟩
simp [constFormalMultilinearSeries_apply hn]
#align has_fpower_series_on_ball_const hasFPowerSeriesOnBall_const
theorem hasFPowerSeriesAt_const {c : F} {e : E} :
HasFPowerSeriesAt (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e :=
⟨⊤, hasFPowerSeriesOnBall_const⟩
#align has_fpower_series_at_const hasFPowerSeriesAt_const
theorem analyticAt_const {v : F} : AnalyticAt 𝕜 (fun _ => v) x :=
⟨constFormalMultilinearSeries 𝕜 E v, hasFPowerSeriesAt_const⟩
#align analytic_at_const analyticAt_const
theorem analyticOn_const {v : F} {s : Set E} : AnalyticOn 𝕜 (fun _ => v) s :=
fun _ _ => analyticAt_const
#align analytic_on_const analyticOn_const
theorem HasFPowerSeriesOnBall.add (hf : HasFPowerSeriesOnBall f pf x r)
(hg : HasFPowerSeriesOnBall g pg x r) : HasFPowerSeriesOnBall (f + g) (pf + pg) x r :=
{ r_le := le_trans (le_min_iff.2 ⟨hf.r_le, hg.r_le⟩) (pf.min_radius_le_radius_add pg)
r_pos := hf.r_pos
hasSum := fun hy => (hf.hasSum hy).add (hg.hasSum hy) }
#align has_fpower_series_on_ball.add HasFPowerSeriesOnBall.add
theorem HasFPowerSeriesAt.add (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x) :
HasFPowerSeriesAt (f + g) (pf + pg) x := by
rcases (hf.eventually.and hg.eventually).exists with ⟨r, hr⟩
exact ⟨r, hr.1.add hr.2⟩
#align has_fpower_series_at.add HasFPowerSeriesAt.add
theorem AnalyticAt.congr (hf : AnalyticAt 𝕜 f x) (hg : f =ᶠ[𝓝 x] g) : AnalyticAt 𝕜 g x :=
let ⟨_, hpf⟩ := hf
(hpf.congr hg).analyticAt
theorem analyticAt_congr (h : f =ᶠ[𝓝 x] g) : AnalyticAt 𝕜 f x ↔ AnalyticAt 𝕜 g x :=
⟨fun hf ↦ hf.congr h, fun hg ↦ hg.congr h.symm⟩
theorem AnalyticAt.add (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) : AnalyticAt 𝕜 (f + g) x :=
let ⟨_, hpf⟩ := hf
let ⟨_, hqf⟩ := hg
(hpf.add hqf).analyticAt
#align analytic_at.add AnalyticAt.add
theorem HasFPowerSeriesOnBall.neg (hf : HasFPowerSeriesOnBall f pf x r) :
HasFPowerSeriesOnBall (-f) (-pf) x r :=
{ r_le := by
rw [pf.radius_neg]
exact hf.r_le
r_pos := hf.r_pos
hasSum := fun hy => (hf.hasSum hy).neg }
#align has_fpower_series_on_ball.neg HasFPowerSeriesOnBall.neg
theorem HasFPowerSeriesAt.neg (hf : HasFPowerSeriesAt f pf x) : HasFPowerSeriesAt (-f) (-pf) x :=
let ⟨_, hrf⟩ := hf
hrf.neg.hasFPowerSeriesAt
#align has_fpower_series_at.neg HasFPowerSeriesAt.neg
theorem AnalyticAt.neg (hf : AnalyticAt 𝕜 f x) : AnalyticAt 𝕜 (-f) x :=
let ⟨_, hpf⟩ := hf
hpf.neg.analyticAt
#align analytic_at.neg AnalyticAt.neg
theorem HasFPowerSeriesOnBall.sub (hf : HasFPowerSeriesOnBall f pf x r)
(hg : HasFPowerSeriesOnBall g pg x r) : HasFPowerSeriesOnBall (f - g) (pf - pg) x r := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align has_fpower_series_on_ball.sub HasFPowerSeriesOnBall.sub
theorem HasFPowerSeriesAt.sub (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x) :
HasFPowerSeriesAt (f - g) (pf - pg) x := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align has_fpower_series_at.sub HasFPowerSeriesAt.sub
theorem AnalyticAt.sub (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) :
AnalyticAt 𝕜 (f - g) x := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align analytic_at.sub AnalyticAt.sub
theorem AnalyticOn.mono {s t : Set E} (hf : AnalyticOn 𝕜 f t) (hst : s ⊆ t) : AnalyticOn 𝕜 f s :=
fun z hz => hf z (hst hz)
#align analytic_on.mono AnalyticOn.mono
theorem AnalyticOn.congr' {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : f =ᶠ[𝓝ˢ s] g) :
AnalyticOn 𝕜 g s :=
fun z hz => (hf z hz).congr (mem_nhdsSet_iff_forall.mp hg z hz)
theorem analyticOn_congr' {s : Set E} (h : f =ᶠ[𝓝ˢ s] g) : AnalyticOn 𝕜 f s ↔ AnalyticOn 𝕜 g s :=
⟨fun hf => hf.congr' h, fun hg => hg.congr' h.symm⟩
theorem AnalyticOn.congr {s : Set E} (hs : IsOpen s) (hf : AnalyticOn 𝕜 f s) (hg : s.EqOn f g) :
AnalyticOn 𝕜 g s :=
hf.congr' $ mem_nhdsSet_iff_forall.mpr
(fun _ hz => eventuallyEq_iff_exists_mem.mpr ⟨s, hs.mem_nhds hz, hg⟩)
theorem analyticOn_congr {s : Set E} (hs : IsOpen s) (h : s.EqOn f g) : AnalyticOn 𝕜 f s ↔
AnalyticOn 𝕜 g s := ⟨fun hf => hf.congr hs h, fun hg => hg.congr hs h.symm⟩
theorem AnalyticOn.add {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
AnalyticOn 𝕜 (f + g) s :=
fun z hz => (hf z hz).add (hg z hz)
#align analytic_on.add AnalyticOn.add
theorem AnalyticOn.sub {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
AnalyticOn 𝕜 (f - g) s :=
fun z hz => (hf z hz).sub (hg z hz)
#align analytic_on.sub AnalyticOn.sub
theorem HasFPowerSeriesOnBall.coeff_zero (hf : HasFPowerSeriesOnBall f pf x r) (v : Fin 0 → E) :
pf 0 v = f x := by
have v_eq : v = fun i => 0 := Subsingleton.elim _ _
have zero_mem : (0 : E) ∈ EMetric.ball (0 : E) r := by simp [hf.r_pos]
have : ∀ i, i ≠ 0 → (pf i fun j => 0) = 0 := by
intro i hi
have : 0 < i := pos_iff_ne_zero.2 hi
exact ContinuousMultilinearMap.map_coord_zero _ (⟨0, this⟩ : Fin i) rfl
have A := (hf.hasSum zero_mem).unique (hasSum_single _ this)
simpa [v_eq] using A.symm
#align has_fpower_series_on_ball.coeff_zero HasFPowerSeriesOnBall.coeff_zero
theorem HasFPowerSeriesAt.coeff_zero (hf : HasFPowerSeriesAt f pf x) (v : Fin 0 → E) :
pf 0 v = f x :=
let ⟨_, hrf⟩ := hf
hrf.coeff_zero v
#align has_fpower_series_at.coeff_zero HasFPowerSeriesAt.coeff_zero
/-- If a function `f` has a power series `p` on a ball and `g` is linear, then `g ∘ f` has the
power series `g ∘ p` on the same ball. -/
theorem ContinuousLinearMap.comp_hasFPowerSeriesOnBall (g : F →L[𝕜] G)
(h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (g ∘ f) (g.compFormalMultilinearSeries p) x r :=
{ r_le := h.r_le.trans (p.radius_le_radius_continuousLinearMap_comp _)
r_pos := h.r_pos
hasSum := fun hy => by
simpa only [ContinuousLinearMap.compFormalMultilinearSeries_apply,
ContinuousLinearMap.compContinuousMultilinearMap_coe, Function.comp_apply] using
g.hasSum (h.hasSum hy) }
#align continuous_linear_map.comp_has_fpower_series_on_ball ContinuousLinearMap.comp_hasFPowerSeriesOnBall
/-- If a function `f` is analytic on a set `s` and `g` is linear, then `g ∘ f` is analytic
on `s`. -/
theorem ContinuousLinearMap.comp_analyticOn {s : Set E} (g : F →L[𝕜] G) (h : AnalyticOn 𝕜 f s) :
AnalyticOn 𝕜 (g ∘ f) s := by
rintro x hx
rcases h x hx with ⟨p, r, hp⟩
exact ⟨g.compFormalMultilinearSeries p, r, g.comp_hasFPowerSeriesOnBall hp⟩
#align continuous_linear_map.comp_analytic_on ContinuousLinearMap.comp_analyticOn
/-- If a function admits a power series expansion, then it is exponentially close to the partial
sums of this power series on strict subdisks of the disk of convergence.
This version provides an upper estimate that decreases both in `‖y‖` and `n`. See also
`HasFPowerSeriesOnBall.uniform_geometric_approx` for a weaker version. -/
theorem HasFPowerSeriesOnBall.uniform_geometric_approx' {r' : ℝ≥0}
(hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n := by
obtain ⟨a, ha, C, hC, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n, ‖p n‖ * (r' : ℝ) ^ n ≤ C * a ^ n :=
p.norm_mul_pow_le_mul_pow_of_lt_radius (h.trans_le hf.r_le)
refine' ⟨a, ha, C / (1 - a), div_pos hC (sub_pos.2 ha.2), fun y hy n => _⟩
have yr' : ‖y‖ < r' := by
rw [ball_zero_eq] at hy
exact hy
have hr'0 : 0 < (r' : ℝ) := (norm_nonneg _).trans_lt yr'
have : y ∈ EMetric.ball (0 : E) r := by
refine' mem_emetric_ball_zero_iff.2 (lt_trans _ h)
exact mod_cast yr'
rw [norm_sub_rev, ← mul_div_right_comm]
have ya : a * (‖y‖ / ↑r') ≤ a :=
mul_le_of_le_one_right ha.1.le (div_le_one_of_le yr'.le r'.coe_nonneg)
suffices ‖p.partialSum n y - f (x + y)‖ ≤ C * (a * (‖y‖ / r')) ^ n / (1 - a * (‖y‖ / r')) by
refine' this.trans _
have : 0 < a := ha.1
gcongr
apply_rules [sub_pos.2, ha.2]
apply norm_sub_le_of_geometric_bound_of_hasSum (ya.trans_lt ha.2) _ (hf.hasSum this)
intro n
calc
‖(p n) fun _ : Fin n => y‖
_ ≤ ‖p n‖ * ∏ _i : Fin n, ‖y‖ := ContinuousMultilinearMap.le_op_norm _ _
_ = ‖p n‖ * (r' : ℝ) ^ n * (‖y‖ / r') ^ n := by field_simp [mul_right_comm]
_ ≤ C * a ^ n * (‖y‖ / r') ^ n := by gcongr ?_ * _; apply hp
_ ≤ C * (a * (‖y‖ / r')) ^ n := by rw [mul_pow, mul_assoc]
#align has_fpower_series_on_ball.uniform_geometric_approx' HasFPowerSeriesOnBall.uniform_geometric_approx'
/-- If a function admits a power series expansion, then it is exponentially close to the partial
sums of this power series on strict subdisks of the disk of convergence. -/
theorem HasFPowerSeriesOnBall.uniform_geometric_approx {r' : ℝ≥0}
(hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
∃ a ∈ Ioo (0 : ℝ) 1,
∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n, ‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n := by
obtain ⟨a, ha, C, hC, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n :=
hf.uniform_geometric_approx' h
refine' ⟨a, ha, C, hC, fun y hy n => (hp y hy n).trans _⟩
have yr' : ‖y‖ < r' := by rwa [ball_zero_eq] at hy
gcongr
exacts [mul_nonneg ha.1.le (div_nonneg (norm_nonneg y) r'.coe_nonneg),
mul_le_of_le_one_right ha.1.le (div_le_one_of_le yr'.le r'.coe_nonneg)]
#align has_fpower_series_on_ball.uniform_geometric_approx HasFPowerSeriesOnBall.uniform_geometric_approx
/-- Taylor formula for an analytic function, `IsBigO` version. -/
theorem HasFPowerSeriesAt.isBigO_sub_partialSum_pow (hf : HasFPowerSeriesAt f p x) (n : ℕ) :
(fun y : E => f (x + y) - p.partialSum n y) =O[𝓝 0] fun y => ‖y‖ ^ n := by
rcases hf with ⟨r, hf⟩
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hf.r_pos with ⟨r', r'0, h⟩
obtain ⟨a, -, C, -, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n :=
hf.uniform_geometric_approx' h
refine' isBigO_iff.2 ⟨C * (a / r') ^ n, _⟩
replace r'0 : 0 < (r' : ℝ); · exact mod_cast r'0
filter_upwards [Metric.ball_mem_nhds (0 : E) r'0] with y hy
simpa [mul_pow, mul_div_assoc, mul_assoc, div_mul_eq_mul_div] using hp y hy n
set_option linter.uppercaseLean3 false in
#align has_fpower_series_at.is_O_sub_partial_sum_pow HasFPowerSeriesAt.isBigO_sub_partialSum_pow
/-- If `f` has formal power series `∑ n, pₙ` on a ball of radius `r`, then for `y, z` in any smaller
ball, the norm of the difference `f y - f z - p 1 (fun _ ↦ y - z)` is bounded above by
`C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖`. This lemma formulates this property using `IsBigO` and
`Filter.principal` on `E × E`. -/
theorem HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal
(hf : HasFPowerSeriesOnBall f p x r) (hr : r' < r) :
(fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) =O[𝓟 (EMetric.ball (x, x) r')]
fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖ := by
lift r' to ℝ≥0 using ne_top_of_lt hr
rcases (zero_le r').eq_or_lt with (rfl | hr'0)
· simp only [isBigO_bot, EMetric.ball_zero, principal_empty, ENNReal.coe_zero]
obtain ⟨a, ha, C, hC : 0 < C, hp⟩ :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n : ℕ, ‖p n‖ * (r' : ℝ) ^ n ≤ C * a ^ n
exact p.norm_mul_pow_le_mul_pow_of_lt_radius (hr.trans_le hf.r_le)
simp only [← le_div_iff (pow_pos (NNReal.coe_pos.2 hr'0) _)] at hp
set L : E × E → ℝ := fun y =>
C * (a / r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * (a / (1 - a) ^ 2 + 2 / (1 - a))
have hL : ∀ y ∈ EMetric.ball (x, x) r', ‖f y.1 - f y.2 - p 1 fun _ => y.1 - y.2‖ ≤ L y := by
intro y hy'
have hy : y ∈ EMetric.ball x r ×ˢ EMetric.ball x r := by
rw [EMetric.ball_prod_same]
exact EMetric.ball_subset_ball hr.le hy'
set A : ℕ → F := fun n => (p n fun _ => y.1 - x) - p n fun _ => y.2 - x
have hA : HasSum (fun n => A (n + 2)) (f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) := by
convert (hasSum_nat_add_iff' 2).2 ((hf.hasSum_sub hy.1).sub (hf.hasSum_sub hy.2)) using 1
rw [Finset.sum_range_succ, Finset.sum_range_one, hf.coeff_zero, hf.coeff_zero, sub_self,
zero_add, ← Subsingleton.pi_single_eq (0 : Fin 1) (y.1 - x), Pi.single,
← Subsingleton.pi_single_eq (0 : Fin 1) (y.2 - x), Pi.single, ← (p 1).map_sub, ← Pi.single,
Subsingleton.pi_single_eq, sub_sub_sub_cancel_right]
rw [EMetric.mem_ball, edist_eq_coe_nnnorm_sub, ENNReal.coe_lt_coe] at hy'
set B : ℕ → ℝ := fun n => C * (a / r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * ((n + 2) * a ^ n)
have hAB : ∀ n, ‖A (n + 2)‖ ≤ B n := fun n =>
calc
‖A (n + 2)‖ ≤ ‖p (n + 2)‖ * ↑(n + 2) * ‖y - (x, x)‖ ^ (n + 1) * ‖y.1 - y.2‖ := by
-- porting note: `pi_norm_const` was `pi_norm_const (_ : E)`
simpa only [Fintype.card_fin, pi_norm_const, Prod.norm_def, Pi.sub_def,
Prod.fst_sub, Prod.snd_sub, sub_sub_sub_cancel_right] using
(p <| n + 2).norm_image_sub_le (fun _ => y.1 - x) fun _ => y.2 - x
_ = ‖p (n + 2)‖ * ‖y - (x, x)‖ ^ n * (↑(n + 2) * ‖y - (x, x)‖ * ‖y.1 - y.2‖) := by
rw [pow_succ ‖y - (x, x)‖]
ring
-- porting note: the two `↑` in `↑r'` are new, without them, Lean fails to synthesize
-- instances `HDiv ℝ ℝ≥0 ?m` or `HMul ℝ ℝ≥0 ?m`
_ ≤ C * a ^ (n + 2) / ↑r' ^ (n + 2)
* ↑r' ^ n * (↑(n + 2) * ‖y - (x, x)‖ * ‖y.1 - y.2‖) := by
have : 0 < a := ha.1
gcongr
· apply hp
· apply hy'.le
_ = B n := by
-- porting note: in the original, `B` was in the `field_simp`, but now Lean does not
-- accept it. The current proof works in Lean 4, but does not in Lean 3.
field_simp [pow_succ]
simp only [mul_assoc, mul_comm, mul_left_comm]
have hBL : HasSum B (L y) := by
apply HasSum.mul_left
simp only [add_mul]
have : ‖a‖ < 1 := by simp only [Real.norm_eq_abs, abs_of_pos ha.1, ha.2]
rw [div_eq_mul_inv, div_eq_mul_inv]
exact (hasSum_coe_mul_geometric_of_norm_lt_1 this).add -- porting note: was `convert`!
((hasSum_geometric_of_norm_lt_1 this).mul_left 2)
exact hA.norm_le_of_bounded hBL hAB
suffices L =O[𝓟 (EMetric.ball (x, x) r')] fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖ by
refine' (IsBigO.of_bound 1 (eventually_principal.2 fun y hy => _)).trans this
rw [one_mul]
exact (hL y hy).trans (le_abs_self _)
simp_rw [mul_right_comm _ (_ * _)] -- porting note: there was an `L` inside the `simp_rw`.
exact (isBigO_refl _ _).const_mul_left _
set_option linter.uppercaseLean3 false in
#align has_fpower_series_on_ball.is_O_image_sub_image_sub_deriv_principal HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal
/-- If `f` has formal power series `∑ n, pₙ` on a ball of radius `r`, then for `y, z` in any smaller
ball, the norm of the difference `f y - f z - p 1 (fun _ ↦ y - z)` is bounded above by
`C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖`. -/
theorem HasFPowerSeriesOnBall.image_sub_sub_deriv_le (hf : HasFPowerSeriesOnBall f p x r)
(hr : r' < r) :
∃ C, ∀ᵉ (y ∈ EMetric.ball x r') (z ∈ EMetric.ball x r'),
‖f y - f z - p 1 fun _ => y - z‖ ≤ C * max ‖y - x‖ ‖z - x‖ * ‖y - z‖ := by
simpa only [isBigO_principal, mul_assoc, norm_mul, norm_norm, Prod.forall, EMetric.mem_ball,
Prod.edist_eq, max_lt_iff, and_imp, @forall_swap (_ < _) E] using
hf.isBigO_image_sub_image_sub_deriv_principal hr
#align has_fpower_series_on_ball.image_sub_sub_deriv_le HasFPowerSeriesOnBall.image_sub_sub_deriv_le
/-- If `f` has formal power series `∑ n, pₙ` at `x`, then
`f y - f z - p 1 (fun _ ↦ y - z) = O(‖(y, z) - (x, x)‖ * ‖y - z‖)` as `(y, z) → (x, x)`.
In particular, `f` is strictly differentiable at `x`. -/
theorem HasFPowerSeriesAt.isBigO_image_sub_norm_mul_norm_sub (hf : HasFPowerSeriesAt f p x) :
(fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) =O[𝓝 (x, x)] fun y =>
‖y - (x, x)‖ * ‖y.1 - y.2‖ := by
rcases hf with ⟨r, hf⟩
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hf.r_pos with ⟨r', r'0, h⟩
refine' (hf.isBigO_image_sub_image_sub_deriv_principal h).mono _
exact le_principal_iff.2 (EMetric.ball_mem_nhds _ r'0)
set_option linter.uppercaseLean3 false in
#align has_fpower_series_at.is_O_image_sub_norm_mul_norm_sub HasFPowerSeriesAt.isBigO_image_sub_norm_mul_norm_sub
/-- If a function admits a power series expansion at `x`, then it is the uniform limit of the
partial sums of this power series on strict subdisks of the disk of convergence, i.e., `f (x + y)`
is the uniform limit of `p.partialSum n y` there. -/
theorem HasFPowerSeriesOnBall.tendstoUniformlyOn {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r)
(h : (r' : ℝ≥0∞) < r) :
TendstoUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
(Metric.ball (0 : E) r') := by
obtain ⟨a, ha, C, -, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n
exact hf.uniform_geometric_approx h
refine' Metric.tendstoUniformlyOn_iff.2 fun ε εpos => _
have L : Tendsto (fun n => (C : ℝ) * a ^ n) atTop (𝓝 ((C : ℝ) * 0)) :=
tendsto_const_nhds.mul (tendsto_pow_atTop_nhds_0_of_lt_1 ha.1.le ha.2)
rw [mul_zero] at L
refine' (L.eventually (gt_mem_nhds εpos)).mono fun n hn y hy => _
rw [dist_eq_norm]
exact (hp y hy n).trans_lt hn
#align has_fpower_series_on_ball.tendsto_uniformly_on HasFPowerSeriesOnBall.tendstoUniformlyOn
/-- If a function admits a power series expansion at `x`, then it is the locally uniform limit of
the partial sums of this power series on the disk of convergence, i.e., `f (x + y)`
is the locally uniform limit of `p.partialSum n y` there. -/
theorem HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn (hf : HasFPowerSeriesOnBall f p x r) :
TendstoLocallyUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
(EMetric.ball (0 : E) r) := by
intro u hu x hx
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hx with ⟨r', xr', hr'⟩
have : EMetric.ball (0 : E) r' ∈ 𝓝 x := IsOpen.mem_nhds EMetric.isOpen_ball xr'
refine' ⟨EMetric.ball (0 : E) r', mem_nhdsWithin_of_mem_nhds this, _⟩
simpa [Metric.emetric_ball_nnreal] using hf.tendstoUniformlyOn hr' u hu
#align has_fpower_series_on_ball.tendsto_locally_uniformly_on HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn
/-- If a function admits a power series expansion at `x`, then it is the uniform limit of the
partial sums of this power series on strict subdisks of the disk of convergence, i.e., `f y`
is the uniform limit of `p.partialSum n (y - x)` there. -/
theorem HasFPowerSeriesOnBall.tendstoUniformlyOn' {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r)
(h : (r' : ℝ≥0∞) < r) :
TendstoUniformlyOn (fun n y => p.partialSum n (y - x)) f atTop (Metric.ball (x : E) r') := by
convert (hf.tendstoUniformlyOn h).comp fun y => y - x using 1
· simp [(· ∘ ·)]
· ext z
simp [dist_eq_norm]
#align has_fpower_series_on_ball.tendsto_uniformly_on' HasFPowerSeriesOnBall.tendstoUniformlyOn'
/-- If a function admits a power series expansion at `x`, then it is the locally uniform limit of
the partial sums of this power series on the disk of convergence, i.e., `f y`
is the locally uniform limit of `p.partialSum n (y - x)` there. -/
theorem HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn' (hf : HasFPowerSeriesOnBall f p x r) :
TendstoLocallyUniformlyOn (fun n y => p.partialSum n (y - x)) f atTop
(EMetric.ball (x : E) r) := by
have A : ContinuousOn (fun y : E => y - x) (EMetric.ball (x : E) r) :=
(continuous_id.sub continuous_const).continuousOn
convert hf.tendstoLocallyUniformlyOn.comp (fun y : E => y - x) _ A using 1
· ext z
simp
· intro z
simp [edist_eq_coe_nnnorm, edist_eq_coe_nnnorm_sub]
#align has_fpower_series_on_ball.tendsto_locally_uniformly_on' HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn'
/-- If a function admits a power series expansion on a disk, then it is continuous there. -/
protected theorem HasFPowerSeriesOnBall.continuousOn (hf : HasFPowerSeriesOnBall f p x r) :
ContinuousOn f (EMetric.ball x r) :=
hf.tendstoLocallyUniformlyOn'.continuousOn <|
eventually_of_forall fun n =>
((p.partialSum_continuous n).comp (continuous_id.sub continuous_const)).continuousOn
#align has_fpower_series_on_ball.continuous_on HasFPowerSeriesOnBall.continuousOn
protected theorem HasFPowerSeriesAt.continuousAt (hf : HasFPowerSeriesAt f p x) :
ContinuousAt f x :=
let ⟨_, hr⟩ := hf
hr.continuousOn.continuousAt (EMetric.ball_mem_nhds x hr.r_pos)
#align has_fpower_series_at.continuous_at HasFPowerSeriesAt.continuousAt
protected theorem AnalyticAt.continuousAt (hf : AnalyticAt 𝕜 f x) : ContinuousAt f x :=
let ⟨_, hp⟩ := hf
hp.continuousAt
#align analytic_at.continuous_at AnalyticAt.continuousAt
protected theorem AnalyticOn.continuousOn {s : Set E} (hf : AnalyticOn 𝕜 f s) : ContinuousOn f s :=
fun x hx => (hf x hx).continuousAt.continuousWithinAt
#align analytic_on.continuous_on AnalyticOn.continuousOn
/-- Analytic everywhere implies continuous -/
theorem AnalyticOn.continuous {f : E → F} (fa : AnalyticOn 𝕜 f univ) : Continuous f := by
rw [continuous_iff_continuousOn_univ]; exact fa.continuousOn
/-- In a complete space, the sum of a converging power series `p` admits `p` as a power series.
This is not totally obvious as we need to check the convergence of the series. -/
protected theorem FormalMultilinearSeries.hasFPowerSeriesOnBall [CompleteSpace F]
(p : FormalMultilinearSeries 𝕜 E F) (h : 0 < p.radius) :
HasFPowerSeriesOnBall p.sum p 0 p.radius :=
{ r_le := le_rfl
r_pos := h
hasSum := fun hy => by
rw [zero_add]
exact p.hasSum hy }
#align formal_multilinear_series.has_fpower_series_on_ball FormalMultilinearSeries.hasFPowerSeriesOnBall
theorem HasFPowerSeriesOnBall.sum (h : HasFPowerSeriesOnBall f p x r) {y : E}
(hy : y ∈ EMetric.ball (0 : E) r) : f (x + y) = p.sum y :=
(h.hasSum hy).tsum_eq.symm
#align has_fpower_series_on_ball.sum HasFPowerSeriesOnBall.sum
/-- The sum of a converging power series is continuous in its disk of convergence. -/
protected theorem FormalMultilinearSeries.continuousOn [CompleteSpace F] :
ContinuousOn p.sum (EMetric.ball 0 p.radius) := by
rcases (zero_le p.radius).eq_or_lt with h | h
· simp [← h, continuousOn_empty]
· exact (p.hasFPowerSeriesOnBall h).continuousOn
#align formal_multilinear_series.continuous_on FormalMultilinearSeries.continuousOn
end
/-!
### Uniqueness of power series
If a function `f : E → F` has two representations as power series at a point `x : E`, corresponding
to formal multilinear series `p₁` and `p₂`, then these representations agree term-by-term. That is,
for any `n : ℕ` and `y : E`, `p₁ n (fun i ↦ y) = p₂ n (fun i ↦ y)`. In the one-dimensional case,
when `f : 𝕜 → E`, the continuous multilinear maps `p₁ n` and `p₂ n` are given by
`ContinuousMultilinearMap.mkPiField`, and hence are determined completely by the value of
`p₁ n (fun i ↦ 1)`, so `p₁ = p₂`. Consequently, the radius of convergence for one series can be
transferred to the other.
-/
section Uniqueness
open ContinuousMultilinearMap
theorem Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero {n : ℕ} {p : E[×n]→L[𝕜] F}
(h : (fun y => p fun _ => y) =O[𝓝 0] fun y => ‖y‖ ^ (n + 1)) (y : E) : (p fun _ => y) = 0 := by
obtain ⟨c, c_pos, hc⟩ := h.exists_pos
obtain ⟨t, ht, t_open, z_mem⟩ := eventually_nhds_iff.mp (isBigOWith_iff.mp hc)
obtain ⟨δ, δ_pos, δε⟩ := (Metric.isOpen_iff.mp t_open) 0 z_mem
clear h hc z_mem
cases' n with n
· exact norm_eq_zero.mp (by
-- porting note: the symmetric difference of the `simpa only` sets:
-- added `Nat.zero_eq, zero_add, pow_one`
-- removed `zero_pow', Ne.def, Nat.one_ne_zero, not_false_iff`
simpa only [Nat.zero_eq, fin0_apply_norm, norm_eq_zero, norm_zero, zero_add, pow_one,
mul_zero, norm_le_zero_iff] using ht 0 (δε (Metric.mem_ball_self δ_pos)))
· refine' Or.elim (Classical.em (y = 0))
(fun hy => by simpa only [hy] using p.map_zero) fun hy => _
replace hy := norm_pos_iff.mpr hy
refine' norm_eq_zero.mp (le_antisymm (le_of_forall_pos_le_add fun ε ε_pos => _) (norm_nonneg _))
have h₀ := _root_.mul_pos c_pos (pow_pos hy (n.succ + 1))
obtain ⟨k, k_pos, k_norm⟩ := NormedField.exists_norm_lt 𝕜
(lt_min (mul_pos δ_pos (inv_pos.mpr hy)) (mul_pos ε_pos (inv_pos.mpr h₀)))
have h₁ : ‖k • y‖ < δ := by
rw [norm_smul]
exact inv_mul_cancel_right₀ hy.ne.symm δ ▸
mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_left _ _)) hy
have h₂ :=
calc
‖p fun _ => k • y‖ ≤ c * ‖k • y‖ ^ (n.succ + 1) := by
-- porting note: now Lean wants `_root_.`
simpa only [norm_pow, _root_.norm_norm] using ht (k • y) (δε (mem_ball_zero_iff.mpr h₁))
--simpa only [norm_pow, norm_norm] using ht (k • y) (δε (mem_ball_zero_iff.mpr h₁))
_ = ‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))) := by
-- porting note: added `Nat.succ_eq_add_one` since otherwise `ring` does not conclude.
simp only [norm_smul, mul_pow, Nat.succ_eq_add_one]
-- porting note: removed `rw [pow_succ]`, since it now becomes superfluous.
ring
have h₃ : ‖k‖ * (c * ‖y‖ ^ (n.succ + 1)) < ε :=
inv_mul_cancel_right₀ h₀.ne.symm ε ▸
mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_right _ _)) h₀
calc
‖p fun _ => y‖ = ‖k⁻¹ ^ n.succ‖ * ‖p fun _ => k • y‖ := by
simpa only [inv_smul_smul₀ (norm_pos_iff.mp k_pos), norm_smul, Finset.prod_const,
Finset.card_fin] using
congr_arg norm (p.map_smul_univ (fun _ : Fin n.succ => k⁻¹) fun _ : Fin n.succ => k • y)
_ ≤ ‖k⁻¹ ^ n.succ‖ * (‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1)))) := by gcongr
_ = ‖(k⁻¹ * k) ^ n.succ‖ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))) := by
rw [← mul_assoc]
simp [norm_mul, mul_pow]
_ ≤ 0 + ε := by
rw [inv_mul_cancel (norm_pos_iff.mp k_pos)]
simpa using h₃.le
set_option linter.uppercaseLean3 false in
#align asymptotics.is_O.continuous_multilinear_map_apply_eq_zero Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero
/-- If a formal multilinear series `p` represents the zero function at `x : E`, then the
terms `p n (fun i ↦ y)` appearing in the sum are zero for any `n : ℕ`, `y : E`. -/
theorem HasFPowerSeriesAt.apply_eq_zero {p : FormalMultilinearSeries 𝕜 E F} {x : E}
(h : HasFPowerSeriesAt 0 p x) (n : ℕ) : ∀ y : E, (p n fun _ => y) = 0 := by
refine' Nat.strong_induction_on n fun k hk => _
have psum_eq : p.partialSum (k + 1) = fun y => p k fun _ => y := by
funext z
refine' Finset.sum_eq_single _ (fun b hb hnb => _) fun hn => _
· have := Finset.mem_range_succ_iff.mp hb
simp only [hk b (this.lt_of_ne hnb), Pi.zero_apply]
· exact False.elim (hn (Finset.mem_range.mpr (lt_add_one k)))
replace h := h.isBigO_sub_partialSum_pow k.succ
simp only [psum_eq, zero_sub, Pi.zero_apply, Asymptotics.isBigO_neg_left] at h
exact h.continuousMultilinearMap_apply_eq_zero
#align has_fpower_series_at.apply_eq_zero HasFPowerSeriesAt.apply_eq_zero
/-- A one-dimensional formal multilinear series representing the zero function is zero. -/
theorem HasFPowerSeriesAt.eq_zero {p : FormalMultilinearSeries 𝕜 𝕜 E} {x : 𝕜}
(h : HasFPowerSeriesAt 0 p x) : p = 0 := by
-- porting note: `funext; ext` was `ext (n x)`
funext n
ext x
rw [← mkPiField_apply_one_eq_self (p n)]
-- porting note: nasty hack, was `simp [h.apply_eq_zero n 1]`
have := Or.intro_right ?_ (h.apply_eq_zero n 1)
simpa using this
#align has_fpower_series_at.eq_zero HasFPowerSeriesAt.eq_zero
/-- One-dimensional formal multilinear series representing the same function are equal. -/
theorem HasFPowerSeriesAt.eq_formalMultilinearSeries {p₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E}
{f : 𝕜 → E} {x : 𝕜} (h₁ : HasFPowerSeriesAt f p₁ x) (h₂ : HasFPowerSeriesAt f p₂ x) : p₁ = p₂ :=
sub_eq_zero.mp (HasFPowerSeriesAt.eq_zero (by simpa only [sub_self] using h₁.sub h₂))
#align has_fpower_series_at.eq_formal_multilinear_series HasFPowerSeriesAt.eq_formalMultilinearSeries
theorem HasFPowerSeriesAt.eq_formalMultilinearSeries_of_eventually
{p q : FormalMultilinearSeries 𝕜 𝕜 E} {f g : 𝕜 → E} {x : 𝕜} (hp : HasFPowerSeriesAt f p x)
(hq : HasFPowerSeriesAt g q x) (heq : ∀ᶠ z in 𝓝 x, f z = g z) : p = q :=
(hp.congr heq).eq_formalMultilinearSeries hq
#align has_fpower_series_at.eq_formal_multilinear_series_of_eventually HasFPowerSeriesAt.eq_formalMultilinearSeries_of_eventually
/-- A one-dimensional formal multilinear series representing a locally zero function is zero. -/
theorem HasFPowerSeriesAt.eq_zero_of_eventually {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E}
{x : 𝕜} (hp : HasFPowerSeriesAt f p x) (hf : f =ᶠ[𝓝 x] 0) : p = 0 :=
(hp.congr hf).eq_zero
#align has_fpower_series_at.eq_zero_of_eventually HasFPowerSeriesAt.eq_zero_of_eventually
/-- If a function `f : 𝕜 → E` has two power series representations at `x`, then the given radii in
which convergence is guaranteed may be interchanged. This can be useful when the formal multilinear
series in one representation has a particularly nice form, but the other has a larger radius. -/
theorem HasFPowerSeriesOnBall.exchange_radius {p₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E}
{r₁ r₂ : ℝ≥0∞} {x : 𝕜} (h₁ : HasFPowerSeriesOnBall f p₁ x r₁)
(h₂ : HasFPowerSeriesOnBall f p₂ x r₂) : HasFPowerSeriesOnBall f p₁ x r₂ :=
h₂.hasFPowerSeriesAt.eq_formalMultilinearSeries h₁.hasFPowerSeriesAt ▸ h₂
#align has_fpower_series_on_ball.exchange_radius HasFPowerSeriesOnBall.exchange_radius
/-- If a function `f : 𝕜 → E` has power series representation `p` on a ball of some radius and for
each positive radius it has some power series representation, then `p` converges to `f` on the whole
`𝕜`. -/
theorem HasFPowerSeriesOnBall.r_eq_top_of_exists {f : 𝕜 → E} {r : ℝ≥0∞} {x : 𝕜}
{p : FormalMultilinearSeries 𝕜 𝕜 E} (h : HasFPowerSeriesOnBall f p x r)
(h' : ∀ (r' : ℝ≥0) (_ : 0 < r'), ∃ p' : FormalMultilinearSeries 𝕜 𝕜 E,
HasFPowerSeriesOnBall f p' x r') :
HasFPowerSeriesOnBall f p x ∞ :=
{ r_le := ENNReal.le_of_forall_pos_nnreal_lt fun r hr _ =>
let ⟨_, hp'⟩ := h' r hr
(h.exchange_radius hp').r_le
r_pos := ENNReal.coe_lt_top
hasSum := fun {y} _ =>
let ⟨r', hr'⟩ := exists_gt ‖y‖₊
let ⟨_, hp'⟩ := h' r' hr'.ne_bot.bot_lt
(h.exchange_radius hp').hasSum <| mem_emetric_ball_zero_iff.mpr (ENNReal.coe_lt_coe.2 hr') }
#align has_fpower_series_on_ball.r_eq_top_of_exists HasFPowerSeriesOnBall.r_eq_top_of_exists
end Uniqueness
/-!
### Changing origin in a power series
If a function is analytic in a disk `D(x, R)`, then it is analytic in any disk contained in that
one. Indeed, one can write
$$
f (x + y + z) = \sum_{n} p_n (y + z)^n = \sum_{n, k} \binom{n}{k} p_n y^{n-k} z^k
= \sum_{k} \Bigl(\sum_{n} \binom{n}{k} p_n y^{n-k}\Bigr) z^k.
$$
The corresponding power series has thus a `k`-th coefficient equal to
$\sum_{n} \binom{n}{k} p_n y^{n-k}$. In the general case where `pₙ` is a multilinear map, this has
to be interpreted suitably: instead of having a binomial coefficient, one should sum over all
possible subsets `s` of `Fin n` of cardinal `k`, and attribute `z` to the indices in `s` and
`y` to the indices outside of `s`.
In this paragraph, we implement this. The new power series is called `p.changeOrigin y`. Then, we
check its convergence and the fact that its sum coincides with the original sum. The outcome of this
discussion is that the set of points where a function is analytic is open.
-/
namespace FormalMultilinearSeries
section
variable (p : FormalMultilinearSeries 𝕜 E F) {x y : E} {r R : ℝ≥0}
/-- A term of `FormalMultilinearSeries.changeOriginSeries`.
Given a formal multilinear series `p` and a point `x` in its ball of convergence,
`p.changeOrigin x` is a formal multilinear series such that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense. Each term of `p.changeOrigin x`
is itself an analytic function of `x` given by the series `p.changeOriginSeries`. Each term in
`changeOriginSeries` is the sum of `changeOriginSeriesTerm`'s over all `s` of cardinality `l`.
The definition is such that `p.changeOriginSeriesTerm k l s hs (fun _ ↦ x) (fun _ ↦ y) =
p (k + l) (s.piecewise (fun _ ↦ x) (fun _ ↦ y))`
-/
def changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
E[×l]→L[𝕜] E[×k]→L[𝕜] F := by
let a := ContinuousMultilinearMap.curryFinFinset 𝕜 E F hs
(by erw [Finset.card_compl, Fintype.card_fin, hs, add_tsub_cancel_right])
exact a (p (k + l))
#align formal_multilinear_series.change_origin_series_term FormalMultilinearSeries.changeOriginSeriesTerm
theorem changeOriginSeriesTerm_apply (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l)
(x y : E) :
(p.changeOriginSeriesTerm k l s hs (fun _ => x) fun _ => y) =
p (k + l) (s.piecewise (fun _ => x) fun _ => y) :=
ContinuousMultilinearMap.curryFinFinset_apply_const _ _ _ _ _
#align formal_multilinear_series.change_origin_series_term_apply FormalMultilinearSeries.changeOriginSeriesTerm_apply
@[simp]
theorem norm_changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
‖p.changeOriginSeriesTerm k l s hs‖ = ‖p (k + l)‖ := by
simp only [changeOriginSeriesTerm, LinearIsometryEquiv.norm_map]
#align formal_multilinear_series.norm_change_origin_series_term FormalMultilinearSeries.norm_changeOriginSeriesTerm
@[simp]
theorem nnnorm_changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
‖p.changeOriginSeriesTerm k l s hs‖₊ = ‖p (k + l)‖₊ := by
simp only [changeOriginSeriesTerm, LinearIsometryEquiv.nnnorm_map]
#align formal_multilinear_series.nnnorm_change_origin_series_term FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm
theorem nnnorm_changeOriginSeriesTerm_apply_le (k l : ℕ) (s : Finset (Fin (k + l)))
(hs : s.card = l) (x y : E) :
‖p.changeOriginSeriesTerm k l s hs (fun _ => x) fun _ => y‖₊ ≤
‖p (k + l)‖₊ * ‖x‖₊ ^ l * ‖y‖₊ ^ k := by
rw [← p.nnnorm_changeOriginSeriesTerm k l s hs, ← Fin.prod_const, ← Fin.prod_const]
apply ContinuousMultilinearMap.le_of_op_nnnorm_le
apply ContinuousMultilinearMap.le_op_nnnorm
#align formal_multilinear_series.nnnorm_change_origin_series_term_apply_le FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm_apply_le
/-- The power series for `f.changeOrigin k`.
Given a formal multilinear series `p` and a point `x` in its ball of convergence,
`p.changeOrigin x` is a formal multilinear series such that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense. Its `k`-th term is the sum of
the series `p.changeOriginSeries k`. -/
def changeOriginSeries (k : ℕ) : FormalMultilinearSeries 𝕜 E (E[×k]→L[𝕜] F) := fun l =>
∑ s : { s : Finset (Fin (k + l)) // Finset.card s = l }, p.changeOriginSeriesTerm k l s s.2
#align formal_multilinear_series.change_origin_series FormalMultilinearSeries.changeOriginSeries
theorem nnnorm_changeOriginSeries_le_tsum (k l : ℕ) :
‖p.changeOriginSeries k l‖₊ ≤
∑' _ : { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + l)‖₊ :=
(nnnorm_sum_le _ (fun t => changeOriginSeriesTerm p k l (Subtype.val t) t.prop)).trans_eq <| by
simp_rw [tsum_fintype, nnnorm_changeOriginSeriesTerm (p := p) (k := k) (l := l)]
#align formal_multilinear_series.nnnorm_change_origin_series_le_tsum FormalMultilinearSeries.nnnorm_changeOriginSeries_le_tsum
theorem nnnorm_changeOriginSeries_apply_le_tsum (k l : ℕ) (x : E) :
‖p.changeOriginSeries k l fun _ => x‖₊ ≤
∑' _ : { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + l)‖₊ * ‖x‖₊ ^ l := by
rw [NNReal.tsum_mul_right, ← Fin.prod_const]
exact (p.changeOriginSeries k l).le_of_op_nnnorm_le _ (p.nnnorm_changeOriginSeries_le_tsum _ _)
#align formal_multilinear_series.nnnorm_change_origin_series_apply_le_tsum FormalMultilinearSeries.nnnorm_changeOriginSeries_apply_le_tsum
/-- Changing the origin of a formal multilinear series `p`, so that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense.
-/
def changeOrigin (x : E) : FormalMultilinearSeries 𝕜 E F :=
fun k => (p.changeOriginSeries k).sum x
#align formal_multilinear_series.change_origin FormalMultilinearSeries.changeOrigin
/-- An auxiliary equivalence useful in the proofs about
`FormalMultilinearSeries.changeOriginSeries`: the set of triples `(k, l, s)`, where `s` is a
`Finset (Fin (k + l))` of cardinality `l` is equivalent to the set of pairs `(n, s)`, where `s` is a
`Finset (Fin n)`.
The forward map sends `(k, l, s)` to `(k + l, s)` and the inverse map sends `(n, s)` to
`(n - Finset.card s, Finset.card s, s)`. The actual definition is less readable because of problems
with non-definitional equalities. -/
@[simps]
def changeOriginIndexEquiv :
(Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) ≃ Σn : ℕ, Finset (Fin n) where
toFun s := ⟨s.1 + s.2.1, s.2.2⟩
invFun s :=
⟨s.1 - s.2.card, s.2.card,
⟨s.2.map
(Fin.castIso <| (tsub_add_cancel_of_le <| card_finset_fin_le s.2).symm).toEquiv.toEmbedding,
Finset.card_map _⟩⟩
left_inv := by
rintro ⟨k, l, ⟨s : Finset (Fin <| k + l), hs : s.card = l⟩⟩
dsimp only [Subtype.coe_mk]
-- Lean can't automatically generalize `k' = k + l - s.card`, `l' = s.card`, so we explicitly
-- formulate the generalized goal
suffices ∀ k' l', k' = k → l' = l → ∀ (hkl : k + l = k' + l') (hs'),
(⟨k', l', ⟨Finset.map (Fin.castIso hkl).toEquiv.toEmbedding s, hs'⟩⟩ :
Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) = ⟨k, l, ⟨s, hs⟩⟩ by
apply this <;> simp only [hs, add_tsub_cancel_right]
rintro _ _ rfl rfl hkl hs'
simp only [Equiv.refl_toEmbedding, Fin.castIso_refl, Finset.map_refl, eq_self_iff_true,
OrderIso.refl_toEquiv, and_self_iff, heq_iff_eq]
right_inv := by
rintro ⟨n, s⟩
simp [tsub_add_cancel_of_le (card_finset_fin_le s), Fin.castIso_to_equiv]
#align formal_multilinear_series.change_origin_index_equiv FormalMultilinearSeries.changeOriginIndexEquiv
theorem changeOriginSeries_summable_aux₁ {r r' : ℝ≥0} (hr : (r + r' : ℝ≥0∞) < p.radius) :
Summable fun s : Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l } =>
‖p (s.1 + s.2.1)‖₊ * r ^ s.2.1 * r' ^ s.1 := by
rw [← changeOriginIndexEquiv.symm.summable_iff]
dsimp only [Function.comp_def, changeOriginIndexEquiv_symm_apply_fst,
changeOriginIndexEquiv_symm_apply_snd_fst]
have : ∀ n : ℕ,
HasSum (fun s : Finset (Fin n) => ‖p (n - s.card + s.card)‖₊ * r ^ s.card * r' ^ (n - s.card))
(‖p n‖₊ * (r + r') ^ n) := by
intro n
-- TODO: why `simp only [tsub_add_cancel_of_le (card_finset_fin_le _)]` fails?
convert_to HasSum (fun s : Finset (Fin n) => ‖p n‖₊ * (r ^ s.card * r' ^ (n - s.card))) _
· ext1 s
rw [tsub_add_cancel_of_le (card_finset_fin_le _), mul_assoc]
rw [← Fin.sum_pow_mul_eq_add_pow]
exact (hasSum_fintype _).mul_left _
refine' NNReal.summable_sigma.2 ⟨fun n => (this n).summable, _⟩
simp only [(this _).tsum_eq]
exact p.summable_nnnorm_mul_pow hr
#align formal_multilinear_series.change_origin_series_summable_aux₁ FormalMultilinearSeries.changeOriginSeries_summable_aux₁
theorem changeOriginSeries_summable_aux₂ (hr : (r : ℝ≥0∞) < p.radius) (k : ℕ) :
Summable fun s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l } =>
‖p (k + s.1)‖₊ * r ^ s.1 := by
rcases ENNReal.lt_iff_exists_add_pos_lt.1 hr with ⟨r', h0, hr'⟩
simpa only [mul_inv_cancel_right₀ (pow_pos h0 _).ne'] using
((NNReal.summable_sigma.1 (p.changeOriginSeries_summable_aux₁ hr')).1 k).mul_right (r' ^ k)⁻¹
#align formal_multilinear_series.change_origin_series_summable_aux₂ FormalMultilinearSeries.changeOriginSeries_summable_aux₂
theorem changeOriginSeries_summable_aux₃ {r : ℝ≥0} (hr : ↑r < p.radius) (k : ℕ) :
Summable fun l : ℕ => ‖p.changeOriginSeries k l‖₊ * r ^ l := by
refine' NNReal.summable_of_le
(fun n => _) (NNReal.summable_sigma.1 <| p.changeOriginSeries_summable_aux₂ hr k).2
simp only [NNReal.tsum_mul_right]
exact mul_le_mul' (p.nnnorm_changeOriginSeries_le_tsum _ _) le_rfl
#align formal_multilinear_series.change_origin_series_summable_aux₃ FormalMultilinearSeries.changeOriginSeries_summable_aux₃
theorem le_changeOriginSeries_radius (k : ℕ) : p.radius ≤ (p.changeOriginSeries k).radius :=
ENNReal.le_of_forall_nnreal_lt fun _r hr =>
le_radius_of_summable_nnnorm _ (p.changeOriginSeries_summable_aux₃ hr k)
#align formal_multilinear_series.le_change_origin_series_radius FormalMultilinearSeries.le_changeOriginSeries_radius
theorem nnnorm_changeOrigin_le (k : ℕ) (h : (‖x‖₊ : ℝ≥0∞) < p.radius) :
‖p.changeOrigin x k‖₊ ≤
∑' s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + s.1)‖₊ * ‖x‖₊ ^ s.1 := by
refine' tsum_of_nnnorm_bounded _ fun l => p.nnnorm_changeOriginSeries_apply_le_tsum k l x
have := p.changeOriginSeries_summable_aux₂ h k
refine' HasSum.sigma this.hasSum fun l => _
exact ((NNReal.summable_sigma.1 this).1 l).hasSum
#align formal_multilinear_series.nnnorm_change_origin_le FormalMultilinearSeries.nnnorm_changeOrigin_le
/-- The radius of convergence of `p.changeOrigin x` is at least `p.radius - ‖x‖`. In other words,
`p.changeOrigin x` is well defined on the largest ball contained in the original ball of
convergence. -/
theorem changeOrigin_radius : p.radius - ‖x‖₊ ≤ (p.changeOrigin x).radius := by
refine' ENNReal.le_of_forall_pos_nnreal_lt fun r _h0 hr => _
rw [lt_tsub_iff_right, add_comm] at hr
have hr' : (‖x‖₊ : ℝ≥0∞) < p.radius := (le_add_right le_rfl).trans_lt hr
apply le_radius_of_summable_nnnorm
have : ∀ k : ℕ,
‖p.changeOrigin x k‖₊ * r ^ k ≤
(∑' s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + s.1)‖₊ * ‖x‖₊ ^ s.1) *
r ^ k :=
fun k => mul_le_mul_right' (p.nnnorm_changeOrigin_le k hr') (r ^ k)
refine' NNReal.summable_of_le this _
simpa only [← NNReal.tsum_mul_right] using
(NNReal.summable_sigma.1 (p.changeOriginSeries_summable_aux₁ hr)).2
#align formal_multilinear_series.change_origin_radius FormalMultilinearSeries.changeOrigin_radius
end
-- From this point on, assume that the space is complete, to make sure that series that converge
-- in norm also converge in `F`.
variable [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x y : E} {r R : ℝ≥0}
theorem hasFPowerSeriesOnBall_changeOrigin (k : ℕ) (hr : 0 < p.radius) :
HasFPowerSeriesOnBall (fun x => p.changeOrigin x k) (p.changeOriginSeries k) 0 p.radius :=
have := p.le_changeOriginSeries_radius k
((p.changeOriginSeries k).hasFPowerSeriesOnBall (hr.trans_le this)).mono hr this
#align formal_multilinear_series.has_fpower_series_on_ball_change_origin FormalMultilinearSeries.hasFPowerSeriesOnBall_changeOrigin
/-- Summing the series `p.changeOrigin x` at a point `y` gives back `p (x + y)`. -/
theorem changeOrigin_eval (h : (‖x‖₊ + ‖y‖₊ : ℝ≥0∞) < p.radius) :
(p.changeOrigin x).sum y = p.sum (x + y) := by
have radius_pos : 0 < p.radius := lt_of_le_of_lt (zero_le _) h
have x_mem_ball : x ∈ EMetric.ball (0 : E) p.radius :=
mem_emetric_ball_zero_iff.2 ((le_add_right le_rfl).trans_lt h)
have y_mem_ball : y ∈ EMetric.ball (0 : E) (p.changeOrigin x).radius := by
refine' mem_emetric_ball_zero_iff.2 (lt_of_lt_of_le _ p.changeOrigin_radius)
rwa [lt_tsub_iff_right, add_comm]
have x_add_y_mem_ball : x + y ∈ EMetric.ball (0 : E) p.radius := by
refine' mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt _ h)
exact mod_cast nnnorm_add_le x y
set f : (Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) → F := fun s =>
p.changeOriginSeriesTerm s.1 s.2.1 s.2.2 s.2.2.2 (fun _ => x) fun _ => y
have hsf : Summable f := by
refine' .of_nnnorm_bounded _ (p.changeOriginSeries_summable_aux₁ h) _
rintro ⟨k, l, s, hs⟩
dsimp only [Subtype.coe_mk]
exact p.nnnorm_changeOriginSeriesTerm_apply_le _ _ _ _ _ _
have hf : HasSum f ((p.changeOrigin x).sum y) := by
refine' HasSum.sigma_of_hasSum ((p.changeOrigin x).summable y_mem_ball).hasSum (fun k => _) hsf
· dsimp only
refine' ContinuousMultilinearMap.hasSum_eval _ _
have := (p.hasFPowerSeriesOnBall_changeOrigin k radius_pos).hasSum x_mem_ball
rw [zero_add] at this
refine' HasSum.sigma_of_hasSum this (fun l => _) _
· simp only [changeOriginSeries, ContinuousMultilinearMap.sum_apply]
apply hasSum_fintype
· refine' .of_nnnorm_bounded _
(p.changeOriginSeries_summable_aux₂ (mem_emetric_ball_zero_iff.1 x_mem_ball) k) fun s => _
refine' (ContinuousMultilinearMap.le_op_nnnorm _ _).trans_eq _
simp
refine' hf.unique (changeOriginIndexEquiv.symm.hasSum_iff.1 _)
refine' HasSum.sigma_of_hasSum
(p.hasSum x_add_y_mem_ball) (fun n => _) (changeOriginIndexEquiv.symm.summable_iff.2 hsf)
erw [(p n).map_add_univ (fun _ => x) fun _ => y]
-- porting note: added explicit function
convert hasSum_fintype (fun c : Finset (Fin n) => f (changeOriginIndexEquiv.symm ⟨n, c⟩))
rename_i s _
dsimp only [changeOriginSeriesTerm, (· ∘ ·), changeOriginIndexEquiv_symm_apply_fst,
changeOriginIndexEquiv_symm_apply_snd_fst, changeOriginIndexEquiv_symm_apply_snd_snd_coe]
rw [ContinuousMultilinearMap.curryFinFinset_apply_const]
have : ∀ (m) (hm : n = m), p n (s.piecewise (fun _ => x) fun _ => y) =
p m ((s.map (Fin.castIso hm).toEquiv.toEmbedding).piecewise (fun _ => x) fun _ => y) := by
rintro m rfl
simp (config := { unfoldPartialApp := true }) [Finset.piecewise]
apply this
#align formal_multilinear_series.change_origin_eval FormalMultilinearSeries.changeOrigin_eval
/-- Power series terms are analytic as we vary the origin -/
theorem analyticAt_changeOrigin (p : FormalMultilinearSeries 𝕜 E F) (rp : p.radius > 0) (n : ℕ) :
AnalyticAt 𝕜 (fun x ↦ p.changeOrigin x n) 0 :=
(FormalMultilinearSeries.hasFPowerSeriesOnBall_changeOrigin p n rp).analyticAt
end FormalMultilinearSeries
section
variable [CompleteSpace F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x y : E} {r : ℝ≥0∞}
/-- If a function admits a power series expansion `p` on a ball `B (x, r)`, then it also admits a
power series on any subball of this ball (even with a different center), given by `p.changeOrigin`.
-/
theorem HasFPowerSeriesOnBall.changeOrigin (hf : HasFPowerSeriesOnBall f p x r)
(h : (‖y‖₊ : ℝ≥0∞) < r) : HasFPowerSeriesOnBall f (p.changeOrigin y) (x + y) (r - ‖y‖₊) :=
{ r_le := by
apply le_trans _ p.changeOrigin_radius
exact tsub_le_tsub hf.r_le le_rfl
r_pos := by simp [h]
hasSum := fun {z} hz => by
have : f (x + y + z) =
FormalMultilinearSeries.sum (FormalMultilinearSeries.changeOrigin p y) z := by
rw [mem_emetric_ball_zero_iff, lt_tsub_iff_right, add_comm] at hz
rw [p.changeOrigin_eval (hz.trans_le hf.r_le), add_assoc, hf.sum]
refine' mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt _ hz)
exact mod_cast nnnorm_add_le y z
rw [this]
apply (p.changeOrigin y).hasSum
refine' EMetric.ball_subset_ball (le_trans _ p.changeOrigin_radius) hz
exact tsub_le_tsub hf.r_le le_rfl }
#align has_fpower_series_on_ball.change_origin HasFPowerSeriesOnBall.changeOrigin
/-- If a function admits a power series expansion `p` on an open ball `B (x, r)`, then
it is analytic at every point of this ball. -/
theorem HasFPowerSeriesOnBall.analyticAt_of_mem (hf : HasFPowerSeriesOnBall f p x r)
(h : y ∈ EMetric.ball x r) : AnalyticAt 𝕜 f y := by
have : (‖y - x‖₊ : ℝ≥0∞) < r := by simpa [edist_eq_coe_nnnorm_sub] using h
have := hf.changeOrigin this
rw [add_sub_cancel'_right] at this
exact this.analyticAt
#align has_fpower_series_on_ball.analytic_at_of_mem HasFPowerSeriesOnBall.analyticAt_of_mem
theorem HasFPowerSeriesOnBall.analyticOn (hf : HasFPowerSeriesOnBall f p x r) :
AnalyticOn 𝕜 f (EMetric.ball x r) :=
fun _y hy => hf.analyticAt_of_mem hy
#align has_fpower_series_on_ball.analytic_on HasFPowerSeriesOnBall.analyticOn
variable (𝕜 f)
/-- For any function `f` from a normed vector space to a Banach space, the set of points `x` such
that `f` is analytic at `x` is open. -/
theorem isOpen_analyticAt : IsOpen { x | AnalyticAt 𝕜 f x } := by
rw [isOpen_iff_mem_nhds]
rintro x ⟨p, r, hr⟩
exact mem_of_superset (EMetric.ball_mem_nhds _ hr.r_pos) fun y hy => hr.analyticAt_of_mem hy
#align is_open_analytic_at isOpen_analyticAt
variable {𝕜}
theorem AnalyticAt.eventually_analyticAt {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∀ᶠ y in 𝓝 x, AnalyticAt 𝕜 f y :=
(isOpen_analyticAt 𝕜 f).mem_nhds h
theorem AnalyticAt.exists_mem_nhds_analyticOn {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∃ s ∈ 𝓝 x, AnalyticOn 𝕜 f s :=
h.eventually_analyticAt.exists_mem
/-- If we're analytic at a point, we're analytic in a nonempty ball -/
theorem AnalyticAt.exists_ball_analyticOn {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∃ r : ℝ, 0 < r ∧ AnalyticOn 𝕜 f (Metric.ball x r) :=
Metric.isOpen_iff.mp (isOpen_analyticAt _ _) _ h
end
section
open FormalMultilinearSeries
variable {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E} {z₀ : 𝕜}
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z)) := by
refine' ⟨fun ⟨r, _, r_pos, h⟩ =>
eventually_of_mem (EMetric.ball_mem_nhds 0 r_pos) fun _ => by simpa using h, _⟩
simp only [Metric.eventually_nhds_iff]
rintro ⟨r, r_pos, h⟩
refine' ⟨p.radius ⊓ r.toNNReal, by simp, _, _⟩
· simp only [r_pos.lt, lt_inf_iff, ENNReal.coe_pos, Real.toNNReal_pos, and_true_iff]
obtain ⟨z, z_pos, le_z⟩ := NormedField.exists_norm_lt 𝕜 r_pos.lt
have : (‖z‖₊ : ENNReal) ≤ p.radius := by
simp only [dist_zero_right] at h
apply FormalMultilinearSeries.le_radius_of_tendsto
convert tendsto_norm.comp (h le_z).summable.tendsto_atTop_zero
funext
|
simp [norm_smul, mul_comm]
|
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z)) := by
refine' ⟨fun ⟨r, _, r_pos, h⟩ =>
eventually_of_mem (EMetric.ball_mem_nhds 0 r_pos) fun _ => by simpa using h, _⟩
simp only [Metric.eventually_nhds_iff]
rintro ⟨r, r_pos, h⟩
refine' ⟨p.radius ⊓ r.toNNReal, by simp, _, _⟩
· simp only [r_pos.lt, lt_inf_iff, ENNReal.coe_pos, Real.toNNReal_pos, and_true_iff]
obtain ⟨z, z_pos, le_z⟩ := NormedField.exists_norm_lt 𝕜 r_pos.lt
have : (‖z‖₊ : ENNReal) ≤ p.radius := by
simp only [dist_zero_right] at h
apply FormalMultilinearSeries.le_radius_of_tendsto
convert tendsto_norm.comp (h le_z).summable.tendsto_atTop_zero
funext
|
Mathlib.Analysis.Analytic.Basic.1430_0.jQw1fRSE1vGpOll
|
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z))
|
Mathlib_Analysis_Analytic_Basic
|
case intro.intro.refine'_1.intro.intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
p : FormalMultilinearSeries 𝕜 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
r : ℝ
r_pos : r > 0
h : ∀ ⦃y : 𝕜⦄, dist y 0 < r → HasSum (fun n => y ^ n • coeff p n) (f (z₀ + y))
z : 𝕜
z_pos : 0 < ‖z‖
le_z : ‖z‖ < r
this : ↑‖z‖₊ ≤ radius p
⊢ 0 < radius p
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Logic.Equiv.Fin
import Mathlib.Topology.Algebra.InfiniteSum.Module
#align_import analysis.analytic.basic from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
/-!
# Analytic functions
A function is analytic in one dimension around `0` if it can be written as a converging power series
`Σ pₙ zⁿ`. This definition can be extended to any dimension (even in infinite dimension) by
requiring that `pₙ` is a continuous `n`-multilinear map. In general, `pₙ` is not unique (in two
dimensions, taking `p₂ (x, y) (x', y') = x y'` or `y x'` gives the same map when applied to a
vector `(x, y) (x, y)`). A way to guarantee uniqueness is to take a symmetric `pₙ`, but this is not
always possible in nonzero characteristic (in characteristic 2, the previous example has no
symmetric representative). Therefore, we do not insist on symmetry or uniqueness in the definition,
and we only require the existence of a converging series.
The general framework is important to say that the exponential map on bounded operators on a Banach
space is analytic, as well as the inverse on invertible operators.
## Main definitions
Let `p` be a formal multilinear series from `E` to `F`, i.e., `p n` is a multilinear map on `E^n`
for `n : ℕ`.
* `p.radius`: the largest `r : ℝ≥0∞` such that `‖p n‖ * r^n` grows subexponentially.
* `p.le_radius_of_bound`, `p.le_radius_of_bound_nnreal`, `p.le_radius_of_isBigO`: if `‖p n‖ * r ^ n`
is bounded above, then `r ≤ p.radius`;
* `p.isLittleO_of_lt_radius`, `p.norm_mul_pow_le_mul_pow_of_lt_radius`,
`p.isLittleO_one_of_lt_radius`,
`p.norm_mul_pow_le_of_lt_radius`, `p.nnnorm_mul_pow_le_of_lt_radius`: if `r < p.radius`, then
`‖p n‖ * r ^ n` tends to zero exponentially;
* `p.lt_radius_of_isBigO`: if `r ≠ 0` and `‖p n‖ * r ^ n = O(a ^ n)` for some `-1 < a < 1`, then
`r < p.radius`;
* `p.partialSum n x`: the sum `∑_{i = 0}^{n-1} pᵢ xⁱ`.
* `p.sum x`: the sum `∑'_{i = 0}^{∞} pᵢ xⁱ`.
Additionally, let `f` be a function from `E` to `F`.
* `HasFPowerSeriesOnBall f p x r`: on the ball of center `x` with radius `r`,
`f (x + y) = ∑'_n pₙ yⁿ`.
* `HasFPowerSeriesAt f p x`: on some ball of center `x` with positive radius, holds
`HasFPowerSeriesOnBall f p x r`.
* `AnalyticAt 𝕜 f x`: there exists a power series `p` such that holds `HasFPowerSeriesAt f p x`.
* `AnalyticOn 𝕜 f s`: the function `f` is analytic at every point of `s`.
We develop the basic properties of these notions, notably:
* If a function admits a power series, it is continuous (see
`HasFPowerSeriesOnBall.continuousOn` and `HasFPowerSeriesAt.continuousAt` and
`AnalyticAt.continuousAt`).
* In a complete space, the sum of a formal power series with positive radius is well defined on the
disk of convergence, see `FormalMultilinearSeries.hasFPowerSeriesOnBall`.
* If a function admits a power series in a ball, then it is analytic at any point `y` of this ball,
and the power series there can be expressed in terms of the initial power series `p` as
`p.changeOrigin y`. See `HasFPowerSeriesOnBall.changeOrigin`. It follows in particular that
the set of points at which a given function is analytic is open, see `isOpen_analyticAt`.
## Implementation details
We only introduce the radius of convergence of a power series, as `p.radius`.
For a power series in finitely many dimensions, there is a finer (directional, coordinate-dependent)
notion, describing the polydisk of convergence. This notion is more specific, and not necessary to
build the general theory. We do not define it here.
-/
noncomputable section
variable {𝕜 E F G : Type*}
open Topology Classical BigOperators NNReal Filter ENNReal
open Set Filter Asymptotics
namespace FormalMultilinearSeries
variable [Ring 𝕜] [AddCommGroup E] [AddCommGroup F] [Module 𝕜 E] [Module 𝕜 F]
variable [TopologicalSpace E] [TopologicalSpace F]
variable [TopologicalAddGroup E] [TopologicalAddGroup F]
variable [ContinuousConstSMul 𝕜 E] [ContinuousConstSMul 𝕜 F]
/-- Given a formal multilinear series `p` and a vector `x`, then `p.sum x` is the sum `Σ pₙ xⁿ`. A
priori, it only behaves well when `‖x‖ < p.radius`. -/
protected def sum (p : FormalMultilinearSeries 𝕜 E F) (x : E) : F :=
∑' n : ℕ, p n fun _ => x
#align formal_multilinear_series.sum FormalMultilinearSeries.sum
/-- Given a formal multilinear series `p` and a vector `x`, then `p.partialSum n x` is the sum
`Σ pₖ xᵏ` for `k ∈ {0,..., n-1}`. -/
def partialSum (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) (x : E) : F :=
∑ k in Finset.range n, p k fun _ : Fin k => x
#align formal_multilinear_series.partial_sum FormalMultilinearSeries.partialSum
/-- The partial sums of a formal multilinear series are continuous. -/
theorem partialSum_continuous (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) :
Continuous (p.partialSum n) := by
unfold partialSum -- Porting note: added
continuity
#align formal_multilinear_series.partial_sum_continuous FormalMultilinearSeries.partialSum_continuous
end FormalMultilinearSeries
/-! ### The radius of a formal multilinear series -/
variable [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F]
[NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G]
namespace FormalMultilinearSeries
variable (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
/-- The radius of a formal multilinear series is the largest `r` such that the sum `Σ ‖pₙ‖ ‖y‖ⁿ`
converges for all `‖y‖ < r`. This implies that `Σ pₙ yⁿ` converges for all `‖y‖ < r`, but these
definitions are *not* equivalent in general. -/
def radius (p : FormalMultilinearSeries 𝕜 E F) : ℝ≥0∞ :=
⨆ (r : ℝ≥0) (C : ℝ) (_ : ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C), (r : ℝ≥0∞)
#align formal_multilinear_series.radius FormalMultilinearSeries.radius
/-- If `‖pₙ‖ rⁿ` is bounded in `n`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_bound (C : ℝ) {r : ℝ≥0} (h : ∀ n : ℕ, ‖p n‖ * (r : ℝ) ^ n ≤ C) :
(r : ℝ≥0∞) ≤ p.radius :=
le_iSup_of_le r <| le_iSup_of_le C <| le_iSup (fun _ => (r : ℝ≥0∞)) h
#align formal_multilinear_series.le_radius_of_bound FormalMultilinearSeries.le_radius_of_bound
/-- If `‖pₙ‖ rⁿ` is bounded in `n`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_bound_nnreal (C : ℝ≥0) {r : ℝ≥0} (h : ∀ n : ℕ, ‖p n‖₊ * r ^ n ≤ C) :
(r : ℝ≥0∞) ≤ p.radius :=
p.le_radius_of_bound C fun n => mod_cast h n
#align formal_multilinear_series.le_radius_of_bound_nnreal FormalMultilinearSeries.le_radius_of_bound_nnreal
/-- If `‖pₙ‖ rⁿ = O(1)`, as `n → ∞`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_isBigO (h : (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] fun _ => (1 : ℝ)) :
↑r ≤ p.radius :=
Exists.elim (isBigO_one_nat_atTop_iff.1 h) fun C hC =>
p.le_radius_of_bound C fun n => (le_abs_self _).trans (hC n)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.le_radius_of_is_O FormalMultilinearSeries.le_radius_of_isBigO
theorem le_radius_of_eventually_le (C) (h : ∀ᶠ n in atTop, ‖p n‖ * (r : ℝ) ^ n ≤ C) :
↑r ≤ p.radius :=
p.le_radius_of_isBigO <| IsBigO.of_bound C <| h.mono fun n hn => by simpa
#align formal_multilinear_series.le_radius_of_eventually_le FormalMultilinearSeries.le_radius_of_eventually_le
theorem le_radius_of_summable_nnnorm (h : Summable fun n => ‖p n‖₊ * r ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_bound_nnreal (∑' n, ‖p n‖₊ * r ^ n) fun _ => le_tsum' h _
#align formal_multilinear_series.le_radius_of_summable_nnnorm FormalMultilinearSeries.le_radius_of_summable_nnnorm
theorem le_radius_of_summable (h : Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_summable_nnnorm <| by
simp only [← coe_nnnorm] at h
exact mod_cast h
#align formal_multilinear_series.le_radius_of_summable FormalMultilinearSeries.le_radius_of_summable
theorem radius_eq_top_of_forall_nnreal_isBigO
(h : ∀ r : ℝ≥0, (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] fun _ => (1 : ℝ)) : p.radius = ∞ :=
ENNReal.eq_top_of_forall_nnreal_le fun r => p.le_radius_of_isBigO (h r)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.radius_eq_top_of_forall_nnreal_is_O FormalMultilinearSeries.radius_eq_top_of_forall_nnreal_isBigO
theorem radius_eq_top_of_eventually_eq_zero (h : ∀ᶠ n in atTop, p n = 0) : p.radius = ∞ :=
p.radius_eq_top_of_forall_nnreal_isBigO fun r =>
(isBigO_zero _ _).congr' (h.mono fun n hn => by simp [hn]) EventuallyEq.rfl
#align formal_multilinear_series.radius_eq_top_of_eventually_eq_zero FormalMultilinearSeries.radius_eq_top_of_eventually_eq_zero
theorem radius_eq_top_of_forall_image_add_eq_zero (n : ℕ) (hn : ∀ m, p (m + n) = 0) :
p.radius = ∞ :=
p.radius_eq_top_of_eventually_eq_zero <|
mem_atTop_sets.2 ⟨n, fun _ hk => tsub_add_cancel_of_le hk ▸ hn _⟩
#align formal_multilinear_series.radius_eq_top_of_forall_image_add_eq_zero FormalMultilinearSeries.radius_eq_top_of_forall_image_add_eq_zero
@[simp]
theorem constFormalMultilinearSeries_radius {v : F} :
(constFormalMultilinearSeries 𝕜 E v).radius = ⊤ :=
(constFormalMultilinearSeries 𝕜 E v).radius_eq_top_of_forall_image_add_eq_zero 1
(by simp [constFormalMultilinearSeries])
#align formal_multilinear_series.const_formal_multilinear_series_radius FormalMultilinearSeries.constFormalMultilinearSeries_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` tends to zero exponentially:
for some `0 < a < 1`, `‖p n‖ rⁿ = o(aⁿ)`. -/
theorem isLittleO_of_lt_radius (h : ↑r < p.radius) :
∃ a ∈ Ioo (0 : ℝ) 1, (fun n => ‖p n‖ * (r : ℝ) ^ n) =o[atTop] (a ^ ·) := by
have := (TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 4
rw [this]
-- Porting note: was
-- rw [(TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 4]
simp only [radius, lt_iSup_iff] at h
rcases h with ⟨t, C, hC, rt⟩
rw [ENNReal.coe_lt_coe, ← NNReal.coe_lt_coe] at rt
have : 0 < (t : ℝ) := r.coe_nonneg.trans_lt rt
rw [← div_lt_one this] at rt
refine' ⟨_, rt, C, Or.inr zero_lt_one, fun n => _⟩
calc
|‖p n‖ * (r : ℝ) ^ n| = ‖p n‖ * (t : ℝ) ^ n * (r / t : ℝ) ^ n := by
field_simp [mul_right_comm, abs_mul]
_ ≤ C * (r / t : ℝ) ^ n := by gcongr; apply hC
#align formal_multilinear_series.is_o_of_lt_radius FormalMultilinearSeries.isLittleO_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ = o(1)`. -/
theorem isLittleO_one_of_lt_radius (h : ↑r < p.radius) :
(fun n => ‖p n‖ * (r : ℝ) ^ n) =o[atTop] (fun _ => 1 : ℕ → ℝ) :=
let ⟨_, ha, hp⟩ := p.isLittleO_of_lt_radius h
hp.trans <| (isLittleO_pow_pow_of_lt_left ha.1.le ha.2).congr (fun _ => rfl) one_pow
#align formal_multilinear_series.is_o_one_of_lt_radius FormalMultilinearSeries.isLittleO_one_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` tends to zero exponentially:
for some `0 < a < 1` and `C > 0`, `‖p n‖ * r ^ n ≤ C * a ^ n`. -/
theorem norm_mul_pow_le_mul_pow_of_lt_radius (h : ↑r < p.radius) :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C * a ^ n := by
-- Porting note: moved out of `rcases`
have := ((TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 5).mp
(p.isLittleO_of_lt_radius h)
rcases this with ⟨a, ha, C, hC, H⟩
exact ⟨a, ha, C, hC, fun n => (le_abs_self _).trans (H n)⟩
#align formal_multilinear_series.norm_mul_pow_le_mul_pow_of_lt_radius FormalMultilinearSeries.norm_mul_pow_le_mul_pow_of_lt_radius
/-- If `r ≠ 0` and `‖pₙ‖ rⁿ = O(aⁿ)` for some `-1 < a < 1`, then `r < p.radius`. -/
theorem lt_radius_of_isBigO (h₀ : r ≠ 0) {a : ℝ} (ha : a ∈ Ioo (-1 : ℝ) 1)
(hp : (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] (a ^ ·)) : ↑r < p.radius := by
-- Porting note: moved out of `rcases`
have := ((TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 2 5)
rcases this.mp ⟨a, ha, hp⟩ with ⟨a, ha, C, hC, hp⟩
rw [← pos_iff_ne_zero, ← NNReal.coe_pos] at h₀
lift a to ℝ≥0 using ha.1.le
have : (r : ℝ) < r / a := by
simpa only [div_one] using (div_lt_div_left h₀ zero_lt_one ha.1).2 ha.2
norm_cast at this
rw [← ENNReal.coe_lt_coe] at this
refine' this.trans_le (p.le_radius_of_bound C fun n => _)
rw [NNReal.coe_div, div_pow, ← mul_div_assoc, div_le_iff (pow_pos ha.1 n)]
exact (le_abs_self _).trans (hp n)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.lt_radius_of_is_O FormalMultilinearSeries.lt_radius_of_isBigO
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem norm_mul_pow_le_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C :=
let ⟨_, ha, C, hC, h⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius h
⟨C, hC, fun n => (h n).trans <| mul_le_of_le_one_right hC.lt.le (pow_le_one _ ha.1.le ha.2.le)⟩
#align formal_multilinear_series.norm_mul_pow_le_of_lt_radius FormalMultilinearSeries.norm_mul_pow_le_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem norm_le_div_pow_of_pos_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h0 : 0 < r) (h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖ ≤ C / (r : ℝ) ^ n :=
let ⟨C, hC, hp⟩ := p.norm_mul_pow_le_of_lt_radius h
⟨C, hC, fun n => Iff.mpr (le_div_iff (pow_pos h0 _)) (hp n)⟩
#align formal_multilinear_series.norm_le_div_pow_of_pos_of_lt_radius FormalMultilinearSeries.norm_le_div_pow_of_pos_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem nnnorm_mul_pow_le_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖₊ * r ^ n ≤ C :=
let ⟨C, hC, hp⟩ := p.norm_mul_pow_le_of_lt_radius h
⟨⟨C, hC.lt.le⟩, hC, mod_cast hp⟩
#align formal_multilinear_series.nnnorm_mul_pow_le_of_lt_radius FormalMultilinearSeries.nnnorm_mul_pow_le_of_lt_radius
theorem le_radius_of_tendsto (p : FormalMultilinearSeries 𝕜 E F) {l : ℝ}
(h : Tendsto (fun n => ‖p n‖ * (r : ℝ) ^ n) atTop (𝓝 l)) : ↑r ≤ p.radius :=
p.le_radius_of_isBigO (h.isBigO_one _)
#align formal_multilinear_series.le_radius_of_tendsto FormalMultilinearSeries.le_radius_of_tendsto
theorem le_radius_of_summable_norm (p : FormalMultilinearSeries 𝕜 E F)
(hs : Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_tendsto hs.tendsto_atTop_zero
#align formal_multilinear_series.le_radius_of_summable_norm FormalMultilinearSeries.le_radius_of_summable_norm
theorem not_summable_norm_of_radius_lt_nnnorm (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(h : p.radius < ‖x‖₊) : ¬Summable fun n => ‖p n‖ * ‖x‖ ^ n :=
fun hs => not_le_of_lt h (p.le_radius_of_summable_norm hs)
#align formal_multilinear_series.not_summable_norm_of_radius_lt_nnnorm FormalMultilinearSeries.not_summable_norm_of_radius_lt_nnnorm
theorem summable_norm_mul_pow (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0} (h : ↑r < p.radius) :
Summable fun n : ℕ => ‖p n‖ * (r : ℝ) ^ n := by
obtain ⟨a, ha : a ∈ Ioo (0 : ℝ) 1, C, - : 0 < C, hp⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius h
exact .of_nonneg_of_le (fun n => mul_nonneg (norm_nonneg _) (pow_nonneg r.coe_nonneg _))
hp ((summable_geometric_of_lt_1 ha.1.le ha.2).mul_left _)
#align formal_multilinear_series.summable_norm_mul_pow FormalMultilinearSeries.summable_norm_mul_pow
theorem summable_norm_apply (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : Summable fun n : ℕ => ‖p n fun _ => x‖ := by
rw [mem_emetric_ball_zero_iff] at hx
refine' .of_nonneg_of_le
(fun _ => norm_nonneg _) (fun n => ((p n).le_op_norm _).trans_eq _) (p.summable_norm_mul_pow hx)
simp
#align formal_multilinear_series.summable_norm_apply FormalMultilinearSeries.summable_norm_apply
theorem summable_nnnorm_mul_pow (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0} (h : ↑r < p.radius) :
Summable fun n : ℕ => ‖p n‖₊ * r ^ n := by
rw [← NNReal.summable_coe]
push_cast
exact p.summable_norm_mul_pow h
#align formal_multilinear_series.summable_nnnorm_mul_pow FormalMultilinearSeries.summable_nnnorm_mul_pow
protected theorem summable [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : Summable fun n : ℕ => p n fun _ => x :=
(p.summable_norm_apply hx).of_norm
#align formal_multilinear_series.summable FormalMultilinearSeries.summable
theorem radius_eq_top_of_summable_norm (p : FormalMultilinearSeries 𝕜 E F)
(hs : ∀ r : ℝ≥0, Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : p.radius = ∞ :=
ENNReal.eq_top_of_forall_nnreal_le fun r => p.le_radius_of_summable_norm (hs r)
#align formal_multilinear_series.radius_eq_top_of_summable_norm FormalMultilinearSeries.radius_eq_top_of_summable_norm
theorem radius_eq_top_iff_summable_norm (p : FormalMultilinearSeries 𝕜 E F) :
p.radius = ∞ ↔ ∀ r : ℝ≥0, Summable fun n => ‖p n‖ * (r : ℝ) ^ n := by
constructor
· intro h r
obtain ⟨a, ha : a ∈ Ioo (0 : ℝ) 1, C, - : 0 < C, hp⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius
(show (r : ℝ≥0∞) < p.radius from h.symm ▸ ENNReal.coe_lt_top)
refine' .of_norm_bounded
(fun n => (C : ℝ) * a ^ n) ((summable_geometric_of_lt_1 ha.1.le ha.2).mul_left _) fun n => _
specialize hp n
rwa [Real.norm_of_nonneg (mul_nonneg (norm_nonneg _) (pow_nonneg r.coe_nonneg n))]
· exact p.radius_eq_top_of_summable_norm
#align formal_multilinear_series.radius_eq_top_iff_summable_norm FormalMultilinearSeries.radius_eq_top_iff_summable_norm
/-- If the radius of `p` is positive, then `‖pₙ‖` grows at most geometrically. -/
theorem le_mul_pow_of_radius_pos (p : FormalMultilinearSeries 𝕜 E F) (h : 0 < p.radius) :
∃ (C r : _) (hC : 0 < C) (_ : 0 < r), ∀ n, ‖p n‖ ≤ C * r ^ n := by
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 h with ⟨r, r0, rlt⟩
have rpos : 0 < (r : ℝ) := by simp [ENNReal.coe_pos.1 r0]
rcases norm_le_div_pow_of_pos_of_lt_radius p rpos rlt with ⟨C, Cpos, hCp⟩
refine' ⟨C, r⁻¹, Cpos, by simp only [inv_pos, rpos], fun n => _⟩
-- Porting note: was `convert`
rw [inv_pow, ← div_eq_mul_inv]
exact hCp n
#align formal_multilinear_series.le_mul_pow_of_radius_pos FormalMultilinearSeries.le_mul_pow_of_radius_pos
/-- The radius of the sum of two formal series is at least the minimum of their two radii. -/
theorem min_radius_le_radius_add (p q : FormalMultilinearSeries 𝕜 E F) :
min p.radius q.radius ≤ (p + q).radius := by
refine' ENNReal.le_of_forall_nnreal_lt fun r hr => _
rw [lt_min_iff] at hr
have := ((p.isLittleO_one_of_lt_radius hr.1).add (q.isLittleO_one_of_lt_radius hr.2)).isBigO
refine' (p + q).le_radius_of_isBigO ((isBigO_of_le _ fun n => _).trans this)
rw [← add_mul, norm_mul, norm_mul, norm_norm]
exact mul_le_mul_of_nonneg_right ((norm_add_le _ _).trans (le_abs_self _)) (norm_nonneg _)
#align formal_multilinear_series.min_radius_le_radius_add FormalMultilinearSeries.min_radius_le_radius_add
@[simp]
theorem radius_neg (p : FormalMultilinearSeries 𝕜 E F) : (-p).radius = p.radius := by
simp only [radius, neg_apply, norm_neg]
#align formal_multilinear_series.radius_neg FormalMultilinearSeries.radius_neg
protected theorem hasSum [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : HasSum (fun n : ℕ => p n fun _ => x) (p.sum x) :=
(p.summable hx).hasSum
#align formal_multilinear_series.has_sum FormalMultilinearSeries.hasSum
theorem radius_le_radius_continuousLinearMap_comp (p : FormalMultilinearSeries 𝕜 E F)
(f : F →L[𝕜] G) : p.radius ≤ (f.compFormalMultilinearSeries p).radius := by
refine' ENNReal.le_of_forall_nnreal_lt fun r hr => _
apply le_radius_of_isBigO
apply (IsBigO.trans_isLittleO _ (p.isLittleO_one_of_lt_radius hr)).isBigO
refine' IsBigO.mul (@IsBigOWith.isBigO _ _ _ _ _ ‖f‖ _ _ _ _) (isBigO_refl _ _)
refine IsBigOWith.of_bound (eventually_of_forall fun n => ?_)
simpa only [norm_norm] using f.norm_compContinuousMultilinearMap_le (p n)
#align formal_multilinear_series.radius_le_radius_continuous_linear_map_comp FormalMultilinearSeries.radius_le_radius_continuousLinearMap_comp
end FormalMultilinearSeries
/-! ### Expanding a function as a power series -/
section
variable {f g : E → F} {p pf pg : FormalMultilinearSeries 𝕜 E F} {x : E} {r r' : ℝ≥0∞}
/-- Given a function `f : E → F` and a formal multilinear series `p`, we say that `f` has `p` as
a power series on the ball of radius `r > 0` around `x` if `f (x + y) = ∑' pₙ yⁿ` for all `‖y‖ < r`.
-/
structure HasFPowerSeriesOnBall (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (x : E) (r : ℝ≥0∞) :
Prop where
r_le : r ≤ p.radius
r_pos : 0 < r
hasSum :
∀ {y}, y ∈ EMetric.ball (0 : E) r → HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y))
#align has_fpower_series_on_ball HasFPowerSeriesOnBall
/-- Given a function `f : E → F` and a formal multilinear series `p`, we say that `f` has `p` as
a power series around `x` if `f (x + y) = ∑' pₙ yⁿ` for all `y` in a neighborhood of `0`. -/
def HasFPowerSeriesAt (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (x : E) :=
∃ r, HasFPowerSeriesOnBall f p x r
#align has_fpower_series_at HasFPowerSeriesAt
variable (𝕜)
/-- Given a function `f : E → F`, we say that `f` is analytic at `x` if it admits a convergent power
series expansion around `x`. -/
def AnalyticAt (f : E → F) (x : E) :=
∃ p : FormalMultilinearSeries 𝕜 E F, HasFPowerSeriesAt f p x
#align analytic_at AnalyticAt
/-- Given a function `f : E → F`, we say that `f` is analytic on a set `s` if it is analytic around
every point of `s`. -/
def AnalyticOn (f : E → F) (s : Set E) :=
∀ x, x ∈ s → AnalyticAt 𝕜 f x
#align analytic_on AnalyticOn
variable {𝕜}
theorem HasFPowerSeriesOnBall.hasFPowerSeriesAt (hf : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesAt f p x :=
⟨r, hf⟩
#align has_fpower_series_on_ball.has_fpower_series_at HasFPowerSeriesOnBall.hasFPowerSeriesAt
theorem HasFPowerSeriesAt.analyticAt (hf : HasFPowerSeriesAt f p x) : AnalyticAt 𝕜 f x :=
⟨p, hf⟩
#align has_fpower_series_at.analytic_at HasFPowerSeriesAt.analyticAt
theorem HasFPowerSeriesOnBall.analyticAt (hf : HasFPowerSeriesOnBall f p x r) : AnalyticAt 𝕜 f x :=
hf.hasFPowerSeriesAt.analyticAt
#align has_fpower_series_on_ball.analytic_at HasFPowerSeriesOnBall.analyticAt
theorem HasFPowerSeriesOnBall.congr (hf : HasFPowerSeriesOnBall f p x r)
(hg : EqOn f g (EMetric.ball x r)) : HasFPowerSeriesOnBall g p x r :=
{ r_le := hf.r_le
r_pos := hf.r_pos
hasSum := fun {y} hy => by
convert hf.hasSum hy using 1
apply hg.symm
simpa [edist_eq_coe_nnnorm_sub] using hy }
#align has_fpower_series_on_ball.congr HasFPowerSeriesOnBall.congr
/-- If a function `f` has a power series `p` around `x`, then the function `z ↦ f (z - y)` has the
same power series around `x + y`. -/
theorem HasFPowerSeriesOnBall.comp_sub (hf : HasFPowerSeriesOnBall f p x r) (y : E) :
HasFPowerSeriesOnBall (fun z => f (z - y)) p (x + y) r :=
{ r_le := hf.r_le
r_pos := hf.r_pos
hasSum := fun {z} hz => by
convert hf.hasSum hz using 2
abel }
#align has_fpower_series_on_ball.comp_sub HasFPowerSeriesOnBall.comp_sub
theorem HasFPowerSeriesOnBall.hasSum_sub (hf : HasFPowerSeriesOnBall f p x r) {y : E}
(hy : y ∈ EMetric.ball x r) : HasSum (fun n : ℕ => p n fun _ => y - x) (f y) := by
have : y - x ∈ EMetric.ball (0 : E) r := by simpa [edist_eq_coe_nnnorm_sub] using hy
simpa only [add_sub_cancel'_right] using hf.hasSum this
#align has_fpower_series_on_ball.has_sum_sub HasFPowerSeriesOnBall.hasSum_sub
theorem HasFPowerSeriesOnBall.radius_pos (hf : HasFPowerSeriesOnBall f p x r) : 0 < p.radius :=
lt_of_lt_of_le hf.r_pos hf.r_le
#align has_fpower_series_on_ball.radius_pos HasFPowerSeriesOnBall.radius_pos
theorem HasFPowerSeriesAt.radius_pos (hf : HasFPowerSeriesAt f p x) : 0 < p.radius :=
let ⟨_, hr⟩ := hf
hr.radius_pos
#align has_fpower_series_at.radius_pos HasFPowerSeriesAt.radius_pos
theorem HasFPowerSeriesOnBall.mono (hf : HasFPowerSeriesOnBall f p x r) (r'_pos : 0 < r')
(hr : r' ≤ r) : HasFPowerSeriesOnBall f p x r' :=
⟨le_trans hr hf.1, r'_pos, fun hy => hf.hasSum (EMetric.ball_subset_ball hr hy)⟩
#align has_fpower_series_on_ball.mono HasFPowerSeriesOnBall.mono
theorem HasFPowerSeriesAt.congr (hf : HasFPowerSeriesAt f p x) (hg : f =ᶠ[𝓝 x] g) :
HasFPowerSeriesAt g p x := by
rcases hf with ⟨r₁, h₁⟩
rcases EMetric.mem_nhds_iff.mp hg with ⟨r₂, h₂pos, h₂⟩
exact ⟨min r₁ r₂,
(h₁.mono (lt_min h₁.r_pos h₂pos) inf_le_left).congr
fun y hy => h₂ (EMetric.ball_subset_ball inf_le_right hy)⟩
#align has_fpower_series_at.congr HasFPowerSeriesAt.congr
protected theorem HasFPowerSeriesAt.eventually (hf : HasFPowerSeriesAt f p x) :
∀ᶠ r : ℝ≥0∞ in 𝓝[>] 0, HasFPowerSeriesOnBall f p x r :=
let ⟨_, hr⟩ := hf
mem_of_superset (Ioo_mem_nhdsWithin_Ioi (left_mem_Ico.2 hr.r_pos)) fun _ hr' =>
hr.mono hr'.1 hr'.2.le
#align has_fpower_series_at.eventually HasFPowerSeriesAt.eventually
theorem HasFPowerSeriesOnBall.eventually_hasSum (hf : HasFPowerSeriesOnBall f p x r) :
∀ᶠ y in 𝓝 0, HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y)) := by
filter_upwards [EMetric.ball_mem_nhds (0 : E) hf.r_pos] using fun _ => hf.hasSum
#align has_fpower_series_on_ball.eventually_has_sum HasFPowerSeriesOnBall.eventually_hasSum
theorem HasFPowerSeriesAt.eventually_hasSum (hf : HasFPowerSeriesAt f p x) :
∀ᶠ y in 𝓝 0, HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y)) :=
let ⟨_, hr⟩ := hf
hr.eventually_hasSum
#align has_fpower_series_at.eventually_has_sum HasFPowerSeriesAt.eventually_hasSum
theorem HasFPowerSeriesOnBall.eventually_hasSum_sub (hf : HasFPowerSeriesOnBall f p x r) :
∀ᶠ y in 𝓝 x, HasSum (fun n : ℕ => p n fun _ : Fin n => y - x) (f y) := by
filter_upwards [EMetric.ball_mem_nhds x hf.r_pos] with y using hf.hasSum_sub
#align has_fpower_series_on_ball.eventually_has_sum_sub HasFPowerSeriesOnBall.eventually_hasSum_sub
theorem HasFPowerSeriesAt.eventually_hasSum_sub (hf : HasFPowerSeriesAt f p x) :
∀ᶠ y in 𝓝 x, HasSum (fun n : ℕ => p n fun _ : Fin n => y - x) (f y) :=
let ⟨_, hr⟩ := hf
hr.eventually_hasSum_sub
#align has_fpower_series_at.eventually_has_sum_sub HasFPowerSeriesAt.eventually_hasSum_sub
theorem HasFPowerSeriesOnBall.eventually_eq_zero
(hf : HasFPowerSeriesOnBall f (0 : FormalMultilinearSeries 𝕜 E F) x r) :
∀ᶠ z in 𝓝 x, f z = 0 := by
filter_upwards [hf.eventually_hasSum_sub] with z hz using hz.unique hasSum_zero
#align has_fpower_series_on_ball.eventually_eq_zero HasFPowerSeriesOnBall.eventually_eq_zero
theorem HasFPowerSeriesAt.eventually_eq_zero
(hf : HasFPowerSeriesAt f (0 : FormalMultilinearSeries 𝕜 E F) x) : ∀ᶠ z in 𝓝 x, f z = 0 :=
let ⟨_, hr⟩ := hf
hr.eventually_eq_zero
#align has_fpower_series_at.eventually_eq_zero HasFPowerSeriesAt.eventually_eq_zero
theorem hasFPowerSeriesOnBall_const {c : F} {e : E} :
HasFPowerSeriesOnBall (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e ⊤ := by
refine' ⟨by simp, WithTop.zero_lt_top, fun _ => hasSum_single 0 fun n hn => _⟩
simp [constFormalMultilinearSeries_apply hn]
#align has_fpower_series_on_ball_const hasFPowerSeriesOnBall_const
theorem hasFPowerSeriesAt_const {c : F} {e : E} :
HasFPowerSeriesAt (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e :=
⟨⊤, hasFPowerSeriesOnBall_const⟩
#align has_fpower_series_at_const hasFPowerSeriesAt_const
theorem analyticAt_const {v : F} : AnalyticAt 𝕜 (fun _ => v) x :=
⟨constFormalMultilinearSeries 𝕜 E v, hasFPowerSeriesAt_const⟩
#align analytic_at_const analyticAt_const
theorem analyticOn_const {v : F} {s : Set E} : AnalyticOn 𝕜 (fun _ => v) s :=
fun _ _ => analyticAt_const
#align analytic_on_const analyticOn_const
theorem HasFPowerSeriesOnBall.add (hf : HasFPowerSeriesOnBall f pf x r)
(hg : HasFPowerSeriesOnBall g pg x r) : HasFPowerSeriesOnBall (f + g) (pf + pg) x r :=
{ r_le := le_trans (le_min_iff.2 ⟨hf.r_le, hg.r_le⟩) (pf.min_radius_le_radius_add pg)
r_pos := hf.r_pos
hasSum := fun hy => (hf.hasSum hy).add (hg.hasSum hy) }
#align has_fpower_series_on_ball.add HasFPowerSeriesOnBall.add
theorem HasFPowerSeriesAt.add (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x) :
HasFPowerSeriesAt (f + g) (pf + pg) x := by
rcases (hf.eventually.and hg.eventually).exists with ⟨r, hr⟩
exact ⟨r, hr.1.add hr.2⟩
#align has_fpower_series_at.add HasFPowerSeriesAt.add
theorem AnalyticAt.congr (hf : AnalyticAt 𝕜 f x) (hg : f =ᶠ[𝓝 x] g) : AnalyticAt 𝕜 g x :=
let ⟨_, hpf⟩ := hf
(hpf.congr hg).analyticAt
theorem analyticAt_congr (h : f =ᶠ[𝓝 x] g) : AnalyticAt 𝕜 f x ↔ AnalyticAt 𝕜 g x :=
⟨fun hf ↦ hf.congr h, fun hg ↦ hg.congr h.symm⟩
theorem AnalyticAt.add (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) : AnalyticAt 𝕜 (f + g) x :=
let ⟨_, hpf⟩ := hf
let ⟨_, hqf⟩ := hg
(hpf.add hqf).analyticAt
#align analytic_at.add AnalyticAt.add
theorem HasFPowerSeriesOnBall.neg (hf : HasFPowerSeriesOnBall f pf x r) :
HasFPowerSeriesOnBall (-f) (-pf) x r :=
{ r_le := by
rw [pf.radius_neg]
exact hf.r_le
r_pos := hf.r_pos
hasSum := fun hy => (hf.hasSum hy).neg }
#align has_fpower_series_on_ball.neg HasFPowerSeriesOnBall.neg
theorem HasFPowerSeriesAt.neg (hf : HasFPowerSeriesAt f pf x) : HasFPowerSeriesAt (-f) (-pf) x :=
let ⟨_, hrf⟩ := hf
hrf.neg.hasFPowerSeriesAt
#align has_fpower_series_at.neg HasFPowerSeriesAt.neg
theorem AnalyticAt.neg (hf : AnalyticAt 𝕜 f x) : AnalyticAt 𝕜 (-f) x :=
let ⟨_, hpf⟩ := hf
hpf.neg.analyticAt
#align analytic_at.neg AnalyticAt.neg
theorem HasFPowerSeriesOnBall.sub (hf : HasFPowerSeriesOnBall f pf x r)
(hg : HasFPowerSeriesOnBall g pg x r) : HasFPowerSeriesOnBall (f - g) (pf - pg) x r := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align has_fpower_series_on_ball.sub HasFPowerSeriesOnBall.sub
theorem HasFPowerSeriesAt.sub (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x) :
HasFPowerSeriesAt (f - g) (pf - pg) x := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align has_fpower_series_at.sub HasFPowerSeriesAt.sub
theorem AnalyticAt.sub (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) :
AnalyticAt 𝕜 (f - g) x := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align analytic_at.sub AnalyticAt.sub
theorem AnalyticOn.mono {s t : Set E} (hf : AnalyticOn 𝕜 f t) (hst : s ⊆ t) : AnalyticOn 𝕜 f s :=
fun z hz => hf z (hst hz)
#align analytic_on.mono AnalyticOn.mono
theorem AnalyticOn.congr' {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : f =ᶠ[𝓝ˢ s] g) :
AnalyticOn 𝕜 g s :=
fun z hz => (hf z hz).congr (mem_nhdsSet_iff_forall.mp hg z hz)
theorem analyticOn_congr' {s : Set E} (h : f =ᶠ[𝓝ˢ s] g) : AnalyticOn 𝕜 f s ↔ AnalyticOn 𝕜 g s :=
⟨fun hf => hf.congr' h, fun hg => hg.congr' h.symm⟩
theorem AnalyticOn.congr {s : Set E} (hs : IsOpen s) (hf : AnalyticOn 𝕜 f s) (hg : s.EqOn f g) :
AnalyticOn 𝕜 g s :=
hf.congr' $ mem_nhdsSet_iff_forall.mpr
(fun _ hz => eventuallyEq_iff_exists_mem.mpr ⟨s, hs.mem_nhds hz, hg⟩)
theorem analyticOn_congr {s : Set E} (hs : IsOpen s) (h : s.EqOn f g) : AnalyticOn 𝕜 f s ↔
AnalyticOn 𝕜 g s := ⟨fun hf => hf.congr hs h, fun hg => hg.congr hs h.symm⟩
theorem AnalyticOn.add {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
AnalyticOn 𝕜 (f + g) s :=
fun z hz => (hf z hz).add (hg z hz)
#align analytic_on.add AnalyticOn.add
theorem AnalyticOn.sub {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
AnalyticOn 𝕜 (f - g) s :=
fun z hz => (hf z hz).sub (hg z hz)
#align analytic_on.sub AnalyticOn.sub
theorem HasFPowerSeriesOnBall.coeff_zero (hf : HasFPowerSeriesOnBall f pf x r) (v : Fin 0 → E) :
pf 0 v = f x := by
have v_eq : v = fun i => 0 := Subsingleton.elim _ _
have zero_mem : (0 : E) ∈ EMetric.ball (0 : E) r := by simp [hf.r_pos]
have : ∀ i, i ≠ 0 → (pf i fun j => 0) = 0 := by
intro i hi
have : 0 < i := pos_iff_ne_zero.2 hi
exact ContinuousMultilinearMap.map_coord_zero _ (⟨0, this⟩ : Fin i) rfl
have A := (hf.hasSum zero_mem).unique (hasSum_single _ this)
simpa [v_eq] using A.symm
#align has_fpower_series_on_ball.coeff_zero HasFPowerSeriesOnBall.coeff_zero
theorem HasFPowerSeriesAt.coeff_zero (hf : HasFPowerSeriesAt f pf x) (v : Fin 0 → E) :
pf 0 v = f x :=
let ⟨_, hrf⟩ := hf
hrf.coeff_zero v
#align has_fpower_series_at.coeff_zero HasFPowerSeriesAt.coeff_zero
/-- If a function `f` has a power series `p` on a ball and `g` is linear, then `g ∘ f` has the
power series `g ∘ p` on the same ball. -/
theorem ContinuousLinearMap.comp_hasFPowerSeriesOnBall (g : F →L[𝕜] G)
(h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (g ∘ f) (g.compFormalMultilinearSeries p) x r :=
{ r_le := h.r_le.trans (p.radius_le_radius_continuousLinearMap_comp _)
r_pos := h.r_pos
hasSum := fun hy => by
simpa only [ContinuousLinearMap.compFormalMultilinearSeries_apply,
ContinuousLinearMap.compContinuousMultilinearMap_coe, Function.comp_apply] using
g.hasSum (h.hasSum hy) }
#align continuous_linear_map.comp_has_fpower_series_on_ball ContinuousLinearMap.comp_hasFPowerSeriesOnBall
/-- If a function `f` is analytic on a set `s` and `g` is linear, then `g ∘ f` is analytic
on `s`. -/
theorem ContinuousLinearMap.comp_analyticOn {s : Set E} (g : F →L[𝕜] G) (h : AnalyticOn 𝕜 f s) :
AnalyticOn 𝕜 (g ∘ f) s := by
rintro x hx
rcases h x hx with ⟨p, r, hp⟩
exact ⟨g.compFormalMultilinearSeries p, r, g.comp_hasFPowerSeriesOnBall hp⟩
#align continuous_linear_map.comp_analytic_on ContinuousLinearMap.comp_analyticOn
/-- If a function admits a power series expansion, then it is exponentially close to the partial
sums of this power series on strict subdisks of the disk of convergence.
This version provides an upper estimate that decreases both in `‖y‖` and `n`. See also
`HasFPowerSeriesOnBall.uniform_geometric_approx` for a weaker version. -/
theorem HasFPowerSeriesOnBall.uniform_geometric_approx' {r' : ℝ≥0}
(hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n := by
obtain ⟨a, ha, C, hC, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n, ‖p n‖ * (r' : ℝ) ^ n ≤ C * a ^ n :=
p.norm_mul_pow_le_mul_pow_of_lt_radius (h.trans_le hf.r_le)
refine' ⟨a, ha, C / (1 - a), div_pos hC (sub_pos.2 ha.2), fun y hy n => _⟩
have yr' : ‖y‖ < r' := by
rw [ball_zero_eq] at hy
exact hy
have hr'0 : 0 < (r' : ℝ) := (norm_nonneg _).trans_lt yr'
have : y ∈ EMetric.ball (0 : E) r := by
refine' mem_emetric_ball_zero_iff.2 (lt_trans _ h)
exact mod_cast yr'
rw [norm_sub_rev, ← mul_div_right_comm]
have ya : a * (‖y‖ / ↑r') ≤ a :=
mul_le_of_le_one_right ha.1.le (div_le_one_of_le yr'.le r'.coe_nonneg)
suffices ‖p.partialSum n y - f (x + y)‖ ≤ C * (a * (‖y‖ / r')) ^ n / (1 - a * (‖y‖ / r')) by
refine' this.trans _
have : 0 < a := ha.1
gcongr
apply_rules [sub_pos.2, ha.2]
apply norm_sub_le_of_geometric_bound_of_hasSum (ya.trans_lt ha.2) _ (hf.hasSum this)
intro n
calc
‖(p n) fun _ : Fin n => y‖
_ ≤ ‖p n‖ * ∏ _i : Fin n, ‖y‖ := ContinuousMultilinearMap.le_op_norm _ _
_ = ‖p n‖ * (r' : ℝ) ^ n * (‖y‖ / r') ^ n := by field_simp [mul_right_comm]
_ ≤ C * a ^ n * (‖y‖ / r') ^ n := by gcongr ?_ * _; apply hp
_ ≤ C * (a * (‖y‖ / r')) ^ n := by rw [mul_pow, mul_assoc]
#align has_fpower_series_on_ball.uniform_geometric_approx' HasFPowerSeriesOnBall.uniform_geometric_approx'
/-- If a function admits a power series expansion, then it is exponentially close to the partial
sums of this power series on strict subdisks of the disk of convergence. -/
theorem HasFPowerSeriesOnBall.uniform_geometric_approx {r' : ℝ≥0}
(hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
∃ a ∈ Ioo (0 : ℝ) 1,
∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n, ‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n := by
obtain ⟨a, ha, C, hC, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n :=
hf.uniform_geometric_approx' h
refine' ⟨a, ha, C, hC, fun y hy n => (hp y hy n).trans _⟩
have yr' : ‖y‖ < r' := by rwa [ball_zero_eq] at hy
gcongr
exacts [mul_nonneg ha.1.le (div_nonneg (norm_nonneg y) r'.coe_nonneg),
mul_le_of_le_one_right ha.1.le (div_le_one_of_le yr'.le r'.coe_nonneg)]
#align has_fpower_series_on_ball.uniform_geometric_approx HasFPowerSeriesOnBall.uniform_geometric_approx
/-- Taylor formula for an analytic function, `IsBigO` version. -/
theorem HasFPowerSeriesAt.isBigO_sub_partialSum_pow (hf : HasFPowerSeriesAt f p x) (n : ℕ) :
(fun y : E => f (x + y) - p.partialSum n y) =O[𝓝 0] fun y => ‖y‖ ^ n := by
rcases hf with ⟨r, hf⟩
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hf.r_pos with ⟨r', r'0, h⟩
obtain ⟨a, -, C, -, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n :=
hf.uniform_geometric_approx' h
refine' isBigO_iff.2 ⟨C * (a / r') ^ n, _⟩
replace r'0 : 0 < (r' : ℝ); · exact mod_cast r'0
filter_upwards [Metric.ball_mem_nhds (0 : E) r'0] with y hy
simpa [mul_pow, mul_div_assoc, mul_assoc, div_mul_eq_mul_div] using hp y hy n
set_option linter.uppercaseLean3 false in
#align has_fpower_series_at.is_O_sub_partial_sum_pow HasFPowerSeriesAt.isBigO_sub_partialSum_pow
/-- If `f` has formal power series `∑ n, pₙ` on a ball of radius `r`, then for `y, z` in any smaller
ball, the norm of the difference `f y - f z - p 1 (fun _ ↦ y - z)` is bounded above by
`C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖`. This lemma formulates this property using `IsBigO` and
`Filter.principal` on `E × E`. -/
theorem HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal
(hf : HasFPowerSeriesOnBall f p x r) (hr : r' < r) :
(fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) =O[𝓟 (EMetric.ball (x, x) r')]
fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖ := by
lift r' to ℝ≥0 using ne_top_of_lt hr
rcases (zero_le r').eq_or_lt with (rfl | hr'0)
· simp only [isBigO_bot, EMetric.ball_zero, principal_empty, ENNReal.coe_zero]
obtain ⟨a, ha, C, hC : 0 < C, hp⟩ :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n : ℕ, ‖p n‖ * (r' : ℝ) ^ n ≤ C * a ^ n
exact p.norm_mul_pow_le_mul_pow_of_lt_radius (hr.trans_le hf.r_le)
simp only [← le_div_iff (pow_pos (NNReal.coe_pos.2 hr'0) _)] at hp
set L : E × E → ℝ := fun y =>
C * (a / r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * (a / (1 - a) ^ 2 + 2 / (1 - a))
have hL : ∀ y ∈ EMetric.ball (x, x) r', ‖f y.1 - f y.2 - p 1 fun _ => y.1 - y.2‖ ≤ L y := by
intro y hy'
have hy : y ∈ EMetric.ball x r ×ˢ EMetric.ball x r := by
rw [EMetric.ball_prod_same]
exact EMetric.ball_subset_ball hr.le hy'
set A : ℕ → F := fun n => (p n fun _ => y.1 - x) - p n fun _ => y.2 - x
have hA : HasSum (fun n => A (n + 2)) (f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) := by
convert (hasSum_nat_add_iff' 2).2 ((hf.hasSum_sub hy.1).sub (hf.hasSum_sub hy.2)) using 1
rw [Finset.sum_range_succ, Finset.sum_range_one, hf.coeff_zero, hf.coeff_zero, sub_self,
zero_add, ← Subsingleton.pi_single_eq (0 : Fin 1) (y.1 - x), Pi.single,
← Subsingleton.pi_single_eq (0 : Fin 1) (y.2 - x), Pi.single, ← (p 1).map_sub, ← Pi.single,
Subsingleton.pi_single_eq, sub_sub_sub_cancel_right]
rw [EMetric.mem_ball, edist_eq_coe_nnnorm_sub, ENNReal.coe_lt_coe] at hy'
set B : ℕ → ℝ := fun n => C * (a / r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * ((n + 2) * a ^ n)
have hAB : ∀ n, ‖A (n + 2)‖ ≤ B n := fun n =>
calc
‖A (n + 2)‖ ≤ ‖p (n + 2)‖ * ↑(n + 2) * ‖y - (x, x)‖ ^ (n + 1) * ‖y.1 - y.2‖ := by
-- porting note: `pi_norm_const` was `pi_norm_const (_ : E)`
simpa only [Fintype.card_fin, pi_norm_const, Prod.norm_def, Pi.sub_def,
Prod.fst_sub, Prod.snd_sub, sub_sub_sub_cancel_right] using
(p <| n + 2).norm_image_sub_le (fun _ => y.1 - x) fun _ => y.2 - x
_ = ‖p (n + 2)‖ * ‖y - (x, x)‖ ^ n * (↑(n + 2) * ‖y - (x, x)‖ * ‖y.1 - y.2‖) := by
rw [pow_succ ‖y - (x, x)‖]
ring
-- porting note: the two `↑` in `↑r'` are new, without them, Lean fails to synthesize
-- instances `HDiv ℝ ℝ≥0 ?m` or `HMul ℝ ℝ≥0 ?m`
_ ≤ C * a ^ (n + 2) / ↑r' ^ (n + 2)
* ↑r' ^ n * (↑(n + 2) * ‖y - (x, x)‖ * ‖y.1 - y.2‖) := by
have : 0 < a := ha.1
gcongr
· apply hp
· apply hy'.le
_ = B n := by
-- porting note: in the original, `B` was in the `field_simp`, but now Lean does not
-- accept it. The current proof works in Lean 4, but does not in Lean 3.
field_simp [pow_succ]
simp only [mul_assoc, mul_comm, mul_left_comm]
have hBL : HasSum B (L y) := by
apply HasSum.mul_left
simp only [add_mul]
have : ‖a‖ < 1 := by simp only [Real.norm_eq_abs, abs_of_pos ha.1, ha.2]
rw [div_eq_mul_inv, div_eq_mul_inv]
exact (hasSum_coe_mul_geometric_of_norm_lt_1 this).add -- porting note: was `convert`!
((hasSum_geometric_of_norm_lt_1 this).mul_left 2)
exact hA.norm_le_of_bounded hBL hAB
suffices L =O[𝓟 (EMetric.ball (x, x) r')] fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖ by
refine' (IsBigO.of_bound 1 (eventually_principal.2 fun y hy => _)).trans this
rw [one_mul]
exact (hL y hy).trans (le_abs_self _)
simp_rw [mul_right_comm _ (_ * _)] -- porting note: there was an `L` inside the `simp_rw`.
exact (isBigO_refl _ _).const_mul_left _
set_option linter.uppercaseLean3 false in
#align has_fpower_series_on_ball.is_O_image_sub_image_sub_deriv_principal HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal
/-- If `f` has formal power series `∑ n, pₙ` on a ball of radius `r`, then for `y, z` in any smaller
ball, the norm of the difference `f y - f z - p 1 (fun _ ↦ y - z)` is bounded above by
`C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖`. -/
theorem HasFPowerSeriesOnBall.image_sub_sub_deriv_le (hf : HasFPowerSeriesOnBall f p x r)
(hr : r' < r) :
∃ C, ∀ᵉ (y ∈ EMetric.ball x r') (z ∈ EMetric.ball x r'),
‖f y - f z - p 1 fun _ => y - z‖ ≤ C * max ‖y - x‖ ‖z - x‖ * ‖y - z‖ := by
simpa only [isBigO_principal, mul_assoc, norm_mul, norm_norm, Prod.forall, EMetric.mem_ball,
Prod.edist_eq, max_lt_iff, and_imp, @forall_swap (_ < _) E] using
hf.isBigO_image_sub_image_sub_deriv_principal hr
#align has_fpower_series_on_ball.image_sub_sub_deriv_le HasFPowerSeriesOnBall.image_sub_sub_deriv_le
/-- If `f` has formal power series `∑ n, pₙ` at `x`, then
`f y - f z - p 1 (fun _ ↦ y - z) = O(‖(y, z) - (x, x)‖ * ‖y - z‖)` as `(y, z) → (x, x)`.
In particular, `f` is strictly differentiable at `x`. -/
theorem HasFPowerSeriesAt.isBigO_image_sub_norm_mul_norm_sub (hf : HasFPowerSeriesAt f p x) :
(fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) =O[𝓝 (x, x)] fun y =>
‖y - (x, x)‖ * ‖y.1 - y.2‖ := by
rcases hf with ⟨r, hf⟩
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hf.r_pos with ⟨r', r'0, h⟩
refine' (hf.isBigO_image_sub_image_sub_deriv_principal h).mono _
exact le_principal_iff.2 (EMetric.ball_mem_nhds _ r'0)
set_option linter.uppercaseLean3 false in
#align has_fpower_series_at.is_O_image_sub_norm_mul_norm_sub HasFPowerSeriesAt.isBigO_image_sub_norm_mul_norm_sub
/-- If a function admits a power series expansion at `x`, then it is the uniform limit of the
partial sums of this power series on strict subdisks of the disk of convergence, i.e., `f (x + y)`
is the uniform limit of `p.partialSum n y` there. -/
theorem HasFPowerSeriesOnBall.tendstoUniformlyOn {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r)
(h : (r' : ℝ≥0∞) < r) :
TendstoUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
(Metric.ball (0 : E) r') := by
obtain ⟨a, ha, C, -, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n
exact hf.uniform_geometric_approx h
refine' Metric.tendstoUniformlyOn_iff.2 fun ε εpos => _
have L : Tendsto (fun n => (C : ℝ) * a ^ n) atTop (𝓝 ((C : ℝ) * 0)) :=
tendsto_const_nhds.mul (tendsto_pow_atTop_nhds_0_of_lt_1 ha.1.le ha.2)
rw [mul_zero] at L
refine' (L.eventually (gt_mem_nhds εpos)).mono fun n hn y hy => _
rw [dist_eq_norm]
exact (hp y hy n).trans_lt hn
#align has_fpower_series_on_ball.tendsto_uniformly_on HasFPowerSeriesOnBall.tendstoUniformlyOn
/-- If a function admits a power series expansion at `x`, then it is the locally uniform limit of
the partial sums of this power series on the disk of convergence, i.e., `f (x + y)`
is the locally uniform limit of `p.partialSum n y` there. -/
theorem HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn (hf : HasFPowerSeriesOnBall f p x r) :
TendstoLocallyUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
(EMetric.ball (0 : E) r) := by
intro u hu x hx
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hx with ⟨r', xr', hr'⟩
have : EMetric.ball (0 : E) r' ∈ 𝓝 x := IsOpen.mem_nhds EMetric.isOpen_ball xr'
refine' ⟨EMetric.ball (0 : E) r', mem_nhdsWithin_of_mem_nhds this, _⟩
simpa [Metric.emetric_ball_nnreal] using hf.tendstoUniformlyOn hr' u hu
#align has_fpower_series_on_ball.tendsto_locally_uniformly_on HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn
/-- If a function admits a power series expansion at `x`, then it is the uniform limit of the
partial sums of this power series on strict subdisks of the disk of convergence, i.e., `f y`
is the uniform limit of `p.partialSum n (y - x)` there. -/
theorem HasFPowerSeriesOnBall.tendstoUniformlyOn' {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r)
(h : (r' : ℝ≥0∞) < r) :
TendstoUniformlyOn (fun n y => p.partialSum n (y - x)) f atTop (Metric.ball (x : E) r') := by
convert (hf.tendstoUniformlyOn h).comp fun y => y - x using 1
· simp [(· ∘ ·)]
· ext z
simp [dist_eq_norm]
#align has_fpower_series_on_ball.tendsto_uniformly_on' HasFPowerSeriesOnBall.tendstoUniformlyOn'
/-- If a function admits a power series expansion at `x`, then it is the locally uniform limit of
the partial sums of this power series on the disk of convergence, i.e., `f y`
is the locally uniform limit of `p.partialSum n (y - x)` there. -/
theorem HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn' (hf : HasFPowerSeriesOnBall f p x r) :
TendstoLocallyUniformlyOn (fun n y => p.partialSum n (y - x)) f atTop
(EMetric.ball (x : E) r) := by
have A : ContinuousOn (fun y : E => y - x) (EMetric.ball (x : E) r) :=
(continuous_id.sub continuous_const).continuousOn
convert hf.tendstoLocallyUniformlyOn.comp (fun y : E => y - x) _ A using 1
· ext z
simp
· intro z
simp [edist_eq_coe_nnnorm, edist_eq_coe_nnnorm_sub]
#align has_fpower_series_on_ball.tendsto_locally_uniformly_on' HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn'
/-- If a function admits a power series expansion on a disk, then it is continuous there. -/
protected theorem HasFPowerSeriesOnBall.continuousOn (hf : HasFPowerSeriesOnBall f p x r) :
ContinuousOn f (EMetric.ball x r) :=
hf.tendstoLocallyUniformlyOn'.continuousOn <|
eventually_of_forall fun n =>
((p.partialSum_continuous n).comp (continuous_id.sub continuous_const)).continuousOn
#align has_fpower_series_on_ball.continuous_on HasFPowerSeriesOnBall.continuousOn
protected theorem HasFPowerSeriesAt.continuousAt (hf : HasFPowerSeriesAt f p x) :
ContinuousAt f x :=
let ⟨_, hr⟩ := hf
hr.continuousOn.continuousAt (EMetric.ball_mem_nhds x hr.r_pos)
#align has_fpower_series_at.continuous_at HasFPowerSeriesAt.continuousAt
protected theorem AnalyticAt.continuousAt (hf : AnalyticAt 𝕜 f x) : ContinuousAt f x :=
let ⟨_, hp⟩ := hf
hp.continuousAt
#align analytic_at.continuous_at AnalyticAt.continuousAt
protected theorem AnalyticOn.continuousOn {s : Set E} (hf : AnalyticOn 𝕜 f s) : ContinuousOn f s :=
fun x hx => (hf x hx).continuousAt.continuousWithinAt
#align analytic_on.continuous_on AnalyticOn.continuousOn
/-- Analytic everywhere implies continuous -/
theorem AnalyticOn.continuous {f : E → F} (fa : AnalyticOn 𝕜 f univ) : Continuous f := by
rw [continuous_iff_continuousOn_univ]; exact fa.continuousOn
/-- In a complete space, the sum of a converging power series `p` admits `p` as a power series.
This is not totally obvious as we need to check the convergence of the series. -/
protected theorem FormalMultilinearSeries.hasFPowerSeriesOnBall [CompleteSpace F]
(p : FormalMultilinearSeries 𝕜 E F) (h : 0 < p.radius) :
HasFPowerSeriesOnBall p.sum p 0 p.radius :=
{ r_le := le_rfl
r_pos := h
hasSum := fun hy => by
rw [zero_add]
exact p.hasSum hy }
#align formal_multilinear_series.has_fpower_series_on_ball FormalMultilinearSeries.hasFPowerSeriesOnBall
theorem HasFPowerSeriesOnBall.sum (h : HasFPowerSeriesOnBall f p x r) {y : E}
(hy : y ∈ EMetric.ball (0 : E) r) : f (x + y) = p.sum y :=
(h.hasSum hy).tsum_eq.symm
#align has_fpower_series_on_ball.sum HasFPowerSeriesOnBall.sum
/-- The sum of a converging power series is continuous in its disk of convergence. -/
protected theorem FormalMultilinearSeries.continuousOn [CompleteSpace F] :
ContinuousOn p.sum (EMetric.ball 0 p.radius) := by
rcases (zero_le p.radius).eq_or_lt with h | h
· simp [← h, continuousOn_empty]
· exact (p.hasFPowerSeriesOnBall h).continuousOn
#align formal_multilinear_series.continuous_on FormalMultilinearSeries.continuousOn
end
/-!
### Uniqueness of power series
If a function `f : E → F` has two representations as power series at a point `x : E`, corresponding
to formal multilinear series `p₁` and `p₂`, then these representations agree term-by-term. That is,
for any `n : ℕ` and `y : E`, `p₁ n (fun i ↦ y) = p₂ n (fun i ↦ y)`. In the one-dimensional case,
when `f : 𝕜 → E`, the continuous multilinear maps `p₁ n` and `p₂ n` are given by
`ContinuousMultilinearMap.mkPiField`, and hence are determined completely by the value of
`p₁ n (fun i ↦ 1)`, so `p₁ = p₂`. Consequently, the radius of convergence for one series can be
transferred to the other.
-/
section Uniqueness
open ContinuousMultilinearMap
theorem Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero {n : ℕ} {p : E[×n]→L[𝕜] F}
(h : (fun y => p fun _ => y) =O[𝓝 0] fun y => ‖y‖ ^ (n + 1)) (y : E) : (p fun _ => y) = 0 := by
obtain ⟨c, c_pos, hc⟩ := h.exists_pos
obtain ⟨t, ht, t_open, z_mem⟩ := eventually_nhds_iff.mp (isBigOWith_iff.mp hc)
obtain ⟨δ, δ_pos, δε⟩ := (Metric.isOpen_iff.mp t_open) 0 z_mem
clear h hc z_mem
cases' n with n
· exact norm_eq_zero.mp (by
-- porting note: the symmetric difference of the `simpa only` sets:
-- added `Nat.zero_eq, zero_add, pow_one`
-- removed `zero_pow', Ne.def, Nat.one_ne_zero, not_false_iff`
simpa only [Nat.zero_eq, fin0_apply_norm, norm_eq_zero, norm_zero, zero_add, pow_one,
mul_zero, norm_le_zero_iff] using ht 0 (δε (Metric.mem_ball_self δ_pos)))
· refine' Or.elim (Classical.em (y = 0))
(fun hy => by simpa only [hy] using p.map_zero) fun hy => _
replace hy := norm_pos_iff.mpr hy
refine' norm_eq_zero.mp (le_antisymm (le_of_forall_pos_le_add fun ε ε_pos => _) (norm_nonneg _))
have h₀ := _root_.mul_pos c_pos (pow_pos hy (n.succ + 1))
obtain ⟨k, k_pos, k_norm⟩ := NormedField.exists_norm_lt 𝕜
(lt_min (mul_pos δ_pos (inv_pos.mpr hy)) (mul_pos ε_pos (inv_pos.mpr h₀)))
have h₁ : ‖k • y‖ < δ := by
rw [norm_smul]
exact inv_mul_cancel_right₀ hy.ne.symm δ ▸
mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_left _ _)) hy
have h₂ :=
calc
‖p fun _ => k • y‖ ≤ c * ‖k • y‖ ^ (n.succ + 1) := by
-- porting note: now Lean wants `_root_.`
simpa only [norm_pow, _root_.norm_norm] using ht (k • y) (δε (mem_ball_zero_iff.mpr h₁))
--simpa only [norm_pow, norm_norm] using ht (k • y) (δε (mem_ball_zero_iff.mpr h₁))
_ = ‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))) := by
-- porting note: added `Nat.succ_eq_add_one` since otherwise `ring` does not conclude.
simp only [norm_smul, mul_pow, Nat.succ_eq_add_one]
-- porting note: removed `rw [pow_succ]`, since it now becomes superfluous.
ring
have h₃ : ‖k‖ * (c * ‖y‖ ^ (n.succ + 1)) < ε :=
inv_mul_cancel_right₀ h₀.ne.symm ε ▸
mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_right _ _)) h₀
calc
‖p fun _ => y‖ = ‖k⁻¹ ^ n.succ‖ * ‖p fun _ => k • y‖ := by
simpa only [inv_smul_smul₀ (norm_pos_iff.mp k_pos), norm_smul, Finset.prod_const,
Finset.card_fin] using
congr_arg norm (p.map_smul_univ (fun _ : Fin n.succ => k⁻¹) fun _ : Fin n.succ => k • y)
_ ≤ ‖k⁻¹ ^ n.succ‖ * (‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1)))) := by gcongr
_ = ‖(k⁻¹ * k) ^ n.succ‖ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))) := by
rw [← mul_assoc]
simp [norm_mul, mul_pow]
_ ≤ 0 + ε := by
rw [inv_mul_cancel (norm_pos_iff.mp k_pos)]
simpa using h₃.le
set_option linter.uppercaseLean3 false in
#align asymptotics.is_O.continuous_multilinear_map_apply_eq_zero Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero
/-- If a formal multilinear series `p` represents the zero function at `x : E`, then the
terms `p n (fun i ↦ y)` appearing in the sum are zero for any `n : ℕ`, `y : E`. -/
theorem HasFPowerSeriesAt.apply_eq_zero {p : FormalMultilinearSeries 𝕜 E F} {x : E}
(h : HasFPowerSeriesAt 0 p x) (n : ℕ) : ∀ y : E, (p n fun _ => y) = 0 := by
refine' Nat.strong_induction_on n fun k hk => _
have psum_eq : p.partialSum (k + 1) = fun y => p k fun _ => y := by
funext z
refine' Finset.sum_eq_single _ (fun b hb hnb => _) fun hn => _
· have := Finset.mem_range_succ_iff.mp hb
simp only [hk b (this.lt_of_ne hnb), Pi.zero_apply]
· exact False.elim (hn (Finset.mem_range.mpr (lt_add_one k)))
replace h := h.isBigO_sub_partialSum_pow k.succ
simp only [psum_eq, zero_sub, Pi.zero_apply, Asymptotics.isBigO_neg_left] at h
exact h.continuousMultilinearMap_apply_eq_zero
#align has_fpower_series_at.apply_eq_zero HasFPowerSeriesAt.apply_eq_zero
/-- A one-dimensional formal multilinear series representing the zero function is zero. -/
theorem HasFPowerSeriesAt.eq_zero {p : FormalMultilinearSeries 𝕜 𝕜 E} {x : 𝕜}
(h : HasFPowerSeriesAt 0 p x) : p = 0 := by
-- porting note: `funext; ext` was `ext (n x)`
funext n
ext x
rw [← mkPiField_apply_one_eq_self (p n)]
-- porting note: nasty hack, was `simp [h.apply_eq_zero n 1]`
have := Or.intro_right ?_ (h.apply_eq_zero n 1)
simpa using this
#align has_fpower_series_at.eq_zero HasFPowerSeriesAt.eq_zero
/-- One-dimensional formal multilinear series representing the same function are equal. -/
theorem HasFPowerSeriesAt.eq_formalMultilinearSeries {p₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E}
{f : 𝕜 → E} {x : 𝕜} (h₁ : HasFPowerSeriesAt f p₁ x) (h₂ : HasFPowerSeriesAt f p₂ x) : p₁ = p₂ :=
sub_eq_zero.mp (HasFPowerSeriesAt.eq_zero (by simpa only [sub_self] using h₁.sub h₂))
#align has_fpower_series_at.eq_formal_multilinear_series HasFPowerSeriesAt.eq_formalMultilinearSeries
theorem HasFPowerSeriesAt.eq_formalMultilinearSeries_of_eventually
{p q : FormalMultilinearSeries 𝕜 𝕜 E} {f g : 𝕜 → E} {x : 𝕜} (hp : HasFPowerSeriesAt f p x)
(hq : HasFPowerSeriesAt g q x) (heq : ∀ᶠ z in 𝓝 x, f z = g z) : p = q :=
(hp.congr heq).eq_formalMultilinearSeries hq
#align has_fpower_series_at.eq_formal_multilinear_series_of_eventually HasFPowerSeriesAt.eq_formalMultilinearSeries_of_eventually
/-- A one-dimensional formal multilinear series representing a locally zero function is zero. -/
theorem HasFPowerSeriesAt.eq_zero_of_eventually {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E}
{x : 𝕜} (hp : HasFPowerSeriesAt f p x) (hf : f =ᶠ[𝓝 x] 0) : p = 0 :=
(hp.congr hf).eq_zero
#align has_fpower_series_at.eq_zero_of_eventually HasFPowerSeriesAt.eq_zero_of_eventually
/-- If a function `f : 𝕜 → E` has two power series representations at `x`, then the given radii in
which convergence is guaranteed may be interchanged. This can be useful when the formal multilinear
series in one representation has a particularly nice form, but the other has a larger radius. -/
theorem HasFPowerSeriesOnBall.exchange_radius {p₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E}
{r₁ r₂ : ℝ≥0∞} {x : 𝕜} (h₁ : HasFPowerSeriesOnBall f p₁ x r₁)
(h₂ : HasFPowerSeriesOnBall f p₂ x r₂) : HasFPowerSeriesOnBall f p₁ x r₂ :=
h₂.hasFPowerSeriesAt.eq_formalMultilinearSeries h₁.hasFPowerSeriesAt ▸ h₂
#align has_fpower_series_on_ball.exchange_radius HasFPowerSeriesOnBall.exchange_radius
/-- If a function `f : 𝕜 → E` has power series representation `p` on a ball of some radius and for
each positive radius it has some power series representation, then `p` converges to `f` on the whole
`𝕜`. -/
theorem HasFPowerSeriesOnBall.r_eq_top_of_exists {f : 𝕜 → E} {r : ℝ≥0∞} {x : 𝕜}
{p : FormalMultilinearSeries 𝕜 𝕜 E} (h : HasFPowerSeriesOnBall f p x r)
(h' : ∀ (r' : ℝ≥0) (_ : 0 < r'), ∃ p' : FormalMultilinearSeries 𝕜 𝕜 E,
HasFPowerSeriesOnBall f p' x r') :
HasFPowerSeriesOnBall f p x ∞ :=
{ r_le := ENNReal.le_of_forall_pos_nnreal_lt fun r hr _ =>
let ⟨_, hp'⟩ := h' r hr
(h.exchange_radius hp').r_le
r_pos := ENNReal.coe_lt_top
hasSum := fun {y} _ =>
let ⟨r', hr'⟩ := exists_gt ‖y‖₊
let ⟨_, hp'⟩ := h' r' hr'.ne_bot.bot_lt
(h.exchange_radius hp').hasSum <| mem_emetric_ball_zero_iff.mpr (ENNReal.coe_lt_coe.2 hr') }
#align has_fpower_series_on_ball.r_eq_top_of_exists HasFPowerSeriesOnBall.r_eq_top_of_exists
end Uniqueness
/-!
### Changing origin in a power series
If a function is analytic in a disk `D(x, R)`, then it is analytic in any disk contained in that
one. Indeed, one can write
$$
f (x + y + z) = \sum_{n} p_n (y + z)^n = \sum_{n, k} \binom{n}{k} p_n y^{n-k} z^k
= \sum_{k} \Bigl(\sum_{n} \binom{n}{k} p_n y^{n-k}\Bigr) z^k.
$$
The corresponding power series has thus a `k`-th coefficient equal to
$\sum_{n} \binom{n}{k} p_n y^{n-k}$. In the general case where `pₙ` is a multilinear map, this has
to be interpreted suitably: instead of having a binomial coefficient, one should sum over all
possible subsets `s` of `Fin n` of cardinal `k`, and attribute `z` to the indices in `s` and
`y` to the indices outside of `s`.
In this paragraph, we implement this. The new power series is called `p.changeOrigin y`. Then, we
check its convergence and the fact that its sum coincides with the original sum. The outcome of this
discussion is that the set of points where a function is analytic is open.
-/
namespace FormalMultilinearSeries
section
variable (p : FormalMultilinearSeries 𝕜 E F) {x y : E} {r R : ℝ≥0}
/-- A term of `FormalMultilinearSeries.changeOriginSeries`.
Given a formal multilinear series `p` and a point `x` in its ball of convergence,
`p.changeOrigin x` is a formal multilinear series such that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense. Each term of `p.changeOrigin x`
is itself an analytic function of `x` given by the series `p.changeOriginSeries`. Each term in
`changeOriginSeries` is the sum of `changeOriginSeriesTerm`'s over all `s` of cardinality `l`.
The definition is such that `p.changeOriginSeriesTerm k l s hs (fun _ ↦ x) (fun _ ↦ y) =
p (k + l) (s.piecewise (fun _ ↦ x) (fun _ ↦ y))`
-/
def changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
E[×l]→L[𝕜] E[×k]→L[𝕜] F := by
let a := ContinuousMultilinearMap.curryFinFinset 𝕜 E F hs
(by erw [Finset.card_compl, Fintype.card_fin, hs, add_tsub_cancel_right])
exact a (p (k + l))
#align formal_multilinear_series.change_origin_series_term FormalMultilinearSeries.changeOriginSeriesTerm
theorem changeOriginSeriesTerm_apply (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l)
(x y : E) :
(p.changeOriginSeriesTerm k l s hs (fun _ => x) fun _ => y) =
p (k + l) (s.piecewise (fun _ => x) fun _ => y) :=
ContinuousMultilinearMap.curryFinFinset_apply_const _ _ _ _ _
#align formal_multilinear_series.change_origin_series_term_apply FormalMultilinearSeries.changeOriginSeriesTerm_apply
@[simp]
theorem norm_changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
‖p.changeOriginSeriesTerm k l s hs‖ = ‖p (k + l)‖ := by
simp only [changeOriginSeriesTerm, LinearIsometryEquiv.norm_map]
#align formal_multilinear_series.norm_change_origin_series_term FormalMultilinearSeries.norm_changeOriginSeriesTerm
@[simp]
theorem nnnorm_changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
‖p.changeOriginSeriesTerm k l s hs‖₊ = ‖p (k + l)‖₊ := by
simp only [changeOriginSeriesTerm, LinearIsometryEquiv.nnnorm_map]
#align formal_multilinear_series.nnnorm_change_origin_series_term FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm
theorem nnnorm_changeOriginSeriesTerm_apply_le (k l : ℕ) (s : Finset (Fin (k + l)))
(hs : s.card = l) (x y : E) :
‖p.changeOriginSeriesTerm k l s hs (fun _ => x) fun _ => y‖₊ ≤
‖p (k + l)‖₊ * ‖x‖₊ ^ l * ‖y‖₊ ^ k := by
rw [← p.nnnorm_changeOriginSeriesTerm k l s hs, ← Fin.prod_const, ← Fin.prod_const]
apply ContinuousMultilinearMap.le_of_op_nnnorm_le
apply ContinuousMultilinearMap.le_op_nnnorm
#align formal_multilinear_series.nnnorm_change_origin_series_term_apply_le FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm_apply_le
/-- The power series for `f.changeOrigin k`.
Given a formal multilinear series `p` and a point `x` in its ball of convergence,
`p.changeOrigin x` is a formal multilinear series such that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense. Its `k`-th term is the sum of
the series `p.changeOriginSeries k`. -/
def changeOriginSeries (k : ℕ) : FormalMultilinearSeries 𝕜 E (E[×k]→L[𝕜] F) := fun l =>
∑ s : { s : Finset (Fin (k + l)) // Finset.card s = l }, p.changeOriginSeriesTerm k l s s.2
#align formal_multilinear_series.change_origin_series FormalMultilinearSeries.changeOriginSeries
theorem nnnorm_changeOriginSeries_le_tsum (k l : ℕ) :
‖p.changeOriginSeries k l‖₊ ≤
∑' _ : { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + l)‖₊ :=
(nnnorm_sum_le _ (fun t => changeOriginSeriesTerm p k l (Subtype.val t) t.prop)).trans_eq <| by
simp_rw [tsum_fintype, nnnorm_changeOriginSeriesTerm (p := p) (k := k) (l := l)]
#align formal_multilinear_series.nnnorm_change_origin_series_le_tsum FormalMultilinearSeries.nnnorm_changeOriginSeries_le_tsum
theorem nnnorm_changeOriginSeries_apply_le_tsum (k l : ℕ) (x : E) :
‖p.changeOriginSeries k l fun _ => x‖₊ ≤
∑' _ : { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + l)‖₊ * ‖x‖₊ ^ l := by
rw [NNReal.tsum_mul_right, ← Fin.prod_const]
exact (p.changeOriginSeries k l).le_of_op_nnnorm_le _ (p.nnnorm_changeOriginSeries_le_tsum _ _)
#align formal_multilinear_series.nnnorm_change_origin_series_apply_le_tsum FormalMultilinearSeries.nnnorm_changeOriginSeries_apply_le_tsum
/-- Changing the origin of a formal multilinear series `p`, so that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense.
-/
def changeOrigin (x : E) : FormalMultilinearSeries 𝕜 E F :=
fun k => (p.changeOriginSeries k).sum x
#align formal_multilinear_series.change_origin FormalMultilinearSeries.changeOrigin
/-- An auxiliary equivalence useful in the proofs about
`FormalMultilinearSeries.changeOriginSeries`: the set of triples `(k, l, s)`, where `s` is a
`Finset (Fin (k + l))` of cardinality `l` is equivalent to the set of pairs `(n, s)`, where `s` is a
`Finset (Fin n)`.
The forward map sends `(k, l, s)` to `(k + l, s)` and the inverse map sends `(n, s)` to
`(n - Finset.card s, Finset.card s, s)`. The actual definition is less readable because of problems
with non-definitional equalities. -/
@[simps]
def changeOriginIndexEquiv :
(Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) ≃ Σn : ℕ, Finset (Fin n) where
toFun s := ⟨s.1 + s.2.1, s.2.2⟩
invFun s :=
⟨s.1 - s.2.card, s.2.card,
⟨s.2.map
(Fin.castIso <| (tsub_add_cancel_of_le <| card_finset_fin_le s.2).symm).toEquiv.toEmbedding,
Finset.card_map _⟩⟩
left_inv := by
rintro ⟨k, l, ⟨s : Finset (Fin <| k + l), hs : s.card = l⟩⟩
dsimp only [Subtype.coe_mk]
-- Lean can't automatically generalize `k' = k + l - s.card`, `l' = s.card`, so we explicitly
-- formulate the generalized goal
suffices ∀ k' l', k' = k → l' = l → ∀ (hkl : k + l = k' + l') (hs'),
(⟨k', l', ⟨Finset.map (Fin.castIso hkl).toEquiv.toEmbedding s, hs'⟩⟩ :
Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) = ⟨k, l, ⟨s, hs⟩⟩ by
apply this <;> simp only [hs, add_tsub_cancel_right]
rintro _ _ rfl rfl hkl hs'
simp only [Equiv.refl_toEmbedding, Fin.castIso_refl, Finset.map_refl, eq_self_iff_true,
OrderIso.refl_toEquiv, and_self_iff, heq_iff_eq]
right_inv := by
rintro ⟨n, s⟩
simp [tsub_add_cancel_of_le (card_finset_fin_le s), Fin.castIso_to_equiv]
#align formal_multilinear_series.change_origin_index_equiv FormalMultilinearSeries.changeOriginIndexEquiv
theorem changeOriginSeries_summable_aux₁ {r r' : ℝ≥0} (hr : (r + r' : ℝ≥0∞) < p.radius) :
Summable fun s : Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l } =>
‖p (s.1 + s.2.1)‖₊ * r ^ s.2.1 * r' ^ s.1 := by
rw [← changeOriginIndexEquiv.symm.summable_iff]
dsimp only [Function.comp_def, changeOriginIndexEquiv_symm_apply_fst,
changeOriginIndexEquiv_symm_apply_snd_fst]
have : ∀ n : ℕ,
HasSum (fun s : Finset (Fin n) => ‖p (n - s.card + s.card)‖₊ * r ^ s.card * r' ^ (n - s.card))
(‖p n‖₊ * (r + r') ^ n) := by
intro n
-- TODO: why `simp only [tsub_add_cancel_of_le (card_finset_fin_le _)]` fails?
convert_to HasSum (fun s : Finset (Fin n) => ‖p n‖₊ * (r ^ s.card * r' ^ (n - s.card))) _
· ext1 s
rw [tsub_add_cancel_of_le (card_finset_fin_le _), mul_assoc]
rw [← Fin.sum_pow_mul_eq_add_pow]
exact (hasSum_fintype _).mul_left _
refine' NNReal.summable_sigma.2 ⟨fun n => (this n).summable, _⟩
simp only [(this _).tsum_eq]
exact p.summable_nnnorm_mul_pow hr
#align formal_multilinear_series.change_origin_series_summable_aux₁ FormalMultilinearSeries.changeOriginSeries_summable_aux₁
theorem changeOriginSeries_summable_aux₂ (hr : (r : ℝ≥0∞) < p.radius) (k : ℕ) :
Summable fun s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l } =>
‖p (k + s.1)‖₊ * r ^ s.1 := by
rcases ENNReal.lt_iff_exists_add_pos_lt.1 hr with ⟨r', h0, hr'⟩
simpa only [mul_inv_cancel_right₀ (pow_pos h0 _).ne'] using
((NNReal.summable_sigma.1 (p.changeOriginSeries_summable_aux₁ hr')).1 k).mul_right (r' ^ k)⁻¹
#align formal_multilinear_series.change_origin_series_summable_aux₂ FormalMultilinearSeries.changeOriginSeries_summable_aux₂
theorem changeOriginSeries_summable_aux₃ {r : ℝ≥0} (hr : ↑r < p.radius) (k : ℕ) :
Summable fun l : ℕ => ‖p.changeOriginSeries k l‖₊ * r ^ l := by
refine' NNReal.summable_of_le
(fun n => _) (NNReal.summable_sigma.1 <| p.changeOriginSeries_summable_aux₂ hr k).2
simp only [NNReal.tsum_mul_right]
exact mul_le_mul' (p.nnnorm_changeOriginSeries_le_tsum _ _) le_rfl
#align formal_multilinear_series.change_origin_series_summable_aux₃ FormalMultilinearSeries.changeOriginSeries_summable_aux₃
theorem le_changeOriginSeries_radius (k : ℕ) : p.radius ≤ (p.changeOriginSeries k).radius :=
ENNReal.le_of_forall_nnreal_lt fun _r hr =>
le_radius_of_summable_nnnorm _ (p.changeOriginSeries_summable_aux₃ hr k)
#align formal_multilinear_series.le_change_origin_series_radius FormalMultilinearSeries.le_changeOriginSeries_radius
theorem nnnorm_changeOrigin_le (k : ℕ) (h : (‖x‖₊ : ℝ≥0∞) < p.radius) :
‖p.changeOrigin x k‖₊ ≤
∑' s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + s.1)‖₊ * ‖x‖₊ ^ s.1 := by
refine' tsum_of_nnnorm_bounded _ fun l => p.nnnorm_changeOriginSeries_apply_le_tsum k l x
have := p.changeOriginSeries_summable_aux₂ h k
refine' HasSum.sigma this.hasSum fun l => _
exact ((NNReal.summable_sigma.1 this).1 l).hasSum
#align formal_multilinear_series.nnnorm_change_origin_le FormalMultilinearSeries.nnnorm_changeOrigin_le
/-- The radius of convergence of `p.changeOrigin x` is at least `p.radius - ‖x‖`. In other words,
`p.changeOrigin x` is well defined on the largest ball contained in the original ball of
convergence. -/
theorem changeOrigin_radius : p.radius - ‖x‖₊ ≤ (p.changeOrigin x).radius := by
refine' ENNReal.le_of_forall_pos_nnreal_lt fun r _h0 hr => _
rw [lt_tsub_iff_right, add_comm] at hr
have hr' : (‖x‖₊ : ℝ≥0∞) < p.radius := (le_add_right le_rfl).trans_lt hr
apply le_radius_of_summable_nnnorm
have : ∀ k : ℕ,
‖p.changeOrigin x k‖₊ * r ^ k ≤
(∑' s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + s.1)‖₊ * ‖x‖₊ ^ s.1) *
r ^ k :=
fun k => mul_le_mul_right' (p.nnnorm_changeOrigin_le k hr') (r ^ k)
refine' NNReal.summable_of_le this _
simpa only [← NNReal.tsum_mul_right] using
(NNReal.summable_sigma.1 (p.changeOriginSeries_summable_aux₁ hr)).2
#align formal_multilinear_series.change_origin_radius FormalMultilinearSeries.changeOrigin_radius
end
-- From this point on, assume that the space is complete, to make sure that series that converge
-- in norm also converge in `F`.
variable [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x y : E} {r R : ℝ≥0}
theorem hasFPowerSeriesOnBall_changeOrigin (k : ℕ) (hr : 0 < p.radius) :
HasFPowerSeriesOnBall (fun x => p.changeOrigin x k) (p.changeOriginSeries k) 0 p.radius :=
have := p.le_changeOriginSeries_radius k
((p.changeOriginSeries k).hasFPowerSeriesOnBall (hr.trans_le this)).mono hr this
#align formal_multilinear_series.has_fpower_series_on_ball_change_origin FormalMultilinearSeries.hasFPowerSeriesOnBall_changeOrigin
/-- Summing the series `p.changeOrigin x` at a point `y` gives back `p (x + y)`. -/
theorem changeOrigin_eval (h : (‖x‖₊ + ‖y‖₊ : ℝ≥0∞) < p.radius) :
(p.changeOrigin x).sum y = p.sum (x + y) := by
have radius_pos : 0 < p.radius := lt_of_le_of_lt (zero_le _) h
have x_mem_ball : x ∈ EMetric.ball (0 : E) p.radius :=
mem_emetric_ball_zero_iff.2 ((le_add_right le_rfl).trans_lt h)
have y_mem_ball : y ∈ EMetric.ball (0 : E) (p.changeOrigin x).radius := by
refine' mem_emetric_ball_zero_iff.2 (lt_of_lt_of_le _ p.changeOrigin_radius)
rwa [lt_tsub_iff_right, add_comm]
have x_add_y_mem_ball : x + y ∈ EMetric.ball (0 : E) p.radius := by
refine' mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt _ h)
exact mod_cast nnnorm_add_le x y
set f : (Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) → F := fun s =>
p.changeOriginSeriesTerm s.1 s.2.1 s.2.2 s.2.2.2 (fun _ => x) fun _ => y
have hsf : Summable f := by
refine' .of_nnnorm_bounded _ (p.changeOriginSeries_summable_aux₁ h) _
rintro ⟨k, l, s, hs⟩
dsimp only [Subtype.coe_mk]
exact p.nnnorm_changeOriginSeriesTerm_apply_le _ _ _ _ _ _
have hf : HasSum f ((p.changeOrigin x).sum y) := by
refine' HasSum.sigma_of_hasSum ((p.changeOrigin x).summable y_mem_ball).hasSum (fun k => _) hsf
· dsimp only
refine' ContinuousMultilinearMap.hasSum_eval _ _
have := (p.hasFPowerSeriesOnBall_changeOrigin k radius_pos).hasSum x_mem_ball
rw [zero_add] at this
refine' HasSum.sigma_of_hasSum this (fun l => _) _
· simp only [changeOriginSeries, ContinuousMultilinearMap.sum_apply]
apply hasSum_fintype
· refine' .of_nnnorm_bounded _
(p.changeOriginSeries_summable_aux₂ (mem_emetric_ball_zero_iff.1 x_mem_ball) k) fun s => _
refine' (ContinuousMultilinearMap.le_op_nnnorm _ _).trans_eq _
simp
refine' hf.unique (changeOriginIndexEquiv.symm.hasSum_iff.1 _)
refine' HasSum.sigma_of_hasSum
(p.hasSum x_add_y_mem_ball) (fun n => _) (changeOriginIndexEquiv.symm.summable_iff.2 hsf)
erw [(p n).map_add_univ (fun _ => x) fun _ => y]
-- porting note: added explicit function
convert hasSum_fintype (fun c : Finset (Fin n) => f (changeOriginIndexEquiv.symm ⟨n, c⟩))
rename_i s _
dsimp only [changeOriginSeriesTerm, (· ∘ ·), changeOriginIndexEquiv_symm_apply_fst,
changeOriginIndexEquiv_symm_apply_snd_fst, changeOriginIndexEquiv_symm_apply_snd_snd_coe]
rw [ContinuousMultilinearMap.curryFinFinset_apply_const]
have : ∀ (m) (hm : n = m), p n (s.piecewise (fun _ => x) fun _ => y) =
p m ((s.map (Fin.castIso hm).toEquiv.toEmbedding).piecewise (fun _ => x) fun _ => y) := by
rintro m rfl
simp (config := { unfoldPartialApp := true }) [Finset.piecewise]
apply this
#align formal_multilinear_series.change_origin_eval FormalMultilinearSeries.changeOrigin_eval
/-- Power series terms are analytic as we vary the origin -/
theorem analyticAt_changeOrigin (p : FormalMultilinearSeries 𝕜 E F) (rp : p.radius > 0) (n : ℕ) :
AnalyticAt 𝕜 (fun x ↦ p.changeOrigin x n) 0 :=
(FormalMultilinearSeries.hasFPowerSeriesOnBall_changeOrigin p n rp).analyticAt
end FormalMultilinearSeries
section
variable [CompleteSpace F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x y : E} {r : ℝ≥0∞}
/-- If a function admits a power series expansion `p` on a ball `B (x, r)`, then it also admits a
power series on any subball of this ball (even with a different center), given by `p.changeOrigin`.
-/
theorem HasFPowerSeriesOnBall.changeOrigin (hf : HasFPowerSeriesOnBall f p x r)
(h : (‖y‖₊ : ℝ≥0∞) < r) : HasFPowerSeriesOnBall f (p.changeOrigin y) (x + y) (r - ‖y‖₊) :=
{ r_le := by
apply le_trans _ p.changeOrigin_radius
exact tsub_le_tsub hf.r_le le_rfl
r_pos := by simp [h]
hasSum := fun {z} hz => by
have : f (x + y + z) =
FormalMultilinearSeries.sum (FormalMultilinearSeries.changeOrigin p y) z := by
rw [mem_emetric_ball_zero_iff, lt_tsub_iff_right, add_comm] at hz
rw [p.changeOrigin_eval (hz.trans_le hf.r_le), add_assoc, hf.sum]
refine' mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt _ hz)
exact mod_cast nnnorm_add_le y z
rw [this]
apply (p.changeOrigin y).hasSum
refine' EMetric.ball_subset_ball (le_trans _ p.changeOrigin_radius) hz
exact tsub_le_tsub hf.r_le le_rfl }
#align has_fpower_series_on_ball.change_origin HasFPowerSeriesOnBall.changeOrigin
/-- If a function admits a power series expansion `p` on an open ball `B (x, r)`, then
it is analytic at every point of this ball. -/
theorem HasFPowerSeriesOnBall.analyticAt_of_mem (hf : HasFPowerSeriesOnBall f p x r)
(h : y ∈ EMetric.ball x r) : AnalyticAt 𝕜 f y := by
have : (‖y - x‖₊ : ℝ≥0∞) < r := by simpa [edist_eq_coe_nnnorm_sub] using h
have := hf.changeOrigin this
rw [add_sub_cancel'_right] at this
exact this.analyticAt
#align has_fpower_series_on_ball.analytic_at_of_mem HasFPowerSeriesOnBall.analyticAt_of_mem
theorem HasFPowerSeriesOnBall.analyticOn (hf : HasFPowerSeriesOnBall f p x r) :
AnalyticOn 𝕜 f (EMetric.ball x r) :=
fun _y hy => hf.analyticAt_of_mem hy
#align has_fpower_series_on_ball.analytic_on HasFPowerSeriesOnBall.analyticOn
variable (𝕜 f)
/-- For any function `f` from a normed vector space to a Banach space, the set of points `x` such
that `f` is analytic at `x` is open. -/
theorem isOpen_analyticAt : IsOpen { x | AnalyticAt 𝕜 f x } := by
rw [isOpen_iff_mem_nhds]
rintro x ⟨p, r, hr⟩
exact mem_of_superset (EMetric.ball_mem_nhds _ hr.r_pos) fun y hy => hr.analyticAt_of_mem hy
#align is_open_analytic_at isOpen_analyticAt
variable {𝕜}
theorem AnalyticAt.eventually_analyticAt {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∀ᶠ y in 𝓝 x, AnalyticAt 𝕜 f y :=
(isOpen_analyticAt 𝕜 f).mem_nhds h
theorem AnalyticAt.exists_mem_nhds_analyticOn {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∃ s ∈ 𝓝 x, AnalyticOn 𝕜 f s :=
h.eventually_analyticAt.exists_mem
/-- If we're analytic at a point, we're analytic in a nonempty ball -/
theorem AnalyticAt.exists_ball_analyticOn {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∃ r : ℝ, 0 < r ∧ AnalyticOn 𝕜 f (Metric.ball x r) :=
Metric.isOpen_iff.mp (isOpen_analyticAt _ _) _ h
end
section
open FormalMultilinearSeries
variable {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E} {z₀ : 𝕜}
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z)) := by
refine' ⟨fun ⟨r, _, r_pos, h⟩ =>
eventually_of_mem (EMetric.ball_mem_nhds 0 r_pos) fun _ => by simpa using h, _⟩
simp only [Metric.eventually_nhds_iff]
rintro ⟨r, r_pos, h⟩
refine' ⟨p.radius ⊓ r.toNNReal, by simp, _, _⟩
· simp only [r_pos.lt, lt_inf_iff, ENNReal.coe_pos, Real.toNNReal_pos, and_true_iff]
obtain ⟨z, z_pos, le_z⟩ := NormedField.exists_norm_lt 𝕜 r_pos.lt
have : (‖z‖₊ : ENNReal) ≤ p.radius := by
simp only [dist_zero_right] at h
apply FormalMultilinearSeries.le_radius_of_tendsto
convert tendsto_norm.comp (h le_z).summable.tendsto_atTop_zero
funext
simp [norm_smul, mul_comm]
|
refine' lt_of_lt_of_le _ this
|
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z)) := by
refine' ⟨fun ⟨r, _, r_pos, h⟩ =>
eventually_of_mem (EMetric.ball_mem_nhds 0 r_pos) fun _ => by simpa using h, _⟩
simp only [Metric.eventually_nhds_iff]
rintro ⟨r, r_pos, h⟩
refine' ⟨p.radius ⊓ r.toNNReal, by simp, _, _⟩
· simp only [r_pos.lt, lt_inf_iff, ENNReal.coe_pos, Real.toNNReal_pos, and_true_iff]
obtain ⟨z, z_pos, le_z⟩ := NormedField.exists_norm_lt 𝕜 r_pos.lt
have : (‖z‖₊ : ENNReal) ≤ p.radius := by
simp only [dist_zero_right] at h
apply FormalMultilinearSeries.le_radius_of_tendsto
convert tendsto_norm.comp (h le_z).summable.tendsto_atTop_zero
funext
simp [norm_smul, mul_comm]
|
Mathlib.Analysis.Analytic.Basic.1430_0.jQw1fRSE1vGpOll
|
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z))
|
Mathlib_Analysis_Analytic_Basic
|
case intro.intro.refine'_1.intro.intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
p : FormalMultilinearSeries 𝕜 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
r : ℝ
r_pos : r > 0
h : ∀ ⦃y : 𝕜⦄, dist y 0 < r → HasSum (fun n => y ^ n • coeff p n) (f (z₀ + y))
z : 𝕜
z_pos : 0 < ‖z‖
le_z : ‖z‖ < r
this : ↑‖z‖₊ ≤ radius p
⊢ 0 < ↑‖z‖₊
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Logic.Equiv.Fin
import Mathlib.Topology.Algebra.InfiniteSum.Module
#align_import analysis.analytic.basic from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
/-!
# Analytic functions
A function is analytic in one dimension around `0` if it can be written as a converging power series
`Σ pₙ zⁿ`. This definition can be extended to any dimension (even in infinite dimension) by
requiring that `pₙ` is a continuous `n`-multilinear map. In general, `pₙ` is not unique (in two
dimensions, taking `p₂ (x, y) (x', y') = x y'` or `y x'` gives the same map when applied to a
vector `(x, y) (x, y)`). A way to guarantee uniqueness is to take a symmetric `pₙ`, but this is not
always possible in nonzero characteristic (in characteristic 2, the previous example has no
symmetric representative). Therefore, we do not insist on symmetry or uniqueness in the definition,
and we only require the existence of a converging series.
The general framework is important to say that the exponential map on bounded operators on a Banach
space is analytic, as well as the inverse on invertible operators.
## Main definitions
Let `p` be a formal multilinear series from `E` to `F`, i.e., `p n` is a multilinear map on `E^n`
for `n : ℕ`.
* `p.radius`: the largest `r : ℝ≥0∞` such that `‖p n‖ * r^n` grows subexponentially.
* `p.le_radius_of_bound`, `p.le_radius_of_bound_nnreal`, `p.le_radius_of_isBigO`: if `‖p n‖ * r ^ n`
is bounded above, then `r ≤ p.radius`;
* `p.isLittleO_of_lt_radius`, `p.norm_mul_pow_le_mul_pow_of_lt_radius`,
`p.isLittleO_one_of_lt_radius`,
`p.norm_mul_pow_le_of_lt_radius`, `p.nnnorm_mul_pow_le_of_lt_radius`: if `r < p.radius`, then
`‖p n‖ * r ^ n` tends to zero exponentially;
* `p.lt_radius_of_isBigO`: if `r ≠ 0` and `‖p n‖ * r ^ n = O(a ^ n)` for some `-1 < a < 1`, then
`r < p.radius`;
* `p.partialSum n x`: the sum `∑_{i = 0}^{n-1} pᵢ xⁱ`.
* `p.sum x`: the sum `∑'_{i = 0}^{∞} pᵢ xⁱ`.
Additionally, let `f` be a function from `E` to `F`.
* `HasFPowerSeriesOnBall f p x r`: on the ball of center `x` with radius `r`,
`f (x + y) = ∑'_n pₙ yⁿ`.
* `HasFPowerSeriesAt f p x`: on some ball of center `x` with positive radius, holds
`HasFPowerSeriesOnBall f p x r`.
* `AnalyticAt 𝕜 f x`: there exists a power series `p` such that holds `HasFPowerSeriesAt f p x`.
* `AnalyticOn 𝕜 f s`: the function `f` is analytic at every point of `s`.
We develop the basic properties of these notions, notably:
* If a function admits a power series, it is continuous (see
`HasFPowerSeriesOnBall.continuousOn` and `HasFPowerSeriesAt.continuousAt` and
`AnalyticAt.continuousAt`).
* In a complete space, the sum of a formal power series with positive radius is well defined on the
disk of convergence, see `FormalMultilinearSeries.hasFPowerSeriesOnBall`.
* If a function admits a power series in a ball, then it is analytic at any point `y` of this ball,
and the power series there can be expressed in terms of the initial power series `p` as
`p.changeOrigin y`. See `HasFPowerSeriesOnBall.changeOrigin`. It follows in particular that
the set of points at which a given function is analytic is open, see `isOpen_analyticAt`.
## Implementation details
We only introduce the radius of convergence of a power series, as `p.radius`.
For a power series in finitely many dimensions, there is a finer (directional, coordinate-dependent)
notion, describing the polydisk of convergence. This notion is more specific, and not necessary to
build the general theory. We do not define it here.
-/
noncomputable section
variable {𝕜 E F G : Type*}
open Topology Classical BigOperators NNReal Filter ENNReal
open Set Filter Asymptotics
namespace FormalMultilinearSeries
variable [Ring 𝕜] [AddCommGroup E] [AddCommGroup F] [Module 𝕜 E] [Module 𝕜 F]
variable [TopologicalSpace E] [TopologicalSpace F]
variable [TopologicalAddGroup E] [TopologicalAddGroup F]
variable [ContinuousConstSMul 𝕜 E] [ContinuousConstSMul 𝕜 F]
/-- Given a formal multilinear series `p` and a vector `x`, then `p.sum x` is the sum `Σ pₙ xⁿ`. A
priori, it only behaves well when `‖x‖ < p.radius`. -/
protected def sum (p : FormalMultilinearSeries 𝕜 E F) (x : E) : F :=
∑' n : ℕ, p n fun _ => x
#align formal_multilinear_series.sum FormalMultilinearSeries.sum
/-- Given a formal multilinear series `p` and a vector `x`, then `p.partialSum n x` is the sum
`Σ pₖ xᵏ` for `k ∈ {0,..., n-1}`. -/
def partialSum (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) (x : E) : F :=
∑ k in Finset.range n, p k fun _ : Fin k => x
#align formal_multilinear_series.partial_sum FormalMultilinearSeries.partialSum
/-- The partial sums of a formal multilinear series are continuous. -/
theorem partialSum_continuous (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) :
Continuous (p.partialSum n) := by
unfold partialSum -- Porting note: added
continuity
#align formal_multilinear_series.partial_sum_continuous FormalMultilinearSeries.partialSum_continuous
end FormalMultilinearSeries
/-! ### The radius of a formal multilinear series -/
variable [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F]
[NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G]
namespace FormalMultilinearSeries
variable (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
/-- The radius of a formal multilinear series is the largest `r` such that the sum `Σ ‖pₙ‖ ‖y‖ⁿ`
converges for all `‖y‖ < r`. This implies that `Σ pₙ yⁿ` converges for all `‖y‖ < r`, but these
definitions are *not* equivalent in general. -/
def radius (p : FormalMultilinearSeries 𝕜 E F) : ℝ≥0∞ :=
⨆ (r : ℝ≥0) (C : ℝ) (_ : ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C), (r : ℝ≥0∞)
#align formal_multilinear_series.radius FormalMultilinearSeries.radius
/-- If `‖pₙ‖ rⁿ` is bounded in `n`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_bound (C : ℝ) {r : ℝ≥0} (h : ∀ n : ℕ, ‖p n‖ * (r : ℝ) ^ n ≤ C) :
(r : ℝ≥0∞) ≤ p.radius :=
le_iSup_of_le r <| le_iSup_of_le C <| le_iSup (fun _ => (r : ℝ≥0∞)) h
#align formal_multilinear_series.le_radius_of_bound FormalMultilinearSeries.le_radius_of_bound
/-- If `‖pₙ‖ rⁿ` is bounded in `n`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_bound_nnreal (C : ℝ≥0) {r : ℝ≥0} (h : ∀ n : ℕ, ‖p n‖₊ * r ^ n ≤ C) :
(r : ℝ≥0∞) ≤ p.radius :=
p.le_radius_of_bound C fun n => mod_cast h n
#align formal_multilinear_series.le_radius_of_bound_nnreal FormalMultilinearSeries.le_radius_of_bound_nnreal
/-- If `‖pₙ‖ rⁿ = O(1)`, as `n → ∞`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_isBigO (h : (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] fun _ => (1 : ℝ)) :
↑r ≤ p.radius :=
Exists.elim (isBigO_one_nat_atTop_iff.1 h) fun C hC =>
p.le_radius_of_bound C fun n => (le_abs_self _).trans (hC n)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.le_radius_of_is_O FormalMultilinearSeries.le_radius_of_isBigO
theorem le_radius_of_eventually_le (C) (h : ∀ᶠ n in atTop, ‖p n‖ * (r : ℝ) ^ n ≤ C) :
↑r ≤ p.radius :=
p.le_radius_of_isBigO <| IsBigO.of_bound C <| h.mono fun n hn => by simpa
#align formal_multilinear_series.le_radius_of_eventually_le FormalMultilinearSeries.le_radius_of_eventually_le
theorem le_radius_of_summable_nnnorm (h : Summable fun n => ‖p n‖₊ * r ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_bound_nnreal (∑' n, ‖p n‖₊ * r ^ n) fun _ => le_tsum' h _
#align formal_multilinear_series.le_radius_of_summable_nnnorm FormalMultilinearSeries.le_radius_of_summable_nnnorm
theorem le_radius_of_summable (h : Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_summable_nnnorm <| by
simp only [← coe_nnnorm] at h
exact mod_cast h
#align formal_multilinear_series.le_radius_of_summable FormalMultilinearSeries.le_radius_of_summable
theorem radius_eq_top_of_forall_nnreal_isBigO
(h : ∀ r : ℝ≥0, (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] fun _ => (1 : ℝ)) : p.radius = ∞ :=
ENNReal.eq_top_of_forall_nnreal_le fun r => p.le_radius_of_isBigO (h r)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.radius_eq_top_of_forall_nnreal_is_O FormalMultilinearSeries.radius_eq_top_of_forall_nnreal_isBigO
theorem radius_eq_top_of_eventually_eq_zero (h : ∀ᶠ n in atTop, p n = 0) : p.radius = ∞ :=
p.radius_eq_top_of_forall_nnreal_isBigO fun r =>
(isBigO_zero _ _).congr' (h.mono fun n hn => by simp [hn]) EventuallyEq.rfl
#align formal_multilinear_series.radius_eq_top_of_eventually_eq_zero FormalMultilinearSeries.radius_eq_top_of_eventually_eq_zero
theorem radius_eq_top_of_forall_image_add_eq_zero (n : ℕ) (hn : ∀ m, p (m + n) = 0) :
p.radius = ∞ :=
p.radius_eq_top_of_eventually_eq_zero <|
mem_atTop_sets.2 ⟨n, fun _ hk => tsub_add_cancel_of_le hk ▸ hn _⟩
#align formal_multilinear_series.radius_eq_top_of_forall_image_add_eq_zero FormalMultilinearSeries.radius_eq_top_of_forall_image_add_eq_zero
@[simp]
theorem constFormalMultilinearSeries_radius {v : F} :
(constFormalMultilinearSeries 𝕜 E v).radius = ⊤ :=
(constFormalMultilinearSeries 𝕜 E v).radius_eq_top_of_forall_image_add_eq_zero 1
(by simp [constFormalMultilinearSeries])
#align formal_multilinear_series.const_formal_multilinear_series_radius FormalMultilinearSeries.constFormalMultilinearSeries_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` tends to zero exponentially:
for some `0 < a < 1`, `‖p n‖ rⁿ = o(aⁿ)`. -/
theorem isLittleO_of_lt_radius (h : ↑r < p.radius) :
∃ a ∈ Ioo (0 : ℝ) 1, (fun n => ‖p n‖ * (r : ℝ) ^ n) =o[atTop] (a ^ ·) := by
have := (TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 4
rw [this]
-- Porting note: was
-- rw [(TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 4]
simp only [radius, lt_iSup_iff] at h
rcases h with ⟨t, C, hC, rt⟩
rw [ENNReal.coe_lt_coe, ← NNReal.coe_lt_coe] at rt
have : 0 < (t : ℝ) := r.coe_nonneg.trans_lt rt
rw [← div_lt_one this] at rt
refine' ⟨_, rt, C, Or.inr zero_lt_one, fun n => _⟩
calc
|‖p n‖ * (r : ℝ) ^ n| = ‖p n‖ * (t : ℝ) ^ n * (r / t : ℝ) ^ n := by
field_simp [mul_right_comm, abs_mul]
_ ≤ C * (r / t : ℝ) ^ n := by gcongr; apply hC
#align formal_multilinear_series.is_o_of_lt_radius FormalMultilinearSeries.isLittleO_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ = o(1)`. -/
theorem isLittleO_one_of_lt_radius (h : ↑r < p.radius) :
(fun n => ‖p n‖ * (r : ℝ) ^ n) =o[atTop] (fun _ => 1 : ℕ → ℝ) :=
let ⟨_, ha, hp⟩ := p.isLittleO_of_lt_radius h
hp.trans <| (isLittleO_pow_pow_of_lt_left ha.1.le ha.2).congr (fun _ => rfl) one_pow
#align formal_multilinear_series.is_o_one_of_lt_radius FormalMultilinearSeries.isLittleO_one_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` tends to zero exponentially:
for some `0 < a < 1` and `C > 0`, `‖p n‖ * r ^ n ≤ C * a ^ n`. -/
theorem norm_mul_pow_le_mul_pow_of_lt_radius (h : ↑r < p.radius) :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C * a ^ n := by
-- Porting note: moved out of `rcases`
have := ((TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 5).mp
(p.isLittleO_of_lt_radius h)
rcases this with ⟨a, ha, C, hC, H⟩
exact ⟨a, ha, C, hC, fun n => (le_abs_self _).trans (H n)⟩
#align formal_multilinear_series.norm_mul_pow_le_mul_pow_of_lt_radius FormalMultilinearSeries.norm_mul_pow_le_mul_pow_of_lt_radius
/-- If `r ≠ 0` and `‖pₙ‖ rⁿ = O(aⁿ)` for some `-1 < a < 1`, then `r < p.radius`. -/
theorem lt_radius_of_isBigO (h₀ : r ≠ 0) {a : ℝ} (ha : a ∈ Ioo (-1 : ℝ) 1)
(hp : (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] (a ^ ·)) : ↑r < p.radius := by
-- Porting note: moved out of `rcases`
have := ((TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 2 5)
rcases this.mp ⟨a, ha, hp⟩ with ⟨a, ha, C, hC, hp⟩
rw [← pos_iff_ne_zero, ← NNReal.coe_pos] at h₀
lift a to ℝ≥0 using ha.1.le
have : (r : ℝ) < r / a := by
simpa only [div_one] using (div_lt_div_left h₀ zero_lt_one ha.1).2 ha.2
norm_cast at this
rw [← ENNReal.coe_lt_coe] at this
refine' this.trans_le (p.le_radius_of_bound C fun n => _)
rw [NNReal.coe_div, div_pow, ← mul_div_assoc, div_le_iff (pow_pos ha.1 n)]
exact (le_abs_self _).trans (hp n)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.lt_radius_of_is_O FormalMultilinearSeries.lt_radius_of_isBigO
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem norm_mul_pow_le_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C :=
let ⟨_, ha, C, hC, h⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius h
⟨C, hC, fun n => (h n).trans <| mul_le_of_le_one_right hC.lt.le (pow_le_one _ ha.1.le ha.2.le)⟩
#align formal_multilinear_series.norm_mul_pow_le_of_lt_radius FormalMultilinearSeries.norm_mul_pow_le_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem norm_le_div_pow_of_pos_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h0 : 0 < r) (h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖ ≤ C / (r : ℝ) ^ n :=
let ⟨C, hC, hp⟩ := p.norm_mul_pow_le_of_lt_radius h
⟨C, hC, fun n => Iff.mpr (le_div_iff (pow_pos h0 _)) (hp n)⟩
#align formal_multilinear_series.norm_le_div_pow_of_pos_of_lt_radius FormalMultilinearSeries.norm_le_div_pow_of_pos_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem nnnorm_mul_pow_le_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖₊ * r ^ n ≤ C :=
let ⟨C, hC, hp⟩ := p.norm_mul_pow_le_of_lt_radius h
⟨⟨C, hC.lt.le⟩, hC, mod_cast hp⟩
#align formal_multilinear_series.nnnorm_mul_pow_le_of_lt_radius FormalMultilinearSeries.nnnorm_mul_pow_le_of_lt_radius
theorem le_radius_of_tendsto (p : FormalMultilinearSeries 𝕜 E F) {l : ℝ}
(h : Tendsto (fun n => ‖p n‖ * (r : ℝ) ^ n) atTop (𝓝 l)) : ↑r ≤ p.radius :=
p.le_radius_of_isBigO (h.isBigO_one _)
#align formal_multilinear_series.le_radius_of_tendsto FormalMultilinearSeries.le_radius_of_tendsto
theorem le_radius_of_summable_norm (p : FormalMultilinearSeries 𝕜 E F)
(hs : Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_tendsto hs.tendsto_atTop_zero
#align formal_multilinear_series.le_radius_of_summable_norm FormalMultilinearSeries.le_radius_of_summable_norm
theorem not_summable_norm_of_radius_lt_nnnorm (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(h : p.radius < ‖x‖₊) : ¬Summable fun n => ‖p n‖ * ‖x‖ ^ n :=
fun hs => not_le_of_lt h (p.le_radius_of_summable_norm hs)
#align formal_multilinear_series.not_summable_norm_of_radius_lt_nnnorm FormalMultilinearSeries.not_summable_norm_of_radius_lt_nnnorm
theorem summable_norm_mul_pow (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0} (h : ↑r < p.radius) :
Summable fun n : ℕ => ‖p n‖ * (r : ℝ) ^ n := by
obtain ⟨a, ha : a ∈ Ioo (0 : ℝ) 1, C, - : 0 < C, hp⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius h
exact .of_nonneg_of_le (fun n => mul_nonneg (norm_nonneg _) (pow_nonneg r.coe_nonneg _))
hp ((summable_geometric_of_lt_1 ha.1.le ha.2).mul_left _)
#align formal_multilinear_series.summable_norm_mul_pow FormalMultilinearSeries.summable_norm_mul_pow
theorem summable_norm_apply (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : Summable fun n : ℕ => ‖p n fun _ => x‖ := by
rw [mem_emetric_ball_zero_iff] at hx
refine' .of_nonneg_of_le
(fun _ => norm_nonneg _) (fun n => ((p n).le_op_norm _).trans_eq _) (p.summable_norm_mul_pow hx)
simp
#align formal_multilinear_series.summable_norm_apply FormalMultilinearSeries.summable_norm_apply
theorem summable_nnnorm_mul_pow (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0} (h : ↑r < p.radius) :
Summable fun n : ℕ => ‖p n‖₊ * r ^ n := by
rw [← NNReal.summable_coe]
push_cast
exact p.summable_norm_mul_pow h
#align formal_multilinear_series.summable_nnnorm_mul_pow FormalMultilinearSeries.summable_nnnorm_mul_pow
protected theorem summable [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : Summable fun n : ℕ => p n fun _ => x :=
(p.summable_norm_apply hx).of_norm
#align formal_multilinear_series.summable FormalMultilinearSeries.summable
theorem radius_eq_top_of_summable_norm (p : FormalMultilinearSeries 𝕜 E F)
(hs : ∀ r : ℝ≥0, Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : p.radius = ∞ :=
ENNReal.eq_top_of_forall_nnreal_le fun r => p.le_radius_of_summable_norm (hs r)
#align formal_multilinear_series.radius_eq_top_of_summable_norm FormalMultilinearSeries.radius_eq_top_of_summable_norm
theorem radius_eq_top_iff_summable_norm (p : FormalMultilinearSeries 𝕜 E F) :
p.radius = ∞ ↔ ∀ r : ℝ≥0, Summable fun n => ‖p n‖ * (r : ℝ) ^ n := by
constructor
· intro h r
obtain ⟨a, ha : a ∈ Ioo (0 : ℝ) 1, C, - : 0 < C, hp⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius
(show (r : ℝ≥0∞) < p.radius from h.symm ▸ ENNReal.coe_lt_top)
refine' .of_norm_bounded
(fun n => (C : ℝ) * a ^ n) ((summable_geometric_of_lt_1 ha.1.le ha.2).mul_left _) fun n => _
specialize hp n
rwa [Real.norm_of_nonneg (mul_nonneg (norm_nonneg _) (pow_nonneg r.coe_nonneg n))]
· exact p.radius_eq_top_of_summable_norm
#align formal_multilinear_series.radius_eq_top_iff_summable_norm FormalMultilinearSeries.radius_eq_top_iff_summable_norm
/-- If the radius of `p` is positive, then `‖pₙ‖` grows at most geometrically. -/
theorem le_mul_pow_of_radius_pos (p : FormalMultilinearSeries 𝕜 E F) (h : 0 < p.radius) :
∃ (C r : _) (hC : 0 < C) (_ : 0 < r), ∀ n, ‖p n‖ ≤ C * r ^ n := by
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 h with ⟨r, r0, rlt⟩
have rpos : 0 < (r : ℝ) := by simp [ENNReal.coe_pos.1 r0]
rcases norm_le_div_pow_of_pos_of_lt_radius p rpos rlt with ⟨C, Cpos, hCp⟩
refine' ⟨C, r⁻¹, Cpos, by simp only [inv_pos, rpos], fun n => _⟩
-- Porting note: was `convert`
rw [inv_pow, ← div_eq_mul_inv]
exact hCp n
#align formal_multilinear_series.le_mul_pow_of_radius_pos FormalMultilinearSeries.le_mul_pow_of_radius_pos
/-- The radius of the sum of two formal series is at least the minimum of their two radii. -/
theorem min_radius_le_radius_add (p q : FormalMultilinearSeries 𝕜 E F) :
min p.radius q.radius ≤ (p + q).radius := by
refine' ENNReal.le_of_forall_nnreal_lt fun r hr => _
rw [lt_min_iff] at hr
have := ((p.isLittleO_one_of_lt_radius hr.1).add (q.isLittleO_one_of_lt_radius hr.2)).isBigO
refine' (p + q).le_radius_of_isBigO ((isBigO_of_le _ fun n => _).trans this)
rw [← add_mul, norm_mul, norm_mul, norm_norm]
exact mul_le_mul_of_nonneg_right ((norm_add_le _ _).trans (le_abs_self _)) (norm_nonneg _)
#align formal_multilinear_series.min_radius_le_radius_add FormalMultilinearSeries.min_radius_le_radius_add
@[simp]
theorem radius_neg (p : FormalMultilinearSeries 𝕜 E F) : (-p).radius = p.radius := by
simp only [radius, neg_apply, norm_neg]
#align formal_multilinear_series.radius_neg FormalMultilinearSeries.radius_neg
protected theorem hasSum [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : HasSum (fun n : ℕ => p n fun _ => x) (p.sum x) :=
(p.summable hx).hasSum
#align formal_multilinear_series.has_sum FormalMultilinearSeries.hasSum
theorem radius_le_radius_continuousLinearMap_comp (p : FormalMultilinearSeries 𝕜 E F)
(f : F →L[𝕜] G) : p.radius ≤ (f.compFormalMultilinearSeries p).radius := by
refine' ENNReal.le_of_forall_nnreal_lt fun r hr => _
apply le_radius_of_isBigO
apply (IsBigO.trans_isLittleO _ (p.isLittleO_one_of_lt_radius hr)).isBigO
refine' IsBigO.mul (@IsBigOWith.isBigO _ _ _ _ _ ‖f‖ _ _ _ _) (isBigO_refl _ _)
refine IsBigOWith.of_bound (eventually_of_forall fun n => ?_)
simpa only [norm_norm] using f.norm_compContinuousMultilinearMap_le (p n)
#align formal_multilinear_series.radius_le_radius_continuous_linear_map_comp FormalMultilinearSeries.radius_le_radius_continuousLinearMap_comp
end FormalMultilinearSeries
/-! ### Expanding a function as a power series -/
section
variable {f g : E → F} {p pf pg : FormalMultilinearSeries 𝕜 E F} {x : E} {r r' : ℝ≥0∞}
/-- Given a function `f : E → F` and a formal multilinear series `p`, we say that `f` has `p` as
a power series on the ball of radius `r > 0` around `x` if `f (x + y) = ∑' pₙ yⁿ` for all `‖y‖ < r`.
-/
structure HasFPowerSeriesOnBall (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (x : E) (r : ℝ≥0∞) :
Prop where
r_le : r ≤ p.radius
r_pos : 0 < r
hasSum :
∀ {y}, y ∈ EMetric.ball (0 : E) r → HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y))
#align has_fpower_series_on_ball HasFPowerSeriesOnBall
/-- Given a function `f : E → F` and a formal multilinear series `p`, we say that `f` has `p` as
a power series around `x` if `f (x + y) = ∑' pₙ yⁿ` for all `y` in a neighborhood of `0`. -/
def HasFPowerSeriesAt (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (x : E) :=
∃ r, HasFPowerSeriesOnBall f p x r
#align has_fpower_series_at HasFPowerSeriesAt
variable (𝕜)
/-- Given a function `f : E → F`, we say that `f` is analytic at `x` if it admits a convergent power
series expansion around `x`. -/
def AnalyticAt (f : E → F) (x : E) :=
∃ p : FormalMultilinearSeries 𝕜 E F, HasFPowerSeriesAt f p x
#align analytic_at AnalyticAt
/-- Given a function `f : E → F`, we say that `f` is analytic on a set `s` if it is analytic around
every point of `s`. -/
def AnalyticOn (f : E → F) (s : Set E) :=
∀ x, x ∈ s → AnalyticAt 𝕜 f x
#align analytic_on AnalyticOn
variable {𝕜}
theorem HasFPowerSeriesOnBall.hasFPowerSeriesAt (hf : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesAt f p x :=
⟨r, hf⟩
#align has_fpower_series_on_ball.has_fpower_series_at HasFPowerSeriesOnBall.hasFPowerSeriesAt
theorem HasFPowerSeriesAt.analyticAt (hf : HasFPowerSeriesAt f p x) : AnalyticAt 𝕜 f x :=
⟨p, hf⟩
#align has_fpower_series_at.analytic_at HasFPowerSeriesAt.analyticAt
theorem HasFPowerSeriesOnBall.analyticAt (hf : HasFPowerSeriesOnBall f p x r) : AnalyticAt 𝕜 f x :=
hf.hasFPowerSeriesAt.analyticAt
#align has_fpower_series_on_ball.analytic_at HasFPowerSeriesOnBall.analyticAt
theorem HasFPowerSeriesOnBall.congr (hf : HasFPowerSeriesOnBall f p x r)
(hg : EqOn f g (EMetric.ball x r)) : HasFPowerSeriesOnBall g p x r :=
{ r_le := hf.r_le
r_pos := hf.r_pos
hasSum := fun {y} hy => by
convert hf.hasSum hy using 1
apply hg.symm
simpa [edist_eq_coe_nnnorm_sub] using hy }
#align has_fpower_series_on_ball.congr HasFPowerSeriesOnBall.congr
/-- If a function `f` has a power series `p` around `x`, then the function `z ↦ f (z - y)` has the
same power series around `x + y`. -/
theorem HasFPowerSeriesOnBall.comp_sub (hf : HasFPowerSeriesOnBall f p x r) (y : E) :
HasFPowerSeriesOnBall (fun z => f (z - y)) p (x + y) r :=
{ r_le := hf.r_le
r_pos := hf.r_pos
hasSum := fun {z} hz => by
convert hf.hasSum hz using 2
abel }
#align has_fpower_series_on_ball.comp_sub HasFPowerSeriesOnBall.comp_sub
theorem HasFPowerSeriesOnBall.hasSum_sub (hf : HasFPowerSeriesOnBall f p x r) {y : E}
(hy : y ∈ EMetric.ball x r) : HasSum (fun n : ℕ => p n fun _ => y - x) (f y) := by
have : y - x ∈ EMetric.ball (0 : E) r := by simpa [edist_eq_coe_nnnorm_sub] using hy
simpa only [add_sub_cancel'_right] using hf.hasSum this
#align has_fpower_series_on_ball.has_sum_sub HasFPowerSeriesOnBall.hasSum_sub
theorem HasFPowerSeriesOnBall.radius_pos (hf : HasFPowerSeriesOnBall f p x r) : 0 < p.radius :=
lt_of_lt_of_le hf.r_pos hf.r_le
#align has_fpower_series_on_ball.radius_pos HasFPowerSeriesOnBall.radius_pos
theorem HasFPowerSeriesAt.radius_pos (hf : HasFPowerSeriesAt f p x) : 0 < p.radius :=
let ⟨_, hr⟩ := hf
hr.radius_pos
#align has_fpower_series_at.radius_pos HasFPowerSeriesAt.radius_pos
theorem HasFPowerSeriesOnBall.mono (hf : HasFPowerSeriesOnBall f p x r) (r'_pos : 0 < r')
(hr : r' ≤ r) : HasFPowerSeriesOnBall f p x r' :=
⟨le_trans hr hf.1, r'_pos, fun hy => hf.hasSum (EMetric.ball_subset_ball hr hy)⟩
#align has_fpower_series_on_ball.mono HasFPowerSeriesOnBall.mono
theorem HasFPowerSeriesAt.congr (hf : HasFPowerSeriesAt f p x) (hg : f =ᶠ[𝓝 x] g) :
HasFPowerSeriesAt g p x := by
rcases hf with ⟨r₁, h₁⟩
rcases EMetric.mem_nhds_iff.mp hg with ⟨r₂, h₂pos, h₂⟩
exact ⟨min r₁ r₂,
(h₁.mono (lt_min h₁.r_pos h₂pos) inf_le_left).congr
fun y hy => h₂ (EMetric.ball_subset_ball inf_le_right hy)⟩
#align has_fpower_series_at.congr HasFPowerSeriesAt.congr
protected theorem HasFPowerSeriesAt.eventually (hf : HasFPowerSeriesAt f p x) :
∀ᶠ r : ℝ≥0∞ in 𝓝[>] 0, HasFPowerSeriesOnBall f p x r :=
let ⟨_, hr⟩ := hf
mem_of_superset (Ioo_mem_nhdsWithin_Ioi (left_mem_Ico.2 hr.r_pos)) fun _ hr' =>
hr.mono hr'.1 hr'.2.le
#align has_fpower_series_at.eventually HasFPowerSeriesAt.eventually
theorem HasFPowerSeriesOnBall.eventually_hasSum (hf : HasFPowerSeriesOnBall f p x r) :
∀ᶠ y in 𝓝 0, HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y)) := by
filter_upwards [EMetric.ball_mem_nhds (0 : E) hf.r_pos] using fun _ => hf.hasSum
#align has_fpower_series_on_ball.eventually_has_sum HasFPowerSeriesOnBall.eventually_hasSum
theorem HasFPowerSeriesAt.eventually_hasSum (hf : HasFPowerSeriesAt f p x) :
∀ᶠ y in 𝓝 0, HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y)) :=
let ⟨_, hr⟩ := hf
hr.eventually_hasSum
#align has_fpower_series_at.eventually_has_sum HasFPowerSeriesAt.eventually_hasSum
theorem HasFPowerSeriesOnBall.eventually_hasSum_sub (hf : HasFPowerSeriesOnBall f p x r) :
∀ᶠ y in 𝓝 x, HasSum (fun n : ℕ => p n fun _ : Fin n => y - x) (f y) := by
filter_upwards [EMetric.ball_mem_nhds x hf.r_pos] with y using hf.hasSum_sub
#align has_fpower_series_on_ball.eventually_has_sum_sub HasFPowerSeriesOnBall.eventually_hasSum_sub
theorem HasFPowerSeriesAt.eventually_hasSum_sub (hf : HasFPowerSeriesAt f p x) :
∀ᶠ y in 𝓝 x, HasSum (fun n : ℕ => p n fun _ : Fin n => y - x) (f y) :=
let ⟨_, hr⟩ := hf
hr.eventually_hasSum_sub
#align has_fpower_series_at.eventually_has_sum_sub HasFPowerSeriesAt.eventually_hasSum_sub
theorem HasFPowerSeriesOnBall.eventually_eq_zero
(hf : HasFPowerSeriesOnBall f (0 : FormalMultilinearSeries 𝕜 E F) x r) :
∀ᶠ z in 𝓝 x, f z = 0 := by
filter_upwards [hf.eventually_hasSum_sub] with z hz using hz.unique hasSum_zero
#align has_fpower_series_on_ball.eventually_eq_zero HasFPowerSeriesOnBall.eventually_eq_zero
theorem HasFPowerSeriesAt.eventually_eq_zero
(hf : HasFPowerSeriesAt f (0 : FormalMultilinearSeries 𝕜 E F) x) : ∀ᶠ z in 𝓝 x, f z = 0 :=
let ⟨_, hr⟩ := hf
hr.eventually_eq_zero
#align has_fpower_series_at.eventually_eq_zero HasFPowerSeriesAt.eventually_eq_zero
theorem hasFPowerSeriesOnBall_const {c : F} {e : E} :
HasFPowerSeriesOnBall (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e ⊤ := by
refine' ⟨by simp, WithTop.zero_lt_top, fun _ => hasSum_single 0 fun n hn => _⟩
simp [constFormalMultilinearSeries_apply hn]
#align has_fpower_series_on_ball_const hasFPowerSeriesOnBall_const
theorem hasFPowerSeriesAt_const {c : F} {e : E} :
HasFPowerSeriesAt (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e :=
⟨⊤, hasFPowerSeriesOnBall_const⟩
#align has_fpower_series_at_const hasFPowerSeriesAt_const
theorem analyticAt_const {v : F} : AnalyticAt 𝕜 (fun _ => v) x :=
⟨constFormalMultilinearSeries 𝕜 E v, hasFPowerSeriesAt_const⟩
#align analytic_at_const analyticAt_const
theorem analyticOn_const {v : F} {s : Set E} : AnalyticOn 𝕜 (fun _ => v) s :=
fun _ _ => analyticAt_const
#align analytic_on_const analyticOn_const
theorem HasFPowerSeriesOnBall.add (hf : HasFPowerSeriesOnBall f pf x r)
(hg : HasFPowerSeriesOnBall g pg x r) : HasFPowerSeriesOnBall (f + g) (pf + pg) x r :=
{ r_le := le_trans (le_min_iff.2 ⟨hf.r_le, hg.r_le⟩) (pf.min_radius_le_radius_add pg)
r_pos := hf.r_pos
hasSum := fun hy => (hf.hasSum hy).add (hg.hasSum hy) }
#align has_fpower_series_on_ball.add HasFPowerSeriesOnBall.add
theorem HasFPowerSeriesAt.add (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x) :
HasFPowerSeriesAt (f + g) (pf + pg) x := by
rcases (hf.eventually.and hg.eventually).exists with ⟨r, hr⟩
exact ⟨r, hr.1.add hr.2⟩
#align has_fpower_series_at.add HasFPowerSeriesAt.add
theorem AnalyticAt.congr (hf : AnalyticAt 𝕜 f x) (hg : f =ᶠ[𝓝 x] g) : AnalyticAt 𝕜 g x :=
let ⟨_, hpf⟩ := hf
(hpf.congr hg).analyticAt
theorem analyticAt_congr (h : f =ᶠ[𝓝 x] g) : AnalyticAt 𝕜 f x ↔ AnalyticAt 𝕜 g x :=
⟨fun hf ↦ hf.congr h, fun hg ↦ hg.congr h.symm⟩
theorem AnalyticAt.add (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) : AnalyticAt 𝕜 (f + g) x :=
let ⟨_, hpf⟩ := hf
let ⟨_, hqf⟩ := hg
(hpf.add hqf).analyticAt
#align analytic_at.add AnalyticAt.add
theorem HasFPowerSeriesOnBall.neg (hf : HasFPowerSeriesOnBall f pf x r) :
HasFPowerSeriesOnBall (-f) (-pf) x r :=
{ r_le := by
rw [pf.radius_neg]
exact hf.r_le
r_pos := hf.r_pos
hasSum := fun hy => (hf.hasSum hy).neg }
#align has_fpower_series_on_ball.neg HasFPowerSeriesOnBall.neg
theorem HasFPowerSeriesAt.neg (hf : HasFPowerSeriesAt f pf x) : HasFPowerSeriesAt (-f) (-pf) x :=
let ⟨_, hrf⟩ := hf
hrf.neg.hasFPowerSeriesAt
#align has_fpower_series_at.neg HasFPowerSeriesAt.neg
theorem AnalyticAt.neg (hf : AnalyticAt 𝕜 f x) : AnalyticAt 𝕜 (-f) x :=
let ⟨_, hpf⟩ := hf
hpf.neg.analyticAt
#align analytic_at.neg AnalyticAt.neg
theorem HasFPowerSeriesOnBall.sub (hf : HasFPowerSeriesOnBall f pf x r)
(hg : HasFPowerSeriesOnBall g pg x r) : HasFPowerSeriesOnBall (f - g) (pf - pg) x r := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align has_fpower_series_on_ball.sub HasFPowerSeriesOnBall.sub
theorem HasFPowerSeriesAt.sub (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x) :
HasFPowerSeriesAt (f - g) (pf - pg) x := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align has_fpower_series_at.sub HasFPowerSeriesAt.sub
theorem AnalyticAt.sub (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) :
AnalyticAt 𝕜 (f - g) x := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align analytic_at.sub AnalyticAt.sub
theorem AnalyticOn.mono {s t : Set E} (hf : AnalyticOn 𝕜 f t) (hst : s ⊆ t) : AnalyticOn 𝕜 f s :=
fun z hz => hf z (hst hz)
#align analytic_on.mono AnalyticOn.mono
theorem AnalyticOn.congr' {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : f =ᶠ[𝓝ˢ s] g) :
AnalyticOn 𝕜 g s :=
fun z hz => (hf z hz).congr (mem_nhdsSet_iff_forall.mp hg z hz)
theorem analyticOn_congr' {s : Set E} (h : f =ᶠ[𝓝ˢ s] g) : AnalyticOn 𝕜 f s ↔ AnalyticOn 𝕜 g s :=
⟨fun hf => hf.congr' h, fun hg => hg.congr' h.symm⟩
theorem AnalyticOn.congr {s : Set E} (hs : IsOpen s) (hf : AnalyticOn 𝕜 f s) (hg : s.EqOn f g) :
AnalyticOn 𝕜 g s :=
hf.congr' $ mem_nhdsSet_iff_forall.mpr
(fun _ hz => eventuallyEq_iff_exists_mem.mpr ⟨s, hs.mem_nhds hz, hg⟩)
theorem analyticOn_congr {s : Set E} (hs : IsOpen s) (h : s.EqOn f g) : AnalyticOn 𝕜 f s ↔
AnalyticOn 𝕜 g s := ⟨fun hf => hf.congr hs h, fun hg => hg.congr hs h.symm⟩
theorem AnalyticOn.add {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
AnalyticOn 𝕜 (f + g) s :=
fun z hz => (hf z hz).add (hg z hz)
#align analytic_on.add AnalyticOn.add
theorem AnalyticOn.sub {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
AnalyticOn 𝕜 (f - g) s :=
fun z hz => (hf z hz).sub (hg z hz)
#align analytic_on.sub AnalyticOn.sub
theorem HasFPowerSeriesOnBall.coeff_zero (hf : HasFPowerSeriesOnBall f pf x r) (v : Fin 0 → E) :
pf 0 v = f x := by
have v_eq : v = fun i => 0 := Subsingleton.elim _ _
have zero_mem : (0 : E) ∈ EMetric.ball (0 : E) r := by simp [hf.r_pos]
have : ∀ i, i ≠ 0 → (pf i fun j => 0) = 0 := by
intro i hi
have : 0 < i := pos_iff_ne_zero.2 hi
exact ContinuousMultilinearMap.map_coord_zero _ (⟨0, this⟩ : Fin i) rfl
have A := (hf.hasSum zero_mem).unique (hasSum_single _ this)
simpa [v_eq] using A.symm
#align has_fpower_series_on_ball.coeff_zero HasFPowerSeriesOnBall.coeff_zero
theorem HasFPowerSeriesAt.coeff_zero (hf : HasFPowerSeriesAt f pf x) (v : Fin 0 → E) :
pf 0 v = f x :=
let ⟨_, hrf⟩ := hf
hrf.coeff_zero v
#align has_fpower_series_at.coeff_zero HasFPowerSeriesAt.coeff_zero
/-- If a function `f` has a power series `p` on a ball and `g` is linear, then `g ∘ f` has the
power series `g ∘ p` on the same ball. -/
theorem ContinuousLinearMap.comp_hasFPowerSeriesOnBall (g : F →L[𝕜] G)
(h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (g ∘ f) (g.compFormalMultilinearSeries p) x r :=
{ r_le := h.r_le.trans (p.radius_le_radius_continuousLinearMap_comp _)
r_pos := h.r_pos
hasSum := fun hy => by
simpa only [ContinuousLinearMap.compFormalMultilinearSeries_apply,
ContinuousLinearMap.compContinuousMultilinearMap_coe, Function.comp_apply] using
g.hasSum (h.hasSum hy) }
#align continuous_linear_map.comp_has_fpower_series_on_ball ContinuousLinearMap.comp_hasFPowerSeriesOnBall
/-- If a function `f` is analytic on a set `s` and `g` is linear, then `g ∘ f` is analytic
on `s`. -/
theorem ContinuousLinearMap.comp_analyticOn {s : Set E} (g : F →L[𝕜] G) (h : AnalyticOn 𝕜 f s) :
AnalyticOn 𝕜 (g ∘ f) s := by
rintro x hx
rcases h x hx with ⟨p, r, hp⟩
exact ⟨g.compFormalMultilinearSeries p, r, g.comp_hasFPowerSeriesOnBall hp⟩
#align continuous_linear_map.comp_analytic_on ContinuousLinearMap.comp_analyticOn
/-- If a function admits a power series expansion, then it is exponentially close to the partial
sums of this power series on strict subdisks of the disk of convergence.
This version provides an upper estimate that decreases both in `‖y‖` and `n`. See also
`HasFPowerSeriesOnBall.uniform_geometric_approx` for a weaker version. -/
theorem HasFPowerSeriesOnBall.uniform_geometric_approx' {r' : ℝ≥0}
(hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n := by
obtain ⟨a, ha, C, hC, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n, ‖p n‖ * (r' : ℝ) ^ n ≤ C * a ^ n :=
p.norm_mul_pow_le_mul_pow_of_lt_radius (h.trans_le hf.r_le)
refine' ⟨a, ha, C / (1 - a), div_pos hC (sub_pos.2 ha.2), fun y hy n => _⟩
have yr' : ‖y‖ < r' := by
rw [ball_zero_eq] at hy
exact hy
have hr'0 : 0 < (r' : ℝ) := (norm_nonneg _).trans_lt yr'
have : y ∈ EMetric.ball (0 : E) r := by
refine' mem_emetric_ball_zero_iff.2 (lt_trans _ h)
exact mod_cast yr'
rw [norm_sub_rev, ← mul_div_right_comm]
have ya : a * (‖y‖ / ↑r') ≤ a :=
mul_le_of_le_one_right ha.1.le (div_le_one_of_le yr'.le r'.coe_nonneg)
suffices ‖p.partialSum n y - f (x + y)‖ ≤ C * (a * (‖y‖ / r')) ^ n / (1 - a * (‖y‖ / r')) by
refine' this.trans _
have : 0 < a := ha.1
gcongr
apply_rules [sub_pos.2, ha.2]
apply norm_sub_le_of_geometric_bound_of_hasSum (ya.trans_lt ha.2) _ (hf.hasSum this)
intro n
calc
‖(p n) fun _ : Fin n => y‖
_ ≤ ‖p n‖ * ∏ _i : Fin n, ‖y‖ := ContinuousMultilinearMap.le_op_norm _ _
_ = ‖p n‖ * (r' : ℝ) ^ n * (‖y‖ / r') ^ n := by field_simp [mul_right_comm]
_ ≤ C * a ^ n * (‖y‖ / r') ^ n := by gcongr ?_ * _; apply hp
_ ≤ C * (a * (‖y‖ / r')) ^ n := by rw [mul_pow, mul_assoc]
#align has_fpower_series_on_ball.uniform_geometric_approx' HasFPowerSeriesOnBall.uniform_geometric_approx'
/-- If a function admits a power series expansion, then it is exponentially close to the partial
sums of this power series on strict subdisks of the disk of convergence. -/
theorem HasFPowerSeriesOnBall.uniform_geometric_approx {r' : ℝ≥0}
(hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
∃ a ∈ Ioo (0 : ℝ) 1,
∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n, ‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n := by
obtain ⟨a, ha, C, hC, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n :=
hf.uniform_geometric_approx' h
refine' ⟨a, ha, C, hC, fun y hy n => (hp y hy n).trans _⟩
have yr' : ‖y‖ < r' := by rwa [ball_zero_eq] at hy
gcongr
exacts [mul_nonneg ha.1.le (div_nonneg (norm_nonneg y) r'.coe_nonneg),
mul_le_of_le_one_right ha.1.le (div_le_one_of_le yr'.le r'.coe_nonneg)]
#align has_fpower_series_on_ball.uniform_geometric_approx HasFPowerSeriesOnBall.uniform_geometric_approx
/-- Taylor formula for an analytic function, `IsBigO` version. -/
theorem HasFPowerSeriesAt.isBigO_sub_partialSum_pow (hf : HasFPowerSeriesAt f p x) (n : ℕ) :
(fun y : E => f (x + y) - p.partialSum n y) =O[𝓝 0] fun y => ‖y‖ ^ n := by
rcases hf with ⟨r, hf⟩
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hf.r_pos with ⟨r', r'0, h⟩
obtain ⟨a, -, C, -, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n :=
hf.uniform_geometric_approx' h
refine' isBigO_iff.2 ⟨C * (a / r') ^ n, _⟩
replace r'0 : 0 < (r' : ℝ); · exact mod_cast r'0
filter_upwards [Metric.ball_mem_nhds (0 : E) r'0] with y hy
simpa [mul_pow, mul_div_assoc, mul_assoc, div_mul_eq_mul_div] using hp y hy n
set_option linter.uppercaseLean3 false in
#align has_fpower_series_at.is_O_sub_partial_sum_pow HasFPowerSeriesAt.isBigO_sub_partialSum_pow
/-- If `f` has formal power series `∑ n, pₙ` on a ball of radius `r`, then for `y, z` in any smaller
ball, the norm of the difference `f y - f z - p 1 (fun _ ↦ y - z)` is bounded above by
`C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖`. This lemma formulates this property using `IsBigO` and
`Filter.principal` on `E × E`. -/
theorem HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal
(hf : HasFPowerSeriesOnBall f p x r) (hr : r' < r) :
(fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) =O[𝓟 (EMetric.ball (x, x) r')]
fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖ := by
lift r' to ℝ≥0 using ne_top_of_lt hr
rcases (zero_le r').eq_or_lt with (rfl | hr'0)
· simp only [isBigO_bot, EMetric.ball_zero, principal_empty, ENNReal.coe_zero]
obtain ⟨a, ha, C, hC : 0 < C, hp⟩ :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n : ℕ, ‖p n‖ * (r' : ℝ) ^ n ≤ C * a ^ n
exact p.norm_mul_pow_le_mul_pow_of_lt_radius (hr.trans_le hf.r_le)
simp only [← le_div_iff (pow_pos (NNReal.coe_pos.2 hr'0) _)] at hp
set L : E × E → ℝ := fun y =>
C * (a / r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * (a / (1 - a) ^ 2 + 2 / (1 - a))
have hL : ∀ y ∈ EMetric.ball (x, x) r', ‖f y.1 - f y.2 - p 1 fun _ => y.1 - y.2‖ ≤ L y := by
intro y hy'
have hy : y ∈ EMetric.ball x r ×ˢ EMetric.ball x r := by
rw [EMetric.ball_prod_same]
exact EMetric.ball_subset_ball hr.le hy'
set A : ℕ → F := fun n => (p n fun _ => y.1 - x) - p n fun _ => y.2 - x
have hA : HasSum (fun n => A (n + 2)) (f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) := by
convert (hasSum_nat_add_iff' 2).2 ((hf.hasSum_sub hy.1).sub (hf.hasSum_sub hy.2)) using 1
rw [Finset.sum_range_succ, Finset.sum_range_one, hf.coeff_zero, hf.coeff_zero, sub_self,
zero_add, ← Subsingleton.pi_single_eq (0 : Fin 1) (y.1 - x), Pi.single,
← Subsingleton.pi_single_eq (0 : Fin 1) (y.2 - x), Pi.single, ← (p 1).map_sub, ← Pi.single,
Subsingleton.pi_single_eq, sub_sub_sub_cancel_right]
rw [EMetric.mem_ball, edist_eq_coe_nnnorm_sub, ENNReal.coe_lt_coe] at hy'
set B : ℕ → ℝ := fun n => C * (a / r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * ((n + 2) * a ^ n)
have hAB : ∀ n, ‖A (n + 2)‖ ≤ B n := fun n =>
calc
‖A (n + 2)‖ ≤ ‖p (n + 2)‖ * ↑(n + 2) * ‖y - (x, x)‖ ^ (n + 1) * ‖y.1 - y.2‖ := by
-- porting note: `pi_norm_const` was `pi_norm_const (_ : E)`
simpa only [Fintype.card_fin, pi_norm_const, Prod.norm_def, Pi.sub_def,
Prod.fst_sub, Prod.snd_sub, sub_sub_sub_cancel_right] using
(p <| n + 2).norm_image_sub_le (fun _ => y.1 - x) fun _ => y.2 - x
_ = ‖p (n + 2)‖ * ‖y - (x, x)‖ ^ n * (↑(n + 2) * ‖y - (x, x)‖ * ‖y.1 - y.2‖) := by
rw [pow_succ ‖y - (x, x)‖]
ring
-- porting note: the two `↑` in `↑r'` are new, without them, Lean fails to synthesize
-- instances `HDiv ℝ ℝ≥0 ?m` or `HMul ℝ ℝ≥0 ?m`
_ ≤ C * a ^ (n + 2) / ↑r' ^ (n + 2)
* ↑r' ^ n * (↑(n + 2) * ‖y - (x, x)‖ * ‖y.1 - y.2‖) := by
have : 0 < a := ha.1
gcongr
· apply hp
· apply hy'.le
_ = B n := by
-- porting note: in the original, `B` was in the `field_simp`, but now Lean does not
-- accept it. The current proof works in Lean 4, but does not in Lean 3.
field_simp [pow_succ]
simp only [mul_assoc, mul_comm, mul_left_comm]
have hBL : HasSum B (L y) := by
apply HasSum.mul_left
simp only [add_mul]
have : ‖a‖ < 1 := by simp only [Real.norm_eq_abs, abs_of_pos ha.1, ha.2]
rw [div_eq_mul_inv, div_eq_mul_inv]
exact (hasSum_coe_mul_geometric_of_norm_lt_1 this).add -- porting note: was `convert`!
((hasSum_geometric_of_norm_lt_1 this).mul_left 2)
exact hA.norm_le_of_bounded hBL hAB
suffices L =O[𝓟 (EMetric.ball (x, x) r')] fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖ by
refine' (IsBigO.of_bound 1 (eventually_principal.2 fun y hy => _)).trans this
rw [one_mul]
exact (hL y hy).trans (le_abs_self _)
simp_rw [mul_right_comm _ (_ * _)] -- porting note: there was an `L` inside the `simp_rw`.
exact (isBigO_refl _ _).const_mul_left _
set_option linter.uppercaseLean3 false in
#align has_fpower_series_on_ball.is_O_image_sub_image_sub_deriv_principal HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal
/-- If `f` has formal power series `∑ n, pₙ` on a ball of radius `r`, then for `y, z` in any smaller
ball, the norm of the difference `f y - f z - p 1 (fun _ ↦ y - z)` is bounded above by
`C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖`. -/
theorem HasFPowerSeriesOnBall.image_sub_sub_deriv_le (hf : HasFPowerSeriesOnBall f p x r)
(hr : r' < r) :
∃ C, ∀ᵉ (y ∈ EMetric.ball x r') (z ∈ EMetric.ball x r'),
‖f y - f z - p 1 fun _ => y - z‖ ≤ C * max ‖y - x‖ ‖z - x‖ * ‖y - z‖ := by
simpa only [isBigO_principal, mul_assoc, norm_mul, norm_norm, Prod.forall, EMetric.mem_ball,
Prod.edist_eq, max_lt_iff, and_imp, @forall_swap (_ < _) E] using
hf.isBigO_image_sub_image_sub_deriv_principal hr
#align has_fpower_series_on_ball.image_sub_sub_deriv_le HasFPowerSeriesOnBall.image_sub_sub_deriv_le
/-- If `f` has formal power series `∑ n, pₙ` at `x`, then
`f y - f z - p 1 (fun _ ↦ y - z) = O(‖(y, z) - (x, x)‖ * ‖y - z‖)` as `(y, z) → (x, x)`.
In particular, `f` is strictly differentiable at `x`. -/
theorem HasFPowerSeriesAt.isBigO_image_sub_norm_mul_norm_sub (hf : HasFPowerSeriesAt f p x) :
(fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) =O[𝓝 (x, x)] fun y =>
‖y - (x, x)‖ * ‖y.1 - y.2‖ := by
rcases hf with ⟨r, hf⟩
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hf.r_pos with ⟨r', r'0, h⟩
refine' (hf.isBigO_image_sub_image_sub_deriv_principal h).mono _
exact le_principal_iff.2 (EMetric.ball_mem_nhds _ r'0)
set_option linter.uppercaseLean3 false in
#align has_fpower_series_at.is_O_image_sub_norm_mul_norm_sub HasFPowerSeriesAt.isBigO_image_sub_norm_mul_norm_sub
/-- If a function admits a power series expansion at `x`, then it is the uniform limit of the
partial sums of this power series on strict subdisks of the disk of convergence, i.e., `f (x + y)`
is the uniform limit of `p.partialSum n y` there. -/
theorem HasFPowerSeriesOnBall.tendstoUniformlyOn {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r)
(h : (r' : ℝ≥0∞) < r) :
TendstoUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
(Metric.ball (0 : E) r') := by
obtain ⟨a, ha, C, -, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n
exact hf.uniform_geometric_approx h
refine' Metric.tendstoUniformlyOn_iff.2 fun ε εpos => _
have L : Tendsto (fun n => (C : ℝ) * a ^ n) atTop (𝓝 ((C : ℝ) * 0)) :=
tendsto_const_nhds.mul (tendsto_pow_atTop_nhds_0_of_lt_1 ha.1.le ha.2)
rw [mul_zero] at L
refine' (L.eventually (gt_mem_nhds εpos)).mono fun n hn y hy => _
rw [dist_eq_norm]
exact (hp y hy n).trans_lt hn
#align has_fpower_series_on_ball.tendsto_uniformly_on HasFPowerSeriesOnBall.tendstoUniformlyOn
/-- If a function admits a power series expansion at `x`, then it is the locally uniform limit of
the partial sums of this power series on the disk of convergence, i.e., `f (x + y)`
is the locally uniform limit of `p.partialSum n y` there. -/
theorem HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn (hf : HasFPowerSeriesOnBall f p x r) :
TendstoLocallyUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
(EMetric.ball (0 : E) r) := by
intro u hu x hx
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hx with ⟨r', xr', hr'⟩
have : EMetric.ball (0 : E) r' ∈ 𝓝 x := IsOpen.mem_nhds EMetric.isOpen_ball xr'
refine' ⟨EMetric.ball (0 : E) r', mem_nhdsWithin_of_mem_nhds this, _⟩
simpa [Metric.emetric_ball_nnreal] using hf.tendstoUniformlyOn hr' u hu
#align has_fpower_series_on_ball.tendsto_locally_uniformly_on HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn
/-- If a function admits a power series expansion at `x`, then it is the uniform limit of the
partial sums of this power series on strict subdisks of the disk of convergence, i.e., `f y`
is the uniform limit of `p.partialSum n (y - x)` there. -/
theorem HasFPowerSeriesOnBall.tendstoUniformlyOn' {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r)
(h : (r' : ℝ≥0∞) < r) :
TendstoUniformlyOn (fun n y => p.partialSum n (y - x)) f atTop (Metric.ball (x : E) r') := by
convert (hf.tendstoUniformlyOn h).comp fun y => y - x using 1
· simp [(· ∘ ·)]
· ext z
simp [dist_eq_norm]
#align has_fpower_series_on_ball.tendsto_uniformly_on' HasFPowerSeriesOnBall.tendstoUniformlyOn'
/-- If a function admits a power series expansion at `x`, then it is the locally uniform limit of
the partial sums of this power series on the disk of convergence, i.e., `f y`
is the locally uniform limit of `p.partialSum n (y - x)` there. -/
theorem HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn' (hf : HasFPowerSeriesOnBall f p x r) :
TendstoLocallyUniformlyOn (fun n y => p.partialSum n (y - x)) f atTop
(EMetric.ball (x : E) r) := by
have A : ContinuousOn (fun y : E => y - x) (EMetric.ball (x : E) r) :=
(continuous_id.sub continuous_const).continuousOn
convert hf.tendstoLocallyUniformlyOn.comp (fun y : E => y - x) _ A using 1
· ext z
simp
· intro z
simp [edist_eq_coe_nnnorm, edist_eq_coe_nnnorm_sub]
#align has_fpower_series_on_ball.tendsto_locally_uniformly_on' HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn'
/-- If a function admits a power series expansion on a disk, then it is continuous there. -/
protected theorem HasFPowerSeriesOnBall.continuousOn (hf : HasFPowerSeriesOnBall f p x r) :
ContinuousOn f (EMetric.ball x r) :=
hf.tendstoLocallyUniformlyOn'.continuousOn <|
eventually_of_forall fun n =>
((p.partialSum_continuous n).comp (continuous_id.sub continuous_const)).continuousOn
#align has_fpower_series_on_ball.continuous_on HasFPowerSeriesOnBall.continuousOn
protected theorem HasFPowerSeriesAt.continuousAt (hf : HasFPowerSeriesAt f p x) :
ContinuousAt f x :=
let ⟨_, hr⟩ := hf
hr.continuousOn.continuousAt (EMetric.ball_mem_nhds x hr.r_pos)
#align has_fpower_series_at.continuous_at HasFPowerSeriesAt.continuousAt
protected theorem AnalyticAt.continuousAt (hf : AnalyticAt 𝕜 f x) : ContinuousAt f x :=
let ⟨_, hp⟩ := hf
hp.continuousAt
#align analytic_at.continuous_at AnalyticAt.continuousAt
protected theorem AnalyticOn.continuousOn {s : Set E} (hf : AnalyticOn 𝕜 f s) : ContinuousOn f s :=
fun x hx => (hf x hx).continuousAt.continuousWithinAt
#align analytic_on.continuous_on AnalyticOn.continuousOn
/-- Analytic everywhere implies continuous -/
theorem AnalyticOn.continuous {f : E → F} (fa : AnalyticOn 𝕜 f univ) : Continuous f := by
rw [continuous_iff_continuousOn_univ]; exact fa.continuousOn
/-- In a complete space, the sum of a converging power series `p` admits `p` as a power series.
This is not totally obvious as we need to check the convergence of the series. -/
protected theorem FormalMultilinearSeries.hasFPowerSeriesOnBall [CompleteSpace F]
(p : FormalMultilinearSeries 𝕜 E F) (h : 0 < p.radius) :
HasFPowerSeriesOnBall p.sum p 0 p.radius :=
{ r_le := le_rfl
r_pos := h
hasSum := fun hy => by
rw [zero_add]
exact p.hasSum hy }
#align formal_multilinear_series.has_fpower_series_on_ball FormalMultilinearSeries.hasFPowerSeriesOnBall
theorem HasFPowerSeriesOnBall.sum (h : HasFPowerSeriesOnBall f p x r) {y : E}
(hy : y ∈ EMetric.ball (0 : E) r) : f (x + y) = p.sum y :=
(h.hasSum hy).tsum_eq.symm
#align has_fpower_series_on_ball.sum HasFPowerSeriesOnBall.sum
/-- The sum of a converging power series is continuous in its disk of convergence. -/
protected theorem FormalMultilinearSeries.continuousOn [CompleteSpace F] :
ContinuousOn p.sum (EMetric.ball 0 p.radius) := by
rcases (zero_le p.radius).eq_or_lt with h | h
· simp [← h, continuousOn_empty]
· exact (p.hasFPowerSeriesOnBall h).continuousOn
#align formal_multilinear_series.continuous_on FormalMultilinearSeries.continuousOn
end
/-!
### Uniqueness of power series
If a function `f : E → F` has two representations as power series at a point `x : E`, corresponding
to formal multilinear series `p₁` and `p₂`, then these representations agree term-by-term. That is,
for any `n : ℕ` and `y : E`, `p₁ n (fun i ↦ y) = p₂ n (fun i ↦ y)`. In the one-dimensional case,
when `f : 𝕜 → E`, the continuous multilinear maps `p₁ n` and `p₂ n` are given by
`ContinuousMultilinearMap.mkPiField`, and hence are determined completely by the value of
`p₁ n (fun i ↦ 1)`, so `p₁ = p₂`. Consequently, the radius of convergence for one series can be
transferred to the other.
-/
section Uniqueness
open ContinuousMultilinearMap
theorem Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero {n : ℕ} {p : E[×n]→L[𝕜] F}
(h : (fun y => p fun _ => y) =O[𝓝 0] fun y => ‖y‖ ^ (n + 1)) (y : E) : (p fun _ => y) = 0 := by
obtain ⟨c, c_pos, hc⟩ := h.exists_pos
obtain ⟨t, ht, t_open, z_mem⟩ := eventually_nhds_iff.mp (isBigOWith_iff.mp hc)
obtain ⟨δ, δ_pos, δε⟩ := (Metric.isOpen_iff.mp t_open) 0 z_mem
clear h hc z_mem
cases' n with n
· exact norm_eq_zero.mp (by
-- porting note: the symmetric difference of the `simpa only` sets:
-- added `Nat.zero_eq, zero_add, pow_one`
-- removed `zero_pow', Ne.def, Nat.one_ne_zero, not_false_iff`
simpa only [Nat.zero_eq, fin0_apply_norm, norm_eq_zero, norm_zero, zero_add, pow_one,
mul_zero, norm_le_zero_iff] using ht 0 (δε (Metric.mem_ball_self δ_pos)))
· refine' Or.elim (Classical.em (y = 0))
(fun hy => by simpa only [hy] using p.map_zero) fun hy => _
replace hy := norm_pos_iff.mpr hy
refine' norm_eq_zero.mp (le_antisymm (le_of_forall_pos_le_add fun ε ε_pos => _) (norm_nonneg _))
have h₀ := _root_.mul_pos c_pos (pow_pos hy (n.succ + 1))
obtain ⟨k, k_pos, k_norm⟩ := NormedField.exists_norm_lt 𝕜
(lt_min (mul_pos δ_pos (inv_pos.mpr hy)) (mul_pos ε_pos (inv_pos.mpr h₀)))
have h₁ : ‖k • y‖ < δ := by
rw [norm_smul]
exact inv_mul_cancel_right₀ hy.ne.symm δ ▸
mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_left _ _)) hy
have h₂ :=
calc
‖p fun _ => k • y‖ ≤ c * ‖k • y‖ ^ (n.succ + 1) := by
-- porting note: now Lean wants `_root_.`
simpa only [norm_pow, _root_.norm_norm] using ht (k • y) (δε (mem_ball_zero_iff.mpr h₁))
--simpa only [norm_pow, norm_norm] using ht (k • y) (δε (mem_ball_zero_iff.mpr h₁))
_ = ‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))) := by
-- porting note: added `Nat.succ_eq_add_one` since otherwise `ring` does not conclude.
simp only [norm_smul, mul_pow, Nat.succ_eq_add_one]
-- porting note: removed `rw [pow_succ]`, since it now becomes superfluous.
ring
have h₃ : ‖k‖ * (c * ‖y‖ ^ (n.succ + 1)) < ε :=
inv_mul_cancel_right₀ h₀.ne.symm ε ▸
mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_right _ _)) h₀
calc
‖p fun _ => y‖ = ‖k⁻¹ ^ n.succ‖ * ‖p fun _ => k • y‖ := by
simpa only [inv_smul_smul₀ (norm_pos_iff.mp k_pos), norm_smul, Finset.prod_const,
Finset.card_fin] using
congr_arg norm (p.map_smul_univ (fun _ : Fin n.succ => k⁻¹) fun _ : Fin n.succ => k • y)
_ ≤ ‖k⁻¹ ^ n.succ‖ * (‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1)))) := by gcongr
_ = ‖(k⁻¹ * k) ^ n.succ‖ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))) := by
rw [← mul_assoc]
simp [norm_mul, mul_pow]
_ ≤ 0 + ε := by
rw [inv_mul_cancel (norm_pos_iff.mp k_pos)]
simpa using h₃.le
set_option linter.uppercaseLean3 false in
#align asymptotics.is_O.continuous_multilinear_map_apply_eq_zero Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero
/-- If a formal multilinear series `p` represents the zero function at `x : E`, then the
terms `p n (fun i ↦ y)` appearing in the sum are zero for any `n : ℕ`, `y : E`. -/
theorem HasFPowerSeriesAt.apply_eq_zero {p : FormalMultilinearSeries 𝕜 E F} {x : E}
(h : HasFPowerSeriesAt 0 p x) (n : ℕ) : ∀ y : E, (p n fun _ => y) = 0 := by
refine' Nat.strong_induction_on n fun k hk => _
have psum_eq : p.partialSum (k + 1) = fun y => p k fun _ => y := by
funext z
refine' Finset.sum_eq_single _ (fun b hb hnb => _) fun hn => _
· have := Finset.mem_range_succ_iff.mp hb
simp only [hk b (this.lt_of_ne hnb), Pi.zero_apply]
· exact False.elim (hn (Finset.mem_range.mpr (lt_add_one k)))
replace h := h.isBigO_sub_partialSum_pow k.succ
simp only [psum_eq, zero_sub, Pi.zero_apply, Asymptotics.isBigO_neg_left] at h
exact h.continuousMultilinearMap_apply_eq_zero
#align has_fpower_series_at.apply_eq_zero HasFPowerSeriesAt.apply_eq_zero
/-- A one-dimensional formal multilinear series representing the zero function is zero. -/
theorem HasFPowerSeriesAt.eq_zero {p : FormalMultilinearSeries 𝕜 𝕜 E} {x : 𝕜}
(h : HasFPowerSeriesAt 0 p x) : p = 0 := by
-- porting note: `funext; ext` was `ext (n x)`
funext n
ext x
rw [← mkPiField_apply_one_eq_self (p n)]
-- porting note: nasty hack, was `simp [h.apply_eq_zero n 1]`
have := Or.intro_right ?_ (h.apply_eq_zero n 1)
simpa using this
#align has_fpower_series_at.eq_zero HasFPowerSeriesAt.eq_zero
/-- One-dimensional formal multilinear series representing the same function are equal. -/
theorem HasFPowerSeriesAt.eq_formalMultilinearSeries {p₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E}
{f : 𝕜 → E} {x : 𝕜} (h₁ : HasFPowerSeriesAt f p₁ x) (h₂ : HasFPowerSeriesAt f p₂ x) : p₁ = p₂ :=
sub_eq_zero.mp (HasFPowerSeriesAt.eq_zero (by simpa only [sub_self] using h₁.sub h₂))
#align has_fpower_series_at.eq_formal_multilinear_series HasFPowerSeriesAt.eq_formalMultilinearSeries
theorem HasFPowerSeriesAt.eq_formalMultilinearSeries_of_eventually
{p q : FormalMultilinearSeries 𝕜 𝕜 E} {f g : 𝕜 → E} {x : 𝕜} (hp : HasFPowerSeriesAt f p x)
(hq : HasFPowerSeriesAt g q x) (heq : ∀ᶠ z in 𝓝 x, f z = g z) : p = q :=
(hp.congr heq).eq_formalMultilinearSeries hq
#align has_fpower_series_at.eq_formal_multilinear_series_of_eventually HasFPowerSeriesAt.eq_formalMultilinearSeries_of_eventually
/-- A one-dimensional formal multilinear series representing a locally zero function is zero. -/
theorem HasFPowerSeriesAt.eq_zero_of_eventually {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E}
{x : 𝕜} (hp : HasFPowerSeriesAt f p x) (hf : f =ᶠ[𝓝 x] 0) : p = 0 :=
(hp.congr hf).eq_zero
#align has_fpower_series_at.eq_zero_of_eventually HasFPowerSeriesAt.eq_zero_of_eventually
/-- If a function `f : 𝕜 → E` has two power series representations at `x`, then the given radii in
which convergence is guaranteed may be interchanged. This can be useful when the formal multilinear
series in one representation has a particularly nice form, but the other has a larger radius. -/
theorem HasFPowerSeriesOnBall.exchange_radius {p₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E}
{r₁ r₂ : ℝ≥0∞} {x : 𝕜} (h₁ : HasFPowerSeriesOnBall f p₁ x r₁)
(h₂ : HasFPowerSeriesOnBall f p₂ x r₂) : HasFPowerSeriesOnBall f p₁ x r₂ :=
h₂.hasFPowerSeriesAt.eq_formalMultilinearSeries h₁.hasFPowerSeriesAt ▸ h₂
#align has_fpower_series_on_ball.exchange_radius HasFPowerSeriesOnBall.exchange_radius
/-- If a function `f : 𝕜 → E` has power series representation `p` on a ball of some radius and for
each positive radius it has some power series representation, then `p` converges to `f` on the whole
`𝕜`. -/
theorem HasFPowerSeriesOnBall.r_eq_top_of_exists {f : 𝕜 → E} {r : ℝ≥0∞} {x : 𝕜}
{p : FormalMultilinearSeries 𝕜 𝕜 E} (h : HasFPowerSeriesOnBall f p x r)
(h' : ∀ (r' : ℝ≥0) (_ : 0 < r'), ∃ p' : FormalMultilinearSeries 𝕜 𝕜 E,
HasFPowerSeriesOnBall f p' x r') :
HasFPowerSeriesOnBall f p x ∞ :=
{ r_le := ENNReal.le_of_forall_pos_nnreal_lt fun r hr _ =>
let ⟨_, hp'⟩ := h' r hr
(h.exchange_radius hp').r_le
r_pos := ENNReal.coe_lt_top
hasSum := fun {y} _ =>
let ⟨r', hr'⟩ := exists_gt ‖y‖₊
let ⟨_, hp'⟩ := h' r' hr'.ne_bot.bot_lt
(h.exchange_radius hp').hasSum <| mem_emetric_ball_zero_iff.mpr (ENNReal.coe_lt_coe.2 hr') }
#align has_fpower_series_on_ball.r_eq_top_of_exists HasFPowerSeriesOnBall.r_eq_top_of_exists
end Uniqueness
/-!
### Changing origin in a power series
If a function is analytic in a disk `D(x, R)`, then it is analytic in any disk contained in that
one. Indeed, one can write
$$
f (x + y + z) = \sum_{n} p_n (y + z)^n = \sum_{n, k} \binom{n}{k} p_n y^{n-k} z^k
= \sum_{k} \Bigl(\sum_{n} \binom{n}{k} p_n y^{n-k}\Bigr) z^k.
$$
The corresponding power series has thus a `k`-th coefficient equal to
$\sum_{n} \binom{n}{k} p_n y^{n-k}$. In the general case where `pₙ` is a multilinear map, this has
to be interpreted suitably: instead of having a binomial coefficient, one should sum over all
possible subsets `s` of `Fin n` of cardinal `k`, and attribute `z` to the indices in `s` and
`y` to the indices outside of `s`.
In this paragraph, we implement this. The new power series is called `p.changeOrigin y`. Then, we
check its convergence and the fact that its sum coincides with the original sum. The outcome of this
discussion is that the set of points where a function is analytic is open.
-/
namespace FormalMultilinearSeries
section
variable (p : FormalMultilinearSeries 𝕜 E F) {x y : E} {r R : ℝ≥0}
/-- A term of `FormalMultilinearSeries.changeOriginSeries`.
Given a formal multilinear series `p` and a point `x` in its ball of convergence,
`p.changeOrigin x` is a formal multilinear series such that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense. Each term of `p.changeOrigin x`
is itself an analytic function of `x` given by the series `p.changeOriginSeries`. Each term in
`changeOriginSeries` is the sum of `changeOriginSeriesTerm`'s over all `s` of cardinality `l`.
The definition is such that `p.changeOriginSeriesTerm k l s hs (fun _ ↦ x) (fun _ ↦ y) =
p (k + l) (s.piecewise (fun _ ↦ x) (fun _ ↦ y))`
-/
def changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
E[×l]→L[𝕜] E[×k]→L[𝕜] F := by
let a := ContinuousMultilinearMap.curryFinFinset 𝕜 E F hs
(by erw [Finset.card_compl, Fintype.card_fin, hs, add_tsub_cancel_right])
exact a (p (k + l))
#align formal_multilinear_series.change_origin_series_term FormalMultilinearSeries.changeOriginSeriesTerm
theorem changeOriginSeriesTerm_apply (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l)
(x y : E) :
(p.changeOriginSeriesTerm k l s hs (fun _ => x) fun _ => y) =
p (k + l) (s.piecewise (fun _ => x) fun _ => y) :=
ContinuousMultilinearMap.curryFinFinset_apply_const _ _ _ _ _
#align formal_multilinear_series.change_origin_series_term_apply FormalMultilinearSeries.changeOriginSeriesTerm_apply
@[simp]
theorem norm_changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
‖p.changeOriginSeriesTerm k l s hs‖ = ‖p (k + l)‖ := by
simp only [changeOriginSeriesTerm, LinearIsometryEquiv.norm_map]
#align formal_multilinear_series.norm_change_origin_series_term FormalMultilinearSeries.norm_changeOriginSeriesTerm
@[simp]
theorem nnnorm_changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
‖p.changeOriginSeriesTerm k l s hs‖₊ = ‖p (k + l)‖₊ := by
simp only [changeOriginSeriesTerm, LinearIsometryEquiv.nnnorm_map]
#align formal_multilinear_series.nnnorm_change_origin_series_term FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm
theorem nnnorm_changeOriginSeriesTerm_apply_le (k l : ℕ) (s : Finset (Fin (k + l)))
(hs : s.card = l) (x y : E) :
‖p.changeOriginSeriesTerm k l s hs (fun _ => x) fun _ => y‖₊ ≤
‖p (k + l)‖₊ * ‖x‖₊ ^ l * ‖y‖₊ ^ k := by
rw [← p.nnnorm_changeOriginSeriesTerm k l s hs, ← Fin.prod_const, ← Fin.prod_const]
apply ContinuousMultilinearMap.le_of_op_nnnorm_le
apply ContinuousMultilinearMap.le_op_nnnorm
#align formal_multilinear_series.nnnorm_change_origin_series_term_apply_le FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm_apply_le
/-- The power series for `f.changeOrigin k`.
Given a formal multilinear series `p` and a point `x` in its ball of convergence,
`p.changeOrigin x` is a formal multilinear series such that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense. Its `k`-th term is the sum of
the series `p.changeOriginSeries k`. -/
def changeOriginSeries (k : ℕ) : FormalMultilinearSeries 𝕜 E (E[×k]→L[𝕜] F) := fun l =>
∑ s : { s : Finset (Fin (k + l)) // Finset.card s = l }, p.changeOriginSeriesTerm k l s s.2
#align formal_multilinear_series.change_origin_series FormalMultilinearSeries.changeOriginSeries
theorem nnnorm_changeOriginSeries_le_tsum (k l : ℕ) :
‖p.changeOriginSeries k l‖₊ ≤
∑' _ : { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + l)‖₊ :=
(nnnorm_sum_le _ (fun t => changeOriginSeriesTerm p k l (Subtype.val t) t.prop)).trans_eq <| by
simp_rw [tsum_fintype, nnnorm_changeOriginSeriesTerm (p := p) (k := k) (l := l)]
#align formal_multilinear_series.nnnorm_change_origin_series_le_tsum FormalMultilinearSeries.nnnorm_changeOriginSeries_le_tsum
theorem nnnorm_changeOriginSeries_apply_le_tsum (k l : ℕ) (x : E) :
‖p.changeOriginSeries k l fun _ => x‖₊ ≤
∑' _ : { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + l)‖₊ * ‖x‖₊ ^ l := by
rw [NNReal.tsum_mul_right, ← Fin.prod_const]
exact (p.changeOriginSeries k l).le_of_op_nnnorm_le _ (p.nnnorm_changeOriginSeries_le_tsum _ _)
#align formal_multilinear_series.nnnorm_change_origin_series_apply_le_tsum FormalMultilinearSeries.nnnorm_changeOriginSeries_apply_le_tsum
/-- Changing the origin of a formal multilinear series `p`, so that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense.
-/
def changeOrigin (x : E) : FormalMultilinearSeries 𝕜 E F :=
fun k => (p.changeOriginSeries k).sum x
#align formal_multilinear_series.change_origin FormalMultilinearSeries.changeOrigin
/-- An auxiliary equivalence useful in the proofs about
`FormalMultilinearSeries.changeOriginSeries`: the set of triples `(k, l, s)`, where `s` is a
`Finset (Fin (k + l))` of cardinality `l` is equivalent to the set of pairs `(n, s)`, where `s` is a
`Finset (Fin n)`.
The forward map sends `(k, l, s)` to `(k + l, s)` and the inverse map sends `(n, s)` to
`(n - Finset.card s, Finset.card s, s)`. The actual definition is less readable because of problems
with non-definitional equalities. -/
@[simps]
def changeOriginIndexEquiv :
(Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) ≃ Σn : ℕ, Finset (Fin n) where
toFun s := ⟨s.1 + s.2.1, s.2.2⟩
invFun s :=
⟨s.1 - s.2.card, s.2.card,
⟨s.2.map
(Fin.castIso <| (tsub_add_cancel_of_le <| card_finset_fin_le s.2).symm).toEquiv.toEmbedding,
Finset.card_map _⟩⟩
left_inv := by
rintro ⟨k, l, ⟨s : Finset (Fin <| k + l), hs : s.card = l⟩⟩
dsimp only [Subtype.coe_mk]
-- Lean can't automatically generalize `k' = k + l - s.card`, `l' = s.card`, so we explicitly
-- formulate the generalized goal
suffices ∀ k' l', k' = k → l' = l → ∀ (hkl : k + l = k' + l') (hs'),
(⟨k', l', ⟨Finset.map (Fin.castIso hkl).toEquiv.toEmbedding s, hs'⟩⟩ :
Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) = ⟨k, l, ⟨s, hs⟩⟩ by
apply this <;> simp only [hs, add_tsub_cancel_right]
rintro _ _ rfl rfl hkl hs'
simp only [Equiv.refl_toEmbedding, Fin.castIso_refl, Finset.map_refl, eq_self_iff_true,
OrderIso.refl_toEquiv, and_self_iff, heq_iff_eq]
right_inv := by
rintro ⟨n, s⟩
simp [tsub_add_cancel_of_le (card_finset_fin_le s), Fin.castIso_to_equiv]
#align formal_multilinear_series.change_origin_index_equiv FormalMultilinearSeries.changeOriginIndexEquiv
theorem changeOriginSeries_summable_aux₁ {r r' : ℝ≥0} (hr : (r + r' : ℝ≥0∞) < p.radius) :
Summable fun s : Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l } =>
‖p (s.1 + s.2.1)‖₊ * r ^ s.2.1 * r' ^ s.1 := by
rw [← changeOriginIndexEquiv.symm.summable_iff]
dsimp only [Function.comp_def, changeOriginIndexEquiv_symm_apply_fst,
changeOriginIndexEquiv_symm_apply_snd_fst]
have : ∀ n : ℕ,
HasSum (fun s : Finset (Fin n) => ‖p (n - s.card + s.card)‖₊ * r ^ s.card * r' ^ (n - s.card))
(‖p n‖₊ * (r + r') ^ n) := by
intro n
-- TODO: why `simp only [tsub_add_cancel_of_le (card_finset_fin_le _)]` fails?
convert_to HasSum (fun s : Finset (Fin n) => ‖p n‖₊ * (r ^ s.card * r' ^ (n - s.card))) _
· ext1 s
rw [tsub_add_cancel_of_le (card_finset_fin_le _), mul_assoc]
rw [← Fin.sum_pow_mul_eq_add_pow]
exact (hasSum_fintype _).mul_left _
refine' NNReal.summable_sigma.2 ⟨fun n => (this n).summable, _⟩
simp only [(this _).tsum_eq]
exact p.summable_nnnorm_mul_pow hr
#align formal_multilinear_series.change_origin_series_summable_aux₁ FormalMultilinearSeries.changeOriginSeries_summable_aux₁
theorem changeOriginSeries_summable_aux₂ (hr : (r : ℝ≥0∞) < p.radius) (k : ℕ) :
Summable fun s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l } =>
‖p (k + s.1)‖₊ * r ^ s.1 := by
rcases ENNReal.lt_iff_exists_add_pos_lt.1 hr with ⟨r', h0, hr'⟩
simpa only [mul_inv_cancel_right₀ (pow_pos h0 _).ne'] using
((NNReal.summable_sigma.1 (p.changeOriginSeries_summable_aux₁ hr')).1 k).mul_right (r' ^ k)⁻¹
#align formal_multilinear_series.change_origin_series_summable_aux₂ FormalMultilinearSeries.changeOriginSeries_summable_aux₂
theorem changeOriginSeries_summable_aux₃ {r : ℝ≥0} (hr : ↑r < p.radius) (k : ℕ) :
Summable fun l : ℕ => ‖p.changeOriginSeries k l‖₊ * r ^ l := by
refine' NNReal.summable_of_le
(fun n => _) (NNReal.summable_sigma.1 <| p.changeOriginSeries_summable_aux₂ hr k).2
simp only [NNReal.tsum_mul_right]
exact mul_le_mul' (p.nnnorm_changeOriginSeries_le_tsum _ _) le_rfl
#align formal_multilinear_series.change_origin_series_summable_aux₃ FormalMultilinearSeries.changeOriginSeries_summable_aux₃
theorem le_changeOriginSeries_radius (k : ℕ) : p.radius ≤ (p.changeOriginSeries k).radius :=
ENNReal.le_of_forall_nnreal_lt fun _r hr =>
le_radius_of_summable_nnnorm _ (p.changeOriginSeries_summable_aux₃ hr k)
#align formal_multilinear_series.le_change_origin_series_radius FormalMultilinearSeries.le_changeOriginSeries_radius
theorem nnnorm_changeOrigin_le (k : ℕ) (h : (‖x‖₊ : ℝ≥0∞) < p.radius) :
‖p.changeOrigin x k‖₊ ≤
∑' s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + s.1)‖₊ * ‖x‖₊ ^ s.1 := by
refine' tsum_of_nnnorm_bounded _ fun l => p.nnnorm_changeOriginSeries_apply_le_tsum k l x
have := p.changeOriginSeries_summable_aux₂ h k
refine' HasSum.sigma this.hasSum fun l => _
exact ((NNReal.summable_sigma.1 this).1 l).hasSum
#align formal_multilinear_series.nnnorm_change_origin_le FormalMultilinearSeries.nnnorm_changeOrigin_le
/-- The radius of convergence of `p.changeOrigin x` is at least `p.radius - ‖x‖`. In other words,
`p.changeOrigin x` is well defined on the largest ball contained in the original ball of
convergence. -/
theorem changeOrigin_radius : p.radius - ‖x‖₊ ≤ (p.changeOrigin x).radius := by
refine' ENNReal.le_of_forall_pos_nnreal_lt fun r _h0 hr => _
rw [lt_tsub_iff_right, add_comm] at hr
have hr' : (‖x‖₊ : ℝ≥0∞) < p.radius := (le_add_right le_rfl).trans_lt hr
apply le_radius_of_summable_nnnorm
have : ∀ k : ℕ,
‖p.changeOrigin x k‖₊ * r ^ k ≤
(∑' s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + s.1)‖₊ * ‖x‖₊ ^ s.1) *
r ^ k :=
fun k => mul_le_mul_right' (p.nnnorm_changeOrigin_le k hr') (r ^ k)
refine' NNReal.summable_of_le this _
simpa only [← NNReal.tsum_mul_right] using
(NNReal.summable_sigma.1 (p.changeOriginSeries_summable_aux₁ hr)).2
#align formal_multilinear_series.change_origin_radius FormalMultilinearSeries.changeOrigin_radius
end
-- From this point on, assume that the space is complete, to make sure that series that converge
-- in norm also converge in `F`.
variable [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x y : E} {r R : ℝ≥0}
theorem hasFPowerSeriesOnBall_changeOrigin (k : ℕ) (hr : 0 < p.radius) :
HasFPowerSeriesOnBall (fun x => p.changeOrigin x k) (p.changeOriginSeries k) 0 p.radius :=
have := p.le_changeOriginSeries_radius k
((p.changeOriginSeries k).hasFPowerSeriesOnBall (hr.trans_le this)).mono hr this
#align formal_multilinear_series.has_fpower_series_on_ball_change_origin FormalMultilinearSeries.hasFPowerSeriesOnBall_changeOrigin
/-- Summing the series `p.changeOrigin x` at a point `y` gives back `p (x + y)`. -/
theorem changeOrigin_eval (h : (‖x‖₊ + ‖y‖₊ : ℝ≥0∞) < p.radius) :
(p.changeOrigin x).sum y = p.sum (x + y) := by
have radius_pos : 0 < p.radius := lt_of_le_of_lt (zero_le _) h
have x_mem_ball : x ∈ EMetric.ball (0 : E) p.radius :=
mem_emetric_ball_zero_iff.2 ((le_add_right le_rfl).trans_lt h)
have y_mem_ball : y ∈ EMetric.ball (0 : E) (p.changeOrigin x).radius := by
refine' mem_emetric_ball_zero_iff.2 (lt_of_lt_of_le _ p.changeOrigin_radius)
rwa [lt_tsub_iff_right, add_comm]
have x_add_y_mem_ball : x + y ∈ EMetric.ball (0 : E) p.radius := by
refine' mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt _ h)
exact mod_cast nnnorm_add_le x y
set f : (Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) → F := fun s =>
p.changeOriginSeriesTerm s.1 s.2.1 s.2.2 s.2.2.2 (fun _ => x) fun _ => y
have hsf : Summable f := by
refine' .of_nnnorm_bounded _ (p.changeOriginSeries_summable_aux₁ h) _
rintro ⟨k, l, s, hs⟩
dsimp only [Subtype.coe_mk]
exact p.nnnorm_changeOriginSeriesTerm_apply_le _ _ _ _ _ _
have hf : HasSum f ((p.changeOrigin x).sum y) := by
refine' HasSum.sigma_of_hasSum ((p.changeOrigin x).summable y_mem_ball).hasSum (fun k => _) hsf
· dsimp only
refine' ContinuousMultilinearMap.hasSum_eval _ _
have := (p.hasFPowerSeriesOnBall_changeOrigin k radius_pos).hasSum x_mem_ball
rw [zero_add] at this
refine' HasSum.sigma_of_hasSum this (fun l => _) _
· simp only [changeOriginSeries, ContinuousMultilinearMap.sum_apply]
apply hasSum_fintype
· refine' .of_nnnorm_bounded _
(p.changeOriginSeries_summable_aux₂ (mem_emetric_ball_zero_iff.1 x_mem_ball) k) fun s => _
refine' (ContinuousMultilinearMap.le_op_nnnorm _ _).trans_eq _
simp
refine' hf.unique (changeOriginIndexEquiv.symm.hasSum_iff.1 _)
refine' HasSum.sigma_of_hasSum
(p.hasSum x_add_y_mem_ball) (fun n => _) (changeOriginIndexEquiv.symm.summable_iff.2 hsf)
erw [(p n).map_add_univ (fun _ => x) fun _ => y]
-- porting note: added explicit function
convert hasSum_fintype (fun c : Finset (Fin n) => f (changeOriginIndexEquiv.symm ⟨n, c⟩))
rename_i s _
dsimp only [changeOriginSeriesTerm, (· ∘ ·), changeOriginIndexEquiv_symm_apply_fst,
changeOriginIndexEquiv_symm_apply_snd_fst, changeOriginIndexEquiv_symm_apply_snd_snd_coe]
rw [ContinuousMultilinearMap.curryFinFinset_apply_const]
have : ∀ (m) (hm : n = m), p n (s.piecewise (fun _ => x) fun _ => y) =
p m ((s.map (Fin.castIso hm).toEquiv.toEmbedding).piecewise (fun _ => x) fun _ => y) := by
rintro m rfl
simp (config := { unfoldPartialApp := true }) [Finset.piecewise]
apply this
#align formal_multilinear_series.change_origin_eval FormalMultilinearSeries.changeOrigin_eval
/-- Power series terms are analytic as we vary the origin -/
theorem analyticAt_changeOrigin (p : FormalMultilinearSeries 𝕜 E F) (rp : p.radius > 0) (n : ℕ) :
AnalyticAt 𝕜 (fun x ↦ p.changeOrigin x n) 0 :=
(FormalMultilinearSeries.hasFPowerSeriesOnBall_changeOrigin p n rp).analyticAt
end FormalMultilinearSeries
section
variable [CompleteSpace F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x y : E} {r : ℝ≥0∞}
/-- If a function admits a power series expansion `p` on a ball `B (x, r)`, then it also admits a
power series on any subball of this ball (even with a different center), given by `p.changeOrigin`.
-/
theorem HasFPowerSeriesOnBall.changeOrigin (hf : HasFPowerSeriesOnBall f p x r)
(h : (‖y‖₊ : ℝ≥0∞) < r) : HasFPowerSeriesOnBall f (p.changeOrigin y) (x + y) (r - ‖y‖₊) :=
{ r_le := by
apply le_trans _ p.changeOrigin_radius
exact tsub_le_tsub hf.r_le le_rfl
r_pos := by simp [h]
hasSum := fun {z} hz => by
have : f (x + y + z) =
FormalMultilinearSeries.sum (FormalMultilinearSeries.changeOrigin p y) z := by
rw [mem_emetric_ball_zero_iff, lt_tsub_iff_right, add_comm] at hz
rw [p.changeOrigin_eval (hz.trans_le hf.r_le), add_assoc, hf.sum]
refine' mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt _ hz)
exact mod_cast nnnorm_add_le y z
rw [this]
apply (p.changeOrigin y).hasSum
refine' EMetric.ball_subset_ball (le_trans _ p.changeOrigin_radius) hz
exact tsub_le_tsub hf.r_le le_rfl }
#align has_fpower_series_on_ball.change_origin HasFPowerSeriesOnBall.changeOrigin
/-- If a function admits a power series expansion `p` on an open ball `B (x, r)`, then
it is analytic at every point of this ball. -/
theorem HasFPowerSeriesOnBall.analyticAt_of_mem (hf : HasFPowerSeriesOnBall f p x r)
(h : y ∈ EMetric.ball x r) : AnalyticAt 𝕜 f y := by
have : (‖y - x‖₊ : ℝ≥0∞) < r := by simpa [edist_eq_coe_nnnorm_sub] using h
have := hf.changeOrigin this
rw [add_sub_cancel'_right] at this
exact this.analyticAt
#align has_fpower_series_on_ball.analytic_at_of_mem HasFPowerSeriesOnBall.analyticAt_of_mem
theorem HasFPowerSeriesOnBall.analyticOn (hf : HasFPowerSeriesOnBall f p x r) :
AnalyticOn 𝕜 f (EMetric.ball x r) :=
fun _y hy => hf.analyticAt_of_mem hy
#align has_fpower_series_on_ball.analytic_on HasFPowerSeriesOnBall.analyticOn
variable (𝕜 f)
/-- For any function `f` from a normed vector space to a Banach space, the set of points `x` such
that `f` is analytic at `x` is open. -/
theorem isOpen_analyticAt : IsOpen { x | AnalyticAt 𝕜 f x } := by
rw [isOpen_iff_mem_nhds]
rintro x ⟨p, r, hr⟩
exact mem_of_superset (EMetric.ball_mem_nhds _ hr.r_pos) fun y hy => hr.analyticAt_of_mem hy
#align is_open_analytic_at isOpen_analyticAt
variable {𝕜}
theorem AnalyticAt.eventually_analyticAt {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∀ᶠ y in 𝓝 x, AnalyticAt 𝕜 f y :=
(isOpen_analyticAt 𝕜 f).mem_nhds h
theorem AnalyticAt.exists_mem_nhds_analyticOn {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∃ s ∈ 𝓝 x, AnalyticOn 𝕜 f s :=
h.eventually_analyticAt.exists_mem
/-- If we're analytic at a point, we're analytic in a nonempty ball -/
theorem AnalyticAt.exists_ball_analyticOn {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∃ r : ℝ, 0 < r ∧ AnalyticOn 𝕜 f (Metric.ball x r) :=
Metric.isOpen_iff.mp (isOpen_analyticAt _ _) _ h
end
section
open FormalMultilinearSeries
variable {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E} {z₀ : 𝕜}
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z)) := by
refine' ⟨fun ⟨r, _, r_pos, h⟩ =>
eventually_of_mem (EMetric.ball_mem_nhds 0 r_pos) fun _ => by simpa using h, _⟩
simp only [Metric.eventually_nhds_iff]
rintro ⟨r, r_pos, h⟩
refine' ⟨p.radius ⊓ r.toNNReal, by simp, _, _⟩
· simp only [r_pos.lt, lt_inf_iff, ENNReal.coe_pos, Real.toNNReal_pos, and_true_iff]
obtain ⟨z, z_pos, le_z⟩ := NormedField.exists_norm_lt 𝕜 r_pos.lt
have : (‖z‖₊ : ENNReal) ≤ p.radius := by
simp only [dist_zero_right] at h
apply FormalMultilinearSeries.le_radius_of_tendsto
convert tendsto_norm.comp (h le_z).summable.tendsto_atTop_zero
funext
simp [norm_smul, mul_comm]
refine' lt_of_lt_of_le _ this
|
simp only [ENNReal.coe_pos]
|
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z)) := by
refine' ⟨fun ⟨r, _, r_pos, h⟩ =>
eventually_of_mem (EMetric.ball_mem_nhds 0 r_pos) fun _ => by simpa using h, _⟩
simp only [Metric.eventually_nhds_iff]
rintro ⟨r, r_pos, h⟩
refine' ⟨p.radius ⊓ r.toNNReal, by simp, _, _⟩
· simp only [r_pos.lt, lt_inf_iff, ENNReal.coe_pos, Real.toNNReal_pos, and_true_iff]
obtain ⟨z, z_pos, le_z⟩ := NormedField.exists_norm_lt 𝕜 r_pos.lt
have : (‖z‖₊ : ENNReal) ≤ p.radius := by
simp only [dist_zero_right] at h
apply FormalMultilinearSeries.le_radius_of_tendsto
convert tendsto_norm.comp (h le_z).summable.tendsto_atTop_zero
funext
simp [norm_smul, mul_comm]
refine' lt_of_lt_of_le _ this
|
Mathlib.Analysis.Analytic.Basic.1430_0.jQw1fRSE1vGpOll
|
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z))
|
Mathlib_Analysis_Analytic_Basic
|
case intro.intro.refine'_1.intro.intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
p : FormalMultilinearSeries 𝕜 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
r : ℝ
r_pos : r > 0
h : ∀ ⦃y : 𝕜⦄, dist y 0 < r → HasSum (fun n => y ^ n • coeff p n) (f (z₀ + y))
z : 𝕜
z_pos : 0 < ‖z‖
le_z : ‖z‖ < r
this : ↑‖z‖₊ ≤ radius p
⊢ 0 < ‖z‖₊
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Logic.Equiv.Fin
import Mathlib.Topology.Algebra.InfiniteSum.Module
#align_import analysis.analytic.basic from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
/-!
# Analytic functions
A function is analytic in one dimension around `0` if it can be written as a converging power series
`Σ pₙ zⁿ`. This definition can be extended to any dimension (even in infinite dimension) by
requiring that `pₙ` is a continuous `n`-multilinear map. In general, `pₙ` is not unique (in two
dimensions, taking `p₂ (x, y) (x', y') = x y'` or `y x'` gives the same map when applied to a
vector `(x, y) (x, y)`). A way to guarantee uniqueness is to take a symmetric `pₙ`, but this is not
always possible in nonzero characteristic (in characteristic 2, the previous example has no
symmetric representative). Therefore, we do not insist on symmetry or uniqueness in the definition,
and we only require the existence of a converging series.
The general framework is important to say that the exponential map on bounded operators on a Banach
space is analytic, as well as the inverse on invertible operators.
## Main definitions
Let `p` be a formal multilinear series from `E` to `F`, i.e., `p n` is a multilinear map on `E^n`
for `n : ℕ`.
* `p.radius`: the largest `r : ℝ≥0∞` such that `‖p n‖ * r^n` grows subexponentially.
* `p.le_radius_of_bound`, `p.le_radius_of_bound_nnreal`, `p.le_radius_of_isBigO`: if `‖p n‖ * r ^ n`
is bounded above, then `r ≤ p.radius`;
* `p.isLittleO_of_lt_radius`, `p.norm_mul_pow_le_mul_pow_of_lt_radius`,
`p.isLittleO_one_of_lt_radius`,
`p.norm_mul_pow_le_of_lt_radius`, `p.nnnorm_mul_pow_le_of_lt_radius`: if `r < p.radius`, then
`‖p n‖ * r ^ n` tends to zero exponentially;
* `p.lt_radius_of_isBigO`: if `r ≠ 0` and `‖p n‖ * r ^ n = O(a ^ n)` for some `-1 < a < 1`, then
`r < p.radius`;
* `p.partialSum n x`: the sum `∑_{i = 0}^{n-1} pᵢ xⁱ`.
* `p.sum x`: the sum `∑'_{i = 0}^{∞} pᵢ xⁱ`.
Additionally, let `f` be a function from `E` to `F`.
* `HasFPowerSeriesOnBall f p x r`: on the ball of center `x` with radius `r`,
`f (x + y) = ∑'_n pₙ yⁿ`.
* `HasFPowerSeriesAt f p x`: on some ball of center `x` with positive radius, holds
`HasFPowerSeriesOnBall f p x r`.
* `AnalyticAt 𝕜 f x`: there exists a power series `p` such that holds `HasFPowerSeriesAt f p x`.
* `AnalyticOn 𝕜 f s`: the function `f` is analytic at every point of `s`.
We develop the basic properties of these notions, notably:
* If a function admits a power series, it is continuous (see
`HasFPowerSeriesOnBall.continuousOn` and `HasFPowerSeriesAt.continuousAt` and
`AnalyticAt.continuousAt`).
* In a complete space, the sum of a formal power series with positive radius is well defined on the
disk of convergence, see `FormalMultilinearSeries.hasFPowerSeriesOnBall`.
* If a function admits a power series in a ball, then it is analytic at any point `y` of this ball,
and the power series there can be expressed in terms of the initial power series `p` as
`p.changeOrigin y`. See `HasFPowerSeriesOnBall.changeOrigin`. It follows in particular that
the set of points at which a given function is analytic is open, see `isOpen_analyticAt`.
## Implementation details
We only introduce the radius of convergence of a power series, as `p.radius`.
For a power series in finitely many dimensions, there is a finer (directional, coordinate-dependent)
notion, describing the polydisk of convergence. This notion is more specific, and not necessary to
build the general theory. We do not define it here.
-/
noncomputable section
variable {𝕜 E F G : Type*}
open Topology Classical BigOperators NNReal Filter ENNReal
open Set Filter Asymptotics
namespace FormalMultilinearSeries
variable [Ring 𝕜] [AddCommGroup E] [AddCommGroup F] [Module 𝕜 E] [Module 𝕜 F]
variable [TopologicalSpace E] [TopologicalSpace F]
variable [TopologicalAddGroup E] [TopologicalAddGroup F]
variable [ContinuousConstSMul 𝕜 E] [ContinuousConstSMul 𝕜 F]
/-- Given a formal multilinear series `p` and a vector `x`, then `p.sum x` is the sum `Σ pₙ xⁿ`. A
priori, it only behaves well when `‖x‖ < p.radius`. -/
protected def sum (p : FormalMultilinearSeries 𝕜 E F) (x : E) : F :=
∑' n : ℕ, p n fun _ => x
#align formal_multilinear_series.sum FormalMultilinearSeries.sum
/-- Given a formal multilinear series `p` and a vector `x`, then `p.partialSum n x` is the sum
`Σ pₖ xᵏ` for `k ∈ {0,..., n-1}`. -/
def partialSum (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) (x : E) : F :=
∑ k in Finset.range n, p k fun _ : Fin k => x
#align formal_multilinear_series.partial_sum FormalMultilinearSeries.partialSum
/-- The partial sums of a formal multilinear series are continuous. -/
theorem partialSum_continuous (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) :
Continuous (p.partialSum n) := by
unfold partialSum -- Porting note: added
continuity
#align formal_multilinear_series.partial_sum_continuous FormalMultilinearSeries.partialSum_continuous
end FormalMultilinearSeries
/-! ### The radius of a formal multilinear series -/
variable [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F]
[NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G]
namespace FormalMultilinearSeries
variable (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
/-- The radius of a formal multilinear series is the largest `r` such that the sum `Σ ‖pₙ‖ ‖y‖ⁿ`
converges for all `‖y‖ < r`. This implies that `Σ pₙ yⁿ` converges for all `‖y‖ < r`, but these
definitions are *not* equivalent in general. -/
def radius (p : FormalMultilinearSeries 𝕜 E F) : ℝ≥0∞ :=
⨆ (r : ℝ≥0) (C : ℝ) (_ : ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C), (r : ℝ≥0∞)
#align formal_multilinear_series.radius FormalMultilinearSeries.radius
/-- If `‖pₙ‖ rⁿ` is bounded in `n`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_bound (C : ℝ) {r : ℝ≥0} (h : ∀ n : ℕ, ‖p n‖ * (r : ℝ) ^ n ≤ C) :
(r : ℝ≥0∞) ≤ p.radius :=
le_iSup_of_le r <| le_iSup_of_le C <| le_iSup (fun _ => (r : ℝ≥0∞)) h
#align formal_multilinear_series.le_radius_of_bound FormalMultilinearSeries.le_radius_of_bound
/-- If `‖pₙ‖ rⁿ` is bounded in `n`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_bound_nnreal (C : ℝ≥0) {r : ℝ≥0} (h : ∀ n : ℕ, ‖p n‖₊ * r ^ n ≤ C) :
(r : ℝ≥0∞) ≤ p.radius :=
p.le_radius_of_bound C fun n => mod_cast h n
#align formal_multilinear_series.le_radius_of_bound_nnreal FormalMultilinearSeries.le_radius_of_bound_nnreal
/-- If `‖pₙ‖ rⁿ = O(1)`, as `n → ∞`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_isBigO (h : (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] fun _ => (1 : ℝ)) :
↑r ≤ p.radius :=
Exists.elim (isBigO_one_nat_atTop_iff.1 h) fun C hC =>
p.le_radius_of_bound C fun n => (le_abs_self _).trans (hC n)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.le_radius_of_is_O FormalMultilinearSeries.le_radius_of_isBigO
theorem le_radius_of_eventually_le (C) (h : ∀ᶠ n in atTop, ‖p n‖ * (r : ℝ) ^ n ≤ C) :
↑r ≤ p.radius :=
p.le_radius_of_isBigO <| IsBigO.of_bound C <| h.mono fun n hn => by simpa
#align formal_multilinear_series.le_radius_of_eventually_le FormalMultilinearSeries.le_radius_of_eventually_le
theorem le_radius_of_summable_nnnorm (h : Summable fun n => ‖p n‖₊ * r ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_bound_nnreal (∑' n, ‖p n‖₊ * r ^ n) fun _ => le_tsum' h _
#align formal_multilinear_series.le_radius_of_summable_nnnorm FormalMultilinearSeries.le_radius_of_summable_nnnorm
theorem le_radius_of_summable (h : Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_summable_nnnorm <| by
simp only [← coe_nnnorm] at h
exact mod_cast h
#align formal_multilinear_series.le_radius_of_summable FormalMultilinearSeries.le_radius_of_summable
theorem radius_eq_top_of_forall_nnreal_isBigO
(h : ∀ r : ℝ≥0, (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] fun _ => (1 : ℝ)) : p.radius = ∞ :=
ENNReal.eq_top_of_forall_nnreal_le fun r => p.le_radius_of_isBigO (h r)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.radius_eq_top_of_forall_nnreal_is_O FormalMultilinearSeries.radius_eq_top_of_forall_nnreal_isBigO
theorem radius_eq_top_of_eventually_eq_zero (h : ∀ᶠ n in atTop, p n = 0) : p.radius = ∞ :=
p.radius_eq_top_of_forall_nnreal_isBigO fun r =>
(isBigO_zero _ _).congr' (h.mono fun n hn => by simp [hn]) EventuallyEq.rfl
#align formal_multilinear_series.radius_eq_top_of_eventually_eq_zero FormalMultilinearSeries.radius_eq_top_of_eventually_eq_zero
theorem radius_eq_top_of_forall_image_add_eq_zero (n : ℕ) (hn : ∀ m, p (m + n) = 0) :
p.radius = ∞ :=
p.radius_eq_top_of_eventually_eq_zero <|
mem_atTop_sets.2 ⟨n, fun _ hk => tsub_add_cancel_of_le hk ▸ hn _⟩
#align formal_multilinear_series.radius_eq_top_of_forall_image_add_eq_zero FormalMultilinearSeries.radius_eq_top_of_forall_image_add_eq_zero
@[simp]
theorem constFormalMultilinearSeries_radius {v : F} :
(constFormalMultilinearSeries 𝕜 E v).radius = ⊤ :=
(constFormalMultilinearSeries 𝕜 E v).radius_eq_top_of_forall_image_add_eq_zero 1
(by simp [constFormalMultilinearSeries])
#align formal_multilinear_series.const_formal_multilinear_series_radius FormalMultilinearSeries.constFormalMultilinearSeries_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` tends to zero exponentially:
for some `0 < a < 1`, `‖p n‖ rⁿ = o(aⁿ)`. -/
theorem isLittleO_of_lt_radius (h : ↑r < p.radius) :
∃ a ∈ Ioo (0 : ℝ) 1, (fun n => ‖p n‖ * (r : ℝ) ^ n) =o[atTop] (a ^ ·) := by
have := (TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 4
rw [this]
-- Porting note: was
-- rw [(TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 4]
simp only [radius, lt_iSup_iff] at h
rcases h with ⟨t, C, hC, rt⟩
rw [ENNReal.coe_lt_coe, ← NNReal.coe_lt_coe] at rt
have : 0 < (t : ℝ) := r.coe_nonneg.trans_lt rt
rw [← div_lt_one this] at rt
refine' ⟨_, rt, C, Or.inr zero_lt_one, fun n => _⟩
calc
|‖p n‖ * (r : ℝ) ^ n| = ‖p n‖ * (t : ℝ) ^ n * (r / t : ℝ) ^ n := by
field_simp [mul_right_comm, abs_mul]
_ ≤ C * (r / t : ℝ) ^ n := by gcongr; apply hC
#align formal_multilinear_series.is_o_of_lt_radius FormalMultilinearSeries.isLittleO_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ = o(1)`. -/
theorem isLittleO_one_of_lt_radius (h : ↑r < p.radius) :
(fun n => ‖p n‖ * (r : ℝ) ^ n) =o[atTop] (fun _ => 1 : ℕ → ℝ) :=
let ⟨_, ha, hp⟩ := p.isLittleO_of_lt_radius h
hp.trans <| (isLittleO_pow_pow_of_lt_left ha.1.le ha.2).congr (fun _ => rfl) one_pow
#align formal_multilinear_series.is_o_one_of_lt_radius FormalMultilinearSeries.isLittleO_one_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` tends to zero exponentially:
for some `0 < a < 1` and `C > 0`, `‖p n‖ * r ^ n ≤ C * a ^ n`. -/
theorem norm_mul_pow_le_mul_pow_of_lt_radius (h : ↑r < p.radius) :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C * a ^ n := by
-- Porting note: moved out of `rcases`
have := ((TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 5).mp
(p.isLittleO_of_lt_radius h)
rcases this with ⟨a, ha, C, hC, H⟩
exact ⟨a, ha, C, hC, fun n => (le_abs_self _).trans (H n)⟩
#align formal_multilinear_series.norm_mul_pow_le_mul_pow_of_lt_radius FormalMultilinearSeries.norm_mul_pow_le_mul_pow_of_lt_radius
/-- If `r ≠ 0` and `‖pₙ‖ rⁿ = O(aⁿ)` for some `-1 < a < 1`, then `r < p.radius`. -/
theorem lt_radius_of_isBigO (h₀ : r ≠ 0) {a : ℝ} (ha : a ∈ Ioo (-1 : ℝ) 1)
(hp : (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] (a ^ ·)) : ↑r < p.radius := by
-- Porting note: moved out of `rcases`
have := ((TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 2 5)
rcases this.mp ⟨a, ha, hp⟩ with ⟨a, ha, C, hC, hp⟩
rw [← pos_iff_ne_zero, ← NNReal.coe_pos] at h₀
lift a to ℝ≥0 using ha.1.le
have : (r : ℝ) < r / a := by
simpa only [div_one] using (div_lt_div_left h₀ zero_lt_one ha.1).2 ha.2
norm_cast at this
rw [← ENNReal.coe_lt_coe] at this
refine' this.trans_le (p.le_radius_of_bound C fun n => _)
rw [NNReal.coe_div, div_pow, ← mul_div_assoc, div_le_iff (pow_pos ha.1 n)]
exact (le_abs_self _).trans (hp n)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.lt_radius_of_is_O FormalMultilinearSeries.lt_radius_of_isBigO
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem norm_mul_pow_le_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C :=
let ⟨_, ha, C, hC, h⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius h
⟨C, hC, fun n => (h n).trans <| mul_le_of_le_one_right hC.lt.le (pow_le_one _ ha.1.le ha.2.le)⟩
#align formal_multilinear_series.norm_mul_pow_le_of_lt_radius FormalMultilinearSeries.norm_mul_pow_le_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem norm_le_div_pow_of_pos_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h0 : 0 < r) (h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖ ≤ C / (r : ℝ) ^ n :=
let ⟨C, hC, hp⟩ := p.norm_mul_pow_le_of_lt_radius h
⟨C, hC, fun n => Iff.mpr (le_div_iff (pow_pos h0 _)) (hp n)⟩
#align formal_multilinear_series.norm_le_div_pow_of_pos_of_lt_radius FormalMultilinearSeries.norm_le_div_pow_of_pos_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem nnnorm_mul_pow_le_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖₊ * r ^ n ≤ C :=
let ⟨C, hC, hp⟩ := p.norm_mul_pow_le_of_lt_radius h
⟨⟨C, hC.lt.le⟩, hC, mod_cast hp⟩
#align formal_multilinear_series.nnnorm_mul_pow_le_of_lt_radius FormalMultilinearSeries.nnnorm_mul_pow_le_of_lt_radius
theorem le_radius_of_tendsto (p : FormalMultilinearSeries 𝕜 E F) {l : ℝ}
(h : Tendsto (fun n => ‖p n‖ * (r : ℝ) ^ n) atTop (𝓝 l)) : ↑r ≤ p.radius :=
p.le_radius_of_isBigO (h.isBigO_one _)
#align formal_multilinear_series.le_radius_of_tendsto FormalMultilinearSeries.le_radius_of_tendsto
theorem le_radius_of_summable_norm (p : FormalMultilinearSeries 𝕜 E F)
(hs : Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_tendsto hs.tendsto_atTop_zero
#align formal_multilinear_series.le_radius_of_summable_norm FormalMultilinearSeries.le_radius_of_summable_norm
theorem not_summable_norm_of_radius_lt_nnnorm (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(h : p.radius < ‖x‖₊) : ¬Summable fun n => ‖p n‖ * ‖x‖ ^ n :=
fun hs => not_le_of_lt h (p.le_radius_of_summable_norm hs)
#align formal_multilinear_series.not_summable_norm_of_radius_lt_nnnorm FormalMultilinearSeries.not_summable_norm_of_radius_lt_nnnorm
theorem summable_norm_mul_pow (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0} (h : ↑r < p.radius) :
Summable fun n : ℕ => ‖p n‖ * (r : ℝ) ^ n := by
obtain ⟨a, ha : a ∈ Ioo (0 : ℝ) 1, C, - : 0 < C, hp⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius h
exact .of_nonneg_of_le (fun n => mul_nonneg (norm_nonneg _) (pow_nonneg r.coe_nonneg _))
hp ((summable_geometric_of_lt_1 ha.1.le ha.2).mul_left _)
#align formal_multilinear_series.summable_norm_mul_pow FormalMultilinearSeries.summable_norm_mul_pow
theorem summable_norm_apply (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : Summable fun n : ℕ => ‖p n fun _ => x‖ := by
rw [mem_emetric_ball_zero_iff] at hx
refine' .of_nonneg_of_le
(fun _ => norm_nonneg _) (fun n => ((p n).le_op_norm _).trans_eq _) (p.summable_norm_mul_pow hx)
simp
#align formal_multilinear_series.summable_norm_apply FormalMultilinearSeries.summable_norm_apply
theorem summable_nnnorm_mul_pow (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0} (h : ↑r < p.radius) :
Summable fun n : ℕ => ‖p n‖₊ * r ^ n := by
rw [← NNReal.summable_coe]
push_cast
exact p.summable_norm_mul_pow h
#align formal_multilinear_series.summable_nnnorm_mul_pow FormalMultilinearSeries.summable_nnnorm_mul_pow
protected theorem summable [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : Summable fun n : ℕ => p n fun _ => x :=
(p.summable_norm_apply hx).of_norm
#align formal_multilinear_series.summable FormalMultilinearSeries.summable
theorem radius_eq_top_of_summable_norm (p : FormalMultilinearSeries 𝕜 E F)
(hs : ∀ r : ℝ≥0, Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : p.radius = ∞ :=
ENNReal.eq_top_of_forall_nnreal_le fun r => p.le_radius_of_summable_norm (hs r)
#align formal_multilinear_series.radius_eq_top_of_summable_norm FormalMultilinearSeries.radius_eq_top_of_summable_norm
theorem radius_eq_top_iff_summable_norm (p : FormalMultilinearSeries 𝕜 E F) :
p.radius = ∞ ↔ ∀ r : ℝ≥0, Summable fun n => ‖p n‖ * (r : ℝ) ^ n := by
constructor
· intro h r
obtain ⟨a, ha : a ∈ Ioo (0 : ℝ) 1, C, - : 0 < C, hp⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius
(show (r : ℝ≥0∞) < p.radius from h.symm ▸ ENNReal.coe_lt_top)
refine' .of_norm_bounded
(fun n => (C : ℝ) * a ^ n) ((summable_geometric_of_lt_1 ha.1.le ha.2).mul_left _) fun n => _
specialize hp n
rwa [Real.norm_of_nonneg (mul_nonneg (norm_nonneg _) (pow_nonneg r.coe_nonneg n))]
· exact p.radius_eq_top_of_summable_norm
#align formal_multilinear_series.radius_eq_top_iff_summable_norm FormalMultilinearSeries.radius_eq_top_iff_summable_norm
/-- If the radius of `p` is positive, then `‖pₙ‖` grows at most geometrically. -/
theorem le_mul_pow_of_radius_pos (p : FormalMultilinearSeries 𝕜 E F) (h : 0 < p.radius) :
∃ (C r : _) (hC : 0 < C) (_ : 0 < r), ∀ n, ‖p n‖ ≤ C * r ^ n := by
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 h with ⟨r, r0, rlt⟩
have rpos : 0 < (r : ℝ) := by simp [ENNReal.coe_pos.1 r0]
rcases norm_le_div_pow_of_pos_of_lt_radius p rpos rlt with ⟨C, Cpos, hCp⟩
refine' ⟨C, r⁻¹, Cpos, by simp only [inv_pos, rpos], fun n => _⟩
-- Porting note: was `convert`
rw [inv_pow, ← div_eq_mul_inv]
exact hCp n
#align formal_multilinear_series.le_mul_pow_of_radius_pos FormalMultilinearSeries.le_mul_pow_of_radius_pos
/-- The radius of the sum of two formal series is at least the minimum of their two radii. -/
theorem min_radius_le_radius_add (p q : FormalMultilinearSeries 𝕜 E F) :
min p.radius q.radius ≤ (p + q).radius := by
refine' ENNReal.le_of_forall_nnreal_lt fun r hr => _
rw [lt_min_iff] at hr
have := ((p.isLittleO_one_of_lt_radius hr.1).add (q.isLittleO_one_of_lt_radius hr.2)).isBigO
refine' (p + q).le_radius_of_isBigO ((isBigO_of_le _ fun n => _).trans this)
rw [← add_mul, norm_mul, norm_mul, norm_norm]
exact mul_le_mul_of_nonneg_right ((norm_add_le _ _).trans (le_abs_self _)) (norm_nonneg _)
#align formal_multilinear_series.min_radius_le_radius_add FormalMultilinearSeries.min_radius_le_radius_add
@[simp]
theorem radius_neg (p : FormalMultilinearSeries 𝕜 E F) : (-p).radius = p.radius := by
simp only [radius, neg_apply, norm_neg]
#align formal_multilinear_series.radius_neg FormalMultilinearSeries.radius_neg
protected theorem hasSum [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : HasSum (fun n : ℕ => p n fun _ => x) (p.sum x) :=
(p.summable hx).hasSum
#align formal_multilinear_series.has_sum FormalMultilinearSeries.hasSum
theorem radius_le_radius_continuousLinearMap_comp (p : FormalMultilinearSeries 𝕜 E F)
(f : F →L[𝕜] G) : p.radius ≤ (f.compFormalMultilinearSeries p).radius := by
refine' ENNReal.le_of_forall_nnreal_lt fun r hr => _
apply le_radius_of_isBigO
apply (IsBigO.trans_isLittleO _ (p.isLittleO_one_of_lt_radius hr)).isBigO
refine' IsBigO.mul (@IsBigOWith.isBigO _ _ _ _ _ ‖f‖ _ _ _ _) (isBigO_refl _ _)
refine IsBigOWith.of_bound (eventually_of_forall fun n => ?_)
simpa only [norm_norm] using f.norm_compContinuousMultilinearMap_le (p n)
#align formal_multilinear_series.radius_le_radius_continuous_linear_map_comp FormalMultilinearSeries.radius_le_radius_continuousLinearMap_comp
end FormalMultilinearSeries
/-! ### Expanding a function as a power series -/
section
variable {f g : E → F} {p pf pg : FormalMultilinearSeries 𝕜 E F} {x : E} {r r' : ℝ≥0∞}
/-- Given a function `f : E → F` and a formal multilinear series `p`, we say that `f` has `p` as
a power series on the ball of radius `r > 0` around `x` if `f (x + y) = ∑' pₙ yⁿ` for all `‖y‖ < r`.
-/
structure HasFPowerSeriesOnBall (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (x : E) (r : ℝ≥0∞) :
Prop where
r_le : r ≤ p.radius
r_pos : 0 < r
hasSum :
∀ {y}, y ∈ EMetric.ball (0 : E) r → HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y))
#align has_fpower_series_on_ball HasFPowerSeriesOnBall
/-- Given a function `f : E → F` and a formal multilinear series `p`, we say that `f` has `p` as
a power series around `x` if `f (x + y) = ∑' pₙ yⁿ` for all `y` in a neighborhood of `0`. -/
def HasFPowerSeriesAt (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (x : E) :=
∃ r, HasFPowerSeriesOnBall f p x r
#align has_fpower_series_at HasFPowerSeriesAt
variable (𝕜)
/-- Given a function `f : E → F`, we say that `f` is analytic at `x` if it admits a convergent power
series expansion around `x`. -/
def AnalyticAt (f : E → F) (x : E) :=
∃ p : FormalMultilinearSeries 𝕜 E F, HasFPowerSeriesAt f p x
#align analytic_at AnalyticAt
/-- Given a function `f : E → F`, we say that `f` is analytic on a set `s` if it is analytic around
every point of `s`. -/
def AnalyticOn (f : E → F) (s : Set E) :=
∀ x, x ∈ s → AnalyticAt 𝕜 f x
#align analytic_on AnalyticOn
variable {𝕜}
theorem HasFPowerSeriesOnBall.hasFPowerSeriesAt (hf : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesAt f p x :=
⟨r, hf⟩
#align has_fpower_series_on_ball.has_fpower_series_at HasFPowerSeriesOnBall.hasFPowerSeriesAt
theorem HasFPowerSeriesAt.analyticAt (hf : HasFPowerSeriesAt f p x) : AnalyticAt 𝕜 f x :=
⟨p, hf⟩
#align has_fpower_series_at.analytic_at HasFPowerSeriesAt.analyticAt
theorem HasFPowerSeriesOnBall.analyticAt (hf : HasFPowerSeriesOnBall f p x r) : AnalyticAt 𝕜 f x :=
hf.hasFPowerSeriesAt.analyticAt
#align has_fpower_series_on_ball.analytic_at HasFPowerSeriesOnBall.analyticAt
theorem HasFPowerSeriesOnBall.congr (hf : HasFPowerSeriesOnBall f p x r)
(hg : EqOn f g (EMetric.ball x r)) : HasFPowerSeriesOnBall g p x r :=
{ r_le := hf.r_le
r_pos := hf.r_pos
hasSum := fun {y} hy => by
convert hf.hasSum hy using 1
apply hg.symm
simpa [edist_eq_coe_nnnorm_sub] using hy }
#align has_fpower_series_on_ball.congr HasFPowerSeriesOnBall.congr
/-- If a function `f` has a power series `p` around `x`, then the function `z ↦ f (z - y)` has the
same power series around `x + y`. -/
theorem HasFPowerSeriesOnBall.comp_sub (hf : HasFPowerSeriesOnBall f p x r) (y : E) :
HasFPowerSeriesOnBall (fun z => f (z - y)) p (x + y) r :=
{ r_le := hf.r_le
r_pos := hf.r_pos
hasSum := fun {z} hz => by
convert hf.hasSum hz using 2
abel }
#align has_fpower_series_on_ball.comp_sub HasFPowerSeriesOnBall.comp_sub
theorem HasFPowerSeriesOnBall.hasSum_sub (hf : HasFPowerSeriesOnBall f p x r) {y : E}
(hy : y ∈ EMetric.ball x r) : HasSum (fun n : ℕ => p n fun _ => y - x) (f y) := by
have : y - x ∈ EMetric.ball (0 : E) r := by simpa [edist_eq_coe_nnnorm_sub] using hy
simpa only [add_sub_cancel'_right] using hf.hasSum this
#align has_fpower_series_on_ball.has_sum_sub HasFPowerSeriesOnBall.hasSum_sub
theorem HasFPowerSeriesOnBall.radius_pos (hf : HasFPowerSeriesOnBall f p x r) : 0 < p.radius :=
lt_of_lt_of_le hf.r_pos hf.r_le
#align has_fpower_series_on_ball.radius_pos HasFPowerSeriesOnBall.radius_pos
theorem HasFPowerSeriesAt.radius_pos (hf : HasFPowerSeriesAt f p x) : 0 < p.radius :=
let ⟨_, hr⟩ := hf
hr.radius_pos
#align has_fpower_series_at.radius_pos HasFPowerSeriesAt.radius_pos
theorem HasFPowerSeriesOnBall.mono (hf : HasFPowerSeriesOnBall f p x r) (r'_pos : 0 < r')
(hr : r' ≤ r) : HasFPowerSeriesOnBall f p x r' :=
⟨le_trans hr hf.1, r'_pos, fun hy => hf.hasSum (EMetric.ball_subset_ball hr hy)⟩
#align has_fpower_series_on_ball.mono HasFPowerSeriesOnBall.mono
theorem HasFPowerSeriesAt.congr (hf : HasFPowerSeriesAt f p x) (hg : f =ᶠ[𝓝 x] g) :
HasFPowerSeriesAt g p x := by
rcases hf with ⟨r₁, h₁⟩
rcases EMetric.mem_nhds_iff.mp hg with ⟨r₂, h₂pos, h₂⟩
exact ⟨min r₁ r₂,
(h₁.mono (lt_min h₁.r_pos h₂pos) inf_le_left).congr
fun y hy => h₂ (EMetric.ball_subset_ball inf_le_right hy)⟩
#align has_fpower_series_at.congr HasFPowerSeriesAt.congr
protected theorem HasFPowerSeriesAt.eventually (hf : HasFPowerSeriesAt f p x) :
∀ᶠ r : ℝ≥0∞ in 𝓝[>] 0, HasFPowerSeriesOnBall f p x r :=
let ⟨_, hr⟩ := hf
mem_of_superset (Ioo_mem_nhdsWithin_Ioi (left_mem_Ico.2 hr.r_pos)) fun _ hr' =>
hr.mono hr'.1 hr'.2.le
#align has_fpower_series_at.eventually HasFPowerSeriesAt.eventually
theorem HasFPowerSeriesOnBall.eventually_hasSum (hf : HasFPowerSeriesOnBall f p x r) :
∀ᶠ y in 𝓝 0, HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y)) := by
filter_upwards [EMetric.ball_mem_nhds (0 : E) hf.r_pos] using fun _ => hf.hasSum
#align has_fpower_series_on_ball.eventually_has_sum HasFPowerSeriesOnBall.eventually_hasSum
theorem HasFPowerSeriesAt.eventually_hasSum (hf : HasFPowerSeriesAt f p x) :
∀ᶠ y in 𝓝 0, HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y)) :=
let ⟨_, hr⟩ := hf
hr.eventually_hasSum
#align has_fpower_series_at.eventually_has_sum HasFPowerSeriesAt.eventually_hasSum
theorem HasFPowerSeriesOnBall.eventually_hasSum_sub (hf : HasFPowerSeriesOnBall f p x r) :
∀ᶠ y in 𝓝 x, HasSum (fun n : ℕ => p n fun _ : Fin n => y - x) (f y) := by
filter_upwards [EMetric.ball_mem_nhds x hf.r_pos] with y using hf.hasSum_sub
#align has_fpower_series_on_ball.eventually_has_sum_sub HasFPowerSeriesOnBall.eventually_hasSum_sub
theorem HasFPowerSeriesAt.eventually_hasSum_sub (hf : HasFPowerSeriesAt f p x) :
∀ᶠ y in 𝓝 x, HasSum (fun n : ℕ => p n fun _ : Fin n => y - x) (f y) :=
let ⟨_, hr⟩ := hf
hr.eventually_hasSum_sub
#align has_fpower_series_at.eventually_has_sum_sub HasFPowerSeriesAt.eventually_hasSum_sub
theorem HasFPowerSeriesOnBall.eventually_eq_zero
(hf : HasFPowerSeriesOnBall f (0 : FormalMultilinearSeries 𝕜 E F) x r) :
∀ᶠ z in 𝓝 x, f z = 0 := by
filter_upwards [hf.eventually_hasSum_sub] with z hz using hz.unique hasSum_zero
#align has_fpower_series_on_ball.eventually_eq_zero HasFPowerSeriesOnBall.eventually_eq_zero
theorem HasFPowerSeriesAt.eventually_eq_zero
(hf : HasFPowerSeriesAt f (0 : FormalMultilinearSeries 𝕜 E F) x) : ∀ᶠ z in 𝓝 x, f z = 0 :=
let ⟨_, hr⟩ := hf
hr.eventually_eq_zero
#align has_fpower_series_at.eventually_eq_zero HasFPowerSeriesAt.eventually_eq_zero
theorem hasFPowerSeriesOnBall_const {c : F} {e : E} :
HasFPowerSeriesOnBall (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e ⊤ := by
refine' ⟨by simp, WithTop.zero_lt_top, fun _ => hasSum_single 0 fun n hn => _⟩
simp [constFormalMultilinearSeries_apply hn]
#align has_fpower_series_on_ball_const hasFPowerSeriesOnBall_const
theorem hasFPowerSeriesAt_const {c : F} {e : E} :
HasFPowerSeriesAt (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e :=
⟨⊤, hasFPowerSeriesOnBall_const⟩
#align has_fpower_series_at_const hasFPowerSeriesAt_const
theorem analyticAt_const {v : F} : AnalyticAt 𝕜 (fun _ => v) x :=
⟨constFormalMultilinearSeries 𝕜 E v, hasFPowerSeriesAt_const⟩
#align analytic_at_const analyticAt_const
theorem analyticOn_const {v : F} {s : Set E} : AnalyticOn 𝕜 (fun _ => v) s :=
fun _ _ => analyticAt_const
#align analytic_on_const analyticOn_const
theorem HasFPowerSeriesOnBall.add (hf : HasFPowerSeriesOnBall f pf x r)
(hg : HasFPowerSeriesOnBall g pg x r) : HasFPowerSeriesOnBall (f + g) (pf + pg) x r :=
{ r_le := le_trans (le_min_iff.2 ⟨hf.r_le, hg.r_le⟩) (pf.min_radius_le_radius_add pg)
r_pos := hf.r_pos
hasSum := fun hy => (hf.hasSum hy).add (hg.hasSum hy) }
#align has_fpower_series_on_ball.add HasFPowerSeriesOnBall.add
theorem HasFPowerSeriesAt.add (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x) :
HasFPowerSeriesAt (f + g) (pf + pg) x := by
rcases (hf.eventually.and hg.eventually).exists with ⟨r, hr⟩
exact ⟨r, hr.1.add hr.2⟩
#align has_fpower_series_at.add HasFPowerSeriesAt.add
theorem AnalyticAt.congr (hf : AnalyticAt 𝕜 f x) (hg : f =ᶠ[𝓝 x] g) : AnalyticAt 𝕜 g x :=
let ⟨_, hpf⟩ := hf
(hpf.congr hg).analyticAt
theorem analyticAt_congr (h : f =ᶠ[𝓝 x] g) : AnalyticAt 𝕜 f x ↔ AnalyticAt 𝕜 g x :=
⟨fun hf ↦ hf.congr h, fun hg ↦ hg.congr h.symm⟩
theorem AnalyticAt.add (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) : AnalyticAt 𝕜 (f + g) x :=
let ⟨_, hpf⟩ := hf
let ⟨_, hqf⟩ := hg
(hpf.add hqf).analyticAt
#align analytic_at.add AnalyticAt.add
theorem HasFPowerSeriesOnBall.neg (hf : HasFPowerSeriesOnBall f pf x r) :
HasFPowerSeriesOnBall (-f) (-pf) x r :=
{ r_le := by
rw [pf.radius_neg]
exact hf.r_le
r_pos := hf.r_pos
hasSum := fun hy => (hf.hasSum hy).neg }
#align has_fpower_series_on_ball.neg HasFPowerSeriesOnBall.neg
theorem HasFPowerSeriesAt.neg (hf : HasFPowerSeriesAt f pf x) : HasFPowerSeriesAt (-f) (-pf) x :=
let ⟨_, hrf⟩ := hf
hrf.neg.hasFPowerSeriesAt
#align has_fpower_series_at.neg HasFPowerSeriesAt.neg
theorem AnalyticAt.neg (hf : AnalyticAt 𝕜 f x) : AnalyticAt 𝕜 (-f) x :=
let ⟨_, hpf⟩ := hf
hpf.neg.analyticAt
#align analytic_at.neg AnalyticAt.neg
theorem HasFPowerSeriesOnBall.sub (hf : HasFPowerSeriesOnBall f pf x r)
(hg : HasFPowerSeriesOnBall g pg x r) : HasFPowerSeriesOnBall (f - g) (pf - pg) x r := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align has_fpower_series_on_ball.sub HasFPowerSeriesOnBall.sub
theorem HasFPowerSeriesAt.sub (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x) :
HasFPowerSeriesAt (f - g) (pf - pg) x := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align has_fpower_series_at.sub HasFPowerSeriesAt.sub
theorem AnalyticAt.sub (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) :
AnalyticAt 𝕜 (f - g) x := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align analytic_at.sub AnalyticAt.sub
theorem AnalyticOn.mono {s t : Set E} (hf : AnalyticOn 𝕜 f t) (hst : s ⊆ t) : AnalyticOn 𝕜 f s :=
fun z hz => hf z (hst hz)
#align analytic_on.mono AnalyticOn.mono
theorem AnalyticOn.congr' {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : f =ᶠ[𝓝ˢ s] g) :
AnalyticOn 𝕜 g s :=
fun z hz => (hf z hz).congr (mem_nhdsSet_iff_forall.mp hg z hz)
theorem analyticOn_congr' {s : Set E} (h : f =ᶠ[𝓝ˢ s] g) : AnalyticOn 𝕜 f s ↔ AnalyticOn 𝕜 g s :=
⟨fun hf => hf.congr' h, fun hg => hg.congr' h.symm⟩
theorem AnalyticOn.congr {s : Set E} (hs : IsOpen s) (hf : AnalyticOn 𝕜 f s) (hg : s.EqOn f g) :
AnalyticOn 𝕜 g s :=
hf.congr' $ mem_nhdsSet_iff_forall.mpr
(fun _ hz => eventuallyEq_iff_exists_mem.mpr ⟨s, hs.mem_nhds hz, hg⟩)
theorem analyticOn_congr {s : Set E} (hs : IsOpen s) (h : s.EqOn f g) : AnalyticOn 𝕜 f s ↔
AnalyticOn 𝕜 g s := ⟨fun hf => hf.congr hs h, fun hg => hg.congr hs h.symm⟩
theorem AnalyticOn.add {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
AnalyticOn 𝕜 (f + g) s :=
fun z hz => (hf z hz).add (hg z hz)
#align analytic_on.add AnalyticOn.add
theorem AnalyticOn.sub {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
AnalyticOn 𝕜 (f - g) s :=
fun z hz => (hf z hz).sub (hg z hz)
#align analytic_on.sub AnalyticOn.sub
theorem HasFPowerSeriesOnBall.coeff_zero (hf : HasFPowerSeriesOnBall f pf x r) (v : Fin 0 → E) :
pf 0 v = f x := by
have v_eq : v = fun i => 0 := Subsingleton.elim _ _
have zero_mem : (0 : E) ∈ EMetric.ball (0 : E) r := by simp [hf.r_pos]
have : ∀ i, i ≠ 0 → (pf i fun j => 0) = 0 := by
intro i hi
have : 0 < i := pos_iff_ne_zero.2 hi
exact ContinuousMultilinearMap.map_coord_zero _ (⟨0, this⟩ : Fin i) rfl
have A := (hf.hasSum zero_mem).unique (hasSum_single _ this)
simpa [v_eq] using A.symm
#align has_fpower_series_on_ball.coeff_zero HasFPowerSeriesOnBall.coeff_zero
theorem HasFPowerSeriesAt.coeff_zero (hf : HasFPowerSeriesAt f pf x) (v : Fin 0 → E) :
pf 0 v = f x :=
let ⟨_, hrf⟩ := hf
hrf.coeff_zero v
#align has_fpower_series_at.coeff_zero HasFPowerSeriesAt.coeff_zero
/-- If a function `f` has a power series `p` on a ball and `g` is linear, then `g ∘ f` has the
power series `g ∘ p` on the same ball. -/
theorem ContinuousLinearMap.comp_hasFPowerSeriesOnBall (g : F →L[𝕜] G)
(h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (g ∘ f) (g.compFormalMultilinearSeries p) x r :=
{ r_le := h.r_le.trans (p.radius_le_radius_continuousLinearMap_comp _)
r_pos := h.r_pos
hasSum := fun hy => by
simpa only [ContinuousLinearMap.compFormalMultilinearSeries_apply,
ContinuousLinearMap.compContinuousMultilinearMap_coe, Function.comp_apply] using
g.hasSum (h.hasSum hy) }
#align continuous_linear_map.comp_has_fpower_series_on_ball ContinuousLinearMap.comp_hasFPowerSeriesOnBall
/-- If a function `f` is analytic on a set `s` and `g` is linear, then `g ∘ f` is analytic
on `s`. -/
theorem ContinuousLinearMap.comp_analyticOn {s : Set E} (g : F →L[𝕜] G) (h : AnalyticOn 𝕜 f s) :
AnalyticOn 𝕜 (g ∘ f) s := by
rintro x hx
rcases h x hx with ⟨p, r, hp⟩
exact ⟨g.compFormalMultilinearSeries p, r, g.comp_hasFPowerSeriesOnBall hp⟩
#align continuous_linear_map.comp_analytic_on ContinuousLinearMap.comp_analyticOn
/-- If a function admits a power series expansion, then it is exponentially close to the partial
sums of this power series on strict subdisks of the disk of convergence.
This version provides an upper estimate that decreases both in `‖y‖` and `n`. See also
`HasFPowerSeriesOnBall.uniform_geometric_approx` for a weaker version. -/
theorem HasFPowerSeriesOnBall.uniform_geometric_approx' {r' : ℝ≥0}
(hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n := by
obtain ⟨a, ha, C, hC, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n, ‖p n‖ * (r' : ℝ) ^ n ≤ C * a ^ n :=
p.norm_mul_pow_le_mul_pow_of_lt_radius (h.trans_le hf.r_le)
refine' ⟨a, ha, C / (1 - a), div_pos hC (sub_pos.2 ha.2), fun y hy n => _⟩
have yr' : ‖y‖ < r' := by
rw [ball_zero_eq] at hy
exact hy
have hr'0 : 0 < (r' : ℝ) := (norm_nonneg _).trans_lt yr'
have : y ∈ EMetric.ball (0 : E) r := by
refine' mem_emetric_ball_zero_iff.2 (lt_trans _ h)
exact mod_cast yr'
rw [norm_sub_rev, ← mul_div_right_comm]
have ya : a * (‖y‖ / ↑r') ≤ a :=
mul_le_of_le_one_right ha.1.le (div_le_one_of_le yr'.le r'.coe_nonneg)
suffices ‖p.partialSum n y - f (x + y)‖ ≤ C * (a * (‖y‖ / r')) ^ n / (1 - a * (‖y‖ / r')) by
refine' this.trans _
have : 0 < a := ha.1
gcongr
apply_rules [sub_pos.2, ha.2]
apply norm_sub_le_of_geometric_bound_of_hasSum (ya.trans_lt ha.2) _ (hf.hasSum this)
intro n
calc
‖(p n) fun _ : Fin n => y‖
_ ≤ ‖p n‖ * ∏ _i : Fin n, ‖y‖ := ContinuousMultilinearMap.le_op_norm _ _
_ = ‖p n‖ * (r' : ℝ) ^ n * (‖y‖ / r') ^ n := by field_simp [mul_right_comm]
_ ≤ C * a ^ n * (‖y‖ / r') ^ n := by gcongr ?_ * _; apply hp
_ ≤ C * (a * (‖y‖ / r')) ^ n := by rw [mul_pow, mul_assoc]
#align has_fpower_series_on_ball.uniform_geometric_approx' HasFPowerSeriesOnBall.uniform_geometric_approx'
/-- If a function admits a power series expansion, then it is exponentially close to the partial
sums of this power series on strict subdisks of the disk of convergence. -/
theorem HasFPowerSeriesOnBall.uniform_geometric_approx {r' : ℝ≥0}
(hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
∃ a ∈ Ioo (0 : ℝ) 1,
∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n, ‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n := by
obtain ⟨a, ha, C, hC, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n :=
hf.uniform_geometric_approx' h
refine' ⟨a, ha, C, hC, fun y hy n => (hp y hy n).trans _⟩
have yr' : ‖y‖ < r' := by rwa [ball_zero_eq] at hy
gcongr
exacts [mul_nonneg ha.1.le (div_nonneg (norm_nonneg y) r'.coe_nonneg),
mul_le_of_le_one_right ha.1.le (div_le_one_of_le yr'.le r'.coe_nonneg)]
#align has_fpower_series_on_ball.uniform_geometric_approx HasFPowerSeriesOnBall.uniform_geometric_approx
/-- Taylor formula for an analytic function, `IsBigO` version. -/
theorem HasFPowerSeriesAt.isBigO_sub_partialSum_pow (hf : HasFPowerSeriesAt f p x) (n : ℕ) :
(fun y : E => f (x + y) - p.partialSum n y) =O[𝓝 0] fun y => ‖y‖ ^ n := by
rcases hf with ⟨r, hf⟩
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hf.r_pos with ⟨r', r'0, h⟩
obtain ⟨a, -, C, -, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n :=
hf.uniform_geometric_approx' h
refine' isBigO_iff.2 ⟨C * (a / r') ^ n, _⟩
replace r'0 : 0 < (r' : ℝ); · exact mod_cast r'0
filter_upwards [Metric.ball_mem_nhds (0 : E) r'0] with y hy
simpa [mul_pow, mul_div_assoc, mul_assoc, div_mul_eq_mul_div] using hp y hy n
set_option linter.uppercaseLean3 false in
#align has_fpower_series_at.is_O_sub_partial_sum_pow HasFPowerSeriesAt.isBigO_sub_partialSum_pow
/-- If `f` has formal power series `∑ n, pₙ` on a ball of radius `r`, then for `y, z` in any smaller
ball, the norm of the difference `f y - f z - p 1 (fun _ ↦ y - z)` is bounded above by
`C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖`. This lemma formulates this property using `IsBigO` and
`Filter.principal` on `E × E`. -/
theorem HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal
(hf : HasFPowerSeriesOnBall f p x r) (hr : r' < r) :
(fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) =O[𝓟 (EMetric.ball (x, x) r')]
fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖ := by
lift r' to ℝ≥0 using ne_top_of_lt hr
rcases (zero_le r').eq_or_lt with (rfl | hr'0)
· simp only [isBigO_bot, EMetric.ball_zero, principal_empty, ENNReal.coe_zero]
obtain ⟨a, ha, C, hC : 0 < C, hp⟩ :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n : ℕ, ‖p n‖ * (r' : ℝ) ^ n ≤ C * a ^ n
exact p.norm_mul_pow_le_mul_pow_of_lt_radius (hr.trans_le hf.r_le)
simp only [← le_div_iff (pow_pos (NNReal.coe_pos.2 hr'0) _)] at hp
set L : E × E → ℝ := fun y =>
C * (a / r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * (a / (1 - a) ^ 2 + 2 / (1 - a))
have hL : ∀ y ∈ EMetric.ball (x, x) r', ‖f y.1 - f y.2 - p 1 fun _ => y.1 - y.2‖ ≤ L y := by
intro y hy'
have hy : y ∈ EMetric.ball x r ×ˢ EMetric.ball x r := by
rw [EMetric.ball_prod_same]
exact EMetric.ball_subset_ball hr.le hy'
set A : ℕ → F := fun n => (p n fun _ => y.1 - x) - p n fun _ => y.2 - x
have hA : HasSum (fun n => A (n + 2)) (f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) := by
convert (hasSum_nat_add_iff' 2).2 ((hf.hasSum_sub hy.1).sub (hf.hasSum_sub hy.2)) using 1
rw [Finset.sum_range_succ, Finset.sum_range_one, hf.coeff_zero, hf.coeff_zero, sub_self,
zero_add, ← Subsingleton.pi_single_eq (0 : Fin 1) (y.1 - x), Pi.single,
← Subsingleton.pi_single_eq (0 : Fin 1) (y.2 - x), Pi.single, ← (p 1).map_sub, ← Pi.single,
Subsingleton.pi_single_eq, sub_sub_sub_cancel_right]
rw [EMetric.mem_ball, edist_eq_coe_nnnorm_sub, ENNReal.coe_lt_coe] at hy'
set B : ℕ → ℝ := fun n => C * (a / r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * ((n + 2) * a ^ n)
have hAB : ∀ n, ‖A (n + 2)‖ ≤ B n := fun n =>
calc
‖A (n + 2)‖ ≤ ‖p (n + 2)‖ * ↑(n + 2) * ‖y - (x, x)‖ ^ (n + 1) * ‖y.1 - y.2‖ := by
-- porting note: `pi_norm_const` was `pi_norm_const (_ : E)`
simpa only [Fintype.card_fin, pi_norm_const, Prod.norm_def, Pi.sub_def,
Prod.fst_sub, Prod.snd_sub, sub_sub_sub_cancel_right] using
(p <| n + 2).norm_image_sub_le (fun _ => y.1 - x) fun _ => y.2 - x
_ = ‖p (n + 2)‖ * ‖y - (x, x)‖ ^ n * (↑(n + 2) * ‖y - (x, x)‖ * ‖y.1 - y.2‖) := by
rw [pow_succ ‖y - (x, x)‖]
ring
-- porting note: the two `↑` in `↑r'` are new, without them, Lean fails to synthesize
-- instances `HDiv ℝ ℝ≥0 ?m` or `HMul ℝ ℝ≥0 ?m`
_ ≤ C * a ^ (n + 2) / ↑r' ^ (n + 2)
* ↑r' ^ n * (↑(n + 2) * ‖y - (x, x)‖ * ‖y.1 - y.2‖) := by
have : 0 < a := ha.1
gcongr
· apply hp
· apply hy'.le
_ = B n := by
-- porting note: in the original, `B` was in the `field_simp`, but now Lean does not
-- accept it. The current proof works in Lean 4, but does not in Lean 3.
field_simp [pow_succ]
simp only [mul_assoc, mul_comm, mul_left_comm]
have hBL : HasSum B (L y) := by
apply HasSum.mul_left
simp only [add_mul]
have : ‖a‖ < 1 := by simp only [Real.norm_eq_abs, abs_of_pos ha.1, ha.2]
rw [div_eq_mul_inv, div_eq_mul_inv]
exact (hasSum_coe_mul_geometric_of_norm_lt_1 this).add -- porting note: was `convert`!
((hasSum_geometric_of_norm_lt_1 this).mul_left 2)
exact hA.norm_le_of_bounded hBL hAB
suffices L =O[𝓟 (EMetric.ball (x, x) r')] fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖ by
refine' (IsBigO.of_bound 1 (eventually_principal.2 fun y hy => _)).trans this
rw [one_mul]
exact (hL y hy).trans (le_abs_self _)
simp_rw [mul_right_comm _ (_ * _)] -- porting note: there was an `L` inside the `simp_rw`.
exact (isBigO_refl _ _).const_mul_left _
set_option linter.uppercaseLean3 false in
#align has_fpower_series_on_ball.is_O_image_sub_image_sub_deriv_principal HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal
/-- If `f` has formal power series `∑ n, pₙ` on a ball of radius `r`, then for `y, z` in any smaller
ball, the norm of the difference `f y - f z - p 1 (fun _ ↦ y - z)` is bounded above by
`C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖`. -/
theorem HasFPowerSeriesOnBall.image_sub_sub_deriv_le (hf : HasFPowerSeriesOnBall f p x r)
(hr : r' < r) :
∃ C, ∀ᵉ (y ∈ EMetric.ball x r') (z ∈ EMetric.ball x r'),
‖f y - f z - p 1 fun _ => y - z‖ ≤ C * max ‖y - x‖ ‖z - x‖ * ‖y - z‖ := by
simpa only [isBigO_principal, mul_assoc, norm_mul, norm_norm, Prod.forall, EMetric.mem_ball,
Prod.edist_eq, max_lt_iff, and_imp, @forall_swap (_ < _) E] using
hf.isBigO_image_sub_image_sub_deriv_principal hr
#align has_fpower_series_on_ball.image_sub_sub_deriv_le HasFPowerSeriesOnBall.image_sub_sub_deriv_le
/-- If `f` has formal power series `∑ n, pₙ` at `x`, then
`f y - f z - p 1 (fun _ ↦ y - z) = O(‖(y, z) - (x, x)‖ * ‖y - z‖)` as `(y, z) → (x, x)`.
In particular, `f` is strictly differentiable at `x`. -/
theorem HasFPowerSeriesAt.isBigO_image_sub_norm_mul_norm_sub (hf : HasFPowerSeriesAt f p x) :
(fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) =O[𝓝 (x, x)] fun y =>
‖y - (x, x)‖ * ‖y.1 - y.2‖ := by
rcases hf with ⟨r, hf⟩
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hf.r_pos with ⟨r', r'0, h⟩
refine' (hf.isBigO_image_sub_image_sub_deriv_principal h).mono _
exact le_principal_iff.2 (EMetric.ball_mem_nhds _ r'0)
set_option linter.uppercaseLean3 false in
#align has_fpower_series_at.is_O_image_sub_norm_mul_norm_sub HasFPowerSeriesAt.isBigO_image_sub_norm_mul_norm_sub
/-- If a function admits a power series expansion at `x`, then it is the uniform limit of the
partial sums of this power series on strict subdisks of the disk of convergence, i.e., `f (x + y)`
is the uniform limit of `p.partialSum n y` there. -/
theorem HasFPowerSeriesOnBall.tendstoUniformlyOn {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r)
(h : (r' : ℝ≥0∞) < r) :
TendstoUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
(Metric.ball (0 : E) r') := by
obtain ⟨a, ha, C, -, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n
exact hf.uniform_geometric_approx h
refine' Metric.tendstoUniformlyOn_iff.2 fun ε εpos => _
have L : Tendsto (fun n => (C : ℝ) * a ^ n) atTop (𝓝 ((C : ℝ) * 0)) :=
tendsto_const_nhds.mul (tendsto_pow_atTop_nhds_0_of_lt_1 ha.1.le ha.2)
rw [mul_zero] at L
refine' (L.eventually (gt_mem_nhds εpos)).mono fun n hn y hy => _
rw [dist_eq_norm]
exact (hp y hy n).trans_lt hn
#align has_fpower_series_on_ball.tendsto_uniformly_on HasFPowerSeriesOnBall.tendstoUniformlyOn
/-- If a function admits a power series expansion at `x`, then it is the locally uniform limit of
the partial sums of this power series on the disk of convergence, i.e., `f (x + y)`
is the locally uniform limit of `p.partialSum n y` there. -/
theorem HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn (hf : HasFPowerSeriesOnBall f p x r) :
TendstoLocallyUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
(EMetric.ball (0 : E) r) := by
intro u hu x hx
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hx with ⟨r', xr', hr'⟩
have : EMetric.ball (0 : E) r' ∈ 𝓝 x := IsOpen.mem_nhds EMetric.isOpen_ball xr'
refine' ⟨EMetric.ball (0 : E) r', mem_nhdsWithin_of_mem_nhds this, _⟩
simpa [Metric.emetric_ball_nnreal] using hf.tendstoUniformlyOn hr' u hu
#align has_fpower_series_on_ball.tendsto_locally_uniformly_on HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn
/-- If a function admits a power series expansion at `x`, then it is the uniform limit of the
partial sums of this power series on strict subdisks of the disk of convergence, i.e., `f y`
is the uniform limit of `p.partialSum n (y - x)` there. -/
theorem HasFPowerSeriesOnBall.tendstoUniformlyOn' {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r)
(h : (r' : ℝ≥0∞) < r) :
TendstoUniformlyOn (fun n y => p.partialSum n (y - x)) f atTop (Metric.ball (x : E) r') := by
convert (hf.tendstoUniformlyOn h).comp fun y => y - x using 1
· simp [(· ∘ ·)]
· ext z
simp [dist_eq_norm]
#align has_fpower_series_on_ball.tendsto_uniformly_on' HasFPowerSeriesOnBall.tendstoUniformlyOn'
/-- If a function admits a power series expansion at `x`, then it is the locally uniform limit of
the partial sums of this power series on the disk of convergence, i.e., `f y`
is the locally uniform limit of `p.partialSum n (y - x)` there. -/
theorem HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn' (hf : HasFPowerSeriesOnBall f p x r) :
TendstoLocallyUniformlyOn (fun n y => p.partialSum n (y - x)) f atTop
(EMetric.ball (x : E) r) := by
have A : ContinuousOn (fun y : E => y - x) (EMetric.ball (x : E) r) :=
(continuous_id.sub continuous_const).continuousOn
convert hf.tendstoLocallyUniformlyOn.comp (fun y : E => y - x) _ A using 1
· ext z
simp
· intro z
simp [edist_eq_coe_nnnorm, edist_eq_coe_nnnorm_sub]
#align has_fpower_series_on_ball.tendsto_locally_uniformly_on' HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn'
/-- If a function admits a power series expansion on a disk, then it is continuous there. -/
protected theorem HasFPowerSeriesOnBall.continuousOn (hf : HasFPowerSeriesOnBall f p x r) :
ContinuousOn f (EMetric.ball x r) :=
hf.tendstoLocallyUniformlyOn'.continuousOn <|
eventually_of_forall fun n =>
((p.partialSum_continuous n).comp (continuous_id.sub continuous_const)).continuousOn
#align has_fpower_series_on_ball.continuous_on HasFPowerSeriesOnBall.continuousOn
protected theorem HasFPowerSeriesAt.continuousAt (hf : HasFPowerSeriesAt f p x) :
ContinuousAt f x :=
let ⟨_, hr⟩ := hf
hr.continuousOn.continuousAt (EMetric.ball_mem_nhds x hr.r_pos)
#align has_fpower_series_at.continuous_at HasFPowerSeriesAt.continuousAt
protected theorem AnalyticAt.continuousAt (hf : AnalyticAt 𝕜 f x) : ContinuousAt f x :=
let ⟨_, hp⟩ := hf
hp.continuousAt
#align analytic_at.continuous_at AnalyticAt.continuousAt
protected theorem AnalyticOn.continuousOn {s : Set E} (hf : AnalyticOn 𝕜 f s) : ContinuousOn f s :=
fun x hx => (hf x hx).continuousAt.continuousWithinAt
#align analytic_on.continuous_on AnalyticOn.continuousOn
/-- Analytic everywhere implies continuous -/
theorem AnalyticOn.continuous {f : E → F} (fa : AnalyticOn 𝕜 f univ) : Continuous f := by
rw [continuous_iff_continuousOn_univ]; exact fa.continuousOn
/-- In a complete space, the sum of a converging power series `p` admits `p` as a power series.
This is not totally obvious as we need to check the convergence of the series. -/
protected theorem FormalMultilinearSeries.hasFPowerSeriesOnBall [CompleteSpace F]
(p : FormalMultilinearSeries 𝕜 E F) (h : 0 < p.radius) :
HasFPowerSeriesOnBall p.sum p 0 p.radius :=
{ r_le := le_rfl
r_pos := h
hasSum := fun hy => by
rw [zero_add]
exact p.hasSum hy }
#align formal_multilinear_series.has_fpower_series_on_ball FormalMultilinearSeries.hasFPowerSeriesOnBall
theorem HasFPowerSeriesOnBall.sum (h : HasFPowerSeriesOnBall f p x r) {y : E}
(hy : y ∈ EMetric.ball (0 : E) r) : f (x + y) = p.sum y :=
(h.hasSum hy).tsum_eq.symm
#align has_fpower_series_on_ball.sum HasFPowerSeriesOnBall.sum
/-- The sum of a converging power series is continuous in its disk of convergence. -/
protected theorem FormalMultilinearSeries.continuousOn [CompleteSpace F] :
ContinuousOn p.sum (EMetric.ball 0 p.radius) := by
rcases (zero_le p.radius).eq_or_lt with h | h
· simp [← h, continuousOn_empty]
· exact (p.hasFPowerSeriesOnBall h).continuousOn
#align formal_multilinear_series.continuous_on FormalMultilinearSeries.continuousOn
end
/-!
### Uniqueness of power series
If a function `f : E → F` has two representations as power series at a point `x : E`, corresponding
to formal multilinear series `p₁` and `p₂`, then these representations agree term-by-term. That is,
for any `n : ℕ` and `y : E`, `p₁ n (fun i ↦ y) = p₂ n (fun i ↦ y)`. In the one-dimensional case,
when `f : 𝕜 → E`, the continuous multilinear maps `p₁ n` and `p₂ n` are given by
`ContinuousMultilinearMap.mkPiField`, and hence are determined completely by the value of
`p₁ n (fun i ↦ 1)`, so `p₁ = p₂`. Consequently, the radius of convergence for one series can be
transferred to the other.
-/
section Uniqueness
open ContinuousMultilinearMap
theorem Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero {n : ℕ} {p : E[×n]→L[𝕜] F}
(h : (fun y => p fun _ => y) =O[𝓝 0] fun y => ‖y‖ ^ (n + 1)) (y : E) : (p fun _ => y) = 0 := by
obtain ⟨c, c_pos, hc⟩ := h.exists_pos
obtain ⟨t, ht, t_open, z_mem⟩ := eventually_nhds_iff.mp (isBigOWith_iff.mp hc)
obtain ⟨δ, δ_pos, δε⟩ := (Metric.isOpen_iff.mp t_open) 0 z_mem
clear h hc z_mem
cases' n with n
· exact norm_eq_zero.mp (by
-- porting note: the symmetric difference of the `simpa only` sets:
-- added `Nat.zero_eq, zero_add, pow_one`
-- removed `zero_pow', Ne.def, Nat.one_ne_zero, not_false_iff`
simpa only [Nat.zero_eq, fin0_apply_norm, norm_eq_zero, norm_zero, zero_add, pow_one,
mul_zero, norm_le_zero_iff] using ht 0 (δε (Metric.mem_ball_self δ_pos)))
· refine' Or.elim (Classical.em (y = 0))
(fun hy => by simpa only [hy] using p.map_zero) fun hy => _
replace hy := norm_pos_iff.mpr hy
refine' norm_eq_zero.mp (le_antisymm (le_of_forall_pos_le_add fun ε ε_pos => _) (norm_nonneg _))
have h₀ := _root_.mul_pos c_pos (pow_pos hy (n.succ + 1))
obtain ⟨k, k_pos, k_norm⟩ := NormedField.exists_norm_lt 𝕜
(lt_min (mul_pos δ_pos (inv_pos.mpr hy)) (mul_pos ε_pos (inv_pos.mpr h₀)))
have h₁ : ‖k • y‖ < δ := by
rw [norm_smul]
exact inv_mul_cancel_right₀ hy.ne.symm δ ▸
mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_left _ _)) hy
have h₂ :=
calc
‖p fun _ => k • y‖ ≤ c * ‖k • y‖ ^ (n.succ + 1) := by
-- porting note: now Lean wants `_root_.`
simpa only [norm_pow, _root_.norm_norm] using ht (k • y) (δε (mem_ball_zero_iff.mpr h₁))
--simpa only [norm_pow, norm_norm] using ht (k • y) (δε (mem_ball_zero_iff.mpr h₁))
_ = ‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))) := by
-- porting note: added `Nat.succ_eq_add_one` since otherwise `ring` does not conclude.
simp only [norm_smul, mul_pow, Nat.succ_eq_add_one]
-- porting note: removed `rw [pow_succ]`, since it now becomes superfluous.
ring
have h₃ : ‖k‖ * (c * ‖y‖ ^ (n.succ + 1)) < ε :=
inv_mul_cancel_right₀ h₀.ne.symm ε ▸
mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_right _ _)) h₀
calc
‖p fun _ => y‖ = ‖k⁻¹ ^ n.succ‖ * ‖p fun _ => k • y‖ := by
simpa only [inv_smul_smul₀ (norm_pos_iff.mp k_pos), norm_smul, Finset.prod_const,
Finset.card_fin] using
congr_arg norm (p.map_smul_univ (fun _ : Fin n.succ => k⁻¹) fun _ : Fin n.succ => k • y)
_ ≤ ‖k⁻¹ ^ n.succ‖ * (‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1)))) := by gcongr
_ = ‖(k⁻¹ * k) ^ n.succ‖ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))) := by
rw [← mul_assoc]
simp [norm_mul, mul_pow]
_ ≤ 0 + ε := by
rw [inv_mul_cancel (norm_pos_iff.mp k_pos)]
simpa using h₃.le
set_option linter.uppercaseLean3 false in
#align asymptotics.is_O.continuous_multilinear_map_apply_eq_zero Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero
/-- If a formal multilinear series `p` represents the zero function at `x : E`, then the
terms `p n (fun i ↦ y)` appearing in the sum are zero for any `n : ℕ`, `y : E`. -/
theorem HasFPowerSeriesAt.apply_eq_zero {p : FormalMultilinearSeries 𝕜 E F} {x : E}
(h : HasFPowerSeriesAt 0 p x) (n : ℕ) : ∀ y : E, (p n fun _ => y) = 0 := by
refine' Nat.strong_induction_on n fun k hk => _
have psum_eq : p.partialSum (k + 1) = fun y => p k fun _ => y := by
funext z
refine' Finset.sum_eq_single _ (fun b hb hnb => _) fun hn => _
· have := Finset.mem_range_succ_iff.mp hb
simp only [hk b (this.lt_of_ne hnb), Pi.zero_apply]
· exact False.elim (hn (Finset.mem_range.mpr (lt_add_one k)))
replace h := h.isBigO_sub_partialSum_pow k.succ
simp only [psum_eq, zero_sub, Pi.zero_apply, Asymptotics.isBigO_neg_left] at h
exact h.continuousMultilinearMap_apply_eq_zero
#align has_fpower_series_at.apply_eq_zero HasFPowerSeriesAt.apply_eq_zero
/-- A one-dimensional formal multilinear series representing the zero function is zero. -/
theorem HasFPowerSeriesAt.eq_zero {p : FormalMultilinearSeries 𝕜 𝕜 E} {x : 𝕜}
(h : HasFPowerSeriesAt 0 p x) : p = 0 := by
-- porting note: `funext; ext` was `ext (n x)`
funext n
ext x
rw [← mkPiField_apply_one_eq_self (p n)]
-- porting note: nasty hack, was `simp [h.apply_eq_zero n 1]`
have := Or.intro_right ?_ (h.apply_eq_zero n 1)
simpa using this
#align has_fpower_series_at.eq_zero HasFPowerSeriesAt.eq_zero
/-- One-dimensional formal multilinear series representing the same function are equal. -/
theorem HasFPowerSeriesAt.eq_formalMultilinearSeries {p₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E}
{f : 𝕜 → E} {x : 𝕜} (h₁ : HasFPowerSeriesAt f p₁ x) (h₂ : HasFPowerSeriesAt f p₂ x) : p₁ = p₂ :=
sub_eq_zero.mp (HasFPowerSeriesAt.eq_zero (by simpa only [sub_self] using h₁.sub h₂))
#align has_fpower_series_at.eq_formal_multilinear_series HasFPowerSeriesAt.eq_formalMultilinearSeries
theorem HasFPowerSeriesAt.eq_formalMultilinearSeries_of_eventually
{p q : FormalMultilinearSeries 𝕜 𝕜 E} {f g : 𝕜 → E} {x : 𝕜} (hp : HasFPowerSeriesAt f p x)
(hq : HasFPowerSeriesAt g q x) (heq : ∀ᶠ z in 𝓝 x, f z = g z) : p = q :=
(hp.congr heq).eq_formalMultilinearSeries hq
#align has_fpower_series_at.eq_formal_multilinear_series_of_eventually HasFPowerSeriesAt.eq_formalMultilinearSeries_of_eventually
/-- A one-dimensional formal multilinear series representing a locally zero function is zero. -/
theorem HasFPowerSeriesAt.eq_zero_of_eventually {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E}
{x : 𝕜} (hp : HasFPowerSeriesAt f p x) (hf : f =ᶠ[𝓝 x] 0) : p = 0 :=
(hp.congr hf).eq_zero
#align has_fpower_series_at.eq_zero_of_eventually HasFPowerSeriesAt.eq_zero_of_eventually
/-- If a function `f : 𝕜 → E` has two power series representations at `x`, then the given radii in
which convergence is guaranteed may be interchanged. This can be useful when the formal multilinear
series in one representation has a particularly nice form, but the other has a larger radius. -/
theorem HasFPowerSeriesOnBall.exchange_radius {p₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E}
{r₁ r₂ : ℝ≥0∞} {x : 𝕜} (h₁ : HasFPowerSeriesOnBall f p₁ x r₁)
(h₂ : HasFPowerSeriesOnBall f p₂ x r₂) : HasFPowerSeriesOnBall f p₁ x r₂ :=
h₂.hasFPowerSeriesAt.eq_formalMultilinearSeries h₁.hasFPowerSeriesAt ▸ h₂
#align has_fpower_series_on_ball.exchange_radius HasFPowerSeriesOnBall.exchange_radius
/-- If a function `f : 𝕜 → E` has power series representation `p` on a ball of some radius and for
each positive radius it has some power series representation, then `p` converges to `f` on the whole
`𝕜`. -/
theorem HasFPowerSeriesOnBall.r_eq_top_of_exists {f : 𝕜 → E} {r : ℝ≥0∞} {x : 𝕜}
{p : FormalMultilinearSeries 𝕜 𝕜 E} (h : HasFPowerSeriesOnBall f p x r)
(h' : ∀ (r' : ℝ≥0) (_ : 0 < r'), ∃ p' : FormalMultilinearSeries 𝕜 𝕜 E,
HasFPowerSeriesOnBall f p' x r') :
HasFPowerSeriesOnBall f p x ∞ :=
{ r_le := ENNReal.le_of_forall_pos_nnreal_lt fun r hr _ =>
let ⟨_, hp'⟩ := h' r hr
(h.exchange_radius hp').r_le
r_pos := ENNReal.coe_lt_top
hasSum := fun {y} _ =>
let ⟨r', hr'⟩ := exists_gt ‖y‖₊
let ⟨_, hp'⟩ := h' r' hr'.ne_bot.bot_lt
(h.exchange_radius hp').hasSum <| mem_emetric_ball_zero_iff.mpr (ENNReal.coe_lt_coe.2 hr') }
#align has_fpower_series_on_ball.r_eq_top_of_exists HasFPowerSeriesOnBall.r_eq_top_of_exists
end Uniqueness
/-!
### Changing origin in a power series
If a function is analytic in a disk `D(x, R)`, then it is analytic in any disk contained in that
one. Indeed, one can write
$$
f (x + y + z) = \sum_{n} p_n (y + z)^n = \sum_{n, k} \binom{n}{k} p_n y^{n-k} z^k
= \sum_{k} \Bigl(\sum_{n} \binom{n}{k} p_n y^{n-k}\Bigr) z^k.
$$
The corresponding power series has thus a `k`-th coefficient equal to
$\sum_{n} \binom{n}{k} p_n y^{n-k}$. In the general case where `pₙ` is a multilinear map, this has
to be interpreted suitably: instead of having a binomial coefficient, one should sum over all
possible subsets `s` of `Fin n` of cardinal `k`, and attribute `z` to the indices in `s` and
`y` to the indices outside of `s`.
In this paragraph, we implement this. The new power series is called `p.changeOrigin y`. Then, we
check its convergence and the fact that its sum coincides with the original sum. The outcome of this
discussion is that the set of points where a function is analytic is open.
-/
namespace FormalMultilinearSeries
section
variable (p : FormalMultilinearSeries 𝕜 E F) {x y : E} {r R : ℝ≥0}
/-- A term of `FormalMultilinearSeries.changeOriginSeries`.
Given a formal multilinear series `p` and a point `x` in its ball of convergence,
`p.changeOrigin x` is a formal multilinear series such that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense. Each term of `p.changeOrigin x`
is itself an analytic function of `x` given by the series `p.changeOriginSeries`. Each term in
`changeOriginSeries` is the sum of `changeOriginSeriesTerm`'s over all `s` of cardinality `l`.
The definition is such that `p.changeOriginSeriesTerm k l s hs (fun _ ↦ x) (fun _ ↦ y) =
p (k + l) (s.piecewise (fun _ ↦ x) (fun _ ↦ y))`
-/
def changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
E[×l]→L[𝕜] E[×k]→L[𝕜] F := by
let a := ContinuousMultilinearMap.curryFinFinset 𝕜 E F hs
(by erw [Finset.card_compl, Fintype.card_fin, hs, add_tsub_cancel_right])
exact a (p (k + l))
#align formal_multilinear_series.change_origin_series_term FormalMultilinearSeries.changeOriginSeriesTerm
theorem changeOriginSeriesTerm_apply (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l)
(x y : E) :
(p.changeOriginSeriesTerm k l s hs (fun _ => x) fun _ => y) =
p (k + l) (s.piecewise (fun _ => x) fun _ => y) :=
ContinuousMultilinearMap.curryFinFinset_apply_const _ _ _ _ _
#align formal_multilinear_series.change_origin_series_term_apply FormalMultilinearSeries.changeOriginSeriesTerm_apply
@[simp]
theorem norm_changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
‖p.changeOriginSeriesTerm k l s hs‖ = ‖p (k + l)‖ := by
simp only [changeOriginSeriesTerm, LinearIsometryEquiv.norm_map]
#align formal_multilinear_series.norm_change_origin_series_term FormalMultilinearSeries.norm_changeOriginSeriesTerm
@[simp]
theorem nnnorm_changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
‖p.changeOriginSeriesTerm k l s hs‖₊ = ‖p (k + l)‖₊ := by
simp only [changeOriginSeriesTerm, LinearIsometryEquiv.nnnorm_map]
#align formal_multilinear_series.nnnorm_change_origin_series_term FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm
theorem nnnorm_changeOriginSeriesTerm_apply_le (k l : ℕ) (s : Finset (Fin (k + l)))
(hs : s.card = l) (x y : E) :
‖p.changeOriginSeriesTerm k l s hs (fun _ => x) fun _ => y‖₊ ≤
‖p (k + l)‖₊ * ‖x‖₊ ^ l * ‖y‖₊ ^ k := by
rw [← p.nnnorm_changeOriginSeriesTerm k l s hs, ← Fin.prod_const, ← Fin.prod_const]
apply ContinuousMultilinearMap.le_of_op_nnnorm_le
apply ContinuousMultilinearMap.le_op_nnnorm
#align formal_multilinear_series.nnnorm_change_origin_series_term_apply_le FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm_apply_le
/-- The power series for `f.changeOrigin k`.
Given a formal multilinear series `p` and a point `x` in its ball of convergence,
`p.changeOrigin x` is a formal multilinear series such that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense. Its `k`-th term is the sum of
the series `p.changeOriginSeries k`. -/
def changeOriginSeries (k : ℕ) : FormalMultilinearSeries 𝕜 E (E[×k]→L[𝕜] F) := fun l =>
∑ s : { s : Finset (Fin (k + l)) // Finset.card s = l }, p.changeOriginSeriesTerm k l s s.2
#align formal_multilinear_series.change_origin_series FormalMultilinearSeries.changeOriginSeries
theorem nnnorm_changeOriginSeries_le_tsum (k l : ℕ) :
‖p.changeOriginSeries k l‖₊ ≤
∑' _ : { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + l)‖₊ :=
(nnnorm_sum_le _ (fun t => changeOriginSeriesTerm p k l (Subtype.val t) t.prop)).trans_eq <| by
simp_rw [tsum_fintype, nnnorm_changeOriginSeriesTerm (p := p) (k := k) (l := l)]
#align formal_multilinear_series.nnnorm_change_origin_series_le_tsum FormalMultilinearSeries.nnnorm_changeOriginSeries_le_tsum
theorem nnnorm_changeOriginSeries_apply_le_tsum (k l : ℕ) (x : E) :
‖p.changeOriginSeries k l fun _ => x‖₊ ≤
∑' _ : { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + l)‖₊ * ‖x‖₊ ^ l := by
rw [NNReal.tsum_mul_right, ← Fin.prod_const]
exact (p.changeOriginSeries k l).le_of_op_nnnorm_le _ (p.nnnorm_changeOriginSeries_le_tsum _ _)
#align formal_multilinear_series.nnnorm_change_origin_series_apply_le_tsum FormalMultilinearSeries.nnnorm_changeOriginSeries_apply_le_tsum
/-- Changing the origin of a formal multilinear series `p`, so that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense.
-/
def changeOrigin (x : E) : FormalMultilinearSeries 𝕜 E F :=
fun k => (p.changeOriginSeries k).sum x
#align formal_multilinear_series.change_origin FormalMultilinearSeries.changeOrigin
/-- An auxiliary equivalence useful in the proofs about
`FormalMultilinearSeries.changeOriginSeries`: the set of triples `(k, l, s)`, where `s` is a
`Finset (Fin (k + l))` of cardinality `l` is equivalent to the set of pairs `(n, s)`, where `s` is a
`Finset (Fin n)`.
The forward map sends `(k, l, s)` to `(k + l, s)` and the inverse map sends `(n, s)` to
`(n - Finset.card s, Finset.card s, s)`. The actual definition is less readable because of problems
with non-definitional equalities. -/
@[simps]
def changeOriginIndexEquiv :
(Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) ≃ Σn : ℕ, Finset (Fin n) where
toFun s := ⟨s.1 + s.2.1, s.2.2⟩
invFun s :=
⟨s.1 - s.2.card, s.2.card,
⟨s.2.map
(Fin.castIso <| (tsub_add_cancel_of_le <| card_finset_fin_le s.2).symm).toEquiv.toEmbedding,
Finset.card_map _⟩⟩
left_inv := by
rintro ⟨k, l, ⟨s : Finset (Fin <| k + l), hs : s.card = l⟩⟩
dsimp only [Subtype.coe_mk]
-- Lean can't automatically generalize `k' = k + l - s.card`, `l' = s.card`, so we explicitly
-- formulate the generalized goal
suffices ∀ k' l', k' = k → l' = l → ∀ (hkl : k + l = k' + l') (hs'),
(⟨k', l', ⟨Finset.map (Fin.castIso hkl).toEquiv.toEmbedding s, hs'⟩⟩ :
Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) = ⟨k, l, ⟨s, hs⟩⟩ by
apply this <;> simp only [hs, add_tsub_cancel_right]
rintro _ _ rfl rfl hkl hs'
simp only [Equiv.refl_toEmbedding, Fin.castIso_refl, Finset.map_refl, eq_self_iff_true,
OrderIso.refl_toEquiv, and_self_iff, heq_iff_eq]
right_inv := by
rintro ⟨n, s⟩
simp [tsub_add_cancel_of_le (card_finset_fin_le s), Fin.castIso_to_equiv]
#align formal_multilinear_series.change_origin_index_equiv FormalMultilinearSeries.changeOriginIndexEquiv
theorem changeOriginSeries_summable_aux₁ {r r' : ℝ≥0} (hr : (r + r' : ℝ≥0∞) < p.radius) :
Summable fun s : Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l } =>
‖p (s.1 + s.2.1)‖₊ * r ^ s.2.1 * r' ^ s.1 := by
rw [← changeOriginIndexEquiv.symm.summable_iff]
dsimp only [Function.comp_def, changeOriginIndexEquiv_symm_apply_fst,
changeOriginIndexEquiv_symm_apply_snd_fst]
have : ∀ n : ℕ,
HasSum (fun s : Finset (Fin n) => ‖p (n - s.card + s.card)‖₊ * r ^ s.card * r' ^ (n - s.card))
(‖p n‖₊ * (r + r') ^ n) := by
intro n
-- TODO: why `simp only [tsub_add_cancel_of_le (card_finset_fin_le _)]` fails?
convert_to HasSum (fun s : Finset (Fin n) => ‖p n‖₊ * (r ^ s.card * r' ^ (n - s.card))) _
· ext1 s
rw [tsub_add_cancel_of_le (card_finset_fin_le _), mul_assoc]
rw [← Fin.sum_pow_mul_eq_add_pow]
exact (hasSum_fintype _).mul_left _
refine' NNReal.summable_sigma.2 ⟨fun n => (this n).summable, _⟩
simp only [(this _).tsum_eq]
exact p.summable_nnnorm_mul_pow hr
#align formal_multilinear_series.change_origin_series_summable_aux₁ FormalMultilinearSeries.changeOriginSeries_summable_aux₁
theorem changeOriginSeries_summable_aux₂ (hr : (r : ℝ≥0∞) < p.radius) (k : ℕ) :
Summable fun s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l } =>
‖p (k + s.1)‖₊ * r ^ s.1 := by
rcases ENNReal.lt_iff_exists_add_pos_lt.1 hr with ⟨r', h0, hr'⟩
simpa only [mul_inv_cancel_right₀ (pow_pos h0 _).ne'] using
((NNReal.summable_sigma.1 (p.changeOriginSeries_summable_aux₁ hr')).1 k).mul_right (r' ^ k)⁻¹
#align formal_multilinear_series.change_origin_series_summable_aux₂ FormalMultilinearSeries.changeOriginSeries_summable_aux₂
theorem changeOriginSeries_summable_aux₃ {r : ℝ≥0} (hr : ↑r < p.radius) (k : ℕ) :
Summable fun l : ℕ => ‖p.changeOriginSeries k l‖₊ * r ^ l := by
refine' NNReal.summable_of_le
(fun n => _) (NNReal.summable_sigma.1 <| p.changeOriginSeries_summable_aux₂ hr k).2
simp only [NNReal.tsum_mul_right]
exact mul_le_mul' (p.nnnorm_changeOriginSeries_le_tsum _ _) le_rfl
#align formal_multilinear_series.change_origin_series_summable_aux₃ FormalMultilinearSeries.changeOriginSeries_summable_aux₃
theorem le_changeOriginSeries_radius (k : ℕ) : p.radius ≤ (p.changeOriginSeries k).radius :=
ENNReal.le_of_forall_nnreal_lt fun _r hr =>
le_radius_of_summable_nnnorm _ (p.changeOriginSeries_summable_aux₃ hr k)
#align formal_multilinear_series.le_change_origin_series_radius FormalMultilinearSeries.le_changeOriginSeries_radius
theorem nnnorm_changeOrigin_le (k : ℕ) (h : (‖x‖₊ : ℝ≥0∞) < p.radius) :
‖p.changeOrigin x k‖₊ ≤
∑' s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + s.1)‖₊ * ‖x‖₊ ^ s.1 := by
refine' tsum_of_nnnorm_bounded _ fun l => p.nnnorm_changeOriginSeries_apply_le_tsum k l x
have := p.changeOriginSeries_summable_aux₂ h k
refine' HasSum.sigma this.hasSum fun l => _
exact ((NNReal.summable_sigma.1 this).1 l).hasSum
#align formal_multilinear_series.nnnorm_change_origin_le FormalMultilinearSeries.nnnorm_changeOrigin_le
/-- The radius of convergence of `p.changeOrigin x` is at least `p.radius - ‖x‖`. In other words,
`p.changeOrigin x` is well defined on the largest ball contained in the original ball of
convergence. -/
theorem changeOrigin_radius : p.radius - ‖x‖₊ ≤ (p.changeOrigin x).radius := by
refine' ENNReal.le_of_forall_pos_nnreal_lt fun r _h0 hr => _
rw [lt_tsub_iff_right, add_comm] at hr
have hr' : (‖x‖₊ : ℝ≥0∞) < p.radius := (le_add_right le_rfl).trans_lt hr
apply le_radius_of_summable_nnnorm
have : ∀ k : ℕ,
‖p.changeOrigin x k‖₊ * r ^ k ≤
(∑' s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + s.1)‖₊ * ‖x‖₊ ^ s.1) *
r ^ k :=
fun k => mul_le_mul_right' (p.nnnorm_changeOrigin_le k hr') (r ^ k)
refine' NNReal.summable_of_le this _
simpa only [← NNReal.tsum_mul_right] using
(NNReal.summable_sigma.1 (p.changeOriginSeries_summable_aux₁ hr)).2
#align formal_multilinear_series.change_origin_radius FormalMultilinearSeries.changeOrigin_radius
end
-- From this point on, assume that the space is complete, to make sure that series that converge
-- in norm also converge in `F`.
variable [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x y : E} {r R : ℝ≥0}
theorem hasFPowerSeriesOnBall_changeOrigin (k : ℕ) (hr : 0 < p.radius) :
HasFPowerSeriesOnBall (fun x => p.changeOrigin x k) (p.changeOriginSeries k) 0 p.radius :=
have := p.le_changeOriginSeries_radius k
((p.changeOriginSeries k).hasFPowerSeriesOnBall (hr.trans_le this)).mono hr this
#align formal_multilinear_series.has_fpower_series_on_ball_change_origin FormalMultilinearSeries.hasFPowerSeriesOnBall_changeOrigin
/-- Summing the series `p.changeOrigin x` at a point `y` gives back `p (x + y)`. -/
theorem changeOrigin_eval (h : (‖x‖₊ + ‖y‖₊ : ℝ≥0∞) < p.radius) :
(p.changeOrigin x).sum y = p.sum (x + y) := by
have radius_pos : 0 < p.radius := lt_of_le_of_lt (zero_le _) h
have x_mem_ball : x ∈ EMetric.ball (0 : E) p.radius :=
mem_emetric_ball_zero_iff.2 ((le_add_right le_rfl).trans_lt h)
have y_mem_ball : y ∈ EMetric.ball (0 : E) (p.changeOrigin x).radius := by
refine' mem_emetric_ball_zero_iff.2 (lt_of_lt_of_le _ p.changeOrigin_radius)
rwa [lt_tsub_iff_right, add_comm]
have x_add_y_mem_ball : x + y ∈ EMetric.ball (0 : E) p.radius := by
refine' mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt _ h)
exact mod_cast nnnorm_add_le x y
set f : (Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) → F := fun s =>
p.changeOriginSeriesTerm s.1 s.2.1 s.2.2 s.2.2.2 (fun _ => x) fun _ => y
have hsf : Summable f := by
refine' .of_nnnorm_bounded _ (p.changeOriginSeries_summable_aux₁ h) _
rintro ⟨k, l, s, hs⟩
dsimp only [Subtype.coe_mk]
exact p.nnnorm_changeOriginSeriesTerm_apply_le _ _ _ _ _ _
have hf : HasSum f ((p.changeOrigin x).sum y) := by
refine' HasSum.sigma_of_hasSum ((p.changeOrigin x).summable y_mem_ball).hasSum (fun k => _) hsf
· dsimp only
refine' ContinuousMultilinearMap.hasSum_eval _ _
have := (p.hasFPowerSeriesOnBall_changeOrigin k radius_pos).hasSum x_mem_ball
rw [zero_add] at this
refine' HasSum.sigma_of_hasSum this (fun l => _) _
· simp only [changeOriginSeries, ContinuousMultilinearMap.sum_apply]
apply hasSum_fintype
· refine' .of_nnnorm_bounded _
(p.changeOriginSeries_summable_aux₂ (mem_emetric_ball_zero_iff.1 x_mem_ball) k) fun s => _
refine' (ContinuousMultilinearMap.le_op_nnnorm _ _).trans_eq _
simp
refine' hf.unique (changeOriginIndexEquiv.symm.hasSum_iff.1 _)
refine' HasSum.sigma_of_hasSum
(p.hasSum x_add_y_mem_ball) (fun n => _) (changeOriginIndexEquiv.symm.summable_iff.2 hsf)
erw [(p n).map_add_univ (fun _ => x) fun _ => y]
-- porting note: added explicit function
convert hasSum_fintype (fun c : Finset (Fin n) => f (changeOriginIndexEquiv.symm ⟨n, c⟩))
rename_i s _
dsimp only [changeOriginSeriesTerm, (· ∘ ·), changeOriginIndexEquiv_symm_apply_fst,
changeOriginIndexEquiv_symm_apply_snd_fst, changeOriginIndexEquiv_symm_apply_snd_snd_coe]
rw [ContinuousMultilinearMap.curryFinFinset_apply_const]
have : ∀ (m) (hm : n = m), p n (s.piecewise (fun _ => x) fun _ => y) =
p m ((s.map (Fin.castIso hm).toEquiv.toEmbedding).piecewise (fun _ => x) fun _ => y) := by
rintro m rfl
simp (config := { unfoldPartialApp := true }) [Finset.piecewise]
apply this
#align formal_multilinear_series.change_origin_eval FormalMultilinearSeries.changeOrigin_eval
/-- Power series terms are analytic as we vary the origin -/
theorem analyticAt_changeOrigin (p : FormalMultilinearSeries 𝕜 E F) (rp : p.radius > 0) (n : ℕ) :
AnalyticAt 𝕜 (fun x ↦ p.changeOrigin x n) 0 :=
(FormalMultilinearSeries.hasFPowerSeriesOnBall_changeOrigin p n rp).analyticAt
end FormalMultilinearSeries
section
variable [CompleteSpace F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x y : E} {r : ℝ≥0∞}
/-- If a function admits a power series expansion `p` on a ball `B (x, r)`, then it also admits a
power series on any subball of this ball (even with a different center), given by `p.changeOrigin`.
-/
theorem HasFPowerSeriesOnBall.changeOrigin (hf : HasFPowerSeriesOnBall f p x r)
(h : (‖y‖₊ : ℝ≥0∞) < r) : HasFPowerSeriesOnBall f (p.changeOrigin y) (x + y) (r - ‖y‖₊) :=
{ r_le := by
apply le_trans _ p.changeOrigin_radius
exact tsub_le_tsub hf.r_le le_rfl
r_pos := by simp [h]
hasSum := fun {z} hz => by
have : f (x + y + z) =
FormalMultilinearSeries.sum (FormalMultilinearSeries.changeOrigin p y) z := by
rw [mem_emetric_ball_zero_iff, lt_tsub_iff_right, add_comm] at hz
rw [p.changeOrigin_eval (hz.trans_le hf.r_le), add_assoc, hf.sum]
refine' mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt _ hz)
exact mod_cast nnnorm_add_le y z
rw [this]
apply (p.changeOrigin y).hasSum
refine' EMetric.ball_subset_ball (le_trans _ p.changeOrigin_radius) hz
exact tsub_le_tsub hf.r_le le_rfl }
#align has_fpower_series_on_ball.change_origin HasFPowerSeriesOnBall.changeOrigin
/-- If a function admits a power series expansion `p` on an open ball `B (x, r)`, then
it is analytic at every point of this ball. -/
theorem HasFPowerSeriesOnBall.analyticAt_of_mem (hf : HasFPowerSeriesOnBall f p x r)
(h : y ∈ EMetric.ball x r) : AnalyticAt 𝕜 f y := by
have : (‖y - x‖₊ : ℝ≥0∞) < r := by simpa [edist_eq_coe_nnnorm_sub] using h
have := hf.changeOrigin this
rw [add_sub_cancel'_right] at this
exact this.analyticAt
#align has_fpower_series_on_ball.analytic_at_of_mem HasFPowerSeriesOnBall.analyticAt_of_mem
theorem HasFPowerSeriesOnBall.analyticOn (hf : HasFPowerSeriesOnBall f p x r) :
AnalyticOn 𝕜 f (EMetric.ball x r) :=
fun _y hy => hf.analyticAt_of_mem hy
#align has_fpower_series_on_ball.analytic_on HasFPowerSeriesOnBall.analyticOn
variable (𝕜 f)
/-- For any function `f` from a normed vector space to a Banach space, the set of points `x` such
that `f` is analytic at `x` is open. -/
theorem isOpen_analyticAt : IsOpen { x | AnalyticAt 𝕜 f x } := by
rw [isOpen_iff_mem_nhds]
rintro x ⟨p, r, hr⟩
exact mem_of_superset (EMetric.ball_mem_nhds _ hr.r_pos) fun y hy => hr.analyticAt_of_mem hy
#align is_open_analytic_at isOpen_analyticAt
variable {𝕜}
theorem AnalyticAt.eventually_analyticAt {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∀ᶠ y in 𝓝 x, AnalyticAt 𝕜 f y :=
(isOpen_analyticAt 𝕜 f).mem_nhds h
theorem AnalyticAt.exists_mem_nhds_analyticOn {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∃ s ∈ 𝓝 x, AnalyticOn 𝕜 f s :=
h.eventually_analyticAt.exists_mem
/-- If we're analytic at a point, we're analytic in a nonempty ball -/
theorem AnalyticAt.exists_ball_analyticOn {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∃ r : ℝ, 0 < r ∧ AnalyticOn 𝕜 f (Metric.ball x r) :=
Metric.isOpen_iff.mp (isOpen_analyticAt _ _) _ h
end
section
open FormalMultilinearSeries
variable {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E} {z₀ : 𝕜}
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z)) := by
refine' ⟨fun ⟨r, _, r_pos, h⟩ =>
eventually_of_mem (EMetric.ball_mem_nhds 0 r_pos) fun _ => by simpa using h, _⟩
simp only [Metric.eventually_nhds_iff]
rintro ⟨r, r_pos, h⟩
refine' ⟨p.radius ⊓ r.toNNReal, by simp, _, _⟩
· simp only [r_pos.lt, lt_inf_iff, ENNReal.coe_pos, Real.toNNReal_pos, and_true_iff]
obtain ⟨z, z_pos, le_z⟩ := NormedField.exists_norm_lt 𝕜 r_pos.lt
have : (‖z‖₊ : ENNReal) ≤ p.radius := by
simp only [dist_zero_right] at h
apply FormalMultilinearSeries.le_radius_of_tendsto
convert tendsto_norm.comp (h le_z).summable.tendsto_atTop_zero
funext
simp [norm_smul, mul_comm]
refine' lt_of_lt_of_le _ this
simp only [ENNReal.coe_pos]
|
exact zero_lt_iff.mpr (nnnorm_ne_zero_iff.mpr (norm_pos_iff.mp z_pos))
|
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z)) := by
refine' ⟨fun ⟨r, _, r_pos, h⟩ =>
eventually_of_mem (EMetric.ball_mem_nhds 0 r_pos) fun _ => by simpa using h, _⟩
simp only [Metric.eventually_nhds_iff]
rintro ⟨r, r_pos, h⟩
refine' ⟨p.radius ⊓ r.toNNReal, by simp, _, _⟩
· simp only [r_pos.lt, lt_inf_iff, ENNReal.coe_pos, Real.toNNReal_pos, and_true_iff]
obtain ⟨z, z_pos, le_z⟩ := NormedField.exists_norm_lt 𝕜 r_pos.lt
have : (‖z‖₊ : ENNReal) ≤ p.radius := by
simp only [dist_zero_right] at h
apply FormalMultilinearSeries.le_radius_of_tendsto
convert tendsto_norm.comp (h le_z).summable.tendsto_atTop_zero
funext
simp [norm_smul, mul_comm]
refine' lt_of_lt_of_le _ this
simp only [ENNReal.coe_pos]
|
Mathlib.Analysis.Analytic.Basic.1430_0.jQw1fRSE1vGpOll
|
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z))
|
Mathlib_Analysis_Analytic_Basic
|
case intro.intro.refine'_2
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
p : FormalMultilinearSeries 𝕜 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
r : ℝ
r_pos : r > 0
h : ∀ ⦃y : 𝕜⦄, dist y 0 < r → HasSum (fun n => y ^ n • coeff p n) (f (z₀ + y))
⊢ ∀ {y : 𝕜}, y ∈ EMetric.ball 0 (radius p ⊓ ↑(Real.toNNReal r)) → HasSum (fun n => (p n) fun x => y) (f (z₀ + y))
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Logic.Equiv.Fin
import Mathlib.Topology.Algebra.InfiniteSum.Module
#align_import analysis.analytic.basic from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
/-!
# Analytic functions
A function is analytic in one dimension around `0` if it can be written as a converging power series
`Σ pₙ zⁿ`. This definition can be extended to any dimension (even in infinite dimension) by
requiring that `pₙ` is a continuous `n`-multilinear map. In general, `pₙ` is not unique (in two
dimensions, taking `p₂ (x, y) (x', y') = x y'` or `y x'` gives the same map when applied to a
vector `(x, y) (x, y)`). A way to guarantee uniqueness is to take a symmetric `pₙ`, but this is not
always possible in nonzero characteristic (in characteristic 2, the previous example has no
symmetric representative). Therefore, we do not insist on symmetry or uniqueness in the definition,
and we only require the existence of a converging series.
The general framework is important to say that the exponential map on bounded operators on a Banach
space is analytic, as well as the inverse on invertible operators.
## Main definitions
Let `p` be a formal multilinear series from `E` to `F`, i.e., `p n` is a multilinear map on `E^n`
for `n : ℕ`.
* `p.radius`: the largest `r : ℝ≥0∞` such that `‖p n‖ * r^n` grows subexponentially.
* `p.le_radius_of_bound`, `p.le_radius_of_bound_nnreal`, `p.le_radius_of_isBigO`: if `‖p n‖ * r ^ n`
is bounded above, then `r ≤ p.radius`;
* `p.isLittleO_of_lt_radius`, `p.norm_mul_pow_le_mul_pow_of_lt_radius`,
`p.isLittleO_one_of_lt_radius`,
`p.norm_mul_pow_le_of_lt_radius`, `p.nnnorm_mul_pow_le_of_lt_radius`: if `r < p.radius`, then
`‖p n‖ * r ^ n` tends to zero exponentially;
* `p.lt_radius_of_isBigO`: if `r ≠ 0` and `‖p n‖ * r ^ n = O(a ^ n)` for some `-1 < a < 1`, then
`r < p.radius`;
* `p.partialSum n x`: the sum `∑_{i = 0}^{n-1} pᵢ xⁱ`.
* `p.sum x`: the sum `∑'_{i = 0}^{∞} pᵢ xⁱ`.
Additionally, let `f` be a function from `E` to `F`.
* `HasFPowerSeriesOnBall f p x r`: on the ball of center `x` with radius `r`,
`f (x + y) = ∑'_n pₙ yⁿ`.
* `HasFPowerSeriesAt f p x`: on some ball of center `x` with positive radius, holds
`HasFPowerSeriesOnBall f p x r`.
* `AnalyticAt 𝕜 f x`: there exists a power series `p` such that holds `HasFPowerSeriesAt f p x`.
* `AnalyticOn 𝕜 f s`: the function `f` is analytic at every point of `s`.
We develop the basic properties of these notions, notably:
* If a function admits a power series, it is continuous (see
`HasFPowerSeriesOnBall.continuousOn` and `HasFPowerSeriesAt.continuousAt` and
`AnalyticAt.continuousAt`).
* In a complete space, the sum of a formal power series with positive radius is well defined on the
disk of convergence, see `FormalMultilinearSeries.hasFPowerSeriesOnBall`.
* If a function admits a power series in a ball, then it is analytic at any point `y` of this ball,
and the power series there can be expressed in terms of the initial power series `p` as
`p.changeOrigin y`. See `HasFPowerSeriesOnBall.changeOrigin`. It follows in particular that
the set of points at which a given function is analytic is open, see `isOpen_analyticAt`.
## Implementation details
We only introduce the radius of convergence of a power series, as `p.radius`.
For a power series in finitely many dimensions, there is a finer (directional, coordinate-dependent)
notion, describing the polydisk of convergence. This notion is more specific, and not necessary to
build the general theory. We do not define it here.
-/
noncomputable section
variable {𝕜 E F G : Type*}
open Topology Classical BigOperators NNReal Filter ENNReal
open Set Filter Asymptotics
namespace FormalMultilinearSeries
variable [Ring 𝕜] [AddCommGroup E] [AddCommGroup F] [Module 𝕜 E] [Module 𝕜 F]
variable [TopologicalSpace E] [TopologicalSpace F]
variable [TopologicalAddGroup E] [TopologicalAddGroup F]
variable [ContinuousConstSMul 𝕜 E] [ContinuousConstSMul 𝕜 F]
/-- Given a formal multilinear series `p` and a vector `x`, then `p.sum x` is the sum `Σ pₙ xⁿ`. A
priori, it only behaves well when `‖x‖ < p.radius`. -/
protected def sum (p : FormalMultilinearSeries 𝕜 E F) (x : E) : F :=
∑' n : ℕ, p n fun _ => x
#align formal_multilinear_series.sum FormalMultilinearSeries.sum
/-- Given a formal multilinear series `p` and a vector `x`, then `p.partialSum n x` is the sum
`Σ pₖ xᵏ` for `k ∈ {0,..., n-1}`. -/
def partialSum (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) (x : E) : F :=
∑ k in Finset.range n, p k fun _ : Fin k => x
#align formal_multilinear_series.partial_sum FormalMultilinearSeries.partialSum
/-- The partial sums of a formal multilinear series are continuous. -/
theorem partialSum_continuous (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) :
Continuous (p.partialSum n) := by
unfold partialSum -- Porting note: added
continuity
#align formal_multilinear_series.partial_sum_continuous FormalMultilinearSeries.partialSum_continuous
end FormalMultilinearSeries
/-! ### The radius of a formal multilinear series -/
variable [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F]
[NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G]
namespace FormalMultilinearSeries
variable (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
/-- The radius of a formal multilinear series is the largest `r` such that the sum `Σ ‖pₙ‖ ‖y‖ⁿ`
converges for all `‖y‖ < r`. This implies that `Σ pₙ yⁿ` converges for all `‖y‖ < r`, but these
definitions are *not* equivalent in general. -/
def radius (p : FormalMultilinearSeries 𝕜 E F) : ℝ≥0∞ :=
⨆ (r : ℝ≥0) (C : ℝ) (_ : ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C), (r : ℝ≥0∞)
#align formal_multilinear_series.radius FormalMultilinearSeries.radius
/-- If `‖pₙ‖ rⁿ` is bounded in `n`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_bound (C : ℝ) {r : ℝ≥0} (h : ∀ n : ℕ, ‖p n‖ * (r : ℝ) ^ n ≤ C) :
(r : ℝ≥0∞) ≤ p.radius :=
le_iSup_of_le r <| le_iSup_of_le C <| le_iSup (fun _ => (r : ℝ≥0∞)) h
#align formal_multilinear_series.le_radius_of_bound FormalMultilinearSeries.le_radius_of_bound
/-- If `‖pₙ‖ rⁿ` is bounded in `n`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_bound_nnreal (C : ℝ≥0) {r : ℝ≥0} (h : ∀ n : ℕ, ‖p n‖₊ * r ^ n ≤ C) :
(r : ℝ≥0∞) ≤ p.radius :=
p.le_radius_of_bound C fun n => mod_cast h n
#align formal_multilinear_series.le_radius_of_bound_nnreal FormalMultilinearSeries.le_radius_of_bound_nnreal
/-- If `‖pₙ‖ rⁿ = O(1)`, as `n → ∞`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_isBigO (h : (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] fun _ => (1 : ℝ)) :
↑r ≤ p.radius :=
Exists.elim (isBigO_one_nat_atTop_iff.1 h) fun C hC =>
p.le_radius_of_bound C fun n => (le_abs_self _).trans (hC n)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.le_radius_of_is_O FormalMultilinearSeries.le_radius_of_isBigO
theorem le_radius_of_eventually_le (C) (h : ∀ᶠ n in atTop, ‖p n‖ * (r : ℝ) ^ n ≤ C) :
↑r ≤ p.radius :=
p.le_radius_of_isBigO <| IsBigO.of_bound C <| h.mono fun n hn => by simpa
#align formal_multilinear_series.le_radius_of_eventually_le FormalMultilinearSeries.le_radius_of_eventually_le
theorem le_radius_of_summable_nnnorm (h : Summable fun n => ‖p n‖₊ * r ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_bound_nnreal (∑' n, ‖p n‖₊ * r ^ n) fun _ => le_tsum' h _
#align formal_multilinear_series.le_radius_of_summable_nnnorm FormalMultilinearSeries.le_radius_of_summable_nnnorm
theorem le_radius_of_summable (h : Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_summable_nnnorm <| by
simp only [← coe_nnnorm] at h
exact mod_cast h
#align formal_multilinear_series.le_radius_of_summable FormalMultilinearSeries.le_radius_of_summable
theorem radius_eq_top_of_forall_nnreal_isBigO
(h : ∀ r : ℝ≥0, (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] fun _ => (1 : ℝ)) : p.radius = ∞ :=
ENNReal.eq_top_of_forall_nnreal_le fun r => p.le_radius_of_isBigO (h r)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.radius_eq_top_of_forall_nnreal_is_O FormalMultilinearSeries.radius_eq_top_of_forall_nnreal_isBigO
theorem radius_eq_top_of_eventually_eq_zero (h : ∀ᶠ n in atTop, p n = 0) : p.radius = ∞ :=
p.radius_eq_top_of_forall_nnreal_isBigO fun r =>
(isBigO_zero _ _).congr' (h.mono fun n hn => by simp [hn]) EventuallyEq.rfl
#align formal_multilinear_series.radius_eq_top_of_eventually_eq_zero FormalMultilinearSeries.radius_eq_top_of_eventually_eq_zero
theorem radius_eq_top_of_forall_image_add_eq_zero (n : ℕ) (hn : ∀ m, p (m + n) = 0) :
p.radius = ∞ :=
p.radius_eq_top_of_eventually_eq_zero <|
mem_atTop_sets.2 ⟨n, fun _ hk => tsub_add_cancel_of_le hk ▸ hn _⟩
#align formal_multilinear_series.radius_eq_top_of_forall_image_add_eq_zero FormalMultilinearSeries.radius_eq_top_of_forall_image_add_eq_zero
@[simp]
theorem constFormalMultilinearSeries_radius {v : F} :
(constFormalMultilinearSeries 𝕜 E v).radius = ⊤ :=
(constFormalMultilinearSeries 𝕜 E v).radius_eq_top_of_forall_image_add_eq_zero 1
(by simp [constFormalMultilinearSeries])
#align formal_multilinear_series.const_formal_multilinear_series_radius FormalMultilinearSeries.constFormalMultilinearSeries_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` tends to zero exponentially:
for some `0 < a < 1`, `‖p n‖ rⁿ = o(aⁿ)`. -/
theorem isLittleO_of_lt_radius (h : ↑r < p.radius) :
∃ a ∈ Ioo (0 : ℝ) 1, (fun n => ‖p n‖ * (r : ℝ) ^ n) =o[atTop] (a ^ ·) := by
have := (TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 4
rw [this]
-- Porting note: was
-- rw [(TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 4]
simp only [radius, lt_iSup_iff] at h
rcases h with ⟨t, C, hC, rt⟩
rw [ENNReal.coe_lt_coe, ← NNReal.coe_lt_coe] at rt
have : 0 < (t : ℝ) := r.coe_nonneg.trans_lt rt
rw [← div_lt_one this] at rt
refine' ⟨_, rt, C, Or.inr zero_lt_one, fun n => _⟩
calc
|‖p n‖ * (r : ℝ) ^ n| = ‖p n‖ * (t : ℝ) ^ n * (r / t : ℝ) ^ n := by
field_simp [mul_right_comm, abs_mul]
_ ≤ C * (r / t : ℝ) ^ n := by gcongr; apply hC
#align formal_multilinear_series.is_o_of_lt_radius FormalMultilinearSeries.isLittleO_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ = o(1)`. -/
theorem isLittleO_one_of_lt_radius (h : ↑r < p.radius) :
(fun n => ‖p n‖ * (r : ℝ) ^ n) =o[atTop] (fun _ => 1 : ℕ → ℝ) :=
let ⟨_, ha, hp⟩ := p.isLittleO_of_lt_radius h
hp.trans <| (isLittleO_pow_pow_of_lt_left ha.1.le ha.2).congr (fun _ => rfl) one_pow
#align formal_multilinear_series.is_o_one_of_lt_radius FormalMultilinearSeries.isLittleO_one_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` tends to zero exponentially:
for some `0 < a < 1` and `C > 0`, `‖p n‖ * r ^ n ≤ C * a ^ n`. -/
theorem norm_mul_pow_le_mul_pow_of_lt_radius (h : ↑r < p.radius) :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C * a ^ n := by
-- Porting note: moved out of `rcases`
have := ((TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 5).mp
(p.isLittleO_of_lt_radius h)
rcases this with ⟨a, ha, C, hC, H⟩
exact ⟨a, ha, C, hC, fun n => (le_abs_self _).trans (H n)⟩
#align formal_multilinear_series.norm_mul_pow_le_mul_pow_of_lt_radius FormalMultilinearSeries.norm_mul_pow_le_mul_pow_of_lt_radius
/-- If `r ≠ 0` and `‖pₙ‖ rⁿ = O(aⁿ)` for some `-1 < a < 1`, then `r < p.radius`. -/
theorem lt_radius_of_isBigO (h₀ : r ≠ 0) {a : ℝ} (ha : a ∈ Ioo (-1 : ℝ) 1)
(hp : (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] (a ^ ·)) : ↑r < p.radius := by
-- Porting note: moved out of `rcases`
have := ((TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 2 5)
rcases this.mp ⟨a, ha, hp⟩ with ⟨a, ha, C, hC, hp⟩
rw [← pos_iff_ne_zero, ← NNReal.coe_pos] at h₀
lift a to ℝ≥0 using ha.1.le
have : (r : ℝ) < r / a := by
simpa only [div_one] using (div_lt_div_left h₀ zero_lt_one ha.1).2 ha.2
norm_cast at this
rw [← ENNReal.coe_lt_coe] at this
refine' this.trans_le (p.le_radius_of_bound C fun n => _)
rw [NNReal.coe_div, div_pow, ← mul_div_assoc, div_le_iff (pow_pos ha.1 n)]
exact (le_abs_self _).trans (hp n)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.lt_radius_of_is_O FormalMultilinearSeries.lt_radius_of_isBigO
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem norm_mul_pow_le_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C :=
let ⟨_, ha, C, hC, h⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius h
⟨C, hC, fun n => (h n).trans <| mul_le_of_le_one_right hC.lt.le (pow_le_one _ ha.1.le ha.2.le)⟩
#align formal_multilinear_series.norm_mul_pow_le_of_lt_radius FormalMultilinearSeries.norm_mul_pow_le_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem norm_le_div_pow_of_pos_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h0 : 0 < r) (h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖ ≤ C / (r : ℝ) ^ n :=
let ⟨C, hC, hp⟩ := p.norm_mul_pow_le_of_lt_radius h
⟨C, hC, fun n => Iff.mpr (le_div_iff (pow_pos h0 _)) (hp n)⟩
#align formal_multilinear_series.norm_le_div_pow_of_pos_of_lt_radius FormalMultilinearSeries.norm_le_div_pow_of_pos_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem nnnorm_mul_pow_le_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖₊ * r ^ n ≤ C :=
let ⟨C, hC, hp⟩ := p.norm_mul_pow_le_of_lt_radius h
⟨⟨C, hC.lt.le⟩, hC, mod_cast hp⟩
#align formal_multilinear_series.nnnorm_mul_pow_le_of_lt_radius FormalMultilinearSeries.nnnorm_mul_pow_le_of_lt_radius
theorem le_radius_of_tendsto (p : FormalMultilinearSeries 𝕜 E F) {l : ℝ}
(h : Tendsto (fun n => ‖p n‖ * (r : ℝ) ^ n) atTop (𝓝 l)) : ↑r ≤ p.radius :=
p.le_radius_of_isBigO (h.isBigO_one _)
#align formal_multilinear_series.le_radius_of_tendsto FormalMultilinearSeries.le_radius_of_tendsto
theorem le_radius_of_summable_norm (p : FormalMultilinearSeries 𝕜 E F)
(hs : Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_tendsto hs.tendsto_atTop_zero
#align formal_multilinear_series.le_radius_of_summable_norm FormalMultilinearSeries.le_radius_of_summable_norm
theorem not_summable_norm_of_radius_lt_nnnorm (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(h : p.radius < ‖x‖₊) : ¬Summable fun n => ‖p n‖ * ‖x‖ ^ n :=
fun hs => not_le_of_lt h (p.le_radius_of_summable_norm hs)
#align formal_multilinear_series.not_summable_norm_of_radius_lt_nnnorm FormalMultilinearSeries.not_summable_norm_of_radius_lt_nnnorm
theorem summable_norm_mul_pow (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0} (h : ↑r < p.radius) :
Summable fun n : ℕ => ‖p n‖ * (r : ℝ) ^ n := by
obtain ⟨a, ha : a ∈ Ioo (0 : ℝ) 1, C, - : 0 < C, hp⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius h
exact .of_nonneg_of_le (fun n => mul_nonneg (norm_nonneg _) (pow_nonneg r.coe_nonneg _))
hp ((summable_geometric_of_lt_1 ha.1.le ha.2).mul_left _)
#align formal_multilinear_series.summable_norm_mul_pow FormalMultilinearSeries.summable_norm_mul_pow
theorem summable_norm_apply (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : Summable fun n : ℕ => ‖p n fun _ => x‖ := by
rw [mem_emetric_ball_zero_iff] at hx
refine' .of_nonneg_of_le
(fun _ => norm_nonneg _) (fun n => ((p n).le_op_norm _).trans_eq _) (p.summable_norm_mul_pow hx)
simp
#align formal_multilinear_series.summable_norm_apply FormalMultilinearSeries.summable_norm_apply
theorem summable_nnnorm_mul_pow (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0} (h : ↑r < p.radius) :
Summable fun n : ℕ => ‖p n‖₊ * r ^ n := by
rw [← NNReal.summable_coe]
push_cast
exact p.summable_norm_mul_pow h
#align formal_multilinear_series.summable_nnnorm_mul_pow FormalMultilinearSeries.summable_nnnorm_mul_pow
protected theorem summable [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : Summable fun n : ℕ => p n fun _ => x :=
(p.summable_norm_apply hx).of_norm
#align formal_multilinear_series.summable FormalMultilinearSeries.summable
theorem radius_eq_top_of_summable_norm (p : FormalMultilinearSeries 𝕜 E F)
(hs : ∀ r : ℝ≥0, Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : p.radius = ∞ :=
ENNReal.eq_top_of_forall_nnreal_le fun r => p.le_radius_of_summable_norm (hs r)
#align formal_multilinear_series.radius_eq_top_of_summable_norm FormalMultilinearSeries.radius_eq_top_of_summable_norm
theorem radius_eq_top_iff_summable_norm (p : FormalMultilinearSeries 𝕜 E F) :
p.radius = ∞ ↔ ∀ r : ℝ≥0, Summable fun n => ‖p n‖ * (r : ℝ) ^ n := by
constructor
· intro h r
obtain ⟨a, ha : a ∈ Ioo (0 : ℝ) 1, C, - : 0 < C, hp⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius
(show (r : ℝ≥0∞) < p.radius from h.symm ▸ ENNReal.coe_lt_top)
refine' .of_norm_bounded
(fun n => (C : ℝ) * a ^ n) ((summable_geometric_of_lt_1 ha.1.le ha.2).mul_left _) fun n => _
specialize hp n
rwa [Real.norm_of_nonneg (mul_nonneg (norm_nonneg _) (pow_nonneg r.coe_nonneg n))]
· exact p.radius_eq_top_of_summable_norm
#align formal_multilinear_series.radius_eq_top_iff_summable_norm FormalMultilinearSeries.radius_eq_top_iff_summable_norm
/-- If the radius of `p` is positive, then `‖pₙ‖` grows at most geometrically. -/
theorem le_mul_pow_of_radius_pos (p : FormalMultilinearSeries 𝕜 E F) (h : 0 < p.radius) :
∃ (C r : _) (hC : 0 < C) (_ : 0 < r), ∀ n, ‖p n‖ ≤ C * r ^ n := by
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 h with ⟨r, r0, rlt⟩
have rpos : 0 < (r : ℝ) := by simp [ENNReal.coe_pos.1 r0]
rcases norm_le_div_pow_of_pos_of_lt_radius p rpos rlt with ⟨C, Cpos, hCp⟩
refine' ⟨C, r⁻¹, Cpos, by simp only [inv_pos, rpos], fun n => _⟩
-- Porting note: was `convert`
rw [inv_pow, ← div_eq_mul_inv]
exact hCp n
#align formal_multilinear_series.le_mul_pow_of_radius_pos FormalMultilinearSeries.le_mul_pow_of_radius_pos
/-- The radius of the sum of two formal series is at least the minimum of their two radii. -/
theorem min_radius_le_radius_add (p q : FormalMultilinearSeries 𝕜 E F) :
min p.radius q.radius ≤ (p + q).radius := by
refine' ENNReal.le_of_forall_nnreal_lt fun r hr => _
rw [lt_min_iff] at hr
have := ((p.isLittleO_one_of_lt_radius hr.1).add (q.isLittleO_one_of_lt_radius hr.2)).isBigO
refine' (p + q).le_radius_of_isBigO ((isBigO_of_le _ fun n => _).trans this)
rw [← add_mul, norm_mul, norm_mul, norm_norm]
exact mul_le_mul_of_nonneg_right ((norm_add_le _ _).trans (le_abs_self _)) (norm_nonneg _)
#align formal_multilinear_series.min_radius_le_radius_add FormalMultilinearSeries.min_radius_le_radius_add
@[simp]
theorem radius_neg (p : FormalMultilinearSeries 𝕜 E F) : (-p).radius = p.radius := by
simp only [radius, neg_apply, norm_neg]
#align formal_multilinear_series.radius_neg FormalMultilinearSeries.radius_neg
protected theorem hasSum [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : HasSum (fun n : ℕ => p n fun _ => x) (p.sum x) :=
(p.summable hx).hasSum
#align formal_multilinear_series.has_sum FormalMultilinearSeries.hasSum
theorem radius_le_radius_continuousLinearMap_comp (p : FormalMultilinearSeries 𝕜 E F)
(f : F →L[𝕜] G) : p.radius ≤ (f.compFormalMultilinearSeries p).radius := by
refine' ENNReal.le_of_forall_nnreal_lt fun r hr => _
apply le_radius_of_isBigO
apply (IsBigO.trans_isLittleO _ (p.isLittleO_one_of_lt_radius hr)).isBigO
refine' IsBigO.mul (@IsBigOWith.isBigO _ _ _ _ _ ‖f‖ _ _ _ _) (isBigO_refl _ _)
refine IsBigOWith.of_bound (eventually_of_forall fun n => ?_)
simpa only [norm_norm] using f.norm_compContinuousMultilinearMap_le (p n)
#align formal_multilinear_series.radius_le_radius_continuous_linear_map_comp FormalMultilinearSeries.radius_le_radius_continuousLinearMap_comp
end FormalMultilinearSeries
/-! ### Expanding a function as a power series -/
section
variable {f g : E → F} {p pf pg : FormalMultilinearSeries 𝕜 E F} {x : E} {r r' : ℝ≥0∞}
/-- Given a function `f : E → F` and a formal multilinear series `p`, we say that `f` has `p` as
a power series on the ball of radius `r > 0` around `x` if `f (x + y) = ∑' pₙ yⁿ` for all `‖y‖ < r`.
-/
structure HasFPowerSeriesOnBall (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (x : E) (r : ℝ≥0∞) :
Prop where
r_le : r ≤ p.radius
r_pos : 0 < r
hasSum :
∀ {y}, y ∈ EMetric.ball (0 : E) r → HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y))
#align has_fpower_series_on_ball HasFPowerSeriesOnBall
/-- Given a function `f : E → F` and a formal multilinear series `p`, we say that `f` has `p` as
a power series around `x` if `f (x + y) = ∑' pₙ yⁿ` for all `y` in a neighborhood of `0`. -/
def HasFPowerSeriesAt (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (x : E) :=
∃ r, HasFPowerSeriesOnBall f p x r
#align has_fpower_series_at HasFPowerSeriesAt
variable (𝕜)
/-- Given a function `f : E → F`, we say that `f` is analytic at `x` if it admits a convergent power
series expansion around `x`. -/
def AnalyticAt (f : E → F) (x : E) :=
∃ p : FormalMultilinearSeries 𝕜 E F, HasFPowerSeriesAt f p x
#align analytic_at AnalyticAt
/-- Given a function `f : E → F`, we say that `f` is analytic on a set `s` if it is analytic around
every point of `s`. -/
def AnalyticOn (f : E → F) (s : Set E) :=
∀ x, x ∈ s → AnalyticAt 𝕜 f x
#align analytic_on AnalyticOn
variable {𝕜}
theorem HasFPowerSeriesOnBall.hasFPowerSeriesAt (hf : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesAt f p x :=
⟨r, hf⟩
#align has_fpower_series_on_ball.has_fpower_series_at HasFPowerSeriesOnBall.hasFPowerSeriesAt
theorem HasFPowerSeriesAt.analyticAt (hf : HasFPowerSeriesAt f p x) : AnalyticAt 𝕜 f x :=
⟨p, hf⟩
#align has_fpower_series_at.analytic_at HasFPowerSeriesAt.analyticAt
theorem HasFPowerSeriesOnBall.analyticAt (hf : HasFPowerSeriesOnBall f p x r) : AnalyticAt 𝕜 f x :=
hf.hasFPowerSeriesAt.analyticAt
#align has_fpower_series_on_ball.analytic_at HasFPowerSeriesOnBall.analyticAt
theorem HasFPowerSeriesOnBall.congr (hf : HasFPowerSeriesOnBall f p x r)
(hg : EqOn f g (EMetric.ball x r)) : HasFPowerSeriesOnBall g p x r :=
{ r_le := hf.r_le
r_pos := hf.r_pos
hasSum := fun {y} hy => by
convert hf.hasSum hy using 1
apply hg.symm
simpa [edist_eq_coe_nnnorm_sub] using hy }
#align has_fpower_series_on_ball.congr HasFPowerSeriesOnBall.congr
/-- If a function `f` has a power series `p` around `x`, then the function `z ↦ f (z - y)` has the
same power series around `x + y`. -/
theorem HasFPowerSeriesOnBall.comp_sub (hf : HasFPowerSeriesOnBall f p x r) (y : E) :
HasFPowerSeriesOnBall (fun z => f (z - y)) p (x + y) r :=
{ r_le := hf.r_le
r_pos := hf.r_pos
hasSum := fun {z} hz => by
convert hf.hasSum hz using 2
abel }
#align has_fpower_series_on_ball.comp_sub HasFPowerSeriesOnBall.comp_sub
theorem HasFPowerSeriesOnBall.hasSum_sub (hf : HasFPowerSeriesOnBall f p x r) {y : E}
(hy : y ∈ EMetric.ball x r) : HasSum (fun n : ℕ => p n fun _ => y - x) (f y) := by
have : y - x ∈ EMetric.ball (0 : E) r := by simpa [edist_eq_coe_nnnorm_sub] using hy
simpa only [add_sub_cancel'_right] using hf.hasSum this
#align has_fpower_series_on_ball.has_sum_sub HasFPowerSeriesOnBall.hasSum_sub
theorem HasFPowerSeriesOnBall.radius_pos (hf : HasFPowerSeriesOnBall f p x r) : 0 < p.radius :=
lt_of_lt_of_le hf.r_pos hf.r_le
#align has_fpower_series_on_ball.radius_pos HasFPowerSeriesOnBall.radius_pos
theorem HasFPowerSeriesAt.radius_pos (hf : HasFPowerSeriesAt f p x) : 0 < p.radius :=
let ⟨_, hr⟩ := hf
hr.radius_pos
#align has_fpower_series_at.radius_pos HasFPowerSeriesAt.radius_pos
theorem HasFPowerSeriesOnBall.mono (hf : HasFPowerSeriesOnBall f p x r) (r'_pos : 0 < r')
(hr : r' ≤ r) : HasFPowerSeriesOnBall f p x r' :=
⟨le_trans hr hf.1, r'_pos, fun hy => hf.hasSum (EMetric.ball_subset_ball hr hy)⟩
#align has_fpower_series_on_ball.mono HasFPowerSeriesOnBall.mono
theorem HasFPowerSeriesAt.congr (hf : HasFPowerSeriesAt f p x) (hg : f =ᶠ[𝓝 x] g) :
HasFPowerSeriesAt g p x := by
rcases hf with ⟨r₁, h₁⟩
rcases EMetric.mem_nhds_iff.mp hg with ⟨r₂, h₂pos, h₂⟩
exact ⟨min r₁ r₂,
(h₁.mono (lt_min h₁.r_pos h₂pos) inf_le_left).congr
fun y hy => h₂ (EMetric.ball_subset_ball inf_le_right hy)⟩
#align has_fpower_series_at.congr HasFPowerSeriesAt.congr
protected theorem HasFPowerSeriesAt.eventually (hf : HasFPowerSeriesAt f p x) :
∀ᶠ r : ℝ≥0∞ in 𝓝[>] 0, HasFPowerSeriesOnBall f p x r :=
let ⟨_, hr⟩ := hf
mem_of_superset (Ioo_mem_nhdsWithin_Ioi (left_mem_Ico.2 hr.r_pos)) fun _ hr' =>
hr.mono hr'.1 hr'.2.le
#align has_fpower_series_at.eventually HasFPowerSeriesAt.eventually
theorem HasFPowerSeriesOnBall.eventually_hasSum (hf : HasFPowerSeriesOnBall f p x r) :
∀ᶠ y in 𝓝 0, HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y)) := by
filter_upwards [EMetric.ball_mem_nhds (0 : E) hf.r_pos] using fun _ => hf.hasSum
#align has_fpower_series_on_ball.eventually_has_sum HasFPowerSeriesOnBall.eventually_hasSum
theorem HasFPowerSeriesAt.eventually_hasSum (hf : HasFPowerSeriesAt f p x) :
∀ᶠ y in 𝓝 0, HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y)) :=
let ⟨_, hr⟩ := hf
hr.eventually_hasSum
#align has_fpower_series_at.eventually_has_sum HasFPowerSeriesAt.eventually_hasSum
theorem HasFPowerSeriesOnBall.eventually_hasSum_sub (hf : HasFPowerSeriesOnBall f p x r) :
∀ᶠ y in 𝓝 x, HasSum (fun n : ℕ => p n fun _ : Fin n => y - x) (f y) := by
filter_upwards [EMetric.ball_mem_nhds x hf.r_pos] with y using hf.hasSum_sub
#align has_fpower_series_on_ball.eventually_has_sum_sub HasFPowerSeriesOnBall.eventually_hasSum_sub
theorem HasFPowerSeriesAt.eventually_hasSum_sub (hf : HasFPowerSeriesAt f p x) :
∀ᶠ y in 𝓝 x, HasSum (fun n : ℕ => p n fun _ : Fin n => y - x) (f y) :=
let ⟨_, hr⟩ := hf
hr.eventually_hasSum_sub
#align has_fpower_series_at.eventually_has_sum_sub HasFPowerSeriesAt.eventually_hasSum_sub
theorem HasFPowerSeriesOnBall.eventually_eq_zero
(hf : HasFPowerSeriesOnBall f (0 : FormalMultilinearSeries 𝕜 E F) x r) :
∀ᶠ z in 𝓝 x, f z = 0 := by
filter_upwards [hf.eventually_hasSum_sub] with z hz using hz.unique hasSum_zero
#align has_fpower_series_on_ball.eventually_eq_zero HasFPowerSeriesOnBall.eventually_eq_zero
theorem HasFPowerSeriesAt.eventually_eq_zero
(hf : HasFPowerSeriesAt f (0 : FormalMultilinearSeries 𝕜 E F) x) : ∀ᶠ z in 𝓝 x, f z = 0 :=
let ⟨_, hr⟩ := hf
hr.eventually_eq_zero
#align has_fpower_series_at.eventually_eq_zero HasFPowerSeriesAt.eventually_eq_zero
theorem hasFPowerSeriesOnBall_const {c : F} {e : E} :
HasFPowerSeriesOnBall (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e ⊤ := by
refine' ⟨by simp, WithTop.zero_lt_top, fun _ => hasSum_single 0 fun n hn => _⟩
simp [constFormalMultilinearSeries_apply hn]
#align has_fpower_series_on_ball_const hasFPowerSeriesOnBall_const
theorem hasFPowerSeriesAt_const {c : F} {e : E} :
HasFPowerSeriesAt (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e :=
⟨⊤, hasFPowerSeriesOnBall_const⟩
#align has_fpower_series_at_const hasFPowerSeriesAt_const
theorem analyticAt_const {v : F} : AnalyticAt 𝕜 (fun _ => v) x :=
⟨constFormalMultilinearSeries 𝕜 E v, hasFPowerSeriesAt_const⟩
#align analytic_at_const analyticAt_const
theorem analyticOn_const {v : F} {s : Set E} : AnalyticOn 𝕜 (fun _ => v) s :=
fun _ _ => analyticAt_const
#align analytic_on_const analyticOn_const
theorem HasFPowerSeriesOnBall.add (hf : HasFPowerSeriesOnBall f pf x r)
(hg : HasFPowerSeriesOnBall g pg x r) : HasFPowerSeriesOnBall (f + g) (pf + pg) x r :=
{ r_le := le_trans (le_min_iff.2 ⟨hf.r_le, hg.r_le⟩) (pf.min_radius_le_radius_add pg)
r_pos := hf.r_pos
hasSum := fun hy => (hf.hasSum hy).add (hg.hasSum hy) }
#align has_fpower_series_on_ball.add HasFPowerSeriesOnBall.add
theorem HasFPowerSeriesAt.add (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x) :
HasFPowerSeriesAt (f + g) (pf + pg) x := by
rcases (hf.eventually.and hg.eventually).exists with ⟨r, hr⟩
exact ⟨r, hr.1.add hr.2⟩
#align has_fpower_series_at.add HasFPowerSeriesAt.add
theorem AnalyticAt.congr (hf : AnalyticAt 𝕜 f x) (hg : f =ᶠ[𝓝 x] g) : AnalyticAt 𝕜 g x :=
let ⟨_, hpf⟩ := hf
(hpf.congr hg).analyticAt
theorem analyticAt_congr (h : f =ᶠ[𝓝 x] g) : AnalyticAt 𝕜 f x ↔ AnalyticAt 𝕜 g x :=
⟨fun hf ↦ hf.congr h, fun hg ↦ hg.congr h.symm⟩
theorem AnalyticAt.add (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) : AnalyticAt 𝕜 (f + g) x :=
let ⟨_, hpf⟩ := hf
let ⟨_, hqf⟩ := hg
(hpf.add hqf).analyticAt
#align analytic_at.add AnalyticAt.add
theorem HasFPowerSeriesOnBall.neg (hf : HasFPowerSeriesOnBall f pf x r) :
HasFPowerSeriesOnBall (-f) (-pf) x r :=
{ r_le := by
rw [pf.radius_neg]
exact hf.r_le
r_pos := hf.r_pos
hasSum := fun hy => (hf.hasSum hy).neg }
#align has_fpower_series_on_ball.neg HasFPowerSeriesOnBall.neg
theorem HasFPowerSeriesAt.neg (hf : HasFPowerSeriesAt f pf x) : HasFPowerSeriesAt (-f) (-pf) x :=
let ⟨_, hrf⟩ := hf
hrf.neg.hasFPowerSeriesAt
#align has_fpower_series_at.neg HasFPowerSeriesAt.neg
theorem AnalyticAt.neg (hf : AnalyticAt 𝕜 f x) : AnalyticAt 𝕜 (-f) x :=
let ⟨_, hpf⟩ := hf
hpf.neg.analyticAt
#align analytic_at.neg AnalyticAt.neg
theorem HasFPowerSeriesOnBall.sub (hf : HasFPowerSeriesOnBall f pf x r)
(hg : HasFPowerSeriesOnBall g pg x r) : HasFPowerSeriesOnBall (f - g) (pf - pg) x r := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align has_fpower_series_on_ball.sub HasFPowerSeriesOnBall.sub
theorem HasFPowerSeriesAt.sub (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x) :
HasFPowerSeriesAt (f - g) (pf - pg) x := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align has_fpower_series_at.sub HasFPowerSeriesAt.sub
theorem AnalyticAt.sub (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) :
AnalyticAt 𝕜 (f - g) x := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align analytic_at.sub AnalyticAt.sub
theorem AnalyticOn.mono {s t : Set E} (hf : AnalyticOn 𝕜 f t) (hst : s ⊆ t) : AnalyticOn 𝕜 f s :=
fun z hz => hf z (hst hz)
#align analytic_on.mono AnalyticOn.mono
theorem AnalyticOn.congr' {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : f =ᶠ[𝓝ˢ s] g) :
AnalyticOn 𝕜 g s :=
fun z hz => (hf z hz).congr (mem_nhdsSet_iff_forall.mp hg z hz)
theorem analyticOn_congr' {s : Set E} (h : f =ᶠ[𝓝ˢ s] g) : AnalyticOn 𝕜 f s ↔ AnalyticOn 𝕜 g s :=
⟨fun hf => hf.congr' h, fun hg => hg.congr' h.symm⟩
theorem AnalyticOn.congr {s : Set E} (hs : IsOpen s) (hf : AnalyticOn 𝕜 f s) (hg : s.EqOn f g) :
AnalyticOn 𝕜 g s :=
hf.congr' $ mem_nhdsSet_iff_forall.mpr
(fun _ hz => eventuallyEq_iff_exists_mem.mpr ⟨s, hs.mem_nhds hz, hg⟩)
theorem analyticOn_congr {s : Set E} (hs : IsOpen s) (h : s.EqOn f g) : AnalyticOn 𝕜 f s ↔
AnalyticOn 𝕜 g s := ⟨fun hf => hf.congr hs h, fun hg => hg.congr hs h.symm⟩
theorem AnalyticOn.add {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
AnalyticOn 𝕜 (f + g) s :=
fun z hz => (hf z hz).add (hg z hz)
#align analytic_on.add AnalyticOn.add
theorem AnalyticOn.sub {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
AnalyticOn 𝕜 (f - g) s :=
fun z hz => (hf z hz).sub (hg z hz)
#align analytic_on.sub AnalyticOn.sub
theorem HasFPowerSeriesOnBall.coeff_zero (hf : HasFPowerSeriesOnBall f pf x r) (v : Fin 0 → E) :
pf 0 v = f x := by
have v_eq : v = fun i => 0 := Subsingleton.elim _ _
have zero_mem : (0 : E) ∈ EMetric.ball (0 : E) r := by simp [hf.r_pos]
have : ∀ i, i ≠ 0 → (pf i fun j => 0) = 0 := by
intro i hi
have : 0 < i := pos_iff_ne_zero.2 hi
exact ContinuousMultilinearMap.map_coord_zero _ (⟨0, this⟩ : Fin i) rfl
have A := (hf.hasSum zero_mem).unique (hasSum_single _ this)
simpa [v_eq] using A.symm
#align has_fpower_series_on_ball.coeff_zero HasFPowerSeriesOnBall.coeff_zero
theorem HasFPowerSeriesAt.coeff_zero (hf : HasFPowerSeriesAt f pf x) (v : Fin 0 → E) :
pf 0 v = f x :=
let ⟨_, hrf⟩ := hf
hrf.coeff_zero v
#align has_fpower_series_at.coeff_zero HasFPowerSeriesAt.coeff_zero
/-- If a function `f` has a power series `p` on a ball and `g` is linear, then `g ∘ f` has the
power series `g ∘ p` on the same ball. -/
theorem ContinuousLinearMap.comp_hasFPowerSeriesOnBall (g : F →L[𝕜] G)
(h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (g ∘ f) (g.compFormalMultilinearSeries p) x r :=
{ r_le := h.r_le.trans (p.radius_le_radius_continuousLinearMap_comp _)
r_pos := h.r_pos
hasSum := fun hy => by
simpa only [ContinuousLinearMap.compFormalMultilinearSeries_apply,
ContinuousLinearMap.compContinuousMultilinearMap_coe, Function.comp_apply] using
g.hasSum (h.hasSum hy) }
#align continuous_linear_map.comp_has_fpower_series_on_ball ContinuousLinearMap.comp_hasFPowerSeriesOnBall
/-- If a function `f` is analytic on a set `s` and `g` is linear, then `g ∘ f` is analytic
on `s`. -/
theorem ContinuousLinearMap.comp_analyticOn {s : Set E} (g : F →L[𝕜] G) (h : AnalyticOn 𝕜 f s) :
AnalyticOn 𝕜 (g ∘ f) s := by
rintro x hx
rcases h x hx with ⟨p, r, hp⟩
exact ⟨g.compFormalMultilinearSeries p, r, g.comp_hasFPowerSeriesOnBall hp⟩
#align continuous_linear_map.comp_analytic_on ContinuousLinearMap.comp_analyticOn
/-- If a function admits a power series expansion, then it is exponentially close to the partial
sums of this power series on strict subdisks of the disk of convergence.
This version provides an upper estimate that decreases both in `‖y‖` and `n`. See also
`HasFPowerSeriesOnBall.uniform_geometric_approx` for a weaker version. -/
theorem HasFPowerSeriesOnBall.uniform_geometric_approx' {r' : ℝ≥0}
(hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n := by
obtain ⟨a, ha, C, hC, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n, ‖p n‖ * (r' : ℝ) ^ n ≤ C * a ^ n :=
p.norm_mul_pow_le_mul_pow_of_lt_radius (h.trans_le hf.r_le)
refine' ⟨a, ha, C / (1 - a), div_pos hC (sub_pos.2 ha.2), fun y hy n => _⟩
have yr' : ‖y‖ < r' := by
rw [ball_zero_eq] at hy
exact hy
have hr'0 : 0 < (r' : ℝ) := (norm_nonneg _).trans_lt yr'
have : y ∈ EMetric.ball (0 : E) r := by
refine' mem_emetric_ball_zero_iff.2 (lt_trans _ h)
exact mod_cast yr'
rw [norm_sub_rev, ← mul_div_right_comm]
have ya : a * (‖y‖ / ↑r') ≤ a :=
mul_le_of_le_one_right ha.1.le (div_le_one_of_le yr'.le r'.coe_nonneg)
suffices ‖p.partialSum n y - f (x + y)‖ ≤ C * (a * (‖y‖ / r')) ^ n / (1 - a * (‖y‖ / r')) by
refine' this.trans _
have : 0 < a := ha.1
gcongr
apply_rules [sub_pos.2, ha.2]
apply norm_sub_le_of_geometric_bound_of_hasSum (ya.trans_lt ha.2) _ (hf.hasSum this)
intro n
calc
‖(p n) fun _ : Fin n => y‖
_ ≤ ‖p n‖ * ∏ _i : Fin n, ‖y‖ := ContinuousMultilinearMap.le_op_norm _ _
_ = ‖p n‖ * (r' : ℝ) ^ n * (‖y‖ / r') ^ n := by field_simp [mul_right_comm]
_ ≤ C * a ^ n * (‖y‖ / r') ^ n := by gcongr ?_ * _; apply hp
_ ≤ C * (a * (‖y‖ / r')) ^ n := by rw [mul_pow, mul_assoc]
#align has_fpower_series_on_ball.uniform_geometric_approx' HasFPowerSeriesOnBall.uniform_geometric_approx'
/-- If a function admits a power series expansion, then it is exponentially close to the partial
sums of this power series on strict subdisks of the disk of convergence. -/
theorem HasFPowerSeriesOnBall.uniform_geometric_approx {r' : ℝ≥0}
(hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
∃ a ∈ Ioo (0 : ℝ) 1,
∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n, ‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n := by
obtain ⟨a, ha, C, hC, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n :=
hf.uniform_geometric_approx' h
refine' ⟨a, ha, C, hC, fun y hy n => (hp y hy n).trans _⟩
have yr' : ‖y‖ < r' := by rwa [ball_zero_eq] at hy
gcongr
exacts [mul_nonneg ha.1.le (div_nonneg (norm_nonneg y) r'.coe_nonneg),
mul_le_of_le_one_right ha.1.le (div_le_one_of_le yr'.le r'.coe_nonneg)]
#align has_fpower_series_on_ball.uniform_geometric_approx HasFPowerSeriesOnBall.uniform_geometric_approx
/-- Taylor formula for an analytic function, `IsBigO` version. -/
theorem HasFPowerSeriesAt.isBigO_sub_partialSum_pow (hf : HasFPowerSeriesAt f p x) (n : ℕ) :
(fun y : E => f (x + y) - p.partialSum n y) =O[𝓝 0] fun y => ‖y‖ ^ n := by
rcases hf with ⟨r, hf⟩
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hf.r_pos with ⟨r', r'0, h⟩
obtain ⟨a, -, C, -, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n :=
hf.uniform_geometric_approx' h
refine' isBigO_iff.2 ⟨C * (a / r') ^ n, _⟩
replace r'0 : 0 < (r' : ℝ); · exact mod_cast r'0
filter_upwards [Metric.ball_mem_nhds (0 : E) r'0] with y hy
simpa [mul_pow, mul_div_assoc, mul_assoc, div_mul_eq_mul_div] using hp y hy n
set_option linter.uppercaseLean3 false in
#align has_fpower_series_at.is_O_sub_partial_sum_pow HasFPowerSeriesAt.isBigO_sub_partialSum_pow
/-- If `f` has formal power series `∑ n, pₙ` on a ball of radius `r`, then for `y, z` in any smaller
ball, the norm of the difference `f y - f z - p 1 (fun _ ↦ y - z)` is bounded above by
`C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖`. This lemma formulates this property using `IsBigO` and
`Filter.principal` on `E × E`. -/
theorem HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal
(hf : HasFPowerSeriesOnBall f p x r) (hr : r' < r) :
(fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) =O[𝓟 (EMetric.ball (x, x) r')]
fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖ := by
lift r' to ℝ≥0 using ne_top_of_lt hr
rcases (zero_le r').eq_or_lt with (rfl | hr'0)
· simp only [isBigO_bot, EMetric.ball_zero, principal_empty, ENNReal.coe_zero]
obtain ⟨a, ha, C, hC : 0 < C, hp⟩ :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n : ℕ, ‖p n‖ * (r' : ℝ) ^ n ≤ C * a ^ n
exact p.norm_mul_pow_le_mul_pow_of_lt_radius (hr.trans_le hf.r_le)
simp only [← le_div_iff (pow_pos (NNReal.coe_pos.2 hr'0) _)] at hp
set L : E × E → ℝ := fun y =>
C * (a / r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * (a / (1 - a) ^ 2 + 2 / (1 - a))
have hL : ∀ y ∈ EMetric.ball (x, x) r', ‖f y.1 - f y.2 - p 1 fun _ => y.1 - y.2‖ ≤ L y := by
intro y hy'
have hy : y ∈ EMetric.ball x r ×ˢ EMetric.ball x r := by
rw [EMetric.ball_prod_same]
exact EMetric.ball_subset_ball hr.le hy'
set A : ℕ → F := fun n => (p n fun _ => y.1 - x) - p n fun _ => y.2 - x
have hA : HasSum (fun n => A (n + 2)) (f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) := by
convert (hasSum_nat_add_iff' 2).2 ((hf.hasSum_sub hy.1).sub (hf.hasSum_sub hy.2)) using 1
rw [Finset.sum_range_succ, Finset.sum_range_one, hf.coeff_zero, hf.coeff_zero, sub_self,
zero_add, ← Subsingleton.pi_single_eq (0 : Fin 1) (y.1 - x), Pi.single,
← Subsingleton.pi_single_eq (0 : Fin 1) (y.2 - x), Pi.single, ← (p 1).map_sub, ← Pi.single,
Subsingleton.pi_single_eq, sub_sub_sub_cancel_right]
rw [EMetric.mem_ball, edist_eq_coe_nnnorm_sub, ENNReal.coe_lt_coe] at hy'
set B : ℕ → ℝ := fun n => C * (a / r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * ((n + 2) * a ^ n)
have hAB : ∀ n, ‖A (n + 2)‖ ≤ B n := fun n =>
calc
‖A (n + 2)‖ ≤ ‖p (n + 2)‖ * ↑(n + 2) * ‖y - (x, x)‖ ^ (n + 1) * ‖y.1 - y.2‖ := by
-- porting note: `pi_norm_const` was `pi_norm_const (_ : E)`
simpa only [Fintype.card_fin, pi_norm_const, Prod.norm_def, Pi.sub_def,
Prod.fst_sub, Prod.snd_sub, sub_sub_sub_cancel_right] using
(p <| n + 2).norm_image_sub_le (fun _ => y.1 - x) fun _ => y.2 - x
_ = ‖p (n + 2)‖ * ‖y - (x, x)‖ ^ n * (↑(n + 2) * ‖y - (x, x)‖ * ‖y.1 - y.2‖) := by
rw [pow_succ ‖y - (x, x)‖]
ring
-- porting note: the two `↑` in `↑r'` are new, without them, Lean fails to synthesize
-- instances `HDiv ℝ ℝ≥0 ?m` or `HMul ℝ ℝ≥0 ?m`
_ ≤ C * a ^ (n + 2) / ↑r' ^ (n + 2)
* ↑r' ^ n * (↑(n + 2) * ‖y - (x, x)‖ * ‖y.1 - y.2‖) := by
have : 0 < a := ha.1
gcongr
· apply hp
· apply hy'.le
_ = B n := by
-- porting note: in the original, `B` was in the `field_simp`, but now Lean does not
-- accept it. The current proof works in Lean 4, but does not in Lean 3.
field_simp [pow_succ]
simp only [mul_assoc, mul_comm, mul_left_comm]
have hBL : HasSum B (L y) := by
apply HasSum.mul_left
simp only [add_mul]
have : ‖a‖ < 1 := by simp only [Real.norm_eq_abs, abs_of_pos ha.1, ha.2]
rw [div_eq_mul_inv, div_eq_mul_inv]
exact (hasSum_coe_mul_geometric_of_norm_lt_1 this).add -- porting note: was `convert`!
((hasSum_geometric_of_norm_lt_1 this).mul_left 2)
exact hA.norm_le_of_bounded hBL hAB
suffices L =O[𝓟 (EMetric.ball (x, x) r')] fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖ by
refine' (IsBigO.of_bound 1 (eventually_principal.2 fun y hy => _)).trans this
rw [one_mul]
exact (hL y hy).trans (le_abs_self _)
simp_rw [mul_right_comm _ (_ * _)] -- porting note: there was an `L` inside the `simp_rw`.
exact (isBigO_refl _ _).const_mul_left _
set_option linter.uppercaseLean3 false in
#align has_fpower_series_on_ball.is_O_image_sub_image_sub_deriv_principal HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal
/-- If `f` has formal power series `∑ n, pₙ` on a ball of radius `r`, then for `y, z` in any smaller
ball, the norm of the difference `f y - f z - p 1 (fun _ ↦ y - z)` is bounded above by
`C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖`. -/
theorem HasFPowerSeriesOnBall.image_sub_sub_deriv_le (hf : HasFPowerSeriesOnBall f p x r)
(hr : r' < r) :
∃ C, ∀ᵉ (y ∈ EMetric.ball x r') (z ∈ EMetric.ball x r'),
‖f y - f z - p 1 fun _ => y - z‖ ≤ C * max ‖y - x‖ ‖z - x‖ * ‖y - z‖ := by
simpa only [isBigO_principal, mul_assoc, norm_mul, norm_norm, Prod.forall, EMetric.mem_ball,
Prod.edist_eq, max_lt_iff, and_imp, @forall_swap (_ < _) E] using
hf.isBigO_image_sub_image_sub_deriv_principal hr
#align has_fpower_series_on_ball.image_sub_sub_deriv_le HasFPowerSeriesOnBall.image_sub_sub_deriv_le
/-- If `f` has formal power series `∑ n, pₙ` at `x`, then
`f y - f z - p 1 (fun _ ↦ y - z) = O(‖(y, z) - (x, x)‖ * ‖y - z‖)` as `(y, z) → (x, x)`.
In particular, `f` is strictly differentiable at `x`. -/
theorem HasFPowerSeriesAt.isBigO_image_sub_norm_mul_norm_sub (hf : HasFPowerSeriesAt f p x) :
(fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) =O[𝓝 (x, x)] fun y =>
‖y - (x, x)‖ * ‖y.1 - y.2‖ := by
rcases hf with ⟨r, hf⟩
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hf.r_pos with ⟨r', r'0, h⟩
refine' (hf.isBigO_image_sub_image_sub_deriv_principal h).mono _
exact le_principal_iff.2 (EMetric.ball_mem_nhds _ r'0)
set_option linter.uppercaseLean3 false in
#align has_fpower_series_at.is_O_image_sub_norm_mul_norm_sub HasFPowerSeriesAt.isBigO_image_sub_norm_mul_norm_sub
/-- If a function admits a power series expansion at `x`, then it is the uniform limit of the
partial sums of this power series on strict subdisks of the disk of convergence, i.e., `f (x + y)`
is the uniform limit of `p.partialSum n y` there. -/
theorem HasFPowerSeriesOnBall.tendstoUniformlyOn {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r)
(h : (r' : ℝ≥0∞) < r) :
TendstoUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
(Metric.ball (0 : E) r') := by
obtain ⟨a, ha, C, -, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n
exact hf.uniform_geometric_approx h
refine' Metric.tendstoUniformlyOn_iff.2 fun ε εpos => _
have L : Tendsto (fun n => (C : ℝ) * a ^ n) atTop (𝓝 ((C : ℝ) * 0)) :=
tendsto_const_nhds.mul (tendsto_pow_atTop_nhds_0_of_lt_1 ha.1.le ha.2)
rw [mul_zero] at L
refine' (L.eventually (gt_mem_nhds εpos)).mono fun n hn y hy => _
rw [dist_eq_norm]
exact (hp y hy n).trans_lt hn
#align has_fpower_series_on_ball.tendsto_uniformly_on HasFPowerSeriesOnBall.tendstoUniformlyOn
/-- If a function admits a power series expansion at `x`, then it is the locally uniform limit of
the partial sums of this power series on the disk of convergence, i.e., `f (x + y)`
is the locally uniform limit of `p.partialSum n y` there. -/
theorem HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn (hf : HasFPowerSeriesOnBall f p x r) :
TendstoLocallyUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
(EMetric.ball (0 : E) r) := by
intro u hu x hx
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hx with ⟨r', xr', hr'⟩
have : EMetric.ball (0 : E) r' ∈ 𝓝 x := IsOpen.mem_nhds EMetric.isOpen_ball xr'
refine' ⟨EMetric.ball (0 : E) r', mem_nhdsWithin_of_mem_nhds this, _⟩
simpa [Metric.emetric_ball_nnreal] using hf.tendstoUniformlyOn hr' u hu
#align has_fpower_series_on_ball.tendsto_locally_uniformly_on HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn
/-- If a function admits a power series expansion at `x`, then it is the uniform limit of the
partial sums of this power series on strict subdisks of the disk of convergence, i.e., `f y`
is the uniform limit of `p.partialSum n (y - x)` there. -/
theorem HasFPowerSeriesOnBall.tendstoUniformlyOn' {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r)
(h : (r' : ℝ≥0∞) < r) :
TendstoUniformlyOn (fun n y => p.partialSum n (y - x)) f atTop (Metric.ball (x : E) r') := by
convert (hf.tendstoUniformlyOn h).comp fun y => y - x using 1
· simp [(· ∘ ·)]
· ext z
simp [dist_eq_norm]
#align has_fpower_series_on_ball.tendsto_uniformly_on' HasFPowerSeriesOnBall.tendstoUniformlyOn'
/-- If a function admits a power series expansion at `x`, then it is the locally uniform limit of
the partial sums of this power series on the disk of convergence, i.e., `f y`
is the locally uniform limit of `p.partialSum n (y - x)` there. -/
theorem HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn' (hf : HasFPowerSeriesOnBall f p x r) :
TendstoLocallyUniformlyOn (fun n y => p.partialSum n (y - x)) f atTop
(EMetric.ball (x : E) r) := by
have A : ContinuousOn (fun y : E => y - x) (EMetric.ball (x : E) r) :=
(continuous_id.sub continuous_const).continuousOn
convert hf.tendstoLocallyUniformlyOn.comp (fun y : E => y - x) _ A using 1
· ext z
simp
· intro z
simp [edist_eq_coe_nnnorm, edist_eq_coe_nnnorm_sub]
#align has_fpower_series_on_ball.tendsto_locally_uniformly_on' HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn'
/-- If a function admits a power series expansion on a disk, then it is continuous there. -/
protected theorem HasFPowerSeriesOnBall.continuousOn (hf : HasFPowerSeriesOnBall f p x r) :
ContinuousOn f (EMetric.ball x r) :=
hf.tendstoLocallyUniformlyOn'.continuousOn <|
eventually_of_forall fun n =>
((p.partialSum_continuous n).comp (continuous_id.sub continuous_const)).continuousOn
#align has_fpower_series_on_ball.continuous_on HasFPowerSeriesOnBall.continuousOn
protected theorem HasFPowerSeriesAt.continuousAt (hf : HasFPowerSeriesAt f p x) :
ContinuousAt f x :=
let ⟨_, hr⟩ := hf
hr.continuousOn.continuousAt (EMetric.ball_mem_nhds x hr.r_pos)
#align has_fpower_series_at.continuous_at HasFPowerSeriesAt.continuousAt
protected theorem AnalyticAt.continuousAt (hf : AnalyticAt 𝕜 f x) : ContinuousAt f x :=
let ⟨_, hp⟩ := hf
hp.continuousAt
#align analytic_at.continuous_at AnalyticAt.continuousAt
protected theorem AnalyticOn.continuousOn {s : Set E} (hf : AnalyticOn 𝕜 f s) : ContinuousOn f s :=
fun x hx => (hf x hx).continuousAt.continuousWithinAt
#align analytic_on.continuous_on AnalyticOn.continuousOn
/-- Analytic everywhere implies continuous -/
theorem AnalyticOn.continuous {f : E → F} (fa : AnalyticOn 𝕜 f univ) : Continuous f := by
rw [continuous_iff_continuousOn_univ]; exact fa.continuousOn
/-- In a complete space, the sum of a converging power series `p` admits `p` as a power series.
This is not totally obvious as we need to check the convergence of the series. -/
protected theorem FormalMultilinearSeries.hasFPowerSeriesOnBall [CompleteSpace F]
(p : FormalMultilinearSeries 𝕜 E F) (h : 0 < p.radius) :
HasFPowerSeriesOnBall p.sum p 0 p.radius :=
{ r_le := le_rfl
r_pos := h
hasSum := fun hy => by
rw [zero_add]
exact p.hasSum hy }
#align formal_multilinear_series.has_fpower_series_on_ball FormalMultilinearSeries.hasFPowerSeriesOnBall
theorem HasFPowerSeriesOnBall.sum (h : HasFPowerSeriesOnBall f p x r) {y : E}
(hy : y ∈ EMetric.ball (0 : E) r) : f (x + y) = p.sum y :=
(h.hasSum hy).tsum_eq.symm
#align has_fpower_series_on_ball.sum HasFPowerSeriesOnBall.sum
/-- The sum of a converging power series is continuous in its disk of convergence. -/
protected theorem FormalMultilinearSeries.continuousOn [CompleteSpace F] :
ContinuousOn p.sum (EMetric.ball 0 p.radius) := by
rcases (zero_le p.radius).eq_or_lt with h | h
· simp [← h, continuousOn_empty]
· exact (p.hasFPowerSeriesOnBall h).continuousOn
#align formal_multilinear_series.continuous_on FormalMultilinearSeries.continuousOn
end
/-!
### Uniqueness of power series
If a function `f : E → F` has two representations as power series at a point `x : E`, corresponding
to formal multilinear series `p₁` and `p₂`, then these representations agree term-by-term. That is,
for any `n : ℕ` and `y : E`, `p₁ n (fun i ↦ y) = p₂ n (fun i ↦ y)`. In the one-dimensional case,
when `f : 𝕜 → E`, the continuous multilinear maps `p₁ n` and `p₂ n` are given by
`ContinuousMultilinearMap.mkPiField`, and hence are determined completely by the value of
`p₁ n (fun i ↦ 1)`, so `p₁ = p₂`. Consequently, the radius of convergence for one series can be
transferred to the other.
-/
section Uniqueness
open ContinuousMultilinearMap
theorem Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero {n : ℕ} {p : E[×n]→L[𝕜] F}
(h : (fun y => p fun _ => y) =O[𝓝 0] fun y => ‖y‖ ^ (n + 1)) (y : E) : (p fun _ => y) = 0 := by
obtain ⟨c, c_pos, hc⟩ := h.exists_pos
obtain ⟨t, ht, t_open, z_mem⟩ := eventually_nhds_iff.mp (isBigOWith_iff.mp hc)
obtain ⟨δ, δ_pos, δε⟩ := (Metric.isOpen_iff.mp t_open) 0 z_mem
clear h hc z_mem
cases' n with n
· exact norm_eq_zero.mp (by
-- porting note: the symmetric difference of the `simpa only` sets:
-- added `Nat.zero_eq, zero_add, pow_one`
-- removed `zero_pow', Ne.def, Nat.one_ne_zero, not_false_iff`
simpa only [Nat.zero_eq, fin0_apply_norm, norm_eq_zero, norm_zero, zero_add, pow_one,
mul_zero, norm_le_zero_iff] using ht 0 (δε (Metric.mem_ball_self δ_pos)))
· refine' Or.elim (Classical.em (y = 0))
(fun hy => by simpa only [hy] using p.map_zero) fun hy => _
replace hy := norm_pos_iff.mpr hy
refine' norm_eq_zero.mp (le_antisymm (le_of_forall_pos_le_add fun ε ε_pos => _) (norm_nonneg _))
have h₀ := _root_.mul_pos c_pos (pow_pos hy (n.succ + 1))
obtain ⟨k, k_pos, k_norm⟩ := NormedField.exists_norm_lt 𝕜
(lt_min (mul_pos δ_pos (inv_pos.mpr hy)) (mul_pos ε_pos (inv_pos.mpr h₀)))
have h₁ : ‖k • y‖ < δ := by
rw [norm_smul]
exact inv_mul_cancel_right₀ hy.ne.symm δ ▸
mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_left _ _)) hy
have h₂ :=
calc
‖p fun _ => k • y‖ ≤ c * ‖k • y‖ ^ (n.succ + 1) := by
-- porting note: now Lean wants `_root_.`
simpa only [norm_pow, _root_.norm_norm] using ht (k • y) (δε (mem_ball_zero_iff.mpr h₁))
--simpa only [norm_pow, norm_norm] using ht (k • y) (δε (mem_ball_zero_iff.mpr h₁))
_ = ‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))) := by
-- porting note: added `Nat.succ_eq_add_one` since otherwise `ring` does not conclude.
simp only [norm_smul, mul_pow, Nat.succ_eq_add_one]
-- porting note: removed `rw [pow_succ]`, since it now becomes superfluous.
ring
have h₃ : ‖k‖ * (c * ‖y‖ ^ (n.succ + 1)) < ε :=
inv_mul_cancel_right₀ h₀.ne.symm ε ▸
mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_right _ _)) h₀
calc
‖p fun _ => y‖ = ‖k⁻¹ ^ n.succ‖ * ‖p fun _ => k • y‖ := by
simpa only [inv_smul_smul₀ (norm_pos_iff.mp k_pos), norm_smul, Finset.prod_const,
Finset.card_fin] using
congr_arg norm (p.map_smul_univ (fun _ : Fin n.succ => k⁻¹) fun _ : Fin n.succ => k • y)
_ ≤ ‖k⁻¹ ^ n.succ‖ * (‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1)))) := by gcongr
_ = ‖(k⁻¹ * k) ^ n.succ‖ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))) := by
rw [← mul_assoc]
simp [norm_mul, mul_pow]
_ ≤ 0 + ε := by
rw [inv_mul_cancel (norm_pos_iff.mp k_pos)]
simpa using h₃.le
set_option linter.uppercaseLean3 false in
#align asymptotics.is_O.continuous_multilinear_map_apply_eq_zero Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero
/-- If a formal multilinear series `p` represents the zero function at `x : E`, then the
terms `p n (fun i ↦ y)` appearing in the sum are zero for any `n : ℕ`, `y : E`. -/
theorem HasFPowerSeriesAt.apply_eq_zero {p : FormalMultilinearSeries 𝕜 E F} {x : E}
(h : HasFPowerSeriesAt 0 p x) (n : ℕ) : ∀ y : E, (p n fun _ => y) = 0 := by
refine' Nat.strong_induction_on n fun k hk => _
have psum_eq : p.partialSum (k + 1) = fun y => p k fun _ => y := by
funext z
refine' Finset.sum_eq_single _ (fun b hb hnb => _) fun hn => _
· have := Finset.mem_range_succ_iff.mp hb
simp only [hk b (this.lt_of_ne hnb), Pi.zero_apply]
· exact False.elim (hn (Finset.mem_range.mpr (lt_add_one k)))
replace h := h.isBigO_sub_partialSum_pow k.succ
simp only [psum_eq, zero_sub, Pi.zero_apply, Asymptotics.isBigO_neg_left] at h
exact h.continuousMultilinearMap_apply_eq_zero
#align has_fpower_series_at.apply_eq_zero HasFPowerSeriesAt.apply_eq_zero
/-- A one-dimensional formal multilinear series representing the zero function is zero. -/
theorem HasFPowerSeriesAt.eq_zero {p : FormalMultilinearSeries 𝕜 𝕜 E} {x : 𝕜}
(h : HasFPowerSeriesAt 0 p x) : p = 0 := by
-- porting note: `funext; ext` was `ext (n x)`
funext n
ext x
rw [← mkPiField_apply_one_eq_self (p n)]
-- porting note: nasty hack, was `simp [h.apply_eq_zero n 1]`
have := Or.intro_right ?_ (h.apply_eq_zero n 1)
simpa using this
#align has_fpower_series_at.eq_zero HasFPowerSeriesAt.eq_zero
/-- One-dimensional formal multilinear series representing the same function are equal. -/
theorem HasFPowerSeriesAt.eq_formalMultilinearSeries {p₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E}
{f : 𝕜 → E} {x : 𝕜} (h₁ : HasFPowerSeriesAt f p₁ x) (h₂ : HasFPowerSeriesAt f p₂ x) : p₁ = p₂ :=
sub_eq_zero.mp (HasFPowerSeriesAt.eq_zero (by simpa only [sub_self] using h₁.sub h₂))
#align has_fpower_series_at.eq_formal_multilinear_series HasFPowerSeriesAt.eq_formalMultilinearSeries
theorem HasFPowerSeriesAt.eq_formalMultilinearSeries_of_eventually
{p q : FormalMultilinearSeries 𝕜 𝕜 E} {f g : 𝕜 → E} {x : 𝕜} (hp : HasFPowerSeriesAt f p x)
(hq : HasFPowerSeriesAt g q x) (heq : ∀ᶠ z in 𝓝 x, f z = g z) : p = q :=
(hp.congr heq).eq_formalMultilinearSeries hq
#align has_fpower_series_at.eq_formal_multilinear_series_of_eventually HasFPowerSeriesAt.eq_formalMultilinearSeries_of_eventually
/-- A one-dimensional formal multilinear series representing a locally zero function is zero. -/
theorem HasFPowerSeriesAt.eq_zero_of_eventually {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E}
{x : 𝕜} (hp : HasFPowerSeriesAt f p x) (hf : f =ᶠ[𝓝 x] 0) : p = 0 :=
(hp.congr hf).eq_zero
#align has_fpower_series_at.eq_zero_of_eventually HasFPowerSeriesAt.eq_zero_of_eventually
/-- If a function `f : 𝕜 → E` has two power series representations at `x`, then the given radii in
which convergence is guaranteed may be interchanged. This can be useful when the formal multilinear
series in one representation has a particularly nice form, but the other has a larger radius. -/
theorem HasFPowerSeriesOnBall.exchange_radius {p₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E}
{r₁ r₂ : ℝ≥0∞} {x : 𝕜} (h₁ : HasFPowerSeriesOnBall f p₁ x r₁)
(h₂ : HasFPowerSeriesOnBall f p₂ x r₂) : HasFPowerSeriesOnBall f p₁ x r₂ :=
h₂.hasFPowerSeriesAt.eq_formalMultilinearSeries h₁.hasFPowerSeriesAt ▸ h₂
#align has_fpower_series_on_ball.exchange_radius HasFPowerSeriesOnBall.exchange_radius
/-- If a function `f : 𝕜 → E` has power series representation `p` on a ball of some radius and for
each positive radius it has some power series representation, then `p` converges to `f` on the whole
`𝕜`. -/
theorem HasFPowerSeriesOnBall.r_eq_top_of_exists {f : 𝕜 → E} {r : ℝ≥0∞} {x : 𝕜}
{p : FormalMultilinearSeries 𝕜 𝕜 E} (h : HasFPowerSeriesOnBall f p x r)
(h' : ∀ (r' : ℝ≥0) (_ : 0 < r'), ∃ p' : FormalMultilinearSeries 𝕜 𝕜 E,
HasFPowerSeriesOnBall f p' x r') :
HasFPowerSeriesOnBall f p x ∞ :=
{ r_le := ENNReal.le_of_forall_pos_nnreal_lt fun r hr _ =>
let ⟨_, hp'⟩ := h' r hr
(h.exchange_radius hp').r_le
r_pos := ENNReal.coe_lt_top
hasSum := fun {y} _ =>
let ⟨r', hr'⟩ := exists_gt ‖y‖₊
let ⟨_, hp'⟩ := h' r' hr'.ne_bot.bot_lt
(h.exchange_radius hp').hasSum <| mem_emetric_ball_zero_iff.mpr (ENNReal.coe_lt_coe.2 hr') }
#align has_fpower_series_on_ball.r_eq_top_of_exists HasFPowerSeriesOnBall.r_eq_top_of_exists
end Uniqueness
/-!
### Changing origin in a power series
If a function is analytic in a disk `D(x, R)`, then it is analytic in any disk contained in that
one. Indeed, one can write
$$
f (x + y + z) = \sum_{n} p_n (y + z)^n = \sum_{n, k} \binom{n}{k} p_n y^{n-k} z^k
= \sum_{k} \Bigl(\sum_{n} \binom{n}{k} p_n y^{n-k}\Bigr) z^k.
$$
The corresponding power series has thus a `k`-th coefficient equal to
$\sum_{n} \binom{n}{k} p_n y^{n-k}$. In the general case where `pₙ` is a multilinear map, this has
to be interpreted suitably: instead of having a binomial coefficient, one should sum over all
possible subsets `s` of `Fin n` of cardinal `k`, and attribute `z` to the indices in `s` and
`y` to the indices outside of `s`.
In this paragraph, we implement this. The new power series is called `p.changeOrigin y`. Then, we
check its convergence and the fact that its sum coincides with the original sum. The outcome of this
discussion is that the set of points where a function is analytic is open.
-/
namespace FormalMultilinearSeries
section
variable (p : FormalMultilinearSeries 𝕜 E F) {x y : E} {r R : ℝ≥0}
/-- A term of `FormalMultilinearSeries.changeOriginSeries`.
Given a formal multilinear series `p` and a point `x` in its ball of convergence,
`p.changeOrigin x` is a formal multilinear series such that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense. Each term of `p.changeOrigin x`
is itself an analytic function of `x` given by the series `p.changeOriginSeries`. Each term in
`changeOriginSeries` is the sum of `changeOriginSeriesTerm`'s over all `s` of cardinality `l`.
The definition is such that `p.changeOriginSeriesTerm k l s hs (fun _ ↦ x) (fun _ ↦ y) =
p (k + l) (s.piecewise (fun _ ↦ x) (fun _ ↦ y))`
-/
def changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
E[×l]→L[𝕜] E[×k]→L[𝕜] F := by
let a := ContinuousMultilinearMap.curryFinFinset 𝕜 E F hs
(by erw [Finset.card_compl, Fintype.card_fin, hs, add_tsub_cancel_right])
exact a (p (k + l))
#align formal_multilinear_series.change_origin_series_term FormalMultilinearSeries.changeOriginSeriesTerm
theorem changeOriginSeriesTerm_apply (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l)
(x y : E) :
(p.changeOriginSeriesTerm k l s hs (fun _ => x) fun _ => y) =
p (k + l) (s.piecewise (fun _ => x) fun _ => y) :=
ContinuousMultilinearMap.curryFinFinset_apply_const _ _ _ _ _
#align formal_multilinear_series.change_origin_series_term_apply FormalMultilinearSeries.changeOriginSeriesTerm_apply
@[simp]
theorem norm_changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
‖p.changeOriginSeriesTerm k l s hs‖ = ‖p (k + l)‖ := by
simp only [changeOriginSeriesTerm, LinearIsometryEquiv.norm_map]
#align formal_multilinear_series.norm_change_origin_series_term FormalMultilinearSeries.norm_changeOriginSeriesTerm
@[simp]
theorem nnnorm_changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
‖p.changeOriginSeriesTerm k l s hs‖₊ = ‖p (k + l)‖₊ := by
simp only [changeOriginSeriesTerm, LinearIsometryEquiv.nnnorm_map]
#align formal_multilinear_series.nnnorm_change_origin_series_term FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm
theorem nnnorm_changeOriginSeriesTerm_apply_le (k l : ℕ) (s : Finset (Fin (k + l)))
(hs : s.card = l) (x y : E) :
‖p.changeOriginSeriesTerm k l s hs (fun _ => x) fun _ => y‖₊ ≤
‖p (k + l)‖₊ * ‖x‖₊ ^ l * ‖y‖₊ ^ k := by
rw [← p.nnnorm_changeOriginSeriesTerm k l s hs, ← Fin.prod_const, ← Fin.prod_const]
apply ContinuousMultilinearMap.le_of_op_nnnorm_le
apply ContinuousMultilinearMap.le_op_nnnorm
#align formal_multilinear_series.nnnorm_change_origin_series_term_apply_le FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm_apply_le
/-- The power series for `f.changeOrigin k`.
Given a formal multilinear series `p` and a point `x` in its ball of convergence,
`p.changeOrigin x` is a formal multilinear series such that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense. Its `k`-th term is the sum of
the series `p.changeOriginSeries k`. -/
def changeOriginSeries (k : ℕ) : FormalMultilinearSeries 𝕜 E (E[×k]→L[𝕜] F) := fun l =>
∑ s : { s : Finset (Fin (k + l)) // Finset.card s = l }, p.changeOriginSeriesTerm k l s s.2
#align formal_multilinear_series.change_origin_series FormalMultilinearSeries.changeOriginSeries
theorem nnnorm_changeOriginSeries_le_tsum (k l : ℕ) :
‖p.changeOriginSeries k l‖₊ ≤
∑' _ : { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + l)‖₊ :=
(nnnorm_sum_le _ (fun t => changeOriginSeriesTerm p k l (Subtype.val t) t.prop)).trans_eq <| by
simp_rw [tsum_fintype, nnnorm_changeOriginSeriesTerm (p := p) (k := k) (l := l)]
#align formal_multilinear_series.nnnorm_change_origin_series_le_tsum FormalMultilinearSeries.nnnorm_changeOriginSeries_le_tsum
theorem nnnorm_changeOriginSeries_apply_le_tsum (k l : ℕ) (x : E) :
‖p.changeOriginSeries k l fun _ => x‖₊ ≤
∑' _ : { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + l)‖₊ * ‖x‖₊ ^ l := by
rw [NNReal.tsum_mul_right, ← Fin.prod_const]
exact (p.changeOriginSeries k l).le_of_op_nnnorm_le _ (p.nnnorm_changeOriginSeries_le_tsum _ _)
#align formal_multilinear_series.nnnorm_change_origin_series_apply_le_tsum FormalMultilinearSeries.nnnorm_changeOriginSeries_apply_le_tsum
/-- Changing the origin of a formal multilinear series `p`, so that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense.
-/
def changeOrigin (x : E) : FormalMultilinearSeries 𝕜 E F :=
fun k => (p.changeOriginSeries k).sum x
#align formal_multilinear_series.change_origin FormalMultilinearSeries.changeOrigin
/-- An auxiliary equivalence useful in the proofs about
`FormalMultilinearSeries.changeOriginSeries`: the set of triples `(k, l, s)`, where `s` is a
`Finset (Fin (k + l))` of cardinality `l` is equivalent to the set of pairs `(n, s)`, where `s` is a
`Finset (Fin n)`.
The forward map sends `(k, l, s)` to `(k + l, s)` and the inverse map sends `(n, s)` to
`(n - Finset.card s, Finset.card s, s)`. The actual definition is less readable because of problems
with non-definitional equalities. -/
@[simps]
def changeOriginIndexEquiv :
(Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) ≃ Σn : ℕ, Finset (Fin n) where
toFun s := ⟨s.1 + s.2.1, s.2.2⟩
invFun s :=
⟨s.1 - s.2.card, s.2.card,
⟨s.2.map
(Fin.castIso <| (tsub_add_cancel_of_le <| card_finset_fin_le s.2).symm).toEquiv.toEmbedding,
Finset.card_map _⟩⟩
left_inv := by
rintro ⟨k, l, ⟨s : Finset (Fin <| k + l), hs : s.card = l⟩⟩
dsimp only [Subtype.coe_mk]
-- Lean can't automatically generalize `k' = k + l - s.card`, `l' = s.card`, so we explicitly
-- formulate the generalized goal
suffices ∀ k' l', k' = k → l' = l → ∀ (hkl : k + l = k' + l') (hs'),
(⟨k', l', ⟨Finset.map (Fin.castIso hkl).toEquiv.toEmbedding s, hs'⟩⟩ :
Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) = ⟨k, l, ⟨s, hs⟩⟩ by
apply this <;> simp only [hs, add_tsub_cancel_right]
rintro _ _ rfl rfl hkl hs'
simp only [Equiv.refl_toEmbedding, Fin.castIso_refl, Finset.map_refl, eq_self_iff_true,
OrderIso.refl_toEquiv, and_self_iff, heq_iff_eq]
right_inv := by
rintro ⟨n, s⟩
simp [tsub_add_cancel_of_le (card_finset_fin_le s), Fin.castIso_to_equiv]
#align formal_multilinear_series.change_origin_index_equiv FormalMultilinearSeries.changeOriginIndexEquiv
theorem changeOriginSeries_summable_aux₁ {r r' : ℝ≥0} (hr : (r + r' : ℝ≥0∞) < p.radius) :
Summable fun s : Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l } =>
‖p (s.1 + s.2.1)‖₊ * r ^ s.2.1 * r' ^ s.1 := by
rw [← changeOriginIndexEquiv.symm.summable_iff]
dsimp only [Function.comp_def, changeOriginIndexEquiv_symm_apply_fst,
changeOriginIndexEquiv_symm_apply_snd_fst]
have : ∀ n : ℕ,
HasSum (fun s : Finset (Fin n) => ‖p (n - s.card + s.card)‖₊ * r ^ s.card * r' ^ (n - s.card))
(‖p n‖₊ * (r + r') ^ n) := by
intro n
-- TODO: why `simp only [tsub_add_cancel_of_le (card_finset_fin_le _)]` fails?
convert_to HasSum (fun s : Finset (Fin n) => ‖p n‖₊ * (r ^ s.card * r' ^ (n - s.card))) _
· ext1 s
rw [tsub_add_cancel_of_le (card_finset_fin_le _), mul_assoc]
rw [← Fin.sum_pow_mul_eq_add_pow]
exact (hasSum_fintype _).mul_left _
refine' NNReal.summable_sigma.2 ⟨fun n => (this n).summable, _⟩
simp only [(this _).tsum_eq]
exact p.summable_nnnorm_mul_pow hr
#align formal_multilinear_series.change_origin_series_summable_aux₁ FormalMultilinearSeries.changeOriginSeries_summable_aux₁
theorem changeOriginSeries_summable_aux₂ (hr : (r : ℝ≥0∞) < p.radius) (k : ℕ) :
Summable fun s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l } =>
‖p (k + s.1)‖₊ * r ^ s.1 := by
rcases ENNReal.lt_iff_exists_add_pos_lt.1 hr with ⟨r', h0, hr'⟩
simpa only [mul_inv_cancel_right₀ (pow_pos h0 _).ne'] using
((NNReal.summable_sigma.1 (p.changeOriginSeries_summable_aux₁ hr')).1 k).mul_right (r' ^ k)⁻¹
#align formal_multilinear_series.change_origin_series_summable_aux₂ FormalMultilinearSeries.changeOriginSeries_summable_aux₂
theorem changeOriginSeries_summable_aux₃ {r : ℝ≥0} (hr : ↑r < p.radius) (k : ℕ) :
Summable fun l : ℕ => ‖p.changeOriginSeries k l‖₊ * r ^ l := by
refine' NNReal.summable_of_le
(fun n => _) (NNReal.summable_sigma.1 <| p.changeOriginSeries_summable_aux₂ hr k).2
simp only [NNReal.tsum_mul_right]
exact mul_le_mul' (p.nnnorm_changeOriginSeries_le_tsum _ _) le_rfl
#align formal_multilinear_series.change_origin_series_summable_aux₃ FormalMultilinearSeries.changeOriginSeries_summable_aux₃
theorem le_changeOriginSeries_radius (k : ℕ) : p.radius ≤ (p.changeOriginSeries k).radius :=
ENNReal.le_of_forall_nnreal_lt fun _r hr =>
le_radius_of_summable_nnnorm _ (p.changeOriginSeries_summable_aux₃ hr k)
#align formal_multilinear_series.le_change_origin_series_radius FormalMultilinearSeries.le_changeOriginSeries_radius
theorem nnnorm_changeOrigin_le (k : ℕ) (h : (‖x‖₊ : ℝ≥0∞) < p.radius) :
‖p.changeOrigin x k‖₊ ≤
∑' s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + s.1)‖₊ * ‖x‖₊ ^ s.1 := by
refine' tsum_of_nnnorm_bounded _ fun l => p.nnnorm_changeOriginSeries_apply_le_tsum k l x
have := p.changeOriginSeries_summable_aux₂ h k
refine' HasSum.sigma this.hasSum fun l => _
exact ((NNReal.summable_sigma.1 this).1 l).hasSum
#align formal_multilinear_series.nnnorm_change_origin_le FormalMultilinearSeries.nnnorm_changeOrigin_le
/-- The radius of convergence of `p.changeOrigin x` is at least `p.radius - ‖x‖`. In other words,
`p.changeOrigin x` is well defined on the largest ball contained in the original ball of
convergence. -/
theorem changeOrigin_radius : p.radius - ‖x‖₊ ≤ (p.changeOrigin x).radius := by
refine' ENNReal.le_of_forall_pos_nnreal_lt fun r _h0 hr => _
rw [lt_tsub_iff_right, add_comm] at hr
have hr' : (‖x‖₊ : ℝ≥0∞) < p.radius := (le_add_right le_rfl).trans_lt hr
apply le_radius_of_summable_nnnorm
have : ∀ k : ℕ,
‖p.changeOrigin x k‖₊ * r ^ k ≤
(∑' s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + s.1)‖₊ * ‖x‖₊ ^ s.1) *
r ^ k :=
fun k => mul_le_mul_right' (p.nnnorm_changeOrigin_le k hr') (r ^ k)
refine' NNReal.summable_of_le this _
simpa only [← NNReal.tsum_mul_right] using
(NNReal.summable_sigma.1 (p.changeOriginSeries_summable_aux₁ hr)).2
#align formal_multilinear_series.change_origin_radius FormalMultilinearSeries.changeOrigin_radius
end
-- From this point on, assume that the space is complete, to make sure that series that converge
-- in norm also converge in `F`.
variable [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x y : E} {r R : ℝ≥0}
theorem hasFPowerSeriesOnBall_changeOrigin (k : ℕ) (hr : 0 < p.radius) :
HasFPowerSeriesOnBall (fun x => p.changeOrigin x k) (p.changeOriginSeries k) 0 p.radius :=
have := p.le_changeOriginSeries_radius k
((p.changeOriginSeries k).hasFPowerSeriesOnBall (hr.trans_le this)).mono hr this
#align formal_multilinear_series.has_fpower_series_on_ball_change_origin FormalMultilinearSeries.hasFPowerSeriesOnBall_changeOrigin
/-- Summing the series `p.changeOrigin x` at a point `y` gives back `p (x + y)`. -/
theorem changeOrigin_eval (h : (‖x‖₊ + ‖y‖₊ : ℝ≥0∞) < p.radius) :
(p.changeOrigin x).sum y = p.sum (x + y) := by
have radius_pos : 0 < p.radius := lt_of_le_of_lt (zero_le _) h
have x_mem_ball : x ∈ EMetric.ball (0 : E) p.radius :=
mem_emetric_ball_zero_iff.2 ((le_add_right le_rfl).trans_lt h)
have y_mem_ball : y ∈ EMetric.ball (0 : E) (p.changeOrigin x).radius := by
refine' mem_emetric_ball_zero_iff.2 (lt_of_lt_of_le _ p.changeOrigin_radius)
rwa [lt_tsub_iff_right, add_comm]
have x_add_y_mem_ball : x + y ∈ EMetric.ball (0 : E) p.radius := by
refine' mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt _ h)
exact mod_cast nnnorm_add_le x y
set f : (Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) → F := fun s =>
p.changeOriginSeriesTerm s.1 s.2.1 s.2.2 s.2.2.2 (fun _ => x) fun _ => y
have hsf : Summable f := by
refine' .of_nnnorm_bounded _ (p.changeOriginSeries_summable_aux₁ h) _
rintro ⟨k, l, s, hs⟩
dsimp only [Subtype.coe_mk]
exact p.nnnorm_changeOriginSeriesTerm_apply_le _ _ _ _ _ _
have hf : HasSum f ((p.changeOrigin x).sum y) := by
refine' HasSum.sigma_of_hasSum ((p.changeOrigin x).summable y_mem_ball).hasSum (fun k => _) hsf
· dsimp only
refine' ContinuousMultilinearMap.hasSum_eval _ _
have := (p.hasFPowerSeriesOnBall_changeOrigin k radius_pos).hasSum x_mem_ball
rw [zero_add] at this
refine' HasSum.sigma_of_hasSum this (fun l => _) _
· simp only [changeOriginSeries, ContinuousMultilinearMap.sum_apply]
apply hasSum_fintype
· refine' .of_nnnorm_bounded _
(p.changeOriginSeries_summable_aux₂ (mem_emetric_ball_zero_iff.1 x_mem_ball) k) fun s => _
refine' (ContinuousMultilinearMap.le_op_nnnorm _ _).trans_eq _
simp
refine' hf.unique (changeOriginIndexEquiv.symm.hasSum_iff.1 _)
refine' HasSum.sigma_of_hasSum
(p.hasSum x_add_y_mem_ball) (fun n => _) (changeOriginIndexEquiv.symm.summable_iff.2 hsf)
erw [(p n).map_add_univ (fun _ => x) fun _ => y]
-- porting note: added explicit function
convert hasSum_fintype (fun c : Finset (Fin n) => f (changeOriginIndexEquiv.symm ⟨n, c⟩))
rename_i s _
dsimp only [changeOriginSeriesTerm, (· ∘ ·), changeOriginIndexEquiv_symm_apply_fst,
changeOriginIndexEquiv_symm_apply_snd_fst, changeOriginIndexEquiv_symm_apply_snd_snd_coe]
rw [ContinuousMultilinearMap.curryFinFinset_apply_const]
have : ∀ (m) (hm : n = m), p n (s.piecewise (fun _ => x) fun _ => y) =
p m ((s.map (Fin.castIso hm).toEquiv.toEmbedding).piecewise (fun _ => x) fun _ => y) := by
rintro m rfl
simp (config := { unfoldPartialApp := true }) [Finset.piecewise]
apply this
#align formal_multilinear_series.change_origin_eval FormalMultilinearSeries.changeOrigin_eval
/-- Power series terms are analytic as we vary the origin -/
theorem analyticAt_changeOrigin (p : FormalMultilinearSeries 𝕜 E F) (rp : p.radius > 0) (n : ℕ) :
AnalyticAt 𝕜 (fun x ↦ p.changeOrigin x n) 0 :=
(FormalMultilinearSeries.hasFPowerSeriesOnBall_changeOrigin p n rp).analyticAt
end FormalMultilinearSeries
section
variable [CompleteSpace F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x y : E} {r : ℝ≥0∞}
/-- If a function admits a power series expansion `p` on a ball `B (x, r)`, then it also admits a
power series on any subball of this ball (even with a different center), given by `p.changeOrigin`.
-/
theorem HasFPowerSeriesOnBall.changeOrigin (hf : HasFPowerSeriesOnBall f p x r)
(h : (‖y‖₊ : ℝ≥0∞) < r) : HasFPowerSeriesOnBall f (p.changeOrigin y) (x + y) (r - ‖y‖₊) :=
{ r_le := by
apply le_trans _ p.changeOrigin_radius
exact tsub_le_tsub hf.r_le le_rfl
r_pos := by simp [h]
hasSum := fun {z} hz => by
have : f (x + y + z) =
FormalMultilinearSeries.sum (FormalMultilinearSeries.changeOrigin p y) z := by
rw [mem_emetric_ball_zero_iff, lt_tsub_iff_right, add_comm] at hz
rw [p.changeOrigin_eval (hz.trans_le hf.r_le), add_assoc, hf.sum]
refine' mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt _ hz)
exact mod_cast nnnorm_add_le y z
rw [this]
apply (p.changeOrigin y).hasSum
refine' EMetric.ball_subset_ball (le_trans _ p.changeOrigin_radius) hz
exact tsub_le_tsub hf.r_le le_rfl }
#align has_fpower_series_on_ball.change_origin HasFPowerSeriesOnBall.changeOrigin
/-- If a function admits a power series expansion `p` on an open ball `B (x, r)`, then
it is analytic at every point of this ball. -/
theorem HasFPowerSeriesOnBall.analyticAt_of_mem (hf : HasFPowerSeriesOnBall f p x r)
(h : y ∈ EMetric.ball x r) : AnalyticAt 𝕜 f y := by
have : (‖y - x‖₊ : ℝ≥0∞) < r := by simpa [edist_eq_coe_nnnorm_sub] using h
have := hf.changeOrigin this
rw [add_sub_cancel'_right] at this
exact this.analyticAt
#align has_fpower_series_on_ball.analytic_at_of_mem HasFPowerSeriesOnBall.analyticAt_of_mem
theorem HasFPowerSeriesOnBall.analyticOn (hf : HasFPowerSeriesOnBall f p x r) :
AnalyticOn 𝕜 f (EMetric.ball x r) :=
fun _y hy => hf.analyticAt_of_mem hy
#align has_fpower_series_on_ball.analytic_on HasFPowerSeriesOnBall.analyticOn
variable (𝕜 f)
/-- For any function `f` from a normed vector space to a Banach space, the set of points `x` such
that `f` is analytic at `x` is open. -/
theorem isOpen_analyticAt : IsOpen { x | AnalyticAt 𝕜 f x } := by
rw [isOpen_iff_mem_nhds]
rintro x ⟨p, r, hr⟩
exact mem_of_superset (EMetric.ball_mem_nhds _ hr.r_pos) fun y hy => hr.analyticAt_of_mem hy
#align is_open_analytic_at isOpen_analyticAt
variable {𝕜}
theorem AnalyticAt.eventually_analyticAt {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∀ᶠ y in 𝓝 x, AnalyticAt 𝕜 f y :=
(isOpen_analyticAt 𝕜 f).mem_nhds h
theorem AnalyticAt.exists_mem_nhds_analyticOn {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∃ s ∈ 𝓝 x, AnalyticOn 𝕜 f s :=
h.eventually_analyticAt.exists_mem
/-- If we're analytic at a point, we're analytic in a nonempty ball -/
theorem AnalyticAt.exists_ball_analyticOn {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∃ r : ℝ, 0 < r ∧ AnalyticOn 𝕜 f (Metric.ball x r) :=
Metric.isOpen_iff.mp (isOpen_analyticAt _ _) _ h
end
section
open FormalMultilinearSeries
variable {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E} {z₀ : 𝕜}
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z)) := by
refine' ⟨fun ⟨r, _, r_pos, h⟩ =>
eventually_of_mem (EMetric.ball_mem_nhds 0 r_pos) fun _ => by simpa using h, _⟩
simp only [Metric.eventually_nhds_iff]
rintro ⟨r, r_pos, h⟩
refine' ⟨p.radius ⊓ r.toNNReal, by simp, _, _⟩
· simp only [r_pos.lt, lt_inf_iff, ENNReal.coe_pos, Real.toNNReal_pos, and_true_iff]
obtain ⟨z, z_pos, le_z⟩ := NormedField.exists_norm_lt 𝕜 r_pos.lt
have : (‖z‖₊ : ENNReal) ≤ p.radius := by
simp only [dist_zero_right] at h
apply FormalMultilinearSeries.le_radius_of_tendsto
convert tendsto_norm.comp (h le_z).summable.tendsto_atTop_zero
funext
simp [norm_smul, mul_comm]
refine' lt_of_lt_of_le _ this
simp only [ENNReal.coe_pos]
exact zero_lt_iff.mpr (nnnorm_ne_zero_iff.mpr (norm_pos_iff.mp z_pos))
·
|
simp only [EMetric.mem_ball, lt_inf_iff, edist_lt_coe, apply_eq_pow_smul_coeff, and_imp,
dist_zero_right] at h ⊢
|
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z)) := by
refine' ⟨fun ⟨r, _, r_pos, h⟩ =>
eventually_of_mem (EMetric.ball_mem_nhds 0 r_pos) fun _ => by simpa using h, _⟩
simp only [Metric.eventually_nhds_iff]
rintro ⟨r, r_pos, h⟩
refine' ⟨p.radius ⊓ r.toNNReal, by simp, _, _⟩
· simp only [r_pos.lt, lt_inf_iff, ENNReal.coe_pos, Real.toNNReal_pos, and_true_iff]
obtain ⟨z, z_pos, le_z⟩ := NormedField.exists_norm_lt 𝕜 r_pos.lt
have : (‖z‖₊ : ENNReal) ≤ p.radius := by
simp only [dist_zero_right] at h
apply FormalMultilinearSeries.le_radius_of_tendsto
convert tendsto_norm.comp (h le_z).summable.tendsto_atTop_zero
funext
simp [norm_smul, mul_comm]
refine' lt_of_lt_of_le _ this
simp only [ENNReal.coe_pos]
exact zero_lt_iff.mpr (nnnorm_ne_zero_iff.mpr (norm_pos_iff.mp z_pos))
·
|
Mathlib.Analysis.Analytic.Basic.1430_0.jQw1fRSE1vGpOll
|
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z))
|
Mathlib_Analysis_Analytic_Basic
|
case intro.intro.refine'_2
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
p : FormalMultilinearSeries 𝕜 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
r : ℝ
r_pos : r > 0
h : ∀ ⦃y : 𝕜⦄, ‖y‖ < r → HasSum (fun n => y ^ n • coeff p n) (f (z₀ + y))
⊢ ∀ {y : 𝕜}, edist y 0 < radius p → nndist y 0 < Real.toNNReal r → HasSum (fun n => y ^ n • coeff p n) (f (z₀ + y))
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Logic.Equiv.Fin
import Mathlib.Topology.Algebra.InfiniteSum.Module
#align_import analysis.analytic.basic from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
/-!
# Analytic functions
A function is analytic in one dimension around `0` if it can be written as a converging power series
`Σ pₙ zⁿ`. This definition can be extended to any dimension (even in infinite dimension) by
requiring that `pₙ` is a continuous `n`-multilinear map. In general, `pₙ` is not unique (in two
dimensions, taking `p₂ (x, y) (x', y') = x y'` or `y x'` gives the same map when applied to a
vector `(x, y) (x, y)`). A way to guarantee uniqueness is to take a symmetric `pₙ`, but this is not
always possible in nonzero characteristic (in characteristic 2, the previous example has no
symmetric representative). Therefore, we do not insist on symmetry or uniqueness in the definition,
and we only require the existence of a converging series.
The general framework is important to say that the exponential map on bounded operators on a Banach
space is analytic, as well as the inverse on invertible operators.
## Main definitions
Let `p` be a formal multilinear series from `E` to `F`, i.e., `p n` is a multilinear map on `E^n`
for `n : ℕ`.
* `p.radius`: the largest `r : ℝ≥0∞` such that `‖p n‖ * r^n` grows subexponentially.
* `p.le_radius_of_bound`, `p.le_radius_of_bound_nnreal`, `p.le_radius_of_isBigO`: if `‖p n‖ * r ^ n`
is bounded above, then `r ≤ p.radius`;
* `p.isLittleO_of_lt_radius`, `p.norm_mul_pow_le_mul_pow_of_lt_radius`,
`p.isLittleO_one_of_lt_radius`,
`p.norm_mul_pow_le_of_lt_radius`, `p.nnnorm_mul_pow_le_of_lt_radius`: if `r < p.radius`, then
`‖p n‖ * r ^ n` tends to zero exponentially;
* `p.lt_radius_of_isBigO`: if `r ≠ 0` and `‖p n‖ * r ^ n = O(a ^ n)` for some `-1 < a < 1`, then
`r < p.radius`;
* `p.partialSum n x`: the sum `∑_{i = 0}^{n-1} pᵢ xⁱ`.
* `p.sum x`: the sum `∑'_{i = 0}^{∞} pᵢ xⁱ`.
Additionally, let `f` be a function from `E` to `F`.
* `HasFPowerSeriesOnBall f p x r`: on the ball of center `x` with radius `r`,
`f (x + y) = ∑'_n pₙ yⁿ`.
* `HasFPowerSeriesAt f p x`: on some ball of center `x` with positive radius, holds
`HasFPowerSeriesOnBall f p x r`.
* `AnalyticAt 𝕜 f x`: there exists a power series `p` such that holds `HasFPowerSeriesAt f p x`.
* `AnalyticOn 𝕜 f s`: the function `f` is analytic at every point of `s`.
We develop the basic properties of these notions, notably:
* If a function admits a power series, it is continuous (see
`HasFPowerSeriesOnBall.continuousOn` and `HasFPowerSeriesAt.continuousAt` and
`AnalyticAt.continuousAt`).
* In a complete space, the sum of a formal power series with positive radius is well defined on the
disk of convergence, see `FormalMultilinearSeries.hasFPowerSeriesOnBall`.
* If a function admits a power series in a ball, then it is analytic at any point `y` of this ball,
and the power series there can be expressed in terms of the initial power series `p` as
`p.changeOrigin y`. See `HasFPowerSeriesOnBall.changeOrigin`. It follows in particular that
the set of points at which a given function is analytic is open, see `isOpen_analyticAt`.
## Implementation details
We only introduce the radius of convergence of a power series, as `p.radius`.
For a power series in finitely many dimensions, there is a finer (directional, coordinate-dependent)
notion, describing the polydisk of convergence. This notion is more specific, and not necessary to
build the general theory. We do not define it here.
-/
noncomputable section
variable {𝕜 E F G : Type*}
open Topology Classical BigOperators NNReal Filter ENNReal
open Set Filter Asymptotics
namespace FormalMultilinearSeries
variable [Ring 𝕜] [AddCommGroup E] [AddCommGroup F] [Module 𝕜 E] [Module 𝕜 F]
variable [TopologicalSpace E] [TopologicalSpace F]
variable [TopologicalAddGroup E] [TopologicalAddGroup F]
variable [ContinuousConstSMul 𝕜 E] [ContinuousConstSMul 𝕜 F]
/-- Given a formal multilinear series `p` and a vector `x`, then `p.sum x` is the sum `Σ pₙ xⁿ`. A
priori, it only behaves well when `‖x‖ < p.radius`. -/
protected def sum (p : FormalMultilinearSeries 𝕜 E F) (x : E) : F :=
∑' n : ℕ, p n fun _ => x
#align formal_multilinear_series.sum FormalMultilinearSeries.sum
/-- Given a formal multilinear series `p` and a vector `x`, then `p.partialSum n x` is the sum
`Σ pₖ xᵏ` for `k ∈ {0,..., n-1}`. -/
def partialSum (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) (x : E) : F :=
∑ k in Finset.range n, p k fun _ : Fin k => x
#align formal_multilinear_series.partial_sum FormalMultilinearSeries.partialSum
/-- The partial sums of a formal multilinear series are continuous. -/
theorem partialSum_continuous (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) :
Continuous (p.partialSum n) := by
unfold partialSum -- Porting note: added
continuity
#align formal_multilinear_series.partial_sum_continuous FormalMultilinearSeries.partialSum_continuous
end FormalMultilinearSeries
/-! ### The radius of a formal multilinear series -/
variable [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F]
[NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G]
namespace FormalMultilinearSeries
variable (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
/-- The radius of a formal multilinear series is the largest `r` such that the sum `Σ ‖pₙ‖ ‖y‖ⁿ`
converges for all `‖y‖ < r`. This implies that `Σ pₙ yⁿ` converges for all `‖y‖ < r`, but these
definitions are *not* equivalent in general. -/
def radius (p : FormalMultilinearSeries 𝕜 E F) : ℝ≥0∞ :=
⨆ (r : ℝ≥0) (C : ℝ) (_ : ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C), (r : ℝ≥0∞)
#align formal_multilinear_series.radius FormalMultilinearSeries.radius
/-- If `‖pₙ‖ rⁿ` is bounded in `n`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_bound (C : ℝ) {r : ℝ≥0} (h : ∀ n : ℕ, ‖p n‖ * (r : ℝ) ^ n ≤ C) :
(r : ℝ≥0∞) ≤ p.radius :=
le_iSup_of_le r <| le_iSup_of_le C <| le_iSup (fun _ => (r : ℝ≥0∞)) h
#align formal_multilinear_series.le_radius_of_bound FormalMultilinearSeries.le_radius_of_bound
/-- If `‖pₙ‖ rⁿ` is bounded in `n`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_bound_nnreal (C : ℝ≥0) {r : ℝ≥0} (h : ∀ n : ℕ, ‖p n‖₊ * r ^ n ≤ C) :
(r : ℝ≥0∞) ≤ p.radius :=
p.le_radius_of_bound C fun n => mod_cast h n
#align formal_multilinear_series.le_radius_of_bound_nnreal FormalMultilinearSeries.le_radius_of_bound_nnreal
/-- If `‖pₙ‖ rⁿ = O(1)`, as `n → ∞`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_isBigO (h : (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] fun _ => (1 : ℝ)) :
↑r ≤ p.radius :=
Exists.elim (isBigO_one_nat_atTop_iff.1 h) fun C hC =>
p.le_radius_of_bound C fun n => (le_abs_self _).trans (hC n)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.le_radius_of_is_O FormalMultilinearSeries.le_radius_of_isBigO
theorem le_radius_of_eventually_le (C) (h : ∀ᶠ n in atTop, ‖p n‖ * (r : ℝ) ^ n ≤ C) :
↑r ≤ p.radius :=
p.le_radius_of_isBigO <| IsBigO.of_bound C <| h.mono fun n hn => by simpa
#align formal_multilinear_series.le_radius_of_eventually_le FormalMultilinearSeries.le_radius_of_eventually_le
theorem le_radius_of_summable_nnnorm (h : Summable fun n => ‖p n‖₊ * r ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_bound_nnreal (∑' n, ‖p n‖₊ * r ^ n) fun _ => le_tsum' h _
#align formal_multilinear_series.le_radius_of_summable_nnnorm FormalMultilinearSeries.le_radius_of_summable_nnnorm
theorem le_radius_of_summable (h : Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_summable_nnnorm <| by
simp only [← coe_nnnorm] at h
exact mod_cast h
#align formal_multilinear_series.le_radius_of_summable FormalMultilinearSeries.le_radius_of_summable
theorem radius_eq_top_of_forall_nnreal_isBigO
(h : ∀ r : ℝ≥0, (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] fun _ => (1 : ℝ)) : p.radius = ∞ :=
ENNReal.eq_top_of_forall_nnreal_le fun r => p.le_radius_of_isBigO (h r)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.radius_eq_top_of_forall_nnreal_is_O FormalMultilinearSeries.radius_eq_top_of_forall_nnreal_isBigO
theorem radius_eq_top_of_eventually_eq_zero (h : ∀ᶠ n in atTop, p n = 0) : p.radius = ∞ :=
p.radius_eq_top_of_forall_nnreal_isBigO fun r =>
(isBigO_zero _ _).congr' (h.mono fun n hn => by simp [hn]) EventuallyEq.rfl
#align formal_multilinear_series.radius_eq_top_of_eventually_eq_zero FormalMultilinearSeries.radius_eq_top_of_eventually_eq_zero
theorem radius_eq_top_of_forall_image_add_eq_zero (n : ℕ) (hn : ∀ m, p (m + n) = 0) :
p.radius = ∞ :=
p.radius_eq_top_of_eventually_eq_zero <|
mem_atTop_sets.2 ⟨n, fun _ hk => tsub_add_cancel_of_le hk ▸ hn _⟩
#align formal_multilinear_series.radius_eq_top_of_forall_image_add_eq_zero FormalMultilinearSeries.radius_eq_top_of_forall_image_add_eq_zero
@[simp]
theorem constFormalMultilinearSeries_radius {v : F} :
(constFormalMultilinearSeries 𝕜 E v).radius = ⊤ :=
(constFormalMultilinearSeries 𝕜 E v).radius_eq_top_of_forall_image_add_eq_zero 1
(by simp [constFormalMultilinearSeries])
#align formal_multilinear_series.const_formal_multilinear_series_radius FormalMultilinearSeries.constFormalMultilinearSeries_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` tends to zero exponentially:
for some `0 < a < 1`, `‖p n‖ rⁿ = o(aⁿ)`. -/
theorem isLittleO_of_lt_radius (h : ↑r < p.radius) :
∃ a ∈ Ioo (0 : ℝ) 1, (fun n => ‖p n‖ * (r : ℝ) ^ n) =o[atTop] (a ^ ·) := by
have := (TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 4
rw [this]
-- Porting note: was
-- rw [(TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 4]
simp only [radius, lt_iSup_iff] at h
rcases h with ⟨t, C, hC, rt⟩
rw [ENNReal.coe_lt_coe, ← NNReal.coe_lt_coe] at rt
have : 0 < (t : ℝ) := r.coe_nonneg.trans_lt rt
rw [← div_lt_one this] at rt
refine' ⟨_, rt, C, Or.inr zero_lt_one, fun n => _⟩
calc
|‖p n‖ * (r : ℝ) ^ n| = ‖p n‖ * (t : ℝ) ^ n * (r / t : ℝ) ^ n := by
field_simp [mul_right_comm, abs_mul]
_ ≤ C * (r / t : ℝ) ^ n := by gcongr; apply hC
#align formal_multilinear_series.is_o_of_lt_radius FormalMultilinearSeries.isLittleO_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ = o(1)`. -/
theorem isLittleO_one_of_lt_radius (h : ↑r < p.radius) :
(fun n => ‖p n‖ * (r : ℝ) ^ n) =o[atTop] (fun _ => 1 : ℕ → ℝ) :=
let ⟨_, ha, hp⟩ := p.isLittleO_of_lt_radius h
hp.trans <| (isLittleO_pow_pow_of_lt_left ha.1.le ha.2).congr (fun _ => rfl) one_pow
#align formal_multilinear_series.is_o_one_of_lt_radius FormalMultilinearSeries.isLittleO_one_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` tends to zero exponentially:
for some `0 < a < 1` and `C > 0`, `‖p n‖ * r ^ n ≤ C * a ^ n`. -/
theorem norm_mul_pow_le_mul_pow_of_lt_radius (h : ↑r < p.radius) :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C * a ^ n := by
-- Porting note: moved out of `rcases`
have := ((TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 5).mp
(p.isLittleO_of_lt_radius h)
rcases this with ⟨a, ha, C, hC, H⟩
exact ⟨a, ha, C, hC, fun n => (le_abs_self _).trans (H n)⟩
#align formal_multilinear_series.norm_mul_pow_le_mul_pow_of_lt_radius FormalMultilinearSeries.norm_mul_pow_le_mul_pow_of_lt_radius
/-- If `r ≠ 0` and `‖pₙ‖ rⁿ = O(aⁿ)` for some `-1 < a < 1`, then `r < p.radius`. -/
theorem lt_radius_of_isBigO (h₀ : r ≠ 0) {a : ℝ} (ha : a ∈ Ioo (-1 : ℝ) 1)
(hp : (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] (a ^ ·)) : ↑r < p.radius := by
-- Porting note: moved out of `rcases`
have := ((TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 2 5)
rcases this.mp ⟨a, ha, hp⟩ with ⟨a, ha, C, hC, hp⟩
rw [← pos_iff_ne_zero, ← NNReal.coe_pos] at h₀
lift a to ℝ≥0 using ha.1.le
have : (r : ℝ) < r / a := by
simpa only [div_one] using (div_lt_div_left h₀ zero_lt_one ha.1).2 ha.2
norm_cast at this
rw [← ENNReal.coe_lt_coe] at this
refine' this.trans_le (p.le_radius_of_bound C fun n => _)
rw [NNReal.coe_div, div_pow, ← mul_div_assoc, div_le_iff (pow_pos ha.1 n)]
exact (le_abs_self _).trans (hp n)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.lt_radius_of_is_O FormalMultilinearSeries.lt_radius_of_isBigO
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem norm_mul_pow_le_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C :=
let ⟨_, ha, C, hC, h⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius h
⟨C, hC, fun n => (h n).trans <| mul_le_of_le_one_right hC.lt.le (pow_le_one _ ha.1.le ha.2.le)⟩
#align formal_multilinear_series.norm_mul_pow_le_of_lt_radius FormalMultilinearSeries.norm_mul_pow_le_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem norm_le_div_pow_of_pos_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h0 : 0 < r) (h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖ ≤ C / (r : ℝ) ^ n :=
let ⟨C, hC, hp⟩ := p.norm_mul_pow_le_of_lt_radius h
⟨C, hC, fun n => Iff.mpr (le_div_iff (pow_pos h0 _)) (hp n)⟩
#align formal_multilinear_series.norm_le_div_pow_of_pos_of_lt_radius FormalMultilinearSeries.norm_le_div_pow_of_pos_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem nnnorm_mul_pow_le_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖₊ * r ^ n ≤ C :=
let ⟨C, hC, hp⟩ := p.norm_mul_pow_le_of_lt_radius h
⟨⟨C, hC.lt.le⟩, hC, mod_cast hp⟩
#align formal_multilinear_series.nnnorm_mul_pow_le_of_lt_radius FormalMultilinearSeries.nnnorm_mul_pow_le_of_lt_radius
theorem le_radius_of_tendsto (p : FormalMultilinearSeries 𝕜 E F) {l : ℝ}
(h : Tendsto (fun n => ‖p n‖ * (r : ℝ) ^ n) atTop (𝓝 l)) : ↑r ≤ p.radius :=
p.le_radius_of_isBigO (h.isBigO_one _)
#align formal_multilinear_series.le_radius_of_tendsto FormalMultilinearSeries.le_radius_of_tendsto
theorem le_radius_of_summable_norm (p : FormalMultilinearSeries 𝕜 E F)
(hs : Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_tendsto hs.tendsto_atTop_zero
#align formal_multilinear_series.le_radius_of_summable_norm FormalMultilinearSeries.le_radius_of_summable_norm
theorem not_summable_norm_of_radius_lt_nnnorm (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(h : p.radius < ‖x‖₊) : ¬Summable fun n => ‖p n‖ * ‖x‖ ^ n :=
fun hs => not_le_of_lt h (p.le_radius_of_summable_norm hs)
#align formal_multilinear_series.not_summable_norm_of_radius_lt_nnnorm FormalMultilinearSeries.not_summable_norm_of_radius_lt_nnnorm
theorem summable_norm_mul_pow (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0} (h : ↑r < p.radius) :
Summable fun n : ℕ => ‖p n‖ * (r : ℝ) ^ n := by
obtain ⟨a, ha : a ∈ Ioo (0 : ℝ) 1, C, - : 0 < C, hp⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius h
exact .of_nonneg_of_le (fun n => mul_nonneg (norm_nonneg _) (pow_nonneg r.coe_nonneg _))
hp ((summable_geometric_of_lt_1 ha.1.le ha.2).mul_left _)
#align formal_multilinear_series.summable_norm_mul_pow FormalMultilinearSeries.summable_norm_mul_pow
theorem summable_norm_apply (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : Summable fun n : ℕ => ‖p n fun _ => x‖ := by
rw [mem_emetric_ball_zero_iff] at hx
refine' .of_nonneg_of_le
(fun _ => norm_nonneg _) (fun n => ((p n).le_op_norm _).trans_eq _) (p.summable_norm_mul_pow hx)
simp
#align formal_multilinear_series.summable_norm_apply FormalMultilinearSeries.summable_norm_apply
theorem summable_nnnorm_mul_pow (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0} (h : ↑r < p.radius) :
Summable fun n : ℕ => ‖p n‖₊ * r ^ n := by
rw [← NNReal.summable_coe]
push_cast
exact p.summable_norm_mul_pow h
#align formal_multilinear_series.summable_nnnorm_mul_pow FormalMultilinearSeries.summable_nnnorm_mul_pow
protected theorem summable [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : Summable fun n : ℕ => p n fun _ => x :=
(p.summable_norm_apply hx).of_norm
#align formal_multilinear_series.summable FormalMultilinearSeries.summable
theorem radius_eq_top_of_summable_norm (p : FormalMultilinearSeries 𝕜 E F)
(hs : ∀ r : ℝ≥0, Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : p.radius = ∞ :=
ENNReal.eq_top_of_forall_nnreal_le fun r => p.le_radius_of_summable_norm (hs r)
#align formal_multilinear_series.radius_eq_top_of_summable_norm FormalMultilinearSeries.radius_eq_top_of_summable_norm
theorem radius_eq_top_iff_summable_norm (p : FormalMultilinearSeries 𝕜 E F) :
p.radius = ∞ ↔ ∀ r : ℝ≥0, Summable fun n => ‖p n‖ * (r : ℝ) ^ n := by
constructor
· intro h r
obtain ⟨a, ha : a ∈ Ioo (0 : ℝ) 1, C, - : 0 < C, hp⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius
(show (r : ℝ≥0∞) < p.radius from h.symm ▸ ENNReal.coe_lt_top)
refine' .of_norm_bounded
(fun n => (C : ℝ) * a ^ n) ((summable_geometric_of_lt_1 ha.1.le ha.2).mul_left _) fun n => _
specialize hp n
rwa [Real.norm_of_nonneg (mul_nonneg (norm_nonneg _) (pow_nonneg r.coe_nonneg n))]
· exact p.radius_eq_top_of_summable_norm
#align formal_multilinear_series.radius_eq_top_iff_summable_norm FormalMultilinearSeries.radius_eq_top_iff_summable_norm
/-- If the radius of `p` is positive, then `‖pₙ‖` grows at most geometrically. -/
theorem le_mul_pow_of_radius_pos (p : FormalMultilinearSeries 𝕜 E F) (h : 0 < p.radius) :
∃ (C r : _) (hC : 0 < C) (_ : 0 < r), ∀ n, ‖p n‖ ≤ C * r ^ n := by
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 h with ⟨r, r0, rlt⟩
have rpos : 0 < (r : ℝ) := by simp [ENNReal.coe_pos.1 r0]
rcases norm_le_div_pow_of_pos_of_lt_radius p rpos rlt with ⟨C, Cpos, hCp⟩
refine' ⟨C, r⁻¹, Cpos, by simp only [inv_pos, rpos], fun n => _⟩
-- Porting note: was `convert`
rw [inv_pow, ← div_eq_mul_inv]
exact hCp n
#align formal_multilinear_series.le_mul_pow_of_radius_pos FormalMultilinearSeries.le_mul_pow_of_radius_pos
/-- The radius of the sum of two formal series is at least the minimum of their two radii. -/
theorem min_radius_le_radius_add (p q : FormalMultilinearSeries 𝕜 E F) :
min p.radius q.radius ≤ (p + q).radius := by
refine' ENNReal.le_of_forall_nnreal_lt fun r hr => _
rw [lt_min_iff] at hr
have := ((p.isLittleO_one_of_lt_radius hr.1).add (q.isLittleO_one_of_lt_radius hr.2)).isBigO
refine' (p + q).le_radius_of_isBigO ((isBigO_of_le _ fun n => _).trans this)
rw [← add_mul, norm_mul, norm_mul, norm_norm]
exact mul_le_mul_of_nonneg_right ((norm_add_le _ _).trans (le_abs_self _)) (norm_nonneg _)
#align formal_multilinear_series.min_radius_le_radius_add FormalMultilinearSeries.min_radius_le_radius_add
@[simp]
theorem radius_neg (p : FormalMultilinearSeries 𝕜 E F) : (-p).radius = p.radius := by
simp only [radius, neg_apply, norm_neg]
#align formal_multilinear_series.radius_neg FormalMultilinearSeries.radius_neg
protected theorem hasSum [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : HasSum (fun n : ℕ => p n fun _ => x) (p.sum x) :=
(p.summable hx).hasSum
#align formal_multilinear_series.has_sum FormalMultilinearSeries.hasSum
theorem radius_le_radius_continuousLinearMap_comp (p : FormalMultilinearSeries 𝕜 E F)
(f : F →L[𝕜] G) : p.radius ≤ (f.compFormalMultilinearSeries p).radius := by
refine' ENNReal.le_of_forall_nnreal_lt fun r hr => _
apply le_radius_of_isBigO
apply (IsBigO.trans_isLittleO _ (p.isLittleO_one_of_lt_radius hr)).isBigO
refine' IsBigO.mul (@IsBigOWith.isBigO _ _ _ _ _ ‖f‖ _ _ _ _) (isBigO_refl _ _)
refine IsBigOWith.of_bound (eventually_of_forall fun n => ?_)
simpa only [norm_norm] using f.norm_compContinuousMultilinearMap_le (p n)
#align formal_multilinear_series.radius_le_radius_continuous_linear_map_comp FormalMultilinearSeries.radius_le_radius_continuousLinearMap_comp
end FormalMultilinearSeries
/-! ### Expanding a function as a power series -/
section
variable {f g : E → F} {p pf pg : FormalMultilinearSeries 𝕜 E F} {x : E} {r r' : ℝ≥0∞}
/-- Given a function `f : E → F` and a formal multilinear series `p`, we say that `f` has `p` as
a power series on the ball of radius `r > 0` around `x` if `f (x + y) = ∑' pₙ yⁿ` for all `‖y‖ < r`.
-/
structure HasFPowerSeriesOnBall (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (x : E) (r : ℝ≥0∞) :
Prop where
r_le : r ≤ p.radius
r_pos : 0 < r
hasSum :
∀ {y}, y ∈ EMetric.ball (0 : E) r → HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y))
#align has_fpower_series_on_ball HasFPowerSeriesOnBall
/-- Given a function `f : E → F` and a formal multilinear series `p`, we say that `f` has `p` as
a power series around `x` if `f (x + y) = ∑' pₙ yⁿ` for all `y` in a neighborhood of `0`. -/
def HasFPowerSeriesAt (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (x : E) :=
∃ r, HasFPowerSeriesOnBall f p x r
#align has_fpower_series_at HasFPowerSeriesAt
variable (𝕜)
/-- Given a function `f : E → F`, we say that `f` is analytic at `x` if it admits a convergent power
series expansion around `x`. -/
def AnalyticAt (f : E → F) (x : E) :=
∃ p : FormalMultilinearSeries 𝕜 E F, HasFPowerSeriesAt f p x
#align analytic_at AnalyticAt
/-- Given a function `f : E → F`, we say that `f` is analytic on a set `s` if it is analytic around
every point of `s`. -/
def AnalyticOn (f : E → F) (s : Set E) :=
∀ x, x ∈ s → AnalyticAt 𝕜 f x
#align analytic_on AnalyticOn
variable {𝕜}
theorem HasFPowerSeriesOnBall.hasFPowerSeriesAt (hf : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesAt f p x :=
⟨r, hf⟩
#align has_fpower_series_on_ball.has_fpower_series_at HasFPowerSeriesOnBall.hasFPowerSeriesAt
theorem HasFPowerSeriesAt.analyticAt (hf : HasFPowerSeriesAt f p x) : AnalyticAt 𝕜 f x :=
⟨p, hf⟩
#align has_fpower_series_at.analytic_at HasFPowerSeriesAt.analyticAt
theorem HasFPowerSeriesOnBall.analyticAt (hf : HasFPowerSeriesOnBall f p x r) : AnalyticAt 𝕜 f x :=
hf.hasFPowerSeriesAt.analyticAt
#align has_fpower_series_on_ball.analytic_at HasFPowerSeriesOnBall.analyticAt
theorem HasFPowerSeriesOnBall.congr (hf : HasFPowerSeriesOnBall f p x r)
(hg : EqOn f g (EMetric.ball x r)) : HasFPowerSeriesOnBall g p x r :=
{ r_le := hf.r_le
r_pos := hf.r_pos
hasSum := fun {y} hy => by
convert hf.hasSum hy using 1
apply hg.symm
simpa [edist_eq_coe_nnnorm_sub] using hy }
#align has_fpower_series_on_ball.congr HasFPowerSeriesOnBall.congr
/-- If a function `f` has a power series `p` around `x`, then the function `z ↦ f (z - y)` has the
same power series around `x + y`. -/
theorem HasFPowerSeriesOnBall.comp_sub (hf : HasFPowerSeriesOnBall f p x r) (y : E) :
HasFPowerSeriesOnBall (fun z => f (z - y)) p (x + y) r :=
{ r_le := hf.r_le
r_pos := hf.r_pos
hasSum := fun {z} hz => by
convert hf.hasSum hz using 2
abel }
#align has_fpower_series_on_ball.comp_sub HasFPowerSeriesOnBall.comp_sub
theorem HasFPowerSeriesOnBall.hasSum_sub (hf : HasFPowerSeriesOnBall f p x r) {y : E}
(hy : y ∈ EMetric.ball x r) : HasSum (fun n : ℕ => p n fun _ => y - x) (f y) := by
have : y - x ∈ EMetric.ball (0 : E) r := by simpa [edist_eq_coe_nnnorm_sub] using hy
simpa only [add_sub_cancel'_right] using hf.hasSum this
#align has_fpower_series_on_ball.has_sum_sub HasFPowerSeriesOnBall.hasSum_sub
theorem HasFPowerSeriesOnBall.radius_pos (hf : HasFPowerSeriesOnBall f p x r) : 0 < p.radius :=
lt_of_lt_of_le hf.r_pos hf.r_le
#align has_fpower_series_on_ball.radius_pos HasFPowerSeriesOnBall.radius_pos
theorem HasFPowerSeriesAt.radius_pos (hf : HasFPowerSeriesAt f p x) : 0 < p.radius :=
let ⟨_, hr⟩ := hf
hr.radius_pos
#align has_fpower_series_at.radius_pos HasFPowerSeriesAt.radius_pos
theorem HasFPowerSeriesOnBall.mono (hf : HasFPowerSeriesOnBall f p x r) (r'_pos : 0 < r')
(hr : r' ≤ r) : HasFPowerSeriesOnBall f p x r' :=
⟨le_trans hr hf.1, r'_pos, fun hy => hf.hasSum (EMetric.ball_subset_ball hr hy)⟩
#align has_fpower_series_on_ball.mono HasFPowerSeriesOnBall.mono
theorem HasFPowerSeriesAt.congr (hf : HasFPowerSeriesAt f p x) (hg : f =ᶠ[𝓝 x] g) :
HasFPowerSeriesAt g p x := by
rcases hf with ⟨r₁, h₁⟩
rcases EMetric.mem_nhds_iff.mp hg with ⟨r₂, h₂pos, h₂⟩
exact ⟨min r₁ r₂,
(h₁.mono (lt_min h₁.r_pos h₂pos) inf_le_left).congr
fun y hy => h₂ (EMetric.ball_subset_ball inf_le_right hy)⟩
#align has_fpower_series_at.congr HasFPowerSeriesAt.congr
protected theorem HasFPowerSeriesAt.eventually (hf : HasFPowerSeriesAt f p x) :
∀ᶠ r : ℝ≥0∞ in 𝓝[>] 0, HasFPowerSeriesOnBall f p x r :=
let ⟨_, hr⟩ := hf
mem_of_superset (Ioo_mem_nhdsWithin_Ioi (left_mem_Ico.2 hr.r_pos)) fun _ hr' =>
hr.mono hr'.1 hr'.2.le
#align has_fpower_series_at.eventually HasFPowerSeriesAt.eventually
theorem HasFPowerSeriesOnBall.eventually_hasSum (hf : HasFPowerSeriesOnBall f p x r) :
∀ᶠ y in 𝓝 0, HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y)) := by
filter_upwards [EMetric.ball_mem_nhds (0 : E) hf.r_pos] using fun _ => hf.hasSum
#align has_fpower_series_on_ball.eventually_has_sum HasFPowerSeriesOnBall.eventually_hasSum
theorem HasFPowerSeriesAt.eventually_hasSum (hf : HasFPowerSeriesAt f p x) :
∀ᶠ y in 𝓝 0, HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y)) :=
let ⟨_, hr⟩ := hf
hr.eventually_hasSum
#align has_fpower_series_at.eventually_has_sum HasFPowerSeriesAt.eventually_hasSum
theorem HasFPowerSeriesOnBall.eventually_hasSum_sub (hf : HasFPowerSeriesOnBall f p x r) :
∀ᶠ y in 𝓝 x, HasSum (fun n : ℕ => p n fun _ : Fin n => y - x) (f y) := by
filter_upwards [EMetric.ball_mem_nhds x hf.r_pos] with y using hf.hasSum_sub
#align has_fpower_series_on_ball.eventually_has_sum_sub HasFPowerSeriesOnBall.eventually_hasSum_sub
theorem HasFPowerSeriesAt.eventually_hasSum_sub (hf : HasFPowerSeriesAt f p x) :
∀ᶠ y in 𝓝 x, HasSum (fun n : ℕ => p n fun _ : Fin n => y - x) (f y) :=
let ⟨_, hr⟩ := hf
hr.eventually_hasSum_sub
#align has_fpower_series_at.eventually_has_sum_sub HasFPowerSeriesAt.eventually_hasSum_sub
theorem HasFPowerSeriesOnBall.eventually_eq_zero
(hf : HasFPowerSeriesOnBall f (0 : FormalMultilinearSeries 𝕜 E F) x r) :
∀ᶠ z in 𝓝 x, f z = 0 := by
filter_upwards [hf.eventually_hasSum_sub] with z hz using hz.unique hasSum_zero
#align has_fpower_series_on_ball.eventually_eq_zero HasFPowerSeriesOnBall.eventually_eq_zero
theorem HasFPowerSeriesAt.eventually_eq_zero
(hf : HasFPowerSeriesAt f (0 : FormalMultilinearSeries 𝕜 E F) x) : ∀ᶠ z in 𝓝 x, f z = 0 :=
let ⟨_, hr⟩ := hf
hr.eventually_eq_zero
#align has_fpower_series_at.eventually_eq_zero HasFPowerSeriesAt.eventually_eq_zero
theorem hasFPowerSeriesOnBall_const {c : F} {e : E} :
HasFPowerSeriesOnBall (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e ⊤ := by
refine' ⟨by simp, WithTop.zero_lt_top, fun _ => hasSum_single 0 fun n hn => _⟩
simp [constFormalMultilinearSeries_apply hn]
#align has_fpower_series_on_ball_const hasFPowerSeriesOnBall_const
theorem hasFPowerSeriesAt_const {c : F} {e : E} :
HasFPowerSeriesAt (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e :=
⟨⊤, hasFPowerSeriesOnBall_const⟩
#align has_fpower_series_at_const hasFPowerSeriesAt_const
theorem analyticAt_const {v : F} : AnalyticAt 𝕜 (fun _ => v) x :=
⟨constFormalMultilinearSeries 𝕜 E v, hasFPowerSeriesAt_const⟩
#align analytic_at_const analyticAt_const
theorem analyticOn_const {v : F} {s : Set E} : AnalyticOn 𝕜 (fun _ => v) s :=
fun _ _ => analyticAt_const
#align analytic_on_const analyticOn_const
theorem HasFPowerSeriesOnBall.add (hf : HasFPowerSeriesOnBall f pf x r)
(hg : HasFPowerSeriesOnBall g pg x r) : HasFPowerSeriesOnBall (f + g) (pf + pg) x r :=
{ r_le := le_trans (le_min_iff.2 ⟨hf.r_le, hg.r_le⟩) (pf.min_radius_le_radius_add pg)
r_pos := hf.r_pos
hasSum := fun hy => (hf.hasSum hy).add (hg.hasSum hy) }
#align has_fpower_series_on_ball.add HasFPowerSeriesOnBall.add
theorem HasFPowerSeriesAt.add (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x) :
HasFPowerSeriesAt (f + g) (pf + pg) x := by
rcases (hf.eventually.and hg.eventually).exists with ⟨r, hr⟩
exact ⟨r, hr.1.add hr.2⟩
#align has_fpower_series_at.add HasFPowerSeriesAt.add
theorem AnalyticAt.congr (hf : AnalyticAt 𝕜 f x) (hg : f =ᶠ[𝓝 x] g) : AnalyticAt 𝕜 g x :=
let ⟨_, hpf⟩ := hf
(hpf.congr hg).analyticAt
theorem analyticAt_congr (h : f =ᶠ[𝓝 x] g) : AnalyticAt 𝕜 f x ↔ AnalyticAt 𝕜 g x :=
⟨fun hf ↦ hf.congr h, fun hg ↦ hg.congr h.symm⟩
theorem AnalyticAt.add (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) : AnalyticAt 𝕜 (f + g) x :=
let ⟨_, hpf⟩ := hf
let ⟨_, hqf⟩ := hg
(hpf.add hqf).analyticAt
#align analytic_at.add AnalyticAt.add
theorem HasFPowerSeriesOnBall.neg (hf : HasFPowerSeriesOnBall f pf x r) :
HasFPowerSeriesOnBall (-f) (-pf) x r :=
{ r_le := by
rw [pf.radius_neg]
exact hf.r_le
r_pos := hf.r_pos
hasSum := fun hy => (hf.hasSum hy).neg }
#align has_fpower_series_on_ball.neg HasFPowerSeriesOnBall.neg
theorem HasFPowerSeriesAt.neg (hf : HasFPowerSeriesAt f pf x) : HasFPowerSeriesAt (-f) (-pf) x :=
let ⟨_, hrf⟩ := hf
hrf.neg.hasFPowerSeriesAt
#align has_fpower_series_at.neg HasFPowerSeriesAt.neg
theorem AnalyticAt.neg (hf : AnalyticAt 𝕜 f x) : AnalyticAt 𝕜 (-f) x :=
let ⟨_, hpf⟩ := hf
hpf.neg.analyticAt
#align analytic_at.neg AnalyticAt.neg
theorem HasFPowerSeriesOnBall.sub (hf : HasFPowerSeriesOnBall f pf x r)
(hg : HasFPowerSeriesOnBall g pg x r) : HasFPowerSeriesOnBall (f - g) (pf - pg) x r := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align has_fpower_series_on_ball.sub HasFPowerSeriesOnBall.sub
theorem HasFPowerSeriesAt.sub (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x) :
HasFPowerSeriesAt (f - g) (pf - pg) x := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align has_fpower_series_at.sub HasFPowerSeriesAt.sub
theorem AnalyticAt.sub (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) :
AnalyticAt 𝕜 (f - g) x := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align analytic_at.sub AnalyticAt.sub
theorem AnalyticOn.mono {s t : Set E} (hf : AnalyticOn 𝕜 f t) (hst : s ⊆ t) : AnalyticOn 𝕜 f s :=
fun z hz => hf z (hst hz)
#align analytic_on.mono AnalyticOn.mono
theorem AnalyticOn.congr' {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : f =ᶠ[𝓝ˢ s] g) :
AnalyticOn 𝕜 g s :=
fun z hz => (hf z hz).congr (mem_nhdsSet_iff_forall.mp hg z hz)
theorem analyticOn_congr' {s : Set E} (h : f =ᶠ[𝓝ˢ s] g) : AnalyticOn 𝕜 f s ↔ AnalyticOn 𝕜 g s :=
⟨fun hf => hf.congr' h, fun hg => hg.congr' h.symm⟩
theorem AnalyticOn.congr {s : Set E} (hs : IsOpen s) (hf : AnalyticOn 𝕜 f s) (hg : s.EqOn f g) :
AnalyticOn 𝕜 g s :=
hf.congr' $ mem_nhdsSet_iff_forall.mpr
(fun _ hz => eventuallyEq_iff_exists_mem.mpr ⟨s, hs.mem_nhds hz, hg⟩)
theorem analyticOn_congr {s : Set E} (hs : IsOpen s) (h : s.EqOn f g) : AnalyticOn 𝕜 f s ↔
AnalyticOn 𝕜 g s := ⟨fun hf => hf.congr hs h, fun hg => hg.congr hs h.symm⟩
theorem AnalyticOn.add {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
AnalyticOn 𝕜 (f + g) s :=
fun z hz => (hf z hz).add (hg z hz)
#align analytic_on.add AnalyticOn.add
theorem AnalyticOn.sub {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
AnalyticOn 𝕜 (f - g) s :=
fun z hz => (hf z hz).sub (hg z hz)
#align analytic_on.sub AnalyticOn.sub
theorem HasFPowerSeriesOnBall.coeff_zero (hf : HasFPowerSeriesOnBall f pf x r) (v : Fin 0 → E) :
pf 0 v = f x := by
have v_eq : v = fun i => 0 := Subsingleton.elim _ _
have zero_mem : (0 : E) ∈ EMetric.ball (0 : E) r := by simp [hf.r_pos]
have : ∀ i, i ≠ 0 → (pf i fun j => 0) = 0 := by
intro i hi
have : 0 < i := pos_iff_ne_zero.2 hi
exact ContinuousMultilinearMap.map_coord_zero _ (⟨0, this⟩ : Fin i) rfl
have A := (hf.hasSum zero_mem).unique (hasSum_single _ this)
simpa [v_eq] using A.symm
#align has_fpower_series_on_ball.coeff_zero HasFPowerSeriesOnBall.coeff_zero
theorem HasFPowerSeriesAt.coeff_zero (hf : HasFPowerSeriesAt f pf x) (v : Fin 0 → E) :
pf 0 v = f x :=
let ⟨_, hrf⟩ := hf
hrf.coeff_zero v
#align has_fpower_series_at.coeff_zero HasFPowerSeriesAt.coeff_zero
/-- If a function `f` has a power series `p` on a ball and `g` is linear, then `g ∘ f` has the
power series `g ∘ p` on the same ball. -/
theorem ContinuousLinearMap.comp_hasFPowerSeriesOnBall (g : F →L[𝕜] G)
(h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (g ∘ f) (g.compFormalMultilinearSeries p) x r :=
{ r_le := h.r_le.trans (p.radius_le_radius_continuousLinearMap_comp _)
r_pos := h.r_pos
hasSum := fun hy => by
simpa only [ContinuousLinearMap.compFormalMultilinearSeries_apply,
ContinuousLinearMap.compContinuousMultilinearMap_coe, Function.comp_apply] using
g.hasSum (h.hasSum hy) }
#align continuous_linear_map.comp_has_fpower_series_on_ball ContinuousLinearMap.comp_hasFPowerSeriesOnBall
/-- If a function `f` is analytic on a set `s` and `g` is linear, then `g ∘ f` is analytic
on `s`. -/
theorem ContinuousLinearMap.comp_analyticOn {s : Set E} (g : F →L[𝕜] G) (h : AnalyticOn 𝕜 f s) :
AnalyticOn 𝕜 (g ∘ f) s := by
rintro x hx
rcases h x hx with ⟨p, r, hp⟩
exact ⟨g.compFormalMultilinearSeries p, r, g.comp_hasFPowerSeriesOnBall hp⟩
#align continuous_linear_map.comp_analytic_on ContinuousLinearMap.comp_analyticOn
/-- If a function admits a power series expansion, then it is exponentially close to the partial
sums of this power series on strict subdisks of the disk of convergence.
This version provides an upper estimate that decreases both in `‖y‖` and `n`. See also
`HasFPowerSeriesOnBall.uniform_geometric_approx` for a weaker version. -/
theorem HasFPowerSeriesOnBall.uniform_geometric_approx' {r' : ℝ≥0}
(hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n := by
obtain ⟨a, ha, C, hC, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n, ‖p n‖ * (r' : ℝ) ^ n ≤ C * a ^ n :=
p.norm_mul_pow_le_mul_pow_of_lt_radius (h.trans_le hf.r_le)
refine' ⟨a, ha, C / (1 - a), div_pos hC (sub_pos.2 ha.2), fun y hy n => _⟩
have yr' : ‖y‖ < r' := by
rw [ball_zero_eq] at hy
exact hy
have hr'0 : 0 < (r' : ℝ) := (norm_nonneg _).trans_lt yr'
have : y ∈ EMetric.ball (0 : E) r := by
refine' mem_emetric_ball_zero_iff.2 (lt_trans _ h)
exact mod_cast yr'
rw [norm_sub_rev, ← mul_div_right_comm]
have ya : a * (‖y‖ / ↑r') ≤ a :=
mul_le_of_le_one_right ha.1.le (div_le_one_of_le yr'.le r'.coe_nonneg)
suffices ‖p.partialSum n y - f (x + y)‖ ≤ C * (a * (‖y‖ / r')) ^ n / (1 - a * (‖y‖ / r')) by
refine' this.trans _
have : 0 < a := ha.1
gcongr
apply_rules [sub_pos.2, ha.2]
apply norm_sub_le_of_geometric_bound_of_hasSum (ya.trans_lt ha.2) _ (hf.hasSum this)
intro n
calc
‖(p n) fun _ : Fin n => y‖
_ ≤ ‖p n‖ * ∏ _i : Fin n, ‖y‖ := ContinuousMultilinearMap.le_op_norm _ _
_ = ‖p n‖ * (r' : ℝ) ^ n * (‖y‖ / r') ^ n := by field_simp [mul_right_comm]
_ ≤ C * a ^ n * (‖y‖ / r') ^ n := by gcongr ?_ * _; apply hp
_ ≤ C * (a * (‖y‖ / r')) ^ n := by rw [mul_pow, mul_assoc]
#align has_fpower_series_on_ball.uniform_geometric_approx' HasFPowerSeriesOnBall.uniform_geometric_approx'
/-- If a function admits a power series expansion, then it is exponentially close to the partial
sums of this power series on strict subdisks of the disk of convergence. -/
theorem HasFPowerSeriesOnBall.uniform_geometric_approx {r' : ℝ≥0}
(hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
∃ a ∈ Ioo (0 : ℝ) 1,
∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n, ‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n := by
obtain ⟨a, ha, C, hC, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n :=
hf.uniform_geometric_approx' h
refine' ⟨a, ha, C, hC, fun y hy n => (hp y hy n).trans _⟩
have yr' : ‖y‖ < r' := by rwa [ball_zero_eq] at hy
gcongr
exacts [mul_nonneg ha.1.le (div_nonneg (norm_nonneg y) r'.coe_nonneg),
mul_le_of_le_one_right ha.1.le (div_le_one_of_le yr'.le r'.coe_nonneg)]
#align has_fpower_series_on_ball.uniform_geometric_approx HasFPowerSeriesOnBall.uniform_geometric_approx
/-- Taylor formula for an analytic function, `IsBigO` version. -/
theorem HasFPowerSeriesAt.isBigO_sub_partialSum_pow (hf : HasFPowerSeriesAt f p x) (n : ℕ) :
(fun y : E => f (x + y) - p.partialSum n y) =O[𝓝 0] fun y => ‖y‖ ^ n := by
rcases hf with ⟨r, hf⟩
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hf.r_pos with ⟨r', r'0, h⟩
obtain ⟨a, -, C, -, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n :=
hf.uniform_geometric_approx' h
refine' isBigO_iff.2 ⟨C * (a / r') ^ n, _⟩
replace r'0 : 0 < (r' : ℝ); · exact mod_cast r'0
filter_upwards [Metric.ball_mem_nhds (0 : E) r'0] with y hy
simpa [mul_pow, mul_div_assoc, mul_assoc, div_mul_eq_mul_div] using hp y hy n
set_option linter.uppercaseLean3 false in
#align has_fpower_series_at.is_O_sub_partial_sum_pow HasFPowerSeriesAt.isBigO_sub_partialSum_pow
/-- If `f` has formal power series `∑ n, pₙ` on a ball of radius `r`, then for `y, z` in any smaller
ball, the norm of the difference `f y - f z - p 1 (fun _ ↦ y - z)` is bounded above by
`C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖`. This lemma formulates this property using `IsBigO` and
`Filter.principal` on `E × E`. -/
theorem HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal
(hf : HasFPowerSeriesOnBall f p x r) (hr : r' < r) :
(fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) =O[𝓟 (EMetric.ball (x, x) r')]
fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖ := by
lift r' to ℝ≥0 using ne_top_of_lt hr
rcases (zero_le r').eq_or_lt with (rfl | hr'0)
· simp only [isBigO_bot, EMetric.ball_zero, principal_empty, ENNReal.coe_zero]
obtain ⟨a, ha, C, hC : 0 < C, hp⟩ :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n : ℕ, ‖p n‖ * (r' : ℝ) ^ n ≤ C * a ^ n
exact p.norm_mul_pow_le_mul_pow_of_lt_radius (hr.trans_le hf.r_le)
simp only [← le_div_iff (pow_pos (NNReal.coe_pos.2 hr'0) _)] at hp
set L : E × E → ℝ := fun y =>
C * (a / r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * (a / (1 - a) ^ 2 + 2 / (1 - a))
have hL : ∀ y ∈ EMetric.ball (x, x) r', ‖f y.1 - f y.2 - p 1 fun _ => y.1 - y.2‖ ≤ L y := by
intro y hy'
have hy : y ∈ EMetric.ball x r ×ˢ EMetric.ball x r := by
rw [EMetric.ball_prod_same]
exact EMetric.ball_subset_ball hr.le hy'
set A : ℕ → F := fun n => (p n fun _ => y.1 - x) - p n fun _ => y.2 - x
have hA : HasSum (fun n => A (n + 2)) (f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) := by
convert (hasSum_nat_add_iff' 2).2 ((hf.hasSum_sub hy.1).sub (hf.hasSum_sub hy.2)) using 1
rw [Finset.sum_range_succ, Finset.sum_range_one, hf.coeff_zero, hf.coeff_zero, sub_self,
zero_add, ← Subsingleton.pi_single_eq (0 : Fin 1) (y.1 - x), Pi.single,
← Subsingleton.pi_single_eq (0 : Fin 1) (y.2 - x), Pi.single, ← (p 1).map_sub, ← Pi.single,
Subsingleton.pi_single_eq, sub_sub_sub_cancel_right]
rw [EMetric.mem_ball, edist_eq_coe_nnnorm_sub, ENNReal.coe_lt_coe] at hy'
set B : ℕ → ℝ := fun n => C * (a / r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * ((n + 2) * a ^ n)
have hAB : ∀ n, ‖A (n + 2)‖ ≤ B n := fun n =>
calc
‖A (n + 2)‖ ≤ ‖p (n + 2)‖ * ↑(n + 2) * ‖y - (x, x)‖ ^ (n + 1) * ‖y.1 - y.2‖ := by
-- porting note: `pi_norm_const` was `pi_norm_const (_ : E)`
simpa only [Fintype.card_fin, pi_norm_const, Prod.norm_def, Pi.sub_def,
Prod.fst_sub, Prod.snd_sub, sub_sub_sub_cancel_right] using
(p <| n + 2).norm_image_sub_le (fun _ => y.1 - x) fun _ => y.2 - x
_ = ‖p (n + 2)‖ * ‖y - (x, x)‖ ^ n * (↑(n + 2) * ‖y - (x, x)‖ * ‖y.1 - y.2‖) := by
rw [pow_succ ‖y - (x, x)‖]
ring
-- porting note: the two `↑` in `↑r'` are new, without them, Lean fails to synthesize
-- instances `HDiv ℝ ℝ≥0 ?m` or `HMul ℝ ℝ≥0 ?m`
_ ≤ C * a ^ (n + 2) / ↑r' ^ (n + 2)
* ↑r' ^ n * (↑(n + 2) * ‖y - (x, x)‖ * ‖y.1 - y.2‖) := by
have : 0 < a := ha.1
gcongr
· apply hp
· apply hy'.le
_ = B n := by
-- porting note: in the original, `B` was in the `field_simp`, but now Lean does not
-- accept it. The current proof works in Lean 4, but does not in Lean 3.
field_simp [pow_succ]
simp only [mul_assoc, mul_comm, mul_left_comm]
have hBL : HasSum B (L y) := by
apply HasSum.mul_left
simp only [add_mul]
have : ‖a‖ < 1 := by simp only [Real.norm_eq_abs, abs_of_pos ha.1, ha.2]
rw [div_eq_mul_inv, div_eq_mul_inv]
exact (hasSum_coe_mul_geometric_of_norm_lt_1 this).add -- porting note: was `convert`!
((hasSum_geometric_of_norm_lt_1 this).mul_left 2)
exact hA.norm_le_of_bounded hBL hAB
suffices L =O[𝓟 (EMetric.ball (x, x) r')] fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖ by
refine' (IsBigO.of_bound 1 (eventually_principal.2 fun y hy => _)).trans this
rw [one_mul]
exact (hL y hy).trans (le_abs_self _)
simp_rw [mul_right_comm _ (_ * _)] -- porting note: there was an `L` inside the `simp_rw`.
exact (isBigO_refl _ _).const_mul_left _
set_option linter.uppercaseLean3 false in
#align has_fpower_series_on_ball.is_O_image_sub_image_sub_deriv_principal HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal
/-- If `f` has formal power series `∑ n, pₙ` on a ball of radius `r`, then for `y, z` in any smaller
ball, the norm of the difference `f y - f z - p 1 (fun _ ↦ y - z)` is bounded above by
`C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖`. -/
theorem HasFPowerSeriesOnBall.image_sub_sub_deriv_le (hf : HasFPowerSeriesOnBall f p x r)
(hr : r' < r) :
∃ C, ∀ᵉ (y ∈ EMetric.ball x r') (z ∈ EMetric.ball x r'),
‖f y - f z - p 1 fun _ => y - z‖ ≤ C * max ‖y - x‖ ‖z - x‖ * ‖y - z‖ := by
simpa only [isBigO_principal, mul_assoc, norm_mul, norm_norm, Prod.forall, EMetric.mem_ball,
Prod.edist_eq, max_lt_iff, and_imp, @forall_swap (_ < _) E] using
hf.isBigO_image_sub_image_sub_deriv_principal hr
#align has_fpower_series_on_ball.image_sub_sub_deriv_le HasFPowerSeriesOnBall.image_sub_sub_deriv_le
/-- If `f` has formal power series `∑ n, pₙ` at `x`, then
`f y - f z - p 1 (fun _ ↦ y - z) = O(‖(y, z) - (x, x)‖ * ‖y - z‖)` as `(y, z) → (x, x)`.
In particular, `f` is strictly differentiable at `x`. -/
theorem HasFPowerSeriesAt.isBigO_image_sub_norm_mul_norm_sub (hf : HasFPowerSeriesAt f p x) :
(fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) =O[𝓝 (x, x)] fun y =>
‖y - (x, x)‖ * ‖y.1 - y.2‖ := by
rcases hf with ⟨r, hf⟩
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hf.r_pos with ⟨r', r'0, h⟩
refine' (hf.isBigO_image_sub_image_sub_deriv_principal h).mono _
exact le_principal_iff.2 (EMetric.ball_mem_nhds _ r'0)
set_option linter.uppercaseLean3 false in
#align has_fpower_series_at.is_O_image_sub_norm_mul_norm_sub HasFPowerSeriesAt.isBigO_image_sub_norm_mul_norm_sub
/-- If a function admits a power series expansion at `x`, then it is the uniform limit of the
partial sums of this power series on strict subdisks of the disk of convergence, i.e., `f (x + y)`
is the uniform limit of `p.partialSum n y` there. -/
theorem HasFPowerSeriesOnBall.tendstoUniformlyOn {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r)
(h : (r' : ℝ≥0∞) < r) :
TendstoUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
(Metric.ball (0 : E) r') := by
obtain ⟨a, ha, C, -, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n
exact hf.uniform_geometric_approx h
refine' Metric.tendstoUniformlyOn_iff.2 fun ε εpos => _
have L : Tendsto (fun n => (C : ℝ) * a ^ n) atTop (𝓝 ((C : ℝ) * 0)) :=
tendsto_const_nhds.mul (tendsto_pow_atTop_nhds_0_of_lt_1 ha.1.le ha.2)
rw [mul_zero] at L
refine' (L.eventually (gt_mem_nhds εpos)).mono fun n hn y hy => _
rw [dist_eq_norm]
exact (hp y hy n).trans_lt hn
#align has_fpower_series_on_ball.tendsto_uniformly_on HasFPowerSeriesOnBall.tendstoUniformlyOn
/-- If a function admits a power series expansion at `x`, then it is the locally uniform limit of
the partial sums of this power series on the disk of convergence, i.e., `f (x + y)`
is the locally uniform limit of `p.partialSum n y` there. -/
theorem HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn (hf : HasFPowerSeriesOnBall f p x r) :
TendstoLocallyUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
(EMetric.ball (0 : E) r) := by
intro u hu x hx
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hx with ⟨r', xr', hr'⟩
have : EMetric.ball (0 : E) r' ∈ 𝓝 x := IsOpen.mem_nhds EMetric.isOpen_ball xr'
refine' ⟨EMetric.ball (0 : E) r', mem_nhdsWithin_of_mem_nhds this, _⟩
simpa [Metric.emetric_ball_nnreal] using hf.tendstoUniformlyOn hr' u hu
#align has_fpower_series_on_ball.tendsto_locally_uniformly_on HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn
/-- If a function admits a power series expansion at `x`, then it is the uniform limit of the
partial sums of this power series on strict subdisks of the disk of convergence, i.e., `f y`
is the uniform limit of `p.partialSum n (y - x)` there. -/
theorem HasFPowerSeriesOnBall.tendstoUniformlyOn' {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r)
(h : (r' : ℝ≥0∞) < r) :
TendstoUniformlyOn (fun n y => p.partialSum n (y - x)) f atTop (Metric.ball (x : E) r') := by
convert (hf.tendstoUniformlyOn h).comp fun y => y - x using 1
· simp [(· ∘ ·)]
· ext z
simp [dist_eq_norm]
#align has_fpower_series_on_ball.tendsto_uniformly_on' HasFPowerSeriesOnBall.tendstoUniformlyOn'
/-- If a function admits a power series expansion at `x`, then it is the locally uniform limit of
the partial sums of this power series on the disk of convergence, i.e., `f y`
is the locally uniform limit of `p.partialSum n (y - x)` there. -/
theorem HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn' (hf : HasFPowerSeriesOnBall f p x r) :
TendstoLocallyUniformlyOn (fun n y => p.partialSum n (y - x)) f atTop
(EMetric.ball (x : E) r) := by
have A : ContinuousOn (fun y : E => y - x) (EMetric.ball (x : E) r) :=
(continuous_id.sub continuous_const).continuousOn
convert hf.tendstoLocallyUniformlyOn.comp (fun y : E => y - x) _ A using 1
· ext z
simp
· intro z
simp [edist_eq_coe_nnnorm, edist_eq_coe_nnnorm_sub]
#align has_fpower_series_on_ball.tendsto_locally_uniformly_on' HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn'
/-- If a function admits a power series expansion on a disk, then it is continuous there. -/
protected theorem HasFPowerSeriesOnBall.continuousOn (hf : HasFPowerSeriesOnBall f p x r) :
ContinuousOn f (EMetric.ball x r) :=
hf.tendstoLocallyUniformlyOn'.continuousOn <|
eventually_of_forall fun n =>
((p.partialSum_continuous n).comp (continuous_id.sub continuous_const)).continuousOn
#align has_fpower_series_on_ball.continuous_on HasFPowerSeriesOnBall.continuousOn
protected theorem HasFPowerSeriesAt.continuousAt (hf : HasFPowerSeriesAt f p x) :
ContinuousAt f x :=
let ⟨_, hr⟩ := hf
hr.continuousOn.continuousAt (EMetric.ball_mem_nhds x hr.r_pos)
#align has_fpower_series_at.continuous_at HasFPowerSeriesAt.continuousAt
protected theorem AnalyticAt.continuousAt (hf : AnalyticAt 𝕜 f x) : ContinuousAt f x :=
let ⟨_, hp⟩ := hf
hp.continuousAt
#align analytic_at.continuous_at AnalyticAt.continuousAt
protected theorem AnalyticOn.continuousOn {s : Set E} (hf : AnalyticOn 𝕜 f s) : ContinuousOn f s :=
fun x hx => (hf x hx).continuousAt.continuousWithinAt
#align analytic_on.continuous_on AnalyticOn.continuousOn
/-- Analytic everywhere implies continuous -/
theorem AnalyticOn.continuous {f : E → F} (fa : AnalyticOn 𝕜 f univ) : Continuous f := by
rw [continuous_iff_continuousOn_univ]; exact fa.continuousOn
/-- In a complete space, the sum of a converging power series `p` admits `p` as a power series.
This is not totally obvious as we need to check the convergence of the series. -/
protected theorem FormalMultilinearSeries.hasFPowerSeriesOnBall [CompleteSpace F]
(p : FormalMultilinearSeries 𝕜 E F) (h : 0 < p.radius) :
HasFPowerSeriesOnBall p.sum p 0 p.radius :=
{ r_le := le_rfl
r_pos := h
hasSum := fun hy => by
rw [zero_add]
exact p.hasSum hy }
#align formal_multilinear_series.has_fpower_series_on_ball FormalMultilinearSeries.hasFPowerSeriesOnBall
theorem HasFPowerSeriesOnBall.sum (h : HasFPowerSeriesOnBall f p x r) {y : E}
(hy : y ∈ EMetric.ball (0 : E) r) : f (x + y) = p.sum y :=
(h.hasSum hy).tsum_eq.symm
#align has_fpower_series_on_ball.sum HasFPowerSeriesOnBall.sum
/-- The sum of a converging power series is continuous in its disk of convergence. -/
protected theorem FormalMultilinearSeries.continuousOn [CompleteSpace F] :
ContinuousOn p.sum (EMetric.ball 0 p.radius) := by
rcases (zero_le p.radius).eq_or_lt with h | h
· simp [← h, continuousOn_empty]
· exact (p.hasFPowerSeriesOnBall h).continuousOn
#align formal_multilinear_series.continuous_on FormalMultilinearSeries.continuousOn
end
/-!
### Uniqueness of power series
If a function `f : E → F` has two representations as power series at a point `x : E`, corresponding
to formal multilinear series `p₁` and `p₂`, then these representations agree term-by-term. That is,
for any `n : ℕ` and `y : E`, `p₁ n (fun i ↦ y) = p₂ n (fun i ↦ y)`. In the one-dimensional case,
when `f : 𝕜 → E`, the continuous multilinear maps `p₁ n` and `p₂ n` are given by
`ContinuousMultilinearMap.mkPiField`, and hence are determined completely by the value of
`p₁ n (fun i ↦ 1)`, so `p₁ = p₂`. Consequently, the radius of convergence for one series can be
transferred to the other.
-/
section Uniqueness
open ContinuousMultilinearMap
theorem Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero {n : ℕ} {p : E[×n]→L[𝕜] F}
(h : (fun y => p fun _ => y) =O[𝓝 0] fun y => ‖y‖ ^ (n + 1)) (y : E) : (p fun _ => y) = 0 := by
obtain ⟨c, c_pos, hc⟩ := h.exists_pos
obtain ⟨t, ht, t_open, z_mem⟩ := eventually_nhds_iff.mp (isBigOWith_iff.mp hc)
obtain ⟨δ, δ_pos, δε⟩ := (Metric.isOpen_iff.mp t_open) 0 z_mem
clear h hc z_mem
cases' n with n
· exact norm_eq_zero.mp (by
-- porting note: the symmetric difference of the `simpa only` sets:
-- added `Nat.zero_eq, zero_add, pow_one`
-- removed `zero_pow', Ne.def, Nat.one_ne_zero, not_false_iff`
simpa only [Nat.zero_eq, fin0_apply_norm, norm_eq_zero, norm_zero, zero_add, pow_one,
mul_zero, norm_le_zero_iff] using ht 0 (δε (Metric.mem_ball_self δ_pos)))
· refine' Or.elim (Classical.em (y = 0))
(fun hy => by simpa only [hy] using p.map_zero) fun hy => _
replace hy := norm_pos_iff.mpr hy
refine' norm_eq_zero.mp (le_antisymm (le_of_forall_pos_le_add fun ε ε_pos => _) (norm_nonneg _))
have h₀ := _root_.mul_pos c_pos (pow_pos hy (n.succ + 1))
obtain ⟨k, k_pos, k_norm⟩ := NormedField.exists_norm_lt 𝕜
(lt_min (mul_pos δ_pos (inv_pos.mpr hy)) (mul_pos ε_pos (inv_pos.mpr h₀)))
have h₁ : ‖k • y‖ < δ := by
rw [norm_smul]
exact inv_mul_cancel_right₀ hy.ne.symm δ ▸
mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_left _ _)) hy
have h₂ :=
calc
‖p fun _ => k • y‖ ≤ c * ‖k • y‖ ^ (n.succ + 1) := by
-- porting note: now Lean wants `_root_.`
simpa only [norm_pow, _root_.norm_norm] using ht (k • y) (δε (mem_ball_zero_iff.mpr h₁))
--simpa only [norm_pow, norm_norm] using ht (k • y) (δε (mem_ball_zero_iff.mpr h₁))
_ = ‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))) := by
-- porting note: added `Nat.succ_eq_add_one` since otherwise `ring` does not conclude.
simp only [norm_smul, mul_pow, Nat.succ_eq_add_one]
-- porting note: removed `rw [pow_succ]`, since it now becomes superfluous.
ring
have h₃ : ‖k‖ * (c * ‖y‖ ^ (n.succ + 1)) < ε :=
inv_mul_cancel_right₀ h₀.ne.symm ε ▸
mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_right _ _)) h₀
calc
‖p fun _ => y‖ = ‖k⁻¹ ^ n.succ‖ * ‖p fun _ => k • y‖ := by
simpa only [inv_smul_smul₀ (norm_pos_iff.mp k_pos), norm_smul, Finset.prod_const,
Finset.card_fin] using
congr_arg norm (p.map_smul_univ (fun _ : Fin n.succ => k⁻¹) fun _ : Fin n.succ => k • y)
_ ≤ ‖k⁻¹ ^ n.succ‖ * (‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1)))) := by gcongr
_ = ‖(k⁻¹ * k) ^ n.succ‖ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))) := by
rw [← mul_assoc]
simp [norm_mul, mul_pow]
_ ≤ 0 + ε := by
rw [inv_mul_cancel (norm_pos_iff.mp k_pos)]
simpa using h₃.le
set_option linter.uppercaseLean3 false in
#align asymptotics.is_O.continuous_multilinear_map_apply_eq_zero Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero
/-- If a formal multilinear series `p` represents the zero function at `x : E`, then the
terms `p n (fun i ↦ y)` appearing in the sum are zero for any `n : ℕ`, `y : E`. -/
theorem HasFPowerSeriesAt.apply_eq_zero {p : FormalMultilinearSeries 𝕜 E F} {x : E}
(h : HasFPowerSeriesAt 0 p x) (n : ℕ) : ∀ y : E, (p n fun _ => y) = 0 := by
refine' Nat.strong_induction_on n fun k hk => _
have psum_eq : p.partialSum (k + 1) = fun y => p k fun _ => y := by
funext z
refine' Finset.sum_eq_single _ (fun b hb hnb => _) fun hn => _
· have := Finset.mem_range_succ_iff.mp hb
simp only [hk b (this.lt_of_ne hnb), Pi.zero_apply]
· exact False.elim (hn (Finset.mem_range.mpr (lt_add_one k)))
replace h := h.isBigO_sub_partialSum_pow k.succ
simp only [psum_eq, zero_sub, Pi.zero_apply, Asymptotics.isBigO_neg_left] at h
exact h.continuousMultilinearMap_apply_eq_zero
#align has_fpower_series_at.apply_eq_zero HasFPowerSeriesAt.apply_eq_zero
/-- A one-dimensional formal multilinear series representing the zero function is zero. -/
theorem HasFPowerSeriesAt.eq_zero {p : FormalMultilinearSeries 𝕜 𝕜 E} {x : 𝕜}
(h : HasFPowerSeriesAt 0 p x) : p = 0 := by
-- porting note: `funext; ext` was `ext (n x)`
funext n
ext x
rw [← mkPiField_apply_one_eq_self (p n)]
-- porting note: nasty hack, was `simp [h.apply_eq_zero n 1]`
have := Or.intro_right ?_ (h.apply_eq_zero n 1)
simpa using this
#align has_fpower_series_at.eq_zero HasFPowerSeriesAt.eq_zero
/-- One-dimensional formal multilinear series representing the same function are equal. -/
theorem HasFPowerSeriesAt.eq_formalMultilinearSeries {p₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E}
{f : 𝕜 → E} {x : 𝕜} (h₁ : HasFPowerSeriesAt f p₁ x) (h₂ : HasFPowerSeriesAt f p₂ x) : p₁ = p₂ :=
sub_eq_zero.mp (HasFPowerSeriesAt.eq_zero (by simpa only [sub_self] using h₁.sub h₂))
#align has_fpower_series_at.eq_formal_multilinear_series HasFPowerSeriesAt.eq_formalMultilinearSeries
theorem HasFPowerSeriesAt.eq_formalMultilinearSeries_of_eventually
{p q : FormalMultilinearSeries 𝕜 𝕜 E} {f g : 𝕜 → E} {x : 𝕜} (hp : HasFPowerSeriesAt f p x)
(hq : HasFPowerSeriesAt g q x) (heq : ∀ᶠ z in 𝓝 x, f z = g z) : p = q :=
(hp.congr heq).eq_formalMultilinearSeries hq
#align has_fpower_series_at.eq_formal_multilinear_series_of_eventually HasFPowerSeriesAt.eq_formalMultilinearSeries_of_eventually
/-- A one-dimensional formal multilinear series representing a locally zero function is zero. -/
theorem HasFPowerSeriesAt.eq_zero_of_eventually {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E}
{x : 𝕜} (hp : HasFPowerSeriesAt f p x) (hf : f =ᶠ[𝓝 x] 0) : p = 0 :=
(hp.congr hf).eq_zero
#align has_fpower_series_at.eq_zero_of_eventually HasFPowerSeriesAt.eq_zero_of_eventually
/-- If a function `f : 𝕜 → E` has two power series representations at `x`, then the given radii in
which convergence is guaranteed may be interchanged. This can be useful when the formal multilinear
series in one representation has a particularly nice form, but the other has a larger radius. -/
theorem HasFPowerSeriesOnBall.exchange_radius {p₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E}
{r₁ r₂ : ℝ≥0∞} {x : 𝕜} (h₁ : HasFPowerSeriesOnBall f p₁ x r₁)
(h₂ : HasFPowerSeriesOnBall f p₂ x r₂) : HasFPowerSeriesOnBall f p₁ x r₂ :=
h₂.hasFPowerSeriesAt.eq_formalMultilinearSeries h₁.hasFPowerSeriesAt ▸ h₂
#align has_fpower_series_on_ball.exchange_radius HasFPowerSeriesOnBall.exchange_radius
/-- If a function `f : 𝕜 → E` has power series representation `p` on a ball of some radius and for
each positive radius it has some power series representation, then `p` converges to `f` on the whole
`𝕜`. -/
theorem HasFPowerSeriesOnBall.r_eq_top_of_exists {f : 𝕜 → E} {r : ℝ≥0∞} {x : 𝕜}
{p : FormalMultilinearSeries 𝕜 𝕜 E} (h : HasFPowerSeriesOnBall f p x r)
(h' : ∀ (r' : ℝ≥0) (_ : 0 < r'), ∃ p' : FormalMultilinearSeries 𝕜 𝕜 E,
HasFPowerSeriesOnBall f p' x r') :
HasFPowerSeriesOnBall f p x ∞ :=
{ r_le := ENNReal.le_of_forall_pos_nnreal_lt fun r hr _ =>
let ⟨_, hp'⟩ := h' r hr
(h.exchange_radius hp').r_le
r_pos := ENNReal.coe_lt_top
hasSum := fun {y} _ =>
let ⟨r', hr'⟩ := exists_gt ‖y‖₊
let ⟨_, hp'⟩ := h' r' hr'.ne_bot.bot_lt
(h.exchange_radius hp').hasSum <| mem_emetric_ball_zero_iff.mpr (ENNReal.coe_lt_coe.2 hr') }
#align has_fpower_series_on_ball.r_eq_top_of_exists HasFPowerSeriesOnBall.r_eq_top_of_exists
end Uniqueness
/-!
### Changing origin in a power series
If a function is analytic in a disk `D(x, R)`, then it is analytic in any disk contained in that
one. Indeed, one can write
$$
f (x + y + z) = \sum_{n} p_n (y + z)^n = \sum_{n, k} \binom{n}{k} p_n y^{n-k} z^k
= \sum_{k} \Bigl(\sum_{n} \binom{n}{k} p_n y^{n-k}\Bigr) z^k.
$$
The corresponding power series has thus a `k`-th coefficient equal to
$\sum_{n} \binom{n}{k} p_n y^{n-k}$. In the general case where `pₙ` is a multilinear map, this has
to be interpreted suitably: instead of having a binomial coefficient, one should sum over all
possible subsets `s` of `Fin n` of cardinal `k`, and attribute `z` to the indices in `s` and
`y` to the indices outside of `s`.
In this paragraph, we implement this. The new power series is called `p.changeOrigin y`. Then, we
check its convergence and the fact that its sum coincides with the original sum. The outcome of this
discussion is that the set of points where a function is analytic is open.
-/
namespace FormalMultilinearSeries
section
variable (p : FormalMultilinearSeries 𝕜 E F) {x y : E} {r R : ℝ≥0}
/-- A term of `FormalMultilinearSeries.changeOriginSeries`.
Given a formal multilinear series `p` and a point `x` in its ball of convergence,
`p.changeOrigin x` is a formal multilinear series such that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense. Each term of `p.changeOrigin x`
is itself an analytic function of `x` given by the series `p.changeOriginSeries`. Each term in
`changeOriginSeries` is the sum of `changeOriginSeriesTerm`'s over all `s` of cardinality `l`.
The definition is such that `p.changeOriginSeriesTerm k l s hs (fun _ ↦ x) (fun _ ↦ y) =
p (k + l) (s.piecewise (fun _ ↦ x) (fun _ ↦ y))`
-/
def changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
E[×l]→L[𝕜] E[×k]→L[𝕜] F := by
let a := ContinuousMultilinearMap.curryFinFinset 𝕜 E F hs
(by erw [Finset.card_compl, Fintype.card_fin, hs, add_tsub_cancel_right])
exact a (p (k + l))
#align formal_multilinear_series.change_origin_series_term FormalMultilinearSeries.changeOriginSeriesTerm
theorem changeOriginSeriesTerm_apply (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l)
(x y : E) :
(p.changeOriginSeriesTerm k l s hs (fun _ => x) fun _ => y) =
p (k + l) (s.piecewise (fun _ => x) fun _ => y) :=
ContinuousMultilinearMap.curryFinFinset_apply_const _ _ _ _ _
#align formal_multilinear_series.change_origin_series_term_apply FormalMultilinearSeries.changeOriginSeriesTerm_apply
@[simp]
theorem norm_changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
‖p.changeOriginSeriesTerm k l s hs‖ = ‖p (k + l)‖ := by
simp only [changeOriginSeriesTerm, LinearIsometryEquiv.norm_map]
#align formal_multilinear_series.norm_change_origin_series_term FormalMultilinearSeries.norm_changeOriginSeriesTerm
@[simp]
theorem nnnorm_changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
‖p.changeOriginSeriesTerm k l s hs‖₊ = ‖p (k + l)‖₊ := by
simp only [changeOriginSeriesTerm, LinearIsometryEquiv.nnnorm_map]
#align formal_multilinear_series.nnnorm_change_origin_series_term FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm
theorem nnnorm_changeOriginSeriesTerm_apply_le (k l : ℕ) (s : Finset (Fin (k + l)))
(hs : s.card = l) (x y : E) :
‖p.changeOriginSeriesTerm k l s hs (fun _ => x) fun _ => y‖₊ ≤
‖p (k + l)‖₊ * ‖x‖₊ ^ l * ‖y‖₊ ^ k := by
rw [← p.nnnorm_changeOriginSeriesTerm k l s hs, ← Fin.prod_const, ← Fin.prod_const]
apply ContinuousMultilinearMap.le_of_op_nnnorm_le
apply ContinuousMultilinearMap.le_op_nnnorm
#align formal_multilinear_series.nnnorm_change_origin_series_term_apply_le FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm_apply_le
/-- The power series for `f.changeOrigin k`.
Given a formal multilinear series `p` and a point `x` in its ball of convergence,
`p.changeOrigin x` is a formal multilinear series such that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense. Its `k`-th term is the sum of
the series `p.changeOriginSeries k`. -/
def changeOriginSeries (k : ℕ) : FormalMultilinearSeries 𝕜 E (E[×k]→L[𝕜] F) := fun l =>
∑ s : { s : Finset (Fin (k + l)) // Finset.card s = l }, p.changeOriginSeriesTerm k l s s.2
#align formal_multilinear_series.change_origin_series FormalMultilinearSeries.changeOriginSeries
theorem nnnorm_changeOriginSeries_le_tsum (k l : ℕ) :
‖p.changeOriginSeries k l‖₊ ≤
∑' _ : { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + l)‖₊ :=
(nnnorm_sum_le _ (fun t => changeOriginSeriesTerm p k l (Subtype.val t) t.prop)).trans_eq <| by
simp_rw [tsum_fintype, nnnorm_changeOriginSeriesTerm (p := p) (k := k) (l := l)]
#align formal_multilinear_series.nnnorm_change_origin_series_le_tsum FormalMultilinearSeries.nnnorm_changeOriginSeries_le_tsum
theorem nnnorm_changeOriginSeries_apply_le_tsum (k l : ℕ) (x : E) :
‖p.changeOriginSeries k l fun _ => x‖₊ ≤
∑' _ : { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + l)‖₊ * ‖x‖₊ ^ l := by
rw [NNReal.tsum_mul_right, ← Fin.prod_const]
exact (p.changeOriginSeries k l).le_of_op_nnnorm_le _ (p.nnnorm_changeOriginSeries_le_tsum _ _)
#align formal_multilinear_series.nnnorm_change_origin_series_apply_le_tsum FormalMultilinearSeries.nnnorm_changeOriginSeries_apply_le_tsum
/-- Changing the origin of a formal multilinear series `p`, so that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense.
-/
def changeOrigin (x : E) : FormalMultilinearSeries 𝕜 E F :=
fun k => (p.changeOriginSeries k).sum x
#align formal_multilinear_series.change_origin FormalMultilinearSeries.changeOrigin
/-- An auxiliary equivalence useful in the proofs about
`FormalMultilinearSeries.changeOriginSeries`: the set of triples `(k, l, s)`, where `s` is a
`Finset (Fin (k + l))` of cardinality `l` is equivalent to the set of pairs `(n, s)`, where `s` is a
`Finset (Fin n)`.
The forward map sends `(k, l, s)` to `(k + l, s)` and the inverse map sends `(n, s)` to
`(n - Finset.card s, Finset.card s, s)`. The actual definition is less readable because of problems
with non-definitional equalities. -/
@[simps]
def changeOriginIndexEquiv :
(Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) ≃ Σn : ℕ, Finset (Fin n) where
toFun s := ⟨s.1 + s.2.1, s.2.2⟩
invFun s :=
⟨s.1 - s.2.card, s.2.card,
⟨s.2.map
(Fin.castIso <| (tsub_add_cancel_of_le <| card_finset_fin_le s.2).symm).toEquiv.toEmbedding,
Finset.card_map _⟩⟩
left_inv := by
rintro ⟨k, l, ⟨s : Finset (Fin <| k + l), hs : s.card = l⟩⟩
dsimp only [Subtype.coe_mk]
-- Lean can't automatically generalize `k' = k + l - s.card`, `l' = s.card`, so we explicitly
-- formulate the generalized goal
suffices ∀ k' l', k' = k → l' = l → ∀ (hkl : k + l = k' + l') (hs'),
(⟨k', l', ⟨Finset.map (Fin.castIso hkl).toEquiv.toEmbedding s, hs'⟩⟩ :
Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) = ⟨k, l, ⟨s, hs⟩⟩ by
apply this <;> simp only [hs, add_tsub_cancel_right]
rintro _ _ rfl rfl hkl hs'
simp only [Equiv.refl_toEmbedding, Fin.castIso_refl, Finset.map_refl, eq_self_iff_true,
OrderIso.refl_toEquiv, and_self_iff, heq_iff_eq]
right_inv := by
rintro ⟨n, s⟩
simp [tsub_add_cancel_of_le (card_finset_fin_le s), Fin.castIso_to_equiv]
#align formal_multilinear_series.change_origin_index_equiv FormalMultilinearSeries.changeOriginIndexEquiv
theorem changeOriginSeries_summable_aux₁ {r r' : ℝ≥0} (hr : (r + r' : ℝ≥0∞) < p.radius) :
Summable fun s : Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l } =>
‖p (s.1 + s.2.1)‖₊ * r ^ s.2.1 * r' ^ s.1 := by
rw [← changeOriginIndexEquiv.symm.summable_iff]
dsimp only [Function.comp_def, changeOriginIndexEquiv_symm_apply_fst,
changeOriginIndexEquiv_symm_apply_snd_fst]
have : ∀ n : ℕ,
HasSum (fun s : Finset (Fin n) => ‖p (n - s.card + s.card)‖₊ * r ^ s.card * r' ^ (n - s.card))
(‖p n‖₊ * (r + r') ^ n) := by
intro n
-- TODO: why `simp only [tsub_add_cancel_of_le (card_finset_fin_le _)]` fails?
convert_to HasSum (fun s : Finset (Fin n) => ‖p n‖₊ * (r ^ s.card * r' ^ (n - s.card))) _
· ext1 s
rw [tsub_add_cancel_of_le (card_finset_fin_le _), mul_assoc]
rw [← Fin.sum_pow_mul_eq_add_pow]
exact (hasSum_fintype _).mul_left _
refine' NNReal.summable_sigma.2 ⟨fun n => (this n).summable, _⟩
simp only [(this _).tsum_eq]
exact p.summable_nnnorm_mul_pow hr
#align formal_multilinear_series.change_origin_series_summable_aux₁ FormalMultilinearSeries.changeOriginSeries_summable_aux₁
theorem changeOriginSeries_summable_aux₂ (hr : (r : ℝ≥0∞) < p.radius) (k : ℕ) :
Summable fun s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l } =>
‖p (k + s.1)‖₊ * r ^ s.1 := by
rcases ENNReal.lt_iff_exists_add_pos_lt.1 hr with ⟨r', h0, hr'⟩
simpa only [mul_inv_cancel_right₀ (pow_pos h0 _).ne'] using
((NNReal.summable_sigma.1 (p.changeOriginSeries_summable_aux₁ hr')).1 k).mul_right (r' ^ k)⁻¹
#align formal_multilinear_series.change_origin_series_summable_aux₂ FormalMultilinearSeries.changeOriginSeries_summable_aux₂
theorem changeOriginSeries_summable_aux₃ {r : ℝ≥0} (hr : ↑r < p.radius) (k : ℕ) :
Summable fun l : ℕ => ‖p.changeOriginSeries k l‖₊ * r ^ l := by
refine' NNReal.summable_of_le
(fun n => _) (NNReal.summable_sigma.1 <| p.changeOriginSeries_summable_aux₂ hr k).2
simp only [NNReal.tsum_mul_right]
exact mul_le_mul' (p.nnnorm_changeOriginSeries_le_tsum _ _) le_rfl
#align formal_multilinear_series.change_origin_series_summable_aux₃ FormalMultilinearSeries.changeOriginSeries_summable_aux₃
theorem le_changeOriginSeries_radius (k : ℕ) : p.radius ≤ (p.changeOriginSeries k).radius :=
ENNReal.le_of_forall_nnreal_lt fun _r hr =>
le_radius_of_summable_nnnorm _ (p.changeOriginSeries_summable_aux₃ hr k)
#align formal_multilinear_series.le_change_origin_series_radius FormalMultilinearSeries.le_changeOriginSeries_radius
theorem nnnorm_changeOrigin_le (k : ℕ) (h : (‖x‖₊ : ℝ≥0∞) < p.radius) :
‖p.changeOrigin x k‖₊ ≤
∑' s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + s.1)‖₊ * ‖x‖₊ ^ s.1 := by
refine' tsum_of_nnnorm_bounded _ fun l => p.nnnorm_changeOriginSeries_apply_le_tsum k l x
have := p.changeOriginSeries_summable_aux₂ h k
refine' HasSum.sigma this.hasSum fun l => _
exact ((NNReal.summable_sigma.1 this).1 l).hasSum
#align formal_multilinear_series.nnnorm_change_origin_le FormalMultilinearSeries.nnnorm_changeOrigin_le
/-- The radius of convergence of `p.changeOrigin x` is at least `p.radius - ‖x‖`. In other words,
`p.changeOrigin x` is well defined on the largest ball contained in the original ball of
convergence. -/
theorem changeOrigin_radius : p.radius - ‖x‖₊ ≤ (p.changeOrigin x).radius := by
refine' ENNReal.le_of_forall_pos_nnreal_lt fun r _h0 hr => _
rw [lt_tsub_iff_right, add_comm] at hr
have hr' : (‖x‖₊ : ℝ≥0∞) < p.radius := (le_add_right le_rfl).trans_lt hr
apply le_radius_of_summable_nnnorm
have : ∀ k : ℕ,
‖p.changeOrigin x k‖₊ * r ^ k ≤
(∑' s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + s.1)‖₊ * ‖x‖₊ ^ s.1) *
r ^ k :=
fun k => mul_le_mul_right' (p.nnnorm_changeOrigin_le k hr') (r ^ k)
refine' NNReal.summable_of_le this _
simpa only [← NNReal.tsum_mul_right] using
(NNReal.summable_sigma.1 (p.changeOriginSeries_summable_aux₁ hr)).2
#align formal_multilinear_series.change_origin_radius FormalMultilinearSeries.changeOrigin_radius
end
-- From this point on, assume that the space is complete, to make sure that series that converge
-- in norm also converge in `F`.
variable [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x y : E} {r R : ℝ≥0}
theorem hasFPowerSeriesOnBall_changeOrigin (k : ℕ) (hr : 0 < p.radius) :
HasFPowerSeriesOnBall (fun x => p.changeOrigin x k) (p.changeOriginSeries k) 0 p.radius :=
have := p.le_changeOriginSeries_radius k
((p.changeOriginSeries k).hasFPowerSeriesOnBall (hr.trans_le this)).mono hr this
#align formal_multilinear_series.has_fpower_series_on_ball_change_origin FormalMultilinearSeries.hasFPowerSeriesOnBall_changeOrigin
/-- Summing the series `p.changeOrigin x` at a point `y` gives back `p (x + y)`. -/
theorem changeOrigin_eval (h : (‖x‖₊ + ‖y‖₊ : ℝ≥0∞) < p.radius) :
(p.changeOrigin x).sum y = p.sum (x + y) := by
have radius_pos : 0 < p.radius := lt_of_le_of_lt (zero_le _) h
have x_mem_ball : x ∈ EMetric.ball (0 : E) p.radius :=
mem_emetric_ball_zero_iff.2 ((le_add_right le_rfl).trans_lt h)
have y_mem_ball : y ∈ EMetric.ball (0 : E) (p.changeOrigin x).radius := by
refine' mem_emetric_ball_zero_iff.2 (lt_of_lt_of_le _ p.changeOrigin_radius)
rwa [lt_tsub_iff_right, add_comm]
have x_add_y_mem_ball : x + y ∈ EMetric.ball (0 : E) p.radius := by
refine' mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt _ h)
exact mod_cast nnnorm_add_le x y
set f : (Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) → F := fun s =>
p.changeOriginSeriesTerm s.1 s.2.1 s.2.2 s.2.2.2 (fun _ => x) fun _ => y
have hsf : Summable f := by
refine' .of_nnnorm_bounded _ (p.changeOriginSeries_summable_aux₁ h) _
rintro ⟨k, l, s, hs⟩
dsimp only [Subtype.coe_mk]
exact p.nnnorm_changeOriginSeriesTerm_apply_le _ _ _ _ _ _
have hf : HasSum f ((p.changeOrigin x).sum y) := by
refine' HasSum.sigma_of_hasSum ((p.changeOrigin x).summable y_mem_ball).hasSum (fun k => _) hsf
· dsimp only
refine' ContinuousMultilinearMap.hasSum_eval _ _
have := (p.hasFPowerSeriesOnBall_changeOrigin k radius_pos).hasSum x_mem_ball
rw [zero_add] at this
refine' HasSum.sigma_of_hasSum this (fun l => _) _
· simp only [changeOriginSeries, ContinuousMultilinearMap.sum_apply]
apply hasSum_fintype
· refine' .of_nnnorm_bounded _
(p.changeOriginSeries_summable_aux₂ (mem_emetric_ball_zero_iff.1 x_mem_ball) k) fun s => _
refine' (ContinuousMultilinearMap.le_op_nnnorm _ _).trans_eq _
simp
refine' hf.unique (changeOriginIndexEquiv.symm.hasSum_iff.1 _)
refine' HasSum.sigma_of_hasSum
(p.hasSum x_add_y_mem_ball) (fun n => _) (changeOriginIndexEquiv.symm.summable_iff.2 hsf)
erw [(p n).map_add_univ (fun _ => x) fun _ => y]
-- porting note: added explicit function
convert hasSum_fintype (fun c : Finset (Fin n) => f (changeOriginIndexEquiv.symm ⟨n, c⟩))
rename_i s _
dsimp only [changeOriginSeriesTerm, (· ∘ ·), changeOriginIndexEquiv_symm_apply_fst,
changeOriginIndexEquiv_symm_apply_snd_fst, changeOriginIndexEquiv_symm_apply_snd_snd_coe]
rw [ContinuousMultilinearMap.curryFinFinset_apply_const]
have : ∀ (m) (hm : n = m), p n (s.piecewise (fun _ => x) fun _ => y) =
p m ((s.map (Fin.castIso hm).toEquiv.toEmbedding).piecewise (fun _ => x) fun _ => y) := by
rintro m rfl
simp (config := { unfoldPartialApp := true }) [Finset.piecewise]
apply this
#align formal_multilinear_series.change_origin_eval FormalMultilinearSeries.changeOrigin_eval
/-- Power series terms are analytic as we vary the origin -/
theorem analyticAt_changeOrigin (p : FormalMultilinearSeries 𝕜 E F) (rp : p.radius > 0) (n : ℕ) :
AnalyticAt 𝕜 (fun x ↦ p.changeOrigin x n) 0 :=
(FormalMultilinearSeries.hasFPowerSeriesOnBall_changeOrigin p n rp).analyticAt
end FormalMultilinearSeries
section
variable [CompleteSpace F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x y : E} {r : ℝ≥0∞}
/-- If a function admits a power series expansion `p` on a ball `B (x, r)`, then it also admits a
power series on any subball of this ball (even with a different center), given by `p.changeOrigin`.
-/
theorem HasFPowerSeriesOnBall.changeOrigin (hf : HasFPowerSeriesOnBall f p x r)
(h : (‖y‖₊ : ℝ≥0∞) < r) : HasFPowerSeriesOnBall f (p.changeOrigin y) (x + y) (r - ‖y‖₊) :=
{ r_le := by
apply le_trans _ p.changeOrigin_radius
exact tsub_le_tsub hf.r_le le_rfl
r_pos := by simp [h]
hasSum := fun {z} hz => by
have : f (x + y + z) =
FormalMultilinearSeries.sum (FormalMultilinearSeries.changeOrigin p y) z := by
rw [mem_emetric_ball_zero_iff, lt_tsub_iff_right, add_comm] at hz
rw [p.changeOrigin_eval (hz.trans_le hf.r_le), add_assoc, hf.sum]
refine' mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt _ hz)
exact mod_cast nnnorm_add_le y z
rw [this]
apply (p.changeOrigin y).hasSum
refine' EMetric.ball_subset_ball (le_trans _ p.changeOrigin_radius) hz
exact tsub_le_tsub hf.r_le le_rfl }
#align has_fpower_series_on_ball.change_origin HasFPowerSeriesOnBall.changeOrigin
/-- If a function admits a power series expansion `p` on an open ball `B (x, r)`, then
it is analytic at every point of this ball. -/
theorem HasFPowerSeriesOnBall.analyticAt_of_mem (hf : HasFPowerSeriesOnBall f p x r)
(h : y ∈ EMetric.ball x r) : AnalyticAt 𝕜 f y := by
have : (‖y - x‖₊ : ℝ≥0∞) < r := by simpa [edist_eq_coe_nnnorm_sub] using h
have := hf.changeOrigin this
rw [add_sub_cancel'_right] at this
exact this.analyticAt
#align has_fpower_series_on_ball.analytic_at_of_mem HasFPowerSeriesOnBall.analyticAt_of_mem
theorem HasFPowerSeriesOnBall.analyticOn (hf : HasFPowerSeriesOnBall f p x r) :
AnalyticOn 𝕜 f (EMetric.ball x r) :=
fun _y hy => hf.analyticAt_of_mem hy
#align has_fpower_series_on_ball.analytic_on HasFPowerSeriesOnBall.analyticOn
variable (𝕜 f)
/-- For any function `f` from a normed vector space to a Banach space, the set of points `x` such
that `f` is analytic at `x` is open. -/
theorem isOpen_analyticAt : IsOpen { x | AnalyticAt 𝕜 f x } := by
rw [isOpen_iff_mem_nhds]
rintro x ⟨p, r, hr⟩
exact mem_of_superset (EMetric.ball_mem_nhds _ hr.r_pos) fun y hy => hr.analyticAt_of_mem hy
#align is_open_analytic_at isOpen_analyticAt
variable {𝕜}
theorem AnalyticAt.eventually_analyticAt {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∀ᶠ y in 𝓝 x, AnalyticAt 𝕜 f y :=
(isOpen_analyticAt 𝕜 f).mem_nhds h
theorem AnalyticAt.exists_mem_nhds_analyticOn {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∃ s ∈ 𝓝 x, AnalyticOn 𝕜 f s :=
h.eventually_analyticAt.exists_mem
/-- If we're analytic at a point, we're analytic in a nonempty ball -/
theorem AnalyticAt.exists_ball_analyticOn {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∃ r : ℝ, 0 < r ∧ AnalyticOn 𝕜 f (Metric.ball x r) :=
Metric.isOpen_iff.mp (isOpen_analyticAt _ _) _ h
end
section
open FormalMultilinearSeries
variable {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E} {z₀ : 𝕜}
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z)) := by
refine' ⟨fun ⟨r, _, r_pos, h⟩ =>
eventually_of_mem (EMetric.ball_mem_nhds 0 r_pos) fun _ => by simpa using h, _⟩
simp only [Metric.eventually_nhds_iff]
rintro ⟨r, r_pos, h⟩
refine' ⟨p.radius ⊓ r.toNNReal, by simp, _, _⟩
· simp only [r_pos.lt, lt_inf_iff, ENNReal.coe_pos, Real.toNNReal_pos, and_true_iff]
obtain ⟨z, z_pos, le_z⟩ := NormedField.exists_norm_lt 𝕜 r_pos.lt
have : (‖z‖₊ : ENNReal) ≤ p.radius := by
simp only [dist_zero_right] at h
apply FormalMultilinearSeries.le_radius_of_tendsto
convert tendsto_norm.comp (h le_z).summable.tendsto_atTop_zero
funext
simp [norm_smul, mul_comm]
refine' lt_of_lt_of_le _ this
simp only [ENNReal.coe_pos]
exact zero_lt_iff.mpr (nnnorm_ne_zero_iff.mpr (norm_pos_iff.mp z_pos))
· simp only [EMetric.mem_ball, lt_inf_iff, edist_lt_coe, apply_eq_pow_smul_coeff, and_imp,
dist_zero_right] at h ⊢
|
refine' fun {y} _ hyr => h _
|
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z)) := by
refine' ⟨fun ⟨r, _, r_pos, h⟩ =>
eventually_of_mem (EMetric.ball_mem_nhds 0 r_pos) fun _ => by simpa using h, _⟩
simp only [Metric.eventually_nhds_iff]
rintro ⟨r, r_pos, h⟩
refine' ⟨p.radius ⊓ r.toNNReal, by simp, _, _⟩
· simp only [r_pos.lt, lt_inf_iff, ENNReal.coe_pos, Real.toNNReal_pos, and_true_iff]
obtain ⟨z, z_pos, le_z⟩ := NormedField.exists_norm_lt 𝕜 r_pos.lt
have : (‖z‖₊ : ENNReal) ≤ p.radius := by
simp only [dist_zero_right] at h
apply FormalMultilinearSeries.le_radius_of_tendsto
convert tendsto_norm.comp (h le_z).summable.tendsto_atTop_zero
funext
simp [norm_smul, mul_comm]
refine' lt_of_lt_of_le _ this
simp only [ENNReal.coe_pos]
exact zero_lt_iff.mpr (nnnorm_ne_zero_iff.mpr (norm_pos_iff.mp z_pos))
· simp only [EMetric.mem_ball, lt_inf_iff, edist_lt_coe, apply_eq_pow_smul_coeff, and_imp,
dist_zero_right] at h ⊢
|
Mathlib.Analysis.Analytic.Basic.1430_0.jQw1fRSE1vGpOll
|
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z))
|
Mathlib_Analysis_Analytic_Basic
|
case intro.intro.refine'_2
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
p : FormalMultilinearSeries 𝕜 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
r : ℝ
r_pos : r > 0
h : ∀ ⦃y : 𝕜⦄, ‖y‖ < r → HasSum (fun n => y ^ n • coeff p n) (f (z₀ + y))
y : 𝕜
x✝ : edist y 0 < radius p
hyr : nndist y 0 < Real.toNNReal r
⊢ ‖y‖ < r
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Logic.Equiv.Fin
import Mathlib.Topology.Algebra.InfiniteSum.Module
#align_import analysis.analytic.basic from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
/-!
# Analytic functions
A function is analytic in one dimension around `0` if it can be written as a converging power series
`Σ pₙ zⁿ`. This definition can be extended to any dimension (even in infinite dimension) by
requiring that `pₙ` is a continuous `n`-multilinear map. In general, `pₙ` is not unique (in two
dimensions, taking `p₂ (x, y) (x', y') = x y'` or `y x'` gives the same map when applied to a
vector `(x, y) (x, y)`). A way to guarantee uniqueness is to take a symmetric `pₙ`, but this is not
always possible in nonzero characteristic (in characteristic 2, the previous example has no
symmetric representative). Therefore, we do not insist on symmetry or uniqueness in the definition,
and we only require the existence of a converging series.
The general framework is important to say that the exponential map on bounded operators on a Banach
space is analytic, as well as the inverse on invertible operators.
## Main definitions
Let `p` be a formal multilinear series from `E` to `F`, i.e., `p n` is a multilinear map on `E^n`
for `n : ℕ`.
* `p.radius`: the largest `r : ℝ≥0∞` such that `‖p n‖ * r^n` grows subexponentially.
* `p.le_radius_of_bound`, `p.le_radius_of_bound_nnreal`, `p.le_radius_of_isBigO`: if `‖p n‖ * r ^ n`
is bounded above, then `r ≤ p.radius`;
* `p.isLittleO_of_lt_radius`, `p.norm_mul_pow_le_mul_pow_of_lt_radius`,
`p.isLittleO_one_of_lt_radius`,
`p.norm_mul_pow_le_of_lt_radius`, `p.nnnorm_mul_pow_le_of_lt_radius`: if `r < p.radius`, then
`‖p n‖ * r ^ n` tends to zero exponentially;
* `p.lt_radius_of_isBigO`: if `r ≠ 0` and `‖p n‖ * r ^ n = O(a ^ n)` for some `-1 < a < 1`, then
`r < p.radius`;
* `p.partialSum n x`: the sum `∑_{i = 0}^{n-1} pᵢ xⁱ`.
* `p.sum x`: the sum `∑'_{i = 0}^{∞} pᵢ xⁱ`.
Additionally, let `f` be a function from `E` to `F`.
* `HasFPowerSeriesOnBall f p x r`: on the ball of center `x` with radius `r`,
`f (x + y) = ∑'_n pₙ yⁿ`.
* `HasFPowerSeriesAt f p x`: on some ball of center `x` with positive radius, holds
`HasFPowerSeriesOnBall f p x r`.
* `AnalyticAt 𝕜 f x`: there exists a power series `p` such that holds `HasFPowerSeriesAt f p x`.
* `AnalyticOn 𝕜 f s`: the function `f` is analytic at every point of `s`.
We develop the basic properties of these notions, notably:
* If a function admits a power series, it is continuous (see
`HasFPowerSeriesOnBall.continuousOn` and `HasFPowerSeriesAt.continuousAt` and
`AnalyticAt.continuousAt`).
* In a complete space, the sum of a formal power series with positive radius is well defined on the
disk of convergence, see `FormalMultilinearSeries.hasFPowerSeriesOnBall`.
* If a function admits a power series in a ball, then it is analytic at any point `y` of this ball,
and the power series there can be expressed in terms of the initial power series `p` as
`p.changeOrigin y`. See `HasFPowerSeriesOnBall.changeOrigin`. It follows in particular that
the set of points at which a given function is analytic is open, see `isOpen_analyticAt`.
## Implementation details
We only introduce the radius of convergence of a power series, as `p.radius`.
For a power series in finitely many dimensions, there is a finer (directional, coordinate-dependent)
notion, describing the polydisk of convergence. This notion is more specific, and not necessary to
build the general theory. We do not define it here.
-/
noncomputable section
variable {𝕜 E F G : Type*}
open Topology Classical BigOperators NNReal Filter ENNReal
open Set Filter Asymptotics
namespace FormalMultilinearSeries
variable [Ring 𝕜] [AddCommGroup E] [AddCommGroup F] [Module 𝕜 E] [Module 𝕜 F]
variable [TopologicalSpace E] [TopologicalSpace F]
variable [TopologicalAddGroup E] [TopologicalAddGroup F]
variable [ContinuousConstSMul 𝕜 E] [ContinuousConstSMul 𝕜 F]
/-- Given a formal multilinear series `p` and a vector `x`, then `p.sum x` is the sum `Σ pₙ xⁿ`. A
priori, it only behaves well when `‖x‖ < p.radius`. -/
protected def sum (p : FormalMultilinearSeries 𝕜 E F) (x : E) : F :=
∑' n : ℕ, p n fun _ => x
#align formal_multilinear_series.sum FormalMultilinearSeries.sum
/-- Given a formal multilinear series `p` and a vector `x`, then `p.partialSum n x` is the sum
`Σ pₖ xᵏ` for `k ∈ {0,..., n-1}`. -/
def partialSum (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) (x : E) : F :=
∑ k in Finset.range n, p k fun _ : Fin k => x
#align formal_multilinear_series.partial_sum FormalMultilinearSeries.partialSum
/-- The partial sums of a formal multilinear series are continuous. -/
theorem partialSum_continuous (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) :
Continuous (p.partialSum n) := by
unfold partialSum -- Porting note: added
continuity
#align formal_multilinear_series.partial_sum_continuous FormalMultilinearSeries.partialSum_continuous
end FormalMultilinearSeries
/-! ### The radius of a formal multilinear series -/
variable [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F]
[NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G]
namespace FormalMultilinearSeries
variable (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
/-- The radius of a formal multilinear series is the largest `r` such that the sum `Σ ‖pₙ‖ ‖y‖ⁿ`
converges for all `‖y‖ < r`. This implies that `Σ pₙ yⁿ` converges for all `‖y‖ < r`, but these
definitions are *not* equivalent in general. -/
def radius (p : FormalMultilinearSeries 𝕜 E F) : ℝ≥0∞ :=
⨆ (r : ℝ≥0) (C : ℝ) (_ : ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C), (r : ℝ≥0∞)
#align formal_multilinear_series.radius FormalMultilinearSeries.radius
/-- If `‖pₙ‖ rⁿ` is bounded in `n`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_bound (C : ℝ) {r : ℝ≥0} (h : ∀ n : ℕ, ‖p n‖ * (r : ℝ) ^ n ≤ C) :
(r : ℝ≥0∞) ≤ p.radius :=
le_iSup_of_le r <| le_iSup_of_le C <| le_iSup (fun _ => (r : ℝ≥0∞)) h
#align formal_multilinear_series.le_radius_of_bound FormalMultilinearSeries.le_radius_of_bound
/-- If `‖pₙ‖ rⁿ` is bounded in `n`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_bound_nnreal (C : ℝ≥0) {r : ℝ≥0} (h : ∀ n : ℕ, ‖p n‖₊ * r ^ n ≤ C) :
(r : ℝ≥0∞) ≤ p.radius :=
p.le_radius_of_bound C fun n => mod_cast h n
#align formal_multilinear_series.le_radius_of_bound_nnreal FormalMultilinearSeries.le_radius_of_bound_nnreal
/-- If `‖pₙ‖ rⁿ = O(1)`, as `n → ∞`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_isBigO (h : (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] fun _ => (1 : ℝ)) :
↑r ≤ p.radius :=
Exists.elim (isBigO_one_nat_atTop_iff.1 h) fun C hC =>
p.le_radius_of_bound C fun n => (le_abs_self _).trans (hC n)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.le_radius_of_is_O FormalMultilinearSeries.le_radius_of_isBigO
theorem le_radius_of_eventually_le (C) (h : ∀ᶠ n in atTop, ‖p n‖ * (r : ℝ) ^ n ≤ C) :
↑r ≤ p.radius :=
p.le_radius_of_isBigO <| IsBigO.of_bound C <| h.mono fun n hn => by simpa
#align formal_multilinear_series.le_radius_of_eventually_le FormalMultilinearSeries.le_radius_of_eventually_le
theorem le_radius_of_summable_nnnorm (h : Summable fun n => ‖p n‖₊ * r ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_bound_nnreal (∑' n, ‖p n‖₊ * r ^ n) fun _ => le_tsum' h _
#align formal_multilinear_series.le_radius_of_summable_nnnorm FormalMultilinearSeries.le_radius_of_summable_nnnorm
theorem le_radius_of_summable (h : Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_summable_nnnorm <| by
simp only [← coe_nnnorm] at h
exact mod_cast h
#align formal_multilinear_series.le_radius_of_summable FormalMultilinearSeries.le_radius_of_summable
theorem radius_eq_top_of_forall_nnreal_isBigO
(h : ∀ r : ℝ≥0, (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] fun _ => (1 : ℝ)) : p.radius = ∞ :=
ENNReal.eq_top_of_forall_nnreal_le fun r => p.le_radius_of_isBigO (h r)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.radius_eq_top_of_forall_nnreal_is_O FormalMultilinearSeries.radius_eq_top_of_forall_nnreal_isBigO
theorem radius_eq_top_of_eventually_eq_zero (h : ∀ᶠ n in atTop, p n = 0) : p.radius = ∞ :=
p.radius_eq_top_of_forall_nnreal_isBigO fun r =>
(isBigO_zero _ _).congr' (h.mono fun n hn => by simp [hn]) EventuallyEq.rfl
#align formal_multilinear_series.radius_eq_top_of_eventually_eq_zero FormalMultilinearSeries.radius_eq_top_of_eventually_eq_zero
theorem radius_eq_top_of_forall_image_add_eq_zero (n : ℕ) (hn : ∀ m, p (m + n) = 0) :
p.radius = ∞ :=
p.radius_eq_top_of_eventually_eq_zero <|
mem_atTop_sets.2 ⟨n, fun _ hk => tsub_add_cancel_of_le hk ▸ hn _⟩
#align formal_multilinear_series.radius_eq_top_of_forall_image_add_eq_zero FormalMultilinearSeries.radius_eq_top_of_forall_image_add_eq_zero
@[simp]
theorem constFormalMultilinearSeries_radius {v : F} :
(constFormalMultilinearSeries 𝕜 E v).radius = ⊤ :=
(constFormalMultilinearSeries 𝕜 E v).radius_eq_top_of_forall_image_add_eq_zero 1
(by simp [constFormalMultilinearSeries])
#align formal_multilinear_series.const_formal_multilinear_series_radius FormalMultilinearSeries.constFormalMultilinearSeries_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` tends to zero exponentially:
for some `0 < a < 1`, `‖p n‖ rⁿ = o(aⁿ)`. -/
theorem isLittleO_of_lt_radius (h : ↑r < p.radius) :
∃ a ∈ Ioo (0 : ℝ) 1, (fun n => ‖p n‖ * (r : ℝ) ^ n) =o[atTop] (a ^ ·) := by
have := (TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 4
rw [this]
-- Porting note: was
-- rw [(TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 4]
simp only [radius, lt_iSup_iff] at h
rcases h with ⟨t, C, hC, rt⟩
rw [ENNReal.coe_lt_coe, ← NNReal.coe_lt_coe] at rt
have : 0 < (t : ℝ) := r.coe_nonneg.trans_lt rt
rw [← div_lt_one this] at rt
refine' ⟨_, rt, C, Or.inr zero_lt_one, fun n => _⟩
calc
|‖p n‖ * (r : ℝ) ^ n| = ‖p n‖ * (t : ℝ) ^ n * (r / t : ℝ) ^ n := by
field_simp [mul_right_comm, abs_mul]
_ ≤ C * (r / t : ℝ) ^ n := by gcongr; apply hC
#align formal_multilinear_series.is_o_of_lt_radius FormalMultilinearSeries.isLittleO_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ = o(1)`. -/
theorem isLittleO_one_of_lt_radius (h : ↑r < p.radius) :
(fun n => ‖p n‖ * (r : ℝ) ^ n) =o[atTop] (fun _ => 1 : ℕ → ℝ) :=
let ⟨_, ha, hp⟩ := p.isLittleO_of_lt_radius h
hp.trans <| (isLittleO_pow_pow_of_lt_left ha.1.le ha.2).congr (fun _ => rfl) one_pow
#align formal_multilinear_series.is_o_one_of_lt_radius FormalMultilinearSeries.isLittleO_one_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` tends to zero exponentially:
for some `0 < a < 1` and `C > 0`, `‖p n‖ * r ^ n ≤ C * a ^ n`. -/
theorem norm_mul_pow_le_mul_pow_of_lt_radius (h : ↑r < p.radius) :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C * a ^ n := by
-- Porting note: moved out of `rcases`
have := ((TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 5).mp
(p.isLittleO_of_lt_radius h)
rcases this with ⟨a, ha, C, hC, H⟩
exact ⟨a, ha, C, hC, fun n => (le_abs_self _).trans (H n)⟩
#align formal_multilinear_series.norm_mul_pow_le_mul_pow_of_lt_radius FormalMultilinearSeries.norm_mul_pow_le_mul_pow_of_lt_radius
/-- If `r ≠ 0` and `‖pₙ‖ rⁿ = O(aⁿ)` for some `-1 < a < 1`, then `r < p.radius`. -/
theorem lt_radius_of_isBigO (h₀ : r ≠ 0) {a : ℝ} (ha : a ∈ Ioo (-1 : ℝ) 1)
(hp : (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] (a ^ ·)) : ↑r < p.radius := by
-- Porting note: moved out of `rcases`
have := ((TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 2 5)
rcases this.mp ⟨a, ha, hp⟩ with ⟨a, ha, C, hC, hp⟩
rw [← pos_iff_ne_zero, ← NNReal.coe_pos] at h₀
lift a to ℝ≥0 using ha.1.le
have : (r : ℝ) < r / a := by
simpa only [div_one] using (div_lt_div_left h₀ zero_lt_one ha.1).2 ha.2
norm_cast at this
rw [← ENNReal.coe_lt_coe] at this
refine' this.trans_le (p.le_radius_of_bound C fun n => _)
rw [NNReal.coe_div, div_pow, ← mul_div_assoc, div_le_iff (pow_pos ha.1 n)]
exact (le_abs_self _).trans (hp n)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.lt_radius_of_is_O FormalMultilinearSeries.lt_radius_of_isBigO
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem norm_mul_pow_le_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C :=
let ⟨_, ha, C, hC, h⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius h
⟨C, hC, fun n => (h n).trans <| mul_le_of_le_one_right hC.lt.le (pow_le_one _ ha.1.le ha.2.le)⟩
#align formal_multilinear_series.norm_mul_pow_le_of_lt_radius FormalMultilinearSeries.norm_mul_pow_le_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem norm_le_div_pow_of_pos_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h0 : 0 < r) (h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖ ≤ C / (r : ℝ) ^ n :=
let ⟨C, hC, hp⟩ := p.norm_mul_pow_le_of_lt_radius h
⟨C, hC, fun n => Iff.mpr (le_div_iff (pow_pos h0 _)) (hp n)⟩
#align formal_multilinear_series.norm_le_div_pow_of_pos_of_lt_radius FormalMultilinearSeries.norm_le_div_pow_of_pos_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem nnnorm_mul_pow_le_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖₊ * r ^ n ≤ C :=
let ⟨C, hC, hp⟩ := p.norm_mul_pow_le_of_lt_radius h
⟨⟨C, hC.lt.le⟩, hC, mod_cast hp⟩
#align formal_multilinear_series.nnnorm_mul_pow_le_of_lt_radius FormalMultilinearSeries.nnnorm_mul_pow_le_of_lt_radius
theorem le_radius_of_tendsto (p : FormalMultilinearSeries 𝕜 E F) {l : ℝ}
(h : Tendsto (fun n => ‖p n‖ * (r : ℝ) ^ n) atTop (𝓝 l)) : ↑r ≤ p.radius :=
p.le_radius_of_isBigO (h.isBigO_one _)
#align formal_multilinear_series.le_radius_of_tendsto FormalMultilinearSeries.le_radius_of_tendsto
theorem le_radius_of_summable_norm (p : FormalMultilinearSeries 𝕜 E F)
(hs : Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_tendsto hs.tendsto_atTop_zero
#align formal_multilinear_series.le_radius_of_summable_norm FormalMultilinearSeries.le_radius_of_summable_norm
theorem not_summable_norm_of_radius_lt_nnnorm (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(h : p.radius < ‖x‖₊) : ¬Summable fun n => ‖p n‖ * ‖x‖ ^ n :=
fun hs => not_le_of_lt h (p.le_radius_of_summable_norm hs)
#align formal_multilinear_series.not_summable_norm_of_radius_lt_nnnorm FormalMultilinearSeries.not_summable_norm_of_radius_lt_nnnorm
theorem summable_norm_mul_pow (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0} (h : ↑r < p.radius) :
Summable fun n : ℕ => ‖p n‖ * (r : ℝ) ^ n := by
obtain ⟨a, ha : a ∈ Ioo (0 : ℝ) 1, C, - : 0 < C, hp⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius h
exact .of_nonneg_of_le (fun n => mul_nonneg (norm_nonneg _) (pow_nonneg r.coe_nonneg _))
hp ((summable_geometric_of_lt_1 ha.1.le ha.2).mul_left _)
#align formal_multilinear_series.summable_norm_mul_pow FormalMultilinearSeries.summable_norm_mul_pow
theorem summable_norm_apply (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : Summable fun n : ℕ => ‖p n fun _ => x‖ := by
rw [mem_emetric_ball_zero_iff] at hx
refine' .of_nonneg_of_le
(fun _ => norm_nonneg _) (fun n => ((p n).le_op_norm _).trans_eq _) (p.summable_norm_mul_pow hx)
simp
#align formal_multilinear_series.summable_norm_apply FormalMultilinearSeries.summable_norm_apply
theorem summable_nnnorm_mul_pow (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0} (h : ↑r < p.radius) :
Summable fun n : ℕ => ‖p n‖₊ * r ^ n := by
rw [← NNReal.summable_coe]
push_cast
exact p.summable_norm_mul_pow h
#align formal_multilinear_series.summable_nnnorm_mul_pow FormalMultilinearSeries.summable_nnnorm_mul_pow
protected theorem summable [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : Summable fun n : ℕ => p n fun _ => x :=
(p.summable_norm_apply hx).of_norm
#align formal_multilinear_series.summable FormalMultilinearSeries.summable
theorem radius_eq_top_of_summable_norm (p : FormalMultilinearSeries 𝕜 E F)
(hs : ∀ r : ℝ≥0, Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : p.radius = ∞ :=
ENNReal.eq_top_of_forall_nnreal_le fun r => p.le_radius_of_summable_norm (hs r)
#align formal_multilinear_series.radius_eq_top_of_summable_norm FormalMultilinearSeries.radius_eq_top_of_summable_norm
theorem radius_eq_top_iff_summable_norm (p : FormalMultilinearSeries 𝕜 E F) :
p.radius = ∞ ↔ ∀ r : ℝ≥0, Summable fun n => ‖p n‖ * (r : ℝ) ^ n := by
constructor
· intro h r
obtain ⟨a, ha : a ∈ Ioo (0 : ℝ) 1, C, - : 0 < C, hp⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius
(show (r : ℝ≥0∞) < p.radius from h.symm ▸ ENNReal.coe_lt_top)
refine' .of_norm_bounded
(fun n => (C : ℝ) * a ^ n) ((summable_geometric_of_lt_1 ha.1.le ha.2).mul_left _) fun n => _
specialize hp n
rwa [Real.norm_of_nonneg (mul_nonneg (norm_nonneg _) (pow_nonneg r.coe_nonneg n))]
· exact p.radius_eq_top_of_summable_norm
#align formal_multilinear_series.radius_eq_top_iff_summable_norm FormalMultilinearSeries.radius_eq_top_iff_summable_norm
/-- If the radius of `p` is positive, then `‖pₙ‖` grows at most geometrically. -/
theorem le_mul_pow_of_radius_pos (p : FormalMultilinearSeries 𝕜 E F) (h : 0 < p.radius) :
∃ (C r : _) (hC : 0 < C) (_ : 0 < r), ∀ n, ‖p n‖ ≤ C * r ^ n := by
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 h with ⟨r, r0, rlt⟩
have rpos : 0 < (r : ℝ) := by simp [ENNReal.coe_pos.1 r0]
rcases norm_le_div_pow_of_pos_of_lt_radius p rpos rlt with ⟨C, Cpos, hCp⟩
refine' ⟨C, r⁻¹, Cpos, by simp only [inv_pos, rpos], fun n => _⟩
-- Porting note: was `convert`
rw [inv_pow, ← div_eq_mul_inv]
exact hCp n
#align formal_multilinear_series.le_mul_pow_of_radius_pos FormalMultilinearSeries.le_mul_pow_of_radius_pos
/-- The radius of the sum of two formal series is at least the minimum of their two radii. -/
theorem min_radius_le_radius_add (p q : FormalMultilinearSeries 𝕜 E F) :
min p.radius q.radius ≤ (p + q).radius := by
refine' ENNReal.le_of_forall_nnreal_lt fun r hr => _
rw [lt_min_iff] at hr
have := ((p.isLittleO_one_of_lt_radius hr.1).add (q.isLittleO_one_of_lt_radius hr.2)).isBigO
refine' (p + q).le_radius_of_isBigO ((isBigO_of_le _ fun n => _).trans this)
rw [← add_mul, norm_mul, norm_mul, norm_norm]
exact mul_le_mul_of_nonneg_right ((norm_add_le _ _).trans (le_abs_self _)) (norm_nonneg _)
#align formal_multilinear_series.min_radius_le_radius_add FormalMultilinearSeries.min_radius_le_radius_add
@[simp]
theorem radius_neg (p : FormalMultilinearSeries 𝕜 E F) : (-p).radius = p.radius := by
simp only [radius, neg_apply, norm_neg]
#align formal_multilinear_series.radius_neg FormalMultilinearSeries.radius_neg
protected theorem hasSum [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : HasSum (fun n : ℕ => p n fun _ => x) (p.sum x) :=
(p.summable hx).hasSum
#align formal_multilinear_series.has_sum FormalMultilinearSeries.hasSum
theorem radius_le_radius_continuousLinearMap_comp (p : FormalMultilinearSeries 𝕜 E F)
(f : F →L[𝕜] G) : p.radius ≤ (f.compFormalMultilinearSeries p).radius := by
refine' ENNReal.le_of_forall_nnreal_lt fun r hr => _
apply le_radius_of_isBigO
apply (IsBigO.trans_isLittleO _ (p.isLittleO_one_of_lt_radius hr)).isBigO
refine' IsBigO.mul (@IsBigOWith.isBigO _ _ _ _ _ ‖f‖ _ _ _ _) (isBigO_refl _ _)
refine IsBigOWith.of_bound (eventually_of_forall fun n => ?_)
simpa only [norm_norm] using f.norm_compContinuousMultilinearMap_le (p n)
#align formal_multilinear_series.radius_le_radius_continuous_linear_map_comp FormalMultilinearSeries.radius_le_radius_continuousLinearMap_comp
end FormalMultilinearSeries
/-! ### Expanding a function as a power series -/
section
variable {f g : E → F} {p pf pg : FormalMultilinearSeries 𝕜 E F} {x : E} {r r' : ℝ≥0∞}
/-- Given a function `f : E → F` and a formal multilinear series `p`, we say that `f` has `p` as
a power series on the ball of radius `r > 0` around `x` if `f (x + y) = ∑' pₙ yⁿ` for all `‖y‖ < r`.
-/
structure HasFPowerSeriesOnBall (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (x : E) (r : ℝ≥0∞) :
Prop where
r_le : r ≤ p.radius
r_pos : 0 < r
hasSum :
∀ {y}, y ∈ EMetric.ball (0 : E) r → HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y))
#align has_fpower_series_on_ball HasFPowerSeriesOnBall
/-- Given a function `f : E → F` and a formal multilinear series `p`, we say that `f` has `p` as
a power series around `x` if `f (x + y) = ∑' pₙ yⁿ` for all `y` in a neighborhood of `0`. -/
def HasFPowerSeriesAt (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (x : E) :=
∃ r, HasFPowerSeriesOnBall f p x r
#align has_fpower_series_at HasFPowerSeriesAt
variable (𝕜)
/-- Given a function `f : E → F`, we say that `f` is analytic at `x` if it admits a convergent power
series expansion around `x`. -/
def AnalyticAt (f : E → F) (x : E) :=
∃ p : FormalMultilinearSeries 𝕜 E F, HasFPowerSeriesAt f p x
#align analytic_at AnalyticAt
/-- Given a function `f : E → F`, we say that `f` is analytic on a set `s` if it is analytic around
every point of `s`. -/
def AnalyticOn (f : E → F) (s : Set E) :=
∀ x, x ∈ s → AnalyticAt 𝕜 f x
#align analytic_on AnalyticOn
variable {𝕜}
theorem HasFPowerSeriesOnBall.hasFPowerSeriesAt (hf : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesAt f p x :=
⟨r, hf⟩
#align has_fpower_series_on_ball.has_fpower_series_at HasFPowerSeriesOnBall.hasFPowerSeriesAt
theorem HasFPowerSeriesAt.analyticAt (hf : HasFPowerSeriesAt f p x) : AnalyticAt 𝕜 f x :=
⟨p, hf⟩
#align has_fpower_series_at.analytic_at HasFPowerSeriesAt.analyticAt
theorem HasFPowerSeriesOnBall.analyticAt (hf : HasFPowerSeriesOnBall f p x r) : AnalyticAt 𝕜 f x :=
hf.hasFPowerSeriesAt.analyticAt
#align has_fpower_series_on_ball.analytic_at HasFPowerSeriesOnBall.analyticAt
theorem HasFPowerSeriesOnBall.congr (hf : HasFPowerSeriesOnBall f p x r)
(hg : EqOn f g (EMetric.ball x r)) : HasFPowerSeriesOnBall g p x r :=
{ r_le := hf.r_le
r_pos := hf.r_pos
hasSum := fun {y} hy => by
convert hf.hasSum hy using 1
apply hg.symm
simpa [edist_eq_coe_nnnorm_sub] using hy }
#align has_fpower_series_on_ball.congr HasFPowerSeriesOnBall.congr
/-- If a function `f` has a power series `p` around `x`, then the function `z ↦ f (z - y)` has the
same power series around `x + y`. -/
theorem HasFPowerSeriesOnBall.comp_sub (hf : HasFPowerSeriesOnBall f p x r) (y : E) :
HasFPowerSeriesOnBall (fun z => f (z - y)) p (x + y) r :=
{ r_le := hf.r_le
r_pos := hf.r_pos
hasSum := fun {z} hz => by
convert hf.hasSum hz using 2
abel }
#align has_fpower_series_on_ball.comp_sub HasFPowerSeriesOnBall.comp_sub
theorem HasFPowerSeriesOnBall.hasSum_sub (hf : HasFPowerSeriesOnBall f p x r) {y : E}
(hy : y ∈ EMetric.ball x r) : HasSum (fun n : ℕ => p n fun _ => y - x) (f y) := by
have : y - x ∈ EMetric.ball (0 : E) r := by simpa [edist_eq_coe_nnnorm_sub] using hy
simpa only [add_sub_cancel'_right] using hf.hasSum this
#align has_fpower_series_on_ball.has_sum_sub HasFPowerSeriesOnBall.hasSum_sub
theorem HasFPowerSeriesOnBall.radius_pos (hf : HasFPowerSeriesOnBall f p x r) : 0 < p.radius :=
lt_of_lt_of_le hf.r_pos hf.r_le
#align has_fpower_series_on_ball.radius_pos HasFPowerSeriesOnBall.radius_pos
theorem HasFPowerSeriesAt.radius_pos (hf : HasFPowerSeriesAt f p x) : 0 < p.radius :=
let ⟨_, hr⟩ := hf
hr.radius_pos
#align has_fpower_series_at.radius_pos HasFPowerSeriesAt.radius_pos
theorem HasFPowerSeriesOnBall.mono (hf : HasFPowerSeriesOnBall f p x r) (r'_pos : 0 < r')
(hr : r' ≤ r) : HasFPowerSeriesOnBall f p x r' :=
⟨le_trans hr hf.1, r'_pos, fun hy => hf.hasSum (EMetric.ball_subset_ball hr hy)⟩
#align has_fpower_series_on_ball.mono HasFPowerSeriesOnBall.mono
theorem HasFPowerSeriesAt.congr (hf : HasFPowerSeriesAt f p x) (hg : f =ᶠ[𝓝 x] g) :
HasFPowerSeriesAt g p x := by
rcases hf with ⟨r₁, h₁⟩
rcases EMetric.mem_nhds_iff.mp hg with ⟨r₂, h₂pos, h₂⟩
exact ⟨min r₁ r₂,
(h₁.mono (lt_min h₁.r_pos h₂pos) inf_le_left).congr
fun y hy => h₂ (EMetric.ball_subset_ball inf_le_right hy)⟩
#align has_fpower_series_at.congr HasFPowerSeriesAt.congr
protected theorem HasFPowerSeriesAt.eventually (hf : HasFPowerSeriesAt f p x) :
∀ᶠ r : ℝ≥0∞ in 𝓝[>] 0, HasFPowerSeriesOnBall f p x r :=
let ⟨_, hr⟩ := hf
mem_of_superset (Ioo_mem_nhdsWithin_Ioi (left_mem_Ico.2 hr.r_pos)) fun _ hr' =>
hr.mono hr'.1 hr'.2.le
#align has_fpower_series_at.eventually HasFPowerSeriesAt.eventually
theorem HasFPowerSeriesOnBall.eventually_hasSum (hf : HasFPowerSeriesOnBall f p x r) :
∀ᶠ y in 𝓝 0, HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y)) := by
filter_upwards [EMetric.ball_mem_nhds (0 : E) hf.r_pos] using fun _ => hf.hasSum
#align has_fpower_series_on_ball.eventually_has_sum HasFPowerSeriesOnBall.eventually_hasSum
theorem HasFPowerSeriesAt.eventually_hasSum (hf : HasFPowerSeriesAt f p x) :
∀ᶠ y in 𝓝 0, HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y)) :=
let ⟨_, hr⟩ := hf
hr.eventually_hasSum
#align has_fpower_series_at.eventually_has_sum HasFPowerSeriesAt.eventually_hasSum
theorem HasFPowerSeriesOnBall.eventually_hasSum_sub (hf : HasFPowerSeriesOnBall f p x r) :
∀ᶠ y in 𝓝 x, HasSum (fun n : ℕ => p n fun _ : Fin n => y - x) (f y) := by
filter_upwards [EMetric.ball_mem_nhds x hf.r_pos] with y using hf.hasSum_sub
#align has_fpower_series_on_ball.eventually_has_sum_sub HasFPowerSeriesOnBall.eventually_hasSum_sub
theorem HasFPowerSeriesAt.eventually_hasSum_sub (hf : HasFPowerSeriesAt f p x) :
∀ᶠ y in 𝓝 x, HasSum (fun n : ℕ => p n fun _ : Fin n => y - x) (f y) :=
let ⟨_, hr⟩ := hf
hr.eventually_hasSum_sub
#align has_fpower_series_at.eventually_has_sum_sub HasFPowerSeriesAt.eventually_hasSum_sub
theorem HasFPowerSeriesOnBall.eventually_eq_zero
(hf : HasFPowerSeriesOnBall f (0 : FormalMultilinearSeries 𝕜 E F) x r) :
∀ᶠ z in 𝓝 x, f z = 0 := by
filter_upwards [hf.eventually_hasSum_sub] with z hz using hz.unique hasSum_zero
#align has_fpower_series_on_ball.eventually_eq_zero HasFPowerSeriesOnBall.eventually_eq_zero
theorem HasFPowerSeriesAt.eventually_eq_zero
(hf : HasFPowerSeriesAt f (0 : FormalMultilinearSeries 𝕜 E F) x) : ∀ᶠ z in 𝓝 x, f z = 0 :=
let ⟨_, hr⟩ := hf
hr.eventually_eq_zero
#align has_fpower_series_at.eventually_eq_zero HasFPowerSeriesAt.eventually_eq_zero
theorem hasFPowerSeriesOnBall_const {c : F} {e : E} :
HasFPowerSeriesOnBall (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e ⊤ := by
refine' ⟨by simp, WithTop.zero_lt_top, fun _ => hasSum_single 0 fun n hn => _⟩
simp [constFormalMultilinearSeries_apply hn]
#align has_fpower_series_on_ball_const hasFPowerSeriesOnBall_const
theorem hasFPowerSeriesAt_const {c : F} {e : E} :
HasFPowerSeriesAt (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e :=
⟨⊤, hasFPowerSeriesOnBall_const⟩
#align has_fpower_series_at_const hasFPowerSeriesAt_const
theorem analyticAt_const {v : F} : AnalyticAt 𝕜 (fun _ => v) x :=
⟨constFormalMultilinearSeries 𝕜 E v, hasFPowerSeriesAt_const⟩
#align analytic_at_const analyticAt_const
theorem analyticOn_const {v : F} {s : Set E} : AnalyticOn 𝕜 (fun _ => v) s :=
fun _ _ => analyticAt_const
#align analytic_on_const analyticOn_const
theorem HasFPowerSeriesOnBall.add (hf : HasFPowerSeriesOnBall f pf x r)
(hg : HasFPowerSeriesOnBall g pg x r) : HasFPowerSeriesOnBall (f + g) (pf + pg) x r :=
{ r_le := le_trans (le_min_iff.2 ⟨hf.r_le, hg.r_le⟩) (pf.min_radius_le_radius_add pg)
r_pos := hf.r_pos
hasSum := fun hy => (hf.hasSum hy).add (hg.hasSum hy) }
#align has_fpower_series_on_ball.add HasFPowerSeriesOnBall.add
theorem HasFPowerSeriesAt.add (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x) :
HasFPowerSeriesAt (f + g) (pf + pg) x := by
rcases (hf.eventually.and hg.eventually).exists with ⟨r, hr⟩
exact ⟨r, hr.1.add hr.2⟩
#align has_fpower_series_at.add HasFPowerSeriesAt.add
theorem AnalyticAt.congr (hf : AnalyticAt 𝕜 f x) (hg : f =ᶠ[𝓝 x] g) : AnalyticAt 𝕜 g x :=
let ⟨_, hpf⟩ := hf
(hpf.congr hg).analyticAt
theorem analyticAt_congr (h : f =ᶠ[𝓝 x] g) : AnalyticAt 𝕜 f x ↔ AnalyticAt 𝕜 g x :=
⟨fun hf ↦ hf.congr h, fun hg ↦ hg.congr h.symm⟩
theorem AnalyticAt.add (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) : AnalyticAt 𝕜 (f + g) x :=
let ⟨_, hpf⟩ := hf
let ⟨_, hqf⟩ := hg
(hpf.add hqf).analyticAt
#align analytic_at.add AnalyticAt.add
theorem HasFPowerSeriesOnBall.neg (hf : HasFPowerSeriesOnBall f pf x r) :
HasFPowerSeriesOnBall (-f) (-pf) x r :=
{ r_le := by
rw [pf.radius_neg]
exact hf.r_le
r_pos := hf.r_pos
hasSum := fun hy => (hf.hasSum hy).neg }
#align has_fpower_series_on_ball.neg HasFPowerSeriesOnBall.neg
theorem HasFPowerSeriesAt.neg (hf : HasFPowerSeriesAt f pf x) : HasFPowerSeriesAt (-f) (-pf) x :=
let ⟨_, hrf⟩ := hf
hrf.neg.hasFPowerSeriesAt
#align has_fpower_series_at.neg HasFPowerSeriesAt.neg
theorem AnalyticAt.neg (hf : AnalyticAt 𝕜 f x) : AnalyticAt 𝕜 (-f) x :=
let ⟨_, hpf⟩ := hf
hpf.neg.analyticAt
#align analytic_at.neg AnalyticAt.neg
theorem HasFPowerSeriesOnBall.sub (hf : HasFPowerSeriesOnBall f pf x r)
(hg : HasFPowerSeriesOnBall g pg x r) : HasFPowerSeriesOnBall (f - g) (pf - pg) x r := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align has_fpower_series_on_ball.sub HasFPowerSeriesOnBall.sub
theorem HasFPowerSeriesAt.sub (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x) :
HasFPowerSeriesAt (f - g) (pf - pg) x := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align has_fpower_series_at.sub HasFPowerSeriesAt.sub
theorem AnalyticAt.sub (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) :
AnalyticAt 𝕜 (f - g) x := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align analytic_at.sub AnalyticAt.sub
theorem AnalyticOn.mono {s t : Set E} (hf : AnalyticOn 𝕜 f t) (hst : s ⊆ t) : AnalyticOn 𝕜 f s :=
fun z hz => hf z (hst hz)
#align analytic_on.mono AnalyticOn.mono
theorem AnalyticOn.congr' {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : f =ᶠ[𝓝ˢ s] g) :
AnalyticOn 𝕜 g s :=
fun z hz => (hf z hz).congr (mem_nhdsSet_iff_forall.mp hg z hz)
theorem analyticOn_congr' {s : Set E} (h : f =ᶠ[𝓝ˢ s] g) : AnalyticOn 𝕜 f s ↔ AnalyticOn 𝕜 g s :=
⟨fun hf => hf.congr' h, fun hg => hg.congr' h.symm⟩
theorem AnalyticOn.congr {s : Set E} (hs : IsOpen s) (hf : AnalyticOn 𝕜 f s) (hg : s.EqOn f g) :
AnalyticOn 𝕜 g s :=
hf.congr' $ mem_nhdsSet_iff_forall.mpr
(fun _ hz => eventuallyEq_iff_exists_mem.mpr ⟨s, hs.mem_nhds hz, hg⟩)
theorem analyticOn_congr {s : Set E} (hs : IsOpen s) (h : s.EqOn f g) : AnalyticOn 𝕜 f s ↔
AnalyticOn 𝕜 g s := ⟨fun hf => hf.congr hs h, fun hg => hg.congr hs h.symm⟩
theorem AnalyticOn.add {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
AnalyticOn 𝕜 (f + g) s :=
fun z hz => (hf z hz).add (hg z hz)
#align analytic_on.add AnalyticOn.add
theorem AnalyticOn.sub {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
AnalyticOn 𝕜 (f - g) s :=
fun z hz => (hf z hz).sub (hg z hz)
#align analytic_on.sub AnalyticOn.sub
theorem HasFPowerSeriesOnBall.coeff_zero (hf : HasFPowerSeriesOnBall f pf x r) (v : Fin 0 → E) :
pf 0 v = f x := by
have v_eq : v = fun i => 0 := Subsingleton.elim _ _
have zero_mem : (0 : E) ∈ EMetric.ball (0 : E) r := by simp [hf.r_pos]
have : ∀ i, i ≠ 0 → (pf i fun j => 0) = 0 := by
intro i hi
have : 0 < i := pos_iff_ne_zero.2 hi
exact ContinuousMultilinearMap.map_coord_zero _ (⟨0, this⟩ : Fin i) rfl
have A := (hf.hasSum zero_mem).unique (hasSum_single _ this)
simpa [v_eq] using A.symm
#align has_fpower_series_on_ball.coeff_zero HasFPowerSeriesOnBall.coeff_zero
theorem HasFPowerSeriesAt.coeff_zero (hf : HasFPowerSeriesAt f pf x) (v : Fin 0 → E) :
pf 0 v = f x :=
let ⟨_, hrf⟩ := hf
hrf.coeff_zero v
#align has_fpower_series_at.coeff_zero HasFPowerSeriesAt.coeff_zero
/-- If a function `f` has a power series `p` on a ball and `g` is linear, then `g ∘ f` has the
power series `g ∘ p` on the same ball. -/
theorem ContinuousLinearMap.comp_hasFPowerSeriesOnBall (g : F →L[𝕜] G)
(h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (g ∘ f) (g.compFormalMultilinearSeries p) x r :=
{ r_le := h.r_le.trans (p.radius_le_radius_continuousLinearMap_comp _)
r_pos := h.r_pos
hasSum := fun hy => by
simpa only [ContinuousLinearMap.compFormalMultilinearSeries_apply,
ContinuousLinearMap.compContinuousMultilinearMap_coe, Function.comp_apply] using
g.hasSum (h.hasSum hy) }
#align continuous_linear_map.comp_has_fpower_series_on_ball ContinuousLinearMap.comp_hasFPowerSeriesOnBall
/-- If a function `f` is analytic on a set `s` and `g` is linear, then `g ∘ f` is analytic
on `s`. -/
theorem ContinuousLinearMap.comp_analyticOn {s : Set E} (g : F →L[𝕜] G) (h : AnalyticOn 𝕜 f s) :
AnalyticOn 𝕜 (g ∘ f) s := by
rintro x hx
rcases h x hx with ⟨p, r, hp⟩
exact ⟨g.compFormalMultilinearSeries p, r, g.comp_hasFPowerSeriesOnBall hp⟩
#align continuous_linear_map.comp_analytic_on ContinuousLinearMap.comp_analyticOn
/-- If a function admits a power series expansion, then it is exponentially close to the partial
sums of this power series on strict subdisks of the disk of convergence.
This version provides an upper estimate that decreases both in `‖y‖` and `n`. See also
`HasFPowerSeriesOnBall.uniform_geometric_approx` for a weaker version. -/
theorem HasFPowerSeriesOnBall.uniform_geometric_approx' {r' : ℝ≥0}
(hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n := by
obtain ⟨a, ha, C, hC, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n, ‖p n‖ * (r' : ℝ) ^ n ≤ C * a ^ n :=
p.norm_mul_pow_le_mul_pow_of_lt_radius (h.trans_le hf.r_le)
refine' ⟨a, ha, C / (1 - a), div_pos hC (sub_pos.2 ha.2), fun y hy n => _⟩
have yr' : ‖y‖ < r' := by
rw [ball_zero_eq] at hy
exact hy
have hr'0 : 0 < (r' : ℝ) := (norm_nonneg _).trans_lt yr'
have : y ∈ EMetric.ball (0 : E) r := by
refine' mem_emetric_ball_zero_iff.2 (lt_trans _ h)
exact mod_cast yr'
rw [norm_sub_rev, ← mul_div_right_comm]
have ya : a * (‖y‖ / ↑r') ≤ a :=
mul_le_of_le_one_right ha.1.le (div_le_one_of_le yr'.le r'.coe_nonneg)
suffices ‖p.partialSum n y - f (x + y)‖ ≤ C * (a * (‖y‖ / r')) ^ n / (1 - a * (‖y‖ / r')) by
refine' this.trans _
have : 0 < a := ha.1
gcongr
apply_rules [sub_pos.2, ha.2]
apply norm_sub_le_of_geometric_bound_of_hasSum (ya.trans_lt ha.2) _ (hf.hasSum this)
intro n
calc
‖(p n) fun _ : Fin n => y‖
_ ≤ ‖p n‖ * ∏ _i : Fin n, ‖y‖ := ContinuousMultilinearMap.le_op_norm _ _
_ = ‖p n‖ * (r' : ℝ) ^ n * (‖y‖ / r') ^ n := by field_simp [mul_right_comm]
_ ≤ C * a ^ n * (‖y‖ / r') ^ n := by gcongr ?_ * _; apply hp
_ ≤ C * (a * (‖y‖ / r')) ^ n := by rw [mul_pow, mul_assoc]
#align has_fpower_series_on_ball.uniform_geometric_approx' HasFPowerSeriesOnBall.uniform_geometric_approx'
/-- If a function admits a power series expansion, then it is exponentially close to the partial
sums of this power series on strict subdisks of the disk of convergence. -/
theorem HasFPowerSeriesOnBall.uniform_geometric_approx {r' : ℝ≥0}
(hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
∃ a ∈ Ioo (0 : ℝ) 1,
∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n, ‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n := by
obtain ⟨a, ha, C, hC, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n :=
hf.uniform_geometric_approx' h
refine' ⟨a, ha, C, hC, fun y hy n => (hp y hy n).trans _⟩
have yr' : ‖y‖ < r' := by rwa [ball_zero_eq] at hy
gcongr
exacts [mul_nonneg ha.1.le (div_nonneg (norm_nonneg y) r'.coe_nonneg),
mul_le_of_le_one_right ha.1.le (div_le_one_of_le yr'.le r'.coe_nonneg)]
#align has_fpower_series_on_ball.uniform_geometric_approx HasFPowerSeriesOnBall.uniform_geometric_approx
/-- Taylor formula for an analytic function, `IsBigO` version. -/
theorem HasFPowerSeriesAt.isBigO_sub_partialSum_pow (hf : HasFPowerSeriesAt f p x) (n : ℕ) :
(fun y : E => f (x + y) - p.partialSum n y) =O[𝓝 0] fun y => ‖y‖ ^ n := by
rcases hf with ⟨r, hf⟩
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hf.r_pos with ⟨r', r'0, h⟩
obtain ⟨a, -, C, -, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n :=
hf.uniform_geometric_approx' h
refine' isBigO_iff.2 ⟨C * (a / r') ^ n, _⟩
replace r'0 : 0 < (r' : ℝ); · exact mod_cast r'0
filter_upwards [Metric.ball_mem_nhds (0 : E) r'0] with y hy
simpa [mul_pow, mul_div_assoc, mul_assoc, div_mul_eq_mul_div] using hp y hy n
set_option linter.uppercaseLean3 false in
#align has_fpower_series_at.is_O_sub_partial_sum_pow HasFPowerSeriesAt.isBigO_sub_partialSum_pow
/-- If `f` has formal power series `∑ n, pₙ` on a ball of radius `r`, then for `y, z` in any smaller
ball, the norm of the difference `f y - f z - p 1 (fun _ ↦ y - z)` is bounded above by
`C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖`. This lemma formulates this property using `IsBigO` and
`Filter.principal` on `E × E`. -/
theorem HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal
(hf : HasFPowerSeriesOnBall f p x r) (hr : r' < r) :
(fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) =O[𝓟 (EMetric.ball (x, x) r')]
fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖ := by
lift r' to ℝ≥0 using ne_top_of_lt hr
rcases (zero_le r').eq_or_lt with (rfl | hr'0)
· simp only [isBigO_bot, EMetric.ball_zero, principal_empty, ENNReal.coe_zero]
obtain ⟨a, ha, C, hC : 0 < C, hp⟩ :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n : ℕ, ‖p n‖ * (r' : ℝ) ^ n ≤ C * a ^ n
exact p.norm_mul_pow_le_mul_pow_of_lt_radius (hr.trans_le hf.r_le)
simp only [← le_div_iff (pow_pos (NNReal.coe_pos.2 hr'0) _)] at hp
set L : E × E → ℝ := fun y =>
C * (a / r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * (a / (1 - a) ^ 2 + 2 / (1 - a))
have hL : ∀ y ∈ EMetric.ball (x, x) r', ‖f y.1 - f y.2 - p 1 fun _ => y.1 - y.2‖ ≤ L y := by
intro y hy'
have hy : y ∈ EMetric.ball x r ×ˢ EMetric.ball x r := by
rw [EMetric.ball_prod_same]
exact EMetric.ball_subset_ball hr.le hy'
set A : ℕ → F := fun n => (p n fun _ => y.1 - x) - p n fun _ => y.2 - x
have hA : HasSum (fun n => A (n + 2)) (f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) := by
convert (hasSum_nat_add_iff' 2).2 ((hf.hasSum_sub hy.1).sub (hf.hasSum_sub hy.2)) using 1
rw [Finset.sum_range_succ, Finset.sum_range_one, hf.coeff_zero, hf.coeff_zero, sub_self,
zero_add, ← Subsingleton.pi_single_eq (0 : Fin 1) (y.1 - x), Pi.single,
← Subsingleton.pi_single_eq (0 : Fin 1) (y.2 - x), Pi.single, ← (p 1).map_sub, ← Pi.single,
Subsingleton.pi_single_eq, sub_sub_sub_cancel_right]
rw [EMetric.mem_ball, edist_eq_coe_nnnorm_sub, ENNReal.coe_lt_coe] at hy'
set B : ℕ → ℝ := fun n => C * (a / r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * ((n + 2) * a ^ n)
have hAB : ∀ n, ‖A (n + 2)‖ ≤ B n := fun n =>
calc
‖A (n + 2)‖ ≤ ‖p (n + 2)‖ * ↑(n + 2) * ‖y - (x, x)‖ ^ (n + 1) * ‖y.1 - y.2‖ := by
-- porting note: `pi_norm_const` was `pi_norm_const (_ : E)`
simpa only [Fintype.card_fin, pi_norm_const, Prod.norm_def, Pi.sub_def,
Prod.fst_sub, Prod.snd_sub, sub_sub_sub_cancel_right] using
(p <| n + 2).norm_image_sub_le (fun _ => y.1 - x) fun _ => y.2 - x
_ = ‖p (n + 2)‖ * ‖y - (x, x)‖ ^ n * (↑(n + 2) * ‖y - (x, x)‖ * ‖y.1 - y.2‖) := by
rw [pow_succ ‖y - (x, x)‖]
ring
-- porting note: the two `↑` in `↑r'` are new, without them, Lean fails to synthesize
-- instances `HDiv ℝ ℝ≥0 ?m` or `HMul ℝ ℝ≥0 ?m`
_ ≤ C * a ^ (n + 2) / ↑r' ^ (n + 2)
* ↑r' ^ n * (↑(n + 2) * ‖y - (x, x)‖ * ‖y.1 - y.2‖) := by
have : 0 < a := ha.1
gcongr
· apply hp
· apply hy'.le
_ = B n := by
-- porting note: in the original, `B` was in the `field_simp`, but now Lean does not
-- accept it. The current proof works in Lean 4, but does not in Lean 3.
field_simp [pow_succ]
simp only [mul_assoc, mul_comm, mul_left_comm]
have hBL : HasSum B (L y) := by
apply HasSum.mul_left
simp only [add_mul]
have : ‖a‖ < 1 := by simp only [Real.norm_eq_abs, abs_of_pos ha.1, ha.2]
rw [div_eq_mul_inv, div_eq_mul_inv]
exact (hasSum_coe_mul_geometric_of_norm_lt_1 this).add -- porting note: was `convert`!
((hasSum_geometric_of_norm_lt_1 this).mul_left 2)
exact hA.norm_le_of_bounded hBL hAB
suffices L =O[𝓟 (EMetric.ball (x, x) r')] fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖ by
refine' (IsBigO.of_bound 1 (eventually_principal.2 fun y hy => _)).trans this
rw [one_mul]
exact (hL y hy).trans (le_abs_self _)
simp_rw [mul_right_comm _ (_ * _)] -- porting note: there was an `L` inside the `simp_rw`.
exact (isBigO_refl _ _).const_mul_left _
set_option linter.uppercaseLean3 false in
#align has_fpower_series_on_ball.is_O_image_sub_image_sub_deriv_principal HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal
/-- If `f` has formal power series `∑ n, pₙ` on a ball of radius `r`, then for `y, z` in any smaller
ball, the norm of the difference `f y - f z - p 1 (fun _ ↦ y - z)` is bounded above by
`C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖`. -/
theorem HasFPowerSeriesOnBall.image_sub_sub_deriv_le (hf : HasFPowerSeriesOnBall f p x r)
(hr : r' < r) :
∃ C, ∀ᵉ (y ∈ EMetric.ball x r') (z ∈ EMetric.ball x r'),
‖f y - f z - p 1 fun _ => y - z‖ ≤ C * max ‖y - x‖ ‖z - x‖ * ‖y - z‖ := by
simpa only [isBigO_principal, mul_assoc, norm_mul, norm_norm, Prod.forall, EMetric.mem_ball,
Prod.edist_eq, max_lt_iff, and_imp, @forall_swap (_ < _) E] using
hf.isBigO_image_sub_image_sub_deriv_principal hr
#align has_fpower_series_on_ball.image_sub_sub_deriv_le HasFPowerSeriesOnBall.image_sub_sub_deriv_le
/-- If `f` has formal power series `∑ n, pₙ` at `x`, then
`f y - f z - p 1 (fun _ ↦ y - z) = O(‖(y, z) - (x, x)‖ * ‖y - z‖)` as `(y, z) → (x, x)`.
In particular, `f` is strictly differentiable at `x`. -/
theorem HasFPowerSeriesAt.isBigO_image_sub_norm_mul_norm_sub (hf : HasFPowerSeriesAt f p x) :
(fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) =O[𝓝 (x, x)] fun y =>
‖y - (x, x)‖ * ‖y.1 - y.2‖ := by
rcases hf with ⟨r, hf⟩
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hf.r_pos with ⟨r', r'0, h⟩
refine' (hf.isBigO_image_sub_image_sub_deriv_principal h).mono _
exact le_principal_iff.2 (EMetric.ball_mem_nhds _ r'0)
set_option linter.uppercaseLean3 false in
#align has_fpower_series_at.is_O_image_sub_norm_mul_norm_sub HasFPowerSeriesAt.isBigO_image_sub_norm_mul_norm_sub
/-- If a function admits a power series expansion at `x`, then it is the uniform limit of the
partial sums of this power series on strict subdisks of the disk of convergence, i.e., `f (x + y)`
is the uniform limit of `p.partialSum n y` there. -/
theorem HasFPowerSeriesOnBall.tendstoUniformlyOn {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r)
(h : (r' : ℝ≥0∞) < r) :
TendstoUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
(Metric.ball (0 : E) r') := by
obtain ⟨a, ha, C, -, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n
exact hf.uniform_geometric_approx h
refine' Metric.tendstoUniformlyOn_iff.2 fun ε εpos => _
have L : Tendsto (fun n => (C : ℝ) * a ^ n) atTop (𝓝 ((C : ℝ) * 0)) :=
tendsto_const_nhds.mul (tendsto_pow_atTop_nhds_0_of_lt_1 ha.1.le ha.2)
rw [mul_zero] at L
refine' (L.eventually (gt_mem_nhds εpos)).mono fun n hn y hy => _
rw [dist_eq_norm]
exact (hp y hy n).trans_lt hn
#align has_fpower_series_on_ball.tendsto_uniformly_on HasFPowerSeriesOnBall.tendstoUniformlyOn
/-- If a function admits a power series expansion at `x`, then it is the locally uniform limit of
the partial sums of this power series on the disk of convergence, i.e., `f (x + y)`
is the locally uniform limit of `p.partialSum n y` there. -/
theorem HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn (hf : HasFPowerSeriesOnBall f p x r) :
TendstoLocallyUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
(EMetric.ball (0 : E) r) := by
intro u hu x hx
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hx with ⟨r', xr', hr'⟩
have : EMetric.ball (0 : E) r' ∈ 𝓝 x := IsOpen.mem_nhds EMetric.isOpen_ball xr'
refine' ⟨EMetric.ball (0 : E) r', mem_nhdsWithin_of_mem_nhds this, _⟩
simpa [Metric.emetric_ball_nnreal] using hf.tendstoUniformlyOn hr' u hu
#align has_fpower_series_on_ball.tendsto_locally_uniformly_on HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn
/-- If a function admits a power series expansion at `x`, then it is the uniform limit of the
partial sums of this power series on strict subdisks of the disk of convergence, i.e., `f y`
is the uniform limit of `p.partialSum n (y - x)` there. -/
theorem HasFPowerSeriesOnBall.tendstoUniformlyOn' {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r)
(h : (r' : ℝ≥0∞) < r) :
TendstoUniformlyOn (fun n y => p.partialSum n (y - x)) f atTop (Metric.ball (x : E) r') := by
convert (hf.tendstoUniformlyOn h).comp fun y => y - x using 1
· simp [(· ∘ ·)]
· ext z
simp [dist_eq_norm]
#align has_fpower_series_on_ball.tendsto_uniformly_on' HasFPowerSeriesOnBall.tendstoUniformlyOn'
/-- If a function admits a power series expansion at `x`, then it is the locally uniform limit of
the partial sums of this power series on the disk of convergence, i.e., `f y`
is the locally uniform limit of `p.partialSum n (y - x)` there. -/
theorem HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn' (hf : HasFPowerSeriesOnBall f p x r) :
TendstoLocallyUniformlyOn (fun n y => p.partialSum n (y - x)) f atTop
(EMetric.ball (x : E) r) := by
have A : ContinuousOn (fun y : E => y - x) (EMetric.ball (x : E) r) :=
(continuous_id.sub continuous_const).continuousOn
convert hf.tendstoLocallyUniformlyOn.comp (fun y : E => y - x) _ A using 1
· ext z
simp
· intro z
simp [edist_eq_coe_nnnorm, edist_eq_coe_nnnorm_sub]
#align has_fpower_series_on_ball.tendsto_locally_uniformly_on' HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn'
/-- If a function admits a power series expansion on a disk, then it is continuous there. -/
protected theorem HasFPowerSeriesOnBall.continuousOn (hf : HasFPowerSeriesOnBall f p x r) :
ContinuousOn f (EMetric.ball x r) :=
hf.tendstoLocallyUniformlyOn'.continuousOn <|
eventually_of_forall fun n =>
((p.partialSum_continuous n).comp (continuous_id.sub continuous_const)).continuousOn
#align has_fpower_series_on_ball.continuous_on HasFPowerSeriesOnBall.continuousOn
protected theorem HasFPowerSeriesAt.continuousAt (hf : HasFPowerSeriesAt f p x) :
ContinuousAt f x :=
let ⟨_, hr⟩ := hf
hr.continuousOn.continuousAt (EMetric.ball_mem_nhds x hr.r_pos)
#align has_fpower_series_at.continuous_at HasFPowerSeriesAt.continuousAt
protected theorem AnalyticAt.continuousAt (hf : AnalyticAt 𝕜 f x) : ContinuousAt f x :=
let ⟨_, hp⟩ := hf
hp.continuousAt
#align analytic_at.continuous_at AnalyticAt.continuousAt
protected theorem AnalyticOn.continuousOn {s : Set E} (hf : AnalyticOn 𝕜 f s) : ContinuousOn f s :=
fun x hx => (hf x hx).continuousAt.continuousWithinAt
#align analytic_on.continuous_on AnalyticOn.continuousOn
/-- Analytic everywhere implies continuous -/
theorem AnalyticOn.continuous {f : E → F} (fa : AnalyticOn 𝕜 f univ) : Continuous f := by
rw [continuous_iff_continuousOn_univ]; exact fa.continuousOn
/-- In a complete space, the sum of a converging power series `p` admits `p` as a power series.
This is not totally obvious as we need to check the convergence of the series. -/
protected theorem FormalMultilinearSeries.hasFPowerSeriesOnBall [CompleteSpace F]
(p : FormalMultilinearSeries 𝕜 E F) (h : 0 < p.radius) :
HasFPowerSeriesOnBall p.sum p 0 p.radius :=
{ r_le := le_rfl
r_pos := h
hasSum := fun hy => by
rw [zero_add]
exact p.hasSum hy }
#align formal_multilinear_series.has_fpower_series_on_ball FormalMultilinearSeries.hasFPowerSeriesOnBall
theorem HasFPowerSeriesOnBall.sum (h : HasFPowerSeriesOnBall f p x r) {y : E}
(hy : y ∈ EMetric.ball (0 : E) r) : f (x + y) = p.sum y :=
(h.hasSum hy).tsum_eq.symm
#align has_fpower_series_on_ball.sum HasFPowerSeriesOnBall.sum
/-- The sum of a converging power series is continuous in its disk of convergence. -/
protected theorem FormalMultilinearSeries.continuousOn [CompleteSpace F] :
ContinuousOn p.sum (EMetric.ball 0 p.radius) := by
rcases (zero_le p.radius).eq_or_lt with h | h
· simp [← h, continuousOn_empty]
· exact (p.hasFPowerSeriesOnBall h).continuousOn
#align formal_multilinear_series.continuous_on FormalMultilinearSeries.continuousOn
end
/-!
### Uniqueness of power series
If a function `f : E → F` has two representations as power series at a point `x : E`, corresponding
to formal multilinear series `p₁` and `p₂`, then these representations agree term-by-term. That is,
for any `n : ℕ` and `y : E`, `p₁ n (fun i ↦ y) = p₂ n (fun i ↦ y)`. In the one-dimensional case,
when `f : 𝕜 → E`, the continuous multilinear maps `p₁ n` and `p₂ n` are given by
`ContinuousMultilinearMap.mkPiField`, and hence are determined completely by the value of
`p₁ n (fun i ↦ 1)`, so `p₁ = p₂`. Consequently, the radius of convergence for one series can be
transferred to the other.
-/
section Uniqueness
open ContinuousMultilinearMap
theorem Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero {n : ℕ} {p : E[×n]→L[𝕜] F}
(h : (fun y => p fun _ => y) =O[𝓝 0] fun y => ‖y‖ ^ (n + 1)) (y : E) : (p fun _ => y) = 0 := by
obtain ⟨c, c_pos, hc⟩ := h.exists_pos
obtain ⟨t, ht, t_open, z_mem⟩ := eventually_nhds_iff.mp (isBigOWith_iff.mp hc)
obtain ⟨δ, δ_pos, δε⟩ := (Metric.isOpen_iff.mp t_open) 0 z_mem
clear h hc z_mem
cases' n with n
· exact norm_eq_zero.mp (by
-- porting note: the symmetric difference of the `simpa only` sets:
-- added `Nat.zero_eq, zero_add, pow_one`
-- removed `zero_pow', Ne.def, Nat.one_ne_zero, not_false_iff`
simpa only [Nat.zero_eq, fin0_apply_norm, norm_eq_zero, norm_zero, zero_add, pow_one,
mul_zero, norm_le_zero_iff] using ht 0 (δε (Metric.mem_ball_self δ_pos)))
· refine' Or.elim (Classical.em (y = 0))
(fun hy => by simpa only [hy] using p.map_zero) fun hy => _
replace hy := norm_pos_iff.mpr hy
refine' norm_eq_zero.mp (le_antisymm (le_of_forall_pos_le_add fun ε ε_pos => _) (norm_nonneg _))
have h₀ := _root_.mul_pos c_pos (pow_pos hy (n.succ + 1))
obtain ⟨k, k_pos, k_norm⟩ := NormedField.exists_norm_lt 𝕜
(lt_min (mul_pos δ_pos (inv_pos.mpr hy)) (mul_pos ε_pos (inv_pos.mpr h₀)))
have h₁ : ‖k • y‖ < δ := by
rw [norm_smul]
exact inv_mul_cancel_right₀ hy.ne.symm δ ▸
mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_left _ _)) hy
have h₂ :=
calc
‖p fun _ => k • y‖ ≤ c * ‖k • y‖ ^ (n.succ + 1) := by
-- porting note: now Lean wants `_root_.`
simpa only [norm_pow, _root_.norm_norm] using ht (k • y) (δε (mem_ball_zero_iff.mpr h₁))
--simpa only [norm_pow, norm_norm] using ht (k • y) (δε (mem_ball_zero_iff.mpr h₁))
_ = ‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))) := by
-- porting note: added `Nat.succ_eq_add_one` since otherwise `ring` does not conclude.
simp only [norm_smul, mul_pow, Nat.succ_eq_add_one]
-- porting note: removed `rw [pow_succ]`, since it now becomes superfluous.
ring
have h₃ : ‖k‖ * (c * ‖y‖ ^ (n.succ + 1)) < ε :=
inv_mul_cancel_right₀ h₀.ne.symm ε ▸
mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_right _ _)) h₀
calc
‖p fun _ => y‖ = ‖k⁻¹ ^ n.succ‖ * ‖p fun _ => k • y‖ := by
simpa only [inv_smul_smul₀ (norm_pos_iff.mp k_pos), norm_smul, Finset.prod_const,
Finset.card_fin] using
congr_arg norm (p.map_smul_univ (fun _ : Fin n.succ => k⁻¹) fun _ : Fin n.succ => k • y)
_ ≤ ‖k⁻¹ ^ n.succ‖ * (‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1)))) := by gcongr
_ = ‖(k⁻¹ * k) ^ n.succ‖ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))) := by
rw [← mul_assoc]
simp [norm_mul, mul_pow]
_ ≤ 0 + ε := by
rw [inv_mul_cancel (norm_pos_iff.mp k_pos)]
simpa using h₃.le
set_option linter.uppercaseLean3 false in
#align asymptotics.is_O.continuous_multilinear_map_apply_eq_zero Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero
/-- If a formal multilinear series `p` represents the zero function at `x : E`, then the
terms `p n (fun i ↦ y)` appearing in the sum are zero for any `n : ℕ`, `y : E`. -/
theorem HasFPowerSeriesAt.apply_eq_zero {p : FormalMultilinearSeries 𝕜 E F} {x : E}
(h : HasFPowerSeriesAt 0 p x) (n : ℕ) : ∀ y : E, (p n fun _ => y) = 0 := by
refine' Nat.strong_induction_on n fun k hk => _
have psum_eq : p.partialSum (k + 1) = fun y => p k fun _ => y := by
funext z
refine' Finset.sum_eq_single _ (fun b hb hnb => _) fun hn => _
· have := Finset.mem_range_succ_iff.mp hb
simp only [hk b (this.lt_of_ne hnb), Pi.zero_apply]
· exact False.elim (hn (Finset.mem_range.mpr (lt_add_one k)))
replace h := h.isBigO_sub_partialSum_pow k.succ
simp only [psum_eq, zero_sub, Pi.zero_apply, Asymptotics.isBigO_neg_left] at h
exact h.continuousMultilinearMap_apply_eq_zero
#align has_fpower_series_at.apply_eq_zero HasFPowerSeriesAt.apply_eq_zero
/-- A one-dimensional formal multilinear series representing the zero function is zero. -/
theorem HasFPowerSeriesAt.eq_zero {p : FormalMultilinearSeries 𝕜 𝕜 E} {x : 𝕜}
(h : HasFPowerSeriesAt 0 p x) : p = 0 := by
-- porting note: `funext; ext` was `ext (n x)`
funext n
ext x
rw [← mkPiField_apply_one_eq_self (p n)]
-- porting note: nasty hack, was `simp [h.apply_eq_zero n 1]`
have := Or.intro_right ?_ (h.apply_eq_zero n 1)
simpa using this
#align has_fpower_series_at.eq_zero HasFPowerSeriesAt.eq_zero
/-- One-dimensional formal multilinear series representing the same function are equal. -/
theorem HasFPowerSeriesAt.eq_formalMultilinearSeries {p₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E}
{f : 𝕜 → E} {x : 𝕜} (h₁ : HasFPowerSeriesAt f p₁ x) (h₂ : HasFPowerSeriesAt f p₂ x) : p₁ = p₂ :=
sub_eq_zero.mp (HasFPowerSeriesAt.eq_zero (by simpa only [sub_self] using h₁.sub h₂))
#align has_fpower_series_at.eq_formal_multilinear_series HasFPowerSeriesAt.eq_formalMultilinearSeries
theorem HasFPowerSeriesAt.eq_formalMultilinearSeries_of_eventually
{p q : FormalMultilinearSeries 𝕜 𝕜 E} {f g : 𝕜 → E} {x : 𝕜} (hp : HasFPowerSeriesAt f p x)
(hq : HasFPowerSeriesAt g q x) (heq : ∀ᶠ z in 𝓝 x, f z = g z) : p = q :=
(hp.congr heq).eq_formalMultilinearSeries hq
#align has_fpower_series_at.eq_formal_multilinear_series_of_eventually HasFPowerSeriesAt.eq_formalMultilinearSeries_of_eventually
/-- A one-dimensional formal multilinear series representing a locally zero function is zero. -/
theorem HasFPowerSeriesAt.eq_zero_of_eventually {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E}
{x : 𝕜} (hp : HasFPowerSeriesAt f p x) (hf : f =ᶠ[𝓝 x] 0) : p = 0 :=
(hp.congr hf).eq_zero
#align has_fpower_series_at.eq_zero_of_eventually HasFPowerSeriesAt.eq_zero_of_eventually
/-- If a function `f : 𝕜 → E` has two power series representations at `x`, then the given radii in
which convergence is guaranteed may be interchanged. This can be useful when the formal multilinear
series in one representation has a particularly nice form, but the other has a larger radius. -/
theorem HasFPowerSeriesOnBall.exchange_radius {p₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E}
{r₁ r₂ : ℝ≥0∞} {x : 𝕜} (h₁ : HasFPowerSeriesOnBall f p₁ x r₁)
(h₂ : HasFPowerSeriesOnBall f p₂ x r₂) : HasFPowerSeriesOnBall f p₁ x r₂ :=
h₂.hasFPowerSeriesAt.eq_formalMultilinearSeries h₁.hasFPowerSeriesAt ▸ h₂
#align has_fpower_series_on_ball.exchange_radius HasFPowerSeriesOnBall.exchange_radius
/-- If a function `f : 𝕜 → E` has power series representation `p` on a ball of some radius and for
each positive radius it has some power series representation, then `p` converges to `f` on the whole
`𝕜`. -/
theorem HasFPowerSeriesOnBall.r_eq_top_of_exists {f : 𝕜 → E} {r : ℝ≥0∞} {x : 𝕜}
{p : FormalMultilinearSeries 𝕜 𝕜 E} (h : HasFPowerSeriesOnBall f p x r)
(h' : ∀ (r' : ℝ≥0) (_ : 0 < r'), ∃ p' : FormalMultilinearSeries 𝕜 𝕜 E,
HasFPowerSeriesOnBall f p' x r') :
HasFPowerSeriesOnBall f p x ∞ :=
{ r_le := ENNReal.le_of_forall_pos_nnreal_lt fun r hr _ =>
let ⟨_, hp'⟩ := h' r hr
(h.exchange_radius hp').r_le
r_pos := ENNReal.coe_lt_top
hasSum := fun {y} _ =>
let ⟨r', hr'⟩ := exists_gt ‖y‖₊
let ⟨_, hp'⟩ := h' r' hr'.ne_bot.bot_lt
(h.exchange_radius hp').hasSum <| mem_emetric_ball_zero_iff.mpr (ENNReal.coe_lt_coe.2 hr') }
#align has_fpower_series_on_ball.r_eq_top_of_exists HasFPowerSeriesOnBall.r_eq_top_of_exists
end Uniqueness
/-!
### Changing origin in a power series
If a function is analytic in a disk `D(x, R)`, then it is analytic in any disk contained in that
one. Indeed, one can write
$$
f (x + y + z) = \sum_{n} p_n (y + z)^n = \sum_{n, k} \binom{n}{k} p_n y^{n-k} z^k
= \sum_{k} \Bigl(\sum_{n} \binom{n}{k} p_n y^{n-k}\Bigr) z^k.
$$
The corresponding power series has thus a `k`-th coefficient equal to
$\sum_{n} \binom{n}{k} p_n y^{n-k}$. In the general case where `pₙ` is a multilinear map, this has
to be interpreted suitably: instead of having a binomial coefficient, one should sum over all
possible subsets `s` of `Fin n` of cardinal `k`, and attribute `z` to the indices in `s` and
`y` to the indices outside of `s`.
In this paragraph, we implement this. The new power series is called `p.changeOrigin y`. Then, we
check its convergence and the fact that its sum coincides with the original sum. The outcome of this
discussion is that the set of points where a function is analytic is open.
-/
namespace FormalMultilinearSeries
section
variable (p : FormalMultilinearSeries 𝕜 E F) {x y : E} {r R : ℝ≥0}
/-- A term of `FormalMultilinearSeries.changeOriginSeries`.
Given a formal multilinear series `p` and a point `x` in its ball of convergence,
`p.changeOrigin x` is a formal multilinear series such that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense. Each term of `p.changeOrigin x`
is itself an analytic function of `x` given by the series `p.changeOriginSeries`. Each term in
`changeOriginSeries` is the sum of `changeOriginSeriesTerm`'s over all `s` of cardinality `l`.
The definition is such that `p.changeOriginSeriesTerm k l s hs (fun _ ↦ x) (fun _ ↦ y) =
p (k + l) (s.piecewise (fun _ ↦ x) (fun _ ↦ y))`
-/
def changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
E[×l]→L[𝕜] E[×k]→L[𝕜] F := by
let a := ContinuousMultilinearMap.curryFinFinset 𝕜 E F hs
(by erw [Finset.card_compl, Fintype.card_fin, hs, add_tsub_cancel_right])
exact a (p (k + l))
#align formal_multilinear_series.change_origin_series_term FormalMultilinearSeries.changeOriginSeriesTerm
theorem changeOriginSeriesTerm_apply (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l)
(x y : E) :
(p.changeOriginSeriesTerm k l s hs (fun _ => x) fun _ => y) =
p (k + l) (s.piecewise (fun _ => x) fun _ => y) :=
ContinuousMultilinearMap.curryFinFinset_apply_const _ _ _ _ _
#align formal_multilinear_series.change_origin_series_term_apply FormalMultilinearSeries.changeOriginSeriesTerm_apply
@[simp]
theorem norm_changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
‖p.changeOriginSeriesTerm k l s hs‖ = ‖p (k + l)‖ := by
simp only [changeOriginSeriesTerm, LinearIsometryEquiv.norm_map]
#align formal_multilinear_series.norm_change_origin_series_term FormalMultilinearSeries.norm_changeOriginSeriesTerm
@[simp]
theorem nnnorm_changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
‖p.changeOriginSeriesTerm k l s hs‖₊ = ‖p (k + l)‖₊ := by
simp only [changeOriginSeriesTerm, LinearIsometryEquiv.nnnorm_map]
#align formal_multilinear_series.nnnorm_change_origin_series_term FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm
theorem nnnorm_changeOriginSeriesTerm_apply_le (k l : ℕ) (s : Finset (Fin (k + l)))
(hs : s.card = l) (x y : E) :
‖p.changeOriginSeriesTerm k l s hs (fun _ => x) fun _ => y‖₊ ≤
‖p (k + l)‖₊ * ‖x‖₊ ^ l * ‖y‖₊ ^ k := by
rw [← p.nnnorm_changeOriginSeriesTerm k l s hs, ← Fin.prod_const, ← Fin.prod_const]
apply ContinuousMultilinearMap.le_of_op_nnnorm_le
apply ContinuousMultilinearMap.le_op_nnnorm
#align formal_multilinear_series.nnnorm_change_origin_series_term_apply_le FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm_apply_le
/-- The power series for `f.changeOrigin k`.
Given a formal multilinear series `p` and a point `x` in its ball of convergence,
`p.changeOrigin x` is a formal multilinear series such that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense. Its `k`-th term is the sum of
the series `p.changeOriginSeries k`. -/
def changeOriginSeries (k : ℕ) : FormalMultilinearSeries 𝕜 E (E[×k]→L[𝕜] F) := fun l =>
∑ s : { s : Finset (Fin (k + l)) // Finset.card s = l }, p.changeOriginSeriesTerm k l s s.2
#align formal_multilinear_series.change_origin_series FormalMultilinearSeries.changeOriginSeries
theorem nnnorm_changeOriginSeries_le_tsum (k l : ℕ) :
‖p.changeOriginSeries k l‖₊ ≤
∑' _ : { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + l)‖₊ :=
(nnnorm_sum_le _ (fun t => changeOriginSeriesTerm p k l (Subtype.val t) t.prop)).trans_eq <| by
simp_rw [tsum_fintype, nnnorm_changeOriginSeriesTerm (p := p) (k := k) (l := l)]
#align formal_multilinear_series.nnnorm_change_origin_series_le_tsum FormalMultilinearSeries.nnnorm_changeOriginSeries_le_tsum
theorem nnnorm_changeOriginSeries_apply_le_tsum (k l : ℕ) (x : E) :
‖p.changeOriginSeries k l fun _ => x‖₊ ≤
∑' _ : { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + l)‖₊ * ‖x‖₊ ^ l := by
rw [NNReal.tsum_mul_right, ← Fin.prod_const]
exact (p.changeOriginSeries k l).le_of_op_nnnorm_le _ (p.nnnorm_changeOriginSeries_le_tsum _ _)
#align formal_multilinear_series.nnnorm_change_origin_series_apply_le_tsum FormalMultilinearSeries.nnnorm_changeOriginSeries_apply_le_tsum
/-- Changing the origin of a formal multilinear series `p`, so that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense.
-/
def changeOrigin (x : E) : FormalMultilinearSeries 𝕜 E F :=
fun k => (p.changeOriginSeries k).sum x
#align formal_multilinear_series.change_origin FormalMultilinearSeries.changeOrigin
/-- An auxiliary equivalence useful in the proofs about
`FormalMultilinearSeries.changeOriginSeries`: the set of triples `(k, l, s)`, where `s` is a
`Finset (Fin (k + l))` of cardinality `l` is equivalent to the set of pairs `(n, s)`, where `s` is a
`Finset (Fin n)`.
The forward map sends `(k, l, s)` to `(k + l, s)` and the inverse map sends `(n, s)` to
`(n - Finset.card s, Finset.card s, s)`. The actual definition is less readable because of problems
with non-definitional equalities. -/
@[simps]
def changeOriginIndexEquiv :
(Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) ≃ Σn : ℕ, Finset (Fin n) where
toFun s := ⟨s.1 + s.2.1, s.2.2⟩
invFun s :=
⟨s.1 - s.2.card, s.2.card,
⟨s.2.map
(Fin.castIso <| (tsub_add_cancel_of_le <| card_finset_fin_le s.2).symm).toEquiv.toEmbedding,
Finset.card_map _⟩⟩
left_inv := by
rintro ⟨k, l, ⟨s : Finset (Fin <| k + l), hs : s.card = l⟩⟩
dsimp only [Subtype.coe_mk]
-- Lean can't automatically generalize `k' = k + l - s.card`, `l' = s.card`, so we explicitly
-- formulate the generalized goal
suffices ∀ k' l', k' = k → l' = l → ∀ (hkl : k + l = k' + l') (hs'),
(⟨k', l', ⟨Finset.map (Fin.castIso hkl).toEquiv.toEmbedding s, hs'⟩⟩ :
Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) = ⟨k, l, ⟨s, hs⟩⟩ by
apply this <;> simp only [hs, add_tsub_cancel_right]
rintro _ _ rfl rfl hkl hs'
simp only [Equiv.refl_toEmbedding, Fin.castIso_refl, Finset.map_refl, eq_self_iff_true,
OrderIso.refl_toEquiv, and_self_iff, heq_iff_eq]
right_inv := by
rintro ⟨n, s⟩
simp [tsub_add_cancel_of_le (card_finset_fin_le s), Fin.castIso_to_equiv]
#align formal_multilinear_series.change_origin_index_equiv FormalMultilinearSeries.changeOriginIndexEquiv
theorem changeOriginSeries_summable_aux₁ {r r' : ℝ≥0} (hr : (r + r' : ℝ≥0∞) < p.radius) :
Summable fun s : Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l } =>
‖p (s.1 + s.2.1)‖₊ * r ^ s.2.1 * r' ^ s.1 := by
rw [← changeOriginIndexEquiv.symm.summable_iff]
dsimp only [Function.comp_def, changeOriginIndexEquiv_symm_apply_fst,
changeOriginIndexEquiv_symm_apply_snd_fst]
have : ∀ n : ℕ,
HasSum (fun s : Finset (Fin n) => ‖p (n - s.card + s.card)‖₊ * r ^ s.card * r' ^ (n - s.card))
(‖p n‖₊ * (r + r') ^ n) := by
intro n
-- TODO: why `simp only [tsub_add_cancel_of_le (card_finset_fin_le _)]` fails?
convert_to HasSum (fun s : Finset (Fin n) => ‖p n‖₊ * (r ^ s.card * r' ^ (n - s.card))) _
· ext1 s
rw [tsub_add_cancel_of_le (card_finset_fin_le _), mul_assoc]
rw [← Fin.sum_pow_mul_eq_add_pow]
exact (hasSum_fintype _).mul_left _
refine' NNReal.summable_sigma.2 ⟨fun n => (this n).summable, _⟩
simp only [(this _).tsum_eq]
exact p.summable_nnnorm_mul_pow hr
#align formal_multilinear_series.change_origin_series_summable_aux₁ FormalMultilinearSeries.changeOriginSeries_summable_aux₁
theorem changeOriginSeries_summable_aux₂ (hr : (r : ℝ≥0∞) < p.radius) (k : ℕ) :
Summable fun s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l } =>
‖p (k + s.1)‖₊ * r ^ s.1 := by
rcases ENNReal.lt_iff_exists_add_pos_lt.1 hr with ⟨r', h0, hr'⟩
simpa only [mul_inv_cancel_right₀ (pow_pos h0 _).ne'] using
((NNReal.summable_sigma.1 (p.changeOriginSeries_summable_aux₁ hr')).1 k).mul_right (r' ^ k)⁻¹
#align formal_multilinear_series.change_origin_series_summable_aux₂ FormalMultilinearSeries.changeOriginSeries_summable_aux₂
theorem changeOriginSeries_summable_aux₃ {r : ℝ≥0} (hr : ↑r < p.radius) (k : ℕ) :
Summable fun l : ℕ => ‖p.changeOriginSeries k l‖₊ * r ^ l := by
refine' NNReal.summable_of_le
(fun n => _) (NNReal.summable_sigma.1 <| p.changeOriginSeries_summable_aux₂ hr k).2
simp only [NNReal.tsum_mul_right]
exact mul_le_mul' (p.nnnorm_changeOriginSeries_le_tsum _ _) le_rfl
#align formal_multilinear_series.change_origin_series_summable_aux₃ FormalMultilinearSeries.changeOriginSeries_summable_aux₃
theorem le_changeOriginSeries_radius (k : ℕ) : p.radius ≤ (p.changeOriginSeries k).radius :=
ENNReal.le_of_forall_nnreal_lt fun _r hr =>
le_radius_of_summable_nnnorm _ (p.changeOriginSeries_summable_aux₃ hr k)
#align formal_multilinear_series.le_change_origin_series_radius FormalMultilinearSeries.le_changeOriginSeries_radius
theorem nnnorm_changeOrigin_le (k : ℕ) (h : (‖x‖₊ : ℝ≥0∞) < p.radius) :
‖p.changeOrigin x k‖₊ ≤
∑' s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + s.1)‖₊ * ‖x‖₊ ^ s.1 := by
refine' tsum_of_nnnorm_bounded _ fun l => p.nnnorm_changeOriginSeries_apply_le_tsum k l x
have := p.changeOriginSeries_summable_aux₂ h k
refine' HasSum.sigma this.hasSum fun l => _
exact ((NNReal.summable_sigma.1 this).1 l).hasSum
#align formal_multilinear_series.nnnorm_change_origin_le FormalMultilinearSeries.nnnorm_changeOrigin_le
/-- The radius of convergence of `p.changeOrigin x` is at least `p.radius - ‖x‖`. In other words,
`p.changeOrigin x` is well defined on the largest ball contained in the original ball of
convergence. -/
theorem changeOrigin_radius : p.radius - ‖x‖₊ ≤ (p.changeOrigin x).radius := by
refine' ENNReal.le_of_forall_pos_nnreal_lt fun r _h0 hr => _
rw [lt_tsub_iff_right, add_comm] at hr
have hr' : (‖x‖₊ : ℝ≥0∞) < p.radius := (le_add_right le_rfl).trans_lt hr
apply le_radius_of_summable_nnnorm
have : ∀ k : ℕ,
‖p.changeOrigin x k‖₊ * r ^ k ≤
(∑' s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + s.1)‖₊ * ‖x‖₊ ^ s.1) *
r ^ k :=
fun k => mul_le_mul_right' (p.nnnorm_changeOrigin_le k hr') (r ^ k)
refine' NNReal.summable_of_le this _
simpa only [← NNReal.tsum_mul_right] using
(NNReal.summable_sigma.1 (p.changeOriginSeries_summable_aux₁ hr)).2
#align formal_multilinear_series.change_origin_radius FormalMultilinearSeries.changeOrigin_radius
end
-- From this point on, assume that the space is complete, to make sure that series that converge
-- in norm also converge in `F`.
variable [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x y : E} {r R : ℝ≥0}
theorem hasFPowerSeriesOnBall_changeOrigin (k : ℕ) (hr : 0 < p.radius) :
HasFPowerSeriesOnBall (fun x => p.changeOrigin x k) (p.changeOriginSeries k) 0 p.radius :=
have := p.le_changeOriginSeries_radius k
((p.changeOriginSeries k).hasFPowerSeriesOnBall (hr.trans_le this)).mono hr this
#align formal_multilinear_series.has_fpower_series_on_ball_change_origin FormalMultilinearSeries.hasFPowerSeriesOnBall_changeOrigin
/-- Summing the series `p.changeOrigin x` at a point `y` gives back `p (x + y)`. -/
theorem changeOrigin_eval (h : (‖x‖₊ + ‖y‖₊ : ℝ≥0∞) < p.radius) :
(p.changeOrigin x).sum y = p.sum (x + y) := by
have radius_pos : 0 < p.radius := lt_of_le_of_lt (zero_le _) h
have x_mem_ball : x ∈ EMetric.ball (0 : E) p.radius :=
mem_emetric_ball_zero_iff.2 ((le_add_right le_rfl).trans_lt h)
have y_mem_ball : y ∈ EMetric.ball (0 : E) (p.changeOrigin x).radius := by
refine' mem_emetric_ball_zero_iff.2 (lt_of_lt_of_le _ p.changeOrigin_radius)
rwa [lt_tsub_iff_right, add_comm]
have x_add_y_mem_ball : x + y ∈ EMetric.ball (0 : E) p.radius := by
refine' mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt _ h)
exact mod_cast nnnorm_add_le x y
set f : (Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) → F := fun s =>
p.changeOriginSeriesTerm s.1 s.2.1 s.2.2 s.2.2.2 (fun _ => x) fun _ => y
have hsf : Summable f := by
refine' .of_nnnorm_bounded _ (p.changeOriginSeries_summable_aux₁ h) _
rintro ⟨k, l, s, hs⟩
dsimp only [Subtype.coe_mk]
exact p.nnnorm_changeOriginSeriesTerm_apply_le _ _ _ _ _ _
have hf : HasSum f ((p.changeOrigin x).sum y) := by
refine' HasSum.sigma_of_hasSum ((p.changeOrigin x).summable y_mem_ball).hasSum (fun k => _) hsf
· dsimp only
refine' ContinuousMultilinearMap.hasSum_eval _ _
have := (p.hasFPowerSeriesOnBall_changeOrigin k radius_pos).hasSum x_mem_ball
rw [zero_add] at this
refine' HasSum.sigma_of_hasSum this (fun l => _) _
· simp only [changeOriginSeries, ContinuousMultilinearMap.sum_apply]
apply hasSum_fintype
· refine' .of_nnnorm_bounded _
(p.changeOriginSeries_summable_aux₂ (mem_emetric_ball_zero_iff.1 x_mem_ball) k) fun s => _
refine' (ContinuousMultilinearMap.le_op_nnnorm _ _).trans_eq _
simp
refine' hf.unique (changeOriginIndexEquiv.symm.hasSum_iff.1 _)
refine' HasSum.sigma_of_hasSum
(p.hasSum x_add_y_mem_ball) (fun n => _) (changeOriginIndexEquiv.symm.summable_iff.2 hsf)
erw [(p n).map_add_univ (fun _ => x) fun _ => y]
-- porting note: added explicit function
convert hasSum_fintype (fun c : Finset (Fin n) => f (changeOriginIndexEquiv.symm ⟨n, c⟩))
rename_i s _
dsimp only [changeOriginSeriesTerm, (· ∘ ·), changeOriginIndexEquiv_symm_apply_fst,
changeOriginIndexEquiv_symm_apply_snd_fst, changeOriginIndexEquiv_symm_apply_snd_snd_coe]
rw [ContinuousMultilinearMap.curryFinFinset_apply_const]
have : ∀ (m) (hm : n = m), p n (s.piecewise (fun _ => x) fun _ => y) =
p m ((s.map (Fin.castIso hm).toEquiv.toEmbedding).piecewise (fun _ => x) fun _ => y) := by
rintro m rfl
simp (config := { unfoldPartialApp := true }) [Finset.piecewise]
apply this
#align formal_multilinear_series.change_origin_eval FormalMultilinearSeries.changeOrigin_eval
/-- Power series terms are analytic as we vary the origin -/
theorem analyticAt_changeOrigin (p : FormalMultilinearSeries 𝕜 E F) (rp : p.radius > 0) (n : ℕ) :
AnalyticAt 𝕜 (fun x ↦ p.changeOrigin x n) 0 :=
(FormalMultilinearSeries.hasFPowerSeriesOnBall_changeOrigin p n rp).analyticAt
end FormalMultilinearSeries
section
variable [CompleteSpace F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x y : E} {r : ℝ≥0∞}
/-- If a function admits a power series expansion `p` on a ball `B (x, r)`, then it also admits a
power series on any subball of this ball (even with a different center), given by `p.changeOrigin`.
-/
theorem HasFPowerSeriesOnBall.changeOrigin (hf : HasFPowerSeriesOnBall f p x r)
(h : (‖y‖₊ : ℝ≥0∞) < r) : HasFPowerSeriesOnBall f (p.changeOrigin y) (x + y) (r - ‖y‖₊) :=
{ r_le := by
apply le_trans _ p.changeOrigin_radius
exact tsub_le_tsub hf.r_le le_rfl
r_pos := by simp [h]
hasSum := fun {z} hz => by
have : f (x + y + z) =
FormalMultilinearSeries.sum (FormalMultilinearSeries.changeOrigin p y) z := by
rw [mem_emetric_ball_zero_iff, lt_tsub_iff_right, add_comm] at hz
rw [p.changeOrigin_eval (hz.trans_le hf.r_le), add_assoc, hf.sum]
refine' mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt _ hz)
exact mod_cast nnnorm_add_le y z
rw [this]
apply (p.changeOrigin y).hasSum
refine' EMetric.ball_subset_ball (le_trans _ p.changeOrigin_radius) hz
exact tsub_le_tsub hf.r_le le_rfl }
#align has_fpower_series_on_ball.change_origin HasFPowerSeriesOnBall.changeOrigin
/-- If a function admits a power series expansion `p` on an open ball `B (x, r)`, then
it is analytic at every point of this ball. -/
theorem HasFPowerSeriesOnBall.analyticAt_of_mem (hf : HasFPowerSeriesOnBall f p x r)
(h : y ∈ EMetric.ball x r) : AnalyticAt 𝕜 f y := by
have : (‖y - x‖₊ : ℝ≥0∞) < r := by simpa [edist_eq_coe_nnnorm_sub] using h
have := hf.changeOrigin this
rw [add_sub_cancel'_right] at this
exact this.analyticAt
#align has_fpower_series_on_ball.analytic_at_of_mem HasFPowerSeriesOnBall.analyticAt_of_mem
theorem HasFPowerSeriesOnBall.analyticOn (hf : HasFPowerSeriesOnBall f p x r) :
AnalyticOn 𝕜 f (EMetric.ball x r) :=
fun _y hy => hf.analyticAt_of_mem hy
#align has_fpower_series_on_ball.analytic_on HasFPowerSeriesOnBall.analyticOn
variable (𝕜 f)
/-- For any function `f` from a normed vector space to a Banach space, the set of points `x` such
that `f` is analytic at `x` is open. -/
theorem isOpen_analyticAt : IsOpen { x | AnalyticAt 𝕜 f x } := by
rw [isOpen_iff_mem_nhds]
rintro x ⟨p, r, hr⟩
exact mem_of_superset (EMetric.ball_mem_nhds _ hr.r_pos) fun y hy => hr.analyticAt_of_mem hy
#align is_open_analytic_at isOpen_analyticAt
variable {𝕜}
theorem AnalyticAt.eventually_analyticAt {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∀ᶠ y in 𝓝 x, AnalyticAt 𝕜 f y :=
(isOpen_analyticAt 𝕜 f).mem_nhds h
theorem AnalyticAt.exists_mem_nhds_analyticOn {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∃ s ∈ 𝓝 x, AnalyticOn 𝕜 f s :=
h.eventually_analyticAt.exists_mem
/-- If we're analytic at a point, we're analytic in a nonempty ball -/
theorem AnalyticAt.exists_ball_analyticOn {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∃ r : ℝ, 0 < r ∧ AnalyticOn 𝕜 f (Metric.ball x r) :=
Metric.isOpen_iff.mp (isOpen_analyticAt _ _) _ h
end
section
open FormalMultilinearSeries
variable {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E} {z₀ : 𝕜}
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z)) := by
refine' ⟨fun ⟨r, _, r_pos, h⟩ =>
eventually_of_mem (EMetric.ball_mem_nhds 0 r_pos) fun _ => by simpa using h, _⟩
simp only [Metric.eventually_nhds_iff]
rintro ⟨r, r_pos, h⟩
refine' ⟨p.radius ⊓ r.toNNReal, by simp, _, _⟩
· simp only [r_pos.lt, lt_inf_iff, ENNReal.coe_pos, Real.toNNReal_pos, and_true_iff]
obtain ⟨z, z_pos, le_z⟩ := NormedField.exists_norm_lt 𝕜 r_pos.lt
have : (‖z‖₊ : ENNReal) ≤ p.radius := by
simp only [dist_zero_right] at h
apply FormalMultilinearSeries.le_radius_of_tendsto
convert tendsto_norm.comp (h le_z).summable.tendsto_atTop_zero
funext
simp [norm_smul, mul_comm]
refine' lt_of_lt_of_le _ this
simp only [ENNReal.coe_pos]
exact zero_lt_iff.mpr (nnnorm_ne_zero_iff.mpr (norm_pos_iff.mp z_pos))
· simp only [EMetric.mem_ball, lt_inf_iff, edist_lt_coe, apply_eq_pow_smul_coeff, and_imp,
dist_zero_right] at h ⊢
refine' fun {y} _ hyr => h _
|
simpa [nndist_eq_nnnorm, Real.lt_toNNReal_iff_coe_lt] using hyr
|
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z)) := by
refine' ⟨fun ⟨r, _, r_pos, h⟩ =>
eventually_of_mem (EMetric.ball_mem_nhds 0 r_pos) fun _ => by simpa using h, _⟩
simp only [Metric.eventually_nhds_iff]
rintro ⟨r, r_pos, h⟩
refine' ⟨p.radius ⊓ r.toNNReal, by simp, _, _⟩
· simp only [r_pos.lt, lt_inf_iff, ENNReal.coe_pos, Real.toNNReal_pos, and_true_iff]
obtain ⟨z, z_pos, le_z⟩ := NormedField.exists_norm_lt 𝕜 r_pos.lt
have : (‖z‖₊ : ENNReal) ≤ p.radius := by
simp only [dist_zero_right] at h
apply FormalMultilinearSeries.le_radius_of_tendsto
convert tendsto_norm.comp (h le_z).summable.tendsto_atTop_zero
funext
simp [norm_smul, mul_comm]
refine' lt_of_lt_of_le _ this
simp only [ENNReal.coe_pos]
exact zero_lt_iff.mpr (nnnorm_ne_zero_iff.mpr (norm_pos_iff.mp z_pos))
· simp only [EMetric.mem_ball, lt_inf_iff, edist_lt_coe, apply_eq_pow_smul_coeff, and_imp,
dist_zero_right] at h ⊢
refine' fun {y} _ hyr => h _
|
Mathlib.Analysis.Analytic.Basic.1430_0.jQw1fRSE1vGpOll
|
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z))
|
Mathlib_Analysis_Analytic_Basic
|
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
p : FormalMultilinearSeries 𝕜 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
⊢ HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ (z : 𝕜) in 𝓝 z₀, HasSum (fun n => (z - z₀) ^ n • coeff p n) (f z)
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Logic.Equiv.Fin
import Mathlib.Topology.Algebra.InfiniteSum.Module
#align_import analysis.analytic.basic from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
/-!
# Analytic functions
A function is analytic in one dimension around `0` if it can be written as a converging power series
`Σ pₙ zⁿ`. This definition can be extended to any dimension (even in infinite dimension) by
requiring that `pₙ` is a continuous `n`-multilinear map. In general, `pₙ` is not unique (in two
dimensions, taking `p₂ (x, y) (x', y') = x y'` or `y x'` gives the same map when applied to a
vector `(x, y) (x, y)`). A way to guarantee uniqueness is to take a symmetric `pₙ`, but this is not
always possible in nonzero characteristic (in characteristic 2, the previous example has no
symmetric representative). Therefore, we do not insist on symmetry or uniqueness in the definition,
and we only require the existence of a converging series.
The general framework is important to say that the exponential map on bounded operators on a Banach
space is analytic, as well as the inverse on invertible operators.
## Main definitions
Let `p` be a formal multilinear series from `E` to `F`, i.e., `p n` is a multilinear map on `E^n`
for `n : ℕ`.
* `p.radius`: the largest `r : ℝ≥0∞` such that `‖p n‖ * r^n` grows subexponentially.
* `p.le_radius_of_bound`, `p.le_radius_of_bound_nnreal`, `p.le_radius_of_isBigO`: if `‖p n‖ * r ^ n`
is bounded above, then `r ≤ p.radius`;
* `p.isLittleO_of_lt_radius`, `p.norm_mul_pow_le_mul_pow_of_lt_radius`,
`p.isLittleO_one_of_lt_radius`,
`p.norm_mul_pow_le_of_lt_radius`, `p.nnnorm_mul_pow_le_of_lt_radius`: if `r < p.radius`, then
`‖p n‖ * r ^ n` tends to zero exponentially;
* `p.lt_radius_of_isBigO`: if `r ≠ 0` and `‖p n‖ * r ^ n = O(a ^ n)` for some `-1 < a < 1`, then
`r < p.radius`;
* `p.partialSum n x`: the sum `∑_{i = 0}^{n-1} pᵢ xⁱ`.
* `p.sum x`: the sum `∑'_{i = 0}^{∞} pᵢ xⁱ`.
Additionally, let `f` be a function from `E` to `F`.
* `HasFPowerSeriesOnBall f p x r`: on the ball of center `x` with radius `r`,
`f (x + y) = ∑'_n pₙ yⁿ`.
* `HasFPowerSeriesAt f p x`: on some ball of center `x` with positive radius, holds
`HasFPowerSeriesOnBall f p x r`.
* `AnalyticAt 𝕜 f x`: there exists a power series `p` such that holds `HasFPowerSeriesAt f p x`.
* `AnalyticOn 𝕜 f s`: the function `f` is analytic at every point of `s`.
We develop the basic properties of these notions, notably:
* If a function admits a power series, it is continuous (see
`HasFPowerSeriesOnBall.continuousOn` and `HasFPowerSeriesAt.continuousAt` and
`AnalyticAt.continuousAt`).
* In a complete space, the sum of a formal power series with positive radius is well defined on the
disk of convergence, see `FormalMultilinearSeries.hasFPowerSeriesOnBall`.
* If a function admits a power series in a ball, then it is analytic at any point `y` of this ball,
and the power series there can be expressed in terms of the initial power series `p` as
`p.changeOrigin y`. See `HasFPowerSeriesOnBall.changeOrigin`. It follows in particular that
the set of points at which a given function is analytic is open, see `isOpen_analyticAt`.
## Implementation details
We only introduce the radius of convergence of a power series, as `p.radius`.
For a power series in finitely many dimensions, there is a finer (directional, coordinate-dependent)
notion, describing the polydisk of convergence. This notion is more specific, and not necessary to
build the general theory. We do not define it here.
-/
noncomputable section
variable {𝕜 E F G : Type*}
open Topology Classical BigOperators NNReal Filter ENNReal
open Set Filter Asymptotics
namespace FormalMultilinearSeries
variable [Ring 𝕜] [AddCommGroup E] [AddCommGroup F] [Module 𝕜 E] [Module 𝕜 F]
variable [TopologicalSpace E] [TopologicalSpace F]
variable [TopologicalAddGroup E] [TopologicalAddGroup F]
variable [ContinuousConstSMul 𝕜 E] [ContinuousConstSMul 𝕜 F]
/-- Given a formal multilinear series `p` and a vector `x`, then `p.sum x` is the sum `Σ pₙ xⁿ`. A
priori, it only behaves well when `‖x‖ < p.radius`. -/
protected def sum (p : FormalMultilinearSeries 𝕜 E F) (x : E) : F :=
∑' n : ℕ, p n fun _ => x
#align formal_multilinear_series.sum FormalMultilinearSeries.sum
/-- Given a formal multilinear series `p` and a vector `x`, then `p.partialSum n x` is the sum
`Σ pₖ xᵏ` for `k ∈ {0,..., n-1}`. -/
def partialSum (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) (x : E) : F :=
∑ k in Finset.range n, p k fun _ : Fin k => x
#align formal_multilinear_series.partial_sum FormalMultilinearSeries.partialSum
/-- The partial sums of a formal multilinear series are continuous. -/
theorem partialSum_continuous (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) :
Continuous (p.partialSum n) := by
unfold partialSum -- Porting note: added
continuity
#align formal_multilinear_series.partial_sum_continuous FormalMultilinearSeries.partialSum_continuous
end FormalMultilinearSeries
/-! ### The radius of a formal multilinear series -/
variable [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F]
[NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G]
namespace FormalMultilinearSeries
variable (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
/-- The radius of a formal multilinear series is the largest `r` such that the sum `Σ ‖pₙ‖ ‖y‖ⁿ`
converges for all `‖y‖ < r`. This implies that `Σ pₙ yⁿ` converges for all `‖y‖ < r`, but these
definitions are *not* equivalent in general. -/
def radius (p : FormalMultilinearSeries 𝕜 E F) : ℝ≥0∞ :=
⨆ (r : ℝ≥0) (C : ℝ) (_ : ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C), (r : ℝ≥0∞)
#align formal_multilinear_series.radius FormalMultilinearSeries.radius
/-- If `‖pₙ‖ rⁿ` is bounded in `n`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_bound (C : ℝ) {r : ℝ≥0} (h : ∀ n : ℕ, ‖p n‖ * (r : ℝ) ^ n ≤ C) :
(r : ℝ≥0∞) ≤ p.radius :=
le_iSup_of_le r <| le_iSup_of_le C <| le_iSup (fun _ => (r : ℝ≥0∞)) h
#align formal_multilinear_series.le_radius_of_bound FormalMultilinearSeries.le_radius_of_bound
/-- If `‖pₙ‖ rⁿ` is bounded in `n`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_bound_nnreal (C : ℝ≥0) {r : ℝ≥0} (h : ∀ n : ℕ, ‖p n‖₊ * r ^ n ≤ C) :
(r : ℝ≥0∞) ≤ p.radius :=
p.le_radius_of_bound C fun n => mod_cast h n
#align formal_multilinear_series.le_radius_of_bound_nnreal FormalMultilinearSeries.le_radius_of_bound_nnreal
/-- If `‖pₙ‖ rⁿ = O(1)`, as `n → ∞`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_isBigO (h : (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] fun _ => (1 : ℝ)) :
↑r ≤ p.radius :=
Exists.elim (isBigO_one_nat_atTop_iff.1 h) fun C hC =>
p.le_radius_of_bound C fun n => (le_abs_self _).trans (hC n)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.le_radius_of_is_O FormalMultilinearSeries.le_radius_of_isBigO
theorem le_radius_of_eventually_le (C) (h : ∀ᶠ n in atTop, ‖p n‖ * (r : ℝ) ^ n ≤ C) :
↑r ≤ p.radius :=
p.le_radius_of_isBigO <| IsBigO.of_bound C <| h.mono fun n hn => by simpa
#align formal_multilinear_series.le_radius_of_eventually_le FormalMultilinearSeries.le_radius_of_eventually_le
theorem le_radius_of_summable_nnnorm (h : Summable fun n => ‖p n‖₊ * r ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_bound_nnreal (∑' n, ‖p n‖₊ * r ^ n) fun _ => le_tsum' h _
#align formal_multilinear_series.le_radius_of_summable_nnnorm FormalMultilinearSeries.le_radius_of_summable_nnnorm
theorem le_radius_of_summable (h : Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_summable_nnnorm <| by
simp only [← coe_nnnorm] at h
exact mod_cast h
#align formal_multilinear_series.le_radius_of_summable FormalMultilinearSeries.le_radius_of_summable
theorem radius_eq_top_of_forall_nnreal_isBigO
(h : ∀ r : ℝ≥0, (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] fun _ => (1 : ℝ)) : p.radius = ∞ :=
ENNReal.eq_top_of_forall_nnreal_le fun r => p.le_radius_of_isBigO (h r)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.radius_eq_top_of_forall_nnreal_is_O FormalMultilinearSeries.radius_eq_top_of_forall_nnreal_isBigO
theorem radius_eq_top_of_eventually_eq_zero (h : ∀ᶠ n in atTop, p n = 0) : p.radius = ∞ :=
p.radius_eq_top_of_forall_nnreal_isBigO fun r =>
(isBigO_zero _ _).congr' (h.mono fun n hn => by simp [hn]) EventuallyEq.rfl
#align formal_multilinear_series.radius_eq_top_of_eventually_eq_zero FormalMultilinearSeries.radius_eq_top_of_eventually_eq_zero
theorem radius_eq_top_of_forall_image_add_eq_zero (n : ℕ) (hn : ∀ m, p (m + n) = 0) :
p.radius = ∞ :=
p.radius_eq_top_of_eventually_eq_zero <|
mem_atTop_sets.2 ⟨n, fun _ hk => tsub_add_cancel_of_le hk ▸ hn _⟩
#align formal_multilinear_series.radius_eq_top_of_forall_image_add_eq_zero FormalMultilinearSeries.radius_eq_top_of_forall_image_add_eq_zero
@[simp]
theorem constFormalMultilinearSeries_radius {v : F} :
(constFormalMultilinearSeries 𝕜 E v).radius = ⊤ :=
(constFormalMultilinearSeries 𝕜 E v).radius_eq_top_of_forall_image_add_eq_zero 1
(by simp [constFormalMultilinearSeries])
#align formal_multilinear_series.const_formal_multilinear_series_radius FormalMultilinearSeries.constFormalMultilinearSeries_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` tends to zero exponentially:
for some `0 < a < 1`, `‖p n‖ rⁿ = o(aⁿ)`. -/
theorem isLittleO_of_lt_radius (h : ↑r < p.radius) :
∃ a ∈ Ioo (0 : ℝ) 1, (fun n => ‖p n‖ * (r : ℝ) ^ n) =o[atTop] (a ^ ·) := by
have := (TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 4
rw [this]
-- Porting note: was
-- rw [(TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 4]
simp only [radius, lt_iSup_iff] at h
rcases h with ⟨t, C, hC, rt⟩
rw [ENNReal.coe_lt_coe, ← NNReal.coe_lt_coe] at rt
have : 0 < (t : ℝ) := r.coe_nonneg.trans_lt rt
rw [← div_lt_one this] at rt
refine' ⟨_, rt, C, Or.inr zero_lt_one, fun n => _⟩
calc
|‖p n‖ * (r : ℝ) ^ n| = ‖p n‖ * (t : ℝ) ^ n * (r / t : ℝ) ^ n := by
field_simp [mul_right_comm, abs_mul]
_ ≤ C * (r / t : ℝ) ^ n := by gcongr; apply hC
#align formal_multilinear_series.is_o_of_lt_radius FormalMultilinearSeries.isLittleO_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ = o(1)`. -/
theorem isLittleO_one_of_lt_radius (h : ↑r < p.radius) :
(fun n => ‖p n‖ * (r : ℝ) ^ n) =o[atTop] (fun _ => 1 : ℕ → ℝ) :=
let ⟨_, ha, hp⟩ := p.isLittleO_of_lt_radius h
hp.trans <| (isLittleO_pow_pow_of_lt_left ha.1.le ha.2).congr (fun _ => rfl) one_pow
#align formal_multilinear_series.is_o_one_of_lt_radius FormalMultilinearSeries.isLittleO_one_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` tends to zero exponentially:
for some `0 < a < 1` and `C > 0`, `‖p n‖ * r ^ n ≤ C * a ^ n`. -/
theorem norm_mul_pow_le_mul_pow_of_lt_radius (h : ↑r < p.radius) :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C * a ^ n := by
-- Porting note: moved out of `rcases`
have := ((TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 5).mp
(p.isLittleO_of_lt_radius h)
rcases this with ⟨a, ha, C, hC, H⟩
exact ⟨a, ha, C, hC, fun n => (le_abs_self _).trans (H n)⟩
#align formal_multilinear_series.norm_mul_pow_le_mul_pow_of_lt_radius FormalMultilinearSeries.norm_mul_pow_le_mul_pow_of_lt_radius
/-- If `r ≠ 0` and `‖pₙ‖ rⁿ = O(aⁿ)` for some `-1 < a < 1`, then `r < p.radius`. -/
theorem lt_radius_of_isBigO (h₀ : r ≠ 0) {a : ℝ} (ha : a ∈ Ioo (-1 : ℝ) 1)
(hp : (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] (a ^ ·)) : ↑r < p.radius := by
-- Porting note: moved out of `rcases`
have := ((TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 2 5)
rcases this.mp ⟨a, ha, hp⟩ with ⟨a, ha, C, hC, hp⟩
rw [← pos_iff_ne_zero, ← NNReal.coe_pos] at h₀
lift a to ℝ≥0 using ha.1.le
have : (r : ℝ) < r / a := by
simpa only [div_one] using (div_lt_div_left h₀ zero_lt_one ha.1).2 ha.2
norm_cast at this
rw [← ENNReal.coe_lt_coe] at this
refine' this.trans_le (p.le_radius_of_bound C fun n => _)
rw [NNReal.coe_div, div_pow, ← mul_div_assoc, div_le_iff (pow_pos ha.1 n)]
exact (le_abs_self _).trans (hp n)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.lt_radius_of_is_O FormalMultilinearSeries.lt_radius_of_isBigO
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem norm_mul_pow_le_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C :=
let ⟨_, ha, C, hC, h⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius h
⟨C, hC, fun n => (h n).trans <| mul_le_of_le_one_right hC.lt.le (pow_le_one _ ha.1.le ha.2.le)⟩
#align formal_multilinear_series.norm_mul_pow_le_of_lt_radius FormalMultilinearSeries.norm_mul_pow_le_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem norm_le_div_pow_of_pos_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h0 : 0 < r) (h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖ ≤ C / (r : ℝ) ^ n :=
let ⟨C, hC, hp⟩ := p.norm_mul_pow_le_of_lt_radius h
⟨C, hC, fun n => Iff.mpr (le_div_iff (pow_pos h0 _)) (hp n)⟩
#align formal_multilinear_series.norm_le_div_pow_of_pos_of_lt_radius FormalMultilinearSeries.norm_le_div_pow_of_pos_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem nnnorm_mul_pow_le_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖₊ * r ^ n ≤ C :=
let ⟨C, hC, hp⟩ := p.norm_mul_pow_le_of_lt_radius h
⟨⟨C, hC.lt.le⟩, hC, mod_cast hp⟩
#align formal_multilinear_series.nnnorm_mul_pow_le_of_lt_radius FormalMultilinearSeries.nnnorm_mul_pow_le_of_lt_radius
theorem le_radius_of_tendsto (p : FormalMultilinearSeries 𝕜 E F) {l : ℝ}
(h : Tendsto (fun n => ‖p n‖ * (r : ℝ) ^ n) atTop (𝓝 l)) : ↑r ≤ p.radius :=
p.le_radius_of_isBigO (h.isBigO_one _)
#align formal_multilinear_series.le_radius_of_tendsto FormalMultilinearSeries.le_radius_of_tendsto
theorem le_radius_of_summable_norm (p : FormalMultilinearSeries 𝕜 E F)
(hs : Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_tendsto hs.tendsto_atTop_zero
#align formal_multilinear_series.le_radius_of_summable_norm FormalMultilinearSeries.le_radius_of_summable_norm
theorem not_summable_norm_of_radius_lt_nnnorm (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(h : p.radius < ‖x‖₊) : ¬Summable fun n => ‖p n‖ * ‖x‖ ^ n :=
fun hs => not_le_of_lt h (p.le_radius_of_summable_norm hs)
#align formal_multilinear_series.not_summable_norm_of_radius_lt_nnnorm FormalMultilinearSeries.not_summable_norm_of_radius_lt_nnnorm
theorem summable_norm_mul_pow (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0} (h : ↑r < p.radius) :
Summable fun n : ℕ => ‖p n‖ * (r : ℝ) ^ n := by
obtain ⟨a, ha : a ∈ Ioo (0 : ℝ) 1, C, - : 0 < C, hp⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius h
exact .of_nonneg_of_le (fun n => mul_nonneg (norm_nonneg _) (pow_nonneg r.coe_nonneg _))
hp ((summable_geometric_of_lt_1 ha.1.le ha.2).mul_left _)
#align formal_multilinear_series.summable_norm_mul_pow FormalMultilinearSeries.summable_norm_mul_pow
theorem summable_norm_apply (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : Summable fun n : ℕ => ‖p n fun _ => x‖ := by
rw [mem_emetric_ball_zero_iff] at hx
refine' .of_nonneg_of_le
(fun _ => norm_nonneg _) (fun n => ((p n).le_op_norm _).trans_eq _) (p.summable_norm_mul_pow hx)
simp
#align formal_multilinear_series.summable_norm_apply FormalMultilinearSeries.summable_norm_apply
theorem summable_nnnorm_mul_pow (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0} (h : ↑r < p.radius) :
Summable fun n : ℕ => ‖p n‖₊ * r ^ n := by
rw [← NNReal.summable_coe]
push_cast
exact p.summable_norm_mul_pow h
#align formal_multilinear_series.summable_nnnorm_mul_pow FormalMultilinearSeries.summable_nnnorm_mul_pow
protected theorem summable [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : Summable fun n : ℕ => p n fun _ => x :=
(p.summable_norm_apply hx).of_norm
#align formal_multilinear_series.summable FormalMultilinearSeries.summable
theorem radius_eq_top_of_summable_norm (p : FormalMultilinearSeries 𝕜 E F)
(hs : ∀ r : ℝ≥0, Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : p.radius = ∞ :=
ENNReal.eq_top_of_forall_nnreal_le fun r => p.le_radius_of_summable_norm (hs r)
#align formal_multilinear_series.radius_eq_top_of_summable_norm FormalMultilinearSeries.radius_eq_top_of_summable_norm
theorem radius_eq_top_iff_summable_norm (p : FormalMultilinearSeries 𝕜 E F) :
p.radius = ∞ ↔ ∀ r : ℝ≥0, Summable fun n => ‖p n‖ * (r : ℝ) ^ n := by
constructor
· intro h r
obtain ⟨a, ha : a ∈ Ioo (0 : ℝ) 1, C, - : 0 < C, hp⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius
(show (r : ℝ≥0∞) < p.radius from h.symm ▸ ENNReal.coe_lt_top)
refine' .of_norm_bounded
(fun n => (C : ℝ) * a ^ n) ((summable_geometric_of_lt_1 ha.1.le ha.2).mul_left _) fun n => _
specialize hp n
rwa [Real.norm_of_nonneg (mul_nonneg (norm_nonneg _) (pow_nonneg r.coe_nonneg n))]
· exact p.radius_eq_top_of_summable_norm
#align formal_multilinear_series.radius_eq_top_iff_summable_norm FormalMultilinearSeries.radius_eq_top_iff_summable_norm
/-- If the radius of `p` is positive, then `‖pₙ‖` grows at most geometrically. -/
theorem le_mul_pow_of_radius_pos (p : FormalMultilinearSeries 𝕜 E F) (h : 0 < p.radius) :
∃ (C r : _) (hC : 0 < C) (_ : 0 < r), ∀ n, ‖p n‖ ≤ C * r ^ n := by
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 h with ⟨r, r0, rlt⟩
have rpos : 0 < (r : ℝ) := by simp [ENNReal.coe_pos.1 r0]
rcases norm_le_div_pow_of_pos_of_lt_radius p rpos rlt with ⟨C, Cpos, hCp⟩
refine' ⟨C, r⁻¹, Cpos, by simp only [inv_pos, rpos], fun n => _⟩
-- Porting note: was `convert`
rw [inv_pow, ← div_eq_mul_inv]
exact hCp n
#align formal_multilinear_series.le_mul_pow_of_radius_pos FormalMultilinearSeries.le_mul_pow_of_radius_pos
/-- The radius of the sum of two formal series is at least the minimum of their two radii. -/
theorem min_radius_le_radius_add (p q : FormalMultilinearSeries 𝕜 E F) :
min p.radius q.radius ≤ (p + q).radius := by
refine' ENNReal.le_of_forall_nnreal_lt fun r hr => _
rw [lt_min_iff] at hr
have := ((p.isLittleO_one_of_lt_radius hr.1).add (q.isLittleO_one_of_lt_radius hr.2)).isBigO
refine' (p + q).le_radius_of_isBigO ((isBigO_of_le _ fun n => _).trans this)
rw [← add_mul, norm_mul, norm_mul, norm_norm]
exact mul_le_mul_of_nonneg_right ((norm_add_le _ _).trans (le_abs_self _)) (norm_nonneg _)
#align formal_multilinear_series.min_radius_le_radius_add FormalMultilinearSeries.min_radius_le_radius_add
@[simp]
theorem radius_neg (p : FormalMultilinearSeries 𝕜 E F) : (-p).radius = p.radius := by
simp only [radius, neg_apply, norm_neg]
#align formal_multilinear_series.radius_neg FormalMultilinearSeries.radius_neg
protected theorem hasSum [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : HasSum (fun n : ℕ => p n fun _ => x) (p.sum x) :=
(p.summable hx).hasSum
#align formal_multilinear_series.has_sum FormalMultilinearSeries.hasSum
theorem radius_le_radius_continuousLinearMap_comp (p : FormalMultilinearSeries 𝕜 E F)
(f : F →L[𝕜] G) : p.radius ≤ (f.compFormalMultilinearSeries p).radius := by
refine' ENNReal.le_of_forall_nnreal_lt fun r hr => _
apply le_radius_of_isBigO
apply (IsBigO.trans_isLittleO _ (p.isLittleO_one_of_lt_radius hr)).isBigO
refine' IsBigO.mul (@IsBigOWith.isBigO _ _ _ _ _ ‖f‖ _ _ _ _) (isBigO_refl _ _)
refine IsBigOWith.of_bound (eventually_of_forall fun n => ?_)
simpa only [norm_norm] using f.norm_compContinuousMultilinearMap_le (p n)
#align formal_multilinear_series.radius_le_radius_continuous_linear_map_comp FormalMultilinearSeries.radius_le_radius_continuousLinearMap_comp
end FormalMultilinearSeries
/-! ### Expanding a function as a power series -/
section
variable {f g : E → F} {p pf pg : FormalMultilinearSeries 𝕜 E F} {x : E} {r r' : ℝ≥0∞}
/-- Given a function `f : E → F` and a formal multilinear series `p`, we say that `f` has `p` as
a power series on the ball of radius `r > 0` around `x` if `f (x + y) = ∑' pₙ yⁿ` for all `‖y‖ < r`.
-/
structure HasFPowerSeriesOnBall (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (x : E) (r : ℝ≥0∞) :
Prop where
r_le : r ≤ p.radius
r_pos : 0 < r
hasSum :
∀ {y}, y ∈ EMetric.ball (0 : E) r → HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y))
#align has_fpower_series_on_ball HasFPowerSeriesOnBall
/-- Given a function `f : E → F` and a formal multilinear series `p`, we say that `f` has `p` as
a power series around `x` if `f (x + y) = ∑' pₙ yⁿ` for all `y` in a neighborhood of `0`. -/
def HasFPowerSeriesAt (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (x : E) :=
∃ r, HasFPowerSeriesOnBall f p x r
#align has_fpower_series_at HasFPowerSeriesAt
variable (𝕜)
/-- Given a function `f : E → F`, we say that `f` is analytic at `x` if it admits a convergent power
series expansion around `x`. -/
def AnalyticAt (f : E → F) (x : E) :=
∃ p : FormalMultilinearSeries 𝕜 E F, HasFPowerSeriesAt f p x
#align analytic_at AnalyticAt
/-- Given a function `f : E → F`, we say that `f` is analytic on a set `s` if it is analytic around
every point of `s`. -/
def AnalyticOn (f : E → F) (s : Set E) :=
∀ x, x ∈ s → AnalyticAt 𝕜 f x
#align analytic_on AnalyticOn
variable {𝕜}
theorem HasFPowerSeriesOnBall.hasFPowerSeriesAt (hf : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesAt f p x :=
⟨r, hf⟩
#align has_fpower_series_on_ball.has_fpower_series_at HasFPowerSeriesOnBall.hasFPowerSeriesAt
theorem HasFPowerSeriesAt.analyticAt (hf : HasFPowerSeriesAt f p x) : AnalyticAt 𝕜 f x :=
⟨p, hf⟩
#align has_fpower_series_at.analytic_at HasFPowerSeriesAt.analyticAt
theorem HasFPowerSeriesOnBall.analyticAt (hf : HasFPowerSeriesOnBall f p x r) : AnalyticAt 𝕜 f x :=
hf.hasFPowerSeriesAt.analyticAt
#align has_fpower_series_on_ball.analytic_at HasFPowerSeriesOnBall.analyticAt
theorem HasFPowerSeriesOnBall.congr (hf : HasFPowerSeriesOnBall f p x r)
(hg : EqOn f g (EMetric.ball x r)) : HasFPowerSeriesOnBall g p x r :=
{ r_le := hf.r_le
r_pos := hf.r_pos
hasSum := fun {y} hy => by
convert hf.hasSum hy using 1
apply hg.symm
simpa [edist_eq_coe_nnnorm_sub] using hy }
#align has_fpower_series_on_ball.congr HasFPowerSeriesOnBall.congr
/-- If a function `f` has a power series `p` around `x`, then the function `z ↦ f (z - y)` has the
same power series around `x + y`. -/
theorem HasFPowerSeriesOnBall.comp_sub (hf : HasFPowerSeriesOnBall f p x r) (y : E) :
HasFPowerSeriesOnBall (fun z => f (z - y)) p (x + y) r :=
{ r_le := hf.r_le
r_pos := hf.r_pos
hasSum := fun {z} hz => by
convert hf.hasSum hz using 2
abel }
#align has_fpower_series_on_ball.comp_sub HasFPowerSeriesOnBall.comp_sub
theorem HasFPowerSeriesOnBall.hasSum_sub (hf : HasFPowerSeriesOnBall f p x r) {y : E}
(hy : y ∈ EMetric.ball x r) : HasSum (fun n : ℕ => p n fun _ => y - x) (f y) := by
have : y - x ∈ EMetric.ball (0 : E) r := by simpa [edist_eq_coe_nnnorm_sub] using hy
simpa only [add_sub_cancel'_right] using hf.hasSum this
#align has_fpower_series_on_ball.has_sum_sub HasFPowerSeriesOnBall.hasSum_sub
theorem HasFPowerSeriesOnBall.radius_pos (hf : HasFPowerSeriesOnBall f p x r) : 0 < p.radius :=
lt_of_lt_of_le hf.r_pos hf.r_le
#align has_fpower_series_on_ball.radius_pos HasFPowerSeriesOnBall.radius_pos
theorem HasFPowerSeriesAt.radius_pos (hf : HasFPowerSeriesAt f p x) : 0 < p.radius :=
let ⟨_, hr⟩ := hf
hr.radius_pos
#align has_fpower_series_at.radius_pos HasFPowerSeriesAt.radius_pos
theorem HasFPowerSeriesOnBall.mono (hf : HasFPowerSeriesOnBall f p x r) (r'_pos : 0 < r')
(hr : r' ≤ r) : HasFPowerSeriesOnBall f p x r' :=
⟨le_trans hr hf.1, r'_pos, fun hy => hf.hasSum (EMetric.ball_subset_ball hr hy)⟩
#align has_fpower_series_on_ball.mono HasFPowerSeriesOnBall.mono
theorem HasFPowerSeriesAt.congr (hf : HasFPowerSeriesAt f p x) (hg : f =ᶠ[𝓝 x] g) :
HasFPowerSeriesAt g p x := by
rcases hf with ⟨r₁, h₁⟩
rcases EMetric.mem_nhds_iff.mp hg with ⟨r₂, h₂pos, h₂⟩
exact ⟨min r₁ r₂,
(h₁.mono (lt_min h₁.r_pos h₂pos) inf_le_left).congr
fun y hy => h₂ (EMetric.ball_subset_ball inf_le_right hy)⟩
#align has_fpower_series_at.congr HasFPowerSeriesAt.congr
protected theorem HasFPowerSeriesAt.eventually (hf : HasFPowerSeriesAt f p x) :
∀ᶠ r : ℝ≥0∞ in 𝓝[>] 0, HasFPowerSeriesOnBall f p x r :=
let ⟨_, hr⟩ := hf
mem_of_superset (Ioo_mem_nhdsWithin_Ioi (left_mem_Ico.2 hr.r_pos)) fun _ hr' =>
hr.mono hr'.1 hr'.2.le
#align has_fpower_series_at.eventually HasFPowerSeriesAt.eventually
theorem HasFPowerSeriesOnBall.eventually_hasSum (hf : HasFPowerSeriesOnBall f p x r) :
∀ᶠ y in 𝓝 0, HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y)) := by
filter_upwards [EMetric.ball_mem_nhds (0 : E) hf.r_pos] using fun _ => hf.hasSum
#align has_fpower_series_on_ball.eventually_has_sum HasFPowerSeriesOnBall.eventually_hasSum
theorem HasFPowerSeriesAt.eventually_hasSum (hf : HasFPowerSeriesAt f p x) :
∀ᶠ y in 𝓝 0, HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y)) :=
let ⟨_, hr⟩ := hf
hr.eventually_hasSum
#align has_fpower_series_at.eventually_has_sum HasFPowerSeriesAt.eventually_hasSum
theorem HasFPowerSeriesOnBall.eventually_hasSum_sub (hf : HasFPowerSeriesOnBall f p x r) :
∀ᶠ y in 𝓝 x, HasSum (fun n : ℕ => p n fun _ : Fin n => y - x) (f y) := by
filter_upwards [EMetric.ball_mem_nhds x hf.r_pos] with y using hf.hasSum_sub
#align has_fpower_series_on_ball.eventually_has_sum_sub HasFPowerSeriesOnBall.eventually_hasSum_sub
theorem HasFPowerSeriesAt.eventually_hasSum_sub (hf : HasFPowerSeriesAt f p x) :
∀ᶠ y in 𝓝 x, HasSum (fun n : ℕ => p n fun _ : Fin n => y - x) (f y) :=
let ⟨_, hr⟩ := hf
hr.eventually_hasSum_sub
#align has_fpower_series_at.eventually_has_sum_sub HasFPowerSeriesAt.eventually_hasSum_sub
theorem HasFPowerSeriesOnBall.eventually_eq_zero
(hf : HasFPowerSeriesOnBall f (0 : FormalMultilinearSeries 𝕜 E F) x r) :
∀ᶠ z in 𝓝 x, f z = 0 := by
filter_upwards [hf.eventually_hasSum_sub] with z hz using hz.unique hasSum_zero
#align has_fpower_series_on_ball.eventually_eq_zero HasFPowerSeriesOnBall.eventually_eq_zero
theorem HasFPowerSeriesAt.eventually_eq_zero
(hf : HasFPowerSeriesAt f (0 : FormalMultilinearSeries 𝕜 E F) x) : ∀ᶠ z in 𝓝 x, f z = 0 :=
let ⟨_, hr⟩ := hf
hr.eventually_eq_zero
#align has_fpower_series_at.eventually_eq_zero HasFPowerSeriesAt.eventually_eq_zero
theorem hasFPowerSeriesOnBall_const {c : F} {e : E} :
HasFPowerSeriesOnBall (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e ⊤ := by
refine' ⟨by simp, WithTop.zero_lt_top, fun _ => hasSum_single 0 fun n hn => _⟩
simp [constFormalMultilinearSeries_apply hn]
#align has_fpower_series_on_ball_const hasFPowerSeriesOnBall_const
theorem hasFPowerSeriesAt_const {c : F} {e : E} :
HasFPowerSeriesAt (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e :=
⟨⊤, hasFPowerSeriesOnBall_const⟩
#align has_fpower_series_at_const hasFPowerSeriesAt_const
theorem analyticAt_const {v : F} : AnalyticAt 𝕜 (fun _ => v) x :=
⟨constFormalMultilinearSeries 𝕜 E v, hasFPowerSeriesAt_const⟩
#align analytic_at_const analyticAt_const
theorem analyticOn_const {v : F} {s : Set E} : AnalyticOn 𝕜 (fun _ => v) s :=
fun _ _ => analyticAt_const
#align analytic_on_const analyticOn_const
theorem HasFPowerSeriesOnBall.add (hf : HasFPowerSeriesOnBall f pf x r)
(hg : HasFPowerSeriesOnBall g pg x r) : HasFPowerSeriesOnBall (f + g) (pf + pg) x r :=
{ r_le := le_trans (le_min_iff.2 ⟨hf.r_le, hg.r_le⟩) (pf.min_radius_le_radius_add pg)
r_pos := hf.r_pos
hasSum := fun hy => (hf.hasSum hy).add (hg.hasSum hy) }
#align has_fpower_series_on_ball.add HasFPowerSeriesOnBall.add
theorem HasFPowerSeriesAt.add (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x) :
HasFPowerSeriesAt (f + g) (pf + pg) x := by
rcases (hf.eventually.and hg.eventually).exists with ⟨r, hr⟩
exact ⟨r, hr.1.add hr.2⟩
#align has_fpower_series_at.add HasFPowerSeriesAt.add
theorem AnalyticAt.congr (hf : AnalyticAt 𝕜 f x) (hg : f =ᶠ[𝓝 x] g) : AnalyticAt 𝕜 g x :=
let ⟨_, hpf⟩ := hf
(hpf.congr hg).analyticAt
theorem analyticAt_congr (h : f =ᶠ[𝓝 x] g) : AnalyticAt 𝕜 f x ↔ AnalyticAt 𝕜 g x :=
⟨fun hf ↦ hf.congr h, fun hg ↦ hg.congr h.symm⟩
theorem AnalyticAt.add (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) : AnalyticAt 𝕜 (f + g) x :=
let ⟨_, hpf⟩ := hf
let ⟨_, hqf⟩ := hg
(hpf.add hqf).analyticAt
#align analytic_at.add AnalyticAt.add
theorem HasFPowerSeriesOnBall.neg (hf : HasFPowerSeriesOnBall f pf x r) :
HasFPowerSeriesOnBall (-f) (-pf) x r :=
{ r_le := by
rw [pf.radius_neg]
exact hf.r_le
r_pos := hf.r_pos
hasSum := fun hy => (hf.hasSum hy).neg }
#align has_fpower_series_on_ball.neg HasFPowerSeriesOnBall.neg
theorem HasFPowerSeriesAt.neg (hf : HasFPowerSeriesAt f pf x) : HasFPowerSeriesAt (-f) (-pf) x :=
let ⟨_, hrf⟩ := hf
hrf.neg.hasFPowerSeriesAt
#align has_fpower_series_at.neg HasFPowerSeriesAt.neg
theorem AnalyticAt.neg (hf : AnalyticAt 𝕜 f x) : AnalyticAt 𝕜 (-f) x :=
let ⟨_, hpf⟩ := hf
hpf.neg.analyticAt
#align analytic_at.neg AnalyticAt.neg
theorem HasFPowerSeriesOnBall.sub (hf : HasFPowerSeriesOnBall f pf x r)
(hg : HasFPowerSeriesOnBall g pg x r) : HasFPowerSeriesOnBall (f - g) (pf - pg) x r := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align has_fpower_series_on_ball.sub HasFPowerSeriesOnBall.sub
theorem HasFPowerSeriesAt.sub (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x) :
HasFPowerSeriesAt (f - g) (pf - pg) x := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align has_fpower_series_at.sub HasFPowerSeriesAt.sub
theorem AnalyticAt.sub (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) :
AnalyticAt 𝕜 (f - g) x := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align analytic_at.sub AnalyticAt.sub
theorem AnalyticOn.mono {s t : Set E} (hf : AnalyticOn 𝕜 f t) (hst : s ⊆ t) : AnalyticOn 𝕜 f s :=
fun z hz => hf z (hst hz)
#align analytic_on.mono AnalyticOn.mono
theorem AnalyticOn.congr' {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : f =ᶠ[𝓝ˢ s] g) :
AnalyticOn 𝕜 g s :=
fun z hz => (hf z hz).congr (mem_nhdsSet_iff_forall.mp hg z hz)
theorem analyticOn_congr' {s : Set E} (h : f =ᶠ[𝓝ˢ s] g) : AnalyticOn 𝕜 f s ↔ AnalyticOn 𝕜 g s :=
⟨fun hf => hf.congr' h, fun hg => hg.congr' h.symm⟩
theorem AnalyticOn.congr {s : Set E} (hs : IsOpen s) (hf : AnalyticOn 𝕜 f s) (hg : s.EqOn f g) :
AnalyticOn 𝕜 g s :=
hf.congr' $ mem_nhdsSet_iff_forall.mpr
(fun _ hz => eventuallyEq_iff_exists_mem.mpr ⟨s, hs.mem_nhds hz, hg⟩)
theorem analyticOn_congr {s : Set E} (hs : IsOpen s) (h : s.EqOn f g) : AnalyticOn 𝕜 f s ↔
AnalyticOn 𝕜 g s := ⟨fun hf => hf.congr hs h, fun hg => hg.congr hs h.symm⟩
theorem AnalyticOn.add {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
AnalyticOn 𝕜 (f + g) s :=
fun z hz => (hf z hz).add (hg z hz)
#align analytic_on.add AnalyticOn.add
theorem AnalyticOn.sub {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
AnalyticOn 𝕜 (f - g) s :=
fun z hz => (hf z hz).sub (hg z hz)
#align analytic_on.sub AnalyticOn.sub
theorem HasFPowerSeriesOnBall.coeff_zero (hf : HasFPowerSeriesOnBall f pf x r) (v : Fin 0 → E) :
pf 0 v = f x := by
have v_eq : v = fun i => 0 := Subsingleton.elim _ _
have zero_mem : (0 : E) ∈ EMetric.ball (0 : E) r := by simp [hf.r_pos]
have : ∀ i, i ≠ 0 → (pf i fun j => 0) = 0 := by
intro i hi
have : 0 < i := pos_iff_ne_zero.2 hi
exact ContinuousMultilinearMap.map_coord_zero _ (⟨0, this⟩ : Fin i) rfl
have A := (hf.hasSum zero_mem).unique (hasSum_single _ this)
simpa [v_eq] using A.symm
#align has_fpower_series_on_ball.coeff_zero HasFPowerSeriesOnBall.coeff_zero
theorem HasFPowerSeriesAt.coeff_zero (hf : HasFPowerSeriesAt f pf x) (v : Fin 0 → E) :
pf 0 v = f x :=
let ⟨_, hrf⟩ := hf
hrf.coeff_zero v
#align has_fpower_series_at.coeff_zero HasFPowerSeriesAt.coeff_zero
/-- If a function `f` has a power series `p` on a ball and `g` is linear, then `g ∘ f` has the
power series `g ∘ p` on the same ball. -/
theorem ContinuousLinearMap.comp_hasFPowerSeriesOnBall (g : F →L[𝕜] G)
(h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (g ∘ f) (g.compFormalMultilinearSeries p) x r :=
{ r_le := h.r_le.trans (p.radius_le_radius_continuousLinearMap_comp _)
r_pos := h.r_pos
hasSum := fun hy => by
simpa only [ContinuousLinearMap.compFormalMultilinearSeries_apply,
ContinuousLinearMap.compContinuousMultilinearMap_coe, Function.comp_apply] using
g.hasSum (h.hasSum hy) }
#align continuous_linear_map.comp_has_fpower_series_on_ball ContinuousLinearMap.comp_hasFPowerSeriesOnBall
/-- If a function `f` is analytic on a set `s` and `g` is linear, then `g ∘ f` is analytic
on `s`. -/
theorem ContinuousLinearMap.comp_analyticOn {s : Set E} (g : F →L[𝕜] G) (h : AnalyticOn 𝕜 f s) :
AnalyticOn 𝕜 (g ∘ f) s := by
rintro x hx
rcases h x hx with ⟨p, r, hp⟩
exact ⟨g.compFormalMultilinearSeries p, r, g.comp_hasFPowerSeriesOnBall hp⟩
#align continuous_linear_map.comp_analytic_on ContinuousLinearMap.comp_analyticOn
/-- If a function admits a power series expansion, then it is exponentially close to the partial
sums of this power series on strict subdisks of the disk of convergence.
This version provides an upper estimate that decreases both in `‖y‖` and `n`. See also
`HasFPowerSeriesOnBall.uniform_geometric_approx` for a weaker version. -/
theorem HasFPowerSeriesOnBall.uniform_geometric_approx' {r' : ℝ≥0}
(hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n := by
obtain ⟨a, ha, C, hC, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n, ‖p n‖ * (r' : ℝ) ^ n ≤ C * a ^ n :=
p.norm_mul_pow_le_mul_pow_of_lt_radius (h.trans_le hf.r_le)
refine' ⟨a, ha, C / (1 - a), div_pos hC (sub_pos.2 ha.2), fun y hy n => _⟩
have yr' : ‖y‖ < r' := by
rw [ball_zero_eq] at hy
exact hy
have hr'0 : 0 < (r' : ℝ) := (norm_nonneg _).trans_lt yr'
have : y ∈ EMetric.ball (0 : E) r := by
refine' mem_emetric_ball_zero_iff.2 (lt_trans _ h)
exact mod_cast yr'
rw [norm_sub_rev, ← mul_div_right_comm]
have ya : a * (‖y‖ / ↑r') ≤ a :=
mul_le_of_le_one_right ha.1.le (div_le_one_of_le yr'.le r'.coe_nonneg)
suffices ‖p.partialSum n y - f (x + y)‖ ≤ C * (a * (‖y‖ / r')) ^ n / (1 - a * (‖y‖ / r')) by
refine' this.trans _
have : 0 < a := ha.1
gcongr
apply_rules [sub_pos.2, ha.2]
apply norm_sub_le_of_geometric_bound_of_hasSum (ya.trans_lt ha.2) _ (hf.hasSum this)
intro n
calc
‖(p n) fun _ : Fin n => y‖
_ ≤ ‖p n‖ * ∏ _i : Fin n, ‖y‖ := ContinuousMultilinearMap.le_op_norm _ _
_ = ‖p n‖ * (r' : ℝ) ^ n * (‖y‖ / r') ^ n := by field_simp [mul_right_comm]
_ ≤ C * a ^ n * (‖y‖ / r') ^ n := by gcongr ?_ * _; apply hp
_ ≤ C * (a * (‖y‖ / r')) ^ n := by rw [mul_pow, mul_assoc]
#align has_fpower_series_on_ball.uniform_geometric_approx' HasFPowerSeriesOnBall.uniform_geometric_approx'
/-- If a function admits a power series expansion, then it is exponentially close to the partial
sums of this power series on strict subdisks of the disk of convergence. -/
theorem HasFPowerSeriesOnBall.uniform_geometric_approx {r' : ℝ≥0}
(hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
∃ a ∈ Ioo (0 : ℝ) 1,
∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n, ‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n := by
obtain ⟨a, ha, C, hC, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n :=
hf.uniform_geometric_approx' h
refine' ⟨a, ha, C, hC, fun y hy n => (hp y hy n).trans _⟩
have yr' : ‖y‖ < r' := by rwa [ball_zero_eq] at hy
gcongr
exacts [mul_nonneg ha.1.le (div_nonneg (norm_nonneg y) r'.coe_nonneg),
mul_le_of_le_one_right ha.1.le (div_le_one_of_le yr'.le r'.coe_nonneg)]
#align has_fpower_series_on_ball.uniform_geometric_approx HasFPowerSeriesOnBall.uniform_geometric_approx
/-- Taylor formula for an analytic function, `IsBigO` version. -/
theorem HasFPowerSeriesAt.isBigO_sub_partialSum_pow (hf : HasFPowerSeriesAt f p x) (n : ℕ) :
(fun y : E => f (x + y) - p.partialSum n y) =O[𝓝 0] fun y => ‖y‖ ^ n := by
rcases hf with ⟨r, hf⟩
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hf.r_pos with ⟨r', r'0, h⟩
obtain ⟨a, -, C, -, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n :=
hf.uniform_geometric_approx' h
refine' isBigO_iff.2 ⟨C * (a / r') ^ n, _⟩
replace r'0 : 0 < (r' : ℝ); · exact mod_cast r'0
filter_upwards [Metric.ball_mem_nhds (0 : E) r'0] with y hy
simpa [mul_pow, mul_div_assoc, mul_assoc, div_mul_eq_mul_div] using hp y hy n
set_option linter.uppercaseLean3 false in
#align has_fpower_series_at.is_O_sub_partial_sum_pow HasFPowerSeriesAt.isBigO_sub_partialSum_pow
/-- If `f` has formal power series `∑ n, pₙ` on a ball of radius `r`, then for `y, z` in any smaller
ball, the norm of the difference `f y - f z - p 1 (fun _ ↦ y - z)` is bounded above by
`C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖`. This lemma formulates this property using `IsBigO` and
`Filter.principal` on `E × E`. -/
theorem HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal
(hf : HasFPowerSeriesOnBall f p x r) (hr : r' < r) :
(fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) =O[𝓟 (EMetric.ball (x, x) r')]
fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖ := by
lift r' to ℝ≥0 using ne_top_of_lt hr
rcases (zero_le r').eq_or_lt with (rfl | hr'0)
· simp only [isBigO_bot, EMetric.ball_zero, principal_empty, ENNReal.coe_zero]
obtain ⟨a, ha, C, hC : 0 < C, hp⟩ :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n : ℕ, ‖p n‖ * (r' : ℝ) ^ n ≤ C * a ^ n
exact p.norm_mul_pow_le_mul_pow_of_lt_radius (hr.trans_le hf.r_le)
simp only [← le_div_iff (pow_pos (NNReal.coe_pos.2 hr'0) _)] at hp
set L : E × E → ℝ := fun y =>
C * (a / r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * (a / (1 - a) ^ 2 + 2 / (1 - a))
have hL : ∀ y ∈ EMetric.ball (x, x) r', ‖f y.1 - f y.2 - p 1 fun _ => y.1 - y.2‖ ≤ L y := by
intro y hy'
have hy : y ∈ EMetric.ball x r ×ˢ EMetric.ball x r := by
rw [EMetric.ball_prod_same]
exact EMetric.ball_subset_ball hr.le hy'
set A : ℕ → F := fun n => (p n fun _ => y.1 - x) - p n fun _ => y.2 - x
have hA : HasSum (fun n => A (n + 2)) (f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) := by
convert (hasSum_nat_add_iff' 2).2 ((hf.hasSum_sub hy.1).sub (hf.hasSum_sub hy.2)) using 1
rw [Finset.sum_range_succ, Finset.sum_range_one, hf.coeff_zero, hf.coeff_zero, sub_self,
zero_add, ← Subsingleton.pi_single_eq (0 : Fin 1) (y.1 - x), Pi.single,
← Subsingleton.pi_single_eq (0 : Fin 1) (y.2 - x), Pi.single, ← (p 1).map_sub, ← Pi.single,
Subsingleton.pi_single_eq, sub_sub_sub_cancel_right]
rw [EMetric.mem_ball, edist_eq_coe_nnnorm_sub, ENNReal.coe_lt_coe] at hy'
set B : ℕ → ℝ := fun n => C * (a / r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * ((n + 2) * a ^ n)
have hAB : ∀ n, ‖A (n + 2)‖ ≤ B n := fun n =>
calc
‖A (n + 2)‖ ≤ ‖p (n + 2)‖ * ↑(n + 2) * ‖y - (x, x)‖ ^ (n + 1) * ‖y.1 - y.2‖ := by
-- porting note: `pi_norm_const` was `pi_norm_const (_ : E)`
simpa only [Fintype.card_fin, pi_norm_const, Prod.norm_def, Pi.sub_def,
Prod.fst_sub, Prod.snd_sub, sub_sub_sub_cancel_right] using
(p <| n + 2).norm_image_sub_le (fun _ => y.1 - x) fun _ => y.2 - x
_ = ‖p (n + 2)‖ * ‖y - (x, x)‖ ^ n * (↑(n + 2) * ‖y - (x, x)‖ * ‖y.1 - y.2‖) := by
rw [pow_succ ‖y - (x, x)‖]
ring
-- porting note: the two `↑` in `↑r'` are new, without them, Lean fails to synthesize
-- instances `HDiv ℝ ℝ≥0 ?m` or `HMul ℝ ℝ≥0 ?m`
_ ≤ C * a ^ (n + 2) / ↑r' ^ (n + 2)
* ↑r' ^ n * (↑(n + 2) * ‖y - (x, x)‖ * ‖y.1 - y.2‖) := by
have : 0 < a := ha.1
gcongr
· apply hp
· apply hy'.le
_ = B n := by
-- porting note: in the original, `B` was in the `field_simp`, but now Lean does not
-- accept it. The current proof works in Lean 4, but does not in Lean 3.
field_simp [pow_succ]
simp only [mul_assoc, mul_comm, mul_left_comm]
have hBL : HasSum B (L y) := by
apply HasSum.mul_left
simp only [add_mul]
have : ‖a‖ < 1 := by simp only [Real.norm_eq_abs, abs_of_pos ha.1, ha.2]
rw [div_eq_mul_inv, div_eq_mul_inv]
exact (hasSum_coe_mul_geometric_of_norm_lt_1 this).add -- porting note: was `convert`!
((hasSum_geometric_of_norm_lt_1 this).mul_left 2)
exact hA.norm_le_of_bounded hBL hAB
suffices L =O[𝓟 (EMetric.ball (x, x) r')] fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖ by
refine' (IsBigO.of_bound 1 (eventually_principal.2 fun y hy => _)).trans this
rw [one_mul]
exact (hL y hy).trans (le_abs_self _)
simp_rw [mul_right_comm _ (_ * _)] -- porting note: there was an `L` inside the `simp_rw`.
exact (isBigO_refl _ _).const_mul_left _
set_option linter.uppercaseLean3 false in
#align has_fpower_series_on_ball.is_O_image_sub_image_sub_deriv_principal HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal
/-- If `f` has formal power series `∑ n, pₙ` on a ball of radius `r`, then for `y, z` in any smaller
ball, the norm of the difference `f y - f z - p 1 (fun _ ↦ y - z)` is bounded above by
`C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖`. -/
theorem HasFPowerSeriesOnBall.image_sub_sub_deriv_le (hf : HasFPowerSeriesOnBall f p x r)
(hr : r' < r) :
∃ C, ∀ᵉ (y ∈ EMetric.ball x r') (z ∈ EMetric.ball x r'),
‖f y - f z - p 1 fun _ => y - z‖ ≤ C * max ‖y - x‖ ‖z - x‖ * ‖y - z‖ := by
simpa only [isBigO_principal, mul_assoc, norm_mul, norm_norm, Prod.forall, EMetric.mem_ball,
Prod.edist_eq, max_lt_iff, and_imp, @forall_swap (_ < _) E] using
hf.isBigO_image_sub_image_sub_deriv_principal hr
#align has_fpower_series_on_ball.image_sub_sub_deriv_le HasFPowerSeriesOnBall.image_sub_sub_deriv_le
/-- If `f` has formal power series `∑ n, pₙ` at `x`, then
`f y - f z - p 1 (fun _ ↦ y - z) = O(‖(y, z) - (x, x)‖ * ‖y - z‖)` as `(y, z) → (x, x)`.
In particular, `f` is strictly differentiable at `x`. -/
theorem HasFPowerSeriesAt.isBigO_image_sub_norm_mul_norm_sub (hf : HasFPowerSeriesAt f p x) :
(fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) =O[𝓝 (x, x)] fun y =>
‖y - (x, x)‖ * ‖y.1 - y.2‖ := by
rcases hf with ⟨r, hf⟩
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hf.r_pos with ⟨r', r'0, h⟩
refine' (hf.isBigO_image_sub_image_sub_deriv_principal h).mono _
exact le_principal_iff.2 (EMetric.ball_mem_nhds _ r'0)
set_option linter.uppercaseLean3 false in
#align has_fpower_series_at.is_O_image_sub_norm_mul_norm_sub HasFPowerSeriesAt.isBigO_image_sub_norm_mul_norm_sub
/-- If a function admits a power series expansion at `x`, then it is the uniform limit of the
partial sums of this power series on strict subdisks of the disk of convergence, i.e., `f (x + y)`
is the uniform limit of `p.partialSum n y` there. -/
theorem HasFPowerSeriesOnBall.tendstoUniformlyOn {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r)
(h : (r' : ℝ≥0∞) < r) :
TendstoUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
(Metric.ball (0 : E) r') := by
obtain ⟨a, ha, C, -, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n
exact hf.uniform_geometric_approx h
refine' Metric.tendstoUniformlyOn_iff.2 fun ε εpos => _
have L : Tendsto (fun n => (C : ℝ) * a ^ n) atTop (𝓝 ((C : ℝ) * 0)) :=
tendsto_const_nhds.mul (tendsto_pow_atTop_nhds_0_of_lt_1 ha.1.le ha.2)
rw [mul_zero] at L
refine' (L.eventually (gt_mem_nhds εpos)).mono fun n hn y hy => _
rw [dist_eq_norm]
exact (hp y hy n).trans_lt hn
#align has_fpower_series_on_ball.tendsto_uniformly_on HasFPowerSeriesOnBall.tendstoUniformlyOn
/-- If a function admits a power series expansion at `x`, then it is the locally uniform limit of
the partial sums of this power series on the disk of convergence, i.e., `f (x + y)`
is the locally uniform limit of `p.partialSum n y` there. -/
theorem HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn (hf : HasFPowerSeriesOnBall f p x r) :
TendstoLocallyUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
(EMetric.ball (0 : E) r) := by
intro u hu x hx
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hx with ⟨r', xr', hr'⟩
have : EMetric.ball (0 : E) r' ∈ 𝓝 x := IsOpen.mem_nhds EMetric.isOpen_ball xr'
refine' ⟨EMetric.ball (0 : E) r', mem_nhdsWithin_of_mem_nhds this, _⟩
simpa [Metric.emetric_ball_nnreal] using hf.tendstoUniformlyOn hr' u hu
#align has_fpower_series_on_ball.tendsto_locally_uniformly_on HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn
/-- If a function admits a power series expansion at `x`, then it is the uniform limit of the
partial sums of this power series on strict subdisks of the disk of convergence, i.e., `f y`
is the uniform limit of `p.partialSum n (y - x)` there. -/
theorem HasFPowerSeriesOnBall.tendstoUniformlyOn' {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r)
(h : (r' : ℝ≥0∞) < r) :
TendstoUniformlyOn (fun n y => p.partialSum n (y - x)) f atTop (Metric.ball (x : E) r') := by
convert (hf.tendstoUniformlyOn h).comp fun y => y - x using 1
· simp [(· ∘ ·)]
· ext z
simp [dist_eq_norm]
#align has_fpower_series_on_ball.tendsto_uniformly_on' HasFPowerSeriesOnBall.tendstoUniformlyOn'
/-- If a function admits a power series expansion at `x`, then it is the locally uniform limit of
the partial sums of this power series on the disk of convergence, i.e., `f y`
is the locally uniform limit of `p.partialSum n (y - x)` there. -/
theorem HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn' (hf : HasFPowerSeriesOnBall f p x r) :
TendstoLocallyUniformlyOn (fun n y => p.partialSum n (y - x)) f atTop
(EMetric.ball (x : E) r) := by
have A : ContinuousOn (fun y : E => y - x) (EMetric.ball (x : E) r) :=
(continuous_id.sub continuous_const).continuousOn
convert hf.tendstoLocallyUniformlyOn.comp (fun y : E => y - x) _ A using 1
· ext z
simp
· intro z
simp [edist_eq_coe_nnnorm, edist_eq_coe_nnnorm_sub]
#align has_fpower_series_on_ball.tendsto_locally_uniformly_on' HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn'
/-- If a function admits a power series expansion on a disk, then it is continuous there. -/
protected theorem HasFPowerSeriesOnBall.continuousOn (hf : HasFPowerSeriesOnBall f p x r) :
ContinuousOn f (EMetric.ball x r) :=
hf.tendstoLocallyUniformlyOn'.continuousOn <|
eventually_of_forall fun n =>
((p.partialSum_continuous n).comp (continuous_id.sub continuous_const)).continuousOn
#align has_fpower_series_on_ball.continuous_on HasFPowerSeriesOnBall.continuousOn
protected theorem HasFPowerSeriesAt.continuousAt (hf : HasFPowerSeriesAt f p x) :
ContinuousAt f x :=
let ⟨_, hr⟩ := hf
hr.continuousOn.continuousAt (EMetric.ball_mem_nhds x hr.r_pos)
#align has_fpower_series_at.continuous_at HasFPowerSeriesAt.continuousAt
protected theorem AnalyticAt.continuousAt (hf : AnalyticAt 𝕜 f x) : ContinuousAt f x :=
let ⟨_, hp⟩ := hf
hp.continuousAt
#align analytic_at.continuous_at AnalyticAt.continuousAt
protected theorem AnalyticOn.continuousOn {s : Set E} (hf : AnalyticOn 𝕜 f s) : ContinuousOn f s :=
fun x hx => (hf x hx).continuousAt.continuousWithinAt
#align analytic_on.continuous_on AnalyticOn.continuousOn
/-- Analytic everywhere implies continuous -/
theorem AnalyticOn.continuous {f : E → F} (fa : AnalyticOn 𝕜 f univ) : Continuous f := by
rw [continuous_iff_continuousOn_univ]; exact fa.continuousOn
/-- In a complete space, the sum of a converging power series `p` admits `p` as a power series.
This is not totally obvious as we need to check the convergence of the series. -/
protected theorem FormalMultilinearSeries.hasFPowerSeriesOnBall [CompleteSpace F]
(p : FormalMultilinearSeries 𝕜 E F) (h : 0 < p.radius) :
HasFPowerSeriesOnBall p.sum p 0 p.radius :=
{ r_le := le_rfl
r_pos := h
hasSum := fun hy => by
rw [zero_add]
exact p.hasSum hy }
#align formal_multilinear_series.has_fpower_series_on_ball FormalMultilinearSeries.hasFPowerSeriesOnBall
theorem HasFPowerSeriesOnBall.sum (h : HasFPowerSeriesOnBall f p x r) {y : E}
(hy : y ∈ EMetric.ball (0 : E) r) : f (x + y) = p.sum y :=
(h.hasSum hy).tsum_eq.symm
#align has_fpower_series_on_ball.sum HasFPowerSeriesOnBall.sum
/-- The sum of a converging power series is continuous in its disk of convergence. -/
protected theorem FormalMultilinearSeries.continuousOn [CompleteSpace F] :
ContinuousOn p.sum (EMetric.ball 0 p.radius) := by
rcases (zero_le p.radius).eq_or_lt with h | h
· simp [← h, continuousOn_empty]
· exact (p.hasFPowerSeriesOnBall h).continuousOn
#align formal_multilinear_series.continuous_on FormalMultilinearSeries.continuousOn
end
/-!
### Uniqueness of power series
If a function `f : E → F` has two representations as power series at a point `x : E`, corresponding
to formal multilinear series `p₁` and `p₂`, then these representations agree term-by-term. That is,
for any `n : ℕ` and `y : E`, `p₁ n (fun i ↦ y) = p₂ n (fun i ↦ y)`. In the one-dimensional case,
when `f : 𝕜 → E`, the continuous multilinear maps `p₁ n` and `p₂ n` are given by
`ContinuousMultilinearMap.mkPiField`, and hence are determined completely by the value of
`p₁ n (fun i ↦ 1)`, so `p₁ = p₂`. Consequently, the radius of convergence for one series can be
transferred to the other.
-/
section Uniqueness
open ContinuousMultilinearMap
theorem Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero {n : ℕ} {p : E[×n]→L[𝕜] F}
(h : (fun y => p fun _ => y) =O[𝓝 0] fun y => ‖y‖ ^ (n + 1)) (y : E) : (p fun _ => y) = 0 := by
obtain ⟨c, c_pos, hc⟩ := h.exists_pos
obtain ⟨t, ht, t_open, z_mem⟩ := eventually_nhds_iff.mp (isBigOWith_iff.mp hc)
obtain ⟨δ, δ_pos, δε⟩ := (Metric.isOpen_iff.mp t_open) 0 z_mem
clear h hc z_mem
cases' n with n
· exact norm_eq_zero.mp (by
-- porting note: the symmetric difference of the `simpa only` sets:
-- added `Nat.zero_eq, zero_add, pow_one`
-- removed `zero_pow', Ne.def, Nat.one_ne_zero, not_false_iff`
simpa only [Nat.zero_eq, fin0_apply_norm, norm_eq_zero, norm_zero, zero_add, pow_one,
mul_zero, norm_le_zero_iff] using ht 0 (δε (Metric.mem_ball_self δ_pos)))
· refine' Or.elim (Classical.em (y = 0))
(fun hy => by simpa only [hy] using p.map_zero) fun hy => _
replace hy := norm_pos_iff.mpr hy
refine' norm_eq_zero.mp (le_antisymm (le_of_forall_pos_le_add fun ε ε_pos => _) (norm_nonneg _))
have h₀ := _root_.mul_pos c_pos (pow_pos hy (n.succ + 1))
obtain ⟨k, k_pos, k_norm⟩ := NormedField.exists_norm_lt 𝕜
(lt_min (mul_pos δ_pos (inv_pos.mpr hy)) (mul_pos ε_pos (inv_pos.mpr h₀)))
have h₁ : ‖k • y‖ < δ := by
rw [norm_smul]
exact inv_mul_cancel_right₀ hy.ne.symm δ ▸
mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_left _ _)) hy
have h₂ :=
calc
‖p fun _ => k • y‖ ≤ c * ‖k • y‖ ^ (n.succ + 1) := by
-- porting note: now Lean wants `_root_.`
simpa only [norm_pow, _root_.norm_norm] using ht (k • y) (δε (mem_ball_zero_iff.mpr h₁))
--simpa only [norm_pow, norm_norm] using ht (k • y) (δε (mem_ball_zero_iff.mpr h₁))
_ = ‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))) := by
-- porting note: added `Nat.succ_eq_add_one` since otherwise `ring` does not conclude.
simp only [norm_smul, mul_pow, Nat.succ_eq_add_one]
-- porting note: removed `rw [pow_succ]`, since it now becomes superfluous.
ring
have h₃ : ‖k‖ * (c * ‖y‖ ^ (n.succ + 1)) < ε :=
inv_mul_cancel_right₀ h₀.ne.symm ε ▸
mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_right _ _)) h₀
calc
‖p fun _ => y‖ = ‖k⁻¹ ^ n.succ‖ * ‖p fun _ => k • y‖ := by
simpa only [inv_smul_smul₀ (norm_pos_iff.mp k_pos), norm_smul, Finset.prod_const,
Finset.card_fin] using
congr_arg norm (p.map_smul_univ (fun _ : Fin n.succ => k⁻¹) fun _ : Fin n.succ => k • y)
_ ≤ ‖k⁻¹ ^ n.succ‖ * (‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1)))) := by gcongr
_ = ‖(k⁻¹ * k) ^ n.succ‖ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))) := by
rw [← mul_assoc]
simp [norm_mul, mul_pow]
_ ≤ 0 + ε := by
rw [inv_mul_cancel (norm_pos_iff.mp k_pos)]
simpa using h₃.le
set_option linter.uppercaseLean3 false in
#align asymptotics.is_O.continuous_multilinear_map_apply_eq_zero Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero
/-- If a formal multilinear series `p` represents the zero function at `x : E`, then the
terms `p n (fun i ↦ y)` appearing in the sum are zero for any `n : ℕ`, `y : E`. -/
theorem HasFPowerSeriesAt.apply_eq_zero {p : FormalMultilinearSeries 𝕜 E F} {x : E}
(h : HasFPowerSeriesAt 0 p x) (n : ℕ) : ∀ y : E, (p n fun _ => y) = 0 := by
refine' Nat.strong_induction_on n fun k hk => _
have psum_eq : p.partialSum (k + 1) = fun y => p k fun _ => y := by
funext z
refine' Finset.sum_eq_single _ (fun b hb hnb => _) fun hn => _
· have := Finset.mem_range_succ_iff.mp hb
simp only [hk b (this.lt_of_ne hnb), Pi.zero_apply]
· exact False.elim (hn (Finset.mem_range.mpr (lt_add_one k)))
replace h := h.isBigO_sub_partialSum_pow k.succ
simp only [psum_eq, zero_sub, Pi.zero_apply, Asymptotics.isBigO_neg_left] at h
exact h.continuousMultilinearMap_apply_eq_zero
#align has_fpower_series_at.apply_eq_zero HasFPowerSeriesAt.apply_eq_zero
/-- A one-dimensional formal multilinear series representing the zero function is zero. -/
theorem HasFPowerSeriesAt.eq_zero {p : FormalMultilinearSeries 𝕜 𝕜 E} {x : 𝕜}
(h : HasFPowerSeriesAt 0 p x) : p = 0 := by
-- porting note: `funext; ext` was `ext (n x)`
funext n
ext x
rw [← mkPiField_apply_one_eq_self (p n)]
-- porting note: nasty hack, was `simp [h.apply_eq_zero n 1]`
have := Or.intro_right ?_ (h.apply_eq_zero n 1)
simpa using this
#align has_fpower_series_at.eq_zero HasFPowerSeriesAt.eq_zero
/-- One-dimensional formal multilinear series representing the same function are equal. -/
theorem HasFPowerSeriesAt.eq_formalMultilinearSeries {p₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E}
{f : 𝕜 → E} {x : 𝕜} (h₁ : HasFPowerSeriesAt f p₁ x) (h₂ : HasFPowerSeriesAt f p₂ x) : p₁ = p₂ :=
sub_eq_zero.mp (HasFPowerSeriesAt.eq_zero (by simpa only [sub_self] using h₁.sub h₂))
#align has_fpower_series_at.eq_formal_multilinear_series HasFPowerSeriesAt.eq_formalMultilinearSeries
theorem HasFPowerSeriesAt.eq_formalMultilinearSeries_of_eventually
{p q : FormalMultilinearSeries 𝕜 𝕜 E} {f g : 𝕜 → E} {x : 𝕜} (hp : HasFPowerSeriesAt f p x)
(hq : HasFPowerSeriesAt g q x) (heq : ∀ᶠ z in 𝓝 x, f z = g z) : p = q :=
(hp.congr heq).eq_formalMultilinearSeries hq
#align has_fpower_series_at.eq_formal_multilinear_series_of_eventually HasFPowerSeriesAt.eq_formalMultilinearSeries_of_eventually
/-- A one-dimensional formal multilinear series representing a locally zero function is zero. -/
theorem HasFPowerSeriesAt.eq_zero_of_eventually {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E}
{x : 𝕜} (hp : HasFPowerSeriesAt f p x) (hf : f =ᶠ[𝓝 x] 0) : p = 0 :=
(hp.congr hf).eq_zero
#align has_fpower_series_at.eq_zero_of_eventually HasFPowerSeriesAt.eq_zero_of_eventually
/-- If a function `f : 𝕜 → E` has two power series representations at `x`, then the given radii in
which convergence is guaranteed may be interchanged. This can be useful when the formal multilinear
series in one representation has a particularly nice form, but the other has a larger radius. -/
theorem HasFPowerSeriesOnBall.exchange_radius {p₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E}
{r₁ r₂ : ℝ≥0∞} {x : 𝕜} (h₁ : HasFPowerSeriesOnBall f p₁ x r₁)
(h₂ : HasFPowerSeriesOnBall f p₂ x r₂) : HasFPowerSeriesOnBall f p₁ x r₂ :=
h₂.hasFPowerSeriesAt.eq_formalMultilinearSeries h₁.hasFPowerSeriesAt ▸ h₂
#align has_fpower_series_on_ball.exchange_radius HasFPowerSeriesOnBall.exchange_radius
/-- If a function `f : 𝕜 → E` has power series representation `p` on a ball of some radius and for
each positive radius it has some power series representation, then `p` converges to `f` on the whole
`𝕜`. -/
theorem HasFPowerSeriesOnBall.r_eq_top_of_exists {f : 𝕜 → E} {r : ℝ≥0∞} {x : 𝕜}
{p : FormalMultilinearSeries 𝕜 𝕜 E} (h : HasFPowerSeriesOnBall f p x r)
(h' : ∀ (r' : ℝ≥0) (_ : 0 < r'), ∃ p' : FormalMultilinearSeries 𝕜 𝕜 E,
HasFPowerSeriesOnBall f p' x r') :
HasFPowerSeriesOnBall f p x ∞ :=
{ r_le := ENNReal.le_of_forall_pos_nnreal_lt fun r hr _ =>
let ⟨_, hp'⟩ := h' r hr
(h.exchange_radius hp').r_le
r_pos := ENNReal.coe_lt_top
hasSum := fun {y} _ =>
let ⟨r', hr'⟩ := exists_gt ‖y‖₊
let ⟨_, hp'⟩ := h' r' hr'.ne_bot.bot_lt
(h.exchange_radius hp').hasSum <| mem_emetric_ball_zero_iff.mpr (ENNReal.coe_lt_coe.2 hr') }
#align has_fpower_series_on_ball.r_eq_top_of_exists HasFPowerSeriesOnBall.r_eq_top_of_exists
end Uniqueness
/-!
### Changing origin in a power series
If a function is analytic in a disk `D(x, R)`, then it is analytic in any disk contained in that
one. Indeed, one can write
$$
f (x + y + z) = \sum_{n} p_n (y + z)^n = \sum_{n, k} \binom{n}{k} p_n y^{n-k} z^k
= \sum_{k} \Bigl(\sum_{n} \binom{n}{k} p_n y^{n-k}\Bigr) z^k.
$$
The corresponding power series has thus a `k`-th coefficient equal to
$\sum_{n} \binom{n}{k} p_n y^{n-k}$. In the general case where `pₙ` is a multilinear map, this has
to be interpreted suitably: instead of having a binomial coefficient, one should sum over all
possible subsets `s` of `Fin n` of cardinal `k`, and attribute `z` to the indices in `s` and
`y` to the indices outside of `s`.
In this paragraph, we implement this. The new power series is called `p.changeOrigin y`. Then, we
check its convergence and the fact that its sum coincides with the original sum. The outcome of this
discussion is that the set of points where a function is analytic is open.
-/
namespace FormalMultilinearSeries
section
variable (p : FormalMultilinearSeries 𝕜 E F) {x y : E} {r R : ℝ≥0}
/-- A term of `FormalMultilinearSeries.changeOriginSeries`.
Given a formal multilinear series `p` and a point `x` in its ball of convergence,
`p.changeOrigin x` is a formal multilinear series such that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense. Each term of `p.changeOrigin x`
is itself an analytic function of `x` given by the series `p.changeOriginSeries`. Each term in
`changeOriginSeries` is the sum of `changeOriginSeriesTerm`'s over all `s` of cardinality `l`.
The definition is such that `p.changeOriginSeriesTerm k l s hs (fun _ ↦ x) (fun _ ↦ y) =
p (k + l) (s.piecewise (fun _ ↦ x) (fun _ ↦ y))`
-/
def changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
E[×l]→L[𝕜] E[×k]→L[𝕜] F := by
let a := ContinuousMultilinearMap.curryFinFinset 𝕜 E F hs
(by erw [Finset.card_compl, Fintype.card_fin, hs, add_tsub_cancel_right])
exact a (p (k + l))
#align formal_multilinear_series.change_origin_series_term FormalMultilinearSeries.changeOriginSeriesTerm
theorem changeOriginSeriesTerm_apply (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l)
(x y : E) :
(p.changeOriginSeriesTerm k l s hs (fun _ => x) fun _ => y) =
p (k + l) (s.piecewise (fun _ => x) fun _ => y) :=
ContinuousMultilinearMap.curryFinFinset_apply_const _ _ _ _ _
#align formal_multilinear_series.change_origin_series_term_apply FormalMultilinearSeries.changeOriginSeriesTerm_apply
@[simp]
theorem norm_changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
‖p.changeOriginSeriesTerm k l s hs‖ = ‖p (k + l)‖ := by
simp only [changeOriginSeriesTerm, LinearIsometryEquiv.norm_map]
#align formal_multilinear_series.norm_change_origin_series_term FormalMultilinearSeries.norm_changeOriginSeriesTerm
@[simp]
theorem nnnorm_changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
‖p.changeOriginSeriesTerm k l s hs‖₊ = ‖p (k + l)‖₊ := by
simp only [changeOriginSeriesTerm, LinearIsometryEquiv.nnnorm_map]
#align formal_multilinear_series.nnnorm_change_origin_series_term FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm
theorem nnnorm_changeOriginSeriesTerm_apply_le (k l : ℕ) (s : Finset (Fin (k + l)))
(hs : s.card = l) (x y : E) :
‖p.changeOriginSeriesTerm k l s hs (fun _ => x) fun _ => y‖₊ ≤
‖p (k + l)‖₊ * ‖x‖₊ ^ l * ‖y‖₊ ^ k := by
rw [← p.nnnorm_changeOriginSeriesTerm k l s hs, ← Fin.prod_const, ← Fin.prod_const]
apply ContinuousMultilinearMap.le_of_op_nnnorm_le
apply ContinuousMultilinearMap.le_op_nnnorm
#align formal_multilinear_series.nnnorm_change_origin_series_term_apply_le FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm_apply_le
/-- The power series for `f.changeOrigin k`.
Given a formal multilinear series `p` and a point `x` in its ball of convergence,
`p.changeOrigin x` is a formal multilinear series such that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense. Its `k`-th term is the sum of
the series `p.changeOriginSeries k`. -/
def changeOriginSeries (k : ℕ) : FormalMultilinearSeries 𝕜 E (E[×k]→L[𝕜] F) := fun l =>
∑ s : { s : Finset (Fin (k + l)) // Finset.card s = l }, p.changeOriginSeriesTerm k l s s.2
#align formal_multilinear_series.change_origin_series FormalMultilinearSeries.changeOriginSeries
theorem nnnorm_changeOriginSeries_le_tsum (k l : ℕ) :
‖p.changeOriginSeries k l‖₊ ≤
∑' _ : { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + l)‖₊ :=
(nnnorm_sum_le _ (fun t => changeOriginSeriesTerm p k l (Subtype.val t) t.prop)).trans_eq <| by
simp_rw [tsum_fintype, nnnorm_changeOriginSeriesTerm (p := p) (k := k) (l := l)]
#align formal_multilinear_series.nnnorm_change_origin_series_le_tsum FormalMultilinearSeries.nnnorm_changeOriginSeries_le_tsum
theorem nnnorm_changeOriginSeries_apply_le_tsum (k l : ℕ) (x : E) :
‖p.changeOriginSeries k l fun _ => x‖₊ ≤
∑' _ : { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + l)‖₊ * ‖x‖₊ ^ l := by
rw [NNReal.tsum_mul_right, ← Fin.prod_const]
exact (p.changeOriginSeries k l).le_of_op_nnnorm_le _ (p.nnnorm_changeOriginSeries_le_tsum _ _)
#align formal_multilinear_series.nnnorm_change_origin_series_apply_le_tsum FormalMultilinearSeries.nnnorm_changeOriginSeries_apply_le_tsum
/-- Changing the origin of a formal multilinear series `p`, so that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense.
-/
def changeOrigin (x : E) : FormalMultilinearSeries 𝕜 E F :=
fun k => (p.changeOriginSeries k).sum x
#align formal_multilinear_series.change_origin FormalMultilinearSeries.changeOrigin
/-- An auxiliary equivalence useful in the proofs about
`FormalMultilinearSeries.changeOriginSeries`: the set of triples `(k, l, s)`, where `s` is a
`Finset (Fin (k + l))` of cardinality `l` is equivalent to the set of pairs `(n, s)`, where `s` is a
`Finset (Fin n)`.
The forward map sends `(k, l, s)` to `(k + l, s)` and the inverse map sends `(n, s)` to
`(n - Finset.card s, Finset.card s, s)`. The actual definition is less readable because of problems
with non-definitional equalities. -/
@[simps]
def changeOriginIndexEquiv :
(Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) ≃ Σn : ℕ, Finset (Fin n) where
toFun s := ⟨s.1 + s.2.1, s.2.2⟩
invFun s :=
⟨s.1 - s.2.card, s.2.card,
⟨s.2.map
(Fin.castIso <| (tsub_add_cancel_of_le <| card_finset_fin_le s.2).symm).toEquiv.toEmbedding,
Finset.card_map _⟩⟩
left_inv := by
rintro ⟨k, l, ⟨s : Finset (Fin <| k + l), hs : s.card = l⟩⟩
dsimp only [Subtype.coe_mk]
-- Lean can't automatically generalize `k' = k + l - s.card`, `l' = s.card`, so we explicitly
-- formulate the generalized goal
suffices ∀ k' l', k' = k → l' = l → ∀ (hkl : k + l = k' + l') (hs'),
(⟨k', l', ⟨Finset.map (Fin.castIso hkl).toEquiv.toEmbedding s, hs'⟩⟩ :
Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) = ⟨k, l, ⟨s, hs⟩⟩ by
apply this <;> simp only [hs, add_tsub_cancel_right]
rintro _ _ rfl rfl hkl hs'
simp only [Equiv.refl_toEmbedding, Fin.castIso_refl, Finset.map_refl, eq_self_iff_true,
OrderIso.refl_toEquiv, and_self_iff, heq_iff_eq]
right_inv := by
rintro ⟨n, s⟩
simp [tsub_add_cancel_of_le (card_finset_fin_le s), Fin.castIso_to_equiv]
#align formal_multilinear_series.change_origin_index_equiv FormalMultilinearSeries.changeOriginIndexEquiv
theorem changeOriginSeries_summable_aux₁ {r r' : ℝ≥0} (hr : (r + r' : ℝ≥0∞) < p.radius) :
Summable fun s : Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l } =>
‖p (s.1 + s.2.1)‖₊ * r ^ s.2.1 * r' ^ s.1 := by
rw [← changeOriginIndexEquiv.symm.summable_iff]
dsimp only [Function.comp_def, changeOriginIndexEquiv_symm_apply_fst,
changeOriginIndexEquiv_symm_apply_snd_fst]
have : ∀ n : ℕ,
HasSum (fun s : Finset (Fin n) => ‖p (n - s.card + s.card)‖₊ * r ^ s.card * r' ^ (n - s.card))
(‖p n‖₊ * (r + r') ^ n) := by
intro n
-- TODO: why `simp only [tsub_add_cancel_of_le (card_finset_fin_le _)]` fails?
convert_to HasSum (fun s : Finset (Fin n) => ‖p n‖₊ * (r ^ s.card * r' ^ (n - s.card))) _
· ext1 s
rw [tsub_add_cancel_of_le (card_finset_fin_le _), mul_assoc]
rw [← Fin.sum_pow_mul_eq_add_pow]
exact (hasSum_fintype _).mul_left _
refine' NNReal.summable_sigma.2 ⟨fun n => (this n).summable, _⟩
simp only [(this _).tsum_eq]
exact p.summable_nnnorm_mul_pow hr
#align formal_multilinear_series.change_origin_series_summable_aux₁ FormalMultilinearSeries.changeOriginSeries_summable_aux₁
theorem changeOriginSeries_summable_aux₂ (hr : (r : ℝ≥0∞) < p.radius) (k : ℕ) :
Summable fun s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l } =>
‖p (k + s.1)‖₊ * r ^ s.1 := by
rcases ENNReal.lt_iff_exists_add_pos_lt.1 hr with ⟨r', h0, hr'⟩
simpa only [mul_inv_cancel_right₀ (pow_pos h0 _).ne'] using
((NNReal.summable_sigma.1 (p.changeOriginSeries_summable_aux₁ hr')).1 k).mul_right (r' ^ k)⁻¹
#align formal_multilinear_series.change_origin_series_summable_aux₂ FormalMultilinearSeries.changeOriginSeries_summable_aux₂
theorem changeOriginSeries_summable_aux₃ {r : ℝ≥0} (hr : ↑r < p.radius) (k : ℕ) :
Summable fun l : ℕ => ‖p.changeOriginSeries k l‖₊ * r ^ l := by
refine' NNReal.summable_of_le
(fun n => _) (NNReal.summable_sigma.1 <| p.changeOriginSeries_summable_aux₂ hr k).2
simp only [NNReal.tsum_mul_right]
exact mul_le_mul' (p.nnnorm_changeOriginSeries_le_tsum _ _) le_rfl
#align formal_multilinear_series.change_origin_series_summable_aux₃ FormalMultilinearSeries.changeOriginSeries_summable_aux₃
theorem le_changeOriginSeries_radius (k : ℕ) : p.radius ≤ (p.changeOriginSeries k).radius :=
ENNReal.le_of_forall_nnreal_lt fun _r hr =>
le_radius_of_summable_nnnorm _ (p.changeOriginSeries_summable_aux₃ hr k)
#align formal_multilinear_series.le_change_origin_series_radius FormalMultilinearSeries.le_changeOriginSeries_radius
theorem nnnorm_changeOrigin_le (k : ℕ) (h : (‖x‖₊ : ℝ≥0∞) < p.radius) :
‖p.changeOrigin x k‖₊ ≤
∑' s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + s.1)‖₊ * ‖x‖₊ ^ s.1 := by
refine' tsum_of_nnnorm_bounded _ fun l => p.nnnorm_changeOriginSeries_apply_le_tsum k l x
have := p.changeOriginSeries_summable_aux₂ h k
refine' HasSum.sigma this.hasSum fun l => _
exact ((NNReal.summable_sigma.1 this).1 l).hasSum
#align formal_multilinear_series.nnnorm_change_origin_le FormalMultilinearSeries.nnnorm_changeOrigin_le
/-- The radius of convergence of `p.changeOrigin x` is at least `p.radius - ‖x‖`. In other words,
`p.changeOrigin x` is well defined on the largest ball contained in the original ball of
convergence. -/
theorem changeOrigin_radius : p.radius - ‖x‖₊ ≤ (p.changeOrigin x).radius := by
refine' ENNReal.le_of_forall_pos_nnreal_lt fun r _h0 hr => _
rw [lt_tsub_iff_right, add_comm] at hr
have hr' : (‖x‖₊ : ℝ≥0∞) < p.radius := (le_add_right le_rfl).trans_lt hr
apply le_radius_of_summable_nnnorm
have : ∀ k : ℕ,
‖p.changeOrigin x k‖₊ * r ^ k ≤
(∑' s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + s.1)‖₊ * ‖x‖₊ ^ s.1) *
r ^ k :=
fun k => mul_le_mul_right' (p.nnnorm_changeOrigin_le k hr') (r ^ k)
refine' NNReal.summable_of_le this _
simpa only [← NNReal.tsum_mul_right] using
(NNReal.summable_sigma.1 (p.changeOriginSeries_summable_aux₁ hr)).2
#align formal_multilinear_series.change_origin_radius FormalMultilinearSeries.changeOrigin_radius
end
-- From this point on, assume that the space is complete, to make sure that series that converge
-- in norm also converge in `F`.
variable [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x y : E} {r R : ℝ≥0}
theorem hasFPowerSeriesOnBall_changeOrigin (k : ℕ) (hr : 0 < p.radius) :
HasFPowerSeriesOnBall (fun x => p.changeOrigin x k) (p.changeOriginSeries k) 0 p.radius :=
have := p.le_changeOriginSeries_radius k
((p.changeOriginSeries k).hasFPowerSeriesOnBall (hr.trans_le this)).mono hr this
#align formal_multilinear_series.has_fpower_series_on_ball_change_origin FormalMultilinearSeries.hasFPowerSeriesOnBall_changeOrigin
/-- Summing the series `p.changeOrigin x` at a point `y` gives back `p (x + y)`. -/
theorem changeOrigin_eval (h : (‖x‖₊ + ‖y‖₊ : ℝ≥0∞) < p.radius) :
(p.changeOrigin x).sum y = p.sum (x + y) := by
have radius_pos : 0 < p.radius := lt_of_le_of_lt (zero_le _) h
have x_mem_ball : x ∈ EMetric.ball (0 : E) p.radius :=
mem_emetric_ball_zero_iff.2 ((le_add_right le_rfl).trans_lt h)
have y_mem_ball : y ∈ EMetric.ball (0 : E) (p.changeOrigin x).radius := by
refine' mem_emetric_ball_zero_iff.2 (lt_of_lt_of_le _ p.changeOrigin_radius)
rwa [lt_tsub_iff_right, add_comm]
have x_add_y_mem_ball : x + y ∈ EMetric.ball (0 : E) p.radius := by
refine' mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt _ h)
exact mod_cast nnnorm_add_le x y
set f : (Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) → F := fun s =>
p.changeOriginSeriesTerm s.1 s.2.1 s.2.2 s.2.2.2 (fun _ => x) fun _ => y
have hsf : Summable f := by
refine' .of_nnnorm_bounded _ (p.changeOriginSeries_summable_aux₁ h) _
rintro ⟨k, l, s, hs⟩
dsimp only [Subtype.coe_mk]
exact p.nnnorm_changeOriginSeriesTerm_apply_le _ _ _ _ _ _
have hf : HasSum f ((p.changeOrigin x).sum y) := by
refine' HasSum.sigma_of_hasSum ((p.changeOrigin x).summable y_mem_ball).hasSum (fun k => _) hsf
· dsimp only
refine' ContinuousMultilinearMap.hasSum_eval _ _
have := (p.hasFPowerSeriesOnBall_changeOrigin k radius_pos).hasSum x_mem_ball
rw [zero_add] at this
refine' HasSum.sigma_of_hasSum this (fun l => _) _
· simp only [changeOriginSeries, ContinuousMultilinearMap.sum_apply]
apply hasSum_fintype
· refine' .of_nnnorm_bounded _
(p.changeOriginSeries_summable_aux₂ (mem_emetric_ball_zero_iff.1 x_mem_ball) k) fun s => _
refine' (ContinuousMultilinearMap.le_op_nnnorm _ _).trans_eq _
simp
refine' hf.unique (changeOriginIndexEquiv.symm.hasSum_iff.1 _)
refine' HasSum.sigma_of_hasSum
(p.hasSum x_add_y_mem_ball) (fun n => _) (changeOriginIndexEquiv.symm.summable_iff.2 hsf)
erw [(p n).map_add_univ (fun _ => x) fun _ => y]
-- porting note: added explicit function
convert hasSum_fintype (fun c : Finset (Fin n) => f (changeOriginIndexEquiv.symm ⟨n, c⟩))
rename_i s _
dsimp only [changeOriginSeriesTerm, (· ∘ ·), changeOriginIndexEquiv_symm_apply_fst,
changeOriginIndexEquiv_symm_apply_snd_fst, changeOriginIndexEquiv_symm_apply_snd_snd_coe]
rw [ContinuousMultilinearMap.curryFinFinset_apply_const]
have : ∀ (m) (hm : n = m), p n (s.piecewise (fun _ => x) fun _ => y) =
p m ((s.map (Fin.castIso hm).toEquiv.toEmbedding).piecewise (fun _ => x) fun _ => y) := by
rintro m rfl
simp (config := { unfoldPartialApp := true }) [Finset.piecewise]
apply this
#align formal_multilinear_series.change_origin_eval FormalMultilinearSeries.changeOrigin_eval
/-- Power series terms are analytic as we vary the origin -/
theorem analyticAt_changeOrigin (p : FormalMultilinearSeries 𝕜 E F) (rp : p.radius > 0) (n : ℕ) :
AnalyticAt 𝕜 (fun x ↦ p.changeOrigin x n) 0 :=
(FormalMultilinearSeries.hasFPowerSeriesOnBall_changeOrigin p n rp).analyticAt
end FormalMultilinearSeries
section
variable [CompleteSpace F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x y : E} {r : ℝ≥0∞}
/-- If a function admits a power series expansion `p` on a ball `B (x, r)`, then it also admits a
power series on any subball of this ball (even with a different center), given by `p.changeOrigin`.
-/
theorem HasFPowerSeriesOnBall.changeOrigin (hf : HasFPowerSeriesOnBall f p x r)
(h : (‖y‖₊ : ℝ≥0∞) < r) : HasFPowerSeriesOnBall f (p.changeOrigin y) (x + y) (r - ‖y‖₊) :=
{ r_le := by
apply le_trans _ p.changeOrigin_radius
exact tsub_le_tsub hf.r_le le_rfl
r_pos := by simp [h]
hasSum := fun {z} hz => by
have : f (x + y + z) =
FormalMultilinearSeries.sum (FormalMultilinearSeries.changeOrigin p y) z := by
rw [mem_emetric_ball_zero_iff, lt_tsub_iff_right, add_comm] at hz
rw [p.changeOrigin_eval (hz.trans_le hf.r_le), add_assoc, hf.sum]
refine' mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt _ hz)
exact mod_cast nnnorm_add_le y z
rw [this]
apply (p.changeOrigin y).hasSum
refine' EMetric.ball_subset_ball (le_trans _ p.changeOrigin_radius) hz
exact tsub_le_tsub hf.r_le le_rfl }
#align has_fpower_series_on_ball.change_origin HasFPowerSeriesOnBall.changeOrigin
/-- If a function admits a power series expansion `p` on an open ball `B (x, r)`, then
it is analytic at every point of this ball. -/
theorem HasFPowerSeriesOnBall.analyticAt_of_mem (hf : HasFPowerSeriesOnBall f p x r)
(h : y ∈ EMetric.ball x r) : AnalyticAt 𝕜 f y := by
have : (‖y - x‖₊ : ℝ≥0∞) < r := by simpa [edist_eq_coe_nnnorm_sub] using h
have := hf.changeOrigin this
rw [add_sub_cancel'_right] at this
exact this.analyticAt
#align has_fpower_series_on_ball.analytic_at_of_mem HasFPowerSeriesOnBall.analyticAt_of_mem
theorem HasFPowerSeriesOnBall.analyticOn (hf : HasFPowerSeriesOnBall f p x r) :
AnalyticOn 𝕜 f (EMetric.ball x r) :=
fun _y hy => hf.analyticAt_of_mem hy
#align has_fpower_series_on_ball.analytic_on HasFPowerSeriesOnBall.analyticOn
variable (𝕜 f)
/-- For any function `f` from a normed vector space to a Banach space, the set of points `x` such
that `f` is analytic at `x` is open. -/
theorem isOpen_analyticAt : IsOpen { x | AnalyticAt 𝕜 f x } := by
rw [isOpen_iff_mem_nhds]
rintro x ⟨p, r, hr⟩
exact mem_of_superset (EMetric.ball_mem_nhds _ hr.r_pos) fun y hy => hr.analyticAt_of_mem hy
#align is_open_analytic_at isOpen_analyticAt
variable {𝕜}
theorem AnalyticAt.eventually_analyticAt {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∀ᶠ y in 𝓝 x, AnalyticAt 𝕜 f y :=
(isOpen_analyticAt 𝕜 f).mem_nhds h
theorem AnalyticAt.exists_mem_nhds_analyticOn {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∃ s ∈ 𝓝 x, AnalyticOn 𝕜 f s :=
h.eventually_analyticAt.exists_mem
/-- If we're analytic at a point, we're analytic in a nonempty ball -/
theorem AnalyticAt.exists_ball_analyticOn {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∃ r : ℝ, 0 < r ∧ AnalyticOn 𝕜 f (Metric.ball x r) :=
Metric.isOpen_iff.mp (isOpen_analyticAt _ _) _ h
end
section
open FormalMultilinearSeries
variable {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E} {z₀ : 𝕜}
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z)) := by
refine' ⟨fun ⟨r, _, r_pos, h⟩ =>
eventually_of_mem (EMetric.ball_mem_nhds 0 r_pos) fun _ => by simpa using h, _⟩
simp only [Metric.eventually_nhds_iff]
rintro ⟨r, r_pos, h⟩
refine' ⟨p.radius ⊓ r.toNNReal, by simp, _, _⟩
· simp only [r_pos.lt, lt_inf_iff, ENNReal.coe_pos, Real.toNNReal_pos, and_true_iff]
obtain ⟨z, z_pos, le_z⟩ := NormedField.exists_norm_lt 𝕜 r_pos.lt
have : (‖z‖₊ : ENNReal) ≤ p.radius := by
simp only [dist_zero_right] at h
apply FormalMultilinearSeries.le_radius_of_tendsto
convert tendsto_norm.comp (h le_z).summable.tendsto_atTop_zero
funext
simp [norm_smul, mul_comm]
refine' lt_of_lt_of_le _ this
simp only [ENNReal.coe_pos]
exact zero_lt_iff.mpr (nnnorm_ne_zero_iff.mpr (norm_pos_iff.mp z_pos))
· simp only [EMetric.mem_ball, lt_inf_iff, edist_lt_coe, apply_eq_pow_smul_coeff, and_imp,
dist_zero_right] at h ⊢
refine' fun {y} _ hyr => h _
simpa [nndist_eq_nnnorm, Real.lt_toNNReal_iff_coe_lt] using hyr
#align has_fpower_series_at_iff hasFPowerSeriesAt_iff
theorem hasFPowerSeriesAt_iff' :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 z₀, HasSum (fun n => (z - z₀) ^ n • p.coeff n) (f z) := by
|
rw [← map_add_left_nhds_zero, eventually_map, hasFPowerSeriesAt_iff]
|
theorem hasFPowerSeriesAt_iff' :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 z₀, HasSum (fun n => (z - z₀) ^ n • p.coeff n) (f z) := by
|
Mathlib.Analysis.Analytic.Basic.1457_0.jQw1fRSE1vGpOll
|
theorem hasFPowerSeriesAt_iff' :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 z₀, HasSum (fun n => (z - z₀) ^ n • p.coeff n) (f z)
|
Mathlib_Analysis_Analytic_Basic
|
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
p : FormalMultilinearSeries 𝕜 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
⊢ (∀ᶠ (z : 𝕜) in 𝓝 0, HasSum (fun n => z ^ n • coeff p n) (f (z₀ + z))) ↔
∀ᶠ (a : 𝕜) in 𝓝 0, HasSum (fun n => (z₀ + a - z₀) ^ n • coeff p n) (f (z₀ + a))
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Logic.Equiv.Fin
import Mathlib.Topology.Algebra.InfiniteSum.Module
#align_import analysis.analytic.basic from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
/-!
# Analytic functions
A function is analytic in one dimension around `0` if it can be written as a converging power series
`Σ pₙ zⁿ`. This definition can be extended to any dimension (even in infinite dimension) by
requiring that `pₙ` is a continuous `n`-multilinear map. In general, `pₙ` is not unique (in two
dimensions, taking `p₂ (x, y) (x', y') = x y'` or `y x'` gives the same map when applied to a
vector `(x, y) (x, y)`). A way to guarantee uniqueness is to take a symmetric `pₙ`, but this is not
always possible in nonzero characteristic (in characteristic 2, the previous example has no
symmetric representative). Therefore, we do not insist on symmetry or uniqueness in the definition,
and we only require the existence of a converging series.
The general framework is important to say that the exponential map on bounded operators on a Banach
space is analytic, as well as the inverse on invertible operators.
## Main definitions
Let `p` be a formal multilinear series from `E` to `F`, i.e., `p n` is a multilinear map on `E^n`
for `n : ℕ`.
* `p.radius`: the largest `r : ℝ≥0∞` such that `‖p n‖ * r^n` grows subexponentially.
* `p.le_radius_of_bound`, `p.le_radius_of_bound_nnreal`, `p.le_radius_of_isBigO`: if `‖p n‖ * r ^ n`
is bounded above, then `r ≤ p.radius`;
* `p.isLittleO_of_lt_radius`, `p.norm_mul_pow_le_mul_pow_of_lt_radius`,
`p.isLittleO_one_of_lt_radius`,
`p.norm_mul_pow_le_of_lt_radius`, `p.nnnorm_mul_pow_le_of_lt_radius`: if `r < p.radius`, then
`‖p n‖ * r ^ n` tends to zero exponentially;
* `p.lt_radius_of_isBigO`: if `r ≠ 0` and `‖p n‖ * r ^ n = O(a ^ n)` for some `-1 < a < 1`, then
`r < p.radius`;
* `p.partialSum n x`: the sum `∑_{i = 0}^{n-1} pᵢ xⁱ`.
* `p.sum x`: the sum `∑'_{i = 0}^{∞} pᵢ xⁱ`.
Additionally, let `f` be a function from `E` to `F`.
* `HasFPowerSeriesOnBall f p x r`: on the ball of center `x` with radius `r`,
`f (x + y) = ∑'_n pₙ yⁿ`.
* `HasFPowerSeriesAt f p x`: on some ball of center `x` with positive radius, holds
`HasFPowerSeriesOnBall f p x r`.
* `AnalyticAt 𝕜 f x`: there exists a power series `p` such that holds `HasFPowerSeriesAt f p x`.
* `AnalyticOn 𝕜 f s`: the function `f` is analytic at every point of `s`.
We develop the basic properties of these notions, notably:
* If a function admits a power series, it is continuous (see
`HasFPowerSeriesOnBall.continuousOn` and `HasFPowerSeriesAt.continuousAt` and
`AnalyticAt.continuousAt`).
* In a complete space, the sum of a formal power series with positive radius is well defined on the
disk of convergence, see `FormalMultilinearSeries.hasFPowerSeriesOnBall`.
* If a function admits a power series in a ball, then it is analytic at any point `y` of this ball,
and the power series there can be expressed in terms of the initial power series `p` as
`p.changeOrigin y`. See `HasFPowerSeriesOnBall.changeOrigin`. It follows in particular that
the set of points at which a given function is analytic is open, see `isOpen_analyticAt`.
## Implementation details
We only introduce the radius of convergence of a power series, as `p.radius`.
For a power series in finitely many dimensions, there is a finer (directional, coordinate-dependent)
notion, describing the polydisk of convergence. This notion is more specific, and not necessary to
build the general theory. We do not define it here.
-/
noncomputable section
variable {𝕜 E F G : Type*}
open Topology Classical BigOperators NNReal Filter ENNReal
open Set Filter Asymptotics
namespace FormalMultilinearSeries
variable [Ring 𝕜] [AddCommGroup E] [AddCommGroup F] [Module 𝕜 E] [Module 𝕜 F]
variable [TopologicalSpace E] [TopologicalSpace F]
variable [TopologicalAddGroup E] [TopologicalAddGroup F]
variable [ContinuousConstSMul 𝕜 E] [ContinuousConstSMul 𝕜 F]
/-- Given a formal multilinear series `p` and a vector `x`, then `p.sum x` is the sum `Σ pₙ xⁿ`. A
priori, it only behaves well when `‖x‖ < p.radius`. -/
protected def sum (p : FormalMultilinearSeries 𝕜 E F) (x : E) : F :=
∑' n : ℕ, p n fun _ => x
#align formal_multilinear_series.sum FormalMultilinearSeries.sum
/-- Given a formal multilinear series `p` and a vector `x`, then `p.partialSum n x` is the sum
`Σ pₖ xᵏ` for `k ∈ {0,..., n-1}`. -/
def partialSum (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) (x : E) : F :=
∑ k in Finset.range n, p k fun _ : Fin k => x
#align formal_multilinear_series.partial_sum FormalMultilinearSeries.partialSum
/-- The partial sums of a formal multilinear series are continuous. -/
theorem partialSum_continuous (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) :
Continuous (p.partialSum n) := by
unfold partialSum -- Porting note: added
continuity
#align formal_multilinear_series.partial_sum_continuous FormalMultilinearSeries.partialSum_continuous
end FormalMultilinearSeries
/-! ### The radius of a formal multilinear series -/
variable [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F]
[NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G]
namespace FormalMultilinearSeries
variable (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
/-- The radius of a formal multilinear series is the largest `r` such that the sum `Σ ‖pₙ‖ ‖y‖ⁿ`
converges for all `‖y‖ < r`. This implies that `Σ pₙ yⁿ` converges for all `‖y‖ < r`, but these
definitions are *not* equivalent in general. -/
def radius (p : FormalMultilinearSeries 𝕜 E F) : ℝ≥0∞ :=
⨆ (r : ℝ≥0) (C : ℝ) (_ : ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C), (r : ℝ≥0∞)
#align formal_multilinear_series.radius FormalMultilinearSeries.radius
/-- If `‖pₙ‖ rⁿ` is bounded in `n`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_bound (C : ℝ) {r : ℝ≥0} (h : ∀ n : ℕ, ‖p n‖ * (r : ℝ) ^ n ≤ C) :
(r : ℝ≥0∞) ≤ p.radius :=
le_iSup_of_le r <| le_iSup_of_le C <| le_iSup (fun _ => (r : ℝ≥0∞)) h
#align formal_multilinear_series.le_radius_of_bound FormalMultilinearSeries.le_radius_of_bound
/-- If `‖pₙ‖ rⁿ` is bounded in `n`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_bound_nnreal (C : ℝ≥0) {r : ℝ≥0} (h : ∀ n : ℕ, ‖p n‖₊ * r ^ n ≤ C) :
(r : ℝ≥0∞) ≤ p.radius :=
p.le_radius_of_bound C fun n => mod_cast h n
#align formal_multilinear_series.le_radius_of_bound_nnreal FormalMultilinearSeries.le_radius_of_bound_nnreal
/-- If `‖pₙ‖ rⁿ = O(1)`, as `n → ∞`, then the radius of `p` is at least `r`. -/
theorem le_radius_of_isBigO (h : (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] fun _ => (1 : ℝ)) :
↑r ≤ p.radius :=
Exists.elim (isBigO_one_nat_atTop_iff.1 h) fun C hC =>
p.le_radius_of_bound C fun n => (le_abs_self _).trans (hC n)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.le_radius_of_is_O FormalMultilinearSeries.le_radius_of_isBigO
theorem le_radius_of_eventually_le (C) (h : ∀ᶠ n in atTop, ‖p n‖ * (r : ℝ) ^ n ≤ C) :
↑r ≤ p.radius :=
p.le_radius_of_isBigO <| IsBigO.of_bound C <| h.mono fun n hn => by simpa
#align formal_multilinear_series.le_radius_of_eventually_le FormalMultilinearSeries.le_radius_of_eventually_le
theorem le_radius_of_summable_nnnorm (h : Summable fun n => ‖p n‖₊ * r ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_bound_nnreal (∑' n, ‖p n‖₊ * r ^ n) fun _ => le_tsum' h _
#align formal_multilinear_series.le_radius_of_summable_nnnorm FormalMultilinearSeries.le_radius_of_summable_nnnorm
theorem le_radius_of_summable (h : Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_summable_nnnorm <| by
simp only [← coe_nnnorm] at h
exact mod_cast h
#align formal_multilinear_series.le_radius_of_summable FormalMultilinearSeries.le_radius_of_summable
theorem radius_eq_top_of_forall_nnreal_isBigO
(h : ∀ r : ℝ≥0, (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] fun _ => (1 : ℝ)) : p.radius = ∞ :=
ENNReal.eq_top_of_forall_nnreal_le fun r => p.le_radius_of_isBigO (h r)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.radius_eq_top_of_forall_nnreal_is_O FormalMultilinearSeries.radius_eq_top_of_forall_nnreal_isBigO
theorem radius_eq_top_of_eventually_eq_zero (h : ∀ᶠ n in atTop, p n = 0) : p.radius = ∞ :=
p.radius_eq_top_of_forall_nnreal_isBigO fun r =>
(isBigO_zero _ _).congr' (h.mono fun n hn => by simp [hn]) EventuallyEq.rfl
#align formal_multilinear_series.radius_eq_top_of_eventually_eq_zero FormalMultilinearSeries.radius_eq_top_of_eventually_eq_zero
theorem radius_eq_top_of_forall_image_add_eq_zero (n : ℕ) (hn : ∀ m, p (m + n) = 0) :
p.radius = ∞ :=
p.radius_eq_top_of_eventually_eq_zero <|
mem_atTop_sets.2 ⟨n, fun _ hk => tsub_add_cancel_of_le hk ▸ hn _⟩
#align formal_multilinear_series.radius_eq_top_of_forall_image_add_eq_zero FormalMultilinearSeries.radius_eq_top_of_forall_image_add_eq_zero
@[simp]
theorem constFormalMultilinearSeries_radius {v : F} :
(constFormalMultilinearSeries 𝕜 E v).radius = ⊤ :=
(constFormalMultilinearSeries 𝕜 E v).radius_eq_top_of_forall_image_add_eq_zero 1
(by simp [constFormalMultilinearSeries])
#align formal_multilinear_series.const_formal_multilinear_series_radius FormalMultilinearSeries.constFormalMultilinearSeries_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` tends to zero exponentially:
for some `0 < a < 1`, `‖p n‖ rⁿ = o(aⁿ)`. -/
theorem isLittleO_of_lt_radius (h : ↑r < p.radius) :
∃ a ∈ Ioo (0 : ℝ) 1, (fun n => ‖p n‖ * (r : ℝ) ^ n) =o[atTop] (a ^ ·) := by
have := (TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 4
rw [this]
-- Porting note: was
-- rw [(TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 4]
simp only [radius, lt_iSup_iff] at h
rcases h with ⟨t, C, hC, rt⟩
rw [ENNReal.coe_lt_coe, ← NNReal.coe_lt_coe] at rt
have : 0 < (t : ℝ) := r.coe_nonneg.trans_lt rt
rw [← div_lt_one this] at rt
refine' ⟨_, rt, C, Or.inr zero_lt_one, fun n => _⟩
calc
|‖p n‖ * (r : ℝ) ^ n| = ‖p n‖ * (t : ℝ) ^ n * (r / t : ℝ) ^ n := by
field_simp [mul_right_comm, abs_mul]
_ ≤ C * (r / t : ℝ) ^ n := by gcongr; apply hC
#align formal_multilinear_series.is_o_of_lt_radius FormalMultilinearSeries.isLittleO_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ = o(1)`. -/
theorem isLittleO_one_of_lt_radius (h : ↑r < p.radius) :
(fun n => ‖p n‖ * (r : ℝ) ^ n) =o[atTop] (fun _ => 1 : ℕ → ℝ) :=
let ⟨_, ha, hp⟩ := p.isLittleO_of_lt_radius h
hp.trans <| (isLittleO_pow_pow_of_lt_left ha.1.le ha.2).congr (fun _ => rfl) one_pow
#align formal_multilinear_series.is_o_one_of_lt_radius FormalMultilinearSeries.isLittleO_one_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` tends to zero exponentially:
for some `0 < a < 1` and `C > 0`, `‖p n‖ * r ^ n ≤ C * a ^ n`. -/
theorem norm_mul_pow_le_mul_pow_of_lt_radius (h : ↑r < p.radius) :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C * a ^ n := by
-- Porting note: moved out of `rcases`
have := ((TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 1 5).mp
(p.isLittleO_of_lt_radius h)
rcases this with ⟨a, ha, C, hC, H⟩
exact ⟨a, ha, C, hC, fun n => (le_abs_self _).trans (H n)⟩
#align formal_multilinear_series.norm_mul_pow_le_mul_pow_of_lt_radius FormalMultilinearSeries.norm_mul_pow_le_mul_pow_of_lt_radius
/-- If `r ≠ 0` and `‖pₙ‖ rⁿ = O(aⁿ)` for some `-1 < a < 1`, then `r < p.radius`. -/
theorem lt_radius_of_isBigO (h₀ : r ≠ 0) {a : ℝ} (ha : a ∈ Ioo (-1 : ℝ) 1)
(hp : (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] (a ^ ·)) : ↑r < p.radius := by
-- Porting note: moved out of `rcases`
have := ((TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 2 5)
rcases this.mp ⟨a, ha, hp⟩ with ⟨a, ha, C, hC, hp⟩
rw [← pos_iff_ne_zero, ← NNReal.coe_pos] at h₀
lift a to ℝ≥0 using ha.1.le
have : (r : ℝ) < r / a := by
simpa only [div_one] using (div_lt_div_left h₀ zero_lt_one ha.1).2 ha.2
norm_cast at this
rw [← ENNReal.coe_lt_coe] at this
refine' this.trans_le (p.le_radius_of_bound C fun n => _)
rw [NNReal.coe_div, div_pow, ← mul_div_assoc, div_le_iff (pow_pos ha.1 n)]
exact (le_abs_self _).trans (hp n)
set_option linter.uppercaseLean3 false in
#align formal_multilinear_series.lt_radius_of_is_O FormalMultilinearSeries.lt_radius_of_isBigO
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem norm_mul_pow_le_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C :=
let ⟨_, ha, C, hC, h⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius h
⟨C, hC, fun n => (h n).trans <| mul_le_of_le_one_right hC.lt.le (pow_le_one _ ha.1.le ha.2.le)⟩
#align formal_multilinear_series.norm_mul_pow_le_of_lt_radius FormalMultilinearSeries.norm_mul_pow_le_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem norm_le_div_pow_of_pos_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h0 : 0 < r) (h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖ ≤ C / (r : ℝ) ^ n :=
let ⟨C, hC, hp⟩ := p.norm_mul_pow_le_of_lt_radius h
⟨C, hC, fun n => Iff.mpr (le_div_iff (pow_pos h0 _)) (hp n)⟩
#align formal_multilinear_series.norm_le_div_pow_of_pos_of_lt_radius FormalMultilinearSeries.norm_le_div_pow_of_pos_of_lt_radius
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem nnnorm_mul_pow_le_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖₊ * r ^ n ≤ C :=
let ⟨C, hC, hp⟩ := p.norm_mul_pow_le_of_lt_radius h
⟨⟨C, hC.lt.le⟩, hC, mod_cast hp⟩
#align formal_multilinear_series.nnnorm_mul_pow_le_of_lt_radius FormalMultilinearSeries.nnnorm_mul_pow_le_of_lt_radius
theorem le_radius_of_tendsto (p : FormalMultilinearSeries 𝕜 E F) {l : ℝ}
(h : Tendsto (fun n => ‖p n‖ * (r : ℝ) ^ n) atTop (𝓝 l)) : ↑r ≤ p.radius :=
p.le_radius_of_isBigO (h.isBigO_one _)
#align formal_multilinear_series.le_radius_of_tendsto FormalMultilinearSeries.le_radius_of_tendsto
theorem le_radius_of_summable_norm (p : FormalMultilinearSeries 𝕜 E F)
(hs : Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : ↑r ≤ p.radius :=
p.le_radius_of_tendsto hs.tendsto_atTop_zero
#align formal_multilinear_series.le_radius_of_summable_norm FormalMultilinearSeries.le_radius_of_summable_norm
theorem not_summable_norm_of_radius_lt_nnnorm (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(h : p.radius < ‖x‖₊) : ¬Summable fun n => ‖p n‖ * ‖x‖ ^ n :=
fun hs => not_le_of_lt h (p.le_radius_of_summable_norm hs)
#align formal_multilinear_series.not_summable_norm_of_radius_lt_nnnorm FormalMultilinearSeries.not_summable_norm_of_radius_lt_nnnorm
theorem summable_norm_mul_pow (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0} (h : ↑r < p.radius) :
Summable fun n : ℕ => ‖p n‖ * (r : ℝ) ^ n := by
obtain ⟨a, ha : a ∈ Ioo (0 : ℝ) 1, C, - : 0 < C, hp⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius h
exact .of_nonneg_of_le (fun n => mul_nonneg (norm_nonneg _) (pow_nonneg r.coe_nonneg _))
hp ((summable_geometric_of_lt_1 ha.1.le ha.2).mul_left _)
#align formal_multilinear_series.summable_norm_mul_pow FormalMultilinearSeries.summable_norm_mul_pow
theorem summable_norm_apply (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : Summable fun n : ℕ => ‖p n fun _ => x‖ := by
rw [mem_emetric_ball_zero_iff] at hx
refine' .of_nonneg_of_le
(fun _ => norm_nonneg _) (fun n => ((p n).le_op_norm _).trans_eq _) (p.summable_norm_mul_pow hx)
simp
#align formal_multilinear_series.summable_norm_apply FormalMultilinearSeries.summable_norm_apply
theorem summable_nnnorm_mul_pow (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0} (h : ↑r < p.radius) :
Summable fun n : ℕ => ‖p n‖₊ * r ^ n := by
rw [← NNReal.summable_coe]
push_cast
exact p.summable_norm_mul_pow h
#align formal_multilinear_series.summable_nnnorm_mul_pow FormalMultilinearSeries.summable_nnnorm_mul_pow
protected theorem summable [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : Summable fun n : ℕ => p n fun _ => x :=
(p.summable_norm_apply hx).of_norm
#align formal_multilinear_series.summable FormalMultilinearSeries.summable
theorem radius_eq_top_of_summable_norm (p : FormalMultilinearSeries 𝕜 E F)
(hs : ∀ r : ℝ≥0, Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : p.radius = ∞ :=
ENNReal.eq_top_of_forall_nnreal_le fun r => p.le_radius_of_summable_norm (hs r)
#align formal_multilinear_series.radius_eq_top_of_summable_norm FormalMultilinearSeries.radius_eq_top_of_summable_norm
theorem radius_eq_top_iff_summable_norm (p : FormalMultilinearSeries 𝕜 E F) :
p.radius = ∞ ↔ ∀ r : ℝ≥0, Summable fun n => ‖p n‖ * (r : ℝ) ^ n := by
constructor
· intro h r
obtain ⟨a, ha : a ∈ Ioo (0 : ℝ) 1, C, - : 0 < C, hp⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius
(show (r : ℝ≥0∞) < p.radius from h.symm ▸ ENNReal.coe_lt_top)
refine' .of_norm_bounded
(fun n => (C : ℝ) * a ^ n) ((summable_geometric_of_lt_1 ha.1.le ha.2).mul_left _) fun n => _
specialize hp n
rwa [Real.norm_of_nonneg (mul_nonneg (norm_nonneg _) (pow_nonneg r.coe_nonneg n))]
· exact p.radius_eq_top_of_summable_norm
#align formal_multilinear_series.radius_eq_top_iff_summable_norm FormalMultilinearSeries.radius_eq_top_iff_summable_norm
/-- If the radius of `p` is positive, then `‖pₙ‖` grows at most geometrically. -/
theorem le_mul_pow_of_radius_pos (p : FormalMultilinearSeries 𝕜 E F) (h : 0 < p.radius) :
∃ (C r : _) (hC : 0 < C) (_ : 0 < r), ∀ n, ‖p n‖ ≤ C * r ^ n := by
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 h with ⟨r, r0, rlt⟩
have rpos : 0 < (r : ℝ) := by simp [ENNReal.coe_pos.1 r0]
rcases norm_le_div_pow_of_pos_of_lt_radius p rpos rlt with ⟨C, Cpos, hCp⟩
refine' ⟨C, r⁻¹, Cpos, by simp only [inv_pos, rpos], fun n => _⟩
-- Porting note: was `convert`
rw [inv_pow, ← div_eq_mul_inv]
exact hCp n
#align formal_multilinear_series.le_mul_pow_of_radius_pos FormalMultilinearSeries.le_mul_pow_of_radius_pos
/-- The radius of the sum of two formal series is at least the minimum of their two radii. -/
theorem min_radius_le_radius_add (p q : FormalMultilinearSeries 𝕜 E F) :
min p.radius q.radius ≤ (p + q).radius := by
refine' ENNReal.le_of_forall_nnreal_lt fun r hr => _
rw [lt_min_iff] at hr
have := ((p.isLittleO_one_of_lt_radius hr.1).add (q.isLittleO_one_of_lt_radius hr.2)).isBigO
refine' (p + q).le_radius_of_isBigO ((isBigO_of_le _ fun n => _).trans this)
rw [← add_mul, norm_mul, norm_mul, norm_norm]
exact mul_le_mul_of_nonneg_right ((norm_add_le _ _).trans (le_abs_self _)) (norm_nonneg _)
#align formal_multilinear_series.min_radius_le_radius_add FormalMultilinearSeries.min_radius_le_radius_add
@[simp]
theorem radius_neg (p : FormalMultilinearSeries 𝕜 E F) : (-p).radius = p.radius := by
simp only [radius, neg_apply, norm_neg]
#align formal_multilinear_series.radius_neg FormalMultilinearSeries.radius_neg
protected theorem hasSum [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : HasSum (fun n : ℕ => p n fun _ => x) (p.sum x) :=
(p.summable hx).hasSum
#align formal_multilinear_series.has_sum FormalMultilinearSeries.hasSum
theorem radius_le_radius_continuousLinearMap_comp (p : FormalMultilinearSeries 𝕜 E F)
(f : F →L[𝕜] G) : p.radius ≤ (f.compFormalMultilinearSeries p).radius := by
refine' ENNReal.le_of_forall_nnreal_lt fun r hr => _
apply le_radius_of_isBigO
apply (IsBigO.trans_isLittleO _ (p.isLittleO_one_of_lt_radius hr)).isBigO
refine' IsBigO.mul (@IsBigOWith.isBigO _ _ _ _ _ ‖f‖ _ _ _ _) (isBigO_refl _ _)
refine IsBigOWith.of_bound (eventually_of_forall fun n => ?_)
simpa only [norm_norm] using f.norm_compContinuousMultilinearMap_le (p n)
#align formal_multilinear_series.radius_le_radius_continuous_linear_map_comp FormalMultilinearSeries.radius_le_radius_continuousLinearMap_comp
end FormalMultilinearSeries
/-! ### Expanding a function as a power series -/
section
variable {f g : E → F} {p pf pg : FormalMultilinearSeries 𝕜 E F} {x : E} {r r' : ℝ≥0∞}
/-- Given a function `f : E → F` and a formal multilinear series `p`, we say that `f` has `p` as
a power series on the ball of radius `r > 0` around `x` if `f (x + y) = ∑' pₙ yⁿ` for all `‖y‖ < r`.
-/
structure HasFPowerSeriesOnBall (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (x : E) (r : ℝ≥0∞) :
Prop where
r_le : r ≤ p.radius
r_pos : 0 < r
hasSum :
∀ {y}, y ∈ EMetric.ball (0 : E) r → HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y))
#align has_fpower_series_on_ball HasFPowerSeriesOnBall
/-- Given a function `f : E → F` and a formal multilinear series `p`, we say that `f` has `p` as
a power series around `x` if `f (x + y) = ∑' pₙ yⁿ` for all `y` in a neighborhood of `0`. -/
def HasFPowerSeriesAt (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (x : E) :=
∃ r, HasFPowerSeriesOnBall f p x r
#align has_fpower_series_at HasFPowerSeriesAt
variable (𝕜)
/-- Given a function `f : E → F`, we say that `f` is analytic at `x` if it admits a convergent power
series expansion around `x`. -/
def AnalyticAt (f : E → F) (x : E) :=
∃ p : FormalMultilinearSeries 𝕜 E F, HasFPowerSeriesAt f p x
#align analytic_at AnalyticAt
/-- Given a function `f : E → F`, we say that `f` is analytic on a set `s` if it is analytic around
every point of `s`. -/
def AnalyticOn (f : E → F) (s : Set E) :=
∀ x, x ∈ s → AnalyticAt 𝕜 f x
#align analytic_on AnalyticOn
variable {𝕜}
theorem HasFPowerSeriesOnBall.hasFPowerSeriesAt (hf : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesAt f p x :=
⟨r, hf⟩
#align has_fpower_series_on_ball.has_fpower_series_at HasFPowerSeriesOnBall.hasFPowerSeriesAt
theorem HasFPowerSeriesAt.analyticAt (hf : HasFPowerSeriesAt f p x) : AnalyticAt 𝕜 f x :=
⟨p, hf⟩
#align has_fpower_series_at.analytic_at HasFPowerSeriesAt.analyticAt
theorem HasFPowerSeriesOnBall.analyticAt (hf : HasFPowerSeriesOnBall f p x r) : AnalyticAt 𝕜 f x :=
hf.hasFPowerSeriesAt.analyticAt
#align has_fpower_series_on_ball.analytic_at HasFPowerSeriesOnBall.analyticAt
theorem HasFPowerSeriesOnBall.congr (hf : HasFPowerSeriesOnBall f p x r)
(hg : EqOn f g (EMetric.ball x r)) : HasFPowerSeriesOnBall g p x r :=
{ r_le := hf.r_le
r_pos := hf.r_pos
hasSum := fun {y} hy => by
convert hf.hasSum hy using 1
apply hg.symm
simpa [edist_eq_coe_nnnorm_sub] using hy }
#align has_fpower_series_on_ball.congr HasFPowerSeriesOnBall.congr
/-- If a function `f` has a power series `p` around `x`, then the function `z ↦ f (z - y)` has the
same power series around `x + y`. -/
theorem HasFPowerSeriesOnBall.comp_sub (hf : HasFPowerSeriesOnBall f p x r) (y : E) :
HasFPowerSeriesOnBall (fun z => f (z - y)) p (x + y) r :=
{ r_le := hf.r_le
r_pos := hf.r_pos
hasSum := fun {z} hz => by
convert hf.hasSum hz using 2
abel }
#align has_fpower_series_on_ball.comp_sub HasFPowerSeriesOnBall.comp_sub
theorem HasFPowerSeriesOnBall.hasSum_sub (hf : HasFPowerSeriesOnBall f p x r) {y : E}
(hy : y ∈ EMetric.ball x r) : HasSum (fun n : ℕ => p n fun _ => y - x) (f y) := by
have : y - x ∈ EMetric.ball (0 : E) r := by simpa [edist_eq_coe_nnnorm_sub] using hy
simpa only [add_sub_cancel'_right] using hf.hasSum this
#align has_fpower_series_on_ball.has_sum_sub HasFPowerSeriesOnBall.hasSum_sub
theorem HasFPowerSeriesOnBall.radius_pos (hf : HasFPowerSeriesOnBall f p x r) : 0 < p.radius :=
lt_of_lt_of_le hf.r_pos hf.r_le
#align has_fpower_series_on_ball.radius_pos HasFPowerSeriesOnBall.radius_pos
theorem HasFPowerSeriesAt.radius_pos (hf : HasFPowerSeriesAt f p x) : 0 < p.radius :=
let ⟨_, hr⟩ := hf
hr.radius_pos
#align has_fpower_series_at.radius_pos HasFPowerSeriesAt.radius_pos
theorem HasFPowerSeriesOnBall.mono (hf : HasFPowerSeriesOnBall f p x r) (r'_pos : 0 < r')
(hr : r' ≤ r) : HasFPowerSeriesOnBall f p x r' :=
⟨le_trans hr hf.1, r'_pos, fun hy => hf.hasSum (EMetric.ball_subset_ball hr hy)⟩
#align has_fpower_series_on_ball.mono HasFPowerSeriesOnBall.mono
theorem HasFPowerSeriesAt.congr (hf : HasFPowerSeriesAt f p x) (hg : f =ᶠ[𝓝 x] g) :
HasFPowerSeriesAt g p x := by
rcases hf with ⟨r₁, h₁⟩
rcases EMetric.mem_nhds_iff.mp hg with ⟨r₂, h₂pos, h₂⟩
exact ⟨min r₁ r₂,
(h₁.mono (lt_min h₁.r_pos h₂pos) inf_le_left).congr
fun y hy => h₂ (EMetric.ball_subset_ball inf_le_right hy)⟩
#align has_fpower_series_at.congr HasFPowerSeriesAt.congr
protected theorem HasFPowerSeriesAt.eventually (hf : HasFPowerSeriesAt f p x) :
∀ᶠ r : ℝ≥0∞ in 𝓝[>] 0, HasFPowerSeriesOnBall f p x r :=
let ⟨_, hr⟩ := hf
mem_of_superset (Ioo_mem_nhdsWithin_Ioi (left_mem_Ico.2 hr.r_pos)) fun _ hr' =>
hr.mono hr'.1 hr'.2.le
#align has_fpower_series_at.eventually HasFPowerSeriesAt.eventually
theorem HasFPowerSeriesOnBall.eventually_hasSum (hf : HasFPowerSeriesOnBall f p x r) :
∀ᶠ y in 𝓝 0, HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y)) := by
filter_upwards [EMetric.ball_mem_nhds (0 : E) hf.r_pos] using fun _ => hf.hasSum
#align has_fpower_series_on_ball.eventually_has_sum HasFPowerSeriesOnBall.eventually_hasSum
theorem HasFPowerSeriesAt.eventually_hasSum (hf : HasFPowerSeriesAt f p x) :
∀ᶠ y in 𝓝 0, HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y)) :=
let ⟨_, hr⟩ := hf
hr.eventually_hasSum
#align has_fpower_series_at.eventually_has_sum HasFPowerSeriesAt.eventually_hasSum
theorem HasFPowerSeriesOnBall.eventually_hasSum_sub (hf : HasFPowerSeriesOnBall f p x r) :
∀ᶠ y in 𝓝 x, HasSum (fun n : ℕ => p n fun _ : Fin n => y - x) (f y) := by
filter_upwards [EMetric.ball_mem_nhds x hf.r_pos] with y using hf.hasSum_sub
#align has_fpower_series_on_ball.eventually_has_sum_sub HasFPowerSeriesOnBall.eventually_hasSum_sub
theorem HasFPowerSeriesAt.eventually_hasSum_sub (hf : HasFPowerSeriesAt f p x) :
∀ᶠ y in 𝓝 x, HasSum (fun n : ℕ => p n fun _ : Fin n => y - x) (f y) :=
let ⟨_, hr⟩ := hf
hr.eventually_hasSum_sub
#align has_fpower_series_at.eventually_has_sum_sub HasFPowerSeriesAt.eventually_hasSum_sub
theorem HasFPowerSeriesOnBall.eventually_eq_zero
(hf : HasFPowerSeriesOnBall f (0 : FormalMultilinearSeries 𝕜 E F) x r) :
∀ᶠ z in 𝓝 x, f z = 0 := by
filter_upwards [hf.eventually_hasSum_sub] with z hz using hz.unique hasSum_zero
#align has_fpower_series_on_ball.eventually_eq_zero HasFPowerSeriesOnBall.eventually_eq_zero
theorem HasFPowerSeriesAt.eventually_eq_zero
(hf : HasFPowerSeriesAt f (0 : FormalMultilinearSeries 𝕜 E F) x) : ∀ᶠ z in 𝓝 x, f z = 0 :=
let ⟨_, hr⟩ := hf
hr.eventually_eq_zero
#align has_fpower_series_at.eventually_eq_zero HasFPowerSeriesAt.eventually_eq_zero
theorem hasFPowerSeriesOnBall_const {c : F} {e : E} :
HasFPowerSeriesOnBall (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e ⊤ := by
refine' ⟨by simp, WithTop.zero_lt_top, fun _ => hasSum_single 0 fun n hn => _⟩
simp [constFormalMultilinearSeries_apply hn]
#align has_fpower_series_on_ball_const hasFPowerSeriesOnBall_const
theorem hasFPowerSeriesAt_const {c : F} {e : E} :
HasFPowerSeriesAt (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e :=
⟨⊤, hasFPowerSeriesOnBall_const⟩
#align has_fpower_series_at_const hasFPowerSeriesAt_const
theorem analyticAt_const {v : F} : AnalyticAt 𝕜 (fun _ => v) x :=
⟨constFormalMultilinearSeries 𝕜 E v, hasFPowerSeriesAt_const⟩
#align analytic_at_const analyticAt_const
theorem analyticOn_const {v : F} {s : Set E} : AnalyticOn 𝕜 (fun _ => v) s :=
fun _ _ => analyticAt_const
#align analytic_on_const analyticOn_const
theorem HasFPowerSeriesOnBall.add (hf : HasFPowerSeriesOnBall f pf x r)
(hg : HasFPowerSeriesOnBall g pg x r) : HasFPowerSeriesOnBall (f + g) (pf + pg) x r :=
{ r_le := le_trans (le_min_iff.2 ⟨hf.r_le, hg.r_le⟩) (pf.min_radius_le_radius_add pg)
r_pos := hf.r_pos
hasSum := fun hy => (hf.hasSum hy).add (hg.hasSum hy) }
#align has_fpower_series_on_ball.add HasFPowerSeriesOnBall.add
theorem HasFPowerSeriesAt.add (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x) :
HasFPowerSeriesAt (f + g) (pf + pg) x := by
rcases (hf.eventually.and hg.eventually).exists with ⟨r, hr⟩
exact ⟨r, hr.1.add hr.2⟩
#align has_fpower_series_at.add HasFPowerSeriesAt.add
theorem AnalyticAt.congr (hf : AnalyticAt 𝕜 f x) (hg : f =ᶠ[𝓝 x] g) : AnalyticAt 𝕜 g x :=
let ⟨_, hpf⟩ := hf
(hpf.congr hg).analyticAt
theorem analyticAt_congr (h : f =ᶠ[𝓝 x] g) : AnalyticAt 𝕜 f x ↔ AnalyticAt 𝕜 g x :=
⟨fun hf ↦ hf.congr h, fun hg ↦ hg.congr h.symm⟩
theorem AnalyticAt.add (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) : AnalyticAt 𝕜 (f + g) x :=
let ⟨_, hpf⟩ := hf
let ⟨_, hqf⟩ := hg
(hpf.add hqf).analyticAt
#align analytic_at.add AnalyticAt.add
theorem HasFPowerSeriesOnBall.neg (hf : HasFPowerSeriesOnBall f pf x r) :
HasFPowerSeriesOnBall (-f) (-pf) x r :=
{ r_le := by
rw [pf.radius_neg]
exact hf.r_le
r_pos := hf.r_pos
hasSum := fun hy => (hf.hasSum hy).neg }
#align has_fpower_series_on_ball.neg HasFPowerSeriesOnBall.neg
theorem HasFPowerSeriesAt.neg (hf : HasFPowerSeriesAt f pf x) : HasFPowerSeriesAt (-f) (-pf) x :=
let ⟨_, hrf⟩ := hf
hrf.neg.hasFPowerSeriesAt
#align has_fpower_series_at.neg HasFPowerSeriesAt.neg
theorem AnalyticAt.neg (hf : AnalyticAt 𝕜 f x) : AnalyticAt 𝕜 (-f) x :=
let ⟨_, hpf⟩ := hf
hpf.neg.analyticAt
#align analytic_at.neg AnalyticAt.neg
theorem HasFPowerSeriesOnBall.sub (hf : HasFPowerSeriesOnBall f pf x r)
(hg : HasFPowerSeriesOnBall g pg x r) : HasFPowerSeriesOnBall (f - g) (pf - pg) x r := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align has_fpower_series_on_ball.sub HasFPowerSeriesOnBall.sub
theorem HasFPowerSeriesAt.sub (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x) :
HasFPowerSeriesAt (f - g) (pf - pg) x := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align has_fpower_series_at.sub HasFPowerSeriesAt.sub
theorem AnalyticAt.sub (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) :
AnalyticAt 𝕜 (f - g) x := by
simpa only [sub_eq_add_neg] using hf.add hg.neg
#align analytic_at.sub AnalyticAt.sub
theorem AnalyticOn.mono {s t : Set E} (hf : AnalyticOn 𝕜 f t) (hst : s ⊆ t) : AnalyticOn 𝕜 f s :=
fun z hz => hf z (hst hz)
#align analytic_on.mono AnalyticOn.mono
theorem AnalyticOn.congr' {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : f =ᶠ[𝓝ˢ s] g) :
AnalyticOn 𝕜 g s :=
fun z hz => (hf z hz).congr (mem_nhdsSet_iff_forall.mp hg z hz)
theorem analyticOn_congr' {s : Set E} (h : f =ᶠ[𝓝ˢ s] g) : AnalyticOn 𝕜 f s ↔ AnalyticOn 𝕜 g s :=
⟨fun hf => hf.congr' h, fun hg => hg.congr' h.symm⟩
theorem AnalyticOn.congr {s : Set E} (hs : IsOpen s) (hf : AnalyticOn 𝕜 f s) (hg : s.EqOn f g) :
AnalyticOn 𝕜 g s :=
hf.congr' $ mem_nhdsSet_iff_forall.mpr
(fun _ hz => eventuallyEq_iff_exists_mem.mpr ⟨s, hs.mem_nhds hz, hg⟩)
theorem analyticOn_congr {s : Set E} (hs : IsOpen s) (h : s.EqOn f g) : AnalyticOn 𝕜 f s ↔
AnalyticOn 𝕜 g s := ⟨fun hf => hf.congr hs h, fun hg => hg.congr hs h.symm⟩
theorem AnalyticOn.add {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
AnalyticOn 𝕜 (f + g) s :=
fun z hz => (hf z hz).add (hg z hz)
#align analytic_on.add AnalyticOn.add
theorem AnalyticOn.sub {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
AnalyticOn 𝕜 (f - g) s :=
fun z hz => (hf z hz).sub (hg z hz)
#align analytic_on.sub AnalyticOn.sub
theorem HasFPowerSeriesOnBall.coeff_zero (hf : HasFPowerSeriesOnBall f pf x r) (v : Fin 0 → E) :
pf 0 v = f x := by
have v_eq : v = fun i => 0 := Subsingleton.elim _ _
have zero_mem : (0 : E) ∈ EMetric.ball (0 : E) r := by simp [hf.r_pos]
have : ∀ i, i ≠ 0 → (pf i fun j => 0) = 0 := by
intro i hi
have : 0 < i := pos_iff_ne_zero.2 hi
exact ContinuousMultilinearMap.map_coord_zero _ (⟨0, this⟩ : Fin i) rfl
have A := (hf.hasSum zero_mem).unique (hasSum_single _ this)
simpa [v_eq] using A.symm
#align has_fpower_series_on_ball.coeff_zero HasFPowerSeriesOnBall.coeff_zero
theorem HasFPowerSeriesAt.coeff_zero (hf : HasFPowerSeriesAt f pf x) (v : Fin 0 → E) :
pf 0 v = f x :=
let ⟨_, hrf⟩ := hf
hrf.coeff_zero v
#align has_fpower_series_at.coeff_zero HasFPowerSeriesAt.coeff_zero
/-- If a function `f` has a power series `p` on a ball and `g` is linear, then `g ∘ f` has the
power series `g ∘ p` on the same ball. -/
theorem ContinuousLinearMap.comp_hasFPowerSeriesOnBall (g : F →L[𝕜] G)
(h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (g ∘ f) (g.compFormalMultilinearSeries p) x r :=
{ r_le := h.r_le.trans (p.radius_le_radius_continuousLinearMap_comp _)
r_pos := h.r_pos
hasSum := fun hy => by
simpa only [ContinuousLinearMap.compFormalMultilinearSeries_apply,
ContinuousLinearMap.compContinuousMultilinearMap_coe, Function.comp_apply] using
g.hasSum (h.hasSum hy) }
#align continuous_linear_map.comp_has_fpower_series_on_ball ContinuousLinearMap.comp_hasFPowerSeriesOnBall
/-- If a function `f` is analytic on a set `s` and `g` is linear, then `g ∘ f` is analytic
on `s`. -/
theorem ContinuousLinearMap.comp_analyticOn {s : Set E} (g : F →L[𝕜] G) (h : AnalyticOn 𝕜 f s) :
AnalyticOn 𝕜 (g ∘ f) s := by
rintro x hx
rcases h x hx with ⟨p, r, hp⟩
exact ⟨g.compFormalMultilinearSeries p, r, g.comp_hasFPowerSeriesOnBall hp⟩
#align continuous_linear_map.comp_analytic_on ContinuousLinearMap.comp_analyticOn
/-- If a function admits a power series expansion, then it is exponentially close to the partial
sums of this power series on strict subdisks of the disk of convergence.
This version provides an upper estimate that decreases both in `‖y‖` and `n`. See also
`HasFPowerSeriesOnBall.uniform_geometric_approx` for a weaker version. -/
theorem HasFPowerSeriesOnBall.uniform_geometric_approx' {r' : ℝ≥0}
(hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n := by
obtain ⟨a, ha, C, hC, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n, ‖p n‖ * (r' : ℝ) ^ n ≤ C * a ^ n :=
p.norm_mul_pow_le_mul_pow_of_lt_radius (h.trans_le hf.r_le)
refine' ⟨a, ha, C / (1 - a), div_pos hC (sub_pos.2 ha.2), fun y hy n => _⟩
have yr' : ‖y‖ < r' := by
rw [ball_zero_eq] at hy
exact hy
have hr'0 : 0 < (r' : ℝ) := (norm_nonneg _).trans_lt yr'
have : y ∈ EMetric.ball (0 : E) r := by
refine' mem_emetric_ball_zero_iff.2 (lt_trans _ h)
exact mod_cast yr'
rw [norm_sub_rev, ← mul_div_right_comm]
have ya : a * (‖y‖ / ↑r') ≤ a :=
mul_le_of_le_one_right ha.1.le (div_le_one_of_le yr'.le r'.coe_nonneg)
suffices ‖p.partialSum n y - f (x + y)‖ ≤ C * (a * (‖y‖ / r')) ^ n / (1 - a * (‖y‖ / r')) by
refine' this.trans _
have : 0 < a := ha.1
gcongr
apply_rules [sub_pos.2, ha.2]
apply norm_sub_le_of_geometric_bound_of_hasSum (ya.trans_lt ha.2) _ (hf.hasSum this)
intro n
calc
‖(p n) fun _ : Fin n => y‖
_ ≤ ‖p n‖ * ∏ _i : Fin n, ‖y‖ := ContinuousMultilinearMap.le_op_norm _ _
_ = ‖p n‖ * (r' : ℝ) ^ n * (‖y‖ / r') ^ n := by field_simp [mul_right_comm]
_ ≤ C * a ^ n * (‖y‖ / r') ^ n := by gcongr ?_ * _; apply hp
_ ≤ C * (a * (‖y‖ / r')) ^ n := by rw [mul_pow, mul_assoc]
#align has_fpower_series_on_ball.uniform_geometric_approx' HasFPowerSeriesOnBall.uniform_geometric_approx'
/-- If a function admits a power series expansion, then it is exponentially close to the partial
sums of this power series on strict subdisks of the disk of convergence. -/
theorem HasFPowerSeriesOnBall.uniform_geometric_approx {r' : ℝ≥0}
(hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
∃ a ∈ Ioo (0 : ℝ) 1,
∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n, ‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n := by
obtain ⟨a, ha, C, hC, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n :=
hf.uniform_geometric_approx' h
refine' ⟨a, ha, C, hC, fun y hy n => (hp y hy n).trans _⟩
have yr' : ‖y‖ < r' := by rwa [ball_zero_eq] at hy
gcongr
exacts [mul_nonneg ha.1.le (div_nonneg (norm_nonneg y) r'.coe_nonneg),
mul_le_of_le_one_right ha.1.le (div_le_one_of_le yr'.le r'.coe_nonneg)]
#align has_fpower_series_on_ball.uniform_geometric_approx HasFPowerSeriesOnBall.uniform_geometric_approx
/-- Taylor formula for an analytic function, `IsBigO` version. -/
theorem HasFPowerSeriesAt.isBigO_sub_partialSum_pow (hf : HasFPowerSeriesAt f p x) (n : ℕ) :
(fun y : E => f (x + y) - p.partialSum n y) =O[𝓝 0] fun y => ‖y‖ ^ n := by
rcases hf with ⟨r, hf⟩
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hf.r_pos with ⟨r', r'0, h⟩
obtain ⟨a, -, C, -, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n :=
hf.uniform_geometric_approx' h
refine' isBigO_iff.2 ⟨C * (a / r') ^ n, _⟩
replace r'0 : 0 < (r' : ℝ); · exact mod_cast r'0
filter_upwards [Metric.ball_mem_nhds (0 : E) r'0] with y hy
simpa [mul_pow, mul_div_assoc, mul_assoc, div_mul_eq_mul_div] using hp y hy n
set_option linter.uppercaseLean3 false in
#align has_fpower_series_at.is_O_sub_partial_sum_pow HasFPowerSeriesAt.isBigO_sub_partialSum_pow
/-- If `f` has formal power series `∑ n, pₙ` on a ball of radius `r`, then for `y, z` in any smaller
ball, the norm of the difference `f y - f z - p 1 (fun _ ↦ y - z)` is bounded above by
`C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖`. This lemma formulates this property using `IsBigO` and
`Filter.principal` on `E × E`. -/
theorem HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal
(hf : HasFPowerSeriesOnBall f p x r) (hr : r' < r) :
(fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) =O[𝓟 (EMetric.ball (x, x) r')]
fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖ := by
lift r' to ℝ≥0 using ne_top_of_lt hr
rcases (zero_le r').eq_or_lt with (rfl | hr'0)
· simp only [isBigO_bot, EMetric.ball_zero, principal_empty, ENNReal.coe_zero]
obtain ⟨a, ha, C, hC : 0 < C, hp⟩ :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n : ℕ, ‖p n‖ * (r' : ℝ) ^ n ≤ C * a ^ n
exact p.norm_mul_pow_le_mul_pow_of_lt_radius (hr.trans_le hf.r_le)
simp only [← le_div_iff (pow_pos (NNReal.coe_pos.2 hr'0) _)] at hp
set L : E × E → ℝ := fun y =>
C * (a / r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * (a / (1 - a) ^ 2 + 2 / (1 - a))
have hL : ∀ y ∈ EMetric.ball (x, x) r', ‖f y.1 - f y.2 - p 1 fun _ => y.1 - y.2‖ ≤ L y := by
intro y hy'
have hy : y ∈ EMetric.ball x r ×ˢ EMetric.ball x r := by
rw [EMetric.ball_prod_same]
exact EMetric.ball_subset_ball hr.le hy'
set A : ℕ → F := fun n => (p n fun _ => y.1 - x) - p n fun _ => y.2 - x
have hA : HasSum (fun n => A (n + 2)) (f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) := by
convert (hasSum_nat_add_iff' 2).2 ((hf.hasSum_sub hy.1).sub (hf.hasSum_sub hy.2)) using 1
rw [Finset.sum_range_succ, Finset.sum_range_one, hf.coeff_zero, hf.coeff_zero, sub_self,
zero_add, ← Subsingleton.pi_single_eq (0 : Fin 1) (y.1 - x), Pi.single,
← Subsingleton.pi_single_eq (0 : Fin 1) (y.2 - x), Pi.single, ← (p 1).map_sub, ← Pi.single,
Subsingleton.pi_single_eq, sub_sub_sub_cancel_right]
rw [EMetric.mem_ball, edist_eq_coe_nnnorm_sub, ENNReal.coe_lt_coe] at hy'
set B : ℕ → ℝ := fun n => C * (a / r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * ((n + 2) * a ^ n)
have hAB : ∀ n, ‖A (n + 2)‖ ≤ B n := fun n =>
calc
‖A (n + 2)‖ ≤ ‖p (n + 2)‖ * ↑(n + 2) * ‖y - (x, x)‖ ^ (n + 1) * ‖y.1 - y.2‖ := by
-- porting note: `pi_norm_const` was `pi_norm_const (_ : E)`
simpa only [Fintype.card_fin, pi_norm_const, Prod.norm_def, Pi.sub_def,
Prod.fst_sub, Prod.snd_sub, sub_sub_sub_cancel_right] using
(p <| n + 2).norm_image_sub_le (fun _ => y.1 - x) fun _ => y.2 - x
_ = ‖p (n + 2)‖ * ‖y - (x, x)‖ ^ n * (↑(n + 2) * ‖y - (x, x)‖ * ‖y.1 - y.2‖) := by
rw [pow_succ ‖y - (x, x)‖]
ring
-- porting note: the two `↑` in `↑r'` are new, without them, Lean fails to synthesize
-- instances `HDiv ℝ ℝ≥0 ?m` or `HMul ℝ ℝ≥0 ?m`
_ ≤ C * a ^ (n + 2) / ↑r' ^ (n + 2)
* ↑r' ^ n * (↑(n + 2) * ‖y - (x, x)‖ * ‖y.1 - y.2‖) := by
have : 0 < a := ha.1
gcongr
· apply hp
· apply hy'.le
_ = B n := by
-- porting note: in the original, `B` was in the `field_simp`, but now Lean does not
-- accept it. The current proof works in Lean 4, but does not in Lean 3.
field_simp [pow_succ]
simp only [mul_assoc, mul_comm, mul_left_comm]
have hBL : HasSum B (L y) := by
apply HasSum.mul_left
simp only [add_mul]
have : ‖a‖ < 1 := by simp only [Real.norm_eq_abs, abs_of_pos ha.1, ha.2]
rw [div_eq_mul_inv, div_eq_mul_inv]
exact (hasSum_coe_mul_geometric_of_norm_lt_1 this).add -- porting note: was `convert`!
((hasSum_geometric_of_norm_lt_1 this).mul_left 2)
exact hA.norm_le_of_bounded hBL hAB
suffices L =O[𝓟 (EMetric.ball (x, x) r')] fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖ by
refine' (IsBigO.of_bound 1 (eventually_principal.2 fun y hy => _)).trans this
rw [one_mul]
exact (hL y hy).trans (le_abs_self _)
simp_rw [mul_right_comm _ (_ * _)] -- porting note: there was an `L` inside the `simp_rw`.
exact (isBigO_refl _ _).const_mul_left _
set_option linter.uppercaseLean3 false in
#align has_fpower_series_on_ball.is_O_image_sub_image_sub_deriv_principal HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal
/-- If `f` has formal power series `∑ n, pₙ` on a ball of radius `r`, then for `y, z` in any smaller
ball, the norm of the difference `f y - f z - p 1 (fun _ ↦ y - z)` is bounded above by
`C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖`. -/
theorem HasFPowerSeriesOnBall.image_sub_sub_deriv_le (hf : HasFPowerSeriesOnBall f p x r)
(hr : r' < r) :
∃ C, ∀ᵉ (y ∈ EMetric.ball x r') (z ∈ EMetric.ball x r'),
‖f y - f z - p 1 fun _ => y - z‖ ≤ C * max ‖y - x‖ ‖z - x‖ * ‖y - z‖ := by
simpa only [isBigO_principal, mul_assoc, norm_mul, norm_norm, Prod.forall, EMetric.mem_ball,
Prod.edist_eq, max_lt_iff, and_imp, @forall_swap (_ < _) E] using
hf.isBigO_image_sub_image_sub_deriv_principal hr
#align has_fpower_series_on_ball.image_sub_sub_deriv_le HasFPowerSeriesOnBall.image_sub_sub_deriv_le
/-- If `f` has formal power series `∑ n, pₙ` at `x`, then
`f y - f z - p 1 (fun _ ↦ y - z) = O(‖(y, z) - (x, x)‖ * ‖y - z‖)` as `(y, z) → (x, x)`.
In particular, `f` is strictly differentiable at `x`. -/
theorem HasFPowerSeriesAt.isBigO_image_sub_norm_mul_norm_sub (hf : HasFPowerSeriesAt f p x) :
(fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) =O[𝓝 (x, x)] fun y =>
‖y - (x, x)‖ * ‖y.1 - y.2‖ := by
rcases hf with ⟨r, hf⟩
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hf.r_pos with ⟨r', r'0, h⟩
refine' (hf.isBigO_image_sub_image_sub_deriv_principal h).mono _
exact le_principal_iff.2 (EMetric.ball_mem_nhds _ r'0)
set_option linter.uppercaseLean3 false in
#align has_fpower_series_at.is_O_image_sub_norm_mul_norm_sub HasFPowerSeriesAt.isBigO_image_sub_norm_mul_norm_sub
/-- If a function admits a power series expansion at `x`, then it is the uniform limit of the
partial sums of this power series on strict subdisks of the disk of convergence, i.e., `f (x + y)`
is the uniform limit of `p.partialSum n y` there. -/
theorem HasFPowerSeriesOnBall.tendstoUniformlyOn {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r)
(h : (r' : ℝ≥0∞) < r) :
TendstoUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
(Metric.ball (0 : E) r') := by
obtain ⟨a, ha, C, -, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n
exact hf.uniform_geometric_approx h
refine' Metric.tendstoUniformlyOn_iff.2 fun ε εpos => _
have L : Tendsto (fun n => (C : ℝ) * a ^ n) atTop (𝓝 ((C : ℝ) * 0)) :=
tendsto_const_nhds.mul (tendsto_pow_atTop_nhds_0_of_lt_1 ha.1.le ha.2)
rw [mul_zero] at L
refine' (L.eventually (gt_mem_nhds εpos)).mono fun n hn y hy => _
rw [dist_eq_norm]
exact (hp y hy n).trans_lt hn
#align has_fpower_series_on_ball.tendsto_uniformly_on HasFPowerSeriesOnBall.tendstoUniformlyOn
/-- If a function admits a power series expansion at `x`, then it is the locally uniform limit of
the partial sums of this power series on the disk of convergence, i.e., `f (x + y)`
is the locally uniform limit of `p.partialSum n y` there. -/
theorem HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn (hf : HasFPowerSeriesOnBall f p x r) :
TendstoLocallyUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
(EMetric.ball (0 : E) r) := by
intro u hu x hx
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hx with ⟨r', xr', hr'⟩
have : EMetric.ball (0 : E) r' ∈ 𝓝 x := IsOpen.mem_nhds EMetric.isOpen_ball xr'
refine' ⟨EMetric.ball (0 : E) r', mem_nhdsWithin_of_mem_nhds this, _⟩
simpa [Metric.emetric_ball_nnreal] using hf.tendstoUniformlyOn hr' u hu
#align has_fpower_series_on_ball.tendsto_locally_uniformly_on HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn
/-- If a function admits a power series expansion at `x`, then it is the uniform limit of the
partial sums of this power series on strict subdisks of the disk of convergence, i.e., `f y`
is the uniform limit of `p.partialSum n (y - x)` there. -/
theorem HasFPowerSeriesOnBall.tendstoUniformlyOn' {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r)
(h : (r' : ℝ≥0∞) < r) :
TendstoUniformlyOn (fun n y => p.partialSum n (y - x)) f atTop (Metric.ball (x : E) r') := by
convert (hf.tendstoUniformlyOn h).comp fun y => y - x using 1
· simp [(· ∘ ·)]
· ext z
simp [dist_eq_norm]
#align has_fpower_series_on_ball.tendsto_uniformly_on' HasFPowerSeriesOnBall.tendstoUniformlyOn'
/-- If a function admits a power series expansion at `x`, then it is the locally uniform limit of
the partial sums of this power series on the disk of convergence, i.e., `f y`
is the locally uniform limit of `p.partialSum n (y - x)` there. -/
theorem HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn' (hf : HasFPowerSeriesOnBall f p x r) :
TendstoLocallyUniformlyOn (fun n y => p.partialSum n (y - x)) f atTop
(EMetric.ball (x : E) r) := by
have A : ContinuousOn (fun y : E => y - x) (EMetric.ball (x : E) r) :=
(continuous_id.sub continuous_const).continuousOn
convert hf.tendstoLocallyUniformlyOn.comp (fun y : E => y - x) _ A using 1
· ext z
simp
· intro z
simp [edist_eq_coe_nnnorm, edist_eq_coe_nnnorm_sub]
#align has_fpower_series_on_ball.tendsto_locally_uniformly_on' HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn'
/-- If a function admits a power series expansion on a disk, then it is continuous there. -/
protected theorem HasFPowerSeriesOnBall.continuousOn (hf : HasFPowerSeriesOnBall f p x r) :
ContinuousOn f (EMetric.ball x r) :=
hf.tendstoLocallyUniformlyOn'.continuousOn <|
eventually_of_forall fun n =>
((p.partialSum_continuous n).comp (continuous_id.sub continuous_const)).continuousOn
#align has_fpower_series_on_ball.continuous_on HasFPowerSeriesOnBall.continuousOn
protected theorem HasFPowerSeriesAt.continuousAt (hf : HasFPowerSeriesAt f p x) :
ContinuousAt f x :=
let ⟨_, hr⟩ := hf
hr.continuousOn.continuousAt (EMetric.ball_mem_nhds x hr.r_pos)
#align has_fpower_series_at.continuous_at HasFPowerSeriesAt.continuousAt
protected theorem AnalyticAt.continuousAt (hf : AnalyticAt 𝕜 f x) : ContinuousAt f x :=
let ⟨_, hp⟩ := hf
hp.continuousAt
#align analytic_at.continuous_at AnalyticAt.continuousAt
protected theorem AnalyticOn.continuousOn {s : Set E} (hf : AnalyticOn 𝕜 f s) : ContinuousOn f s :=
fun x hx => (hf x hx).continuousAt.continuousWithinAt
#align analytic_on.continuous_on AnalyticOn.continuousOn
/-- Analytic everywhere implies continuous -/
theorem AnalyticOn.continuous {f : E → F} (fa : AnalyticOn 𝕜 f univ) : Continuous f := by
rw [continuous_iff_continuousOn_univ]; exact fa.continuousOn
/-- In a complete space, the sum of a converging power series `p` admits `p` as a power series.
This is not totally obvious as we need to check the convergence of the series. -/
protected theorem FormalMultilinearSeries.hasFPowerSeriesOnBall [CompleteSpace F]
(p : FormalMultilinearSeries 𝕜 E F) (h : 0 < p.radius) :
HasFPowerSeriesOnBall p.sum p 0 p.radius :=
{ r_le := le_rfl
r_pos := h
hasSum := fun hy => by
rw [zero_add]
exact p.hasSum hy }
#align formal_multilinear_series.has_fpower_series_on_ball FormalMultilinearSeries.hasFPowerSeriesOnBall
theorem HasFPowerSeriesOnBall.sum (h : HasFPowerSeriesOnBall f p x r) {y : E}
(hy : y ∈ EMetric.ball (0 : E) r) : f (x + y) = p.sum y :=
(h.hasSum hy).tsum_eq.symm
#align has_fpower_series_on_ball.sum HasFPowerSeriesOnBall.sum
/-- The sum of a converging power series is continuous in its disk of convergence. -/
protected theorem FormalMultilinearSeries.continuousOn [CompleteSpace F] :
ContinuousOn p.sum (EMetric.ball 0 p.radius) := by
rcases (zero_le p.radius).eq_or_lt with h | h
· simp [← h, continuousOn_empty]
· exact (p.hasFPowerSeriesOnBall h).continuousOn
#align formal_multilinear_series.continuous_on FormalMultilinearSeries.continuousOn
end
/-!
### Uniqueness of power series
If a function `f : E → F` has two representations as power series at a point `x : E`, corresponding
to formal multilinear series `p₁` and `p₂`, then these representations agree term-by-term. That is,
for any `n : ℕ` and `y : E`, `p₁ n (fun i ↦ y) = p₂ n (fun i ↦ y)`. In the one-dimensional case,
when `f : 𝕜 → E`, the continuous multilinear maps `p₁ n` and `p₂ n` are given by
`ContinuousMultilinearMap.mkPiField`, and hence are determined completely by the value of
`p₁ n (fun i ↦ 1)`, so `p₁ = p₂`. Consequently, the radius of convergence for one series can be
transferred to the other.
-/
section Uniqueness
open ContinuousMultilinearMap
theorem Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero {n : ℕ} {p : E[×n]→L[𝕜] F}
(h : (fun y => p fun _ => y) =O[𝓝 0] fun y => ‖y‖ ^ (n + 1)) (y : E) : (p fun _ => y) = 0 := by
obtain ⟨c, c_pos, hc⟩ := h.exists_pos
obtain ⟨t, ht, t_open, z_mem⟩ := eventually_nhds_iff.mp (isBigOWith_iff.mp hc)
obtain ⟨δ, δ_pos, δε⟩ := (Metric.isOpen_iff.mp t_open) 0 z_mem
clear h hc z_mem
cases' n with n
· exact norm_eq_zero.mp (by
-- porting note: the symmetric difference of the `simpa only` sets:
-- added `Nat.zero_eq, zero_add, pow_one`
-- removed `zero_pow', Ne.def, Nat.one_ne_zero, not_false_iff`
simpa only [Nat.zero_eq, fin0_apply_norm, norm_eq_zero, norm_zero, zero_add, pow_one,
mul_zero, norm_le_zero_iff] using ht 0 (δε (Metric.mem_ball_self δ_pos)))
· refine' Or.elim (Classical.em (y = 0))
(fun hy => by simpa only [hy] using p.map_zero) fun hy => _
replace hy := norm_pos_iff.mpr hy
refine' norm_eq_zero.mp (le_antisymm (le_of_forall_pos_le_add fun ε ε_pos => _) (norm_nonneg _))
have h₀ := _root_.mul_pos c_pos (pow_pos hy (n.succ + 1))
obtain ⟨k, k_pos, k_norm⟩ := NormedField.exists_norm_lt 𝕜
(lt_min (mul_pos δ_pos (inv_pos.mpr hy)) (mul_pos ε_pos (inv_pos.mpr h₀)))
have h₁ : ‖k • y‖ < δ := by
rw [norm_smul]
exact inv_mul_cancel_right₀ hy.ne.symm δ ▸
mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_left _ _)) hy
have h₂ :=
calc
‖p fun _ => k • y‖ ≤ c * ‖k • y‖ ^ (n.succ + 1) := by
-- porting note: now Lean wants `_root_.`
simpa only [norm_pow, _root_.norm_norm] using ht (k • y) (δε (mem_ball_zero_iff.mpr h₁))
--simpa only [norm_pow, norm_norm] using ht (k • y) (δε (mem_ball_zero_iff.mpr h₁))
_ = ‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))) := by
-- porting note: added `Nat.succ_eq_add_one` since otherwise `ring` does not conclude.
simp only [norm_smul, mul_pow, Nat.succ_eq_add_one]
-- porting note: removed `rw [pow_succ]`, since it now becomes superfluous.
ring
have h₃ : ‖k‖ * (c * ‖y‖ ^ (n.succ + 1)) < ε :=
inv_mul_cancel_right₀ h₀.ne.symm ε ▸
mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_right _ _)) h₀
calc
‖p fun _ => y‖ = ‖k⁻¹ ^ n.succ‖ * ‖p fun _ => k • y‖ := by
simpa only [inv_smul_smul₀ (norm_pos_iff.mp k_pos), norm_smul, Finset.prod_const,
Finset.card_fin] using
congr_arg norm (p.map_smul_univ (fun _ : Fin n.succ => k⁻¹) fun _ : Fin n.succ => k • y)
_ ≤ ‖k⁻¹ ^ n.succ‖ * (‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1)))) := by gcongr
_ = ‖(k⁻¹ * k) ^ n.succ‖ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))) := by
rw [← mul_assoc]
simp [norm_mul, mul_pow]
_ ≤ 0 + ε := by
rw [inv_mul_cancel (norm_pos_iff.mp k_pos)]
simpa using h₃.le
set_option linter.uppercaseLean3 false in
#align asymptotics.is_O.continuous_multilinear_map_apply_eq_zero Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero
/-- If a formal multilinear series `p` represents the zero function at `x : E`, then the
terms `p n (fun i ↦ y)` appearing in the sum are zero for any `n : ℕ`, `y : E`. -/
theorem HasFPowerSeriesAt.apply_eq_zero {p : FormalMultilinearSeries 𝕜 E F} {x : E}
(h : HasFPowerSeriesAt 0 p x) (n : ℕ) : ∀ y : E, (p n fun _ => y) = 0 := by
refine' Nat.strong_induction_on n fun k hk => _
have psum_eq : p.partialSum (k + 1) = fun y => p k fun _ => y := by
funext z
refine' Finset.sum_eq_single _ (fun b hb hnb => _) fun hn => _
· have := Finset.mem_range_succ_iff.mp hb
simp only [hk b (this.lt_of_ne hnb), Pi.zero_apply]
· exact False.elim (hn (Finset.mem_range.mpr (lt_add_one k)))
replace h := h.isBigO_sub_partialSum_pow k.succ
simp only [psum_eq, zero_sub, Pi.zero_apply, Asymptotics.isBigO_neg_left] at h
exact h.continuousMultilinearMap_apply_eq_zero
#align has_fpower_series_at.apply_eq_zero HasFPowerSeriesAt.apply_eq_zero
/-- A one-dimensional formal multilinear series representing the zero function is zero. -/
theorem HasFPowerSeriesAt.eq_zero {p : FormalMultilinearSeries 𝕜 𝕜 E} {x : 𝕜}
(h : HasFPowerSeriesAt 0 p x) : p = 0 := by
-- porting note: `funext; ext` was `ext (n x)`
funext n
ext x
rw [← mkPiField_apply_one_eq_self (p n)]
-- porting note: nasty hack, was `simp [h.apply_eq_zero n 1]`
have := Or.intro_right ?_ (h.apply_eq_zero n 1)
simpa using this
#align has_fpower_series_at.eq_zero HasFPowerSeriesAt.eq_zero
/-- One-dimensional formal multilinear series representing the same function are equal. -/
theorem HasFPowerSeriesAt.eq_formalMultilinearSeries {p₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E}
{f : 𝕜 → E} {x : 𝕜} (h₁ : HasFPowerSeriesAt f p₁ x) (h₂ : HasFPowerSeriesAt f p₂ x) : p₁ = p₂ :=
sub_eq_zero.mp (HasFPowerSeriesAt.eq_zero (by simpa only [sub_self] using h₁.sub h₂))
#align has_fpower_series_at.eq_formal_multilinear_series HasFPowerSeriesAt.eq_formalMultilinearSeries
theorem HasFPowerSeriesAt.eq_formalMultilinearSeries_of_eventually
{p q : FormalMultilinearSeries 𝕜 𝕜 E} {f g : 𝕜 → E} {x : 𝕜} (hp : HasFPowerSeriesAt f p x)
(hq : HasFPowerSeriesAt g q x) (heq : ∀ᶠ z in 𝓝 x, f z = g z) : p = q :=
(hp.congr heq).eq_formalMultilinearSeries hq
#align has_fpower_series_at.eq_formal_multilinear_series_of_eventually HasFPowerSeriesAt.eq_formalMultilinearSeries_of_eventually
/-- A one-dimensional formal multilinear series representing a locally zero function is zero. -/
theorem HasFPowerSeriesAt.eq_zero_of_eventually {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E}
{x : 𝕜} (hp : HasFPowerSeriesAt f p x) (hf : f =ᶠ[𝓝 x] 0) : p = 0 :=
(hp.congr hf).eq_zero
#align has_fpower_series_at.eq_zero_of_eventually HasFPowerSeriesAt.eq_zero_of_eventually
/-- If a function `f : 𝕜 → E` has two power series representations at `x`, then the given radii in
which convergence is guaranteed may be interchanged. This can be useful when the formal multilinear
series in one representation has a particularly nice form, but the other has a larger radius. -/
theorem HasFPowerSeriesOnBall.exchange_radius {p₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E}
{r₁ r₂ : ℝ≥0∞} {x : 𝕜} (h₁ : HasFPowerSeriesOnBall f p₁ x r₁)
(h₂ : HasFPowerSeriesOnBall f p₂ x r₂) : HasFPowerSeriesOnBall f p₁ x r₂ :=
h₂.hasFPowerSeriesAt.eq_formalMultilinearSeries h₁.hasFPowerSeriesAt ▸ h₂
#align has_fpower_series_on_ball.exchange_radius HasFPowerSeriesOnBall.exchange_radius
/-- If a function `f : 𝕜 → E` has power series representation `p` on a ball of some radius and for
each positive radius it has some power series representation, then `p` converges to `f` on the whole
`𝕜`. -/
theorem HasFPowerSeriesOnBall.r_eq_top_of_exists {f : 𝕜 → E} {r : ℝ≥0∞} {x : 𝕜}
{p : FormalMultilinearSeries 𝕜 𝕜 E} (h : HasFPowerSeriesOnBall f p x r)
(h' : ∀ (r' : ℝ≥0) (_ : 0 < r'), ∃ p' : FormalMultilinearSeries 𝕜 𝕜 E,
HasFPowerSeriesOnBall f p' x r') :
HasFPowerSeriesOnBall f p x ∞ :=
{ r_le := ENNReal.le_of_forall_pos_nnreal_lt fun r hr _ =>
let ⟨_, hp'⟩ := h' r hr
(h.exchange_radius hp').r_le
r_pos := ENNReal.coe_lt_top
hasSum := fun {y} _ =>
let ⟨r', hr'⟩ := exists_gt ‖y‖₊
let ⟨_, hp'⟩ := h' r' hr'.ne_bot.bot_lt
(h.exchange_radius hp').hasSum <| mem_emetric_ball_zero_iff.mpr (ENNReal.coe_lt_coe.2 hr') }
#align has_fpower_series_on_ball.r_eq_top_of_exists HasFPowerSeriesOnBall.r_eq_top_of_exists
end Uniqueness
/-!
### Changing origin in a power series
If a function is analytic in a disk `D(x, R)`, then it is analytic in any disk contained in that
one. Indeed, one can write
$$
f (x + y + z) = \sum_{n} p_n (y + z)^n = \sum_{n, k} \binom{n}{k} p_n y^{n-k} z^k
= \sum_{k} \Bigl(\sum_{n} \binom{n}{k} p_n y^{n-k}\Bigr) z^k.
$$
The corresponding power series has thus a `k`-th coefficient equal to
$\sum_{n} \binom{n}{k} p_n y^{n-k}$. In the general case where `pₙ` is a multilinear map, this has
to be interpreted suitably: instead of having a binomial coefficient, one should sum over all
possible subsets `s` of `Fin n` of cardinal `k`, and attribute `z` to the indices in `s` and
`y` to the indices outside of `s`.
In this paragraph, we implement this. The new power series is called `p.changeOrigin y`. Then, we
check its convergence and the fact that its sum coincides with the original sum. The outcome of this
discussion is that the set of points where a function is analytic is open.
-/
namespace FormalMultilinearSeries
section
variable (p : FormalMultilinearSeries 𝕜 E F) {x y : E} {r R : ℝ≥0}
/-- A term of `FormalMultilinearSeries.changeOriginSeries`.
Given a formal multilinear series `p` and a point `x` in its ball of convergence,
`p.changeOrigin x` is a formal multilinear series such that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense. Each term of `p.changeOrigin x`
is itself an analytic function of `x` given by the series `p.changeOriginSeries`. Each term in
`changeOriginSeries` is the sum of `changeOriginSeriesTerm`'s over all `s` of cardinality `l`.
The definition is such that `p.changeOriginSeriesTerm k l s hs (fun _ ↦ x) (fun _ ↦ y) =
p (k + l) (s.piecewise (fun _ ↦ x) (fun _ ↦ y))`
-/
def changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
E[×l]→L[𝕜] E[×k]→L[𝕜] F := by
let a := ContinuousMultilinearMap.curryFinFinset 𝕜 E F hs
(by erw [Finset.card_compl, Fintype.card_fin, hs, add_tsub_cancel_right])
exact a (p (k + l))
#align formal_multilinear_series.change_origin_series_term FormalMultilinearSeries.changeOriginSeriesTerm
theorem changeOriginSeriesTerm_apply (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l)
(x y : E) :
(p.changeOriginSeriesTerm k l s hs (fun _ => x) fun _ => y) =
p (k + l) (s.piecewise (fun _ => x) fun _ => y) :=
ContinuousMultilinearMap.curryFinFinset_apply_const _ _ _ _ _
#align formal_multilinear_series.change_origin_series_term_apply FormalMultilinearSeries.changeOriginSeriesTerm_apply
@[simp]
theorem norm_changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
‖p.changeOriginSeriesTerm k l s hs‖ = ‖p (k + l)‖ := by
simp only [changeOriginSeriesTerm, LinearIsometryEquiv.norm_map]
#align formal_multilinear_series.norm_change_origin_series_term FormalMultilinearSeries.norm_changeOriginSeriesTerm
@[simp]
theorem nnnorm_changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
‖p.changeOriginSeriesTerm k l s hs‖₊ = ‖p (k + l)‖₊ := by
simp only [changeOriginSeriesTerm, LinearIsometryEquiv.nnnorm_map]
#align formal_multilinear_series.nnnorm_change_origin_series_term FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm
theorem nnnorm_changeOriginSeriesTerm_apply_le (k l : ℕ) (s : Finset (Fin (k + l)))
(hs : s.card = l) (x y : E) :
‖p.changeOriginSeriesTerm k l s hs (fun _ => x) fun _ => y‖₊ ≤
‖p (k + l)‖₊ * ‖x‖₊ ^ l * ‖y‖₊ ^ k := by
rw [← p.nnnorm_changeOriginSeriesTerm k l s hs, ← Fin.prod_const, ← Fin.prod_const]
apply ContinuousMultilinearMap.le_of_op_nnnorm_le
apply ContinuousMultilinearMap.le_op_nnnorm
#align formal_multilinear_series.nnnorm_change_origin_series_term_apply_le FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm_apply_le
/-- The power series for `f.changeOrigin k`.
Given a formal multilinear series `p` and a point `x` in its ball of convergence,
`p.changeOrigin x` is a formal multilinear series such that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense. Its `k`-th term is the sum of
the series `p.changeOriginSeries k`. -/
def changeOriginSeries (k : ℕ) : FormalMultilinearSeries 𝕜 E (E[×k]→L[𝕜] F) := fun l =>
∑ s : { s : Finset (Fin (k + l)) // Finset.card s = l }, p.changeOriginSeriesTerm k l s s.2
#align formal_multilinear_series.change_origin_series FormalMultilinearSeries.changeOriginSeries
theorem nnnorm_changeOriginSeries_le_tsum (k l : ℕ) :
‖p.changeOriginSeries k l‖₊ ≤
∑' _ : { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + l)‖₊ :=
(nnnorm_sum_le _ (fun t => changeOriginSeriesTerm p k l (Subtype.val t) t.prop)).trans_eq <| by
simp_rw [tsum_fintype, nnnorm_changeOriginSeriesTerm (p := p) (k := k) (l := l)]
#align formal_multilinear_series.nnnorm_change_origin_series_le_tsum FormalMultilinearSeries.nnnorm_changeOriginSeries_le_tsum
theorem nnnorm_changeOriginSeries_apply_le_tsum (k l : ℕ) (x : E) :
‖p.changeOriginSeries k l fun _ => x‖₊ ≤
∑' _ : { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + l)‖₊ * ‖x‖₊ ^ l := by
rw [NNReal.tsum_mul_right, ← Fin.prod_const]
exact (p.changeOriginSeries k l).le_of_op_nnnorm_le _ (p.nnnorm_changeOriginSeries_le_tsum _ _)
#align formal_multilinear_series.nnnorm_change_origin_series_apply_le_tsum FormalMultilinearSeries.nnnorm_changeOriginSeries_apply_le_tsum
/-- Changing the origin of a formal multilinear series `p`, so that
`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense.
-/
def changeOrigin (x : E) : FormalMultilinearSeries 𝕜 E F :=
fun k => (p.changeOriginSeries k).sum x
#align formal_multilinear_series.change_origin FormalMultilinearSeries.changeOrigin
/-- An auxiliary equivalence useful in the proofs about
`FormalMultilinearSeries.changeOriginSeries`: the set of triples `(k, l, s)`, where `s` is a
`Finset (Fin (k + l))` of cardinality `l` is equivalent to the set of pairs `(n, s)`, where `s` is a
`Finset (Fin n)`.
The forward map sends `(k, l, s)` to `(k + l, s)` and the inverse map sends `(n, s)` to
`(n - Finset.card s, Finset.card s, s)`. The actual definition is less readable because of problems
with non-definitional equalities. -/
@[simps]
def changeOriginIndexEquiv :
(Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) ≃ Σn : ℕ, Finset (Fin n) where
toFun s := ⟨s.1 + s.2.1, s.2.2⟩
invFun s :=
⟨s.1 - s.2.card, s.2.card,
⟨s.2.map
(Fin.castIso <| (tsub_add_cancel_of_le <| card_finset_fin_le s.2).symm).toEquiv.toEmbedding,
Finset.card_map _⟩⟩
left_inv := by
rintro ⟨k, l, ⟨s : Finset (Fin <| k + l), hs : s.card = l⟩⟩
dsimp only [Subtype.coe_mk]
-- Lean can't automatically generalize `k' = k + l - s.card`, `l' = s.card`, so we explicitly
-- formulate the generalized goal
suffices ∀ k' l', k' = k → l' = l → ∀ (hkl : k + l = k' + l') (hs'),
(⟨k', l', ⟨Finset.map (Fin.castIso hkl).toEquiv.toEmbedding s, hs'⟩⟩ :
Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) = ⟨k, l, ⟨s, hs⟩⟩ by
apply this <;> simp only [hs, add_tsub_cancel_right]
rintro _ _ rfl rfl hkl hs'
simp only [Equiv.refl_toEmbedding, Fin.castIso_refl, Finset.map_refl, eq_self_iff_true,
OrderIso.refl_toEquiv, and_self_iff, heq_iff_eq]
right_inv := by
rintro ⟨n, s⟩
simp [tsub_add_cancel_of_le (card_finset_fin_le s), Fin.castIso_to_equiv]
#align formal_multilinear_series.change_origin_index_equiv FormalMultilinearSeries.changeOriginIndexEquiv
theorem changeOriginSeries_summable_aux₁ {r r' : ℝ≥0} (hr : (r + r' : ℝ≥0∞) < p.radius) :
Summable fun s : Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l } =>
‖p (s.1 + s.2.1)‖₊ * r ^ s.2.1 * r' ^ s.1 := by
rw [← changeOriginIndexEquiv.symm.summable_iff]
dsimp only [Function.comp_def, changeOriginIndexEquiv_symm_apply_fst,
changeOriginIndexEquiv_symm_apply_snd_fst]
have : ∀ n : ℕ,
HasSum (fun s : Finset (Fin n) => ‖p (n - s.card + s.card)‖₊ * r ^ s.card * r' ^ (n - s.card))
(‖p n‖₊ * (r + r') ^ n) := by
intro n
-- TODO: why `simp only [tsub_add_cancel_of_le (card_finset_fin_le _)]` fails?
convert_to HasSum (fun s : Finset (Fin n) => ‖p n‖₊ * (r ^ s.card * r' ^ (n - s.card))) _
· ext1 s
rw [tsub_add_cancel_of_le (card_finset_fin_le _), mul_assoc]
rw [← Fin.sum_pow_mul_eq_add_pow]
exact (hasSum_fintype _).mul_left _
refine' NNReal.summable_sigma.2 ⟨fun n => (this n).summable, _⟩
simp only [(this _).tsum_eq]
exact p.summable_nnnorm_mul_pow hr
#align formal_multilinear_series.change_origin_series_summable_aux₁ FormalMultilinearSeries.changeOriginSeries_summable_aux₁
theorem changeOriginSeries_summable_aux₂ (hr : (r : ℝ≥0∞) < p.radius) (k : ℕ) :
Summable fun s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l } =>
‖p (k + s.1)‖₊ * r ^ s.1 := by
rcases ENNReal.lt_iff_exists_add_pos_lt.1 hr with ⟨r', h0, hr'⟩
simpa only [mul_inv_cancel_right₀ (pow_pos h0 _).ne'] using
((NNReal.summable_sigma.1 (p.changeOriginSeries_summable_aux₁ hr')).1 k).mul_right (r' ^ k)⁻¹
#align formal_multilinear_series.change_origin_series_summable_aux₂ FormalMultilinearSeries.changeOriginSeries_summable_aux₂
theorem changeOriginSeries_summable_aux₃ {r : ℝ≥0} (hr : ↑r < p.radius) (k : ℕ) :
Summable fun l : ℕ => ‖p.changeOriginSeries k l‖₊ * r ^ l := by
refine' NNReal.summable_of_le
(fun n => _) (NNReal.summable_sigma.1 <| p.changeOriginSeries_summable_aux₂ hr k).2
simp only [NNReal.tsum_mul_right]
exact mul_le_mul' (p.nnnorm_changeOriginSeries_le_tsum _ _) le_rfl
#align formal_multilinear_series.change_origin_series_summable_aux₃ FormalMultilinearSeries.changeOriginSeries_summable_aux₃
theorem le_changeOriginSeries_radius (k : ℕ) : p.radius ≤ (p.changeOriginSeries k).radius :=
ENNReal.le_of_forall_nnreal_lt fun _r hr =>
le_radius_of_summable_nnnorm _ (p.changeOriginSeries_summable_aux₃ hr k)
#align formal_multilinear_series.le_change_origin_series_radius FormalMultilinearSeries.le_changeOriginSeries_radius
theorem nnnorm_changeOrigin_le (k : ℕ) (h : (‖x‖₊ : ℝ≥0∞) < p.radius) :
‖p.changeOrigin x k‖₊ ≤
∑' s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + s.1)‖₊ * ‖x‖₊ ^ s.1 := by
refine' tsum_of_nnnorm_bounded _ fun l => p.nnnorm_changeOriginSeries_apply_le_tsum k l x
have := p.changeOriginSeries_summable_aux₂ h k
refine' HasSum.sigma this.hasSum fun l => _
exact ((NNReal.summable_sigma.1 this).1 l).hasSum
#align formal_multilinear_series.nnnorm_change_origin_le FormalMultilinearSeries.nnnorm_changeOrigin_le
/-- The radius of convergence of `p.changeOrigin x` is at least `p.radius - ‖x‖`. In other words,
`p.changeOrigin x` is well defined on the largest ball contained in the original ball of
convergence. -/
theorem changeOrigin_radius : p.radius - ‖x‖₊ ≤ (p.changeOrigin x).radius := by
refine' ENNReal.le_of_forall_pos_nnreal_lt fun r _h0 hr => _
rw [lt_tsub_iff_right, add_comm] at hr
have hr' : (‖x‖₊ : ℝ≥0∞) < p.radius := (le_add_right le_rfl).trans_lt hr
apply le_radius_of_summable_nnnorm
have : ∀ k : ℕ,
‖p.changeOrigin x k‖₊ * r ^ k ≤
(∑' s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + s.1)‖₊ * ‖x‖₊ ^ s.1) *
r ^ k :=
fun k => mul_le_mul_right' (p.nnnorm_changeOrigin_le k hr') (r ^ k)
refine' NNReal.summable_of_le this _
simpa only [← NNReal.tsum_mul_right] using
(NNReal.summable_sigma.1 (p.changeOriginSeries_summable_aux₁ hr)).2
#align formal_multilinear_series.change_origin_radius FormalMultilinearSeries.changeOrigin_radius
end
-- From this point on, assume that the space is complete, to make sure that series that converge
-- in norm also converge in `F`.
variable [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x y : E} {r R : ℝ≥0}
theorem hasFPowerSeriesOnBall_changeOrigin (k : ℕ) (hr : 0 < p.radius) :
HasFPowerSeriesOnBall (fun x => p.changeOrigin x k) (p.changeOriginSeries k) 0 p.radius :=
have := p.le_changeOriginSeries_radius k
((p.changeOriginSeries k).hasFPowerSeriesOnBall (hr.trans_le this)).mono hr this
#align formal_multilinear_series.has_fpower_series_on_ball_change_origin FormalMultilinearSeries.hasFPowerSeriesOnBall_changeOrigin
/-- Summing the series `p.changeOrigin x` at a point `y` gives back `p (x + y)`. -/
theorem changeOrigin_eval (h : (‖x‖₊ + ‖y‖₊ : ℝ≥0∞) < p.radius) :
(p.changeOrigin x).sum y = p.sum (x + y) := by
have radius_pos : 0 < p.radius := lt_of_le_of_lt (zero_le _) h
have x_mem_ball : x ∈ EMetric.ball (0 : E) p.radius :=
mem_emetric_ball_zero_iff.2 ((le_add_right le_rfl).trans_lt h)
have y_mem_ball : y ∈ EMetric.ball (0 : E) (p.changeOrigin x).radius := by
refine' mem_emetric_ball_zero_iff.2 (lt_of_lt_of_le _ p.changeOrigin_radius)
rwa [lt_tsub_iff_right, add_comm]
have x_add_y_mem_ball : x + y ∈ EMetric.ball (0 : E) p.radius := by
refine' mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt _ h)
exact mod_cast nnnorm_add_le x y
set f : (Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) → F := fun s =>
p.changeOriginSeriesTerm s.1 s.2.1 s.2.2 s.2.2.2 (fun _ => x) fun _ => y
have hsf : Summable f := by
refine' .of_nnnorm_bounded _ (p.changeOriginSeries_summable_aux₁ h) _
rintro ⟨k, l, s, hs⟩
dsimp only [Subtype.coe_mk]
exact p.nnnorm_changeOriginSeriesTerm_apply_le _ _ _ _ _ _
have hf : HasSum f ((p.changeOrigin x).sum y) := by
refine' HasSum.sigma_of_hasSum ((p.changeOrigin x).summable y_mem_ball).hasSum (fun k => _) hsf
· dsimp only
refine' ContinuousMultilinearMap.hasSum_eval _ _
have := (p.hasFPowerSeriesOnBall_changeOrigin k radius_pos).hasSum x_mem_ball
rw [zero_add] at this
refine' HasSum.sigma_of_hasSum this (fun l => _) _
· simp only [changeOriginSeries, ContinuousMultilinearMap.sum_apply]
apply hasSum_fintype
· refine' .of_nnnorm_bounded _
(p.changeOriginSeries_summable_aux₂ (mem_emetric_ball_zero_iff.1 x_mem_ball) k) fun s => _
refine' (ContinuousMultilinearMap.le_op_nnnorm _ _).trans_eq _
simp
refine' hf.unique (changeOriginIndexEquiv.symm.hasSum_iff.1 _)
refine' HasSum.sigma_of_hasSum
(p.hasSum x_add_y_mem_ball) (fun n => _) (changeOriginIndexEquiv.symm.summable_iff.2 hsf)
erw [(p n).map_add_univ (fun _ => x) fun _ => y]
-- porting note: added explicit function
convert hasSum_fintype (fun c : Finset (Fin n) => f (changeOriginIndexEquiv.symm ⟨n, c⟩))
rename_i s _
dsimp only [changeOriginSeriesTerm, (· ∘ ·), changeOriginIndexEquiv_symm_apply_fst,
changeOriginIndexEquiv_symm_apply_snd_fst, changeOriginIndexEquiv_symm_apply_snd_snd_coe]
rw [ContinuousMultilinearMap.curryFinFinset_apply_const]
have : ∀ (m) (hm : n = m), p n (s.piecewise (fun _ => x) fun _ => y) =
p m ((s.map (Fin.castIso hm).toEquiv.toEmbedding).piecewise (fun _ => x) fun _ => y) := by
rintro m rfl
simp (config := { unfoldPartialApp := true }) [Finset.piecewise]
apply this
#align formal_multilinear_series.change_origin_eval FormalMultilinearSeries.changeOrigin_eval
/-- Power series terms are analytic as we vary the origin -/
theorem analyticAt_changeOrigin (p : FormalMultilinearSeries 𝕜 E F) (rp : p.radius > 0) (n : ℕ) :
AnalyticAt 𝕜 (fun x ↦ p.changeOrigin x n) 0 :=
(FormalMultilinearSeries.hasFPowerSeriesOnBall_changeOrigin p n rp).analyticAt
end FormalMultilinearSeries
section
variable [CompleteSpace F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x y : E} {r : ℝ≥0∞}
/-- If a function admits a power series expansion `p` on a ball `B (x, r)`, then it also admits a
power series on any subball of this ball (even with a different center), given by `p.changeOrigin`.
-/
theorem HasFPowerSeriesOnBall.changeOrigin (hf : HasFPowerSeriesOnBall f p x r)
(h : (‖y‖₊ : ℝ≥0∞) < r) : HasFPowerSeriesOnBall f (p.changeOrigin y) (x + y) (r - ‖y‖₊) :=
{ r_le := by
apply le_trans _ p.changeOrigin_radius
exact tsub_le_tsub hf.r_le le_rfl
r_pos := by simp [h]
hasSum := fun {z} hz => by
have : f (x + y + z) =
FormalMultilinearSeries.sum (FormalMultilinearSeries.changeOrigin p y) z := by
rw [mem_emetric_ball_zero_iff, lt_tsub_iff_right, add_comm] at hz
rw [p.changeOrigin_eval (hz.trans_le hf.r_le), add_assoc, hf.sum]
refine' mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt _ hz)
exact mod_cast nnnorm_add_le y z
rw [this]
apply (p.changeOrigin y).hasSum
refine' EMetric.ball_subset_ball (le_trans _ p.changeOrigin_radius) hz
exact tsub_le_tsub hf.r_le le_rfl }
#align has_fpower_series_on_ball.change_origin HasFPowerSeriesOnBall.changeOrigin
/-- If a function admits a power series expansion `p` on an open ball `B (x, r)`, then
it is analytic at every point of this ball. -/
theorem HasFPowerSeriesOnBall.analyticAt_of_mem (hf : HasFPowerSeriesOnBall f p x r)
(h : y ∈ EMetric.ball x r) : AnalyticAt 𝕜 f y := by
have : (‖y - x‖₊ : ℝ≥0∞) < r := by simpa [edist_eq_coe_nnnorm_sub] using h
have := hf.changeOrigin this
rw [add_sub_cancel'_right] at this
exact this.analyticAt
#align has_fpower_series_on_ball.analytic_at_of_mem HasFPowerSeriesOnBall.analyticAt_of_mem
theorem HasFPowerSeriesOnBall.analyticOn (hf : HasFPowerSeriesOnBall f p x r) :
AnalyticOn 𝕜 f (EMetric.ball x r) :=
fun _y hy => hf.analyticAt_of_mem hy
#align has_fpower_series_on_ball.analytic_on HasFPowerSeriesOnBall.analyticOn
variable (𝕜 f)
/-- For any function `f` from a normed vector space to a Banach space, the set of points `x` such
that `f` is analytic at `x` is open. -/
theorem isOpen_analyticAt : IsOpen { x | AnalyticAt 𝕜 f x } := by
rw [isOpen_iff_mem_nhds]
rintro x ⟨p, r, hr⟩
exact mem_of_superset (EMetric.ball_mem_nhds _ hr.r_pos) fun y hy => hr.analyticAt_of_mem hy
#align is_open_analytic_at isOpen_analyticAt
variable {𝕜}
theorem AnalyticAt.eventually_analyticAt {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∀ᶠ y in 𝓝 x, AnalyticAt 𝕜 f y :=
(isOpen_analyticAt 𝕜 f).mem_nhds h
theorem AnalyticAt.exists_mem_nhds_analyticOn {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∃ s ∈ 𝓝 x, AnalyticOn 𝕜 f s :=
h.eventually_analyticAt.exists_mem
/-- If we're analytic at a point, we're analytic in a nonempty ball -/
theorem AnalyticAt.exists_ball_analyticOn {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
∃ r : ℝ, 0 < r ∧ AnalyticOn 𝕜 f (Metric.ball x r) :=
Metric.isOpen_iff.mp (isOpen_analyticAt _ _) _ h
end
section
open FormalMultilinearSeries
variable {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E} {z₀ : 𝕜}
/-- A function `f : 𝕜 → E` has `p` as power series expansion at a point `z₀` iff it is the sum of
`p` in a neighborhood of `z₀`. This makes some proofs easier by hiding the fact that
`HasFPowerSeriesAt` depends on `p.radius`. -/
theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z)) := by
refine' ⟨fun ⟨r, _, r_pos, h⟩ =>
eventually_of_mem (EMetric.ball_mem_nhds 0 r_pos) fun _ => by simpa using h, _⟩
simp only [Metric.eventually_nhds_iff]
rintro ⟨r, r_pos, h⟩
refine' ⟨p.radius ⊓ r.toNNReal, by simp, _, _⟩
· simp only [r_pos.lt, lt_inf_iff, ENNReal.coe_pos, Real.toNNReal_pos, and_true_iff]
obtain ⟨z, z_pos, le_z⟩ := NormedField.exists_norm_lt 𝕜 r_pos.lt
have : (‖z‖₊ : ENNReal) ≤ p.radius := by
simp only [dist_zero_right] at h
apply FormalMultilinearSeries.le_radius_of_tendsto
convert tendsto_norm.comp (h le_z).summable.tendsto_atTop_zero
funext
simp [norm_smul, mul_comm]
refine' lt_of_lt_of_le _ this
simp only [ENNReal.coe_pos]
exact zero_lt_iff.mpr (nnnorm_ne_zero_iff.mpr (norm_pos_iff.mp z_pos))
· simp only [EMetric.mem_ball, lt_inf_iff, edist_lt_coe, apply_eq_pow_smul_coeff, and_imp,
dist_zero_right] at h ⊢
refine' fun {y} _ hyr => h _
simpa [nndist_eq_nnnorm, Real.lt_toNNReal_iff_coe_lt] using hyr
#align has_fpower_series_at_iff hasFPowerSeriesAt_iff
theorem hasFPowerSeriesAt_iff' :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 z₀, HasSum (fun n => (z - z₀) ^ n • p.coeff n) (f z) := by
rw [← map_add_left_nhds_zero, eventually_map, hasFPowerSeriesAt_iff]
|
simp_rw [add_sub_cancel']
|
theorem hasFPowerSeriesAt_iff' :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 z₀, HasSum (fun n => (z - z₀) ^ n • p.coeff n) (f z) := by
rw [← map_add_left_nhds_zero, eventually_map, hasFPowerSeriesAt_iff]
|
Mathlib.Analysis.Analytic.Basic.1457_0.jQw1fRSE1vGpOll
|
theorem hasFPowerSeriesAt_iff' :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 z₀, HasSum (fun n => (z - z₀) ^ n • p.coeff n) (f z)
|
Mathlib_Analysis_Analytic_Basic
|
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝ : CommRing R
x : 𝕎 R
k : ℕ
h : ∀ i < k + 1, coeff x i = 0
⊢ verschiebung (shift x (Nat.succ k)) = shift x k
|
/-
Copyright (c) 2022 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.Identities
#align_import ring_theory.witt_vector.domain from "leanprover-community/mathlib"@"b1d911acd60ab198808e853292106ee352b648ea"
/-!
# Witt vectors over a domain
This file builds to the proof `WittVector.instIsDomain`,
an instance that says if `R` is an integral domain, then so is `𝕎 R`.
It depends on the API around iterated applications
of `WittVector.verschiebung` and `WittVector.frobenius`
found in `Identities.lean`.
The [proof sketch](https://math.stackexchange.com/questions/4117247/ring-of-witt-vectors-over-an-integral-domain/4118723#4118723)
goes as follows:
any nonzero $x$ is an iterated application of $V$
to some vector $w_x$ whose 0th component is nonzero (`WittVector.verschiebung_nonzero`).
Known identities (`WittVector.iterate_verschiebung_mul`) allow us to transform
the product of two such $x$ and $y$
to the form $V^{m+n}\left(F^n(w_x) \cdot F^m(w_y)\right)$,
the 0th component of which must be nonzero.
## Main declarations
* `WittVector.iterate_verschiebung_mul_coeff` : an identity from [Haze09]
* `WittVector.instIsDomain`
-/
noncomputable section
open scoped Classical
namespace WittVector
open Function
variable {p : ℕ} {R : Type*}
local notation "𝕎" => WittVector p -- type as `\bbW`
/-!
## The `shift` operator
-/
/--
`WittVector.verschiebung` translates the entries of a Witt vector upward, inserting 0s in the gaps.
`WittVector.shift` does the opposite, removing the first entries.
This is mainly useful as an auxiliary construction for `WittVector.verschiebung_nonzero`.
-/
def shift (x : 𝕎 R) (n : ℕ) : 𝕎 R :=
@mk' p R fun i => x.coeff (n + i)
#align witt_vector.shift WittVector.shift
theorem shift_coeff (x : 𝕎 R) (n k : ℕ) : (x.shift n).coeff k = x.coeff (n + k) :=
rfl
#align witt_vector.shift_coeff WittVector.shift_coeff
variable [hp : Fact p.Prime] [CommRing R]
theorem verschiebung_shift (x : 𝕎 R) (k : ℕ) (h : ∀ i < k + 1, x.coeff i = 0) :
verschiebung (x.shift k.succ) = x.shift k := by
|
ext ⟨j⟩
|
theorem verschiebung_shift (x : 𝕎 R) (k : ℕ) (h : ∀ i < k + 1, x.coeff i = 0) :
verschiebung (x.shift k.succ) = x.shift k := by
|
Mathlib.RingTheory.WittVector.Domain.69_0.4uLlcZNQ2uiRcjJ
|
theorem verschiebung_shift (x : 𝕎 R) (k : ℕ) (h : ∀ i < k + 1, x.coeff i = 0) :
verschiebung (x.shift k.succ) = x.shift k
|
Mathlib_RingTheory_WittVector_Domain
|
case h.zero
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝ : CommRing R
x : 𝕎 R
k : ℕ
h : ∀ i < k + 1, coeff x i = 0
⊢ coeff (verschiebung (shift x (Nat.succ k))) Nat.zero = coeff (shift x k) Nat.zero
|
/-
Copyright (c) 2022 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.Identities
#align_import ring_theory.witt_vector.domain from "leanprover-community/mathlib"@"b1d911acd60ab198808e853292106ee352b648ea"
/-!
# Witt vectors over a domain
This file builds to the proof `WittVector.instIsDomain`,
an instance that says if `R` is an integral domain, then so is `𝕎 R`.
It depends on the API around iterated applications
of `WittVector.verschiebung` and `WittVector.frobenius`
found in `Identities.lean`.
The [proof sketch](https://math.stackexchange.com/questions/4117247/ring-of-witt-vectors-over-an-integral-domain/4118723#4118723)
goes as follows:
any nonzero $x$ is an iterated application of $V$
to some vector $w_x$ whose 0th component is nonzero (`WittVector.verschiebung_nonzero`).
Known identities (`WittVector.iterate_verschiebung_mul`) allow us to transform
the product of two such $x$ and $y$
to the form $V^{m+n}\left(F^n(w_x) \cdot F^m(w_y)\right)$,
the 0th component of which must be nonzero.
## Main declarations
* `WittVector.iterate_verschiebung_mul_coeff` : an identity from [Haze09]
* `WittVector.instIsDomain`
-/
noncomputable section
open scoped Classical
namespace WittVector
open Function
variable {p : ℕ} {R : Type*}
local notation "𝕎" => WittVector p -- type as `\bbW`
/-!
## The `shift` operator
-/
/--
`WittVector.verschiebung` translates the entries of a Witt vector upward, inserting 0s in the gaps.
`WittVector.shift` does the opposite, removing the first entries.
This is mainly useful as an auxiliary construction for `WittVector.verschiebung_nonzero`.
-/
def shift (x : 𝕎 R) (n : ℕ) : 𝕎 R :=
@mk' p R fun i => x.coeff (n + i)
#align witt_vector.shift WittVector.shift
theorem shift_coeff (x : 𝕎 R) (n k : ℕ) : (x.shift n).coeff k = x.coeff (n + k) :=
rfl
#align witt_vector.shift_coeff WittVector.shift_coeff
variable [hp : Fact p.Prime] [CommRing R]
theorem verschiebung_shift (x : 𝕎 R) (k : ℕ) (h : ∀ i < k + 1, x.coeff i = 0) :
verschiebung (x.shift k.succ) = x.shift k := by
ext ⟨j⟩
·
|
rw [verschiebung_coeff_zero, shift_coeff, h]
|
theorem verschiebung_shift (x : 𝕎 R) (k : ℕ) (h : ∀ i < k + 1, x.coeff i = 0) :
verschiebung (x.shift k.succ) = x.shift k := by
ext ⟨j⟩
·
|
Mathlib.RingTheory.WittVector.Domain.69_0.4uLlcZNQ2uiRcjJ
|
theorem verschiebung_shift (x : 𝕎 R) (k : ℕ) (h : ∀ i < k + 1, x.coeff i = 0) :
verschiebung (x.shift k.succ) = x.shift k
|
Mathlib_RingTheory_WittVector_Domain
|
case h.zero.a
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝ : CommRing R
x : 𝕎 R
k : ℕ
h : ∀ i < k + 1, coeff x i = 0
⊢ k + Nat.zero < k + 1
|
/-
Copyright (c) 2022 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.Identities
#align_import ring_theory.witt_vector.domain from "leanprover-community/mathlib"@"b1d911acd60ab198808e853292106ee352b648ea"
/-!
# Witt vectors over a domain
This file builds to the proof `WittVector.instIsDomain`,
an instance that says if `R` is an integral domain, then so is `𝕎 R`.
It depends on the API around iterated applications
of `WittVector.verschiebung` and `WittVector.frobenius`
found in `Identities.lean`.
The [proof sketch](https://math.stackexchange.com/questions/4117247/ring-of-witt-vectors-over-an-integral-domain/4118723#4118723)
goes as follows:
any nonzero $x$ is an iterated application of $V$
to some vector $w_x$ whose 0th component is nonzero (`WittVector.verschiebung_nonzero`).
Known identities (`WittVector.iterate_verschiebung_mul`) allow us to transform
the product of two such $x$ and $y$
to the form $V^{m+n}\left(F^n(w_x) \cdot F^m(w_y)\right)$,
the 0th component of which must be nonzero.
## Main declarations
* `WittVector.iterate_verschiebung_mul_coeff` : an identity from [Haze09]
* `WittVector.instIsDomain`
-/
noncomputable section
open scoped Classical
namespace WittVector
open Function
variable {p : ℕ} {R : Type*}
local notation "𝕎" => WittVector p -- type as `\bbW`
/-!
## The `shift` operator
-/
/--
`WittVector.verschiebung` translates the entries of a Witt vector upward, inserting 0s in the gaps.
`WittVector.shift` does the opposite, removing the first entries.
This is mainly useful as an auxiliary construction for `WittVector.verschiebung_nonzero`.
-/
def shift (x : 𝕎 R) (n : ℕ) : 𝕎 R :=
@mk' p R fun i => x.coeff (n + i)
#align witt_vector.shift WittVector.shift
theorem shift_coeff (x : 𝕎 R) (n k : ℕ) : (x.shift n).coeff k = x.coeff (n + k) :=
rfl
#align witt_vector.shift_coeff WittVector.shift_coeff
variable [hp : Fact p.Prime] [CommRing R]
theorem verschiebung_shift (x : 𝕎 R) (k : ℕ) (h : ∀ i < k + 1, x.coeff i = 0) :
verschiebung (x.shift k.succ) = x.shift k := by
ext ⟨j⟩
· rw [verschiebung_coeff_zero, shift_coeff, h]
|
apply Nat.lt_succ_self
|
theorem verschiebung_shift (x : 𝕎 R) (k : ℕ) (h : ∀ i < k + 1, x.coeff i = 0) :
verschiebung (x.shift k.succ) = x.shift k := by
ext ⟨j⟩
· rw [verschiebung_coeff_zero, shift_coeff, h]
|
Mathlib.RingTheory.WittVector.Domain.69_0.4uLlcZNQ2uiRcjJ
|
theorem verschiebung_shift (x : 𝕎 R) (k : ℕ) (h : ∀ i < k + 1, x.coeff i = 0) :
verschiebung (x.shift k.succ) = x.shift k
|
Mathlib_RingTheory_WittVector_Domain
|
case h.succ
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝ : CommRing R
x : 𝕎 R
k : ℕ
h : ∀ i < k + 1, coeff x i = 0
n✝ : ℕ
⊢ coeff (verschiebung (shift x (Nat.succ k))) (Nat.succ n✝) = coeff (shift x k) (Nat.succ n✝)
|
/-
Copyright (c) 2022 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.Identities
#align_import ring_theory.witt_vector.domain from "leanprover-community/mathlib"@"b1d911acd60ab198808e853292106ee352b648ea"
/-!
# Witt vectors over a domain
This file builds to the proof `WittVector.instIsDomain`,
an instance that says if `R` is an integral domain, then so is `𝕎 R`.
It depends on the API around iterated applications
of `WittVector.verschiebung` and `WittVector.frobenius`
found in `Identities.lean`.
The [proof sketch](https://math.stackexchange.com/questions/4117247/ring-of-witt-vectors-over-an-integral-domain/4118723#4118723)
goes as follows:
any nonzero $x$ is an iterated application of $V$
to some vector $w_x$ whose 0th component is nonzero (`WittVector.verschiebung_nonzero`).
Known identities (`WittVector.iterate_verschiebung_mul`) allow us to transform
the product of two such $x$ and $y$
to the form $V^{m+n}\left(F^n(w_x) \cdot F^m(w_y)\right)$,
the 0th component of which must be nonzero.
## Main declarations
* `WittVector.iterate_verschiebung_mul_coeff` : an identity from [Haze09]
* `WittVector.instIsDomain`
-/
noncomputable section
open scoped Classical
namespace WittVector
open Function
variable {p : ℕ} {R : Type*}
local notation "𝕎" => WittVector p -- type as `\bbW`
/-!
## The `shift` operator
-/
/--
`WittVector.verschiebung` translates the entries of a Witt vector upward, inserting 0s in the gaps.
`WittVector.shift` does the opposite, removing the first entries.
This is mainly useful as an auxiliary construction for `WittVector.verschiebung_nonzero`.
-/
def shift (x : 𝕎 R) (n : ℕ) : 𝕎 R :=
@mk' p R fun i => x.coeff (n + i)
#align witt_vector.shift WittVector.shift
theorem shift_coeff (x : 𝕎 R) (n k : ℕ) : (x.shift n).coeff k = x.coeff (n + k) :=
rfl
#align witt_vector.shift_coeff WittVector.shift_coeff
variable [hp : Fact p.Prime] [CommRing R]
theorem verschiebung_shift (x : 𝕎 R) (k : ℕ) (h : ∀ i < k + 1, x.coeff i = 0) :
verschiebung (x.shift k.succ) = x.shift k := by
ext ⟨j⟩
· rw [verschiebung_coeff_zero, shift_coeff, h]
apply Nat.lt_succ_self
·
|
simp only [verschiebung_coeff_succ, shift]
|
theorem verschiebung_shift (x : 𝕎 R) (k : ℕ) (h : ∀ i < k + 1, x.coeff i = 0) :
verschiebung (x.shift k.succ) = x.shift k := by
ext ⟨j⟩
· rw [verschiebung_coeff_zero, shift_coeff, h]
apply Nat.lt_succ_self
·
|
Mathlib.RingTheory.WittVector.Domain.69_0.4uLlcZNQ2uiRcjJ
|
theorem verschiebung_shift (x : 𝕎 R) (k : ℕ) (h : ∀ i < k + 1, x.coeff i = 0) :
verschiebung (x.shift k.succ) = x.shift k
|
Mathlib_RingTheory_WittVector_Domain
|
case h.succ
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝ : CommRing R
x : 𝕎 R
k : ℕ
h : ∀ i < k + 1, coeff x i = 0
n✝ : ℕ
⊢ coeff x (Nat.succ k + n✝) = coeff x (k + Nat.succ n✝)
|
/-
Copyright (c) 2022 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.Identities
#align_import ring_theory.witt_vector.domain from "leanprover-community/mathlib"@"b1d911acd60ab198808e853292106ee352b648ea"
/-!
# Witt vectors over a domain
This file builds to the proof `WittVector.instIsDomain`,
an instance that says if `R` is an integral domain, then so is `𝕎 R`.
It depends on the API around iterated applications
of `WittVector.verschiebung` and `WittVector.frobenius`
found in `Identities.lean`.
The [proof sketch](https://math.stackexchange.com/questions/4117247/ring-of-witt-vectors-over-an-integral-domain/4118723#4118723)
goes as follows:
any nonzero $x$ is an iterated application of $V$
to some vector $w_x$ whose 0th component is nonzero (`WittVector.verschiebung_nonzero`).
Known identities (`WittVector.iterate_verschiebung_mul`) allow us to transform
the product of two such $x$ and $y$
to the form $V^{m+n}\left(F^n(w_x) \cdot F^m(w_y)\right)$,
the 0th component of which must be nonzero.
## Main declarations
* `WittVector.iterate_verschiebung_mul_coeff` : an identity from [Haze09]
* `WittVector.instIsDomain`
-/
noncomputable section
open scoped Classical
namespace WittVector
open Function
variable {p : ℕ} {R : Type*}
local notation "𝕎" => WittVector p -- type as `\bbW`
/-!
## The `shift` operator
-/
/--
`WittVector.verschiebung` translates the entries of a Witt vector upward, inserting 0s in the gaps.
`WittVector.shift` does the opposite, removing the first entries.
This is mainly useful as an auxiliary construction for `WittVector.verschiebung_nonzero`.
-/
def shift (x : 𝕎 R) (n : ℕ) : 𝕎 R :=
@mk' p R fun i => x.coeff (n + i)
#align witt_vector.shift WittVector.shift
theorem shift_coeff (x : 𝕎 R) (n k : ℕ) : (x.shift n).coeff k = x.coeff (n + k) :=
rfl
#align witt_vector.shift_coeff WittVector.shift_coeff
variable [hp : Fact p.Prime] [CommRing R]
theorem verschiebung_shift (x : 𝕎 R) (k : ℕ) (h : ∀ i < k + 1, x.coeff i = 0) :
verschiebung (x.shift k.succ) = x.shift k := by
ext ⟨j⟩
· rw [verschiebung_coeff_zero, shift_coeff, h]
apply Nat.lt_succ_self
· simp only [verschiebung_coeff_succ, shift]
|
congr 1
|
theorem verschiebung_shift (x : 𝕎 R) (k : ℕ) (h : ∀ i < k + 1, x.coeff i = 0) :
verschiebung (x.shift k.succ) = x.shift k := by
ext ⟨j⟩
· rw [verschiebung_coeff_zero, shift_coeff, h]
apply Nat.lt_succ_self
· simp only [verschiebung_coeff_succ, shift]
|
Mathlib.RingTheory.WittVector.Domain.69_0.4uLlcZNQ2uiRcjJ
|
theorem verschiebung_shift (x : 𝕎 R) (k : ℕ) (h : ∀ i < k + 1, x.coeff i = 0) :
verschiebung (x.shift k.succ) = x.shift k
|
Mathlib_RingTheory_WittVector_Domain
|
case h.succ.e_a
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝ : CommRing R
x : 𝕎 R
k : ℕ
h : ∀ i < k + 1, coeff x i = 0
n✝ : ℕ
⊢ Nat.succ k + n✝ = k + Nat.succ n✝
|
/-
Copyright (c) 2022 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.Identities
#align_import ring_theory.witt_vector.domain from "leanprover-community/mathlib"@"b1d911acd60ab198808e853292106ee352b648ea"
/-!
# Witt vectors over a domain
This file builds to the proof `WittVector.instIsDomain`,
an instance that says if `R` is an integral domain, then so is `𝕎 R`.
It depends on the API around iterated applications
of `WittVector.verschiebung` and `WittVector.frobenius`
found in `Identities.lean`.
The [proof sketch](https://math.stackexchange.com/questions/4117247/ring-of-witt-vectors-over-an-integral-domain/4118723#4118723)
goes as follows:
any nonzero $x$ is an iterated application of $V$
to some vector $w_x$ whose 0th component is nonzero (`WittVector.verschiebung_nonzero`).
Known identities (`WittVector.iterate_verschiebung_mul`) allow us to transform
the product of two such $x$ and $y$
to the form $V^{m+n}\left(F^n(w_x) \cdot F^m(w_y)\right)$,
the 0th component of which must be nonzero.
## Main declarations
* `WittVector.iterate_verschiebung_mul_coeff` : an identity from [Haze09]
* `WittVector.instIsDomain`
-/
noncomputable section
open scoped Classical
namespace WittVector
open Function
variable {p : ℕ} {R : Type*}
local notation "𝕎" => WittVector p -- type as `\bbW`
/-!
## The `shift` operator
-/
/--
`WittVector.verschiebung` translates the entries of a Witt vector upward, inserting 0s in the gaps.
`WittVector.shift` does the opposite, removing the first entries.
This is mainly useful as an auxiliary construction for `WittVector.verschiebung_nonzero`.
-/
def shift (x : 𝕎 R) (n : ℕ) : 𝕎 R :=
@mk' p R fun i => x.coeff (n + i)
#align witt_vector.shift WittVector.shift
theorem shift_coeff (x : 𝕎 R) (n k : ℕ) : (x.shift n).coeff k = x.coeff (n + k) :=
rfl
#align witt_vector.shift_coeff WittVector.shift_coeff
variable [hp : Fact p.Prime] [CommRing R]
theorem verschiebung_shift (x : 𝕎 R) (k : ℕ) (h : ∀ i < k + 1, x.coeff i = 0) :
verschiebung (x.shift k.succ) = x.shift k := by
ext ⟨j⟩
· rw [verschiebung_coeff_zero, shift_coeff, h]
apply Nat.lt_succ_self
· simp only [verschiebung_coeff_succ, shift]
congr 1
|
rw [Nat.add_succ, add_comm, Nat.add_succ, add_comm]
|
theorem verschiebung_shift (x : 𝕎 R) (k : ℕ) (h : ∀ i < k + 1, x.coeff i = 0) :
verschiebung (x.shift k.succ) = x.shift k := by
ext ⟨j⟩
· rw [verschiebung_coeff_zero, shift_coeff, h]
apply Nat.lt_succ_self
· simp only [verschiebung_coeff_succ, shift]
congr 1
|
Mathlib.RingTheory.WittVector.Domain.69_0.4uLlcZNQ2uiRcjJ
|
theorem verschiebung_shift (x : 𝕎 R) (k : ℕ) (h : ∀ i < k + 1, x.coeff i = 0) :
verschiebung (x.shift k.succ) = x.shift k
|
Mathlib_RingTheory_WittVector_Domain
|
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝ : CommRing R
x : 𝕎 R
n : ℕ
h : ∀ i < n, coeff x i = 0
⊢ x = (⇑verschiebung)^[n] (shift x n)
|
/-
Copyright (c) 2022 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.Identities
#align_import ring_theory.witt_vector.domain from "leanprover-community/mathlib"@"b1d911acd60ab198808e853292106ee352b648ea"
/-!
# Witt vectors over a domain
This file builds to the proof `WittVector.instIsDomain`,
an instance that says if `R` is an integral domain, then so is `𝕎 R`.
It depends on the API around iterated applications
of `WittVector.verschiebung` and `WittVector.frobenius`
found in `Identities.lean`.
The [proof sketch](https://math.stackexchange.com/questions/4117247/ring-of-witt-vectors-over-an-integral-domain/4118723#4118723)
goes as follows:
any nonzero $x$ is an iterated application of $V$
to some vector $w_x$ whose 0th component is nonzero (`WittVector.verschiebung_nonzero`).
Known identities (`WittVector.iterate_verschiebung_mul`) allow us to transform
the product of two such $x$ and $y$
to the form $V^{m+n}\left(F^n(w_x) \cdot F^m(w_y)\right)$,
the 0th component of which must be nonzero.
## Main declarations
* `WittVector.iterate_verschiebung_mul_coeff` : an identity from [Haze09]
* `WittVector.instIsDomain`
-/
noncomputable section
open scoped Classical
namespace WittVector
open Function
variable {p : ℕ} {R : Type*}
local notation "𝕎" => WittVector p -- type as `\bbW`
/-!
## The `shift` operator
-/
/--
`WittVector.verschiebung` translates the entries of a Witt vector upward, inserting 0s in the gaps.
`WittVector.shift` does the opposite, removing the first entries.
This is mainly useful as an auxiliary construction for `WittVector.verschiebung_nonzero`.
-/
def shift (x : 𝕎 R) (n : ℕ) : 𝕎 R :=
@mk' p R fun i => x.coeff (n + i)
#align witt_vector.shift WittVector.shift
theorem shift_coeff (x : 𝕎 R) (n k : ℕ) : (x.shift n).coeff k = x.coeff (n + k) :=
rfl
#align witt_vector.shift_coeff WittVector.shift_coeff
variable [hp : Fact p.Prime] [CommRing R]
theorem verschiebung_shift (x : 𝕎 R) (k : ℕ) (h : ∀ i < k + 1, x.coeff i = 0) :
verschiebung (x.shift k.succ) = x.shift k := by
ext ⟨j⟩
· rw [verschiebung_coeff_zero, shift_coeff, h]
apply Nat.lt_succ_self
· simp only [verschiebung_coeff_succ, shift]
congr 1
rw [Nat.add_succ, add_comm, Nat.add_succ, add_comm]
#align witt_vector.verschiebung_shift WittVector.verschiebung_shift
theorem eq_iterate_verschiebung {x : 𝕎 R} {n : ℕ} (h : ∀ i < n, x.coeff i = 0) :
x = verschiebung^[n] (x.shift n) := by
|
induction' n with k ih
|
theorem eq_iterate_verschiebung {x : 𝕎 R} {n : ℕ} (h : ∀ i < n, x.coeff i = 0) :
x = verschiebung^[n] (x.shift n) := by
|
Mathlib.RingTheory.WittVector.Domain.79_0.4uLlcZNQ2uiRcjJ
|
theorem eq_iterate_verschiebung {x : 𝕎 R} {n : ℕ} (h : ∀ i < n, x.coeff i = 0) :
x = verschiebung^[n] (x.shift n)
|
Mathlib_RingTheory_WittVector_Domain
|
case zero
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝ : CommRing R
x : 𝕎 R
h : ∀ i < Nat.zero, coeff x i = 0
⊢ x = (⇑verschiebung)^[Nat.zero] (shift x Nat.zero)
|
/-
Copyright (c) 2022 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.Identities
#align_import ring_theory.witt_vector.domain from "leanprover-community/mathlib"@"b1d911acd60ab198808e853292106ee352b648ea"
/-!
# Witt vectors over a domain
This file builds to the proof `WittVector.instIsDomain`,
an instance that says if `R` is an integral domain, then so is `𝕎 R`.
It depends on the API around iterated applications
of `WittVector.verschiebung` and `WittVector.frobenius`
found in `Identities.lean`.
The [proof sketch](https://math.stackexchange.com/questions/4117247/ring-of-witt-vectors-over-an-integral-domain/4118723#4118723)
goes as follows:
any nonzero $x$ is an iterated application of $V$
to some vector $w_x$ whose 0th component is nonzero (`WittVector.verschiebung_nonzero`).
Known identities (`WittVector.iterate_verschiebung_mul`) allow us to transform
the product of two such $x$ and $y$
to the form $V^{m+n}\left(F^n(w_x) \cdot F^m(w_y)\right)$,
the 0th component of which must be nonzero.
## Main declarations
* `WittVector.iterate_verschiebung_mul_coeff` : an identity from [Haze09]
* `WittVector.instIsDomain`
-/
noncomputable section
open scoped Classical
namespace WittVector
open Function
variable {p : ℕ} {R : Type*}
local notation "𝕎" => WittVector p -- type as `\bbW`
/-!
## The `shift` operator
-/
/--
`WittVector.verschiebung` translates the entries of a Witt vector upward, inserting 0s in the gaps.
`WittVector.shift` does the opposite, removing the first entries.
This is mainly useful as an auxiliary construction for `WittVector.verschiebung_nonzero`.
-/
def shift (x : 𝕎 R) (n : ℕ) : 𝕎 R :=
@mk' p R fun i => x.coeff (n + i)
#align witt_vector.shift WittVector.shift
theorem shift_coeff (x : 𝕎 R) (n k : ℕ) : (x.shift n).coeff k = x.coeff (n + k) :=
rfl
#align witt_vector.shift_coeff WittVector.shift_coeff
variable [hp : Fact p.Prime] [CommRing R]
theorem verschiebung_shift (x : 𝕎 R) (k : ℕ) (h : ∀ i < k + 1, x.coeff i = 0) :
verschiebung (x.shift k.succ) = x.shift k := by
ext ⟨j⟩
· rw [verschiebung_coeff_zero, shift_coeff, h]
apply Nat.lt_succ_self
· simp only [verschiebung_coeff_succ, shift]
congr 1
rw [Nat.add_succ, add_comm, Nat.add_succ, add_comm]
#align witt_vector.verschiebung_shift WittVector.verschiebung_shift
theorem eq_iterate_verschiebung {x : 𝕎 R} {n : ℕ} (h : ∀ i < n, x.coeff i = 0) :
x = verschiebung^[n] (x.shift n) := by
induction' n with k ih
·
|
cases x
|
theorem eq_iterate_verschiebung {x : 𝕎 R} {n : ℕ} (h : ∀ i < n, x.coeff i = 0) :
x = verschiebung^[n] (x.shift n) := by
induction' n with k ih
·
|
Mathlib.RingTheory.WittVector.Domain.79_0.4uLlcZNQ2uiRcjJ
|
theorem eq_iterate_verschiebung {x : 𝕎 R} {n : ℕ} (h : ∀ i < n, x.coeff i = 0) :
x = verschiebung^[n] (x.shift n)
|
Mathlib_RingTheory_WittVector_Domain
|
case zero.mk'
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝ : CommRing R
coeff✝ : ℕ → R
h : ∀ i < Nat.zero, coeff { coeff := coeff✝ } i = 0
⊢ { coeff := coeff✝ } = (⇑verschiebung)^[Nat.zero] (shift { coeff := coeff✝ } Nat.zero)
|
/-
Copyright (c) 2022 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.Identities
#align_import ring_theory.witt_vector.domain from "leanprover-community/mathlib"@"b1d911acd60ab198808e853292106ee352b648ea"
/-!
# Witt vectors over a domain
This file builds to the proof `WittVector.instIsDomain`,
an instance that says if `R` is an integral domain, then so is `𝕎 R`.
It depends on the API around iterated applications
of `WittVector.verschiebung` and `WittVector.frobenius`
found in `Identities.lean`.
The [proof sketch](https://math.stackexchange.com/questions/4117247/ring-of-witt-vectors-over-an-integral-domain/4118723#4118723)
goes as follows:
any nonzero $x$ is an iterated application of $V$
to some vector $w_x$ whose 0th component is nonzero (`WittVector.verschiebung_nonzero`).
Known identities (`WittVector.iterate_verschiebung_mul`) allow us to transform
the product of two such $x$ and $y$
to the form $V^{m+n}\left(F^n(w_x) \cdot F^m(w_y)\right)$,
the 0th component of which must be nonzero.
## Main declarations
* `WittVector.iterate_verschiebung_mul_coeff` : an identity from [Haze09]
* `WittVector.instIsDomain`
-/
noncomputable section
open scoped Classical
namespace WittVector
open Function
variable {p : ℕ} {R : Type*}
local notation "𝕎" => WittVector p -- type as `\bbW`
/-!
## The `shift` operator
-/
/--
`WittVector.verschiebung` translates the entries of a Witt vector upward, inserting 0s in the gaps.
`WittVector.shift` does the opposite, removing the first entries.
This is mainly useful as an auxiliary construction for `WittVector.verschiebung_nonzero`.
-/
def shift (x : 𝕎 R) (n : ℕ) : 𝕎 R :=
@mk' p R fun i => x.coeff (n + i)
#align witt_vector.shift WittVector.shift
theorem shift_coeff (x : 𝕎 R) (n k : ℕ) : (x.shift n).coeff k = x.coeff (n + k) :=
rfl
#align witt_vector.shift_coeff WittVector.shift_coeff
variable [hp : Fact p.Prime] [CommRing R]
theorem verschiebung_shift (x : 𝕎 R) (k : ℕ) (h : ∀ i < k + 1, x.coeff i = 0) :
verschiebung (x.shift k.succ) = x.shift k := by
ext ⟨j⟩
· rw [verschiebung_coeff_zero, shift_coeff, h]
apply Nat.lt_succ_self
· simp only [verschiebung_coeff_succ, shift]
congr 1
rw [Nat.add_succ, add_comm, Nat.add_succ, add_comm]
#align witt_vector.verschiebung_shift WittVector.verschiebung_shift
theorem eq_iterate_verschiebung {x : 𝕎 R} {n : ℕ} (h : ∀ i < n, x.coeff i = 0) :
x = verschiebung^[n] (x.shift n) := by
induction' n with k ih
· cases x;
|
simp [shift]
|
theorem eq_iterate_verschiebung {x : 𝕎 R} {n : ℕ} (h : ∀ i < n, x.coeff i = 0) :
x = verschiebung^[n] (x.shift n) := by
induction' n with k ih
· cases x;
|
Mathlib.RingTheory.WittVector.Domain.79_0.4uLlcZNQ2uiRcjJ
|
theorem eq_iterate_verschiebung {x : 𝕎 R} {n : ℕ} (h : ∀ i < n, x.coeff i = 0) :
x = verschiebung^[n] (x.shift n)
|
Mathlib_RingTheory_WittVector_Domain
|
case succ
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝ : CommRing R
x : 𝕎 R
k : ℕ
ih : (∀ i < k, coeff x i = 0) → x = (⇑verschiebung)^[k] (shift x k)
h : ∀ i < Nat.succ k, coeff x i = 0
⊢ x = (⇑verschiebung)^[Nat.succ k] (shift x (Nat.succ k))
|
/-
Copyright (c) 2022 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.Identities
#align_import ring_theory.witt_vector.domain from "leanprover-community/mathlib"@"b1d911acd60ab198808e853292106ee352b648ea"
/-!
# Witt vectors over a domain
This file builds to the proof `WittVector.instIsDomain`,
an instance that says if `R` is an integral domain, then so is `𝕎 R`.
It depends on the API around iterated applications
of `WittVector.verschiebung` and `WittVector.frobenius`
found in `Identities.lean`.
The [proof sketch](https://math.stackexchange.com/questions/4117247/ring-of-witt-vectors-over-an-integral-domain/4118723#4118723)
goes as follows:
any nonzero $x$ is an iterated application of $V$
to some vector $w_x$ whose 0th component is nonzero (`WittVector.verschiebung_nonzero`).
Known identities (`WittVector.iterate_verschiebung_mul`) allow us to transform
the product of two such $x$ and $y$
to the form $V^{m+n}\left(F^n(w_x) \cdot F^m(w_y)\right)$,
the 0th component of which must be nonzero.
## Main declarations
* `WittVector.iterate_verschiebung_mul_coeff` : an identity from [Haze09]
* `WittVector.instIsDomain`
-/
noncomputable section
open scoped Classical
namespace WittVector
open Function
variable {p : ℕ} {R : Type*}
local notation "𝕎" => WittVector p -- type as `\bbW`
/-!
## The `shift` operator
-/
/--
`WittVector.verschiebung` translates the entries of a Witt vector upward, inserting 0s in the gaps.
`WittVector.shift` does the opposite, removing the first entries.
This is mainly useful as an auxiliary construction for `WittVector.verschiebung_nonzero`.
-/
def shift (x : 𝕎 R) (n : ℕ) : 𝕎 R :=
@mk' p R fun i => x.coeff (n + i)
#align witt_vector.shift WittVector.shift
theorem shift_coeff (x : 𝕎 R) (n k : ℕ) : (x.shift n).coeff k = x.coeff (n + k) :=
rfl
#align witt_vector.shift_coeff WittVector.shift_coeff
variable [hp : Fact p.Prime] [CommRing R]
theorem verschiebung_shift (x : 𝕎 R) (k : ℕ) (h : ∀ i < k + 1, x.coeff i = 0) :
verschiebung (x.shift k.succ) = x.shift k := by
ext ⟨j⟩
· rw [verschiebung_coeff_zero, shift_coeff, h]
apply Nat.lt_succ_self
· simp only [verschiebung_coeff_succ, shift]
congr 1
rw [Nat.add_succ, add_comm, Nat.add_succ, add_comm]
#align witt_vector.verschiebung_shift WittVector.verschiebung_shift
theorem eq_iterate_verschiebung {x : 𝕎 R} {n : ℕ} (h : ∀ i < n, x.coeff i = 0) :
x = verschiebung^[n] (x.shift n) := by
induction' n with k ih
· cases x; simp [shift]
·
|
dsimp
|
theorem eq_iterate_verschiebung {x : 𝕎 R} {n : ℕ} (h : ∀ i < n, x.coeff i = 0) :
x = verschiebung^[n] (x.shift n) := by
induction' n with k ih
· cases x; simp [shift]
·
|
Mathlib.RingTheory.WittVector.Domain.79_0.4uLlcZNQ2uiRcjJ
|
theorem eq_iterate_verschiebung {x : 𝕎 R} {n : ℕ} (h : ∀ i < n, x.coeff i = 0) :
x = verschiebung^[n] (x.shift n)
|
Mathlib_RingTheory_WittVector_Domain
|
case succ
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝ : CommRing R
x : 𝕎 R
k : ℕ
ih : (∀ i < k, coeff x i = 0) → x = (⇑verschiebung)^[k] (shift x k)
h : ∀ i < Nat.succ k, coeff x i = 0
⊢ x = (⇑verschiebung)^[k] (verschiebung (shift x (Nat.succ k)))
|
/-
Copyright (c) 2022 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.Identities
#align_import ring_theory.witt_vector.domain from "leanprover-community/mathlib"@"b1d911acd60ab198808e853292106ee352b648ea"
/-!
# Witt vectors over a domain
This file builds to the proof `WittVector.instIsDomain`,
an instance that says if `R` is an integral domain, then so is `𝕎 R`.
It depends on the API around iterated applications
of `WittVector.verschiebung` and `WittVector.frobenius`
found in `Identities.lean`.
The [proof sketch](https://math.stackexchange.com/questions/4117247/ring-of-witt-vectors-over-an-integral-domain/4118723#4118723)
goes as follows:
any nonzero $x$ is an iterated application of $V$
to some vector $w_x$ whose 0th component is nonzero (`WittVector.verschiebung_nonzero`).
Known identities (`WittVector.iterate_verschiebung_mul`) allow us to transform
the product of two such $x$ and $y$
to the form $V^{m+n}\left(F^n(w_x) \cdot F^m(w_y)\right)$,
the 0th component of which must be nonzero.
## Main declarations
* `WittVector.iterate_verschiebung_mul_coeff` : an identity from [Haze09]
* `WittVector.instIsDomain`
-/
noncomputable section
open scoped Classical
namespace WittVector
open Function
variable {p : ℕ} {R : Type*}
local notation "𝕎" => WittVector p -- type as `\bbW`
/-!
## The `shift` operator
-/
/--
`WittVector.verschiebung` translates the entries of a Witt vector upward, inserting 0s in the gaps.
`WittVector.shift` does the opposite, removing the first entries.
This is mainly useful as an auxiliary construction for `WittVector.verschiebung_nonzero`.
-/
def shift (x : 𝕎 R) (n : ℕ) : 𝕎 R :=
@mk' p R fun i => x.coeff (n + i)
#align witt_vector.shift WittVector.shift
theorem shift_coeff (x : 𝕎 R) (n k : ℕ) : (x.shift n).coeff k = x.coeff (n + k) :=
rfl
#align witt_vector.shift_coeff WittVector.shift_coeff
variable [hp : Fact p.Prime] [CommRing R]
theorem verschiebung_shift (x : 𝕎 R) (k : ℕ) (h : ∀ i < k + 1, x.coeff i = 0) :
verschiebung (x.shift k.succ) = x.shift k := by
ext ⟨j⟩
· rw [verschiebung_coeff_zero, shift_coeff, h]
apply Nat.lt_succ_self
· simp only [verschiebung_coeff_succ, shift]
congr 1
rw [Nat.add_succ, add_comm, Nat.add_succ, add_comm]
#align witt_vector.verschiebung_shift WittVector.verschiebung_shift
theorem eq_iterate_verschiebung {x : 𝕎 R} {n : ℕ} (h : ∀ i < n, x.coeff i = 0) :
x = verschiebung^[n] (x.shift n) := by
induction' n with k ih
· cases x; simp [shift]
· dsimp;
|
rw [verschiebung_shift]
|
theorem eq_iterate_verschiebung {x : 𝕎 R} {n : ℕ} (h : ∀ i < n, x.coeff i = 0) :
x = verschiebung^[n] (x.shift n) := by
induction' n with k ih
· cases x; simp [shift]
· dsimp;
|
Mathlib.RingTheory.WittVector.Domain.79_0.4uLlcZNQ2uiRcjJ
|
theorem eq_iterate_verschiebung {x : 𝕎 R} {n : ℕ} (h : ∀ i < n, x.coeff i = 0) :
x = verschiebung^[n] (x.shift n)
|
Mathlib_RingTheory_WittVector_Domain
|
case succ
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝ : CommRing R
x : 𝕎 R
k : ℕ
ih : (∀ i < k, coeff x i = 0) → x = (⇑verschiebung)^[k] (shift x k)
h : ∀ i < Nat.succ k, coeff x i = 0
⊢ x = (⇑verschiebung)^[k] (shift x k)
|
/-
Copyright (c) 2022 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.Identities
#align_import ring_theory.witt_vector.domain from "leanprover-community/mathlib"@"b1d911acd60ab198808e853292106ee352b648ea"
/-!
# Witt vectors over a domain
This file builds to the proof `WittVector.instIsDomain`,
an instance that says if `R` is an integral domain, then so is `𝕎 R`.
It depends on the API around iterated applications
of `WittVector.verschiebung` and `WittVector.frobenius`
found in `Identities.lean`.
The [proof sketch](https://math.stackexchange.com/questions/4117247/ring-of-witt-vectors-over-an-integral-domain/4118723#4118723)
goes as follows:
any nonzero $x$ is an iterated application of $V$
to some vector $w_x$ whose 0th component is nonzero (`WittVector.verschiebung_nonzero`).
Known identities (`WittVector.iterate_verschiebung_mul`) allow us to transform
the product of two such $x$ and $y$
to the form $V^{m+n}\left(F^n(w_x) \cdot F^m(w_y)\right)$,
the 0th component of which must be nonzero.
## Main declarations
* `WittVector.iterate_verschiebung_mul_coeff` : an identity from [Haze09]
* `WittVector.instIsDomain`
-/
noncomputable section
open scoped Classical
namespace WittVector
open Function
variable {p : ℕ} {R : Type*}
local notation "𝕎" => WittVector p -- type as `\bbW`
/-!
## The `shift` operator
-/
/--
`WittVector.verschiebung` translates the entries of a Witt vector upward, inserting 0s in the gaps.
`WittVector.shift` does the opposite, removing the first entries.
This is mainly useful as an auxiliary construction for `WittVector.verschiebung_nonzero`.
-/
def shift (x : 𝕎 R) (n : ℕ) : 𝕎 R :=
@mk' p R fun i => x.coeff (n + i)
#align witt_vector.shift WittVector.shift
theorem shift_coeff (x : 𝕎 R) (n k : ℕ) : (x.shift n).coeff k = x.coeff (n + k) :=
rfl
#align witt_vector.shift_coeff WittVector.shift_coeff
variable [hp : Fact p.Prime] [CommRing R]
theorem verschiebung_shift (x : 𝕎 R) (k : ℕ) (h : ∀ i < k + 1, x.coeff i = 0) :
verschiebung (x.shift k.succ) = x.shift k := by
ext ⟨j⟩
· rw [verschiebung_coeff_zero, shift_coeff, h]
apply Nat.lt_succ_self
· simp only [verschiebung_coeff_succ, shift]
congr 1
rw [Nat.add_succ, add_comm, Nat.add_succ, add_comm]
#align witt_vector.verschiebung_shift WittVector.verschiebung_shift
theorem eq_iterate_verschiebung {x : 𝕎 R} {n : ℕ} (h : ∀ i < n, x.coeff i = 0) :
x = verschiebung^[n] (x.shift n) := by
induction' n with k ih
· cases x; simp [shift]
· dsimp; rw [verschiebung_shift]
·
|
exact ih fun i hi => h _ (hi.trans (Nat.lt_succ_self _))
|
theorem eq_iterate_verschiebung {x : 𝕎 R} {n : ℕ} (h : ∀ i < n, x.coeff i = 0) :
x = verschiebung^[n] (x.shift n) := by
induction' n with k ih
· cases x; simp [shift]
· dsimp; rw [verschiebung_shift]
·
|
Mathlib.RingTheory.WittVector.Domain.79_0.4uLlcZNQ2uiRcjJ
|
theorem eq_iterate_verschiebung {x : 𝕎 R} {n : ℕ} (h : ∀ i < n, x.coeff i = 0) :
x = verschiebung^[n] (x.shift n)
|
Mathlib_RingTheory_WittVector_Domain
|
case succ.h
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝ : CommRing R
x : 𝕎 R
k : ℕ
ih : (∀ i < k, coeff x i = 0) → x = (⇑verschiebung)^[k] (shift x k)
h : ∀ i < Nat.succ k, coeff x i = 0
⊢ ∀ i < k + 1, coeff x i = 0
|
/-
Copyright (c) 2022 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.Identities
#align_import ring_theory.witt_vector.domain from "leanprover-community/mathlib"@"b1d911acd60ab198808e853292106ee352b648ea"
/-!
# Witt vectors over a domain
This file builds to the proof `WittVector.instIsDomain`,
an instance that says if `R` is an integral domain, then so is `𝕎 R`.
It depends on the API around iterated applications
of `WittVector.verschiebung` and `WittVector.frobenius`
found in `Identities.lean`.
The [proof sketch](https://math.stackexchange.com/questions/4117247/ring-of-witt-vectors-over-an-integral-domain/4118723#4118723)
goes as follows:
any nonzero $x$ is an iterated application of $V$
to some vector $w_x$ whose 0th component is nonzero (`WittVector.verschiebung_nonzero`).
Known identities (`WittVector.iterate_verschiebung_mul`) allow us to transform
the product of two such $x$ and $y$
to the form $V^{m+n}\left(F^n(w_x) \cdot F^m(w_y)\right)$,
the 0th component of which must be nonzero.
## Main declarations
* `WittVector.iterate_verschiebung_mul_coeff` : an identity from [Haze09]
* `WittVector.instIsDomain`
-/
noncomputable section
open scoped Classical
namespace WittVector
open Function
variable {p : ℕ} {R : Type*}
local notation "𝕎" => WittVector p -- type as `\bbW`
/-!
## The `shift` operator
-/
/--
`WittVector.verschiebung` translates the entries of a Witt vector upward, inserting 0s in the gaps.
`WittVector.shift` does the opposite, removing the first entries.
This is mainly useful as an auxiliary construction for `WittVector.verschiebung_nonzero`.
-/
def shift (x : 𝕎 R) (n : ℕ) : 𝕎 R :=
@mk' p R fun i => x.coeff (n + i)
#align witt_vector.shift WittVector.shift
theorem shift_coeff (x : 𝕎 R) (n k : ℕ) : (x.shift n).coeff k = x.coeff (n + k) :=
rfl
#align witt_vector.shift_coeff WittVector.shift_coeff
variable [hp : Fact p.Prime] [CommRing R]
theorem verschiebung_shift (x : 𝕎 R) (k : ℕ) (h : ∀ i < k + 1, x.coeff i = 0) :
verschiebung (x.shift k.succ) = x.shift k := by
ext ⟨j⟩
· rw [verschiebung_coeff_zero, shift_coeff, h]
apply Nat.lt_succ_self
· simp only [verschiebung_coeff_succ, shift]
congr 1
rw [Nat.add_succ, add_comm, Nat.add_succ, add_comm]
#align witt_vector.verschiebung_shift WittVector.verschiebung_shift
theorem eq_iterate_verschiebung {x : 𝕎 R} {n : ℕ} (h : ∀ i < n, x.coeff i = 0) :
x = verschiebung^[n] (x.shift n) := by
induction' n with k ih
· cases x; simp [shift]
· dsimp; rw [verschiebung_shift]
· exact ih fun i hi => h _ (hi.trans (Nat.lt_succ_self _))
·
|
exact h
|
theorem eq_iterate_verschiebung {x : 𝕎 R} {n : ℕ} (h : ∀ i < n, x.coeff i = 0) :
x = verschiebung^[n] (x.shift n) := by
induction' n with k ih
· cases x; simp [shift]
· dsimp; rw [verschiebung_shift]
· exact ih fun i hi => h _ (hi.trans (Nat.lt_succ_self _))
·
|
Mathlib.RingTheory.WittVector.Domain.79_0.4uLlcZNQ2uiRcjJ
|
theorem eq_iterate_verschiebung {x : 𝕎 R} {n : ℕ} (h : ∀ i < n, x.coeff i = 0) :
x = verschiebung^[n] (x.shift n)
|
Mathlib_RingTheory_WittVector_Domain
|
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝ : CommRing R
x : 𝕎 R
hx : x ≠ 0
⊢ ∃ n x', coeff x' 0 ≠ 0 ∧ x = (⇑verschiebung)^[n] x'
|
/-
Copyright (c) 2022 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.Identities
#align_import ring_theory.witt_vector.domain from "leanprover-community/mathlib"@"b1d911acd60ab198808e853292106ee352b648ea"
/-!
# Witt vectors over a domain
This file builds to the proof `WittVector.instIsDomain`,
an instance that says if `R` is an integral domain, then so is `𝕎 R`.
It depends on the API around iterated applications
of `WittVector.verschiebung` and `WittVector.frobenius`
found in `Identities.lean`.
The [proof sketch](https://math.stackexchange.com/questions/4117247/ring-of-witt-vectors-over-an-integral-domain/4118723#4118723)
goes as follows:
any nonzero $x$ is an iterated application of $V$
to some vector $w_x$ whose 0th component is nonzero (`WittVector.verschiebung_nonzero`).
Known identities (`WittVector.iterate_verschiebung_mul`) allow us to transform
the product of two such $x$ and $y$
to the form $V^{m+n}\left(F^n(w_x) \cdot F^m(w_y)\right)$,
the 0th component of which must be nonzero.
## Main declarations
* `WittVector.iterate_verschiebung_mul_coeff` : an identity from [Haze09]
* `WittVector.instIsDomain`
-/
noncomputable section
open scoped Classical
namespace WittVector
open Function
variable {p : ℕ} {R : Type*}
local notation "𝕎" => WittVector p -- type as `\bbW`
/-!
## The `shift` operator
-/
/--
`WittVector.verschiebung` translates the entries of a Witt vector upward, inserting 0s in the gaps.
`WittVector.shift` does the opposite, removing the first entries.
This is mainly useful as an auxiliary construction for `WittVector.verschiebung_nonzero`.
-/
def shift (x : 𝕎 R) (n : ℕ) : 𝕎 R :=
@mk' p R fun i => x.coeff (n + i)
#align witt_vector.shift WittVector.shift
theorem shift_coeff (x : 𝕎 R) (n k : ℕ) : (x.shift n).coeff k = x.coeff (n + k) :=
rfl
#align witt_vector.shift_coeff WittVector.shift_coeff
variable [hp : Fact p.Prime] [CommRing R]
theorem verschiebung_shift (x : 𝕎 R) (k : ℕ) (h : ∀ i < k + 1, x.coeff i = 0) :
verschiebung (x.shift k.succ) = x.shift k := by
ext ⟨j⟩
· rw [verschiebung_coeff_zero, shift_coeff, h]
apply Nat.lt_succ_self
· simp only [verschiebung_coeff_succ, shift]
congr 1
rw [Nat.add_succ, add_comm, Nat.add_succ, add_comm]
#align witt_vector.verschiebung_shift WittVector.verschiebung_shift
theorem eq_iterate_verschiebung {x : 𝕎 R} {n : ℕ} (h : ∀ i < n, x.coeff i = 0) :
x = verschiebung^[n] (x.shift n) := by
induction' n with k ih
· cases x; simp [shift]
· dsimp; rw [verschiebung_shift]
· exact ih fun i hi => h _ (hi.trans (Nat.lt_succ_self _))
· exact h
#align witt_vector.eq_iterate_verschiebung WittVector.eq_iterate_verschiebung
theorem verschiebung_nonzero {x : 𝕎 R} (hx : x ≠ 0) :
∃ n : ℕ, ∃ x' : 𝕎 R, x'.coeff 0 ≠ 0 ∧ x = verschiebung^[n] x' := by
|
have hex : ∃ k : ℕ, x.coeff k ≠ 0 := by
by_contra! hall
apply hx
ext i
simp only [hall, zero_coeff]
|
theorem verschiebung_nonzero {x : 𝕎 R} (hx : x ≠ 0) :
∃ n : ℕ, ∃ x' : 𝕎 R, x'.coeff 0 ≠ 0 ∧ x = verschiebung^[n] x' := by
|
Mathlib.RingTheory.WittVector.Domain.88_0.4uLlcZNQ2uiRcjJ
|
theorem verschiebung_nonzero {x : 𝕎 R} (hx : x ≠ 0) :
∃ n : ℕ, ∃ x' : 𝕎 R, x'.coeff 0 ≠ 0 ∧ x = verschiebung^[n] x'
|
Mathlib_RingTheory_WittVector_Domain
|
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝ : CommRing R
x : 𝕎 R
hx : x ≠ 0
⊢ ∃ k, coeff x k ≠ 0
|
/-
Copyright (c) 2022 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.Identities
#align_import ring_theory.witt_vector.domain from "leanprover-community/mathlib"@"b1d911acd60ab198808e853292106ee352b648ea"
/-!
# Witt vectors over a domain
This file builds to the proof `WittVector.instIsDomain`,
an instance that says if `R` is an integral domain, then so is `𝕎 R`.
It depends on the API around iterated applications
of `WittVector.verschiebung` and `WittVector.frobenius`
found in `Identities.lean`.
The [proof sketch](https://math.stackexchange.com/questions/4117247/ring-of-witt-vectors-over-an-integral-domain/4118723#4118723)
goes as follows:
any nonzero $x$ is an iterated application of $V$
to some vector $w_x$ whose 0th component is nonzero (`WittVector.verschiebung_nonzero`).
Known identities (`WittVector.iterate_verschiebung_mul`) allow us to transform
the product of two such $x$ and $y$
to the form $V^{m+n}\left(F^n(w_x) \cdot F^m(w_y)\right)$,
the 0th component of which must be nonzero.
## Main declarations
* `WittVector.iterate_verschiebung_mul_coeff` : an identity from [Haze09]
* `WittVector.instIsDomain`
-/
noncomputable section
open scoped Classical
namespace WittVector
open Function
variable {p : ℕ} {R : Type*}
local notation "𝕎" => WittVector p -- type as `\bbW`
/-!
## The `shift` operator
-/
/--
`WittVector.verschiebung` translates the entries of a Witt vector upward, inserting 0s in the gaps.
`WittVector.shift` does the opposite, removing the first entries.
This is mainly useful as an auxiliary construction for `WittVector.verschiebung_nonzero`.
-/
def shift (x : 𝕎 R) (n : ℕ) : 𝕎 R :=
@mk' p R fun i => x.coeff (n + i)
#align witt_vector.shift WittVector.shift
theorem shift_coeff (x : 𝕎 R) (n k : ℕ) : (x.shift n).coeff k = x.coeff (n + k) :=
rfl
#align witt_vector.shift_coeff WittVector.shift_coeff
variable [hp : Fact p.Prime] [CommRing R]
theorem verschiebung_shift (x : 𝕎 R) (k : ℕ) (h : ∀ i < k + 1, x.coeff i = 0) :
verschiebung (x.shift k.succ) = x.shift k := by
ext ⟨j⟩
· rw [verschiebung_coeff_zero, shift_coeff, h]
apply Nat.lt_succ_self
· simp only [verschiebung_coeff_succ, shift]
congr 1
rw [Nat.add_succ, add_comm, Nat.add_succ, add_comm]
#align witt_vector.verschiebung_shift WittVector.verschiebung_shift
theorem eq_iterate_verschiebung {x : 𝕎 R} {n : ℕ} (h : ∀ i < n, x.coeff i = 0) :
x = verschiebung^[n] (x.shift n) := by
induction' n with k ih
· cases x; simp [shift]
· dsimp; rw [verschiebung_shift]
· exact ih fun i hi => h _ (hi.trans (Nat.lt_succ_self _))
· exact h
#align witt_vector.eq_iterate_verschiebung WittVector.eq_iterate_verschiebung
theorem verschiebung_nonzero {x : 𝕎 R} (hx : x ≠ 0) :
∃ n : ℕ, ∃ x' : 𝕎 R, x'.coeff 0 ≠ 0 ∧ x = verschiebung^[n] x' := by
have hex : ∃ k : ℕ, x.coeff k ≠ 0 := by
|
by_contra! hall
|
theorem verschiebung_nonzero {x : 𝕎 R} (hx : x ≠ 0) :
∃ n : ℕ, ∃ x' : 𝕎 R, x'.coeff 0 ≠ 0 ∧ x = verschiebung^[n] x' := by
have hex : ∃ k : ℕ, x.coeff k ≠ 0 := by
|
Mathlib.RingTheory.WittVector.Domain.88_0.4uLlcZNQ2uiRcjJ
|
theorem verschiebung_nonzero {x : 𝕎 R} (hx : x ≠ 0) :
∃ n : ℕ, ∃ x' : 𝕎 R, x'.coeff 0 ≠ 0 ∧ x = verschiebung^[n] x'
|
Mathlib_RingTheory_WittVector_Domain
|
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝ : CommRing R
x : 𝕎 R
hx : x ≠ 0
hall : ∀ (k : ℕ), coeff x k = 0
⊢ False
|
/-
Copyright (c) 2022 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.Identities
#align_import ring_theory.witt_vector.domain from "leanprover-community/mathlib"@"b1d911acd60ab198808e853292106ee352b648ea"
/-!
# Witt vectors over a domain
This file builds to the proof `WittVector.instIsDomain`,
an instance that says if `R` is an integral domain, then so is `𝕎 R`.
It depends on the API around iterated applications
of `WittVector.verschiebung` and `WittVector.frobenius`
found in `Identities.lean`.
The [proof sketch](https://math.stackexchange.com/questions/4117247/ring-of-witt-vectors-over-an-integral-domain/4118723#4118723)
goes as follows:
any nonzero $x$ is an iterated application of $V$
to some vector $w_x$ whose 0th component is nonzero (`WittVector.verschiebung_nonzero`).
Known identities (`WittVector.iterate_verschiebung_mul`) allow us to transform
the product of two such $x$ and $y$
to the form $V^{m+n}\left(F^n(w_x) \cdot F^m(w_y)\right)$,
the 0th component of which must be nonzero.
## Main declarations
* `WittVector.iterate_verschiebung_mul_coeff` : an identity from [Haze09]
* `WittVector.instIsDomain`
-/
noncomputable section
open scoped Classical
namespace WittVector
open Function
variable {p : ℕ} {R : Type*}
local notation "𝕎" => WittVector p -- type as `\bbW`
/-!
## The `shift` operator
-/
/--
`WittVector.verschiebung` translates the entries of a Witt vector upward, inserting 0s in the gaps.
`WittVector.shift` does the opposite, removing the first entries.
This is mainly useful as an auxiliary construction for `WittVector.verschiebung_nonzero`.
-/
def shift (x : 𝕎 R) (n : ℕ) : 𝕎 R :=
@mk' p R fun i => x.coeff (n + i)
#align witt_vector.shift WittVector.shift
theorem shift_coeff (x : 𝕎 R) (n k : ℕ) : (x.shift n).coeff k = x.coeff (n + k) :=
rfl
#align witt_vector.shift_coeff WittVector.shift_coeff
variable [hp : Fact p.Prime] [CommRing R]
theorem verschiebung_shift (x : 𝕎 R) (k : ℕ) (h : ∀ i < k + 1, x.coeff i = 0) :
verschiebung (x.shift k.succ) = x.shift k := by
ext ⟨j⟩
· rw [verschiebung_coeff_zero, shift_coeff, h]
apply Nat.lt_succ_self
· simp only [verschiebung_coeff_succ, shift]
congr 1
rw [Nat.add_succ, add_comm, Nat.add_succ, add_comm]
#align witt_vector.verschiebung_shift WittVector.verschiebung_shift
theorem eq_iterate_verschiebung {x : 𝕎 R} {n : ℕ} (h : ∀ i < n, x.coeff i = 0) :
x = verschiebung^[n] (x.shift n) := by
induction' n with k ih
· cases x; simp [shift]
· dsimp; rw [verschiebung_shift]
· exact ih fun i hi => h _ (hi.trans (Nat.lt_succ_self _))
· exact h
#align witt_vector.eq_iterate_verschiebung WittVector.eq_iterate_verschiebung
theorem verschiebung_nonzero {x : 𝕎 R} (hx : x ≠ 0) :
∃ n : ℕ, ∃ x' : 𝕎 R, x'.coeff 0 ≠ 0 ∧ x = verschiebung^[n] x' := by
have hex : ∃ k : ℕ, x.coeff k ≠ 0 := by
by_contra! hall
|
apply hx
|
theorem verschiebung_nonzero {x : 𝕎 R} (hx : x ≠ 0) :
∃ n : ℕ, ∃ x' : 𝕎 R, x'.coeff 0 ≠ 0 ∧ x = verschiebung^[n] x' := by
have hex : ∃ k : ℕ, x.coeff k ≠ 0 := by
by_contra! hall
|
Mathlib.RingTheory.WittVector.Domain.88_0.4uLlcZNQ2uiRcjJ
|
theorem verschiebung_nonzero {x : 𝕎 R} (hx : x ≠ 0) :
∃ n : ℕ, ∃ x' : 𝕎 R, x'.coeff 0 ≠ 0 ∧ x = verschiebung^[n] x'
|
Mathlib_RingTheory_WittVector_Domain
|
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝ : CommRing R
x : 𝕎 R
hx : x ≠ 0
hall : ∀ (k : ℕ), coeff x k = 0
⊢ x = 0
|
/-
Copyright (c) 2022 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.Identities
#align_import ring_theory.witt_vector.domain from "leanprover-community/mathlib"@"b1d911acd60ab198808e853292106ee352b648ea"
/-!
# Witt vectors over a domain
This file builds to the proof `WittVector.instIsDomain`,
an instance that says if `R` is an integral domain, then so is `𝕎 R`.
It depends on the API around iterated applications
of `WittVector.verschiebung` and `WittVector.frobenius`
found in `Identities.lean`.
The [proof sketch](https://math.stackexchange.com/questions/4117247/ring-of-witt-vectors-over-an-integral-domain/4118723#4118723)
goes as follows:
any nonzero $x$ is an iterated application of $V$
to some vector $w_x$ whose 0th component is nonzero (`WittVector.verschiebung_nonzero`).
Known identities (`WittVector.iterate_verschiebung_mul`) allow us to transform
the product of two such $x$ and $y$
to the form $V^{m+n}\left(F^n(w_x) \cdot F^m(w_y)\right)$,
the 0th component of which must be nonzero.
## Main declarations
* `WittVector.iterate_verschiebung_mul_coeff` : an identity from [Haze09]
* `WittVector.instIsDomain`
-/
noncomputable section
open scoped Classical
namespace WittVector
open Function
variable {p : ℕ} {R : Type*}
local notation "𝕎" => WittVector p -- type as `\bbW`
/-!
## The `shift` operator
-/
/--
`WittVector.verschiebung` translates the entries of a Witt vector upward, inserting 0s in the gaps.
`WittVector.shift` does the opposite, removing the first entries.
This is mainly useful as an auxiliary construction for `WittVector.verschiebung_nonzero`.
-/
def shift (x : 𝕎 R) (n : ℕ) : 𝕎 R :=
@mk' p R fun i => x.coeff (n + i)
#align witt_vector.shift WittVector.shift
theorem shift_coeff (x : 𝕎 R) (n k : ℕ) : (x.shift n).coeff k = x.coeff (n + k) :=
rfl
#align witt_vector.shift_coeff WittVector.shift_coeff
variable [hp : Fact p.Prime] [CommRing R]
theorem verschiebung_shift (x : 𝕎 R) (k : ℕ) (h : ∀ i < k + 1, x.coeff i = 0) :
verschiebung (x.shift k.succ) = x.shift k := by
ext ⟨j⟩
· rw [verschiebung_coeff_zero, shift_coeff, h]
apply Nat.lt_succ_self
· simp only [verschiebung_coeff_succ, shift]
congr 1
rw [Nat.add_succ, add_comm, Nat.add_succ, add_comm]
#align witt_vector.verschiebung_shift WittVector.verschiebung_shift
theorem eq_iterate_verschiebung {x : 𝕎 R} {n : ℕ} (h : ∀ i < n, x.coeff i = 0) :
x = verschiebung^[n] (x.shift n) := by
induction' n with k ih
· cases x; simp [shift]
· dsimp; rw [verschiebung_shift]
· exact ih fun i hi => h _ (hi.trans (Nat.lt_succ_self _))
· exact h
#align witt_vector.eq_iterate_verschiebung WittVector.eq_iterate_verschiebung
theorem verschiebung_nonzero {x : 𝕎 R} (hx : x ≠ 0) :
∃ n : ℕ, ∃ x' : 𝕎 R, x'.coeff 0 ≠ 0 ∧ x = verschiebung^[n] x' := by
have hex : ∃ k : ℕ, x.coeff k ≠ 0 := by
by_contra! hall
apply hx
|
ext i
|
theorem verschiebung_nonzero {x : 𝕎 R} (hx : x ≠ 0) :
∃ n : ℕ, ∃ x' : 𝕎 R, x'.coeff 0 ≠ 0 ∧ x = verschiebung^[n] x' := by
have hex : ∃ k : ℕ, x.coeff k ≠ 0 := by
by_contra! hall
apply hx
|
Mathlib.RingTheory.WittVector.Domain.88_0.4uLlcZNQ2uiRcjJ
|
theorem verschiebung_nonzero {x : 𝕎 R} (hx : x ≠ 0) :
∃ n : ℕ, ∃ x' : 𝕎 R, x'.coeff 0 ≠ 0 ∧ x = verschiebung^[n] x'
|
Mathlib_RingTheory_WittVector_Domain
|
case h
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝ : CommRing R
x : 𝕎 R
hx : x ≠ 0
hall : ∀ (k : ℕ), coeff x k = 0
i : ℕ
⊢ coeff x i = coeff 0 i
|
/-
Copyright (c) 2022 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.Identities
#align_import ring_theory.witt_vector.domain from "leanprover-community/mathlib"@"b1d911acd60ab198808e853292106ee352b648ea"
/-!
# Witt vectors over a domain
This file builds to the proof `WittVector.instIsDomain`,
an instance that says if `R` is an integral domain, then so is `𝕎 R`.
It depends on the API around iterated applications
of `WittVector.verschiebung` and `WittVector.frobenius`
found in `Identities.lean`.
The [proof sketch](https://math.stackexchange.com/questions/4117247/ring-of-witt-vectors-over-an-integral-domain/4118723#4118723)
goes as follows:
any nonzero $x$ is an iterated application of $V$
to some vector $w_x$ whose 0th component is nonzero (`WittVector.verschiebung_nonzero`).
Known identities (`WittVector.iterate_verschiebung_mul`) allow us to transform
the product of two such $x$ and $y$
to the form $V^{m+n}\left(F^n(w_x) \cdot F^m(w_y)\right)$,
the 0th component of which must be nonzero.
## Main declarations
* `WittVector.iterate_verschiebung_mul_coeff` : an identity from [Haze09]
* `WittVector.instIsDomain`
-/
noncomputable section
open scoped Classical
namespace WittVector
open Function
variable {p : ℕ} {R : Type*}
local notation "𝕎" => WittVector p -- type as `\bbW`
/-!
## The `shift` operator
-/
/--
`WittVector.verschiebung` translates the entries of a Witt vector upward, inserting 0s in the gaps.
`WittVector.shift` does the opposite, removing the first entries.
This is mainly useful as an auxiliary construction for `WittVector.verschiebung_nonzero`.
-/
def shift (x : 𝕎 R) (n : ℕ) : 𝕎 R :=
@mk' p R fun i => x.coeff (n + i)
#align witt_vector.shift WittVector.shift
theorem shift_coeff (x : 𝕎 R) (n k : ℕ) : (x.shift n).coeff k = x.coeff (n + k) :=
rfl
#align witt_vector.shift_coeff WittVector.shift_coeff
variable [hp : Fact p.Prime] [CommRing R]
theorem verschiebung_shift (x : 𝕎 R) (k : ℕ) (h : ∀ i < k + 1, x.coeff i = 0) :
verschiebung (x.shift k.succ) = x.shift k := by
ext ⟨j⟩
· rw [verschiebung_coeff_zero, shift_coeff, h]
apply Nat.lt_succ_self
· simp only [verschiebung_coeff_succ, shift]
congr 1
rw [Nat.add_succ, add_comm, Nat.add_succ, add_comm]
#align witt_vector.verschiebung_shift WittVector.verschiebung_shift
theorem eq_iterate_verschiebung {x : 𝕎 R} {n : ℕ} (h : ∀ i < n, x.coeff i = 0) :
x = verschiebung^[n] (x.shift n) := by
induction' n with k ih
· cases x; simp [shift]
· dsimp; rw [verschiebung_shift]
· exact ih fun i hi => h _ (hi.trans (Nat.lt_succ_self _))
· exact h
#align witt_vector.eq_iterate_verschiebung WittVector.eq_iterate_verschiebung
theorem verschiebung_nonzero {x : 𝕎 R} (hx : x ≠ 0) :
∃ n : ℕ, ∃ x' : 𝕎 R, x'.coeff 0 ≠ 0 ∧ x = verschiebung^[n] x' := by
have hex : ∃ k : ℕ, x.coeff k ≠ 0 := by
by_contra! hall
apply hx
ext i
|
simp only [hall, zero_coeff]
|
theorem verschiebung_nonzero {x : 𝕎 R} (hx : x ≠ 0) :
∃ n : ℕ, ∃ x' : 𝕎 R, x'.coeff 0 ≠ 0 ∧ x = verschiebung^[n] x' := by
have hex : ∃ k : ℕ, x.coeff k ≠ 0 := by
by_contra! hall
apply hx
ext i
|
Mathlib.RingTheory.WittVector.Domain.88_0.4uLlcZNQ2uiRcjJ
|
theorem verschiebung_nonzero {x : 𝕎 R} (hx : x ≠ 0) :
∃ n : ℕ, ∃ x' : 𝕎 R, x'.coeff 0 ≠ 0 ∧ x = verschiebung^[n] x'
|
Mathlib_RingTheory_WittVector_Domain
|
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝ : CommRing R
x : 𝕎 R
hx : x ≠ 0
hex : ∃ k, coeff x k ≠ 0
⊢ ∃ n x', coeff x' 0 ≠ 0 ∧ x = (⇑verschiebung)^[n] x'
|
/-
Copyright (c) 2022 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.Identities
#align_import ring_theory.witt_vector.domain from "leanprover-community/mathlib"@"b1d911acd60ab198808e853292106ee352b648ea"
/-!
# Witt vectors over a domain
This file builds to the proof `WittVector.instIsDomain`,
an instance that says if `R` is an integral domain, then so is `𝕎 R`.
It depends on the API around iterated applications
of `WittVector.verschiebung` and `WittVector.frobenius`
found in `Identities.lean`.
The [proof sketch](https://math.stackexchange.com/questions/4117247/ring-of-witt-vectors-over-an-integral-domain/4118723#4118723)
goes as follows:
any nonzero $x$ is an iterated application of $V$
to some vector $w_x$ whose 0th component is nonzero (`WittVector.verschiebung_nonzero`).
Known identities (`WittVector.iterate_verschiebung_mul`) allow us to transform
the product of two such $x$ and $y$
to the form $V^{m+n}\left(F^n(w_x) \cdot F^m(w_y)\right)$,
the 0th component of which must be nonzero.
## Main declarations
* `WittVector.iterate_verschiebung_mul_coeff` : an identity from [Haze09]
* `WittVector.instIsDomain`
-/
noncomputable section
open scoped Classical
namespace WittVector
open Function
variable {p : ℕ} {R : Type*}
local notation "𝕎" => WittVector p -- type as `\bbW`
/-!
## The `shift` operator
-/
/--
`WittVector.verschiebung` translates the entries of a Witt vector upward, inserting 0s in the gaps.
`WittVector.shift` does the opposite, removing the first entries.
This is mainly useful as an auxiliary construction for `WittVector.verschiebung_nonzero`.
-/
def shift (x : 𝕎 R) (n : ℕ) : 𝕎 R :=
@mk' p R fun i => x.coeff (n + i)
#align witt_vector.shift WittVector.shift
theorem shift_coeff (x : 𝕎 R) (n k : ℕ) : (x.shift n).coeff k = x.coeff (n + k) :=
rfl
#align witt_vector.shift_coeff WittVector.shift_coeff
variable [hp : Fact p.Prime] [CommRing R]
theorem verschiebung_shift (x : 𝕎 R) (k : ℕ) (h : ∀ i < k + 1, x.coeff i = 0) :
verschiebung (x.shift k.succ) = x.shift k := by
ext ⟨j⟩
· rw [verschiebung_coeff_zero, shift_coeff, h]
apply Nat.lt_succ_self
· simp only [verschiebung_coeff_succ, shift]
congr 1
rw [Nat.add_succ, add_comm, Nat.add_succ, add_comm]
#align witt_vector.verschiebung_shift WittVector.verschiebung_shift
theorem eq_iterate_verschiebung {x : 𝕎 R} {n : ℕ} (h : ∀ i < n, x.coeff i = 0) :
x = verschiebung^[n] (x.shift n) := by
induction' n with k ih
· cases x; simp [shift]
· dsimp; rw [verschiebung_shift]
· exact ih fun i hi => h _ (hi.trans (Nat.lt_succ_self _))
· exact h
#align witt_vector.eq_iterate_verschiebung WittVector.eq_iterate_verschiebung
theorem verschiebung_nonzero {x : 𝕎 R} (hx : x ≠ 0) :
∃ n : ℕ, ∃ x' : 𝕎 R, x'.coeff 0 ≠ 0 ∧ x = verschiebung^[n] x' := by
have hex : ∃ k : ℕ, x.coeff k ≠ 0 := by
by_contra! hall
apply hx
ext i
simp only [hall, zero_coeff]
|
let n := Nat.find hex
|
theorem verschiebung_nonzero {x : 𝕎 R} (hx : x ≠ 0) :
∃ n : ℕ, ∃ x' : 𝕎 R, x'.coeff 0 ≠ 0 ∧ x = verschiebung^[n] x' := by
have hex : ∃ k : ℕ, x.coeff k ≠ 0 := by
by_contra! hall
apply hx
ext i
simp only [hall, zero_coeff]
|
Mathlib.RingTheory.WittVector.Domain.88_0.4uLlcZNQ2uiRcjJ
|
theorem verschiebung_nonzero {x : 𝕎 R} (hx : x ≠ 0) :
∃ n : ℕ, ∃ x' : 𝕎 R, x'.coeff 0 ≠ 0 ∧ x = verschiebung^[n] x'
|
Mathlib_RingTheory_WittVector_Domain
|
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝ : CommRing R
x : 𝕎 R
hx : x ≠ 0
hex : ∃ k, coeff x k ≠ 0
n : ℕ := Nat.find hex
⊢ ∃ n x', coeff x' 0 ≠ 0 ∧ x = (⇑verschiebung)^[n] x'
|
/-
Copyright (c) 2022 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.Identities
#align_import ring_theory.witt_vector.domain from "leanprover-community/mathlib"@"b1d911acd60ab198808e853292106ee352b648ea"
/-!
# Witt vectors over a domain
This file builds to the proof `WittVector.instIsDomain`,
an instance that says if `R` is an integral domain, then so is `𝕎 R`.
It depends on the API around iterated applications
of `WittVector.verschiebung` and `WittVector.frobenius`
found in `Identities.lean`.
The [proof sketch](https://math.stackexchange.com/questions/4117247/ring-of-witt-vectors-over-an-integral-domain/4118723#4118723)
goes as follows:
any nonzero $x$ is an iterated application of $V$
to some vector $w_x$ whose 0th component is nonzero (`WittVector.verschiebung_nonzero`).
Known identities (`WittVector.iterate_verschiebung_mul`) allow us to transform
the product of two such $x$ and $y$
to the form $V^{m+n}\left(F^n(w_x) \cdot F^m(w_y)\right)$,
the 0th component of which must be nonzero.
## Main declarations
* `WittVector.iterate_verschiebung_mul_coeff` : an identity from [Haze09]
* `WittVector.instIsDomain`
-/
noncomputable section
open scoped Classical
namespace WittVector
open Function
variable {p : ℕ} {R : Type*}
local notation "𝕎" => WittVector p -- type as `\bbW`
/-!
## The `shift` operator
-/
/--
`WittVector.verschiebung` translates the entries of a Witt vector upward, inserting 0s in the gaps.
`WittVector.shift` does the opposite, removing the first entries.
This is mainly useful as an auxiliary construction for `WittVector.verschiebung_nonzero`.
-/
def shift (x : 𝕎 R) (n : ℕ) : 𝕎 R :=
@mk' p R fun i => x.coeff (n + i)
#align witt_vector.shift WittVector.shift
theorem shift_coeff (x : 𝕎 R) (n k : ℕ) : (x.shift n).coeff k = x.coeff (n + k) :=
rfl
#align witt_vector.shift_coeff WittVector.shift_coeff
variable [hp : Fact p.Prime] [CommRing R]
theorem verschiebung_shift (x : 𝕎 R) (k : ℕ) (h : ∀ i < k + 1, x.coeff i = 0) :
verschiebung (x.shift k.succ) = x.shift k := by
ext ⟨j⟩
· rw [verschiebung_coeff_zero, shift_coeff, h]
apply Nat.lt_succ_self
· simp only [verschiebung_coeff_succ, shift]
congr 1
rw [Nat.add_succ, add_comm, Nat.add_succ, add_comm]
#align witt_vector.verschiebung_shift WittVector.verschiebung_shift
theorem eq_iterate_verschiebung {x : 𝕎 R} {n : ℕ} (h : ∀ i < n, x.coeff i = 0) :
x = verschiebung^[n] (x.shift n) := by
induction' n with k ih
· cases x; simp [shift]
· dsimp; rw [verschiebung_shift]
· exact ih fun i hi => h _ (hi.trans (Nat.lt_succ_self _))
· exact h
#align witt_vector.eq_iterate_verschiebung WittVector.eq_iterate_verschiebung
theorem verschiebung_nonzero {x : 𝕎 R} (hx : x ≠ 0) :
∃ n : ℕ, ∃ x' : 𝕎 R, x'.coeff 0 ≠ 0 ∧ x = verschiebung^[n] x' := by
have hex : ∃ k : ℕ, x.coeff k ≠ 0 := by
by_contra! hall
apply hx
ext i
simp only [hall, zero_coeff]
let n := Nat.find hex
|
use n, x.shift n
|
theorem verschiebung_nonzero {x : 𝕎 R} (hx : x ≠ 0) :
∃ n : ℕ, ∃ x' : 𝕎 R, x'.coeff 0 ≠ 0 ∧ x = verschiebung^[n] x' := by
have hex : ∃ k : ℕ, x.coeff k ≠ 0 := by
by_contra! hall
apply hx
ext i
simp only [hall, zero_coeff]
let n := Nat.find hex
|
Mathlib.RingTheory.WittVector.Domain.88_0.4uLlcZNQ2uiRcjJ
|
theorem verschiebung_nonzero {x : 𝕎 R} (hx : x ≠ 0) :
∃ n : ℕ, ∃ x' : 𝕎 R, x'.coeff 0 ≠ 0 ∧ x = verschiebung^[n] x'
|
Mathlib_RingTheory_WittVector_Domain
|
case h
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝ : CommRing R
x : 𝕎 R
hx : x ≠ 0
hex : ∃ k, coeff x k ≠ 0
n : ℕ := Nat.find hex
⊢ coeff (shift x n) 0 ≠ 0 ∧ x = (⇑verschiebung)^[n] (shift x n)
|
/-
Copyright (c) 2022 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.Identities
#align_import ring_theory.witt_vector.domain from "leanprover-community/mathlib"@"b1d911acd60ab198808e853292106ee352b648ea"
/-!
# Witt vectors over a domain
This file builds to the proof `WittVector.instIsDomain`,
an instance that says if `R` is an integral domain, then so is `𝕎 R`.
It depends on the API around iterated applications
of `WittVector.verschiebung` and `WittVector.frobenius`
found in `Identities.lean`.
The [proof sketch](https://math.stackexchange.com/questions/4117247/ring-of-witt-vectors-over-an-integral-domain/4118723#4118723)
goes as follows:
any nonzero $x$ is an iterated application of $V$
to some vector $w_x$ whose 0th component is nonzero (`WittVector.verschiebung_nonzero`).
Known identities (`WittVector.iterate_verschiebung_mul`) allow us to transform
the product of two such $x$ and $y$
to the form $V^{m+n}\left(F^n(w_x) \cdot F^m(w_y)\right)$,
the 0th component of which must be nonzero.
## Main declarations
* `WittVector.iterate_verschiebung_mul_coeff` : an identity from [Haze09]
* `WittVector.instIsDomain`
-/
noncomputable section
open scoped Classical
namespace WittVector
open Function
variable {p : ℕ} {R : Type*}
local notation "𝕎" => WittVector p -- type as `\bbW`
/-!
## The `shift` operator
-/
/--
`WittVector.verschiebung` translates the entries of a Witt vector upward, inserting 0s in the gaps.
`WittVector.shift` does the opposite, removing the first entries.
This is mainly useful as an auxiliary construction for `WittVector.verschiebung_nonzero`.
-/
def shift (x : 𝕎 R) (n : ℕ) : 𝕎 R :=
@mk' p R fun i => x.coeff (n + i)
#align witt_vector.shift WittVector.shift
theorem shift_coeff (x : 𝕎 R) (n k : ℕ) : (x.shift n).coeff k = x.coeff (n + k) :=
rfl
#align witt_vector.shift_coeff WittVector.shift_coeff
variable [hp : Fact p.Prime] [CommRing R]
theorem verschiebung_shift (x : 𝕎 R) (k : ℕ) (h : ∀ i < k + 1, x.coeff i = 0) :
verschiebung (x.shift k.succ) = x.shift k := by
ext ⟨j⟩
· rw [verschiebung_coeff_zero, shift_coeff, h]
apply Nat.lt_succ_self
· simp only [verschiebung_coeff_succ, shift]
congr 1
rw [Nat.add_succ, add_comm, Nat.add_succ, add_comm]
#align witt_vector.verschiebung_shift WittVector.verschiebung_shift
theorem eq_iterate_verschiebung {x : 𝕎 R} {n : ℕ} (h : ∀ i < n, x.coeff i = 0) :
x = verschiebung^[n] (x.shift n) := by
induction' n with k ih
· cases x; simp [shift]
· dsimp; rw [verschiebung_shift]
· exact ih fun i hi => h _ (hi.trans (Nat.lt_succ_self _))
· exact h
#align witt_vector.eq_iterate_verschiebung WittVector.eq_iterate_verschiebung
theorem verschiebung_nonzero {x : 𝕎 R} (hx : x ≠ 0) :
∃ n : ℕ, ∃ x' : 𝕎 R, x'.coeff 0 ≠ 0 ∧ x = verschiebung^[n] x' := by
have hex : ∃ k : ℕ, x.coeff k ≠ 0 := by
by_contra! hall
apply hx
ext i
simp only [hall, zero_coeff]
let n := Nat.find hex
use n, x.shift n
|
refine' ⟨Nat.find_spec hex, eq_iterate_verschiebung fun i hi => not_not.mp _⟩
|
theorem verschiebung_nonzero {x : 𝕎 R} (hx : x ≠ 0) :
∃ n : ℕ, ∃ x' : 𝕎 R, x'.coeff 0 ≠ 0 ∧ x = verschiebung^[n] x' := by
have hex : ∃ k : ℕ, x.coeff k ≠ 0 := by
by_contra! hall
apply hx
ext i
simp only [hall, zero_coeff]
let n := Nat.find hex
use n, x.shift n
|
Mathlib.RingTheory.WittVector.Domain.88_0.4uLlcZNQ2uiRcjJ
|
theorem verschiebung_nonzero {x : 𝕎 R} (hx : x ≠ 0) :
∃ n : ℕ, ∃ x' : 𝕎 R, x'.coeff 0 ≠ 0 ∧ x = verschiebung^[n] x'
|
Mathlib_RingTheory_WittVector_Domain
|
case h
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝ : CommRing R
x : 𝕎 R
hx : x ≠ 0
hex : ∃ k, coeff x k ≠ 0
n : ℕ := Nat.find hex
i : ℕ
hi : i < n
⊢ ¬¬coeff x i = 0
|
/-
Copyright (c) 2022 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.Identities
#align_import ring_theory.witt_vector.domain from "leanprover-community/mathlib"@"b1d911acd60ab198808e853292106ee352b648ea"
/-!
# Witt vectors over a domain
This file builds to the proof `WittVector.instIsDomain`,
an instance that says if `R` is an integral domain, then so is `𝕎 R`.
It depends on the API around iterated applications
of `WittVector.verschiebung` and `WittVector.frobenius`
found in `Identities.lean`.
The [proof sketch](https://math.stackexchange.com/questions/4117247/ring-of-witt-vectors-over-an-integral-domain/4118723#4118723)
goes as follows:
any nonzero $x$ is an iterated application of $V$
to some vector $w_x$ whose 0th component is nonzero (`WittVector.verschiebung_nonzero`).
Known identities (`WittVector.iterate_verschiebung_mul`) allow us to transform
the product of two such $x$ and $y$
to the form $V^{m+n}\left(F^n(w_x) \cdot F^m(w_y)\right)$,
the 0th component of which must be nonzero.
## Main declarations
* `WittVector.iterate_verschiebung_mul_coeff` : an identity from [Haze09]
* `WittVector.instIsDomain`
-/
noncomputable section
open scoped Classical
namespace WittVector
open Function
variable {p : ℕ} {R : Type*}
local notation "𝕎" => WittVector p -- type as `\bbW`
/-!
## The `shift` operator
-/
/--
`WittVector.verschiebung` translates the entries of a Witt vector upward, inserting 0s in the gaps.
`WittVector.shift` does the opposite, removing the first entries.
This is mainly useful as an auxiliary construction for `WittVector.verschiebung_nonzero`.
-/
def shift (x : 𝕎 R) (n : ℕ) : 𝕎 R :=
@mk' p R fun i => x.coeff (n + i)
#align witt_vector.shift WittVector.shift
theorem shift_coeff (x : 𝕎 R) (n k : ℕ) : (x.shift n).coeff k = x.coeff (n + k) :=
rfl
#align witt_vector.shift_coeff WittVector.shift_coeff
variable [hp : Fact p.Prime] [CommRing R]
theorem verschiebung_shift (x : 𝕎 R) (k : ℕ) (h : ∀ i < k + 1, x.coeff i = 0) :
verschiebung (x.shift k.succ) = x.shift k := by
ext ⟨j⟩
· rw [verschiebung_coeff_zero, shift_coeff, h]
apply Nat.lt_succ_self
· simp only [verschiebung_coeff_succ, shift]
congr 1
rw [Nat.add_succ, add_comm, Nat.add_succ, add_comm]
#align witt_vector.verschiebung_shift WittVector.verschiebung_shift
theorem eq_iterate_verschiebung {x : 𝕎 R} {n : ℕ} (h : ∀ i < n, x.coeff i = 0) :
x = verschiebung^[n] (x.shift n) := by
induction' n with k ih
· cases x; simp [shift]
· dsimp; rw [verschiebung_shift]
· exact ih fun i hi => h _ (hi.trans (Nat.lt_succ_self _))
· exact h
#align witt_vector.eq_iterate_verschiebung WittVector.eq_iterate_verschiebung
theorem verschiebung_nonzero {x : 𝕎 R} (hx : x ≠ 0) :
∃ n : ℕ, ∃ x' : 𝕎 R, x'.coeff 0 ≠ 0 ∧ x = verschiebung^[n] x' := by
have hex : ∃ k : ℕ, x.coeff k ≠ 0 := by
by_contra! hall
apply hx
ext i
simp only [hall, zero_coeff]
let n := Nat.find hex
use n, x.shift n
refine' ⟨Nat.find_spec hex, eq_iterate_verschiebung fun i hi => not_not.mp _⟩
|
exact Nat.find_min hex hi
|
theorem verschiebung_nonzero {x : 𝕎 R} (hx : x ≠ 0) :
∃ n : ℕ, ∃ x' : 𝕎 R, x'.coeff 0 ≠ 0 ∧ x = verschiebung^[n] x' := by
have hex : ∃ k : ℕ, x.coeff k ≠ 0 := by
by_contra! hall
apply hx
ext i
simp only [hall, zero_coeff]
let n := Nat.find hex
use n, x.shift n
refine' ⟨Nat.find_spec hex, eq_iterate_verschiebung fun i hi => not_not.mp _⟩
|
Mathlib.RingTheory.WittVector.Domain.88_0.4uLlcZNQ2uiRcjJ
|
theorem verschiebung_nonzero {x : 𝕎 R} (hx : x ≠ 0) :
∃ n : ℕ, ∃ x' : 𝕎 R, x'.coeff 0 ≠ 0 ∧ x = verschiebung^[n] x'
|
Mathlib_RingTheory_WittVector_Domain
|
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝² : CommRing R
inst✝¹ : CharP R p
inst✝ : NoZeroDivisors R
x y : 𝕎 R
⊢ x * y = 0 → x = 0 ∨ y = 0
|
/-
Copyright (c) 2022 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.Identities
#align_import ring_theory.witt_vector.domain from "leanprover-community/mathlib"@"b1d911acd60ab198808e853292106ee352b648ea"
/-!
# Witt vectors over a domain
This file builds to the proof `WittVector.instIsDomain`,
an instance that says if `R` is an integral domain, then so is `𝕎 R`.
It depends on the API around iterated applications
of `WittVector.verschiebung` and `WittVector.frobenius`
found in `Identities.lean`.
The [proof sketch](https://math.stackexchange.com/questions/4117247/ring-of-witt-vectors-over-an-integral-domain/4118723#4118723)
goes as follows:
any nonzero $x$ is an iterated application of $V$
to some vector $w_x$ whose 0th component is nonzero (`WittVector.verschiebung_nonzero`).
Known identities (`WittVector.iterate_verschiebung_mul`) allow us to transform
the product of two such $x$ and $y$
to the form $V^{m+n}\left(F^n(w_x) \cdot F^m(w_y)\right)$,
the 0th component of which must be nonzero.
## Main declarations
* `WittVector.iterate_verschiebung_mul_coeff` : an identity from [Haze09]
* `WittVector.instIsDomain`
-/
noncomputable section
open scoped Classical
namespace WittVector
open Function
variable {p : ℕ} {R : Type*}
local notation "𝕎" => WittVector p -- type as `\bbW`
/-!
## The `shift` operator
-/
/--
`WittVector.verschiebung` translates the entries of a Witt vector upward, inserting 0s in the gaps.
`WittVector.shift` does the opposite, removing the first entries.
This is mainly useful as an auxiliary construction for `WittVector.verschiebung_nonzero`.
-/
def shift (x : 𝕎 R) (n : ℕ) : 𝕎 R :=
@mk' p R fun i => x.coeff (n + i)
#align witt_vector.shift WittVector.shift
theorem shift_coeff (x : 𝕎 R) (n k : ℕ) : (x.shift n).coeff k = x.coeff (n + k) :=
rfl
#align witt_vector.shift_coeff WittVector.shift_coeff
variable [hp : Fact p.Prime] [CommRing R]
theorem verschiebung_shift (x : 𝕎 R) (k : ℕ) (h : ∀ i < k + 1, x.coeff i = 0) :
verschiebung (x.shift k.succ) = x.shift k := by
ext ⟨j⟩
· rw [verschiebung_coeff_zero, shift_coeff, h]
apply Nat.lt_succ_self
· simp only [verschiebung_coeff_succ, shift]
congr 1
rw [Nat.add_succ, add_comm, Nat.add_succ, add_comm]
#align witt_vector.verschiebung_shift WittVector.verschiebung_shift
theorem eq_iterate_verschiebung {x : 𝕎 R} {n : ℕ} (h : ∀ i < n, x.coeff i = 0) :
x = verschiebung^[n] (x.shift n) := by
induction' n with k ih
· cases x; simp [shift]
· dsimp; rw [verschiebung_shift]
· exact ih fun i hi => h _ (hi.trans (Nat.lt_succ_self _))
· exact h
#align witt_vector.eq_iterate_verschiebung WittVector.eq_iterate_verschiebung
theorem verschiebung_nonzero {x : 𝕎 R} (hx : x ≠ 0) :
∃ n : ℕ, ∃ x' : 𝕎 R, x'.coeff 0 ≠ 0 ∧ x = verschiebung^[n] x' := by
have hex : ∃ k : ℕ, x.coeff k ≠ 0 := by
by_contra! hall
apply hx
ext i
simp only [hall, zero_coeff]
let n := Nat.find hex
use n, x.shift n
refine' ⟨Nat.find_spec hex, eq_iterate_verschiebung fun i hi => not_not.mp _⟩
exact Nat.find_min hex hi
#align witt_vector.verschiebung_nonzero WittVector.verschiebung_nonzero
/-!
## Witt vectors over a domain
If `R` is an integral domain, then so is `𝕎 R`.
This argument is adapted from
<https://math.stackexchange.com/questions/4117247/ring-of-witt-vectors-over-an-integral-domain/4118723#4118723>.
-/
instance [CharP R p] [NoZeroDivisors R] : NoZeroDivisors (𝕎 R) :=
⟨fun {x y} => by
|
contrapose!
|
instance [CharP R p] [NoZeroDivisors R] : NoZeroDivisors (𝕎 R) :=
⟨fun {x y} => by
|
Mathlib.RingTheory.WittVector.Domain.110_0.4uLlcZNQ2uiRcjJ
|
instance [CharP R p] [NoZeroDivisors R] : NoZeroDivisors (𝕎 R)
|
Mathlib_RingTheory_WittVector_Domain
|
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝² : CommRing R
inst✝¹ : CharP R p
inst✝ : NoZeroDivisors R
x y : 𝕎 R
⊢ x ≠ 0 ∧ y ≠ 0 → x * y ≠ 0
|
/-
Copyright (c) 2022 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.Identities
#align_import ring_theory.witt_vector.domain from "leanprover-community/mathlib"@"b1d911acd60ab198808e853292106ee352b648ea"
/-!
# Witt vectors over a domain
This file builds to the proof `WittVector.instIsDomain`,
an instance that says if `R` is an integral domain, then so is `𝕎 R`.
It depends on the API around iterated applications
of `WittVector.verschiebung` and `WittVector.frobenius`
found in `Identities.lean`.
The [proof sketch](https://math.stackexchange.com/questions/4117247/ring-of-witt-vectors-over-an-integral-domain/4118723#4118723)
goes as follows:
any nonzero $x$ is an iterated application of $V$
to some vector $w_x$ whose 0th component is nonzero (`WittVector.verschiebung_nonzero`).
Known identities (`WittVector.iterate_verschiebung_mul`) allow us to transform
the product of two such $x$ and $y$
to the form $V^{m+n}\left(F^n(w_x) \cdot F^m(w_y)\right)$,
the 0th component of which must be nonzero.
## Main declarations
* `WittVector.iterate_verschiebung_mul_coeff` : an identity from [Haze09]
* `WittVector.instIsDomain`
-/
noncomputable section
open scoped Classical
namespace WittVector
open Function
variable {p : ℕ} {R : Type*}
local notation "𝕎" => WittVector p -- type as `\bbW`
/-!
## The `shift` operator
-/
/--
`WittVector.verschiebung` translates the entries of a Witt vector upward, inserting 0s in the gaps.
`WittVector.shift` does the opposite, removing the first entries.
This is mainly useful as an auxiliary construction for `WittVector.verschiebung_nonzero`.
-/
def shift (x : 𝕎 R) (n : ℕ) : 𝕎 R :=
@mk' p R fun i => x.coeff (n + i)
#align witt_vector.shift WittVector.shift
theorem shift_coeff (x : 𝕎 R) (n k : ℕ) : (x.shift n).coeff k = x.coeff (n + k) :=
rfl
#align witt_vector.shift_coeff WittVector.shift_coeff
variable [hp : Fact p.Prime] [CommRing R]
theorem verschiebung_shift (x : 𝕎 R) (k : ℕ) (h : ∀ i < k + 1, x.coeff i = 0) :
verschiebung (x.shift k.succ) = x.shift k := by
ext ⟨j⟩
· rw [verschiebung_coeff_zero, shift_coeff, h]
apply Nat.lt_succ_self
· simp only [verschiebung_coeff_succ, shift]
congr 1
rw [Nat.add_succ, add_comm, Nat.add_succ, add_comm]
#align witt_vector.verschiebung_shift WittVector.verschiebung_shift
theorem eq_iterate_verschiebung {x : 𝕎 R} {n : ℕ} (h : ∀ i < n, x.coeff i = 0) :
x = verschiebung^[n] (x.shift n) := by
induction' n with k ih
· cases x; simp [shift]
· dsimp; rw [verschiebung_shift]
· exact ih fun i hi => h _ (hi.trans (Nat.lt_succ_self _))
· exact h
#align witt_vector.eq_iterate_verschiebung WittVector.eq_iterate_verschiebung
theorem verschiebung_nonzero {x : 𝕎 R} (hx : x ≠ 0) :
∃ n : ℕ, ∃ x' : 𝕎 R, x'.coeff 0 ≠ 0 ∧ x = verschiebung^[n] x' := by
have hex : ∃ k : ℕ, x.coeff k ≠ 0 := by
by_contra! hall
apply hx
ext i
simp only [hall, zero_coeff]
let n := Nat.find hex
use n, x.shift n
refine' ⟨Nat.find_spec hex, eq_iterate_verschiebung fun i hi => not_not.mp _⟩
exact Nat.find_min hex hi
#align witt_vector.verschiebung_nonzero WittVector.verschiebung_nonzero
/-!
## Witt vectors over a domain
If `R` is an integral domain, then so is `𝕎 R`.
This argument is adapted from
<https://math.stackexchange.com/questions/4117247/ring-of-witt-vectors-over-an-integral-domain/4118723#4118723>.
-/
instance [CharP R p] [NoZeroDivisors R] : NoZeroDivisors (𝕎 R) :=
⟨fun {x y} => by
contrapose!
|
rintro ⟨ha, hb⟩
|
instance [CharP R p] [NoZeroDivisors R] : NoZeroDivisors (𝕎 R) :=
⟨fun {x y} => by
contrapose!
|
Mathlib.RingTheory.WittVector.Domain.110_0.4uLlcZNQ2uiRcjJ
|
instance [CharP R p] [NoZeroDivisors R] : NoZeroDivisors (𝕎 R)
|
Mathlib_RingTheory_WittVector_Domain
|
case intro
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝² : CommRing R
inst✝¹ : CharP R p
inst✝ : NoZeroDivisors R
x y : 𝕎 R
ha : x ≠ 0
hb : y ≠ 0
⊢ x * y ≠ 0
|
/-
Copyright (c) 2022 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.Identities
#align_import ring_theory.witt_vector.domain from "leanprover-community/mathlib"@"b1d911acd60ab198808e853292106ee352b648ea"
/-!
# Witt vectors over a domain
This file builds to the proof `WittVector.instIsDomain`,
an instance that says if `R` is an integral domain, then so is `𝕎 R`.
It depends on the API around iterated applications
of `WittVector.verschiebung` and `WittVector.frobenius`
found in `Identities.lean`.
The [proof sketch](https://math.stackexchange.com/questions/4117247/ring-of-witt-vectors-over-an-integral-domain/4118723#4118723)
goes as follows:
any nonzero $x$ is an iterated application of $V$
to some vector $w_x$ whose 0th component is nonzero (`WittVector.verschiebung_nonzero`).
Known identities (`WittVector.iterate_verschiebung_mul`) allow us to transform
the product of two such $x$ and $y$
to the form $V^{m+n}\left(F^n(w_x) \cdot F^m(w_y)\right)$,
the 0th component of which must be nonzero.
## Main declarations
* `WittVector.iterate_verschiebung_mul_coeff` : an identity from [Haze09]
* `WittVector.instIsDomain`
-/
noncomputable section
open scoped Classical
namespace WittVector
open Function
variable {p : ℕ} {R : Type*}
local notation "𝕎" => WittVector p -- type as `\bbW`
/-!
## The `shift` operator
-/
/--
`WittVector.verschiebung` translates the entries of a Witt vector upward, inserting 0s in the gaps.
`WittVector.shift` does the opposite, removing the first entries.
This is mainly useful as an auxiliary construction for `WittVector.verschiebung_nonzero`.
-/
def shift (x : 𝕎 R) (n : ℕ) : 𝕎 R :=
@mk' p R fun i => x.coeff (n + i)
#align witt_vector.shift WittVector.shift
theorem shift_coeff (x : 𝕎 R) (n k : ℕ) : (x.shift n).coeff k = x.coeff (n + k) :=
rfl
#align witt_vector.shift_coeff WittVector.shift_coeff
variable [hp : Fact p.Prime] [CommRing R]
theorem verschiebung_shift (x : 𝕎 R) (k : ℕ) (h : ∀ i < k + 1, x.coeff i = 0) :
verschiebung (x.shift k.succ) = x.shift k := by
ext ⟨j⟩
· rw [verschiebung_coeff_zero, shift_coeff, h]
apply Nat.lt_succ_self
· simp only [verschiebung_coeff_succ, shift]
congr 1
rw [Nat.add_succ, add_comm, Nat.add_succ, add_comm]
#align witt_vector.verschiebung_shift WittVector.verschiebung_shift
theorem eq_iterate_verschiebung {x : 𝕎 R} {n : ℕ} (h : ∀ i < n, x.coeff i = 0) :
x = verschiebung^[n] (x.shift n) := by
induction' n with k ih
· cases x; simp [shift]
· dsimp; rw [verschiebung_shift]
· exact ih fun i hi => h _ (hi.trans (Nat.lt_succ_self _))
· exact h
#align witt_vector.eq_iterate_verschiebung WittVector.eq_iterate_verschiebung
theorem verschiebung_nonzero {x : 𝕎 R} (hx : x ≠ 0) :
∃ n : ℕ, ∃ x' : 𝕎 R, x'.coeff 0 ≠ 0 ∧ x = verschiebung^[n] x' := by
have hex : ∃ k : ℕ, x.coeff k ≠ 0 := by
by_contra! hall
apply hx
ext i
simp only [hall, zero_coeff]
let n := Nat.find hex
use n, x.shift n
refine' ⟨Nat.find_spec hex, eq_iterate_verschiebung fun i hi => not_not.mp _⟩
exact Nat.find_min hex hi
#align witt_vector.verschiebung_nonzero WittVector.verschiebung_nonzero
/-!
## Witt vectors over a domain
If `R` is an integral domain, then so is `𝕎 R`.
This argument is adapted from
<https://math.stackexchange.com/questions/4117247/ring-of-witt-vectors-over-an-integral-domain/4118723#4118723>.
-/
instance [CharP R p] [NoZeroDivisors R] : NoZeroDivisors (𝕎 R) :=
⟨fun {x y} => by
contrapose!
rintro ⟨ha, hb⟩
|
rcases verschiebung_nonzero ha with ⟨na, wa, hwa0, rfl⟩
|
instance [CharP R p] [NoZeroDivisors R] : NoZeroDivisors (𝕎 R) :=
⟨fun {x y} => by
contrapose!
rintro ⟨ha, hb⟩
|
Mathlib.RingTheory.WittVector.Domain.110_0.4uLlcZNQ2uiRcjJ
|
instance [CharP R p] [NoZeroDivisors R] : NoZeroDivisors (𝕎 R)
|
Mathlib_RingTheory_WittVector_Domain
|
case intro.intro.intro.intro
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝² : CommRing R
inst✝¹ : CharP R p
inst✝ : NoZeroDivisors R
y : 𝕎 R
hb : y ≠ 0
na : ℕ
wa : 𝕎 R
hwa0 : coeff wa 0 ≠ 0
ha : (⇑verschiebung)^[na] wa ≠ 0
⊢ (⇑verschiebung)^[na] wa * y ≠ 0
|
/-
Copyright (c) 2022 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.Identities
#align_import ring_theory.witt_vector.domain from "leanprover-community/mathlib"@"b1d911acd60ab198808e853292106ee352b648ea"
/-!
# Witt vectors over a domain
This file builds to the proof `WittVector.instIsDomain`,
an instance that says if `R` is an integral domain, then so is `𝕎 R`.
It depends on the API around iterated applications
of `WittVector.verschiebung` and `WittVector.frobenius`
found in `Identities.lean`.
The [proof sketch](https://math.stackexchange.com/questions/4117247/ring-of-witt-vectors-over-an-integral-domain/4118723#4118723)
goes as follows:
any nonzero $x$ is an iterated application of $V$
to some vector $w_x$ whose 0th component is nonzero (`WittVector.verschiebung_nonzero`).
Known identities (`WittVector.iterate_verschiebung_mul`) allow us to transform
the product of two such $x$ and $y$
to the form $V^{m+n}\left(F^n(w_x) \cdot F^m(w_y)\right)$,
the 0th component of which must be nonzero.
## Main declarations
* `WittVector.iterate_verschiebung_mul_coeff` : an identity from [Haze09]
* `WittVector.instIsDomain`
-/
noncomputable section
open scoped Classical
namespace WittVector
open Function
variable {p : ℕ} {R : Type*}
local notation "𝕎" => WittVector p -- type as `\bbW`
/-!
## The `shift` operator
-/
/--
`WittVector.verschiebung` translates the entries of a Witt vector upward, inserting 0s in the gaps.
`WittVector.shift` does the opposite, removing the first entries.
This is mainly useful as an auxiliary construction for `WittVector.verschiebung_nonzero`.
-/
def shift (x : 𝕎 R) (n : ℕ) : 𝕎 R :=
@mk' p R fun i => x.coeff (n + i)
#align witt_vector.shift WittVector.shift
theorem shift_coeff (x : 𝕎 R) (n k : ℕ) : (x.shift n).coeff k = x.coeff (n + k) :=
rfl
#align witt_vector.shift_coeff WittVector.shift_coeff
variable [hp : Fact p.Prime] [CommRing R]
theorem verschiebung_shift (x : 𝕎 R) (k : ℕ) (h : ∀ i < k + 1, x.coeff i = 0) :
verschiebung (x.shift k.succ) = x.shift k := by
ext ⟨j⟩
· rw [verschiebung_coeff_zero, shift_coeff, h]
apply Nat.lt_succ_self
· simp only [verschiebung_coeff_succ, shift]
congr 1
rw [Nat.add_succ, add_comm, Nat.add_succ, add_comm]
#align witt_vector.verschiebung_shift WittVector.verschiebung_shift
theorem eq_iterate_verschiebung {x : 𝕎 R} {n : ℕ} (h : ∀ i < n, x.coeff i = 0) :
x = verschiebung^[n] (x.shift n) := by
induction' n with k ih
· cases x; simp [shift]
· dsimp; rw [verschiebung_shift]
· exact ih fun i hi => h _ (hi.trans (Nat.lt_succ_self _))
· exact h
#align witt_vector.eq_iterate_verschiebung WittVector.eq_iterate_verschiebung
theorem verschiebung_nonzero {x : 𝕎 R} (hx : x ≠ 0) :
∃ n : ℕ, ∃ x' : 𝕎 R, x'.coeff 0 ≠ 0 ∧ x = verschiebung^[n] x' := by
have hex : ∃ k : ℕ, x.coeff k ≠ 0 := by
by_contra! hall
apply hx
ext i
simp only [hall, zero_coeff]
let n := Nat.find hex
use n, x.shift n
refine' ⟨Nat.find_spec hex, eq_iterate_verschiebung fun i hi => not_not.mp _⟩
exact Nat.find_min hex hi
#align witt_vector.verschiebung_nonzero WittVector.verschiebung_nonzero
/-!
## Witt vectors over a domain
If `R` is an integral domain, then so is `𝕎 R`.
This argument is adapted from
<https://math.stackexchange.com/questions/4117247/ring-of-witt-vectors-over-an-integral-domain/4118723#4118723>.
-/
instance [CharP R p] [NoZeroDivisors R] : NoZeroDivisors (𝕎 R) :=
⟨fun {x y} => by
contrapose!
rintro ⟨ha, hb⟩
rcases verschiebung_nonzero ha with ⟨na, wa, hwa0, rfl⟩
|
rcases verschiebung_nonzero hb with ⟨nb, wb, hwb0, rfl⟩
|
instance [CharP R p] [NoZeroDivisors R] : NoZeroDivisors (𝕎 R) :=
⟨fun {x y} => by
contrapose!
rintro ⟨ha, hb⟩
rcases verschiebung_nonzero ha with ⟨na, wa, hwa0, rfl⟩
|
Mathlib.RingTheory.WittVector.Domain.110_0.4uLlcZNQ2uiRcjJ
|
instance [CharP R p] [NoZeroDivisors R] : NoZeroDivisors (𝕎 R)
|
Mathlib_RingTheory_WittVector_Domain
|
case intro.intro.intro.intro.intro.intro.intro
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝² : CommRing R
inst✝¹ : CharP R p
inst✝ : NoZeroDivisors R
na : ℕ
wa : 𝕎 R
hwa0 : coeff wa 0 ≠ 0
ha : (⇑verschiebung)^[na] wa ≠ 0
nb : ℕ
wb : 𝕎 R
hwb0 : coeff wb 0 ≠ 0
hb : (⇑verschiebung)^[nb] wb ≠ 0
⊢ (⇑verschiebung)^[na] wa * (⇑verschiebung)^[nb] wb ≠ 0
|
/-
Copyright (c) 2022 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.Identities
#align_import ring_theory.witt_vector.domain from "leanprover-community/mathlib"@"b1d911acd60ab198808e853292106ee352b648ea"
/-!
# Witt vectors over a domain
This file builds to the proof `WittVector.instIsDomain`,
an instance that says if `R` is an integral domain, then so is `𝕎 R`.
It depends on the API around iterated applications
of `WittVector.verschiebung` and `WittVector.frobenius`
found in `Identities.lean`.
The [proof sketch](https://math.stackexchange.com/questions/4117247/ring-of-witt-vectors-over-an-integral-domain/4118723#4118723)
goes as follows:
any nonzero $x$ is an iterated application of $V$
to some vector $w_x$ whose 0th component is nonzero (`WittVector.verschiebung_nonzero`).
Known identities (`WittVector.iterate_verschiebung_mul`) allow us to transform
the product of two such $x$ and $y$
to the form $V^{m+n}\left(F^n(w_x) \cdot F^m(w_y)\right)$,
the 0th component of which must be nonzero.
## Main declarations
* `WittVector.iterate_verschiebung_mul_coeff` : an identity from [Haze09]
* `WittVector.instIsDomain`
-/
noncomputable section
open scoped Classical
namespace WittVector
open Function
variable {p : ℕ} {R : Type*}
local notation "𝕎" => WittVector p -- type as `\bbW`
/-!
## The `shift` operator
-/
/--
`WittVector.verschiebung` translates the entries of a Witt vector upward, inserting 0s in the gaps.
`WittVector.shift` does the opposite, removing the first entries.
This is mainly useful as an auxiliary construction for `WittVector.verschiebung_nonzero`.
-/
def shift (x : 𝕎 R) (n : ℕ) : 𝕎 R :=
@mk' p R fun i => x.coeff (n + i)
#align witt_vector.shift WittVector.shift
theorem shift_coeff (x : 𝕎 R) (n k : ℕ) : (x.shift n).coeff k = x.coeff (n + k) :=
rfl
#align witt_vector.shift_coeff WittVector.shift_coeff
variable [hp : Fact p.Prime] [CommRing R]
theorem verschiebung_shift (x : 𝕎 R) (k : ℕ) (h : ∀ i < k + 1, x.coeff i = 0) :
verschiebung (x.shift k.succ) = x.shift k := by
ext ⟨j⟩
· rw [verschiebung_coeff_zero, shift_coeff, h]
apply Nat.lt_succ_self
· simp only [verschiebung_coeff_succ, shift]
congr 1
rw [Nat.add_succ, add_comm, Nat.add_succ, add_comm]
#align witt_vector.verschiebung_shift WittVector.verschiebung_shift
theorem eq_iterate_verschiebung {x : 𝕎 R} {n : ℕ} (h : ∀ i < n, x.coeff i = 0) :
x = verschiebung^[n] (x.shift n) := by
induction' n with k ih
· cases x; simp [shift]
· dsimp; rw [verschiebung_shift]
· exact ih fun i hi => h _ (hi.trans (Nat.lt_succ_self _))
· exact h
#align witt_vector.eq_iterate_verschiebung WittVector.eq_iterate_verschiebung
theorem verschiebung_nonzero {x : 𝕎 R} (hx : x ≠ 0) :
∃ n : ℕ, ∃ x' : 𝕎 R, x'.coeff 0 ≠ 0 ∧ x = verschiebung^[n] x' := by
have hex : ∃ k : ℕ, x.coeff k ≠ 0 := by
by_contra! hall
apply hx
ext i
simp only [hall, zero_coeff]
let n := Nat.find hex
use n, x.shift n
refine' ⟨Nat.find_spec hex, eq_iterate_verschiebung fun i hi => not_not.mp _⟩
exact Nat.find_min hex hi
#align witt_vector.verschiebung_nonzero WittVector.verschiebung_nonzero
/-!
## Witt vectors over a domain
If `R` is an integral domain, then so is `𝕎 R`.
This argument is adapted from
<https://math.stackexchange.com/questions/4117247/ring-of-witt-vectors-over-an-integral-domain/4118723#4118723>.
-/
instance [CharP R p] [NoZeroDivisors R] : NoZeroDivisors (𝕎 R) :=
⟨fun {x y} => by
contrapose!
rintro ⟨ha, hb⟩
rcases verschiebung_nonzero ha with ⟨na, wa, hwa0, rfl⟩
rcases verschiebung_nonzero hb with ⟨nb, wb, hwb0, rfl⟩
|
refine' ne_of_apply_ne (fun x => x.coeff (na + nb)) _
|
instance [CharP R p] [NoZeroDivisors R] : NoZeroDivisors (𝕎 R) :=
⟨fun {x y} => by
contrapose!
rintro ⟨ha, hb⟩
rcases verschiebung_nonzero ha with ⟨na, wa, hwa0, rfl⟩
rcases verschiebung_nonzero hb with ⟨nb, wb, hwb0, rfl⟩
|
Mathlib.RingTheory.WittVector.Domain.110_0.4uLlcZNQ2uiRcjJ
|
instance [CharP R p] [NoZeroDivisors R] : NoZeroDivisors (𝕎 R)
|
Mathlib_RingTheory_WittVector_Domain
|
case intro.intro.intro.intro.intro.intro.intro
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝² : CommRing R
inst✝¹ : CharP R p
inst✝ : NoZeroDivisors R
na : ℕ
wa : 𝕎 R
hwa0 : coeff wa 0 ≠ 0
ha : (⇑verschiebung)^[na] wa ≠ 0
nb : ℕ
wb : 𝕎 R
hwb0 : coeff wb 0 ≠ 0
hb : (⇑verschiebung)^[nb] wb ≠ 0
⊢ (fun x => coeff x (na + nb)) ((⇑verschiebung)^[na] wa * (⇑verschiebung)^[nb] wb) ≠ (fun x => coeff x (na + nb)) 0
|
/-
Copyright (c) 2022 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.Identities
#align_import ring_theory.witt_vector.domain from "leanprover-community/mathlib"@"b1d911acd60ab198808e853292106ee352b648ea"
/-!
# Witt vectors over a domain
This file builds to the proof `WittVector.instIsDomain`,
an instance that says if `R` is an integral domain, then so is `𝕎 R`.
It depends on the API around iterated applications
of `WittVector.verschiebung` and `WittVector.frobenius`
found in `Identities.lean`.
The [proof sketch](https://math.stackexchange.com/questions/4117247/ring-of-witt-vectors-over-an-integral-domain/4118723#4118723)
goes as follows:
any nonzero $x$ is an iterated application of $V$
to some vector $w_x$ whose 0th component is nonzero (`WittVector.verschiebung_nonzero`).
Known identities (`WittVector.iterate_verschiebung_mul`) allow us to transform
the product of two such $x$ and $y$
to the form $V^{m+n}\left(F^n(w_x) \cdot F^m(w_y)\right)$,
the 0th component of which must be nonzero.
## Main declarations
* `WittVector.iterate_verschiebung_mul_coeff` : an identity from [Haze09]
* `WittVector.instIsDomain`
-/
noncomputable section
open scoped Classical
namespace WittVector
open Function
variable {p : ℕ} {R : Type*}
local notation "𝕎" => WittVector p -- type as `\bbW`
/-!
## The `shift` operator
-/
/--
`WittVector.verschiebung` translates the entries of a Witt vector upward, inserting 0s in the gaps.
`WittVector.shift` does the opposite, removing the first entries.
This is mainly useful as an auxiliary construction for `WittVector.verschiebung_nonzero`.
-/
def shift (x : 𝕎 R) (n : ℕ) : 𝕎 R :=
@mk' p R fun i => x.coeff (n + i)
#align witt_vector.shift WittVector.shift
theorem shift_coeff (x : 𝕎 R) (n k : ℕ) : (x.shift n).coeff k = x.coeff (n + k) :=
rfl
#align witt_vector.shift_coeff WittVector.shift_coeff
variable [hp : Fact p.Prime] [CommRing R]
theorem verschiebung_shift (x : 𝕎 R) (k : ℕ) (h : ∀ i < k + 1, x.coeff i = 0) :
verschiebung (x.shift k.succ) = x.shift k := by
ext ⟨j⟩
· rw [verschiebung_coeff_zero, shift_coeff, h]
apply Nat.lt_succ_self
· simp only [verschiebung_coeff_succ, shift]
congr 1
rw [Nat.add_succ, add_comm, Nat.add_succ, add_comm]
#align witt_vector.verschiebung_shift WittVector.verschiebung_shift
theorem eq_iterate_verschiebung {x : 𝕎 R} {n : ℕ} (h : ∀ i < n, x.coeff i = 0) :
x = verschiebung^[n] (x.shift n) := by
induction' n with k ih
· cases x; simp [shift]
· dsimp; rw [verschiebung_shift]
· exact ih fun i hi => h _ (hi.trans (Nat.lt_succ_self _))
· exact h
#align witt_vector.eq_iterate_verschiebung WittVector.eq_iterate_verschiebung
theorem verschiebung_nonzero {x : 𝕎 R} (hx : x ≠ 0) :
∃ n : ℕ, ∃ x' : 𝕎 R, x'.coeff 0 ≠ 0 ∧ x = verschiebung^[n] x' := by
have hex : ∃ k : ℕ, x.coeff k ≠ 0 := by
by_contra! hall
apply hx
ext i
simp only [hall, zero_coeff]
let n := Nat.find hex
use n, x.shift n
refine' ⟨Nat.find_spec hex, eq_iterate_verschiebung fun i hi => not_not.mp _⟩
exact Nat.find_min hex hi
#align witt_vector.verschiebung_nonzero WittVector.verschiebung_nonzero
/-!
## Witt vectors over a domain
If `R` is an integral domain, then so is `𝕎 R`.
This argument is adapted from
<https://math.stackexchange.com/questions/4117247/ring-of-witt-vectors-over-an-integral-domain/4118723#4118723>.
-/
instance [CharP R p] [NoZeroDivisors R] : NoZeroDivisors (𝕎 R) :=
⟨fun {x y} => by
contrapose!
rintro ⟨ha, hb⟩
rcases verschiebung_nonzero ha with ⟨na, wa, hwa0, rfl⟩
rcases verschiebung_nonzero hb with ⟨nb, wb, hwb0, rfl⟩
refine' ne_of_apply_ne (fun x => x.coeff (na + nb)) _
|
dsimp only
|
instance [CharP R p] [NoZeroDivisors R] : NoZeroDivisors (𝕎 R) :=
⟨fun {x y} => by
contrapose!
rintro ⟨ha, hb⟩
rcases verschiebung_nonzero ha with ⟨na, wa, hwa0, rfl⟩
rcases verschiebung_nonzero hb with ⟨nb, wb, hwb0, rfl⟩
refine' ne_of_apply_ne (fun x => x.coeff (na + nb)) _
|
Mathlib.RingTheory.WittVector.Domain.110_0.4uLlcZNQ2uiRcjJ
|
instance [CharP R p] [NoZeroDivisors R] : NoZeroDivisors (𝕎 R)
|
Mathlib_RingTheory_WittVector_Domain
|
case intro.intro.intro.intro.intro.intro.intro
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝² : CommRing R
inst✝¹ : CharP R p
inst✝ : NoZeroDivisors R
na : ℕ
wa : 𝕎 R
hwa0 : coeff wa 0 ≠ 0
ha : (⇑verschiebung)^[na] wa ≠ 0
nb : ℕ
wb : 𝕎 R
hwb0 : coeff wb 0 ≠ 0
hb : (⇑verschiebung)^[nb] wb ≠ 0
⊢ coeff ((⇑verschiebung)^[na] wa * (⇑verschiebung)^[nb] wb) (na + nb) ≠ coeff 0 (na + nb)
|
/-
Copyright (c) 2022 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.Identities
#align_import ring_theory.witt_vector.domain from "leanprover-community/mathlib"@"b1d911acd60ab198808e853292106ee352b648ea"
/-!
# Witt vectors over a domain
This file builds to the proof `WittVector.instIsDomain`,
an instance that says if `R` is an integral domain, then so is `𝕎 R`.
It depends on the API around iterated applications
of `WittVector.verschiebung` and `WittVector.frobenius`
found in `Identities.lean`.
The [proof sketch](https://math.stackexchange.com/questions/4117247/ring-of-witt-vectors-over-an-integral-domain/4118723#4118723)
goes as follows:
any nonzero $x$ is an iterated application of $V$
to some vector $w_x$ whose 0th component is nonzero (`WittVector.verschiebung_nonzero`).
Known identities (`WittVector.iterate_verschiebung_mul`) allow us to transform
the product of two such $x$ and $y$
to the form $V^{m+n}\left(F^n(w_x) \cdot F^m(w_y)\right)$,
the 0th component of which must be nonzero.
## Main declarations
* `WittVector.iterate_verschiebung_mul_coeff` : an identity from [Haze09]
* `WittVector.instIsDomain`
-/
noncomputable section
open scoped Classical
namespace WittVector
open Function
variable {p : ℕ} {R : Type*}
local notation "𝕎" => WittVector p -- type as `\bbW`
/-!
## The `shift` operator
-/
/--
`WittVector.verschiebung` translates the entries of a Witt vector upward, inserting 0s in the gaps.
`WittVector.shift` does the opposite, removing the first entries.
This is mainly useful as an auxiliary construction for `WittVector.verschiebung_nonzero`.
-/
def shift (x : 𝕎 R) (n : ℕ) : 𝕎 R :=
@mk' p R fun i => x.coeff (n + i)
#align witt_vector.shift WittVector.shift
theorem shift_coeff (x : 𝕎 R) (n k : ℕ) : (x.shift n).coeff k = x.coeff (n + k) :=
rfl
#align witt_vector.shift_coeff WittVector.shift_coeff
variable [hp : Fact p.Prime] [CommRing R]
theorem verschiebung_shift (x : 𝕎 R) (k : ℕ) (h : ∀ i < k + 1, x.coeff i = 0) :
verschiebung (x.shift k.succ) = x.shift k := by
ext ⟨j⟩
· rw [verschiebung_coeff_zero, shift_coeff, h]
apply Nat.lt_succ_self
· simp only [verschiebung_coeff_succ, shift]
congr 1
rw [Nat.add_succ, add_comm, Nat.add_succ, add_comm]
#align witt_vector.verschiebung_shift WittVector.verschiebung_shift
theorem eq_iterate_verschiebung {x : 𝕎 R} {n : ℕ} (h : ∀ i < n, x.coeff i = 0) :
x = verschiebung^[n] (x.shift n) := by
induction' n with k ih
· cases x; simp [shift]
· dsimp; rw [verschiebung_shift]
· exact ih fun i hi => h _ (hi.trans (Nat.lt_succ_self _))
· exact h
#align witt_vector.eq_iterate_verschiebung WittVector.eq_iterate_verschiebung
theorem verschiebung_nonzero {x : 𝕎 R} (hx : x ≠ 0) :
∃ n : ℕ, ∃ x' : 𝕎 R, x'.coeff 0 ≠ 0 ∧ x = verschiebung^[n] x' := by
have hex : ∃ k : ℕ, x.coeff k ≠ 0 := by
by_contra! hall
apply hx
ext i
simp only [hall, zero_coeff]
let n := Nat.find hex
use n, x.shift n
refine' ⟨Nat.find_spec hex, eq_iterate_verschiebung fun i hi => not_not.mp _⟩
exact Nat.find_min hex hi
#align witt_vector.verschiebung_nonzero WittVector.verschiebung_nonzero
/-!
## Witt vectors over a domain
If `R` is an integral domain, then so is `𝕎 R`.
This argument is adapted from
<https://math.stackexchange.com/questions/4117247/ring-of-witt-vectors-over-an-integral-domain/4118723#4118723>.
-/
instance [CharP R p] [NoZeroDivisors R] : NoZeroDivisors (𝕎 R) :=
⟨fun {x y} => by
contrapose!
rintro ⟨ha, hb⟩
rcases verschiebung_nonzero ha with ⟨na, wa, hwa0, rfl⟩
rcases verschiebung_nonzero hb with ⟨nb, wb, hwb0, rfl⟩
refine' ne_of_apply_ne (fun x => x.coeff (na + nb)) _
dsimp only
|
rw [iterate_verschiebung_mul_coeff, zero_coeff]
|
instance [CharP R p] [NoZeroDivisors R] : NoZeroDivisors (𝕎 R) :=
⟨fun {x y} => by
contrapose!
rintro ⟨ha, hb⟩
rcases verschiebung_nonzero ha with ⟨na, wa, hwa0, rfl⟩
rcases verschiebung_nonzero hb with ⟨nb, wb, hwb0, rfl⟩
refine' ne_of_apply_ne (fun x => x.coeff (na + nb)) _
dsimp only
|
Mathlib.RingTheory.WittVector.Domain.110_0.4uLlcZNQ2uiRcjJ
|
instance [CharP R p] [NoZeroDivisors R] : NoZeroDivisors (𝕎 R)
|
Mathlib_RingTheory_WittVector_Domain
|
case intro.intro.intro.intro.intro.intro.intro
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝² : CommRing R
inst✝¹ : CharP R p
inst✝ : NoZeroDivisors R
na : ℕ
wa : 𝕎 R
hwa0 : coeff wa 0 ≠ 0
ha : (⇑verschiebung)^[na] wa ≠ 0
nb : ℕ
wb : 𝕎 R
hwb0 : coeff wb 0 ≠ 0
hb : (⇑verschiebung)^[nb] wb ≠ 0
⊢ coeff wa 0 ^ p ^ nb * coeff wb 0 ^ p ^ na ≠ 0
|
/-
Copyright (c) 2022 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.Identities
#align_import ring_theory.witt_vector.domain from "leanprover-community/mathlib"@"b1d911acd60ab198808e853292106ee352b648ea"
/-!
# Witt vectors over a domain
This file builds to the proof `WittVector.instIsDomain`,
an instance that says if `R` is an integral domain, then so is `𝕎 R`.
It depends on the API around iterated applications
of `WittVector.verschiebung` and `WittVector.frobenius`
found in `Identities.lean`.
The [proof sketch](https://math.stackexchange.com/questions/4117247/ring-of-witt-vectors-over-an-integral-domain/4118723#4118723)
goes as follows:
any nonzero $x$ is an iterated application of $V$
to some vector $w_x$ whose 0th component is nonzero (`WittVector.verschiebung_nonzero`).
Known identities (`WittVector.iterate_verschiebung_mul`) allow us to transform
the product of two such $x$ and $y$
to the form $V^{m+n}\left(F^n(w_x) \cdot F^m(w_y)\right)$,
the 0th component of which must be nonzero.
## Main declarations
* `WittVector.iterate_verschiebung_mul_coeff` : an identity from [Haze09]
* `WittVector.instIsDomain`
-/
noncomputable section
open scoped Classical
namespace WittVector
open Function
variable {p : ℕ} {R : Type*}
local notation "𝕎" => WittVector p -- type as `\bbW`
/-!
## The `shift` operator
-/
/--
`WittVector.verschiebung` translates the entries of a Witt vector upward, inserting 0s in the gaps.
`WittVector.shift` does the opposite, removing the first entries.
This is mainly useful as an auxiliary construction for `WittVector.verschiebung_nonzero`.
-/
def shift (x : 𝕎 R) (n : ℕ) : 𝕎 R :=
@mk' p R fun i => x.coeff (n + i)
#align witt_vector.shift WittVector.shift
theorem shift_coeff (x : 𝕎 R) (n k : ℕ) : (x.shift n).coeff k = x.coeff (n + k) :=
rfl
#align witt_vector.shift_coeff WittVector.shift_coeff
variable [hp : Fact p.Prime] [CommRing R]
theorem verschiebung_shift (x : 𝕎 R) (k : ℕ) (h : ∀ i < k + 1, x.coeff i = 0) :
verschiebung (x.shift k.succ) = x.shift k := by
ext ⟨j⟩
· rw [verschiebung_coeff_zero, shift_coeff, h]
apply Nat.lt_succ_self
· simp only [verschiebung_coeff_succ, shift]
congr 1
rw [Nat.add_succ, add_comm, Nat.add_succ, add_comm]
#align witt_vector.verschiebung_shift WittVector.verschiebung_shift
theorem eq_iterate_verschiebung {x : 𝕎 R} {n : ℕ} (h : ∀ i < n, x.coeff i = 0) :
x = verschiebung^[n] (x.shift n) := by
induction' n with k ih
· cases x; simp [shift]
· dsimp; rw [verschiebung_shift]
· exact ih fun i hi => h _ (hi.trans (Nat.lt_succ_self _))
· exact h
#align witt_vector.eq_iterate_verschiebung WittVector.eq_iterate_verschiebung
theorem verschiebung_nonzero {x : 𝕎 R} (hx : x ≠ 0) :
∃ n : ℕ, ∃ x' : 𝕎 R, x'.coeff 0 ≠ 0 ∧ x = verschiebung^[n] x' := by
have hex : ∃ k : ℕ, x.coeff k ≠ 0 := by
by_contra! hall
apply hx
ext i
simp only [hall, zero_coeff]
let n := Nat.find hex
use n, x.shift n
refine' ⟨Nat.find_spec hex, eq_iterate_verschiebung fun i hi => not_not.mp _⟩
exact Nat.find_min hex hi
#align witt_vector.verschiebung_nonzero WittVector.verschiebung_nonzero
/-!
## Witt vectors over a domain
If `R` is an integral domain, then so is `𝕎 R`.
This argument is adapted from
<https://math.stackexchange.com/questions/4117247/ring-of-witt-vectors-over-an-integral-domain/4118723#4118723>.
-/
instance [CharP R p] [NoZeroDivisors R] : NoZeroDivisors (𝕎 R) :=
⟨fun {x y} => by
contrapose!
rintro ⟨ha, hb⟩
rcases verschiebung_nonzero ha with ⟨na, wa, hwa0, rfl⟩
rcases verschiebung_nonzero hb with ⟨nb, wb, hwb0, rfl⟩
refine' ne_of_apply_ne (fun x => x.coeff (na + nb)) _
dsimp only
rw [iterate_verschiebung_mul_coeff, zero_coeff]
|
exact mul_ne_zero (pow_ne_zero _ hwa0) (pow_ne_zero _ hwb0)
|
instance [CharP R p] [NoZeroDivisors R] : NoZeroDivisors (𝕎 R) :=
⟨fun {x y} => by
contrapose!
rintro ⟨ha, hb⟩
rcases verschiebung_nonzero ha with ⟨na, wa, hwa0, rfl⟩
rcases verschiebung_nonzero hb with ⟨nb, wb, hwb0, rfl⟩
refine' ne_of_apply_ne (fun x => x.coeff (na + nb)) _
dsimp only
rw [iterate_verschiebung_mul_coeff, zero_coeff]
|
Mathlib.RingTheory.WittVector.Domain.110_0.4uLlcZNQ2uiRcjJ
|
instance [CharP R p] [NoZeroDivisors R] : NoZeroDivisors (𝕎 R)
|
Mathlib_RingTheory_WittVector_Domain
|
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I : Ideal R
x : R
hx : x ∈ jacobson I
y : R
hxy : I ⊔ span {y * x + 1} = ⊤
p : R
hpi : p ∈ I
q : R
hq : q ∈ span {y * x + 1}
hpq : p + q = 1
r : R
hr : r * (y * x + 1) = q
⊢ r * y * x + r - 1 ∈ I
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
/-!
# Jacobson radical
The Jacobson radical of a ring `R` is defined to be the intersection of all maximal ideals of `R`.
This is similar to how the nilradical is equal to the intersection of all prime ideals of `R`.
We can extend the idea of the nilradical to ideals of `R`,
by letting the radical of an ideal `I` be the intersection of prime ideals containing `I`.
Under this extension, the original nilradical is the radical of the zero ideal `⊥`.
Here we define the Jacobson radical of an ideal `I` in a similar way,
as the intersection of maximal ideals containing `I`.
## Main definitions
Let `R` be a commutative ring, and `I` be an ideal of `R`
* `Ideal.jacobson I` is the jacobson radical, i.e. the infimum of all maximal ideals containing I.
* `Ideal.IsLocal I` is the proposition that the jacobson radical of `I` is itself a maximal ideal
## Main statements
* `mem_jacobson_iff` gives a characterization of members of the jacobson of I
* `Ideal.isLocal_of_isMaximal_radical`: if the radical of I is maximal then so is the jacobson
radical
## Tags
Jacobson, Jacobson radical, Local Ideal
-/
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}
open Polynomial
section Jacobson
section Ring
variable [Ring R] [Ring S] {I : Ideal R}
/-- The Jacobson radical of `I` is the infimum of all maximal (left) ideals containing `I`. -/
def jacobson (I : Ideal R) : Ideal R :=
sInf { J : Ideal R | I ≤ J ∧ IsMaximal J }
#align ideal.jacobson Ideal.jacobson
theorem le_jacobson : I ≤ jacobson I := fun _ hx => mem_sInf.mpr fun _ hJ => hJ.left hx
#align ideal.le_jacobson Ideal.le_jacobson
@[simp]
theorem jacobson_idem : jacobson (jacobson I) = jacobson I :=
le_antisymm (sInf_le_sInf fun _ hJ => ⟨sInf_le hJ, hJ.2⟩) le_jacobson
#align ideal.jacobson_idem Ideal.jacobson_idem
@[simp]
theorem jacobson_top : jacobson (⊤ : Ideal R) = ⊤ :=
eq_top_iff.2 le_jacobson
#align ideal.jacobson_top Ideal.jacobson_top
@[simp]
theorem jacobson_eq_top_iff : jacobson I = ⊤ ↔ I = ⊤ :=
⟨fun H =>
by_contradiction fun hi => let ⟨M, hm, him⟩ := exists_le_maximal I hi
lt_top_iff_ne_top.1
(lt_of_le_of_lt (show jacobson I ≤ M from sInf_le ⟨him, hm⟩) <|
lt_top_iff_ne_top.2 hm.ne_top) H,
fun H => eq_top_iff.2 <| le_sInf fun _ ⟨hij, _⟩ => H ▸ hij⟩
#align ideal.jacobson_eq_top_iff Ideal.jacobson_eq_top_iff
theorem jacobson_eq_bot : jacobson I = ⊥ → I = ⊥ := fun h => eq_bot_iff.mpr (h ▸ le_jacobson)
#align ideal.jacobson_eq_bot Ideal.jacobson_eq_bot
theorem jacobson_eq_self_of_isMaximal [H : IsMaximal I] : I.jacobson = I :=
le_antisymm (sInf_le ⟨le_of_eq rfl, H⟩) le_jacobson
#align ideal.jacobson_eq_self_of_is_maximal Ideal.jacobson_eq_self_of_isMaximal
instance (priority := 100) jacobson.isMaximal [H : IsMaximal I] : IsMaximal (jacobson I) :=
⟨⟨fun htop => H.1.1 (jacobson_eq_top_iff.1 htop), fun _ hJ =>
H.1.2 _ (lt_of_le_of_lt le_jacobson hJ)⟩⟩
#align ideal.jacobson.is_maximal Ideal.jacobson.isMaximal
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I :=
⟨fun hx y =>
by_cases
(fun hxy : I ⊔ span {y * x + 1} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy)
let ⟨r, hr⟩ := mem_span_singleton'.1 hq
⟨r, by
-- Porting note : supply `mul_add_one` with explicit variables
|
rw [mul_assoc, ← mul_add_one r (y * x), hr, ← hpq, ← neg_sub, add_sub_cancel]
|
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I :=
⟨fun hx y =>
by_cases
(fun hxy : I ⊔ span {y * x + 1} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy)
let ⟨r, hr⟩ := mem_span_singleton'.1 hq
⟨r, by
-- Porting note : supply `mul_add_one` with explicit variables
|
Mathlib.RingTheory.JacobsonIdeal.99_0.Lz0MgLQMj1bGzuN
|
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I
|
Mathlib_RingTheory_JacobsonIdeal
|
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I : Ideal R
x : R
hx : x ∈ jacobson I
y : R
hxy : I ⊔ span {y * x + 1} = ⊤
p : R
hpi : p ∈ I
q : R
hq : q ∈ span {y * x + 1}
hpq : p + q = 1
r : R
hr : r * (y * x + 1) = q
⊢ -p ∈ I
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
/-!
# Jacobson radical
The Jacobson radical of a ring `R` is defined to be the intersection of all maximal ideals of `R`.
This is similar to how the nilradical is equal to the intersection of all prime ideals of `R`.
We can extend the idea of the nilradical to ideals of `R`,
by letting the radical of an ideal `I` be the intersection of prime ideals containing `I`.
Under this extension, the original nilradical is the radical of the zero ideal `⊥`.
Here we define the Jacobson radical of an ideal `I` in a similar way,
as the intersection of maximal ideals containing `I`.
## Main definitions
Let `R` be a commutative ring, and `I` be an ideal of `R`
* `Ideal.jacobson I` is the jacobson radical, i.e. the infimum of all maximal ideals containing I.
* `Ideal.IsLocal I` is the proposition that the jacobson radical of `I` is itself a maximal ideal
## Main statements
* `mem_jacobson_iff` gives a characterization of members of the jacobson of I
* `Ideal.isLocal_of_isMaximal_radical`: if the radical of I is maximal then so is the jacobson
radical
## Tags
Jacobson, Jacobson radical, Local Ideal
-/
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}
open Polynomial
section Jacobson
section Ring
variable [Ring R] [Ring S] {I : Ideal R}
/-- The Jacobson radical of `I` is the infimum of all maximal (left) ideals containing `I`. -/
def jacobson (I : Ideal R) : Ideal R :=
sInf { J : Ideal R | I ≤ J ∧ IsMaximal J }
#align ideal.jacobson Ideal.jacobson
theorem le_jacobson : I ≤ jacobson I := fun _ hx => mem_sInf.mpr fun _ hJ => hJ.left hx
#align ideal.le_jacobson Ideal.le_jacobson
@[simp]
theorem jacobson_idem : jacobson (jacobson I) = jacobson I :=
le_antisymm (sInf_le_sInf fun _ hJ => ⟨sInf_le hJ, hJ.2⟩) le_jacobson
#align ideal.jacobson_idem Ideal.jacobson_idem
@[simp]
theorem jacobson_top : jacobson (⊤ : Ideal R) = ⊤ :=
eq_top_iff.2 le_jacobson
#align ideal.jacobson_top Ideal.jacobson_top
@[simp]
theorem jacobson_eq_top_iff : jacobson I = ⊤ ↔ I = ⊤ :=
⟨fun H =>
by_contradiction fun hi => let ⟨M, hm, him⟩ := exists_le_maximal I hi
lt_top_iff_ne_top.1
(lt_of_le_of_lt (show jacobson I ≤ M from sInf_le ⟨him, hm⟩) <|
lt_top_iff_ne_top.2 hm.ne_top) H,
fun H => eq_top_iff.2 <| le_sInf fun _ ⟨hij, _⟩ => H ▸ hij⟩
#align ideal.jacobson_eq_top_iff Ideal.jacobson_eq_top_iff
theorem jacobson_eq_bot : jacobson I = ⊥ → I = ⊥ := fun h => eq_bot_iff.mpr (h ▸ le_jacobson)
#align ideal.jacobson_eq_bot Ideal.jacobson_eq_bot
theorem jacobson_eq_self_of_isMaximal [H : IsMaximal I] : I.jacobson = I :=
le_antisymm (sInf_le ⟨le_of_eq rfl, H⟩) le_jacobson
#align ideal.jacobson_eq_self_of_is_maximal Ideal.jacobson_eq_self_of_isMaximal
instance (priority := 100) jacobson.isMaximal [H : IsMaximal I] : IsMaximal (jacobson I) :=
⟨⟨fun htop => H.1.1 (jacobson_eq_top_iff.1 htop), fun _ hJ =>
H.1.2 _ (lt_of_le_of_lt le_jacobson hJ)⟩⟩
#align ideal.jacobson.is_maximal Ideal.jacobson.isMaximal
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I :=
⟨fun hx y =>
by_cases
(fun hxy : I ⊔ span {y * x + 1} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy)
let ⟨r, hr⟩ := mem_span_singleton'.1 hq
⟨r, by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one r (y * x), hr, ← hpq, ← neg_sub, add_sub_cancel]
|
exact I.neg_mem hpi
|
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I :=
⟨fun hx y =>
by_cases
(fun hxy : I ⊔ span {y * x + 1} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy)
let ⟨r, hr⟩ := mem_span_singleton'.1 hq
⟨r, by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one r (y * x), hr, ← hpq, ← neg_sub, add_sub_cancel]
|
Mathlib.RingTheory.JacobsonIdeal.99_0.Lz0MgLQMj1bGzuN
|
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I
|
Mathlib_RingTheory_JacobsonIdeal
|
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I : Ideal R
x : R
hx : ∀ (y : R), ∃ z, z * y * x + z - 1 ∈ I
M : Ideal R
x✝ : M ∈ {J | I ≤ J ∧ IsMaximal J}
him : I ≤ M
hm : IsMaximal M
hxm : x ∉ M
y i : R
hi : i ∈ M
df : y * x + i = 1
z : R
hz : z * -y * x + z - 1 ∈ I
⊢ z * -y * x + z ∈ M
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
/-!
# Jacobson radical
The Jacobson radical of a ring `R` is defined to be the intersection of all maximal ideals of `R`.
This is similar to how the nilradical is equal to the intersection of all prime ideals of `R`.
We can extend the idea of the nilradical to ideals of `R`,
by letting the radical of an ideal `I` be the intersection of prime ideals containing `I`.
Under this extension, the original nilradical is the radical of the zero ideal `⊥`.
Here we define the Jacobson radical of an ideal `I` in a similar way,
as the intersection of maximal ideals containing `I`.
## Main definitions
Let `R` be a commutative ring, and `I` be an ideal of `R`
* `Ideal.jacobson I` is the jacobson radical, i.e. the infimum of all maximal ideals containing I.
* `Ideal.IsLocal I` is the proposition that the jacobson radical of `I` is itself a maximal ideal
## Main statements
* `mem_jacobson_iff` gives a characterization of members of the jacobson of I
* `Ideal.isLocal_of_isMaximal_radical`: if the radical of I is maximal then so is the jacobson
radical
## Tags
Jacobson, Jacobson radical, Local Ideal
-/
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}
open Polynomial
section Jacobson
section Ring
variable [Ring R] [Ring S] {I : Ideal R}
/-- The Jacobson radical of `I` is the infimum of all maximal (left) ideals containing `I`. -/
def jacobson (I : Ideal R) : Ideal R :=
sInf { J : Ideal R | I ≤ J ∧ IsMaximal J }
#align ideal.jacobson Ideal.jacobson
theorem le_jacobson : I ≤ jacobson I := fun _ hx => mem_sInf.mpr fun _ hJ => hJ.left hx
#align ideal.le_jacobson Ideal.le_jacobson
@[simp]
theorem jacobson_idem : jacobson (jacobson I) = jacobson I :=
le_antisymm (sInf_le_sInf fun _ hJ => ⟨sInf_le hJ, hJ.2⟩) le_jacobson
#align ideal.jacobson_idem Ideal.jacobson_idem
@[simp]
theorem jacobson_top : jacobson (⊤ : Ideal R) = ⊤ :=
eq_top_iff.2 le_jacobson
#align ideal.jacobson_top Ideal.jacobson_top
@[simp]
theorem jacobson_eq_top_iff : jacobson I = ⊤ ↔ I = ⊤ :=
⟨fun H =>
by_contradiction fun hi => let ⟨M, hm, him⟩ := exists_le_maximal I hi
lt_top_iff_ne_top.1
(lt_of_le_of_lt (show jacobson I ≤ M from sInf_le ⟨him, hm⟩) <|
lt_top_iff_ne_top.2 hm.ne_top) H,
fun H => eq_top_iff.2 <| le_sInf fun _ ⟨hij, _⟩ => H ▸ hij⟩
#align ideal.jacobson_eq_top_iff Ideal.jacobson_eq_top_iff
theorem jacobson_eq_bot : jacobson I = ⊥ → I = ⊥ := fun h => eq_bot_iff.mpr (h ▸ le_jacobson)
#align ideal.jacobson_eq_bot Ideal.jacobson_eq_bot
theorem jacobson_eq_self_of_isMaximal [H : IsMaximal I] : I.jacobson = I :=
le_antisymm (sInf_le ⟨le_of_eq rfl, H⟩) le_jacobson
#align ideal.jacobson_eq_self_of_is_maximal Ideal.jacobson_eq_self_of_isMaximal
instance (priority := 100) jacobson.isMaximal [H : IsMaximal I] : IsMaximal (jacobson I) :=
⟨⟨fun htop => H.1.1 (jacobson_eq_top_iff.1 htop), fun _ hJ =>
H.1.2 _ (lt_of_le_of_lt le_jacobson hJ)⟩⟩
#align ideal.jacobson.is_maximal Ideal.jacobson.isMaximal
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I :=
⟨fun hx y =>
by_cases
(fun hxy : I ⊔ span {y * x + 1} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy)
let ⟨r, hr⟩ := mem_span_singleton'.1 hq
⟨r, by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one r (y * x), hr, ← hpq, ← neg_sub, add_sub_cancel]
exact I.neg_mem hpi⟩)
fun hxy : I ⊔ span {y * x + 1} ≠ ⊤ => let ⟨M, hm1, hm2⟩ := exists_le_maximal _ hxy
suffices x ∉ M from (this <| mem_sInf.1 hx ⟨le_trans le_sup_left hm2, hm1⟩).elim
fun hxm => hm1.1.1 <| (eq_top_iff_one _).2 <| add_sub_cancel' (y * x) 1 ▸
M.sub_mem (le_sup_right.trans hm2 <| subset_span rfl) (M.mul_mem_left _ hxm),
fun hx => mem_sInf.2 fun M ⟨him, hm⟩ => by_contradiction fun hxm =>
let ⟨y, i, hi, df⟩ := hm.exists_inv hxm
let ⟨z, hz⟩ := hx (-y)
hm.1.1 <| (eq_top_iff_one _).2 <| sub_sub_cancel (z * -y * x + z) 1 ▸
M.sub_mem (by
-- Porting note : supply `mul_add_one` with explicit variables
|
rw [mul_assoc, ← mul_add_one z, neg_mul, ← sub_eq_iff_eq_add.mpr df.symm, neg_sub,
sub_add_cancel]
|
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I :=
⟨fun hx y =>
by_cases
(fun hxy : I ⊔ span {y * x + 1} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy)
let ⟨r, hr⟩ := mem_span_singleton'.1 hq
⟨r, by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one r (y * x), hr, ← hpq, ← neg_sub, add_sub_cancel]
exact I.neg_mem hpi⟩)
fun hxy : I ⊔ span {y * x + 1} ≠ ⊤ => let ⟨M, hm1, hm2⟩ := exists_le_maximal _ hxy
suffices x ∉ M from (this <| mem_sInf.1 hx ⟨le_trans le_sup_left hm2, hm1⟩).elim
fun hxm => hm1.1.1 <| (eq_top_iff_one _).2 <| add_sub_cancel' (y * x) 1 ▸
M.sub_mem (le_sup_right.trans hm2 <| subset_span rfl) (M.mul_mem_left _ hxm),
fun hx => mem_sInf.2 fun M ⟨him, hm⟩ => by_contradiction fun hxm =>
let ⟨y, i, hi, df⟩ := hm.exists_inv hxm
let ⟨z, hz⟩ := hx (-y)
hm.1.1 <| (eq_top_iff_one _).2 <| sub_sub_cancel (z * -y * x + z) 1 ▸
M.sub_mem (by
-- Porting note : supply `mul_add_one` with explicit variables
|
Mathlib.RingTheory.JacobsonIdeal.99_0.Lz0MgLQMj1bGzuN
|
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I
|
Mathlib_RingTheory_JacobsonIdeal
|
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I : Ideal R
x : R
hx : ∀ (y : R), ∃ z, z * y * x + z - 1 ∈ I
M : Ideal R
x✝ : M ∈ {J | I ≤ J ∧ IsMaximal J}
him : I ≤ M
hm : IsMaximal M
hxm : x ∉ M
y i : R
hi : i ∈ M
df : y * x + i = 1
z : R
hz : z * -y * x + z - 1 ∈ I
⊢ z * i ∈ M
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
/-!
# Jacobson radical
The Jacobson radical of a ring `R` is defined to be the intersection of all maximal ideals of `R`.
This is similar to how the nilradical is equal to the intersection of all prime ideals of `R`.
We can extend the idea of the nilradical to ideals of `R`,
by letting the radical of an ideal `I` be the intersection of prime ideals containing `I`.
Under this extension, the original nilradical is the radical of the zero ideal `⊥`.
Here we define the Jacobson radical of an ideal `I` in a similar way,
as the intersection of maximal ideals containing `I`.
## Main definitions
Let `R` be a commutative ring, and `I` be an ideal of `R`
* `Ideal.jacobson I` is the jacobson radical, i.e. the infimum of all maximal ideals containing I.
* `Ideal.IsLocal I` is the proposition that the jacobson radical of `I` is itself a maximal ideal
## Main statements
* `mem_jacobson_iff` gives a characterization of members of the jacobson of I
* `Ideal.isLocal_of_isMaximal_radical`: if the radical of I is maximal then so is the jacobson
radical
## Tags
Jacobson, Jacobson radical, Local Ideal
-/
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}
open Polynomial
section Jacobson
section Ring
variable [Ring R] [Ring S] {I : Ideal R}
/-- The Jacobson radical of `I` is the infimum of all maximal (left) ideals containing `I`. -/
def jacobson (I : Ideal R) : Ideal R :=
sInf { J : Ideal R | I ≤ J ∧ IsMaximal J }
#align ideal.jacobson Ideal.jacobson
theorem le_jacobson : I ≤ jacobson I := fun _ hx => mem_sInf.mpr fun _ hJ => hJ.left hx
#align ideal.le_jacobson Ideal.le_jacobson
@[simp]
theorem jacobson_idem : jacobson (jacobson I) = jacobson I :=
le_antisymm (sInf_le_sInf fun _ hJ => ⟨sInf_le hJ, hJ.2⟩) le_jacobson
#align ideal.jacobson_idem Ideal.jacobson_idem
@[simp]
theorem jacobson_top : jacobson (⊤ : Ideal R) = ⊤ :=
eq_top_iff.2 le_jacobson
#align ideal.jacobson_top Ideal.jacobson_top
@[simp]
theorem jacobson_eq_top_iff : jacobson I = ⊤ ↔ I = ⊤ :=
⟨fun H =>
by_contradiction fun hi => let ⟨M, hm, him⟩ := exists_le_maximal I hi
lt_top_iff_ne_top.1
(lt_of_le_of_lt (show jacobson I ≤ M from sInf_le ⟨him, hm⟩) <|
lt_top_iff_ne_top.2 hm.ne_top) H,
fun H => eq_top_iff.2 <| le_sInf fun _ ⟨hij, _⟩ => H ▸ hij⟩
#align ideal.jacobson_eq_top_iff Ideal.jacobson_eq_top_iff
theorem jacobson_eq_bot : jacobson I = ⊥ → I = ⊥ := fun h => eq_bot_iff.mpr (h ▸ le_jacobson)
#align ideal.jacobson_eq_bot Ideal.jacobson_eq_bot
theorem jacobson_eq_self_of_isMaximal [H : IsMaximal I] : I.jacobson = I :=
le_antisymm (sInf_le ⟨le_of_eq rfl, H⟩) le_jacobson
#align ideal.jacobson_eq_self_of_is_maximal Ideal.jacobson_eq_self_of_isMaximal
instance (priority := 100) jacobson.isMaximal [H : IsMaximal I] : IsMaximal (jacobson I) :=
⟨⟨fun htop => H.1.1 (jacobson_eq_top_iff.1 htop), fun _ hJ =>
H.1.2 _ (lt_of_le_of_lt le_jacobson hJ)⟩⟩
#align ideal.jacobson.is_maximal Ideal.jacobson.isMaximal
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I :=
⟨fun hx y =>
by_cases
(fun hxy : I ⊔ span {y * x + 1} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy)
let ⟨r, hr⟩ := mem_span_singleton'.1 hq
⟨r, by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one r (y * x), hr, ← hpq, ← neg_sub, add_sub_cancel]
exact I.neg_mem hpi⟩)
fun hxy : I ⊔ span {y * x + 1} ≠ ⊤ => let ⟨M, hm1, hm2⟩ := exists_le_maximal _ hxy
suffices x ∉ M from (this <| mem_sInf.1 hx ⟨le_trans le_sup_left hm2, hm1⟩).elim
fun hxm => hm1.1.1 <| (eq_top_iff_one _).2 <| add_sub_cancel' (y * x) 1 ▸
M.sub_mem (le_sup_right.trans hm2 <| subset_span rfl) (M.mul_mem_left _ hxm),
fun hx => mem_sInf.2 fun M ⟨him, hm⟩ => by_contradiction fun hxm =>
let ⟨y, i, hi, df⟩ := hm.exists_inv hxm
let ⟨z, hz⟩ := hx (-y)
hm.1.1 <| (eq_top_iff_one _).2 <| sub_sub_cancel (z * -y * x + z) 1 ▸
M.sub_mem (by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one z, neg_mul, ← sub_eq_iff_eq_add.mpr df.symm, neg_sub,
sub_add_cancel]
|
exact M.mul_mem_left _ hi
|
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I :=
⟨fun hx y =>
by_cases
(fun hxy : I ⊔ span {y * x + 1} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy)
let ⟨r, hr⟩ := mem_span_singleton'.1 hq
⟨r, by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one r (y * x), hr, ← hpq, ← neg_sub, add_sub_cancel]
exact I.neg_mem hpi⟩)
fun hxy : I ⊔ span {y * x + 1} ≠ ⊤ => let ⟨M, hm1, hm2⟩ := exists_le_maximal _ hxy
suffices x ∉ M from (this <| mem_sInf.1 hx ⟨le_trans le_sup_left hm2, hm1⟩).elim
fun hxm => hm1.1.1 <| (eq_top_iff_one _).2 <| add_sub_cancel' (y * x) 1 ▸
M.sub_mem (le_sup_right.trans hm2 <| subset_span rfl) (M.mul_mem_left _ hxm),
fun hx => mem_sInf.2 fun M ⟨him, hm⟩ => by_contradiction fun hxm =>
let ⟨y, i, hi, df⟩ := hm.exists_inv hxm
let ⟨z, hz⟩ := hx (-y)
hm.1.1 <| (eq_top_iff_one _).2 <| sub_sub_cancel (z * -y * x + z) 1 ▸
M.sub_mem (by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one z, neg_mul, ← sub_eq_iff_eq_add.mpr df.symm, neg_sub,
sub_add_cancel]
|
Mathlib.RingTheory.JacobsonIdeal.99_0.Lz0MgLQMj1bGzuN
|
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I
|
Mathlib_RingTheory_JacobsonIdeal
|
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I✝ I : Ideal R
r : R
h : r - 1 ∈ jacobson I
⊢ ∃ s, s * r - 1 ∈ I
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
/-!
# Jacobson radical
The Jacobson radical of a ring `R` is defined to be the intersection of all maximal ideals of `R`.
This is similar to how the nilradical is equal to the intersection of all prime ideals of `R`.
We can extend the idea of the nilradical to ideals of `R`,
by letting the radical of an ideal `I` be the intersection of prime ideals containing `I`.
Under this extension, the original nilradical is the radical of the zero ideal `⊥`.
Here we define the Jacobson radical of an ideal `I` in a similar way,
as the intersection of maximal ideals containing `I`.
## Main definitions
Let `R` be a commutative ring, and `I` be an ideal of `R`
* `Ideal.jacobson I` is the jacobson radical, i.e. the infimum of all maximal ideals containing I.
* `Ideal.IsLocal I` is the proposition that the jacobson radical of `I` is itself a maximal ideal
## Main statements
* `mem_jacobson_iff` gives a characterization of members of the jacobson of I
* `Ideal.isLocal_of_isMaximal_radical`: if the radical of I is maximal then so is the jacobson
radical
## Tags
Jacobson, Jacobson radical, Local Ideal
-/
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}
open Polynomial
section Jacobson
section Ring
variable [Ring R] [Ring S] {I : Ideal R}
/-- The Jacobson radical of `I` is the infimum of all maximal (left) ideals containing `I`. -/
def jacobson (I : Ideal R) : Ideal R :=
sInf { J : Ideal R | I ≤ J ∧ IsMaximal J }
#align ideal.jacobson Ideal.jacobson
theorem le_jacobson : I ≤ jacobson I := fun _ hx => mem_sInf.mpr fun _ hJ => hJ.left hx
#align ideal.le_jacobson Ideal.le_jacobson
@[simp]
theorem jacobson_idem : jacobson (jacobson I) = jacobson I :=
le_antisymm (sInf_le_sInf fun _ hJ => ⟨sInf_le hJ, hJ.2⟩) le_jacobson
#align ideal.jacobson_idem Ideal.jacobson_idem
@[simp]
theorem jacobson_top : jacobson (⊤ : Ideal R) = ⊤ :=
eq_top_iff.2 le_jacobson
#align ideal.jacobson_top Ideal.jacobson_top
@[simp]
theorem jacobson_eq_top_iff : jacobson I = ⊤ ↔ I = ⊤ :=
⟨fun H =>
by_contradiction fun hi => let ⟨M, hm, him⟩ := exists_le_maximal I hi
lt_top_iff_ne_top.1
(lt_of_le_of_lt (show jacobson I ≤ M from sInf_le ⟨him, hm⟩) <|
lt_top_iff_ne_top.2 hm.ne_top) H,
fun H => eq_top_iff.2 <| le_sInf fun _ ⟨hij, _⟩ => H ▸ hij⟩
#align ideal.jacobson_eq_top_iff Ideal.jacobson_eq_top_iff
theorem jacobson_eq_bot : jacobson I = ⊥ → I = ⊥ := fun h => eq_bot_iff.mpr (h ▸ le_jacobson)
#align ideal.jacobson_eq_bot Ideal.jacobson_eq_bot
theorem jacobson_eq_self_of_isMaximal [H : IsMaximal I] : I.jacobson = I :=
le_antisymm (sInf_le ⟨le_of_eq rfl, H⟩) le_jacobson
#align ideal.jacobson_eq_self_of_is_maximal Ideal.jacobson_eq_self_of_isMaximal
instance (priority := 100) jacobson.isMaximal [H : IsMaximal I] : IsMaximal (jacobson I) :=
⟨⟨fun htop => H.1.1 (jacobson_eq_top_iff.1 htop), fun _ hJ =>
H.1.2 _ (lt_of_le_of_lt le_jacobson hJ)⟩⟩
#align ideal.jacobson.is_maximal Ideal.jacobson.isMaximal
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I :=
⟨fun hx y =>
by_cases
(fun hxy : I ⊔ span {y * x + 1} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy)
let ⟨r, hr⟩ := mem_span_singleton'.1 hq
⟨r, by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one r (y * x), hr, ← hpq, ← neg_sub, add_sub_cancel]
exact I.neg_mem hpi⟩)
fun hxy : I ⊔ span {y * x + 1} ≠ ⊤ => let ⟨M, hm1, hm2⟩ := exists_le_maximal _ hxy
suffices x ∉ M from (this <| mem_sInf.1 hx ⟨le_trans le_sup_left hm2, hm1⟩).elim
fun hxm => hm1.1.1 <| (eq_top_iff_one _).2 <| add_sub_cancel' (y * x) 1 ▸
M.sub_mem (le_sup_right.trans hm2 <| subset_span rfl) (M.mul_mem_left _ hxm),
fun hx => mem_sInf.2 fun M ⟨him, hm⟩ => by_contradiction fun hxm =>
let ⟨y, i, hi, df⟩ := hm.exists_inv hxm
let ⟨z, hz⟩ := hx (-y)
hm.1.1 <| (eq_top_iff_one _).2 <| sub_sub_cancel (z * -y * x + z) 1 ▸
M.sub_mem (by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one z, neg_mul, ← sub_eq_iff_eq_add.mpr df.symm, neg_sub,
sub_add_cancel]
exact M.mul_mem_left _ hi) <| him hz⟩
#align ideal.mem_jacobson_iff Ideal.mem_jacobson_iff
theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
∃ s, s * r - 1 ∈ I := by
|
cases' mem_jacobson_iff.1 h 1 with s hs
|
theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
∃ s, s * r - 1 ∈ I := by
|
Mathlib.RingTheory.JacobsonIdeal.124_0.Lz0MgLQMj1bGzuN
|
theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
∃ s, s * r - 1 ∈ I
|
Mathlib_RingTheory_JacobsonIdeal
|
case intro
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I✝ I : Ideal R
r : R
h : r - 1 ∈ jacobson I
s : R
hs : s * 1 * (r - 1) + s - 1 ∈ I
⊢ ∃ s, s * r - 1 ∈ I
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
/-!
# Jacobson radical
The Jacobson radical of a ring `R` is defined to be the intersection of all maximal ideals of `R`.
This is similar to how the nilradical is equal to the intersection of all prime ideals of `R`.
We can extend the idea of the nilradical to ideals of `R`,
by letting the radical of an ideal `I` be the intersection of prime ideals containing `I`.
Under this extension, the original nilradical is the radical of the zero ideal `⊥`.
Here we define the Jacobson radical of an ideal `I` in a similar way,
as the intersection of maximal ideals containing `I`.
## Main definitions
Let `R` be a commutative ring, and `I` be an ideal of `R`
* `Ideal.jacobson I` is the jacobson radical, i.e. the infimum of all maximal ideals containing I.
* `Ideal.IsLocal I` is the proposition that the jacobson radical of `I` is itself a maximal ideal
## Main statements
* `mem_jacobson_iff` gives a characterization of members of the jacobson of I
* `Ideal.isLocal_of_isMaximal_radical`: if the radical of I is maximal then so is the jacobson
radical
## Tags
Jacobson, Jacobson radical, Local Ideal
-/
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}
open Polynomial
section Jacobson
section Ring
variable [Ring R] [Ring S] {I : Ideal R}
/-- The Jacobson radical of `I` is the infimum of all maximal (left) ideals containing `I`. -/
def jacobson (I : Ideal R) : Ideal R :=
sInf { J : Ideal R | I ≤ J ∧ IsMaximal J }
#align ideal.jacobson Ideal.jacobson
theorem le_jacobson : I ≤ jacobson I := fun _ hx => mem_sInf.mpr fun _ hJ => hJ.left hx
#align ideal.le_jacobson Ideal.le_jacobson
@[simp]
theorem jacobson_idem : jacobson (jacobson I) = jacobson I :=
le_antisymm (sInf_le_sInf fun _ hJ => ⟨sInf_le hJ, hJ.2⟩) le_jacobson
#align ideal.jacobson_idem Ideal.jacobson_idem
@[simp]
theorem jacobson_top : jacobson (⊤ : Ideal R) = ⊤ :=
eq_top_iff.2 le_jacobson
#align ideal.jacobson_top Ideal.jacobson_top
@[simp]
theorem jacobson_eq_top_iff : jacobson I = ⊤ ↔ I = ⊤ :=
⟨fun H =>
by_contradiction fun hi => let ⟨M, hm, him⟩ := exists_le_maximal I hi
lt_top_iff_ne_top.1
(lt_of_le_of_lt (show jacobson I ≤ M from sInf_le ⟨him, hm⟩) <|
lt_top_iff_ne_top.2 hm.ne_top) H,
fun H => eq_top_iff.2 <| le_sInf fun _ ⟨hij, _⟩ => H ▸ hij⟩
#align ideal.jacobson_eq_top_iff Ideal.jacobson_eq_top_iff
theorem jacobson_eq_bot : jacobson I = ⊥ → I = ⊥ := fun h => eq_bot_iff.mpr (h ▸ le_jacobson)
#align ideal.jacobson_eq_bot Ideal.jacobson_eq_bot
theorem jacobson_eq_self_of_isMaximal [H : IsMaximal I] : I.jacobson = I :=
le_antisymm (sInf_le ⟨le_of_eq rfl, H⟩) le_jacobson
#align ideal.jacobson_eq_self_of_is_maximal Ideal.jacobson_eq_self_of_isMaximal
instance (priority := 100) jacobson.isMaximal [H : IsMaximal I] : IsMaximal (jacobson I) :=
⟨⟨fun htop => H.1.1 (jacobson_eq_top_iff.1 htop), fun _ hJ =>
H.1.2 _ (lt_of_le_of_lt le_jacobson hJ)⟩⟩
#align ideal.jacobson.is_maximal Ideal.jacobson.isMaximal
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I :=
⟨fun hx y =>
by_cases
(fun hxy : I ⊔ span {y * x + 1} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy)
let ⟨r, hr⟩ := mem_span_singleton'.1 hq
⟨r, by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one r (y * x), hr, ← hpq, ← neg_sub, add_sub_cancel]
exact I.neg_mem hpi⟩)
fun hxy : I ⊔ span {y * x + 1} ≠ ⊤ => let ⟨M, hm1, hm2⟩ := exists_le_maximal _ hxy
suffices x ∉ M from (this <| mem_sInf.1 hx ⟨le_trans le_sup_left hm2, hm1⟩).elim
fun hxm => hm1.1.1 <| (eq_top_iff_one _).2 <| add_sub_cancel' (y * x) 1 ▸
M.sub_mem (le_sup_right.trans hm2 <| subset_span rfl) (M.mul_mem_left _ hxm),
fun hx => mem_sInf.2 fun M ⟨him, hm⟩ => by_contradiction fun hxm =>
let ⟨y, i, hi, df⟩ := hm.exists_inv hxm
let ⟨z, hz⟩ := hx (-y)
hm.1.1 <| (eq_top_iff_one _).2 <| sub_sub_cancel (z * -y * x + z) 1 ▸
M.sub_mem (by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one z, neg_mul, ← sub_eq_iff_eq_add.mpr df.symm, neg_sub,
sub_add_cancel]
exact M.mul_mem_left _ hi) <| him hz⟩
#align ideal.mem_jacobson_iff Ideal.mem_jacobson_iff
theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
∃ s, s * r - 1 ∈ I := by
cases' mem_jacobson_iff.1 h 1 with s hs
|
use s
|
theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
∃ s, s * r - 1 ∈ I := by
cases' mem_jacobson_iff.1 h 1 with s hs
|
Mathlib.RingTheory.JacobsonIdeal.124_0.Lz0MgLQMj1bGzuN
|
theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
∃ s, s * r - 1 ∈ I
|
Mathlib_RingTheory_JacobsonIdeal
|
case h
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I✝ I : Ideal R
r : R
h : r - 1 ∈ jacobson I
s : R
hs : s * 1 * (r - 1) + s - 1 ∈ I
⊢ s * r - 1 ∈ I
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
/-!
# Jacobson radical
The Jacobson radical of a ring `R` is defined to be the intersection of all maximal ideals of `R`.
This is similar to how the nilradical is equal to the intersection of all prime ideals of `R`.
We can extend the idea of the nilradical to ideals of `R`,
by letting the radical of an ideal `I` be the intersection of prime ideals containing `I`.
Under this extension, the original nilradical is the radical of the zero ideal `⊥`.
Here we define the Jacobson radical of an ideal `I` in a similar way,
as the intersection of maximal ideals containing `I`.
## Main definitions
Let `R` be a commutative ring, and `I` be an ideal of `R`
* `Ideal.jacobson I` is the jacobson radical, i.e. the infimum of all maximal ideals containing I.
* `Ideal.IsLocal I` is the proposition that the jacobson radical of `I` is itself a maximal ideal
## Main statements
* `mem_jacobson_iff` gives a characterization of members of the jacobson of I
* `Ideal.isLocal_of_isMaximal_radical`: if the radical of I is maximal then so is the jacobson
radical
## Tags
Jacobson, Jacobson radical, Local Ideal
-/
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}
open Polynomial
section Jacobson
section Ring
variable [Ring R] [Ring S] {I : Ideal R}
/-- The Jacobson radical of `I` is the infimum of all maximal (left) ideals containing `I`. -/
def jacobson (I : Ideal R) : Ideal R :=
sInf { J : Ideal R | I ≤ J ∧ IsMaximal J }
#align ideal.jacobson Ideal.jacobson
theorem le_jacobson : I ≤ jacobson I := fun _ hx => mem_sInf.mpr fun _ hJ => hJ.left hx
#align ideal.le_jacobson Ideal.le_jacobson
@[simp]
theorem jacobson_idem : jacobson (jacobson I) = jacobson I :=
le_antisymm (sInf_le_sInf fun _ hJ => ⟨sInf_le hJ, hJ.2⟩) le_jacobson
#align ideal.jacobson_idem Ideal.jacobson_idem
@[simp]
theorem jacobson_top : jacobson (⊤ : Ideal R) = ⊤ :=
eq_top_iff.2 le_jacobson
#align ideal.jacobson_top Ideal.jacobson_top
@[simp]
theorem jacobson_eq_top_iff : jacobson I = ⊤ ↔ I = ⊤ :=
⟨fun H =>
by_contradiction fun hi => let ⟨M, hm, him⟩ := exists_le_maximal I hi
lt_top_iff_ne_top.1
(lt_of_le_of_lt (show jacobson I ≤ M from sInf_le ⟨him, hm⟩) <|
lt_top_iff_ne_top.2 hm.ne_top) H,
fun H => eq_top_iff.2 <| le_sInf fun _ ⟨hij, _⟩ => H ▸ hij⟩
#align ideal.jacobson_eq_top_iff Ideal.jacobson_eq_top_iff
theorem jacobson_eq_bot : jacobson I = ⊥ → I = ⊥ := fun h => eq_bot_iff.mpr (h ▸ le_jacobson)
#align ideal.jacobson_eq_bot Ideal.jacobson_eq_bot
theorem jacobson_eq_self_of_isMaximal [H : IsMaximal I] : I.jacobson = I :=
le_antisymm (sInf_le ⟨le_of_eq rfl, H⟩) le_jacobson
#align ideal.jacobson_eq_self_of_is_maximal Ideal.jacobson_eq_self_of_isMaximal
instance (priority := 100) jacobson.isMaximal [H : IsMaximal I] : IsMaximal (jacobson I) :=
⟨⟨fun htop => H.1.1 (jacobson_eq_top_iff.1 htop), fun _ hJ =>
H.1.2 _ (lt_of_le_of_lt le_jacobson hJ)⟩⟩
#align ideal.jacobson.is_maximal Ideal.jacobson.isMaximal
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I :=
⟨fun hx y =>
by_cases
(fun hxy : I ⊔ span {y * x + 1} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy)
let ⟨r, hr⟩ := mem_span_singleton'.1 hq
⟨r, by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one r (y * x), hr, ← hpq, ← neg_sub, add_sub_cancel]
exact I.neg_mem hpi⟩)
fun hxy : I ⊔ span {y * x + 1} ≠ ⊤ => let ⟨M, hm1, hm2⟩ := exists_le_maximal _ hxy
suffices x ∉ M from (this <| mem_sInf.1 hx ⟨le_trans le_sup_left hm2, hm1⟩).elim
fun hxm => hm1.1.1 <| (eq_top_iff_one _).2 <| add_sub_cancel' (y * x) 1 ▸
M.sub_mem (le_sup_right.trans hm2 <| subset_span rfl) (M.mul_mem_left _ hxm),
fun hx => mem_sInf.2 fun M ⟨him, hm⟩ => by_contradiction fun hxm =>
let ⟨y, i, hi, df⟩ := hm.exists_inv hxm
let ⟨z, hz⟩ := hx (-y)
hm.1.1 <| (eq_top_iff_one _).2 <| sub_sub_cancel (z * -y * x + z) 1 ▸
M.sub_mem (by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one z, neg_mul, ← sub_eq_iff_eq_add.mpr df.symm, neg_sub,
sub_add_cancel]
exact M.mul_mem_left _ hi) <| him hz⟩
#align ideal.mem_jacobson_iff Ideal.mem_jacobson_iff
theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
∃ s, s * r - 1 ∈ I := by
cases' mem_jacobson_iff.1 h 1 with s hs
use s
|
simpa [mul_sub] using hs
|
theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
∃ s, s * r - 1 ∈ I := by
cases' mem_jacobson_iff.1 h 1 with s hs
use s
|
Mathlib.RingTheory.JacobsonIdeal.124_0.Lz0MgLQMj1bGzuN
|
theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
∃ s, s * r - 1 ∈ I
|
Mathlib_RingTheory_JacobsonIdeal
|
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I : Ideal R
⊢ jacobson I = I ↔ ∃ M, (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
/-!
# Jacobson radical
The Jacobson radical of a ring `R` is defined to be the intersection of all maximal ideals of `R`.
This is similar to how the nilradical is equal to the intersection of all prime ideals of `R`.
We can extend the idea of the nilradical to ideals of `R`,
by letting the radical of an ideal `I` be the intersection of prime ideals containing `I`.
Under this extension, the original nilradical is the radical of the zero ideal `⊥`.
Here we define the Jacobson radical of an ideal `I` in a similar way,
as the intersection of maximal ideals containing `I`.
## Main definitions
Let `R` be a commutative ring, and `I` be an ideal of `R`
* `Ideal.jacobson I` is the jacobson radical, i.e. the infimum of all maximal ideals containing I.
* `Ideal.IsLocal I` is the proposition that the jacobson radical of `I` is itself a maximal ideal
## Main statements
* `mem_jacobson_iff` gives a characterization of members of the jacobson of I
* `Ideal.isLocal_of_isMaximal_radical`: if the radical of I is maximal then so is the jacobson
radical
## Tags
Jacobson, Jacobson radical, Local Ideal
-/
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}
open Polynomial
section Jacobson
section Ring
variable [Ring R] [Ring S] {I : Ideal R}
/-- The Jacobson radical of `I` is the infimum of all maximal (left) ideals containing `I`. -/
def jacobson (I : Ideal R) : Ideal R :=
sInf { J : Ideal R | I ≤ J ∧ IsMaximal J }
#align ideal.jacobson Ideal.jacobson
theorem le_jacobson : I ≤ jacobson I := fun _ hx => mem_sInf.mpr fun _ hJ => hJ.left hx
#align ideal.le_jacobson Ideal.le_jacobson
@[simp]
theorem jacobson_idem : jacobson (jacobson I) = jacobson I :=
le_antisymm (sInf_le_sInf fun _ hJ => ⟨sInf_le hJ, hJ.2⟩) le_jacobson
#align ideal.jacobson_idem Ideal.jacobson_idem
@[simp]
theorem jacobson_top : jacobson (⊤ : Ideal R) = ⊤ :=
eq_top_iff.2 le_jacobson
#align ideal.jacobson_top Ideal.jacobson_top
@[simp]
theorem jacobson_eq_top_iff : jacobson I = ⊤ ↔ I = ⊤ :=
⟨fun H =>
by_contradiction fun hi => let ⟨M, hm, him⟩ := exists_le_maximal I hi
lt_top_iff_ne_top.1
(lt_of_le_of_lt (show jacobson I ≤ M from sInf_le ⟨him, hm⟩) <|
lt_top_iff_ne_top.2 hm.ne_top) H,
fun H => eq_top_iff.2 <| le_sInf fun _ ⟨hij, _⟩ => H ▸ hij⟩
#align ideal.jacobson_eq_top_iff Ideal.jacobson_eq_top_iff
theorem jacobson_eq_bot : jacobson I = ⊥ → I = ⊥ := fun h => eq_bot_iff.mpr (h ▸ le_jacobson)
#align ideal.jacobson_eq_bot Ideal.jacobson_eq_bot
theorem jacobson_eq_self_of_isMaximal [H : IsMaximal I] : I.jacobson = I :=
le_antisymm (sInf_le ⟨le_of_eq rfl, H⟩) le_jacobson
#align ideal.jacobson_eq_self_of_is_maximal Ideal.jacobson_eq_self_of_isMaximal
instance (priority := 100) jacobson.isMaximal [H : IsMaximal I] : IsMaximal (jacobson I) :=
⟨⟨fun htop => H.1.1 (jacobson_eq_top_iff.1 htop), fun _ hJ =>
H.1.2 _ (lt_of_le_of_lt le_jacobson hJ)⟩⟩
#align ideal.jacobson.is_maximal Ideal.jacobson.isMaximal
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I :=
⟨fun hx y =>
by_cases
(fun hxy : I ⊔ span {y * x + 1} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy)
let ⟨r, hr⟩ := mem_span_singleton'.1 hq
⟨r, by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one r (y * x), hr, ← hpq, ← neg_sub, add_sub_cancel]
exact I.neg_mem hpi⟩)
fun hxy : I ⊔ span {y * x + 1} ≠ ⊤ => let ⟨M, hm1, hm2⟩ := exists_le_maximal _ hxy
suffices x ∉ M from (this <| mem_sInf.1 hx ⟨le_trans le_sup_left hm2, hm1⟩).elim
fun hxm => hm1.1.1 <| (eq_top_iff_one _).2 <| add_sub_cancel' (y * x) 1 ▸
M.sub_mem (le_sup_right.trans hm2 <| subset_span rfl) (M.mul_mem_left _ hxm),
fun hx => mem_sInf.2 fun M ⟨him, hm⟩ => by_contradiction fun hxm =>
let ⟨y, i, hi, df⟩ := hm.exists_inv hxm
let ⟨z, hz⟩ := hx (-y)
hm.1.1 <| (eq_top_iff_one _).2 <| sub_sub_cancel (z * -y * x + z) 1 ▸
M.sub_mem (by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one z, neg_mul, ← sub_eq_iff_eq_add.mpr df.symm, neg_sub,
sub_add_cancel]
exact M.mul_mem_left _ hi) <| him hz⟩
#align ideal.mem_jacobson_iff Ideal.mem_jacobson_iff
theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
∃ s, s * r - 1 ∈ I := by
cases' mem_jacobson_iff.1 h 1 with s hs
use s
simpa [mul_sub] using hs
#align ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson Ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M := by
|
use fun hI => ⟨{ J : Ideal R | I ≤ J ∧ J.IsMaximal }, ⟨fun _ hJ => Or.inl hJ.right, hI.symm⟩⟩
|
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M := by
|
Mathlib.RingTheory.JacobsonIdeal.131_0.Lz0MgLQMj1bGzuN
|
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M
|
Mathlib_RingTheory_JacobsonIdeal
|
case mpr
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I : Ideal R
⊢ (∃ M, (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M) → jacobson I = I
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
/-!
# Jacobson radical
The Jacobson radical of a ring `R` is defined to be the intersection of all maximal ideals of `R`.
This is similar to how the nilradical is equal to the intersection of all prime ideals of `R`.
We can extend the idea of the nilradical to ideals of `R`,
by letting the radical of an ideal `I` be the intersection of prime ideals containing `I`.
Under this extension, the original nilradical is the radical of the zero ideal `⊥`.
Here we define the Jacobson radical of an ideal `I` in a similar way,
as the intersection of maximal ideals containing `I`.
## Main definitions
Let `R` be a commutative ring, and `I` be an ideal of `R`
* `Ideal.jacobson I` is the jacobson radical, i.e. the infimum of all maximal ideals containing I.
* `Ideal.IsLocal I` is the proposition that the jacobson radical of `I` is itself a maximal ideal
## Main statements
* `mem_jacobson_iff` gives a characterization of members of the jacobson of I
* `Ideal.isLocal_of_isMaximal_radical`: if the radical of I is maximal then so is the jacobson
radical
## Tags
Jacobson, Jacobson radical, Local Ideal
-/
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}
open Polynomial
section Jacobson
section Ring
variable [Ring R] [Ring S] {I : Ideal R}
/-- The Jacobson radical of `I` is the infimum of all maximal (left) ideals containing `I`. -/
def jacobson (I : Ideal R) : Ideal R :=
sInf { J : Ideal R | I ≤ J ∧ IsMaximal J }
#align ideal.jacobson Ideal.jacobson
theorem le_jacobson : I ≤ jacobson I := fun _ hx => mem_sInf.mpr fun _ hJ => hJ.left hx
#align ideal.le_jacobson Ideal.le_jacobson
@[simp]
theorem jacobson_idem : jacobson (jacobson I) = jacobson I :=
le_antisymm (sInf_le_sInf fun _ hJ => ⟨sInf_le hJ, hJ.2⟩) le_jacobson
#align ideal.jacobson_idem Ideal.jacobson_idem
@[simp]
theorem jacobson_top : jacobson (⊤ : Ideal R) = ⊤ :=
eq_top_iff.2 le_jacobson
#align ideal.jacobson_top Ideal.jacobson_top
@[simp]
theorem jacobson_eq_top_iff : jacobson I = ⊤ ↔ I = ⊤ :=
⟨fun H =>
by_contradiction fun hi => let ⟨M, hm, him⟩ := exists_le_maximal I hi
lt_top_iff_ne_top.1
(lt_of_le_of_lt (show jacobson I ≤ M from sInf_le ⟨him, hm⟩) <|
lt_top_iff_ne_top.2 hm.ne_top) H,
fun H => eq_top_iff.2 <| le_sInf fun _ ⟨hij, _⟩ => H ▸ hij⟩
#align ideal.jacobson_eq_top_iff Ideal.jacobson_eq_top_iff
theorem jacobson_eq_bot : jacobson I = ⊥ → I = ⊥ := fun h => eq_bot_iff.mpr (h ▸ le_jacobson)
#align ideal.jacobson_eq_bot Ideal.jacobson_eq_bot
theorem jacobson_eq_self_of_isMaximal [H : IsMaximal I] : I.jacobson = I :=
le_antisymm (sInf_le ⟨le_of_eq rfl, H⟩) le_jacobson
#align ideal.jacobson_eq_self_of_is_maximal Ideal.jacobson_eq_self_of_isMaximal
instance (priority := 100) jacobson.isMaximal [H : IsMaximal I] : IsMaximal (jacobson I) :=
⟨⟨fun htop => H.1.1 (jacobson_eq_top_iff.1 htop), fun _ hJ =>
H.1.2 _ (lt_of_le_of_lt le_jacobson hJ)⟩⟩
#align ideal.jacobson.is_maximal Ideal.jacobson.isMaximal
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I :=
⟨fun hx y =>
by_cases
(fun hxy : I ⊔ span {y * x + 1} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy)
let ⟨r, hr⟩ := mem_span_singleton'.1 hq
⟨r, by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one r (y * x), hr, ← hpq, ← neg_sub, add_sub_cancel]
exact I.neg_mem hpi⟩)
fun hxy : I ⊔ span {y * x + 1} ≠ ⊤ => let ⟨M, hm1, hm2⟩ := exists_le_maximal _ hxy
suffices x ∉ M from (this <| mem_sInf.1 hx ⟨le_trans le_sup_left hm2, hm1⟩).elim
fun hxm => hm1.1.1 <| (eq_top_iff_one _).2 <| add_sub_cancel' (y * x) 1 ▸
M.sub_mem (le_sup_right.trans hm2 <| subset_span rfl) (M.mul_mem_left _ hxm),
fun hx => mem_sInf.2 fun M ⟨him, hm⟩ => by_contradiction fun hxm =>
let ⟨y, i, hi, df⟩ := hm.exists_inv hxm
let ⟨z, hz⟩ := hx (-y)
hm.1.1 <| (eq_top_iff_one _).2 <| sub_sub_cancel (z * -y * x + z) 1 ▸
M.sub_mem (by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one z, neg_mul, ← sub_eq_iff_eq_add.mpr df.symm, neg_sub,
sub_add_cancel]
exact M.mul_mem_left _ hi) <| him hz⟩
#align ideal.mem_jacobson_iff Ideal.mem_jacobson_iff
theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
∃ s, s * r - 1 ∈ I := by
cases' mem_jacobson_iff.1 h 1 with s hs
use s
simpa [mul_sub] using hs
#align ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson Ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M := by
use fun hI => ⟨{ J : Ideal R | I ≤ J ∧ J.IsMaximal }, ⟨fun _ hJ => Or.inl hJ.right, hI.symm⟩⟩
|
rintro ⟨M, hM, hInf⟩
|
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M := by
use fun hI => ⟨{ J : Ideal R | I ≤ J ∧ J.IsMaximal }, ⟨fun _ hJ => Or.inl hJ.right, hI.symm⟩⟩
|
Mathlib.RingTheory.JacobsonIdeal.131_0.Lz0MgLQMj1bGzuN
|
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M
|
Mathlib_RingTheory_JacobsonIdeal
|
case mpr.intro.intro
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I : Ideal R
M : Set (Ideal R)
hM : ∀ J ∈ M, IsMaximal J ∨ J = ⊤
hInf : I = sInf M
⊢ jacobson I = I
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
/-!
# Jacobson radical
The Jacobson radical of a ring `R` is defined to be the intersection of all maximal ideals of `R`.
This is similar to how the nilradical is equal to the intersection of all prime ideals of `R`.
We can extend the idea of the nilradical to ideals of `R`,
by letting the radical of an ideal `I` be the intersection of prime ideals containing `I`.
Under this extension, the original nilradical is the radical of the zero ideal `⊥`.
Here we define the Jacobson radical of an ideal `I` in a similar way,
as the intersection of maximal ideals containing `I`.
## Main definitions
Let `R` be a commutative ring, and `I` be an ideal of `R`
* `Ideal.jacobson I` is the jacobson radical, i.e. the infimum of all maximal ideals containing I.
* `Ideal.IsLocal I` is the proposition that the jacobson radical of `I` is itself a maximal ideal
## Main statements
* `mem_jacobson_iff` gives a characterization of members of the jacobson of I
* `Ideal.isLocal_of_isMaximal_radical`: if the radical of I is maximal then so is the jacobson
radical
## Tags
Jacobson, Jacobson radical, Local Ideal
-/
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}
open Polynomial
section Jacobson
section Ring
variable [Ring R] [Ring S] {I : Ideal R}
/-- The Jacobson radical of `I` is the infimum of all maximal (left) ideals containing `I`. -/
def jacobson (I : Ideal R) : Ideal R :=
sInf { J : Ideal R | I ≤ J ∧ IsMaximal J }
#align ideal.jacobson Ideal.jacobson
theorem le_jacobson : I ≤ jacobson I := fun _ hx => mem_sInf.mpr fun _ hJ => hJ.left hx
#align ideal.le_jacobson Ideal.le_jacobson
@[simp]
theorem jacobson_idem : jacobson (jacobson I) = jacobson I :=
le_antisymm (sInf_le_sInf fun _ hJ => ⟨sInf_le hJ, hJ.2⟩) le_jacobson
#align ideal.jacobson_idem Ideal.jacobson_idem
@[simp]
theorem jacobson_top : jacobson (⊤ : Ideal R) = ⊤ :=
eq_top_iff.2 le_jacobson
#align ideal.jacobson_top Ideal.jacobson_top
@[simp]
theorem jacobson_eq_top_iff : jacobson I = ⊤ ↔ I = ⊤ :=
⟨fun H =>
by_contradiction fun hi => let ⟨M, hm, him⟩ := exists_le_maximal I hi
lt_top_iff_ne_top.1
(lt_of_le_of_lt (show jacobson I ≤ M from sInf_le ⟨him, hm⟩) <|
lt_top_iff_ne_top.2 hm.ne_top) H,
fun H => eq_top_iff.2 <| le_sInf fun _ ⟨hij, _⟩ => H ▸ hij⟩
#align ideal.jacobson_eq_top_iff Ideal.jacobson_eq_top_iff
theorem jacobson_eq_bot : jacobson I = ⊥ → I = ⊥ := fun h => eq_bot_iff.mpr (h ▸ le_jacobson)
#align ideal.jacobson_eq_bot Ideal.jacobson_eq_bot
theorem jacobson_eq_self_of_isMaximal [H : IsMaximal I] : I.jacobson = I :=
le_antisymm (sInf_le ⟨le_of_eq rfl, H⟩) le_jacobson
#align ideal.jacobson_eq_self_of_is_maximal Ideal.jacobson_eq_self_of_isMaximal
instance (priority := 100) jacobson.isMaximal [H : IsMaximal I] : IsMaximal (jacobson I) :=
⟨⟨fun htop => H.1.1 (jacobson_eq_top_iff.1 htop), fun _ hJ =>
H.1.2 _ (lt_of_le_of_lt le_jacobson hJ)⟩⟩
#align ideal.jacobson.is_maximal Ideal.jacobson.isMaximal
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I :=
⟨fun hx y =>
by_cases
(fun hxy : I ⊔ span {y * x + 1} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy)
let ⟨r, hr⟩ := mem_span_singleton'.1 hq
⟨r, by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one r (y * x), hr, ← hpq, ← neg_sub, add_sub_cancel]
exact I.neg_mem hpi⟩)
fun hxy : I ⊔ span {y * x + 1} ≠ ⊤ => let ⟨M, hm1, hm2⟩ := exists_le_maximal _ hxy
suffices x ∉ M from (this <| mem_sInf.1 hx ⟨le_trans le_sup_left hm2, hm1⟩).elim
fun hxm => hm1.1.1 <| (eq_top_iff_one _).2 <| add_sub_cancel' (y * x) 1 ▸
M.sub_mem (le_sup_right.trans hm2 <| subset_span rfl) (M.mul_mem_left _ hxm),
fun hx => mem_sInf.2 fun M ⟨him, hm⟩ => by_contradiction fun hxm =>
let ⟨y, i, hi, df⟩ := hm.exists_inv hxm
let ⟨z, hz⟩ := hx (-y)
hm.1.1 <| (eq_top_iff_one _).2 <| sub_sub_cancel (z * -y * x + z) 1 ▸
M.sub_mem (by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one z, neg_mul, ← sub_eq_iff_eq_add.mpr df.symm, neg_sub,
sub_add_cancel]
exact M.mul_mem_left _ hi) <| him hz⟩
#align ideal.mem_jacobson_iff Ideal.mem_jacobson_iff
theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
∃ s, s * r - 1 ∈ I := by
cases' mem_jacobson_iff.1 h 1 with s hs
use s
simpa [mul_sub] using hs
#align ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson Ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M := by
use fun hI => ⟨{ J : Ideal R | I ≤ J ∧ J.IsMaximal }, ⟨fun _ hJ => Or.inl hJ.right, hI.symm⟩⟩
rintro ⟨M, hM, hInf⟩
|
refine le_antisymm (fun x hx => ?_) le_jacobson
|
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M := by
use fun hI => ⟨{ J : Ideal R | I ≤ J ∧ J.IsMaximal }, ⟨fun _ hJ => Or.inl hJ.right, hI.symm⟩⟩
rintro ⟨M, hM, hInf⟩
|
Mathlib.RingTheory.JacobsonIdeal.131_0.Lz0MgLQMj1bGzuN
|
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M
|
Mathlib_RingTheory_JacobsonIdeal
|
case mpr.intro.intro
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I : Ideal R
M : Set (Ideal R)
hM : ∀ J ∈ M, IsMaximal J ∨ J = ⊤
hInf : I = sInf M
x : R
hx : x ∈ jacobson I
⊢ x ∈ I
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
/-!
# Jacobson radical
The Jacobson radical of a ring `R` is defined to be the intersection of all maximal ideals of `R`.
This is similar to how the nilradical is equal to the intersection of all prime ideals of `R`.
We can extend the idea of the nilradical to ideals of `R`,
by letting the radical of an ideal `I` be the intersection of prime ideals containing `I`.
Under this extension, the original nilradical is the radical of the zero ideal `⊥`.
Here we define the Jacobson radical of an ideal `I` in a similar way,
as the intersection of maximal ideals containing `I`.
## Main definitions
Let `R` be a commutative ring, and `I` be an ideal of `R`
* `Ideal.jacobson I` is the jacobson radical, i.e. the infimum of all maximal ideals containing I.
* `Ideal.IsLocal I` is the proposition that the jacobson radical of `I` is itself a maximal ideal
## Main statements
* `mem_jacobson_iff` gives a characterization of members of the jacobson of I
* `Ideal.isLocal_of_isMaximal_radical`: if the radical of I is maximal then so is the jacobson
radical
## Tags
Jacobson, Jacobson radical, Local Ideal
-/
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}
open Polynomial
section Jacobson
section Ring
variable [Ring R] [Ring S] {I : Ideal R}
/-- The Jacobson radical of `I` is the infimum of all maximal (left) ideals containing `I`. -/
def jacobson (I : Ideal R) : Ideal R :=
sInf { J : Ideal R | I ≤ J ∧ IsMaximal J }
#align ideal.jacobson Ideal.jacobson
theorem le_jacobson : I ≤ jacobson I := fun _ hx => mem_sInf.mpr fun _ hJ => hJ.left hx
#align ideal.le_jacobson Ideal.le_jacobson
@[simp]
theorem jacobson_idem : jacobson (jacobson I) = jacobson I :=
le_antisymm (sInf_le_sInf fun _ hJ => ⟨sInf_le hJ, hJ.2⟩) le_jacobson
#align ideal.jacobson_idem Ideal.jacobson_idem
@[simp]
theorem jacobson_top : jacobson (⊤ : Ideal R) = ⊤ :=
eq_top_iff.2 le_jacobson
#align ideal.jacobson_top Ideal.jacobson_top
@[simp]
theorem jacobson_eq_top_iff : jacobson I = ⊤ ↔ I = ⊤ :=
⟨fun H =>
by_contradiction fun hi => let ⟨M, hm, him⟩ := exists_le_maximal I hi
lt_top_iff_ne_top.1
(lt_of_le_of_lt (show jacobson I ≤ M from sInf_le ⟨him, hm⟩) <|
lt_top_iff_ne_top.2 hm.ne_top) H,
fun H => eq_top_iff.2 <| le_sInf fun _ ⟨hij, _⟩ => H ▸ hij⟩
#align ideal.jacobson_eq_top_iff Ideal.jacobson_eq_top_iff
theorem jacobson_eq_bot : jacobson I = ⊥ → I = ⊥ := fun h => eq_bot_iff.mpr (h ▸ le_jacobson)
#align ideal.jacobson_eq_bot Ideal.jacobson_eq_bot
theorem jacobson_eq_self_of_isMaximal [H : IsMaximal I] : I.jacobson = I :=
le_antisymm (sInf_le ⟨le_of_eq rfl, H⟩) le_jacobson
#align ideal.jacobson_eq_self_of_is_maximal Ideal.jacobson_eq_self_of_isMaximal
instance (priority := 100) jacobson.isMaximal [H : IsMaximal I] : IsMaximal (jacobson I) :=
⟨⟨fun htop => H.1.1 (jacobson_eq_top_iff.1 htop), fun _ hJ =>
H.1.2 _ (lt_of_le_of_lt le_jacobson hJ)⟩⟩
#align ideal.jacobson.is_maximal Ideal.jacobson.isMaximal
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I :=
⟨fun hx y =>
by_cases
(fun hxy : I ⊔ span {y * x + 1} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy)
let ⟨r, hr⟩ := mem_span_singleton'.1 hq
⟨r, by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one r (y * x), hr, ← hpq, ← neg_sub, add_sub_cancel]
exact I.neg_mem hpi⟩)
fun hxy : I ⊔ span {y * x + 1} ≠ ⊤ => let ⟨M, hm1, hm2⟩ := exists_le_maximal _ hxy
suffices x ∉ M from (this <| mem_sInf.1 hx ⟨le_trans le_sup_left hm2, hm1⟩).elim
fun hxm => hm1.1.1 <| (eq_top_iff_one _).2 <| add_sub_cancel' (y * x) 1 ▸
M.sub_mem (le_sup_right.trans hm2 <| subset_span rfl) (M.mul_mem_left _ hxm),
fun hx => mem_sInf.2 fun M ⟨him, hm⟩ => by_contradiction fun hxm =>
let ⟨y, i, hi, df⟩ := hm.exists_inv hxm
let ⟨z, hz⟩ := hx (-y)
hm.1.1 <| (eq_top_iff_one _).2 <| sub_sub_cancel (z * -y * x + z) 1 ▸
M.sub_mem (by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one z, neg_mul, ← sub_eq_iff_eq_add.mpr df.symm, neg_sub,
sub_add_cancel]
exact M.mul_mem_left _ hi) <| him hz⟩
#align ideal.mem_jacobson_iff Ideal.mem_jacobson_iff
theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
∃ s, s * r - 1 ∈ I := by
cases' mem_jacobson_iff.1 h 1 with s hs
use s
simpa [mul_sub] using hs
#align ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson Ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M := by
use fun hI => ⟨{ J : Ideal R | I ≤ J ∧ J.IsMaximal }, ⟨fun _ hJ => Or.inl hJ.right, hI.symm⟩⟩
rintro ⟨M, hM, hInf⟩
refine le_antisymm (fun x hx => ?_) le_jacobson
|
rw [hInf, mem_sInf]
|
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M := by
use fun hI => ⟨{ J : Ideal R | I ≤ J ∧ J.IsMaximal }, ⟨fun _ hJ => Or.inl hJ.right, hI.symm⟩⟩
rintro ⟨M, hM, hInf⟩
refine le_antisymm (fun x hx => ?_) le_jacobson
|
Mathlib.RingTheory.JacobsonIdeal.131_0.Lz0MgLQMj1bGzuN
|
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M
|
Mathlib_RingTheory_JacobsonIdeal
|
case mpr.intro.intro
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I : Ideal R
M : Set (Ideal R)
hM : ∀ J ∈ M, IsMaximal J ∨ J = ⊤
hInf : I = sInf M
x : R
hx : x ∈ jacobson I
⊢ ∀ ⦃I : Ideal R⦄, I ∈ M → x ∈ I
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
/-!
# Jacobson radical
The Jacobson radical of a ring `R` is defined to be the intersection of all maximal ideals of `R`.
This is similar to how the nilradical is equal to the intersection of all prime ideals of `R`.
We can extend the idea of the nilradical to ideals of `R`,
by letting the radical of an ideal `I` be the intersection of prime ideals containing `I`.
Under this extension, the original nilradical is the radical of the zero ideal `⊥`.
Here we define the Jacobson radical of an ideal `I` in a similar way,
as the intersection of maximal ideals containing `I`.
## Main definitions
Let `R` be a commutative ring, and `I` be an ideal of `R`
* `Ideal.jacobson I` is the jacobson radical, i.e. the infimum of all maximal ideals containing I.
* `Ideal.IsLocal I` is the proposition that the jacobson radical of `I` is itself a maximal ideal
## Main statements
* `mem_jacobson_iff` gives a characterization of members of the jacobson of I
* `Ideal.isLocal_of_isMaximal_radical`: if the radical of I is maximal then so is the jacobson
radical
## Tags
Jacobson, Jacobson radical, Local Ideal
-/
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}
open Polynomial
section Jacobson
section Ring
variable [Ring R] [Ring S] {I : Ideal R}
/-- The Jacobson radical of `I` is the infimum of all maximal (left) ideals containing `I`. -/
def jacobson (I : Ideal R) : Ideal R :=
sInf { J : Ideal R | I ≤ J ∧ IsMaximal J }
#align ideal.jacobson Ideal.jacobson
theorem le_jacobson : I ≤ jacobson I := fun _ hx => mem_sInf.mpr fun _ hJ => hJ.left hx
#align ideal.le_jacobson Ideal.le_jacobson
@[simp]
theorem jacobson_idem : jacobson (jacobson I) = jacobson I :=
le_antisymm (sInf_le_sInf fun _ hJ => ⟨sInf_le hJ, hJ.2⟩) le_jacobson
#align ideal.jacobson_idem Ideal.jacobson_idem
@[simp]
theorem jacobson_top : jacobson (⊤ : Ideal R) = ⊤ :=
eq_top_iff.2 le_jacobson
#align ideal.jacobson_top Ideal.jacobson_top
@[simp]
theorem jacobson_eq_top_iff : jacobson I = ⊤ ↔ I = ⊤ :=
⟨fun H =>
by_contradiction fun hi => let ⟨M, hm, him⟩ := exists_le_maximal I hi
lt_top_iff_ne_top.1
(lt_of_le_of_lt (show jacobson I ≤ M from sInf_le ⟨him, hm⟩) <|
lt_top_iff_ne_top.2 hm.ne_top) H,
fun H => eq_top_iff.2 <| le_sInf fun _ ⟨hij, _⟩ => H ▸ hij⟩
#align ideal.jacobson_eq_top_iff Ideal.jacobson_eq_top_iff
theorem jacobson_eq_bot : jacobson I = ⊥ → I = ⊥ := fun h => eq_bot_iff.mpr (h ▸ le_jacobson)
#align ideal.jacobson_eq_bot Ideal.jacobson_eq_bot
theorem jacobson_eq_self_of_isMaximal [H : IsMaximal I] : I.jacobson = I :=
le_antisymm (sInf_le ⟨le_of_eq rfl, H⟩) le_jacobson
#align ideal.jacobson_eq_self_of_is_maximal Ideal.jacobson_eq_self_of_isMaximal
instance (priority := 100) jacobson.isMaximal [H : IsMaximal I] : IsMaximal (jacobson I) :=
⟨⟨fun htop => H.1.1 (jacobson_eq_top_iff.1 htop), fun _ hJ =>
H.1.2 _ (lt_of_le_of_lt le_jacobson hJ)⟩⟩
#align ideal.jacobson.is_maximal Ideal.jacobson.isMaximal
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I :=
⟨fun hx y =>
by_cases
(fun hxy : I ⊔ span {y * x + 1} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy)
let ⟨r, hr⟩ := mem_span_singleton'.1 hq
⟨r, by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one r (y * x), hr, ← hpq, ← neg_sub, add_sub_cancel]
exact I.neg_mem hpi⟩)
fun hxy : I ⊔ span {y * x + 1} ≠ ⊤ => let ⟨M, hm1, hm2⟩ := exists_le_maximal _ hxy
suffices x ∉ M from (this <| mem_sInf.1 hx ⟨le_trans le_sup_left hm2, hm1⟩).elim
fun hxm => hm1.1.1 <| (eq_top_iff_one _).2 <| add_sub_cancel' (y * x) 1 ▸
M.sub_mem (le_sup_right.trans hm2 <| subset_span rfl) (M.mul_mem_left _ hxm),
fun hx => mem_sInf.2 fun M ⟨him, hm⟩ => by_contradiction fun hxm =>
let ⟨y, i, hi, df⟩ := hm.exists_inv hxm
let ⟨z, hz⟩ := hx (-y)
hm.1.1 <| (eq_top_iff_one _).2 <| sub_sub_cancel (z * -y * x + z) 1 ▸
M.sub_mem (by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one z, neg_mul, ← sub_eq_iff_eq_add.mpr df.symm, neg_sub,
sub_add_cancel]
exact M.mul_mem_left _ hi) <| him hz⟩
#align ideal.mem_jacobson_iff Ideal.mem_jacobson_iff
theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
∃ s, s * r - 1 ∈ I := by
cases' mem_jacobson_iff.1 h 1 with s hs
use s
simpa [mul_sub] using hs
#align ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson Ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M := by
use fun hI => ⟨{ J : Ideal R | I ≤ J ∧ J.IsMaximal }, ⟨fun _ hJ => Or.inl hJ.right, hI.symm⟩⟩
rintro ⟨M, hM, hInf⟩
refine le_antisymm (fun x hx => ?_) le_jacobson
rw [hInf, mem_sInf]
|
intro I hI
|
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M := by
use fun hI => ⟨{ J : Ideal R | I ≤ J ∧ J.IsMaximal }, ⟨fun _ hJ => Or.inl hJ.right, hI.symm⟩⟩
rintro ⟨M, hM, hInf⟩
refine le_antisymm (fun x hx => ?_) le_jacobson
rw [hInf, mem_sInf]
|
Mathlib.RingTheory.JacobsonIdeal.131_0.Lz0MgLQMj1bGzuN
|
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M
|
Mathlib_RingTheory_JacobsonIdeal
|
case mpr.intro.intro
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I✝ : Ideal R
M : Set (Ideal R)
hM : ∀ J ∈ M, IsMaximal J ∨ J = ⊤
hInf : I✝ = sInf M
x : R
hx : x ∈ jacobson I✝
I : Ideal R
hI : I ∈ M
⊢ x ∈ I
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
/-!
# Jacobson radical
The Jacobson radical of a ring `R` is defined to be the intersection of all maximal ideals of `R`.
This is similar to how the nilradical is equal to the intersection of all prime ideals of `R`.
We can extend the idea of the nilradical to ideals of `R`,
by letting the radical of an ideal `I` be the intersection of prime ideals containing `I`.
Under this extension, the original nilradical is the radical of the zero ideal `⊥`.
Here we define the Jacobson radical of an ideal `I` in a similar way,
as the intersection of maximal ideals containing `I`.
## Main definitions
Let `R` be a commutative ring, and `I` be an ideal of `R`
* `Ideal.jacobson I` is the jacobson radical, i.e. the infimum of all maximal ideals containing I.
* `Ideal.IsLocal I` is the proposition that the jacobson radical of `I` is itself a maximal ideal
## Main statements
* `mem_jacobson_iff` gives a characterization of members of the jacobson of I
* `Ideal.isLocal_of_isMaximal_radical`: if the radical of I is maximal then so is the jacobson
radical
## Tags
Jacobson, Jacobson radical, Local Ideal
-/
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}
open Polynomial
section Jacobson
section Ring
variable [Ring R] [Ring S] {I : Ideal R}
/-- The Jacobson radical of `I` is the infimum of all maximal (left) ideals containing `I`. -/
def jacobson (I : Ideal R) : Ideal R :=
sInf { J : Ideal R | I ≤ J ∧ IsMaximal J }
#align ideal.jacobson Ideal.jacobson
theorem le_jacobson : I ≤ jacobson I := fun _ hx => mem_sInf.mpr fun _ hJ => hJ.left hx
#align ideal.le_jacobson Ideal.le_jacobson
@[simp]
theorem jacobson_idem : jacobson (jacobson I) = jacobson I :=
le_antisymm (sInf_le_sInf fun _ hJ => ⟨sInf_le hJ, hJ.2⟩) le_jacobson
#align ideal.jacobson_idem Ideal.jacobson_idem
@[simp]
theorem jacobson_top : jacobson (⊤ : Ideal R) = ⊤ :=
eq_top_iff.2 le_jacobson
#align ideal.jacobson_top Ideal.jacobson_top
@[simp]
theorem jacobson_eq_top_iff : jacobson I = ⊤ ↔ I = ⊤ :=
⟨fun H =>
by_contradiction fun hi => let ⟨M, hm, him⟩ := exists_le_maximal I hi
lt_top_iff_ne_top.1
(lt_of_le_of_lt (show jacobson I ≤ M from sInf_le ⟨him, hm⟩) <|
lt_top_iff_ne_top.2 hm.ne_top) H,
fun H => eq_top_iff.2 <| le_sInf fun _ ⟨hij, _⟩ => H ▸ hij⟩
#align ideal.jacobson_eq_top_iff Ideal.jacobson_eq_top_iff
theorem jacobson_eq_bot : jacobson I = ⊥ → I = ⊥ := fun h => eq_bot_iff.mpr (h ▸ le_jacobson)
#align ideal.jacobson_eq_bot Ideal.jacobson_eq_bot
theorem jacobson_eq_self_of_isMaximal [H : IsMaximal I] : I.jacobson = I :=
le_antisymm (sInf_le ⟨le_of_eq rfl, H⟩) le_jacobson
#align ideal.jacobson_eq_self_of_is_maximal Ideal.jacobson_eq_self_of_isMaximal
instance (priority := 100) jacobson.isMaximal [H : IsMaximal I] : IsMaximal (jacobson I) :=
⟨⟨fun htop => H.1.1 (jacobson_eq_top_iff.1 htop), fun _ hJ =>
H.1.2 _ (lt_of_le_of_lt le_jacobson hJ)⟩⟩
#align ideal.jacobson.is_maximal Ideal.jacobson.isMaximal
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I :=
⟨fun hx y =>
by_cases
(fun hxy : I ⊔ span {y * x + 1} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy)
let ⟨r, hr⟩ := mem_span_singleton'.1 hq
⟨r, by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one r (y * x), hr, ← hpq, ← neg_sub, add_sub_cancel]
exact I.neg_mem hpi⟩)
fun hxy : I ⊔ span {y * x + 1} ≠ ⊤ => let ⟨M, hm1, hm2⟩ := exists_le_maximal _ hxy
suffices x ∉ M from (this <| mem_sInf.1 hx ⟨le_trans le_sup_left hm2, hm1⟩).elim
fun hxm => hm1.1.1 <| (eq_top_iff_one _).2 <| add_sub_cancel' (y * x) 1 ▸
M.sub_mem (le_sup_right.trans hm2 <| subset_span rfl) (M.mul_mem_left _ hxm),
fun hx => mem_sInf.2 fun M ⟨him, hm⟩ => by_contradiction fun hxm =>
let ⟨y, i, hi, df⟩ := hm.exists_inv hxm
let ⟨z, hz⟩ := hx (-y)
hm.1.1 <| (eq_top_iff_one _).2 <| sub_sub_cancel (z * -y * x + z) 1 ▸
M.sub_mem (by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one z, neg_mul, ← sub_eq_iff_eq_add.mpr df.symm, neg_sub,
sub_add_cancel]
exact M.mul_mem_left _ hi) <| him hz⟩
#align ideal.mem_jacobson_iff Ideal.mem_jacobson_iff
theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
∃ s, s * r - 1 ∈ I := by
cases' mem_jacobson_iff.1 h 1 with s hs
use s
simpa [mul_sub] using hs
#align ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson Ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M := by
use fun hI => ⟨{ J : Ideal R | I ≤ J ∧ J.IsMaximal }, ⟨fun _ hJ => Or.inl hJ.right, hI.symm⟩⟩
rintro ⟨M, hM, hInf⟩
refine le_antisymm (fun x hx => ?_) le_jacobson
rw [hInf, mem_sInf]
intro I hI
|
cases' hM I hI with is_max is_top
|
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M := by
use fun hI => ⟨{ J : Ideal R | I ≤ J ∧ J.IsMaximal }, ⟨fun _ hJ => Or.inl hJ.right, hI.symm⟩⟩
rintro ⟨M, hM, hInf⟩
refine le_antisymm (fun x hx => ?_) le_jacobson
rw [hInf, mem_sInf]
intro I hI
|
Mathlib.RingTheory.JacobsonIdeal.131_0.Lz0MgLQMj1bGzuN
|
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M
|
Mathlib_RingTheory_JacobsonIdeal
|
case mpr.intro.intro.inl
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I✝ : Ideal R
M : Set (Ideal R)
hM : ∀ J ∈ M, IsMaximal J ∨ J = ⊤
hInf : I✝ = sInf M
x : R
hx : x ∈ jacobson I✝
I : Ideal R
hI : I ∈ M
is_max : IsMaximal I
⊢ x ∈ I
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
/-!
# Jacobson radical
The Jacobson radical of a ring `R` is defined to be the intersection of all maximal ideals of `R`.
This is similar to how the nilradical is equal to the intersection of all prime ideals of `R`.
We can extend the idea of the nilradical to ideals of `R`,
by letting the radical of an ideal `I` be the intersection of prime ideals containing `I`.
Under this extension, the original nilradical is the radical of the zero ideal `⊥`.
Here we define the Jacobson radical of an ideal `I` in a similar way,
as the intersection of maximal ideals containing `I`.
## Main definitions
Let `R` be a commutative ring, and `I` be an ideal of `R`
* `Ideal.jacobson I` is the jacobson radical, i.e. the infimum of all maximal ideals containing I.
* `Ideal.IsLocal I` is the proposition that the jacobson radical of `I` is itself a maximal ideal
## Main statements
* `mem_jacobson_iff` gives a characterization of members of the jacobson of I
* `Ideal.isLocal_of_isMaximal_radical`: if the radical of I is maximal then so is the jacobson
radical
## Tags
Jacobson, Jacobson radical, Local Ideal
-/
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}
open Polynomial
section Jacobson
section Ring
variable [Ring R] [Ring S] {I : Ideal R}
/-- The Jacobson radical of `I` is the infimum of all maximal (left) ideals containing `I`. -/
def jacobson (I : Ideal R) : Ideal R :=
sInf { J : Ideal R | I ≤ J ∧ IsMaximal J }
#align ideal.jacobson Ideal.jacobson
theorem le_jacobson : I ≤ jacobson I := fun _ hx => mem_sInf.mpr fun _ hJ => hJ.left hx
#align ideal.le_jacobson Ideal.le_jacobson
@[simp]
theorem jacobson_idem : jacobson (jacobson I) = jacobson I :=
le_antisymm (sInf_le_sInf fun _ hJ => ⟨sInf_le hJ, hJ.2⟩) le_jacobson
#align ideal.jacobson_idem Ideal.jacobson_idem
@[simp]
theorem jacobson_top : jacobson (⊤ : Ideal R) = ⊤ :=
eq_top_iff.2 le_jacobson
#align ideal.jacobson_top Ideal.jacobson_top
@[simp]
theorem jacobson_eq_top_iff : jacobson I = ⊤ ↔ I = ⊤ :=
⟨fun H =>
by_contradiction fun hi => let ⟨M, hm, him⟩ := exists_le_maximal I hi
lt_top_iff_ne_top.1
(lt_of_le_of_lt (show jacobson I ≤ M from sInf_le ⟨him, hm⟩) <|
lt_top_iff_ne_top.2 hm.ne_top) H,
fun H => eq_top_iff.2 <| le_sInf fun _ ⟨hij, _⟩ => H ▸ hij⟩
#align ideal.jacobson_eq_top_iff Ideal.jacobson_eq_top_iff
theorem jacobson_eq_bot : jacobson I = ⊥ → I = ⊥ := fun h => eq_bot_iff.mpr (h ▸ le_jacobson)
#align ideal.jacobson_eq_bot Ideal.jacobson_eq_bot
theorem jacobson_eq_self_of_isMaximal [H : IsMaximal I] : I.jacobson = I :=
le_antisymm (sInf_le ⟨le_of_eq rfl, H⟩) le_jacobson
#align ideal.jacobson_eq_self_of_is_maximal Ideal.jacobson_eq_self_of_isMaximal
instance (priority := 100) jacobson.isMaximal [H : IsMaximal I] : IsMaximal (jacobson I) :=
⟨⟨fun htop => H.1.1 (jacobson_eq_top_iff.1 htop), fun _ hJ =>
H.1.2 _ (lt_of_le_of_lt le_jacobson hJ)⟩⟩
#align ideal.jacobson.is_maximal Ideal.jacobson.isMaximal
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I :=
⟨fun hx y =>
by_cases
(fun hxy : I ⊔ span {y * x + 1} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy)
let ⟨r, hr⟩ := mem_span_singleton'.1 hq
⟨r, by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one r (y * x), hr, ← hpq, ← neg_sub, add_sub_cancel]
exact I.neg_mem hpi⟩)
fun hxy : I ⊔ span {y * x + 1} ≠ ⊤ => let ⟨M, hm1, hm2⟩ := exists_le_maximal _ hxy
suffices x ∉ M from (this <| mem_sInf.1 hx ⟨le_trans le_sup_left hm2, hm1⟩).elim
fun hxm => hm1.1.1 <| (eq_top_iff_one _).2 <| add_sub_cancel' (y * x) 1 ▸
M.sub_mem (le_sup_right.trans hm2 <| subset_span rfl) (M.mul_mem_left _ hxm),
fun hx => mem_sInf.2 fun M ⟨him, hm⟩ => by_contradiction fun hxm =>
let ⟨y, i, hi, df⟩ := hm.exists_inv hxm
let ⟨z, hz⟩ := hx (-y)
hm.1.1 <| (eq_top_iff_one _).2 <| sub_sub_cancel (z * -y * x + z) 1 ▸
M.sub_mem (by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one z, neg_mul, ← sub_eq_iff_eq_add.mpr df.symm, neg_sub,
sub_add_cancel]
exact M.mul_mem_left _ hi) <| him hz⟩
#align ideal.mem_jacobson_iff Ideal.mem_jacobson_iff
theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
∃ s, s * r - 1 ∈ I := by
cases' mem_jacobson_iff.1 h 1 with s hs
use s
simpa [mul_sub] using hs
#align ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson Ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M := by
use fun hI => ⟨{ J : Ideal R | I ≤ J ∧ J.IsMaximal }, ⟨fun _ hJ => Or.inl hJ.right, hI.symm⟩⟩
rintro ⟨M, hM, hInf⟩
refine le_antisymm (fun x hx => ?_) le_jacobson
rw [hInf, mem_sInf]
intro I hI
cases' hM I hI with is_max is_top
·
|
exact (mem_sInf.1 hx) ⟨le_sInf_iff.1 (le_of_eq hInf) I hI, is_max⟩
|
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M := by
use fun hI => ⟨{ J : Ideal R | I ≤ J ∧ J.IsMaximal }, ⟨fun _ hJ => Or.inl hJ.right, hI.symm⟩⟩
rintro ⟨M, hM, hInf⟩
refine le_antisymm (fun x hx => ?_) le_jacobson
rw [hInf, mem_sInf]
intro I hI
cases' hM I hI with is_max is_top
·
|
Mathlib.RingTheory.JacobsonIdeal.131_0.Lz0MgLQMj1bGzuN
|
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M
|
Mathlib_RingTheory_JacobsonIdeal
|
case mpr.intro.intro.inr
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I✝ : Ideal R
M : Set (Ideal R)
hM : ∀ J ∈ M, IsMaximal J ∨ J = ⊤
hInf : I✝ = sInf M
x : R
hx : x ∈ jacobson I✝
I : Ideal R
hI : I ∈ M
is_top : I = ⊤
⊢ x ∈ I
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
/-!
# Jacobson radical
The Jacobson radical of a ring `R` is defined to be the intersection of all maximal ideals of `R`.
This is similar to how the nilradical is equal to the intersection of all prime ideals of `R`.
We can extend the idea of the nilradical to ideals of `R`,
by letting the radical of an ideal `I` be the intersection of prime ideals containing `I`.
Under this extension, the original nilradical is the radical of the zero ideal `⊥`.
Here we define the Jacobson radical of an ideal `I` in a similar way,
as the intersection of maximal ideals containing `I`.
## Main definitions
Let `R` be a commutative ring, and `I` be an ideal of `R`
* `Ideal.jacobson I` is the jacobson radical, i.e. the infimum of all maximal ideals containing I.
* `Ideal.IsLocal I` is the proposition that the jacobson radical of `I` is itself a maximal ideal
## Main statements
* `mem_jacobson_iff` gives a characterization of members of the jacobson of I
* `Ideal.isLocal_of_isMaximal_radical`: if the radical of I is maximal then so is the jacobson
radical
## Tags
Jacobson, Jacobson radical, Local Ideal
-/
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}
open Polynomial
section Jacobson
section Ring
variable [Ring R] [Ring S] {I : Ideal R}
/-- The Jacobson radical of `I` is the infimum of all maximal (left) ideals containing `I`. -/
def jacobson (I : Ideal R) : Ideal R :=
sInf { J : Ideal R | I ≤ J ∧ IsMaximal J }
#align ideal.jacobson Ideal.jacobson
theorem le_jacobson : I ≤ jacobson I := fun _ hx => mem_sInf.mpr fun _ hJ => hJ.left hx
#align ideal.le_jacobson Ideal.le_jacobson
@[simp]
theorem jacobson_idem : jacobson (jacobson I) = jacobson I :=
le_antisymm (sInf_le_sInf fun _ hJ => ⟨sInf_le hJ, hJ.2⟩) le_jacobson
#align ideal.jacobson_idem Ideal.jacobson_idem
@[simp]
theorem jacobson_top : jacobson (⊤ : Ideal R) = ⊤ :=
eq_top_iff.2 le_jacobson
#align ideal.jacobson_top Ideal.jacobson_top
@[simp]
theorem jacobson_eq_top_iff : jacobson I = ⊤ ↔ I = ⊤ :=
⟨fun H =>
by_contradiction fun hi => let ⟨M, hm, him⟩ := exists_le_maximal I hi
lt_top_iff_ne_top.1
(lt_of_le_of_lt (show jacobson I ≤ M from sInf_le ⟨him, hm⟩) <|
lt_top_iff_ne_top.2 hm.ne_top) H,
fun H => eq_top_iff.2 <| le_sInf fun _ ⟨hij, _⟩ => H ▸ hij⟩
#align ideal.jacobson_eq_top_iff Ideal.jacobson_eq_top_iff
theorem jacobson_eq_bot : jacobson I = ⊥ → I = ⊥ := fun h => eq_bot_iff.mpr (h ▸ le_jacobson)
#align ideal.jacobson_eq_bot Ideal.jacobson_eq_bot
theorem jacobson_eq_self_of_isMaximal [H : IsMaximal I] : I.jacobson = I :=
le_antisymm (sInf_le ⟨le_of_eq rfl, H⟩) le_jacobson
#align ideal.jacobson_eq_self_of_is_maximal Ideal.jacobson_eq_self_of_isMaximal
instance (priority := 100) jacobson.isMaximal [H : IsMaximal I] : IsMaximal (jacobson I) :=
⟨⟨fun htop => H.1.1 (jacobson_eq_top_iff.1 htop), fun _ hJ =>
H.1.2 _ (lt_of_le_of_lt le_jacobson hJ)⟩⟩
#align ideal.jacobson.is_maximal Ideal.jacobson.isMaximal
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I :=
⟨fun hx y =>
by_cases
(fun hxy : I ⊔ span {y * x + 1} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy)
let ⟨r, hr⟩ := mem_span_singleton'.1 hq
⟨r, by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one r (y * x), hr, ← hpq, ← neg_sub, add_sub_cancel]
exact I.neg_mem hpi⟩)
fun hxy : I ⊔ span {y * x + 1} ≠ ⊤ => let ⟨M, hm1, hm2⟩ := exists_le_maximal _ hxy
suffices x ∉ M from (this <| mem_sInf.1 hx ⟨le_trans le_sup_left hm2, hm1⟩).elim
fun hxm => hm1.1.1 <| (eq_top_iff_one _).2 <| add_sub_cancel' (y * x) 1 ▸
M.sub_mem (le_sup_right.trans hm2 <| subset_span rfl) (M.mul_mem_left _ hxm),
fun hx => mem_sInf.2 fun M ⟨him, hm⟩ => by_contradiction fun hxm =>
let ⟨y, i, hi, df⟩ := hm.exists_inv hxm
let ⟨z, hz⟩ := hx (-y)
hm.1.1 <| (eq_top_iff_one _).2 <| sub_sub_cancel (z * -y * x + z) 1 ▸
M.sub_mem (by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one z, neg_mul, ← sub_eq_iff_eq_add.mpr df.symm, neg_sub,
sub_add_cancel]
exact M.mul_mem_left _ hi) <| him hz⟩
#align ideal.mem_jacobson_iff Ideal.mem_jacobson_iff
theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
∃ s, s * r - 1 ∈ I := by
cases' mem_jacobson_iff.1 h 1 with s hs
use s
simpa [mul_sub] using hs
#align ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson Ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M := by
use fun hI => ⟨{ J : Ideal R | I ≤ J ∧ J.IsMaximal }, ⟨fun _ hJ => Or.inl hJ.right, hI.symm⟩⟩
rintro ⟨M, hM, hInf⟩
refine le_antisymm (fun x hx => ?_) le_jacobson
rw [hInf, mem_sInf]
intro I hI
cases' hM I hI with is_max is_top
· exact (mem_sInf.1 hx) ⟨le_sInf_iff.1 (le_of_eq hInf) I hI, is_max⟩
·
|
exact is_top.symm ▸ Submodule.mem_top
|
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M := by
use fun hI => ⟨{ J : Ideal R | I ≤ J ∧ J.IsMaximal }, ⟨fun _ hJ => Or.inl hJ.right, hI.symm⟩⟩
rintro ⟨M, hM, hInf⟩
refine le_antisymm (fun x hx => ?_) le_jacobson
rw [hInf, mem_sInf]
intro I hI
cases' hM I hI with is_max is_top
· exact (mem_sInf.1 hx) ⟨le_sInf_iff.1 (le_of_eq hInf) I hI, is_max⟩
·
|
Mathlib.RingTheory.JacobsonIdeal.131_0.Lz0MgLQMj1bGzuN
|
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M
|
Mathlib_RingTheory_JacobsonIdeal
|
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I : Ideal R
⊢ jacobson I = I ↔ ∀ x ∉ I, ∃ M, (I ≤ M ∧ IsMaximal M) ∧ x ∉ M
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
/-!
# Jacobson radical
The Jacobson radical of a ring `R` is defined to be the intersection of all maximal ideals of `R`.
This is similar to how the nilradical is equal to the intersection of all prime ideals of `R`.
We can extend the idea of the nilradical to ideals of `R`,
by letting the radical of an ideal `I` be the intersection of prime ideals containing `I`.
Under this extension, the original nilradical is the radical of the zero ideal `⊥`.
Here we define the Jacobson radical of an ideal `I` in a similar way,
as the intersection of maximal ideals containing `I`.
## Main definitions
Let `R` be a commutative ring, and `I` be an ideal of `R`
* `Ideal.jacobson I` is the jacobson radical, i.e. the infimum of all maximal ideals containing I.
* `Ideal.IsLocal I` is the proposition that the jacobson radical of `I` is itself a maximal ideal
## Main statements
* `mem_jacobson_iff` gives a characterization of members of the jacobson of I
* `Ideal.isLocal_of_isMaximal_radical`: if the radical of I is maximal then so is the jacobson
radical
## Tags
Jacobson, Jacobson radical, Local Ideal
-/
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}
open Polynomial
section Jacobson
section Ring
variable [Ring R] [Ring S] {I : Ideal R}
/-- The Jacobson radical of `I` is the infimum of all maximal (left) ideals containing `I`. -/
def jacobson (I : Ideal R) : Ideal R :=
sInf { J : Ideal R | I ≤ J ∧ IsMaximal J }
#align ideal.jacobson Ideal.jacobson
theorem le_jacobson : I ≤ jacobson I := fun _ hx => mem_sInf.mpr fun _ hJ => hJ.left hx
#align ideal.le_jacobson Ideal.le_jacobson
@[simp]
theorem jacobson_idem : jacobson (jacobson I) = jacobson I :=
le_antisymm (sInf_le_sInf fun _ hJ => ⟨sInf_le hJ, hJ.2⟩) le_jacobson
#align ideal.jacobson_idem Ideal.jacobson_idem
@[simp]
theorem jacobson_top : jacobson (⊤ : Ideal R) = ⊤ :=
eq_top_iff.2 le_jacobson
#align ideal.jacobson_top Ideal.jacobson_top
@[simp]
theorem jacobson_eq_top_iff : jacobson I = ⊤ ↔ I = ⊤ :=
⟨fun H =>
by_contradiction fun hi => let ⟨M, hm, him⟩ := exists_le_maximal I hi
lt_top_iff_ne_top.1
(lt_of_le_of_lt (show jacobson I ≤ M from sInf_le ⟨him, hm⟩) <|
lt_top_iff_ne_top.2 hm.ne_top) H,
fun H => eq_top_iff.2 <| le_sInf fun _ ⟨hij, _⟩ => H ▸ hij⟩
#align ideal.jacobson_eq_top_iff Ideal.jacobson_eq_top_iff
theorem jacobson_eq_bot : jacobson I = ⊥ → I = ⊥ := fun h => eq_bot_iff.mpr (h ▸ le_jacobson)
#align ideal.jacobson_eq_bot Ideal.jacobson_eq_bot
theorem jacobson_eq_self_of_isMaximal [H : IsMaximal I] : I.jacobson = I :=
le_antisymm (sInf_le ⟨le_of_eq rfl, H⟩) le_jacobson
#align ideal.jacobson_eq_self_of_is_maximal Ideal.jacobson_eq_self_of_isMaximal
instance (priority := 100) jacobson.isMaximal [H : IsMaximal I] : IsMaximal (jacobson I) :=
⟨⟨fun htop => H.1.1 (jacobson_eq_top_iff.1 htop), fun _ hJ =>
H.1.2 _ (lt_of_le_of_lt le_jacobson hJ)⟩⟩
#align ideal.jacobson.is_maximal Ideal.jacobson.isMaximal
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I :=
⟨fun hx y =>
by_cases
(fun hxy : I ⊔ span {y * x + 1} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy)
let ⟨r, hr⟩ := mem_span_singleton'.1 hq
⟨r, by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one r (y * x), hr, ← hpq, ← neg_sub, add_sub_cancel]
exact I.neg_mem hpi⟩)
fun hxy : I ⊔ span {y * x + 1} ≠ ⊤ => let ⟨M, hm1, hm2⟩ := exists_le_maximal _ hxy
suffices x ∉ M from (this <| mem_sInf.1 hx ⟨le_trans le_sup_left hm2, hm1⟩).elim
fun hxm => hm1.1.1 <| (eq_top_iff_one _).2 <| add_sub_cancel' (y * x) 1 ▸
M.sub_mem (le_sup_right.trans hm2 <| subset_span rfl) (M.mul_mem_left _ hxm),
fun hx => mem_sInf.2 fun M ⟨him, hm⟩ => by_contradiction fun hxm =>
let ⟨y, i, hi, df⟩ := hm.exists_inv hxm
let ⟨z, hz⟩ := hx (-y)
hm.1.1 <| (eq_top_iff_one _).2 <| sub_sub_cancel (z * -y * x + z) 1 ▸
M.sub_mem (by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one z, neg_mul, ← sub_eq_iff_eq_add.mpr df.symm, neg_sub,
sub_add_cancel]
exact M.mul_mem_left _ hi) <| him hz⟩
#align ideal.mem_jacobson_iff Ideal.mem_jacobson_iff
theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
∃ s, s * r - 1 ∈ I := by
cases' mem_jacobson_iff.1 h 1 with s hs
use s
simpa [mul_sub] using hs
#align ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson Ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M := by
use fun hI => ⟨{ J : Ideal R | I ≤ J ∧ J.IsMaximal }, ⟨fun _ hJ => Or.inl hJ.right, hI.symm⟩⟩
rintro ⟨M, hM, hInf⟩
refine le_antisymm (fun x hx => ?_) le_jacobson
rw [hInf, mem_sInf]
intro I hI
cases' hM I hI with is_max is_top
· exact (mem_sInf.1 hx) ⟨le_sInf_iff.1 (le_of_eq hInf) I hI, is_max⟩
· exact is_top.symm ▸ Submodule.mem_top
#align ideal.eq_jacobson_iff_Inf_maximal Ideal.eq_jacobson_iff_sInf_maximal
theorem eq_jacobson_iff_sInf_maximal' :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, ∀ (K : Ideal R), J < K → K = ⊤) ∧ I = sInf M :=
eq_jacobson_iff_sInf_maximal.trans
⟨fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ K hK =>
Or.recOn (hM.1 J hJ) (fun h => h.1.2 K hK) fun h => eq_top_iff.2 (le_of_lt (h ▸ hK)),
hM.2⟩⟩,
fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ =>
Or.recOn (Classical.em (J = ⊤)) (fun h => Or.inr h) fun h => Or.inl ⟨⟨h, hM.1 J hJ⟩⟩,
hM.2⟩⟩⟩
#align ideal.eq_jacobson_iff_Inf_maximal' Ideal.eq_jacobson_iff_sInf_maximal'
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M := by
|
constructor
|
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M := by
|
Mathlib.RingTheory.JacobsonIdeal.162_0.Lz0MgLQMj1bGzuN
|
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M
|
Mathlib_RingTheory_JacobsonIdeal
|
case mp
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I : Ideal R
⊢ jacobson I = I → ∀ x ∉ I, ∃ M, (I ≤ M ∧ IsMaximal M) ∧ x ∉ M
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
/-!
# Jacobson radical
The Jacobson radical of a ring `R` is defined to be the intersection of all maximal ideals of `R`.
This is similar to how the nilradical is equal to the intersection of all prime ideals of `R`.
We can extend the idea of the nilradical to ideals of `R`,
by letting the radical of an ideal `I` be the intersection of prime ideals containing `I`.
Under this extension, the original nilradical is the radical of the zero ideal `⊥`.
Here we define the Jacobson radical of an ideal `I` in a similar way,
as the intersection of maximal ideals containing `I`.
## Main definitions
Let `R` be a commutative ring, and `I` be an ideal of `R`
* `Ideal.jacobson I` is the jacobson radical, i.e. the infimum of all maximal ideals containing I.
* `Ideal.IsLocal I` is the proposition that the jacobson radical of `I` is itself a maximal ideal
## Main statements
* `mem_jacobson_iff` gives a characterization of members of the jacobson of I
* `Ideal.isLocal_of_isMaximal_radical`: if the radical of I is maximal then so is the jacobson
radical
## Tags
Jacobson, Jacobson radical, Local Ideal
-/
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}
open Polynomial
section Jacobson
section Ring
variable [Ring R] [Ring S] {I : Ideal R}
/-- The Jacobson radical of `I` is the infimum of all maximal (left) ideals containing `I`. -/
def jacobson (I : Ideal R) : Ideal R :=
sInf { J : Ideal R | I ≤ J ∧ IsMaximal J }
#align ideal.jacobson Ideal.jacobson
theorem le_jacobson : I ≤ jacobson I := fun _ hx => mem_sInf.mpr fun _ hJ => hJ.left hx
#align ideal.le_jacobson Ideal.le_jacobson
@[simp]
theorem jacobson_idem : jacobson (jacobson I) = jacobson I :=
le_antisymm (sInf_le_sInf fun _ hJ => ⟨sInf_le hJ, hJ.2⟩) le_jacobson
#align ideal.jacobson_idem Ideal.jacobson_idem
@[simp]
theorem jacobson_top : jacobson (⊤ : Ideal R) = ⊤ :=
eq_top_iff.2 le_jacobson
#align ideal.jacobson_top Ideal.jacobson_top
@[simp]
theorem jacobson_eq_top_iff : jacobson I = ⊤ ↔ I = ⊤ :=
⟨fun H =>
by_contradiction fun hi => let ⟨M, hm, him⟩ := exists_le_maximal I hi
lt_top_iff_ne_top.1
(lt_of_le_of_lt (show jacobson I ≤ M from sInf_le ⟨him, hm⟩) <|
lt_top_iff_ne_top.2 hm.ne_top) H,
fun H => eq_top_iff.2 <| le_sInf fun _ ⟨hij, _⟩ => H ▸ hij⟩
#align ideal.jacobson_eq_top_iff Ideal.jacobson_eq_top_iff
theorem jacobson_eq_bot : jacobson I = ⊥ → I = ⊥ := fun h => eq_bot_iff.mpr (h ▸ le_jacobson)
#align ideal.jacobson_eq_bot Ideal.jacobson_eq_bot
theorem jacobson_eq_self_of_isMaximal [H : IsMaximal I] : I.jacobson = I :=
le_antisymm (sInf_le ⟨le_of_eq rfl, H⟩) le_jacobson
#align ideal.jacobson_eq_self_of_is_maximal Ideal.jacobson_eq_self_of_isMaximal
instance (priority := 100) jacobson.isMaximal [H : IsMaximal I] : IsMaximal (jacobson I) :=
⟨⟨fun htop => H.1.1 (jacobson_eq_top_iff.1 htop), fun _ hJ =>
H.1.2 _ (lt_of_le_of_lt le_jacobson hJ)⟩⟩
#align ideal.jacobson.is_maximal Ideal.jacobson.isMaximal
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I :=
⟨fun hx y =>
by_cases
(fun hxy : I ⊔ span {y * x + 1} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy)
let ⟨r, hr⟩ := mem_span_singleton'.1 hq
⟨r, by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one r (y * x), hr, ← hpq, ← neg_sub, add_sub_cancel]
exact I.neg_mem hpi⟩)
fun hxy : I ⊔ span {y * x + 1} ≠ ⊤ => let ⟨M, hm1, hm2⟩ := exists_le_maximal _ hxy
suffices x ∉ M from (this <| mem_sInf.1 hx ⟨le_trans le_sup_left hm2, hm1⟩).elim
fun hxm => hm1.1.1 <| (eq_top_iff_one _).2 <| add_sub_cancel' (y * x) 1 ▸
M.sub_mem (le_sup_right.trans hm2 <| subset_span rfl) (M.mul_mem_left _ hxm),
fun hx => mem_sInf.2 fun M ⟨him, hm⟩ => by_contradiction fun hxm =>
let ⟨y, i, hi, df⟩ := hm.exists_inv hxm
let ⟨z, hz⟩ := hx (-y)
hm.1.1 <| (eq_top_iff_one _).2 <| sub_sub_cancel (z * -y * x + z) 1 ▸
M.sub_mem (by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one z, neg_mul, ← sub_eq_iff_eq_add.mpr df.symm, neg_sub,
sub_add_cancel]
exact M.mul_mem_left _ hi) <| him hz⟩
#align ideal.mem_jacobson_iff Ideal.mem_jacobson_iff
theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
∃ s, s * r - 1 ∈ I := by
cases' mem_jacobson_iff.1 h 1 with s hs
use s
simpa [mul_sub] using hs
#align ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson Ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M := by
use fun hI => ⟨{ J : Ideal R | I ≤ J ∧ J.IsMaximal }, ⟨fun _ hJ => Or.inl hJ.right, hI.symm⟩⟩
rintro ⟨M, hM, hInf⟩
refine le_antisymm (fun x hx => ?_) le_jacobson
rw [hInf, mem_sInf]
intro I hI
cases' hM I hI with is_max is_top
· exact (mem_sInf.1 hx) ⟨le_sInf_iff.1 (le_of_eq hInf) I hI, is_max⟩
· exact is_top.symm ▸ Submodule.mem_top
#align ideal.eq_jacobson_iff_Inf_maximal Ideal.eq_jacobson_iff_sInf_maximal
theorem eq_jacobson_iff_sInf_maximal' :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, ∀ (K : Ideal R), J < K → K = ⊤) ∧ I = sInf M :=
eq_jacobson_iff_sInf_maximal.trans
⟨fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ K hK =>
Or.recOn (hM.1 J hJ) (fun h => h.1.2 K hK) fun h => eq_top_iff.2 (le_of_lt (h ▸ hK)),
hM.2⟩⟩,
fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ =>
Or.recOn (Classical.em (J = ⊤)) (fun h => Or.inr h) fun h => Or.inl ⟨⟨h, hM.1 J hJ⟩⟩,
hM.2⟩⟩⟩
#align ideal.eq_jacobson_iff_Inf_maximal' Ideal.eq_jacobson_iff_sInf_maximal'
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M := by
constructor
·
|
intro h x hx
|
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M := by
constructor
·
|
Mathlib.RingTheory.JacobsonIdeal.162_0.Lz0MgLQMj1bGzuN
|
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M
|
Mathlib_RingTheory_JacobsonIdeal
|
case mp
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I : Ideal R
h : jacobson I = I
x : R
hx : x ∉ I
⊢ ∃ M, (I ≤ M ∧ IsMaximal M) ∧ x ∉ M
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
/-!
# Jacobson radical
The Jacobson radical of a ring `R` is defined to be the intersection of all maximal ideals of `R`.
This is similar to how the nilradical is equal to the intersection of all prime ideals of `R`.
We can extend the idea of the nilradical to ideals of `R`,
by letting the radical of an ideal `I` be the intersection of prime ideals containing `I`.
Under this extension, the original nilradical is the radical of the zero ideal `⊥`.
Here we define the Jacobson radical of an ideal `I` in a similar way,
as the intersection of maximal ideals containing `I`.
## Main definitions
Let `R` be a commutative ring, and `I` be an ideal of `R`
* `Ideal.jacobson I` is the jacobson radical, i.e. the infimum of all maximal ideals containing I.
* `Ideal.IsLocal I` is the proposition that the jacobson radical of `I` is itself a maximal ideal
## Main statements
* `mem_jacobson_iff` gives a characterization of members of the jacobson of I
* `Ideal.isLocal_of_isMaximal_radical`: if the radical of I is maximal then so is the jacobson
radical
## Tags
Jacobson, Jacobson radical, Local Ideal
-/
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}
open Polynomial
section Jacobson
section Ring
variable [Ring R] [Ring S] {I : Ideal R}
/-- The Jacobson radical of `I` is the infimum of all maximal (left) ideals containing `I`. -/
def jacobson (I : Ideal R) : Ideal R :=
sInf { J : Ideal R | I ≤ J ∧ IsMaximal J }
#align ideal.jacobson Ideal.jacobson
theorem le_jacobson : I ≤ jacobson I := fun _ hx => mem_sInf.mpr fun _ hJ => hJ.left hx
#align ideal.le_jacobson Ideal.le_jacobson
@[simp]
theorem jacobson_idem : jacobson (jacobson I) = jacobson I :=
le_antisymm (sInf_le_sInf fun _ hJ => ⟨sInf_le hJ, hJ.2⟩) le_jacobson
#align ideal.jacobson_idem Ideal.jacobson_idem
@[simp]
theorem jacobson_top : jacobson (⊤ : Ideal R) = ⊤ :=
eq_top_iff.2 le_jacobson
#align ideal.jacobson_top Ideal.jacobson_top
@[simp]
theorem jacobson_eq_top_iff : jacobson I = ⊤ ↔ I = ⊤ :=
⟨fun H =>
by_contradiction fun hi => let ⟨M, hm, him⟩ := exists_le_maximal I hi
lt_top_iff_ne_top.1
(lt_of_le_of_lt (show jacobson I ≤ M from sInf_le ⟨him, hm⟩) <|
lt_top_iff_ne_top.2 hm.ne_top) H,
fun H => eq_top_iff.2 <| le_sInf fun _ ⟨hij, _⟩ => H ▸ hij⟩
#align ideal.jacobson_eq_top_iff Ideal.jacobson_eq_top_iff
theorem jacobson_eq_bot : jacobson I = ⊥ → I = ⊥ := fun h => eq_bot_iff.mpr (h ▸ le_jacobson)
#align ideal.jacobson_eq_bot Ideal.jacobson_eq_bot
theorem jacobson_eq_self_of_isMaximal [H : IsMaximal I] : I.jacobson = I :=
le_antisymm (sInf_le ⟨le_of_eq rfl, H⟩) le_jacobson
#align ideal.jacobson_eq_self_of_is_maximal Ideal.jacobson_eq_self_of_isMaximal
instance (priority := 100) jacobson.isMaximal [H : IsMaximal I] : IsMaximal (jacobson I) :=
⟨⟨fun htop => H.1.1 (jacobson_eq_top_iff.1 htop), fun _ hJ =>
H.1.2 _ (lt_of_le_of_lt le_jacobson hJ)⟩⟩
#align ideal.jacobson.is_maximal Ideal.jacobson.isMaximal
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I :=
⟨fun hx y =>
by_cases
(fun hxy : I ⊔ span {y * x + 1} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy)
let ⟨r, hr⟩ := mem_span_singleton'.1 hq
⟨r, by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one r (y * x), hr, ← hpq, ← neg_sub, add_sub_cancel]
exact I.neg_mem hpi⟩)
fun hxy : I ⊔ span {y * x + 1} ≠ ⊤ => let ⟨M, hm1, hm2⟩ := exists_le_maximal _ hxy
suffices x ∉ M from (this <| mem_sInf.1 hx ⟨le_trans le_sup_left hm2, hm1⟩).elim
fun hxm => hm1.1.1 <| (eq_top_iff_one _).2 <| add_sub_cancel' (y * x) 1 ▸
M.sub_mem (le_sup_right.trans hm2 <| subset_span rfl) (M.mul_mem_left _ hxm),
fun hx => mem_sInf.2 fun M ⟨him, hm⟩ => by_contradiction fun hxm =>
let ⟨y, i, hi, df⟩ := hm.exists_inv hxm
let ⟨z, hz⟩ := hx (-y)
hm.1.1 <| (eq_top_iff_one _).2 <| sub_sub_cancel (z * -y * x + z) 1 ▸
M.sub_mem (by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one z, neg_mul, ← sub_eq_iff_eq_add.mpr df.symm, neg_sub,
sub_add_cancel]
exact M.mul_mem_left _ hi) <| him hz⟩
#align ideal.mem_jacobson_iff Ideal.mem_jacobson_iff
theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
∃ s, s * r - 1 ∈ I := by
cases' mem_jacobson_iff.1 h 1 with s hs
use s
simpa [mul_sub] using hs
#align ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson Ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M := by
use fun hI => ⟨{ J : Ideal R | I ≤ J ∧ J.IsMaximal }, ⟨fun _ hJ => Or.inl hJ.right, hI.symm⟩⟩
rintro ⟨M, hM, hInf⟩
refine le_antisymm (fun x hx => ?_) le_jacobson
rw [hInf, mem_sInf]
intro I hI
cases' hM I hI with is_max is_top
· exact (mem_sInf.1 hx) ⟨le_sInf_iff.1 (le_of_eq hInf) I hI, is_max⟩
· exact is_top.symm ▸ Submodule.mem_top
#align ideal.eq_jacobson_iff_Inf_maximal Ideal.eq_jacobson_iff_sInf_maximal
theorem eq_jacobson_iff_sInf_maximal' :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, ∀ (K : Ideal R), J < K → K = ⊤) ∧ I = sInf M :=
eq_jacobson_iff_sInf_maximal.trans
⟨fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ K hK =>
Or.recOn (hM.1 J hJ) (fun h => h.1.2 K hK) fun h => eq_top_iff.2 (le_of_lt (h ▸ hK)),
hM.2⟩⟩,
fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ =>
Or.recOn (Classical.em (J = ⊤)) (fun h => Or.inr h) fun h => Or.inl ⟨⟨h, hM.1 J hJ⟩⟩,
hM.2⟩⟩⟩
#align ideal.eq_jacobson_iff_Inf_maximal' Ideal.eq_jacobson_iff_sInf_maximal'
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M := by
constructor
· intro h x hx
|
erw [← h, mem_sInf] at hx
|
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M := by
constructor
· intro h x hx
|
Mathlib.RingTheory.JacobsonIdeal.162_0.Lz0MgLQMj1bGzuN
|
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M
|
Mathlib_RingTheory_JacobsonIdeal
|
case mp
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I : Ideal R
h : jacobson I = I
x : R
hx : ¬∀ ⦃I_1 : Ideal R⦄, I_1 ∈ {J | I ≤ J ∧ IsMaximal J} → x ∈ I_1
⊢ ∃ M, (I ≤ M ∧ IsMaximal M) ∧ x ∉ M
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
/-!
# Jacobson radical
The Jacobson radical of a ring `R` is defined to be the intersection of all maximal ideals of `R`.
This is similar to how the nilradical is equal to the intersection of all prime ideals of `R`.
We can extend the idea of the nilradical to ideals of `R`,
by letting the radical of an ideal `I` be the intersection of prime ideals containing `I`.
Under this extension, the original nilradical is the radical of the zero ideal `⊥`.
Here we define the Jacobson radical of an ideal `I` in a similar way,
as the intersection of maximal ideals containing `I`.
## Main definitions
Let `R` be a commutative ring, and `I` be an ideal of `R`
* `Ideal.jacobson I` is the jacobson radical, i.e. the infimum of all maximal ideals containing I.
* `Ideal.IsLocal I` is the proposition that the jacobson radical of `I` is itself a maximal ideal
## Main statements
* `mem_jacobson_iff` gives a characterization of members of the jacobson of I
* `Ideal.isLocal_of_isMaximal_radical`: if the radical of I is maximal then so is the jacobson
radical
## Tags
Jacobson, Jacobson radical, Local Ideal
-/
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}
open Polynomial
section Jacobson
section Ring
variable [Ring R] [Ring S] {I : Ideal R}
/-- The Jacobson radical of `I` is the infimum of all maximal (left) ideals containing `I`. -/
def jacobson (I : Ideal R) : Ideal R :=
sInf { J : Ideal R | I ≤ J ∧ IsMaximal J }
#align ideal.jacobson Ideal.jacobson
theorem le_jacobson : I ≤ jacobson I := fun _ hx => mem_sInf.mpr fun _ hJ => hJ.left hx
#align ideal.le_jacobson Ideal.le_jacobson
@[simp]
theorem jacobson_idem : jacobson (jacobson I) = jacobson I :=
le_antisymm (sInf_le_sInf fun _ hJ => ⟨sInf_le hJ, hJ.2⟩) le_jacobson
#align ideal.jacobson_idem Ideal.jacobson_idem
@[simp]
theorem jacobson_top : jacobson (⊤ : Ideal R) = ⊤ :=
eq_top_iff.2 le_jacobson
#align ideal.jacobson_top Ideal.jacobson_top
@[simp]
theorem jacobson_eq_top_iff : jacobson I = ⊤ ↔ I = ⊤ :=
⟨fun H =>
by_contradiction fun hi => let ⟨M, hm, him⟩ := exists_le_maximal I hi
lt_top_iff_ne_top.1
(lt_of_le_of_lt (show jacobson I ≤ M from sInf_le ⟨him, hm⟩) <|
lt_top_iff_ne_top.2 hm.ne_top) H,
fun H => eq_top_iff.2 <| le_sInf fun _ ⟨hij, _⟩ => H ▸ hij⟩
#align ideal.jacobson_eq_top_iff Ideal.jacobson_eq_top_iff
theorem jacobson_eq_bot : jacobson I = ⊥ → I = ⊥ := fun h => eq_bot_iff.mpr (h ▸ le_jacobson)
#align ideal.jacobson_eq_bot Ideal.jacobson_eq_bot
theorem jacobson_eq_self_of_isMaximal [H : IsMaximal I] : I.jacobson = I :=
le_antisymm (sInf_le ⟨le_of_eq rfl, H⟩) le_jacobson
#align ideal.jacobson_eq_self_of_is_maximal Ideal.jacobson_eq_self_of_isMaximal
instance (priority := 100) jacobson.isMaximal [H : IsMaximal I] : IsMaximal (jacobson I) :=
⟨⟨fun htop => H.1.1 (jacobson_eq_top_iff.1 htop), fun _ hJ =>
H.1.2 _ (lt_of_le_of_lt le_jacobson hJ)⟩⟩
#align ideal.jacobson.is_maximal Ideal.jacobson.isMaximal
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I :=
⟨fun hx y =>
by_cases
(fun hxy : I ⊔ span {y * x + 1} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy)
let ⟨r, hr⟩ := mem_span_singleton'.1 hq
⟨r, by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one r (y * x), hr, ← hpq, ← neg_sub, add_sub_cancel]
exact I.neg_mem hpi⟩)
fun hxy : I ⊔ span {y * x + 1} ≠ ⊤ => let ⟨M, hm1, hm2⟩ := exists_le_maximal _ hxy
suffices x ∉ M from (this <| mem_sInf.1 hx ⟨le_trans le_sup_left hm2, hm1⟩).elim
fun hxm => hm1.1.1 <| (eq_top_iff_one _).2 <| add_sub_cancel' (y * x) 1 ▸
M.sub_mem (le_sup_right.trans hm2 <| subset_span rfl) (M.mul_mem_left _ hxm),
fun hx => mem_sInf.2 fun M ⟨him, hm⟩ => by_contradiction fun hxm =>
let ⟨y, i, hi, df⟩ := hm.exists_inv hxm
let ⟨z, hz⟩ := hx (-y)
hm.1.1 <| (eq_top_iff_one _).2 <| sub_sub_cancel (z * -y * x + z) 1 ▸
M.sub_mem (by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one z, neg_mul, ← sub_eq_iff_eq_add.mpr df.symm, neg_sub,
sub_add_cancel]
exact M.mul_mem_left _ hi) <| him hz⟩
#align ideal.mem_jacobson_iff Ideal.mem_jacobson_iff
theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
∃ s, s * r - 1 ∈ I := by
cases' mem_jacobson_iff.1 h 1 with s hs
use s
simpa [mul_sub] using hs
#align ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson Ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M := by
use fun hI => ⟨{ J : Ideal R | I ≤ J ∧ J.IsMaximal }, ⟨fun _ hJ => Or.inl hJ.right, hI.symm⟩⟩
rintro ⟨M, hM, hInf⟩
refine le_antisymm (fun x hx => ?_) le_jacobson
rw [hInf, mem_sInf]
intro I hI
cases' hM I hI with is_max is_top
· exact (mem_sInf.1 hx) ⟨le_sInf_iff.1 (le_of_eq hInf) I hI, is_max⟩
· exact is_top.symm ▸ Submodule.mem_top
#align ideal.eq_jacobson_iff_Inf_maximal Ideal.eq_jacobson_iff_sInf_maximal
theorem eq_jacobson_iff_sInf_maximal' :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, ∀ (K : Ideal R), J < K → K = ⊤) ∧ I = sInf M :=
eq_jacobson_iff_sInf_maximal.trans
⟨fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ K hK =>
Or.recOn (hM.1 J hJ) (fun h => h.1.2 K hK) fun h => eq_top_iff.2 (le_of_lt (h ▸ hK)),
hM.2⟩⟩,
fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ =>
Or.recOn (Classical.em (J = ⊤)) (fun h => Or.inr h) fun h => Or.inl ⟨⟨h, hM.1 J hJ⟩⟩,
hM.2⟩⟩⟩
#align ideal.eq_jacobson_iff_Inf_maximal' Ideal.eq_jacobson_iff_sInf_maximal'
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M := by
constructor
· intro h x hx
erw [← h, mem_sInf] at hx
|
push_neg at hx
|
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M := by
constructor
· intro h x hx
erw [← h, mem_sInf] at hx
|
Mathlib.RingTheory.JacobsonIdeal.162_0.Lz0MgLQMj1bGzuN
|
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M
|
Mathlib_RingTheory_JacobsonIdeal
|
case mp
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I : Ideal R
h : jacobson I = I
x : R
hx : ∃ I_1 ∈ {J | I ≤ J ∧ IsMaximal J}, x ∉ I_1
⊢ ∃ M, (I ≤ M ∧ IsMaximal M) ∧ x ∉ M
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
/-!
# Jacobson radical
The Jacobson radical of a ring `R` is defined to be the intersection of all maximal ideals of `R`.
This is similar to how the nilradical is equal to the intersection of all prime ideals of `R`.
We can extend the idea of the nilradical to ideals of `R`,
by letting the radical of an ideal `I` be the intersection of prime ideals containing `I`.
Under this extension, the original nilradical is the radical of the zero ideal `⊥`.
Here we define the Jacobson radical of an ideal `I` in a similar way,
as the intersection of maximal ideals containing `I`.
## Main definitions
Let `R` be a commutative ring, and `I` be an ideal of `R`
* `Ideal.jacobson I` is the jacobson radical, i.e. the infimum of all maximal ideals containing I.
* `Ideal.IsLocal I` is the proposition that the jacobson radical of `I` is itself a maximal ideal
## Main statements
* `mem_jacobson_iff` gives a characterization of members of the jacobson of I
* `Ideal.isLocal_of_isMaximal_radical`: if the radical of I is maximal then so is the jacobson
radical
## Tags
Jacobson, Jacobson radical, Local Ideal
-/
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}
open Polynomial
section Jacobson
section Ring
variable [Ring R] [Ring S] {I : Ideal R}
/-- The Jacobson radical of `I` is the infimum of all maximal (left) ideals containing `I`. -/
def jacobson (I : Ideal R) : Ideal R :=
sInf { J : Ideal R | I ≤ J ∧ IsMaximal J }
#align ideal.jacobson Ideal.jacobson
theorem le_jacobson : I ≤ jacobson I := fun _ hx => mem_sInf.mpr fun _ hJ => hJ.left hx
#align ideal.le_jacobson Ideal.le_jacobson
@[simp]
theorem jacobson_idem : jacobson (jacobson I) = jacobson I :=
le_antisymm (sInf_le_sInf fun _ hJ => ⟨sInf_le hJ, hJ.2⟩) le_jacobson
#align ideal.jacobson_idem Ideal.jacobson_idem
@[simp]
theorem jacobson_top : jacobson (⊤ : Ideal R) = ⊤ :=
eq_top_iff.2 le_jacobson
#align ideal.jacobson_top Ideal.jacobson_top
@[simp]
theorem jacobson_eq_top_iff : jacobson I = ⊤ ↔ I = ⊤ :=
⟨fun H =>
by_contradiction fun hi => let ⟨M, hm, him⟩ := exists_le_maximal I hi
lt_top_iff_ne_top.1
(lt_of_le_of_lt (show jacobson I ≤ M from sInf_le ⟨him, hm⟩) <|
lt_top_iff_ne_top.2 hm.ne_top) H,
fun H => eq_top_iff.2 <| le_sInf fun _ ⟨hij, _⟩ => H ▸ hij⟩
#align ideal.jacobson_eq_top_iff Ideal.jacobson_eq_top_iff
theorem jacobson_eq_bot : jacobson I = ⊥ → I = ⊥ := fun h => eq_bot_iff.mpr (h ▸ le_jacobson)
#align ideal.jacobson_eq_bot Ideal.jacobson_eq_bot
theorem jacobson_eq_self_of_isMaximal [H : IsMaximal I] : I.jacobson = I :=
le_antisymm (sInf_le ⟨le_of_eq rfl, H⟩) le_jacobson
#align ideal.jacobson_eq_self_of_is_maximal Ideal.jacobson_eq_self_of_isMaximal
instance (priority := 100) jacobson.isMaximal [H : IsMaximal I] : IsMaximal (jacobson I) :=
⟨⟨fun htop => H.1.1 (jacobson_eq_top_iff.1 htop), fun _ hJ =>
H.1.2 _ (lt_of_le_of_lt le_jacobson hJ)⟩⟩
#align ideal.jacobson.is_maximal Ideal.jacobson.isMaximal
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I :=
⟨fun hx y =>
by_cases
(fun hxy : I ⊔ span {y * x + 1} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy)
let ⟨r, hr⟩ := mem_span_singleton'.1 hq
⟨r, by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one r (y * x), hr, ← hpq, ← neg_sub, add_sub_cancel]
exact I.neg_mem hpi⟩)
fun hxy : I ⊔ span {y * x + 1} ≠ ⊤ => let ⟨M, hm1, hm2⟩ := exists_le_maximal _ hxy
suffices x ∉ M from (this <| mem_sInf.1 hx ⟨le_trans le_sup_left hm2, hm1⟩).elim
fun hxm => hm1.1.1 <| (eq_top_iff_one _).2 <| add_sub_cancel' (y * x) 1 ▸
M.sub_mem (le_sup_right.trans hm2 <| subset_span rfl) (M.mul_mem_left _ hxm),
fun hx => mem_sInf.2 fun M ⟨him, hm⟩ => by_contradiction fun hxm =>
let ⟨y, i, hi, df⟩ := hm.exists_inv hxm
let ⟨z, hz⟩ := hx (-y)
hm.1.1 <| (eq_top_iff_one _).2 <| sub_sub_cancel (z * -y * x + z) 1 ▸
M.sub_mem (by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one z, neg_mul, ← sub_eq_iff_eq_add.mpr df.symm, neg_sub,
sub_add_cancel]
exact M.mul_mem_left _ hi) <| him hz⟩
#align ideal.mem_jacobson_iff Ideal.mem_jacobson_iff
theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
∃ s, s * r - 1 ∈ I := by
cases' mem_jacobson_iff.1 h 1 with s hs
use s
simpa [mul_sub] using hs
#align ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson Ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M := by
use fun hI => ⟨{ J : Ideal R | I ≤ J ∧ J.IsMaximal }, ⟨fun _ hJ => Or.inl hJ.right, hI.symm⟩⟩
rintro ⟨M, hM, hInf⟩
refine le_antisymm (fun x hx => ?_) le_jacobson
rw [hInf, mem_sInf]
intro I hI
cases' hM I hI with is_max is_top
· exact (mem_sInf.1 hx) ⟨le_sInf_iff.1 (le_of_eq hInf) I hI, is_max⟩
· exact is_top.symm ▸ Submodule.mem_top
#align ideal.eq_jacobson_iff_Inf_maximal Ideal.eq_jacobson_iff_sInf_maximal
theorem eq_jacobson_iff_sInf_maximal' :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, ∀ (K : Ideal R), J < K → K = ⊤) ∧ I = sInf M :=
eq_jacobson_iff_sInf_maximal.trans
⟨fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ K hK =>
Or.recOn (hM.1 J hJ) (fun h => h.1.2 K hK) fun h => eq_top_iff.2 (le_of_lt (h ▸ hK)),
hM.2⟩⟩,
fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ =>
Or.recOn (Classical.em (J = ⊤)) (fun h => Or.inr h) fun h => Or.inl ⟨⟨h, hM.1 J hJ⟩⟩,
hM.2⟩⟩⟩
#align ideal.eq_jacobson_iff_Inf_maximal' Ideal.eq_jacobson_iff_sInf_maximal'
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M := by
constructor
· intro h x hx
erw [← h, mem_sInf] at hx
push_neg at hx
|
exact hx
|
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M := by
constructor
· intro h x hx
erw [← h, mem_sInf] at hx
push_neg at hx
|
Mathlib.RingTheory.JacobsonIdeal.162_0.Lz0MgLQMj1bGzuN
|
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M
|
Mathlib_RingTheory_JacobsonIdeal
|
case mpr
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I : Ideal R
⊢ (∀ x ∉ I, ∃ M, (I ≤ M ∧ IsMaximal M) ∧ x ∉ M) → jacobson I = I
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
/-!
# Jacobson radical
The Jacobson radical of a ring `R` is defined to be the intersection of all maximal ideals of `R`.
This is similar to how the nilradical is equal to the intersection of all prime ideals of `R`.
We can extend the idea of the nilradical to ideals of `R`,
by letting the radical of an ideal `I` be the intersection of prime ideals containing `I`.
Under this extension, the original nilradical is the radical of the zero ideal `⊥`.
Here we define the Jacobson radical of an ideal `I` in a similar way,
as the intersection of maximal ideals containing `I`.
## Main definitions
Let `R` be a commutative ring, and `I` be an ideal of `R`
* `Ideal.jacobson I` is the jacobson radical, i.e. the infimum of all maximal ideals containing I.
* `Ideal.IsLocal I` is the proposition that the jacobson radical of `I` is itself a maximal ideal
## Main statements
* `mem_jacobson_iff` gives a characterization of members of the jacobson of I
* `Ideal.isLocal_of_isMaximal_radical`: if the radical of I is maximal then so is the jacobson
radical
## Tags
Jacobson, Jacobson radical, Local Ideal
-/
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}
open Polynomial
section Jacobson
section Ring
variable [Ring R] [Ring S] {I : Ideal R}
/-- The Jacobson radical of `I` is the infimum of all maximal (left) ideals containing `I`. -/
def jacobson (I : Ideal R) : Ideal R :=
sInf { J : Ideal R | I ≤ J ∧ IsMaximal J }
#align ideal.jacobson Ideal.jacobson
theorem le_jacobson : I ≤ jacobson I := fun _ hx => mem_sInf.mpr fun _ hJ => hJ.left hx
#align ideal.le_jacobson Ideal.le_jacobson
@[simp]
theorem jacobson_idem : jacobson (jacobson I) = jacobson I :=
le_antisymm (sInf_le_sInf fun _ hJ => ⟨sInf_le hJ, hJ.2⟩) le_jacobson
#align ideal.jacobson_idem Ideal.jacobson_idem
@[simp]
theorem jacobson_top : jacobson (⊤ : Ideal R) = ⊤ :=
eq_top_iff.2 le_jacobson
#align ideal.jacobson_top Ideal.jacobson_top
@[simp]
theorem jacobson_eq_top_iff : jacobson I = ⊤ ↔ I = ⊤ :=
⟨fun H =>
by_contradiction fun hi => let ⟨M, hm, him⟩ := exists_le_maximal I hi
lt_top_iff_ne_top.1
(lt_of_le_of_lt (show jacobson I ≤ M from sInf_le ⟨him, hm⟩) <|
lt_top_iff_ne_top.2 hm.ne_top) H,
fun H => eq_top_iff.2 <| le_sInf fun _ ⟨hij, _⟩ => H ▸ hij⟩
#align ideal.jacobson_eq_top_iff Ideal.jacobson_eq_top_iff
theorem jacobson_eq_bot : jacobson I = ⊥ → I = ⊥ := fun h => eq_bot_iff.mpr (h ▸ le_jacobson)
#align ideal.jacobson_eq_bot Ideal.jacobson_eq_bot
theorem jacobson_eq_self_of_isMaximal [H : IsMaximal I] : I.jacobson = I :=
le_antisymm (sInf_le ⟨le_of_eq rfl, H⟩) le_jacobson
#align ideal.jacobson_eq_self_of_is_maximal Ideal.jacobson_eq_self_of_isMaximal
instance (priority := 100) jacobson.isMaximal [H : IsMaximal I] : IsMaximal (jacobson I) :=
⟨⟨fun htop => H.1.1 (jacobson_eq_top_iff.1 htop), fun _ hJ =>
H.1.2 _ (lt_of_le_of_lt le_jacobson hJ)⟩⟩
#align ideal.jacobson.is_maximal Ideal.jacobson.isMaximal
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I :=
⟨fun hx y =>
by_cases
(fun hxy : I ⊔ span {y * x + 1} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy)
let ⟨r, hr⟩ := mem_span_singleton'.1 hq
⟨r, by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one r (y * x), hr, ← hpq, ← neg_sub, add_sub_cancel]
exact I.neg_mem hpi⟩)
fun hxy : I ⊔ span {y * x + 1} ≠ ⊤ => let ⟨M, hm1, hm2⟩ := exists_le_maximal _ hxy
suffices x ∉ M from (this <| mem_sInf.1 hx ⟨le_trans le_sup_left hm2, hm1⟩).elim
fun hxm => hm1.1.1 <| (eq_top_iff_one _).2 <| add_sub_cancel' (y * x) 1 ▸
M.sub_mem (le_sup_right.trans hm2 <| subset_span rfl) (M.mul_mem_left _ hxm),
fun hx => mem_sInf.2 fun M ⟨him, hm⟩ => by_contradiction fun hxm =>
let ⟨y, i, hi, df⟩ := hm.exists_inv hxm
let ⟨z, hz⟩ := hx (-y)
hm.1.1 <| (eq_top_iff_one _).2 <| sub_sub_cancel (z * -y * x + z) 1 ▸
M.sub_mem (by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one z, neg_mul, ← sub_eq_iff_eq_add.mpr df.symm, neg_sub,
sub_add_cancel]
exact M.mul_mem_left _ hi) <| him hz⟩
#align ideal.mem_jacobson_iff Ideal.mem_jacobson_iff
theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
∃ s, s * r - 1 ∈ I := by
cases' mem_jacobson_iff.1 h 1 with s hs
use s
simpa [mul_sub] using hs
#align ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson Ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M := by
use fun hI => ⟨{ J : Ideal R | I ≤ J ∧ J.IsMaximal }, ⟨fun _ hJ => Or.inl hJ.right, hI.symm⟩⟩
rintro ⟨M, hM, hInf⟩
refine le_antisymm (fun x hx => ?_) le_jacobson
rw [hInf, mem_sInf]
intro I hI
cases' hM I hI with is_max is_top
· exact (mem_sInf.1 hx) ⟨le_sInf_iff.1 (le_of_eq hInf) I hI, is_max⟩
· exact is_top.symm ▸ Submodule.mem_top
#align ideal.eq_jacobson_iff_Inf_maximal Ideal.eq_jacobson_iff_sInf_maximal
theorem eq_jacobson_iff_sInf_maximal' :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, ∀ (K : Ideal R), J < K → K = ⊤) ∧ I = sInf M :=
eq_jacobson_iff_sInf_maximal.trans
⟨fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ K hK =>
Or.recOn (hM.1 J hJ) (fun h => h.1.2 K hK) fun h => eq_top_iff.2 (le_of_lt (h ▸ hK)),
hM.2⟩⟩,
fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ =>
Or.recOn (Classical.em (J = ⊤)) (fun h => Or.inr h) fun h => Or.inl ⟨⟨h, hM.1 J hJ⟩⟩,
hM.2⟩⟩⟩
#align ideal.eq_jacobson_iff_Inf_maximal' Ideal.eq_jacobson_iff_sInf_maximal'
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M := by
constructor
· intro h x hx
erw [← h, mem_sInf] at hx
push_neg at hx
exact hx
·
|
refine fun h => le_antisymm (fun x hx => ?_) le_jacobson
|
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M := by
constructor
· intro h x hx
erw [← h, mem_sInf] at hx
push_neg at hx
exact hx
·
|
Mathlib.RingTheory.JacobsonIdeal.162_0.Lz0MgLQMj1bGzuN
|
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M
|
Mathlib_RingTheory_JacobsonIdeal
|
case mpr
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I : Ideal R
h : ∀ x ∉ I, ∃ M, (I ≤ M ∧ IsMaximal M) ∧ x ∉ M
x : R
hx : x ∈ jacobson I
⊢ x ∈ I
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
/-!
# Jacobson radical
The Jacobson radical of a ring `R` is defined to be the intersection of all maximal ideals of `R`.
This is similar to how the nilradical is equal to the intersection of all prime ideals of `R`.
We can extend the idea of the nilradical to ideals of `R`,
by letting the radical of an ideal `I` be the intersection of prime ideals containing `I`.
Under this extension, the original nilradical is the radical of the zero ideal `⊥`.
Here we define the Jacobson radical of an ideal `I` in a similar way,
as the intersection of maximal ideals containing `I`.
## Main definitions
Let `R` be a commutative ring, and `I` be an ideal of `R`
* `Ideal.jacobson I` is the jacobson radical, i.e. the infimum of all maximal ideals containing I.
* `Ideal.IsLocal I` is the proposition that the jacobson radical of `I` is itself a maximal ideal
## Main statements
* `mem_jacobson_iff` gives a characterization of members of the jacobson of I
* `Ideal.isLocal_of_isMaximal_radical`: if the radical of I is maximal then so is the jacobson
radical
## Tags
Jacobson, Jacobson radical, Local Ideal
-/
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}
open Polynomial
section Jacobson
section Ring
variable [Ring R] [Ring S] {I : Ideal R}
/-- The Jacobson radical of `I` is the infimum of all maximal (left) ideals containing `I`. -/
def jacobson (I : Ideal R) : Ideal R :=
sInf { J : Ideal R | I ≤ J ∧ IsMaximal J }
#align ideal.jacobson Ideal.jacobson
theorem le_jacobson : I ≤ jacobson I := fun _ hx => mem_sInf.mpr fun _ hJ => hJ.left hx
#align ideal.le_jacobson Ideal.le_jacobson
@[simp]
theorem jacobson_idem : jacobson (jacobson I) = jacobson I :=
le_antisymm (sInf_le_sInf fun _ hJ => ⟨sInf_le hJ, hJ.2⟩) le_jacobson
#align ideal.jacobson_idem Ideal.jacobson_idem
@[simp]
theorem jacobson_top : jacobson (⊤ : Ideal R) = ⊤ :=
eq_top_iff.2 le_jacobson
#align ideal.jacobson_top Ideal.jacobson_top
@[simp]
theorem jacobson_eq_top_iff : jacobson I = ⊤ ↔ I = ⊤ :=
⟨fun H =>
by_contradiction fun hi => let ⟨M, hm, him⟩ := exists_le_maximal I hi
lt_top_iff_ne_top.1
(lt_of_le_of_lt (show jacobson I ≤ M from sInf_le ⟨him, hm⟩) <|
lt_top_iff_ne_top.2 hm.ne_top) H,
fun H => eq_top_iff.2 <| le_sInf fun _ ⟨hij, _⟩ => H ▸ hij⟩
#align ideal.jacobson_eq_top_iff Ideal.jacobson_eq_top_iff
theorem jacobson_eq_bot : jacobson I = ⊥ → I = ⊥ := fun h => eq_bot_iff.mpr (h ▸ le_jacobson)
#align ideal.jacobson_eq_bot Ideal.jacobson_eq_bot
theorem jacobson_eq_self_of_isMaximal [H : IsMaximal I] : I.jacobson = I :=
le_antisymm (sInf_le ⟨le_of_eq rfl, H⟩) le_jacobson
#align ideal.jacobson_eq_self_of_is_maximal Ideal.jacobson_eq_self_of_isMaximal
instance (priority := 100) jacobson.isMaximal [H : IsMaximal I] : IsMaximal (jacobson I) :=
⟨⟨fun htop => H.1.1 (jacobson_eq_top_iff.1 htop), fun _ hJ =>
H.1.2 _ (lt_of_le_of_lt le_jacobson hJ)⟩⟩
#align ideal.jacobson.is_maximal Ideal.jacobson.isMaximal
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I :=
⟨fun hx y =>
by_cases
(fun hxy : I ⊔ span {y * x + 1} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy)
let ⟨r, hr⟩ := mem_span_singleton'.1 hq
⟨r, by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one r (y * x), hr, ← hpq, ← neg_sub, add_sub_cancel]
exact I.neg_mem hpi⟩)
fun hxy : I ⊔ span {y * x + 1} ≠ ⊤ => let ⟨M, hm1, hm2⟩ := exists_le_maximal _ hxy
suffices x ∉ M from (this <| mem_sInf.1 hx ⟨le_trans le_sup_left hm2, hm1⟩).elim
fun hxm => hm1.1.1 <| (eq_top_iff_one _).2 <| add_sub_cancel' (y * x) 1 ▸
M.sub_mem (le_sup_right.trans hm2 <| subset_span rfl) (M.mul_mem_left _ hxm),
fun hx => mem_sInf.2 fun M ⟨him, hm⟩ => by_contradiction fun hxm =>
let ⟨y, i, hi, df⟩ := hm.exists_inv hxm
let ⟨z, hz⟩ := hx (-y)
hm.1.1 <| (eq_top_iff_one _).2 <| sub_sub_cancel (z * -y * x + z) 1 ▸
M.sub_mem (by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one z, neg_mul, ← sub_eq_iff_eq_add.mpr df.symm, neg_sub,
sub_add_cancel]
exact M.mul_mem_left _ hi) <| him hz⟩
#align ideal.mem_jacobson_iff Ideal.mem_jacobson_iff
theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
∃ s, s * r - 1 ∈ I := by
cases' mem_jacobson_iff.1 h 1 with s hs
use s
simpa [mul_sub] using hs
#align ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson Ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M := by
use fun hI => ⟨{ J : Ideal R | I ≤ J ∧ J.IsMaximal }, ⟨fun _ hJ => Or.inl hJ.right, hI.symm⟩⟩
rintro ⟨M, hM, hInf⟩
refine le_antisymm (fun x hx => ?_) le_jacobson
rw [hInf, mem_sInf]
intro I hI
cases' hM I hI with is_max is_top
· exact (mem_sInf.1 hx) ⟨le_sInf_iff.1 (le_of_eq hInf) I hI, is_max⟩
· exact is_top.symm ▸ Submodule.mem_top
#align ideal.eq_jacobson_iff_Inf_maximal Ideal.eq_jacobson_iff_sInf_maximal
theorem eq_jacobson_iff_sInf_maximal' :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, ∀ (K : Ideal R), J < K → K = ⊤) ∧ I = sInf M :=
eq_jacobson_iff_sInf_maximal.trans
⟨fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ K hK =>
Or.recOn (hM.1 J hJ) (fun h => h.1.2 K hK) fun h => eq_top_iff.2 (le_of_lt (h ▸ hK)),
hM.2⟩⟩,
fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ =>
Or.recOn (Classical.em (J = ⊤)) (fun h => Or.inr h) fun h => Or.inl ⟨⟨h, hM.1 J hJ⟩⟩,
hM.2⟩⟩⟩
#align ideal.eq_jacobson_iff_Inf_maximal' Ideal.eq_jacobson_iff_sInf_maximal'
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M := by
constructor
· intro h x hx
erw [← h, mem_sInf] at hx
push_neg at hx
exact hx
· refine fun h => le_antisymm (fun x hx => ?_) le_jacobson
|
contrapose hx
|
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M := by
constructor
· intro h x hx
erw [← h, mem_sInf] at hx
push_neg at hx
exact hx
· refine fun h => le_antisymm (fun x hx => ?_) le_jacobson
|
Mathlib.RingTheory.JacobsonIdeal.162_0.Lz0MgLQMj1bGzuN
|
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M
|
Mathlib_RingTheory_JacobsonIdeal
|
case mpr
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I : Ideal R
h : ∀ x ∉ I, ∃ M, (I ≤ M ∧ IsMaximal M) ∧ x ∉ M
x : R
hx : x ∉ I
⊢ x ∉ jacobson I
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
/-!
# Jacobson radical
The Jacobson radical of a ring `R` is defined to be the intersection of all maximal ideals of `R`.
This is similar to how the nilradical is equal to the intersection of all prime ideals of `R`.
We can extend the idea of the nilradical to ideals of `R`,
by letting the radical of an ideal `I` be the intersection of prime ideals containing `I`.
Under this extension, the original nilradical is the radical of the zero ideal `⊥`.
Here we define the Jacobson radical of an ideal `I` in a similar way,
as the intersection of maximal ideals containing `I`.
## Main definitions
Let `R` be a commutative ring, and `I` be an ideal of `R`
* `Ideal.jacobson I` is the jacobson radical, i.e. the infimum of all maximal ideals containing I.
* `Ideal.IsLocal I` is the proposition that the jacobson radical of `I` is itself a maximal ideal
## Main statements
* `mem_jacobson_iff` gives a characterization of members of the jacobson of I
* `Ideal.isLocal_of_isMaximal_radical`: if the radical of I is maximal then so is the jacobson
radical
## Tags
Jacobson, Jacobson radical, Local Ideal
-/
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}
open Polynomial
section Jacobson
section Ring
variable [Ring R] [Ring S] {I : Ideal R}
/-- The Jacobson radical of `I` is the infimum of all maximal (left) ideals containing `I`. -/
def jacobson (I : Ideal R) : Ideal R :=
sInf { J : Ideal R | I ≤ J ∧ IsMaximal J }
#align ideal.jacobson Ideal.jacobson
theorem le_jacobson : I ≤ jacobson I := fun _ hx => mem_sInf.mpr fun _ hJ => hJ.left hx
#align ideal.le_jacobson Ideal.le_jacobson
@[simp]
theorem jacobson_idem : jacobson (jacobson I) = jacobson I :=
le_antisymm (sInf_le_sInf fun _ hJ => ⟨sInf_le hJ, hJ.2⟩) le_jacobson
#align ideal.jacobson_idem Ideal.jacobson_idem
@[simp]
theorem jacobson_top : jacobson (⊤ : Ideal R) = ⊤ :=
eq_top_iff.2 le_jacobson
#align ideal.jacobson_top Ideal.jacobson_top
@[simp]
theorem jacobson_eq_top_iff : jacobson I = ⊤ ↔ I = ⊤ :=
⟨fun H =>
by_contradiction fun hi => let ⟨M, hm, him⟩ := exists_le_maximal I hi
lt_top_iff_ne_top.1
(lt_of_le_of_lt (show jacobson I ≤ M from sInf_le ⟨him, hm⟩) <|
lt_top_iff_ne_top.2 hm.ne_top) H,
fun H => eq_top_iff.2 <| le_sInf fun _ ⟨hij, _⟩ => H ▸ hij⟩
#align ideal.jacobson_eq_top_iff Ideal.jacobson_eq_top_iff
theorem jacobson_eq_bot : jacobson I = ⊥ → I = ⊥ := fun h => eq_bot_iff.mpr (h ▸ le_jacobson)
#align ideal.jacobson_eq_bot Ideal.jacobson_eq_bot
theorem jacobson_eq_self_of_isMaximal [H : IsMaximal I] : I.jacobson = I :=
le_antisymm (sInf_le ⟨le_of_eq rfl, H⟩) le_jacobson
#align ideal.jacobson_eq_self_of_is_maximal Ideal.jacobson_eq_self_of_isMaximal
instance (priority := 100) jacobson.isMaximal [H : IsMaximal I] : IsMaximal (jacobson I) :=
⟨⟨fun htop => H.1.1 (jacobson_eq_top_iff.1 htop), fun _ hJ =>
H.1.2 _ (lt_of_le_of_lt le_jacobson hJ)⟩⟩
#align ideal.jacobson.is_maximal Ideal.jacobson.isMaximal
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I :=
⟨fun hx y =>
by_cases
(fun hxy : I ⊔ span {y * x + 1} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy)
let ⟨r, hr⟩ := mem_span_singleton'.1 hq
⟨r, by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one r (y * x), hr, ← hpq, ← neg_sub, add_sub_cancel]
exact I.neg_mem hpi⟩)
fun hxy : I ⊔ span {y * x + 1} ≠ ⊤ => let ⟨M, hm1, hm2⟩ := exists_le_maximal _ hxy
suffices x ∉ M from (this <| mem_sInf.1 hx ⟨le_trans le_sup_left hm2, hm1⟩).elim
fun hxm => hm1.1.1 <| (eq_top_iff_one _).2 <| add_sub_cancel' (y * x) 1 ▸
M.sub_mem (le_sup_right.trans hm2 <| subset_span rfl) (M.mul_mem_left _ hxm),
fun hx => mem_sInf.2 fun M ⟨him, hm⟩ => by_contradiction fun hxm =>
let ⟨y, i, hi, df⟩ := hm.exists_inv hxm
let ⟨z, hz⟩ := hx (-y)
hm.1.1 <| (eq_top_iff_one _).2 <| sub_sub_cancel (z * -y * x + z) 1 ▸
M.sub_mem (by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one z, neg_mul, ← sub_eq_iff_eq_add.mpr df.symm, neg_sub,
sub_add_cancel]
exact M.mul_mem_left _ hi) <| him hz⟩
#align ideal.mem_jacobson_iff Ideal.mem_jacobson_iff
theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
∃ s, s * r - 1 ∈ I := by
cases' mem_jacobson_iff.1 h 1 with s hs
use s
simpa [mul_sub] using hs
#align ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson Ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M := by
use fun hI => ⟨{ J : Ideal R | I ≤ J ∧ J.IsMaximal }, ⟨fun _ hJ => Or.inl hJ.right, hI.symm⟩⟩
rintro ⟨M, hM, hInf⟩
refine le_antisymm (fun x hx => ?_) le_jacobson
rw [hInf, mem_sInf]
intro I hI
cases' hM I hI with is_max is_top
· exact (mem_sInf.1 hx) ⟨le_sInf_iff.1 (le_of_eq hInf) I hI, is_max⟩
· exact is_top.symm ▸ Submodule.mem_top
#align ideal.eq_jacobson_iff_Inf_maximal Ideal.eq_jacobson_iff_sInf_maximal
theorem eq_jacobson_iff_sInf_maximal' :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, ∀ (K : Ideal R), J < K → K = ⊤) ∧ I = sInf M :=
eq_jacobson_iff_sInf_maximal.trans
⟨fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ K hK =>
Or.recOn (hM.1 J hJ) (fun h => h.1.2 K hK) fun h => eq_top_iff.2 (le_of_lt (h ▸ hK)),
hM.2⟩⟩,
fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ =>
Or.recOn (Classical.em (J = ⊤)) (fun h => Or.inr h) fun h => Or.inl ⟨⟨h, hM.1 J hJ⟩⟩,
hM.2⟩⟩⟩
#align ideal.eq_jacobson_iff_Inf_maximal' Ideal.eq_jacobson_iff_sInf_maximal'
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M := by
constructor
· intro h x hx
erw [← h, mem_sInf] at hx
push_neg at hx
exact hx
· refine fun h => le_antisymm (fun x hx => ?_) le_jacobson
contrapose hx
|
erw [mem_sInf]
|
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M := by
constructor
· intro h x hx
erw [← h, mem_sInf] at hx
push_neg at hx
exact hx
· refine fun h => le_antisymm (fun x hx => ?_) le_jacobson
contrapose hx
|
Mathlib.RingTheory.JacobsonIdeal.162_0.Lz0MgLQMj1bGzuN
|
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M
|
Mathlib_RingTheory_JacobsonIdeal
|
case mpr
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I : Ideal R
h : ∀ x ∉ I, ∃ M, (I ≤ M ∧ IsMaximal M) ∧ x ∉ M
x : R
hx : x ∉ I
⊢ ¬∀ ⦃I_1 : Ideal R⦄, I_1 ∈ {J | I ≤ J ∧ IsMaximal J} → x ∈ I_1
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
/-!
# Jacobson radical
The Jacobson radical of a ring `R` is defined to be the intersection of all maximal ideals of `R`.
This is similar to how the nilradical is equal to the intersection of all prime ideals of `R`.
We can extend the idea of the nilradical to ideals of `R`,
by letting the radical of an ideal `I` be the intersection of prime ideals containing `I`.
Under this extension, the original nilradical is the radical of the zero ideal `⊥`.
Here we define the Jacobson radical of an ideal `I` in a similar way,
as the intersection of maximal ideals containing `I`.
## Main definitions
Let `R` be a commutative ring, and `I` be an ideal of `R`
* `Ideal.jacobson I` is the jacobson radical, i.e. the infimum of all maximal ideals containing I.
* `Ideal.IsLocal I` is the proposition that the jacobson radical of `I` is itself a maximal ideal
## Main statements
* `mem_jacobson_iff` gives a characterization of members of the jacobson of I
* `Ideal.isLocal_of_isMaximal_radical`: if the radical of I is maximal then so is the jacobson
radical
## Tags
Jacobson, Jacobson radical, Local Ideal
-/
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}
open Polynomial
section Jacobson
section Ring
variable [Ring R] [Ring S] {I : Ideal R}
/-- The Jacobson radical of `I` is the infimum of all maximal (left) ideals containing `I`. -/
def jacobson (I : Ideal R) : Ideal R :=
sInf { J : Ideal R | I ≤ J ∧ IsMaximal J }
#align ideal.jacobson Ideal.jacobson
theorem le_jacobson : I ≤ jacobson I := fun _ hx => mem_sInf.mpr fun _ hJ => hJ.left hx
#align ideal.le_jacobson Ideal.le_jacobson
@[simp]
theorem jacobson_idem : jacobson (jacobson I) = jacobson I :=
le_antisymm (sInf_le_sInf fun _ hJ => ⟨sInf_le hJ, hJ.2⟩) le_jacobson
#align ideal.jacobson_idem Ideal.jacobson_idem
@[simp]
theorem jacobson_top : jacobson (⊤ : Ideal R) = ⊤ :=
eq_top_iff.2 le_jacobson
#align ideal.jacobson_top Ideal.jacobson_top
@[simp]
theorem jacobson_eq_top_iff : jacobson I = ⊤ ↔ I = ⊤ :=
⟨fun H =>
by_contradiction fun hi => let ⟨M, hm, him⟩ := exists_le_maximal I hi
lt_top_iff_ne_top.1
(lt_of_le_of_lt (show jacobson I ≤ M from sInf_le ⟨him, hm⟩) <|
lt_top_iff_ne_top.2 hm.ne_top) H,
fun H => eq_top_iff.2 <| le_sInf fun _ ⟨hij, _⟩ => H ▸ hij⟩
#align ideal.jacobson_eq_top_iff Ideal.jacobson_eq_top_iff
theorem jacobson_eq_bot : jacobson I = ⊥ → I = ⊥ := fun h => eq_bot_iff.mpr (h ▸ le_jacobson)
#align ideal.jacobson_eq_bot Ideal.jacobson_eq_bot
theorem jacobson_eq_self_of_isMaximal [H : IsMaximal I] : I.jacobson = I :=
le_antisymm (sInf_le ⟨le_of_eq rfl, H⟩) le_jacobson
#align ideal.jacobson_eq_self_of_is_maximal Ideal.jacobson_eq_self_of_isMaximal
instance (priority := 100) jacobson.isMaximal [H : IsMaximal I] : IsMaximal (jacobson I) :=
⟨⟨fun htop => H.1.1 (jacobson_eq_top_iff.1 htop), fun _ hJ =>
H.1.2 _ (lt_of_le_of_lt le_jacobson hJ)⟩⟩
#align ideal.jacobson.is_maximal Ideal.jacobson.isMaximal
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I :=
⟨fun hx y =>
by_cases
(fun hxy : I ⊔ span {y * x + 1} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy)
let ⟨r, hr⟩ := mem_span_singleton'.1 hq
⟨r, by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one r (y * x), hr, ← hpq, ← neg_sub, add_sub_cancel]
exact I.neg_mem hpi⟩)
fun hxy : I ⊔ span {y * x + 1} ≠ ⊤ => let ⟨M, hm1, hm2⟩ := exists_le_maximal _ hxy
suffices x ∉ M from (this <| mem_sInf.1 hx ⟨le_trans le_sup_left hm2, hm1⟩).elim
fun hxm => hm1.1.1 <| (eq_top_iff_one _).2 <| add_sub_cancel' (y * x) 1 ▸
M.sub_mem (le_sup_right.trans hm2 <| subset_span rfl) (M.mul_mem_left _ hxm),
fun hx => mem_sInf.2 fun M ⟨him, hm⟩ => by_contradiction fun hxm =>
let ⟨y, i, hi, df⟩ := hm.exists_inv hxm
let ⟨z, hz⟩ := hx (-y)
hm.1.1 <| (eq_top_iff_one _).2 <| sub_sub_cancel (z * -y * x + z) 1 ▸
M.sub_mem (by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one z, neg_mul, ← sub_eq_iff_eq_add.mpr df.symm, neg_sub,
sub_add_cancel]
exact M.mul_mem_left _ hi) <| him hz⟩
#align ideal.mem_jacobson_iff Ideal.mem_jacobson_iff
theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
∃ s, s * r - 1 ∈ I := by
cases' mem_jacobson_iff.1 h 1 with s hs
use s
simpa [mul_sub] using hs
#align ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson Ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M := by
use fun hI => ⟨{ J : Ideal R | I ≤ J ∧ J.IsMaximal }, ⟨fun _ hJ => Or.inl hJ.right, hI.symm⟩⟩
rintro ⟨M, hM, hInf⟩
refine le_antisymm (fun x hx => ?_) le_jacobson
rw [hInf, mem_sInf]
intro I hI
cases' hM I hI with is_max is_top
· exact (mem_sInf.1 hx) ⟨le_sInf_iff.1 (le_of_eq hInf) I hI, is_max⟩
· exact is_top.symm ▸ Submodule.mem_top
#align ideal.eq_jacobson_iff_Inf_maximal Ideal.eq_jacobson_iff_sInf_maximal
theorem eq_jacobson_iff_sInf_maximal' :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, ∀ (K : Ideal R), J < K → K = ⊤) ∧ I = sInf M :=
eq_jacobson_iff_sInf_maximal.trans
⟨fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ K hK =>
Or.recOn (hM.1 J hJ) (fun h => h.1.2 K hK) fun h => eq_top_iff.2 (le_of_lt (h ▸ hK)),
hM.2⟩⟩,
fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ =>
Or.recOn (Classical.em (J = ⊤)) (fun h => Or.inr h) fun h => Or.inl ⟨⟨h, hM.1 J hJ⟩⟩,
hM.2⟩⟩⟩
#align ideal.eq_jacobson_iff_Inf_maximal' Ideal.eq_jacobson_iff_sInf_maximal'
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M := by
constructor
· intro h x hx
erw [← h, mem_sInf] at hx
push_neg at hx
exact hx
· refine fun h => le_antisymm (fun x hx => ?_) le_jacobson
contrapose hx
erw [mem_sInf]
|
push_neg
|
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M := by
constructor
· intro h x hx
erw [← h, mem_sInf] at hx
push_neg at hx
exact hx
· refine fun h => le_antisymm (fun x hx => ?_) le_jacobson
contrapose hx
erw [mem_sInf]
|
Mathlib.RingTheory.JacobsonIdeal.162_0.Lz0MgLQMj1bGzuN
|
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M
|
Mathlib_RingTheory_JacobsonIdeal
|
case mpr
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I : Ideal R
h : ∀ x ∉ I, ∃ M, (I ≤ M ∧ IsMaximal M) ∧ x ∉ M
x : R
hx : x ∉ I
⊢ ∃ I_1 ∈ {J | I ≤ J ∧ IsMaximal J}, x ∉ I_1
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
/-!
# Jacobson radical
The Jacobson radical of a ring `R` is defined to be the intersection of all maximal ideals of `R`.
This is similar to how the nilradical is equal to the intersection of all prime ideals of `R`.
We can extend the idea of the nilradical to ideals of `R`,
by letting the radical of an ideal `I` be the intersection of prime ideals containing `I`.
Under this extension, the original nilradical is the radical of the zero ideal `⊥`.
Here we define the Jacobson radical of an ideal `I` in a similar way,
as the intersection of maximal ideals containing `I`.
## Main definitions
Let `R` be a commutative ring, and `I` be an ideal of `R`
* `Ideal.jacobson I` is the jacobson radical, i.e. the infimum of all maximal ideals containing I.
* `Ideal.IsLocal I` is the proposition that the jacobson radical of `I` is itself a maximal ideal
## Main statements
* `mem_jacobson_iff` gives a characterization of members of the jacobson of I
* `Ideal.isLocal_of_isMaximal_radical`: if the radical of I is maximal then so is the jacobson
radical
## Tags
Jacobson, Jacobson radical, Local Ideal
-/
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}
open Polynomial
section Jacobson
section Ring
variable [Ring R] [Ring S] {I : Ideal R}
/-- The Jacobson radical of `I` is the infimum of all maximal (left) ideals containing `I`. -/
def jacobson (I : Ideal R) : Ideal R :=
sInf { J : Ideal R | I ≤ J ∧ IsMaximal J }
#align ideal.jacobson Ideal.jacobson
theorem le_jacobson : I ≤ jacobson I := fun _ hx => mem_sInf.mpr fun _ hJ => hJ.left hx
#align ideal.le_jacobson Ideal.le_jacobson
@[simp]
theorem jacobson_idem : jacobson (jacobson I) = jacobson I :=
le_antisymm (sInf_le_sInf fun _ hJ => ⟨sInf_le hJ, hJ.2⟩) le_jacobson
#align ideal.jacobson_idem Ideal.jacobson_idem
@[simp]
theorem jacobson_top : jacobson (⊤ : Ideal R) = ⊤ :=
eq_top_iff.2 le_jacobson
#align ideal.jacobson_top Ideal.jacobson_top
@[simp]
theorem jacobson_eq_top_iff : jacobson I = ⊤ ↔ I = ⊤ :=
⟨fun H =>
by_contradiction fun hi => let ⟨M, hm, him⟩ := exists_le_maximal I hi
lt_top_iff_ne_top.1
(lt_of_le_of_lt (show jacobson I ≤ M from sInf_le ⟨him, hm⟩) <|
lt_top_iff_ne_top.2 hm.ne_top) H,
fun H => eq_top_iff.2 <| le_sInf fun _ ⟨hij, _⟩ => H ▸ hij⟩
#align ideal.jacobson_eq_top_iff Ideal.jacobson_eq_top_iff
theorem jacobson_eq_bot : jacobson I = ⊥ → I = ⊥ := fun h => eq_bot_iff.mpr (h ▸ le_jacobson)
#align ideal.jacobson_eq_bot Ideal.jacobson_eq_bot
theorem jacobson_eq_self_of_isMaximal [H : IsMaximal I] : I.jacobson = I :=
le_antisymm (sInf_le ⟨le_of_eq rfl, H⟩) le_jacobson
#align ideal.jacobson_eq_self_of_is_maximal Ideal.jacobson_eq_self_of_isMaximal
instance (priority := 100) jacobson.isMaximal [H : IsMaximal I] : IsMaximal (jacobson I) :=
⟨⟨fun htop => H.1.1 (jacobson_eq_top_iff.1 htop), fun _ hJ =>
H.1.2 _ (lt_of_le_of_lt le_jacobson hJ)⟩⟩
#align ideal.jacobson.is_maximal Ideal.jacobson.isMaximal
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I :=
⟨fun hx y =>
by_cases
(fun hxy : I ⊔ span {y * x + 1} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy)
let ⟨r, hr⟩ := mem_span_singleton'.1 hq
⟨r, by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one r (y * x), hr, ← hpq, ← neg_sub, add_sub_cancel]
exact I.neg_mem hpi⟩)
fun hxy : I ⊔ span {y * x + 1} ≠ ⊤ => let ⟨M, hm1, hm2⟩ := exists_le_maximal _ hxy
suffices x ∉ M from (this <| mem_sInf.1 hx ⟨le_trans le_sup_left hm2, hm1⟩).elim
fun hxm => hm1.1.1 <| (eq_top_iff_one _).2 <| add_sub_cancel' (y * x) 1 ▸
M.sub_mem (le_sup_right.trans hm2 <| subset_span rfl) (M.mul_mem_left _ hxm),
fun hx => mem_sInf.2 fun M ⟨him, hm⟩ => by_contradiction fun hxm =>
let ⟨y, i, hi, df⟩ := hm.exists_inv hxm
let ⟨z, hz⟩ := hx (-y)
hm.1.1 <| (eq_top_iff_one _).2 <| sub_sub_cancel (z * -y * x + z) 1 ▸
M.sub_mem (by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one z, neg_mul, ← sub_eq_iff_eq_add.mpr df.symm, neg_sub,
sub_add_cancel]
exact M.mul_mem_left _ hi) <| him hz⟩
#align ideal.mem_jacobson_iff Ideal.mem_jacobson_iff
theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
∃ s, s * r - 1 ∈ I := by
cases' mem_jacobson_iff.1 h 1 with s hs
use s
simpa [mul_sub] using hs
#align ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson Ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M := by
use fun hI => ⟨{ J : Ideal R | I ≤ J ∧ J.IsMaximal }, ⟨fun _ hJ => Or.inl hJ.right, hI.symm⟩⟩
rintro ⟨M, hM, hInf⟩
refine le_antisymm (fun x hx => ?_) le_jacobson
rw [hInf, mem_sInf]
intro I hI
cases' hM I hI with is_max is_top
· exact (mem_sInf.1 hx) ⟨le_sInf_iff.1 (le_of_eq hInf) I hI, is_max⟩
· exact is_top.symm ▸ Submodule.mem_top
#align ideal.eq_jacobson_iff_Inf_maximal Ideal.eq_jacobson_iff_sInf_maximal
theorem eq_jacobson_iff_sInf_maximal' :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, ∀ (K : Ideal R), J < K → K = ⊤) ∧ I = sInf M :=
eq_jacobson_iff_sInf_maximal.trans
⟨fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ K hK =>
Or.recOn (hM.1 J hJ) (fun h => h.1.2 K hK) fun h => eq_top_iff.2 (le_of_lt (h ▸ hK)),
hM.2⟩⟩,
fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ =>
Or.recOn (Classical.em (J = ⊤)) (fun h => Or.inr h) fun h => Or.inl ⟨⟨h, hM.1 J hJ⟩⟩,
hM.2⟩⟩⟩
#align ideal.eq_jacobson_iff_Inf_maximal' Ideal.eq_jacobson_iff_sInf_maximal'
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M := by
constructor
· intro h x hx
erw [← h, mem_sInf] at hx
push_neg at hx
exact hx
· refine fun h => le_antisymm (fun x hx => ?_) le_jacobson
contrapose hx
erw [mem_sInf]
push_neg
|
exact h x hx
|
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M := by
constructor
· intro h x hx
erw [← h, mem_sInf] at hx
push_neg at hx
exact hx
· refine fun h => le_antisymm (fun x hx => ?_) le_jacobson
contrapose hx
erw [mem_sInf]
push_neg
|
Mathlib.RingTheory.JacobsonIdeal.162_0.Lz0MgLQMj1bGzuN
|
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M
|
Mathlib_RingTheory_JacobsonIdeal
|
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I : Ideal R
f : R →+* S
hf : Function.Surjective ⇑f
⊢ RingHom.ker f ≤ I → map f (jacobson I) = jacobson (map f I)
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
/-!
# Jacobson radical
The Jacobson radical of a ring `R` is defined to be the intersection of all maximal ideals of `R`.
This is similar to how the nilradical is equal to the intersection of all prime ideals of `R`.
We can extend the idea of the nilradical to ideals of `R`,
by letting the radical of an ideal `I` be the intersection of prime ideals containing `I`.
Under this extension, the original nilradical is the radical of the zero ideal `⊥`.
Here we define the Jacobson radical of an ideal `I` in a similar way,
as the intersection of maximal ideals containing `I`.
## Main definitions
Let `R` be a commutative ring, and `I` be an ideal of `R`
* `Ideal.jacobson I` is the jacobson radical, i.e. the infimum of all maximal ideals containing I.
* `Ideal.IsLocal I` is the proposition that the jacobson radical of `I` is itself a maximal ideal
## Main statements
* `mem_jacobson_iff` gives a characterization of members of the jacobson of I
* `Ideal.isLocal_of_isMaximal_radical`: if the radical of I is maximal then so is the jacobson
radical
## Tags
Jacobson, Jacobson radical, Local Ideal
-/
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}
open Polynomial
section Jacobson
section Ring
variable [Ring R] [Ring S] {I : Ideal R}
/-- The Jacobson radical of `I` is the infimum of all maximal (left) ideals containing `I`. -/
def jacobson (I : Ideal R) : Ideal R :=
sInf { J : Ideal R | I ≤ J ∧ IsMaximal J }
#align ideal.jacobson Ideal.jacobson
theorem le_jacobson : I ≤ jacobson I := fun _ hx => mem_sInf.mpr fun _ hJ => hJ.left hx
#align ideal.le_jacobson Ideal.le_jacobson
@[simp]
theorem jacobson_idem : jacobson (jacobson I) = jacobson I :=
le_antisymm (sInf_le_sInf fun _ hJ => ⟨sInf_le hJ, hJ.2⟩) le_jacobson
#align ideal.jacobson_idem Ideal.jacobson_idem
@[simp]
theorem jacobson_top : jacobson (⊤ : Ideal R) = ⊤ :=
eq_top_iff.2 le_jacobson
#align ideal.jacobson_top Ideal.jacobson_top
@[simp]
theorem jacobson_eq_top_iff : jacobson I = ⊤ ↔ I = ⊤ :=
⟨fun H =>
by_contradiction fun hi => let ⟨M, hm, him⟩ := exists_le_maximal I hi
lt_top_iff_ne_top.1
(lt_of_le_of_lt (show jacobson I ≤ M from sInf_le ⟨him, hm⟩) <|
lt_top_iff_ne_top.2 hm.ne_top) H,
fun H => eq_top_iff.2 <| le_sInf fun _ ⟨hij, _⟩ => H ▸ hij⟩
#align ideal.jacobson_eq_top_iff Ideal.jacobson_eq_top_iff
theorem jacobson_eq_bot : jacobson I = ⊥ → I = ⊥ := fun h => eq_bot_iff.mpr (h ▸ le_jacobson)
#align ideal.jacobson_eq_bot Ideal.jacobson_eq_bot
theorem jacobson_eq_self_of_isMaximal [H : IsMaximal I] : I.jacobson = I :=
le_antisymm (sInf_le ⟨le_of_eq rfl, H⟩) le_jacobson
#align ideal.jacobson_eq_self_of_is_maximal Ideal.jacobson_eq_self_of_isMaximal
instance (priority := 100) jacobson.isMaximal [H : IsMaximal I] : IsMaximal (jacobson I) :=
⟨⟨fun htop => H.1.1 (jacobson_eq_top_iff.1 htop), fun _ hJ =>
H.1.2 _ (lt_of_le_of_lt le_jacobson hJ)⟩⟩
#align ideal.jacobson.is_maximal Ideal.jacobson.isMaximal
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I :=
⟨fun hx y =>
by_cases
(fun hxy : I ⊔ span {y * x + 1} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy)
let ⟨r, hr⟩ := mem_span_singleton'.1 hq
⟨r, by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one r (y * x), hr, ← hpq, ← neg_sub, add_sub_cancel]
exact I.neg_mem hpi⟩)
fun hxy : I ⊔ span {y * x + 1} ≠ ⊤ => let ⟨M, hm1, hm2⟩ := exists_le_maximal _ hxy
suffices x ∉ M from (this <| mem_sInf.1 hx ⟨le_trans le_sup_left hm2, hm1⟩).elim
fun hxm => hm1.1.1 <| (eq_top_iff_one _).2 <| add_sub_cancel' (y * x) 1 ▸
M.sub_mem (le_sup_right.trans hm2 <| subset_span rfl) (M.mul_mem_left _ hxm),
fun hx => mem_sInf.2 fun M ⟨him, hm⟩ => by_contradiction fun hxm =>
let ⟨y, i, hi, df⟩ := hm.exists_inv hxm
let ⟨z, hz⟩ := hx (-y)
hm.1.1 <| (eq_top_iff_one _).2 <| sub_sub_cancel (z * -y * x + z) 1 ▸
M.sub_mem (by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one z, neg_mul, ← sub_eq_iff_eq_add.mpr df.symm, neg_sub,
sub_add_cancel]
exact M.mul_mem_left _ hi) <| him hz⟩
#align ideal.mem_jacobson_iff Ideal.mem_jacobson_iff
theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
∃ s, s * r - 1 ∈ I := by
cases' mem_jacobson_iff.1 h 1 with s hs
use s
simpa [mul_sub] using hs
#align ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson Ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M := by
use fun hI => ⟨{ J : Ideal R | I ≤ J ∧ J.IsMaximal }, ⟨fun _ hJ => Or.inl hJ.right, hI.symm⟩⟩
rintro ⟨M, hM, hInf⟩
refine le_antisymm (fun x hx => ?_) le_jacobson
rw [hInf, mem_sInf]
intro I hI
cases' hM I hI with is_max is_top
· exact (mem_sInf.1 hx) ⟨le_sInf_iff.1 (le_of_eq hInf) I hI, is_max⟩
· exact is_top.symm ▸ Submodule.mem_top
#align ideal.eq_jacobson_iff_Inf_maximal Ideal.eq_jacobson_iff_sInf_maximal
theorem eq_jacobson_iff_sInf_maximal' :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, ∀ (K : Ideal R), J < K → K = ⊤) ∧ I = sInf M :=
eq_jacobson_iff_sInf_maximal.trans
⟨fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ K hK =>
Or.recOn (hM.1 J hJ) (fun h => h.1.2 K hK) fun h => eq_top_iff.2 (le_of_lt (h ▸ hK)),
hM.2⟩⟩,
fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ =>
Or.recOn (Classical.em (J = ⊤)) (fun h => Or.inr h) fun h => Or.inl ⟨⟨h, hM.1 J hJ⟩⟩,
hM.2⟩⟩⟩
#align ideal.eq_jacobson_iff_Inf_maximal' Ideal.eq_jacobson_iff_sInf_maximal'
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M := by
constructor
· intro h x hx
erw [← h, mem_sInf] at hx
push_neg at hx
exact hx
· refine fun h => le_antisymm (fun x hx => ?_) le_jacobson
contrapose hx
erw [mem_sInf]
push_neg
exact h x hx
#align ideal.eq_jacobson_iff_not_mem Ideal.eq_jacobson_iff_not_mem
theorem map_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) :
RingHom.ker f ≤ I → map f I.jacobson = (map f I).jacobson := by
|
intro h
|
theorem map_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) :
RingHom.ker f ≤ I → map f I.jacobson = (map f I).jacobson := by
|
Mathlib.RingTheory.JacobsonIdeal.178_0.Lz0MgLQMj1bGzuN
|
theorem map_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) :
RingHom.ker f ≤ I → map f I.jacobson = (map f I).jacobson
|
Mathlib_RingTheory_JacobsonIdeal
|
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I : Ideal R
f : R →+* S
hf : Function.Surjective ⇑f
h : RingHom.ker f ≤ I
⊢ map f (jacobson I) = jacobson (map f I)
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
/-!
# Jacobson radical
The Jacobson radical of a ring `R` is defined to be the intersection of all maximal ideals of `R`.
This is similar to how the nilradical is equal to the intersection of all prime ideals of `R`.
We can extend the idea of the nilradical to ideals of `R`,
by letting the radical of an ideal `I` be the intersection of prime ideals containing `I`.
Under this extension, the original nilradical is the radical of the zero ideal `⊥`.
Here we define the Jacobson radical of an ideal `I` in a similar way,
as the intersection of maximal ideals containing `I`.
## Main definitions
Let `R` be a commutative ring, and `I` be an ideal of `R`
* `Ideal.jacobson I` is the jacobson radical, i.e. the infimum of all maximal ideals containing I.
* `Ideal.IsLocal I` is the proposition that the jacobson radical of `I` is itself a maximal ideal
## Main statements
* `mem_jacobson_iff` gives a characterization of members of the jacobson of I
* `Ideal.isLocal_of_isMaximal_radical`: if the radical of I is maximal then so is the jacobson
radical
## Tags
Jacobson, Jacobson radical, Local Ideal
-/
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}
open Polynomial
section Jacobson
section Ring
variable [Ring R] [Ring S] {I : Ideal R}
/-- The Jacobson radical of `I` is the infimum of all maximal (left) ideals containing `I`. -/
def jacobson (I : Ideal R) : Ideal R :=
sInf { J : Ideal R | I ≤ J ∧ IsMaximal J }
#align ideal.jacobson Ideal.jacobson
theorem le_jacobson : I ≤ jacobson I := fun _ hx => mem_sInf.mpr fun _ hJ => hJ.left hx
#align ideal.le_jacobson Ideal.le_jacobson
@[simp]
theorem jacobson_idem : jacobson (jacobson I) = jacobson I :=
le_antisymm (sInf_le_sInf fun _ hJ => ⟨sInf_le hJ, hJ.2⟩) le_jacobson
#align ideal.jacobson_idem Ideal.jacobson_idem
@[simp]
theorem jacobson_top : jacobson (⊤ : Ideal R) = ⊤ :=
eq_top_iff.2 le_jacobson
#align ideal.jacobson_top Ideal.jacobson_top
@[simp]
theorem jacobson_eq_top_iff : jacobson I = ⊤ ↔ I = ⊤ :=
⟨fun H =>
by_contradiction fun hi => let ⟨M, hm, him⟩ := exists_le_maximal I hi
lt_top_iff_ne_top.1
(lt_of_le_of_lt (show jacobson I ≤ M from sInf_le ⟨him, hm⟩) <|
lt_top_iff_ne_top.2 hm.ne_top) H,
fun H => eq_top_iff.2 <| le_sInf fun _ ⟨hij, _⟩ => H ▸ hij⟩
#align ideal.jacobson_eq_top_iff Ideal.jacobson_eq_top_iff
theorem jacobson_eq_bot : jacobson I = ⊥ → I = ⊥ := fun h => eq_bot_iff.mpr (h ▸ le_jacobson)
#align ideal.jacobson_eq_bot Ideal.jacobson_eq_bot
theorem jacobson_eq_self_of_isMaximal [H : IsMaximal I] : I.jacobson = I :=
le_antisymm (sInf_le ⟨le_of_eq rfl, H⟩) le_jacobson
#align ideal.jacobson_eq_self_of_is_maximal Ideal.jacobson_eq_self_of_isMaximal
instance (priority := 100) jacobson.isMaximal [H : IsMaximal I] : IsMaximal (jacobson I) :=
⟨⟨fun htop => H.1.1 (jacobson_eq_top_iff.1 htop), fun _ hJ =>
H.1.2 _ (lt_of_le_of_lt le_jacobson hJ)⟩⟩
#align ideal.jacobson.is_maximal Ideal.jacobson.isMaximal
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I :=
⟨fun hx y =>
by_cases
(fun hxy : I ⊔ span {y * x + 1} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy)
let ⟨r, hr⟩ := mem_span_singleton'.1 hq
⟨r, by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one r (y * x), hr, ← hpq, ← neg_sub, add_sub_cancel]
exact I.neg_mem hpi⟩)
fun hxy : I ⊔ span {y * x + 1} ≠ ⊤ => let ⟨M, hm1, hm2⟩ := exists_le_maximal _ hxy
suffices x ∉ M from (this <| mem_sInf.1 hx ⟨le_trans le_sup_left hm2, hm1⟩).elim
fun hxm => hm1.1.1 <| (eq_top_iff_one _).2 <| add_sub_cancel' (y * x) 1 ▸
M.sub_mem (le_sup_right.trans hm2 <| subset_span rfl) (M.mul_mem_left _ hxm),
fun hx => mem_sInf.2 fun M ⟨him, hm⟩ => by_contradiction fun hxm =>
let ⟨y, i, hi, df⟩ := hm.exists_inv hxm
let ⟨z, hz⟩ := hx (-y)
hm.1.1 <| (eq_top_iff_one _).2 <| sub_sub_cancel (z * -y * x + z) 1 ▸
M.sub_mem (by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one z, neg_mul, ← sub_eq_iff_eq_add.mpr df.symm, neg_sub,
sub_add_cancel]
exact M.mul_mem_left _ hi) <| him hz⟩
#align ideal.mem_jacobson_iff Ideal.mem_jacobson_iff
theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
∃ s, s * r - 1 ∈ I := by
cases' mem_jacobson_iff.1 h 1 with s hs
use s
simpa [mul_sub] using hs
#align ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson Ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M := by
use fun hI => ⟨{ J : Ideal R | I ≤ J ∧ J.IsMaximal }, ⟨fun _ hJ => Or.inl hJ.right, hI.symm⟩⟩
rintro ⟨M, hM, hInf⟩
refine le_antisymm (fun x hx => ?_) le_jacobson
rw [hInf, mem_sInf]
intro I hI
cases' hM I hI with is_max is_top
· exact (mem_sInf.1 hx) ⟨le_sInf_iff.1 (le_of_eq hInf) I hI, is_max⟩
· exact is_top.symm ▸ Submodule.mem_top
#align ideal.eq_jacobson_iff_Inf_maximal Ideal.eq_jacobson_iff_sInf_maximal
theorem eq_jacobson_iff_sInf_maximal' :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, ∀ (K : Ideal R), J < K → K = ⊤) ∧ I = sInf M :=
eq_jacobson_iff_sInf_maximal.trans
⟨fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ K hK =>
Or.recOn (hM.1 J hJ) (fun h => h.1.2 K hK) fun h => eq_top_iff.2 (le_of_lt (h ▸ hK)),
hM.2⟩⟩,
fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ =>
Or.recOn (Classical.em (J = ⊤)) (fun h => Or.inr h) fun h => Or.inl ⟨⟨h, hM.1 J hJ⟩⟩,
hM.2⟩⟩⟩
#align ideal.eq_jacobson_iff_Inf_maximal' Ideal.eq_jacobson_iff_sInf_maximal'
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M := by
constructor
· intro h x hx
erw [← h, mem_sInf] at hx
push_neg at hx
exact hx
· refine fun h => le_antisymm (fun x hx => ?_) le_jacobson
contrapose hx
erw [mem_sInf]
push_neg
exact h x hx
#align ideal.eq_jacobson_iff_not_mem Ideal.eq_jacobson_iff_not_mem
theorem map_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) :
RingHom.ker f ≤ I → map f I.jacobson = (map f I).jacobson := by
intro h
|
unfold Ideal.jacobson
|
theorem map_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) :
RingHom.ker f ≤ I → map f I.jacobson = (map f I).jacobson := by
intro h
|
Mathlib.RingTheory.JacobsonIdeal.178_0.Lz0MgLQMj1bGzuN
|
theorem map_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) :
RingHom.ker f ≤ I → map f I.jacobson = (map f I).jacobson
|
Mathlib_RingTheory_JacobsonIdeal
|
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I : Ideal R
f : R →+* S
hf : Function.Surjective ⇑f
h : RingHom.ker f ≤ I
⊢ map f (sInf {J | I ≤ J ∧ IsMaximal J}) = sInf {J | map f I ≤ J ∧ IsMaximal J}
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
/-!
# Jacobson radical
The Jacobson radical of a ring `R` is defined to be the intersection of all maximal ideals of `R`.
This is similar to how the nilradical is equal to the intersection of all prime ideals of `R`.
We can extend the idea of the nilradical to ideals of `R`,
by letting the radical of an ideal `I` be the intersection of prime ideals containing `I`.
Under this extension, the original nilradical is the radical of the zero ideal `⊥`.
Here we define the Jacobson radical of an ideal `I` in a similar way,
as the intersection of maximal ideals containing `I`.
## Main definitions
Let `R` be a commutative ring, and `I` be an ideal of `R`
* `Ideal.jacobson I` is the jacobson radical, i.e. the infimum of all maximal ideals containing I.
* `Ideal.IsLocal I` is the proposition that the jacobson radical of `I` is itself a maximal ideal
## Main statements
* `mem_jacobson_iff` gives a characterization of members of the jacobson of I
* `Ideal.isLocal_of_isMaximal_radical`: if the radical of I is maximal then so is the jacobson
radical
## Tags
Jacobson, Jacobson radical, Local Ideal
-/
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}
open Polynomial
section Jacobson
section Ring
variable [Ring R] [Ring S] {I : Ideal R}
/-- The Jacobson radical of `I` is the infimum of all maximal (left) ideals containing `I`. -/
def jacobson (I : Ideal R) : Ideal R :=
sInf { J : Ideal R | I ≤ J ∧ IsMaximal J }
#align ideal.jacobson Ideal.jacobson
theorem le_jacobson : I ≤ jacobson I := fun _ hx => mem_sInf.mpr fun _ hJ => hJ.left hx
#align ideal.le_jacobson Ideal.le_jacobson
@[simp]
theorem jacobson_idem : jacobson (jacobson I) = jacobson I :=
le_antisymm (sInf_le_sInf fun _ hJ => ⟨sInf_le hJ, hJ.2⟩) le_jacobson
#align ideal.jacobson_idem Ideal.jacobson_idem
@[simp]
theorem jacobson_top : jacobson (⊤ : Ideal R) = ⊤ :=
eq_top_iff.2 le_jacobson
#align ideal.jacobson_top Ideal.jacobson_top
@[simp]
theorem jacobson_eq_top_iff : jacobson I = ⊤ ↔ I = ⊤ :=
⟨fun H =>
by_contradiction fun hi => let ⟨M, hm, him⟩ := exists_le_maximal I hi
lt_top_iff_ne_top.1
(lt_of_le_of_lt (show jacobson I ≤ M from sInf_le ⟨him, hm⟩) <|
lt_top_iff_ne_top.2 hm.ne_top) H,
fun H => eq_top_iff.2 <| le_sInf fun _ ⟨hij, _⟩ => H ▸ hij⟩
#align ideal.jacobson_eq_top_iff Ideal.jacobson_eq_top_iff
theorem jacobson_eq_bot : jacobson I = ⊥ → I = ⊥ := fun h => eq_bot_iff.mpr (h ▸ le_jacobson)
#align ideal.jacobson_eq_bot Ideal.jacobson_eq_bot
theorem jacobson_eq_self_of_isMaximal [H : IsMaximal I] : I.jacobson = I :=
le_antisymm (sInf_le ⟨le_of_eq rfl, H⟩) le_jacobson
#align ideal.jacobson_eq_self_of_is_maximal Ideal.jacobson_eq_self_of_isMaximal
instance (priority := 100) jacobson.isMaximal [H : IsMaximal I] : IsMaximal (jacobson I) :=
⟨⟨fun htop => H.1.1 (jacobson_eq_top_iff.1 htop), fun _ hJ =>
H.1.2 _ (lt_of_le_of_lt le_jacobson hJ)⟩⟩
#align ideal.jacobson.is_maximal Ideal.jacobson.isMaximal
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I :=
⟨fun hx y =>
by_cases
(fun hxy : I ⊔ span {y * x + 1} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy)
let ⟨r, hr⟩ := mem_span_singleton'.1 hq
⟨r, by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one r (y * x), hr, ← hpq, ← neg_sub, add_sub_cancel]
exact I.neg_mem hpi⟩)
fun hxy : I ⊔ span {y * x + 1} ≠ ⊤ => let ⟨M, hm1, hm2⟩ := exists_le_maximal _ hxy
suffices x ∉ M from (this <| mem_sInf.1 hx ⟨le_trans le_sup_left hm2, hm1⟩).elim
fun hxm => hm1.1.1 <| (eq_top_iff_one _).2 <| add_sub_cancel' (y * x) 1 ▸
M.sub_mem (le_sup_right.trans hm2 <| subset_span rfl) (M.mul_mem_left _ hxm),
fun hx => mem_sInf.2 fun M ⟨him, hm⟩ => by_contradiction fun hxm =>
let ⟨y, i, hi, df⟩ := hm.exists_inv hxm
let ⟨z, hz⟩ := hx (-y)
hm.1.1 <| (eq_top_iff_one _).2 <| sub_sub_cancel (z * -y * x + z) 1 ▸
M.sub_mem (by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one z, neg_mul, ← sub_eq_iff_eq_add.mpr df.symm, neg_sub,
sub_add_cancel]
exact M.mul_mem_left _ hi) <| him hz⟩
#align ideal.mem_jacobson_iff Ideal.mem_jacobson_iff
theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
∃ s, s * r - 1 ∈ I := by
cases' mem_jacobson_iff.1 h 1 with s hs
use s
simpa [mul_sub] using hs
#align ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson Ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M := by
use fun hI => ⟨{ J : Ideal R | I ≤ J ∧ J.IsMaximal }, ⟨fun _ hJ => Or.inl hJ.right, hI.symm⟩⟩
rintro ⟨M, hM, hInf⟩
refine le_antisymm (fun x hx => ?_) le_jacobson
rw [hInf, mem_sInf]
intro I hI
cases' hM I hI with is_max is_top
· exact (mem_sInf.1 hx) ⟨le_sInf_iff.1 (le_of_eq hInf) I hI, is_max⟩
· exact is_top.symm ▸ Submodule.mem_top
#align ideal.eq_jacobson_iff_Inf_maximal Ideal.eq_jacobson_iff_sInf_maximal
theorem eq_jacobson_iff_sInf_maximal' :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, ∀ (K : Ideal R), J < K → K = ⊤) ∧ I = sInf M :=
eq_jacobson_iff_sInf_maximal.trans
⟨fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ K hK =>
Or.recOn (hM.1 J hJ) (fun h => h.1.2 K hK) fun h => eq_top_iff.2 (le_of_lt (h ▸ hK)),
hM.2⟩⟩,
fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ =>
Or.recOn (Classical.em (J = ⊤)) (fun h => Or.inr h) fun h => Or.inl ⟨⟨h, hM.1 J hJ⟩⟩,
hM.2⟩⟩⟩
#align ideal.eq_jacobson_iff_Inf_maximal' Ideal.eq_jacobson_iff_sInf_maximal'
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M := by
constructor
· intro h x hx
erw [← h, mem_sInf] at hx
push_neg at hx
exact hx
· refine fun h => le_antisymm (fun x hx => ?_) le_jacobson
contrapose hx
erw [mem_sInf]
push_neg
exact h x hx
#align ideal.eq_jacobson_iff_not_mem Ideal.eq_jacobson_iff_not_mem
theorem map_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) :
RingHom.ker f ≤ I → map f I.jacobson = (map f I).jacobson := by
intro h
unfold Ideal.jacobson
-- porting note : dot notation for `RingHom.ker` does not work
|
have : ∀ J ∈ { J : Ideal R | I ≤ J ∧ J.IsMaximal }, RingHom.ker f ≤ J :=
fun J hJ => le_trans h hJ.left
|
theorem map_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) :
RingHom.ker f ≤ I → map f I.jacobson = (map f I).jacobson := by
intro h
unfold Ideal.jacobson
-- porting note : dot notation for `RingHom.ker` does not work
|
Mathlib.RingTheory.JacobsonIdeal.178_0.Lz0MgLQMj1bGzuN
|
theorem map_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) :
RingHom.ker f ≤ I → map f I.jacobson = (map f I).jacobson
|
Mathlib_RingTheory_JacobsonIdeal
|
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I : Ideal R
f : R →+* S
hf : Function.Surjective ⇑f
h : RingHom.ker f ≤ I
this : ∀ J ∈ {J | I ≤ J ∧ IsMaximal J}, RingHom.ker f ≤ J
⊢ map f (sInf {J | I ≤ J ∧ IsMaximal J}) = sInf {J | map f I ≤ J ∧ IsMaximal J}
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
/-!
# Jacobson radical
The Jacobson radical of a ring `R` is defined to be the intersection of all maximal ideals of `R`.
This is similar to how the nilradical is equal to the intersection of all prime ideals of `R`.
We can extend the idea of the nilradical to ideals of `R`,
by letting the radical of an ideal `I` be the intersection of prime ideals containing `I`.
Under this extension, the original nilradical is the radical of the zero ideal `⊥`.
Here we define the Jacobson radical of an ideal `I` in a similar way,
as the intersection of maximal ideals containing `I`.
## Main definitions
Let `R` be a commutative ring, and `I` be an ideal of `R`
* `Ideal.jacobson I` is the jacobson radical, i.e. the infimum of all maximal ideals containing I.
* `Ideal.IsLocal I` is the proposition that the jacobson radical of `I` is itself a maximal ideal
## Main statements
* `mem_jacobson_iff` gives a characterization of members of the jacobson of I
* `Ideal.isLocal_of_isMaximal_radical`: if the radical of I is maximal then so is the jacobson
radical
## Tags
Jacobson, Jacobson radical, Local Ideal
-/
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}
open Polynomial
section Jacobson
section Ring
variable [Ring R] [Ring S] {I : Ideal R}
/-- The Jacobson radical of `I` is the infimum of all maximal (left) ideals containing `I`. -/
def jacobson (I : Ideal R) : Ideal R :=
sInf { J : Ideal R | I ≤ J ∧ IsMaximal J }
#align ideal.jacobson Ideal.jacobson
theorem le_jacobson : I ≤ jacobson I := fun _ hx => mem_sInf.mpr fun _ hJ => hJ.left hx
#align ideal.le_jacobson Ideal.le_jacobson
@[simp]
theorem jacobson_idem : jacobson (jacobson I) = jacobson I :=
le_antisymm (sInf_le_sInf fun _ hJ => ⟨sInf_le hJ, hJ.2⟩) le_jacobson
#align ideal.jacobson_idem Ideal.jacobson_idem
@[simp]
theorem jacobson_top : jacobson (⊤ : Ideal R) = ⊤ :=
eq_top_iff.2 le_jacobson
#align ideal.jacobson_top Ideal.jacobson_top
@[simp]
theorem jacobson_eq_top_iff : jacobson I = ⊤ ↔ I = ⊤ :=
⟨fun H =>
by_contradiction fun hi => let ⟨M, hm, him⟩ := exists_le_maximal I hi
lt_top_iff_ne_top.1
(lt_of_le_of_lt (show jacobson I ≤ M from sInf_le ⟨him, hm⟩) <|
lt_top_iff_ne_top.2 hm.ne_top) H,
fun H => eq_top_iff.2 <| le_sInf fun _ ⟨hij, _⟩ => H ▸ hij⟩
#align ideal.jacobson_eq_top_iff Ideal.jacobson_eq_top_iff
theorem jacobson_eq_bot : jacobson I = ⊥ → I = ⊥ := fun h => eq_bot_iff.mpr (h ▸ le_jacobson)
#align ideal.jacobson_eq_bot Ideal.jacobson_eq_bot
theorem jacobson_eq_self_of_isMaximal [H : IsMaximal I] : I.jacobson = I :=
le_antisymm (sInf_le ⟨le_of_eq rfl, H⟩) le_jacobson
#align ideal.jacobson_eq_self_of_is_maximal Ideal.jacobson_eq_self_of_isMaximal
instance (priority := 100) jacobson.isMaximal [H : IsMaximal I] : IsMaximal (jacobson I) :=
⟨⟨fun htop => H.1.1 (jacobson_eq_top_iff.1 htop), fun _ hJ =>
H.1.2 _ (lt_of_le_of_lt le_jacobson hJ)⟩⟩
#align ideal.jacobson.is_maximal Ideal.jacobson.isMaximal
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I :=
⟨fun hx y =>
by_cases
(fun hxy : I ⊔ span {y * x + 1} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy)
let ⟨r, hr⟩ := mem_span_singleton'.1 hq
⟨r, by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one r (y * x), hr, ← hpq, ← neg_sub, add_sub_cancel]
exact I.neg_mem hpi⟩)
fun hxy : I ⊔ span {y * x + 1} ≠ ⊤ => let ⟨M, hm1, hm2⟩ := exists_le_maximal _ hxy
suffices x ∉ M from (this <| mem_sInf.1 hx ⟨le_trans le_sup_left hm2, hm1⟩).elim
fun hxm => hm1.1.1 <| (eq_top_iff_one _).2 <| add_sub_cancel' (y * x) 1 ▸
M.sub_mem (le_sup_right.trans hm2 <| subset_span rfl) (M.mul_mem_left _ hxm),
fun hx => mem_sInf.2 fun M ⟨him, hm⟩ => by_contradiction fun hxm =>
let ⟨y, i, hi, df⟩ := hm.exists_inv hxm
let ⟨z, hz⟩ := hx (-y)
hm.1.1 <| (eq_top_iff_one _).2 <| sub_sub_cancel (z * -y * x + z) 1 ▸
M.sub_mem (by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one z, neg_mul, ← sub_eq_iff_eq_add.mpr df.symm, neg_sub,
sub_add_cancel]
exact M.mul_mem_left _ hi) <| him hz⟩
#align ideal.mem_jacobson_iff Ideal.mem_jacobson_iff
theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
∃ s, s * r - 1 ∈ I := by
cases' mem_jacobson_iff.1 h 1 with s hs
use s
simpa [mul_sub] using hs
#align ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson Ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M := by
use fun hI => ⟨{ J : Ideal R | I ≤ J ∧ J.IsMaximal }, ⟨fun _ hJ => Or.inl hJ.right, hI.symm⟩⟩
rintro ⟨M, hM, hInf⟩
refine le_antisymm (fun x hx => ?_) le_jacobson
rw [hInf, mem_sInf]
intro I hI
cases' hM I hI with is_max is_top
· exact (mem_sInf.1 hx) ⟨le_sInf_iff.1 (le_of_eq hInf) I hI, is_max⟩
· exact is_top.symm ▸ Submodule.mem_top
#align ideal.eq_jacobson_iff_Inf_maximal Ideal.eq_jacobson_iff_sInf_maximal
theorem eq_jacobson_iff_sInf_maximal' :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, ∀ (K : Ideal R), J < K → K = ⊤) ∧ I = sInf M :=
eq_jacobson_iff_sInf_maximal.trans
⟨fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ K hK =>
Or.recOn (hM.1 J hJ) (fun h => h.1.2 K hK) fun h => eq_top_iff.2 (le_of_lt (h ▸ hK)),
hM.2⟩⟩,
fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ =>
Or.recOn (Classical.em (J = ⊤)) (fun h => Or.inr h) fun h => Or.inl ⟨⟨h, hM.1 J hJ⟩⟩,
hM.2⟩⟩⟩
#align ideal.eq_jacobson_iff_Inf_maximal' Ideal.eq_jacobson_iff_sInf_maximal'
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M := by
constructor
· intro h x hx
erw [← h, mem_sInf] at hx
push_neg at hx
exact hx
· refine fun h => le_antisymm (fun x hx => ?_) le_jacobson
contrapose hx
erw [mem_sInf]
push_neg
exact h x hx
#align ideal.eq_jacobson_iff_not_mem Ideal.eq_jacobson_iff_not_mem
theorem map_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) :
RingHom.ker f ≤ I → map f I.jacobson = (map f I).jacobson := by
intro h
unfold Ideal.jacobson
-- porting note : dot notation for `RingHom.ker` does not work
have : ∀ J ∈ { J : Ideal R | I ≤ J ∧ J.IsMaximal }, RingHom.ker f ≤ J :=
fun J hJ => le_trans h hJ.left
|
refine Trans.trans (map_sInf hf this) (le_antisymm ?_ ?_)
|
theorem map_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) :
RingHom.ker f ≤ I → map f I.jacobson = (map f I).jacobson := by
intro h
unfold Ideal.jacobson
-- porting note : dot notation for `RingHom.ker` does not work
have : ∀ J ∈ { J : Ideal R | I ≤ J ∧ J.IsMaximal }, RingHom.ker f ≤ J :=
fun J hJ => le_trans h hJ.left
|
Mathlib.RingTheory.JacobsonIdeal.178_0.Lz0MgLQMj1bGzuN
|
theorem map_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) :
RingHom.ker f ≤ I → map f I.jacobson = (map f I).jacobson
|
Mathlib_RingTheory_JacobsonIdeal
|
case refine_1
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I : Ideal R
f : R →+* S
hf : Function.Surjective ⇑f
h : RingHom.ker f ≤ I
this : ∀ J ∈ {J | I ≤ J ∧ IsMaximal J}, RingHom.ker f ≤ J
⊢ sInf (map f '' {J | I ≤ J ∧ IsMaximal J}) ≤ sInf {J | map f I ≤ J ∧ IsMaximal J}
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
/-!
# Jacobson radical
The Jacobson radical of a ring `R` is defined to be the intersection of all maximal ideals of `R`.
This is similar to how the nilradical is equal to the intersection of all prime ideals of `R`.
We can extend the idea of the nilradical to ideals of `R`,
by letting the radical of an ideal `I` be the intersection of prime ideals containing `I`.
Under this extension, the original nilradical is the radical of the zero ideal `⊥`.
Here we define the Jacobson radical of an ideal `I` in a similar way,
as the intersection of maximal ideals containing `I`.
## Main definitions
Let `R` be a commutative ring, and `I` be an ideal of `R`
* `Ideal.jacobson I` is the jacobson radical, i.e. the infimum of all maximal ideals containing I.
* `Ideal.IsLocal I` is the proposition that the jacobson radical of `I` is itself a maximal ideal
## Main statements
* `mem_jacobson_iff` gives a characterization of members of the jacobson of I
* `Ideal.isLocal_of_isMaximal_radical`: if the radical of I is maximal then so is the jacobson
radical
## Tags
Jacobson, Jacobson radical, Local Ideal
-/
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}
open Polynomial
section Jacobson
section Ring
variable [Ring R] [Ring S] {I : Ideal R}
/-- The Jacobson radical of `I` is the infimum of all maximal (left) ideals containing `I`. -/
def jacobson (I : Ideal R) : Ideal R :=
sInf { J : Ideal R | I ≤ J ∧ IsMaximal J }
#align ideal.jacobson Ideal.jacobson
theorem le_jacobson : I ≤ jacobson I := fun _ hx => mem_sInf.mpr fun _ hJ => hJ.left hx
#align ideal.le_jacobson Ideal.le_jacobson
@[simp]
theorem jacobson_idem : jacobson (jacobson I) = jacobson I :=
le_antisymm (sInf_le_sInf fun _ hJ => ⟨sInf_le hJ, hJ.2⟩) le_jacobson
#align ideal.jacobson_idem Ideal.jacobson_idem
@[simp]
theorem jacobson_top : jacobson (⊤ : Ideal R) = ⊤ :=
eq_top_iff.2 le_jacobson
#align ideal.jacobson_top Ideal.jacobson_top
@[simp]
theorem jacobson_eq_top_iff : jacobson I = ⊤ ↔ I = ⊤ :=
⟨fun H =>
by_contradiction fun hi => let ⟨M, hm, him⟩ := exists_le_maximal I hi
lt_top_iff_ne_top.1
(lt_of_le_of_lt (show jacobson I ≤ M from sInf_le ⟨him, hm⟩) <|
lt_top_iff_ne_top.2 hm.ne_top) H,
fun H => eq_top_iff.2 <| le_sInf fun _ ⟨hij, _⟩ => H ▸ hij⟩
#align ideal.jacobson_eq_top_iff Ideal.jacobson_eq_top_iff
theorem jacobson_eq_bot : jacobson I = ⊥ → I = ⊥ := fun h => eq_bot_iff.mpr (h ▸ le_jacobson)
#align ideal.jacobson_eq_bot Ideal.jacobson_eq_bot
theorem jacobson_eq_self_of_isMaximal [H : IsMaximal I] : I.jacobson = I :=
le_antisymm (sInf_le ⟨le_of_eq rfl, H⟩) le_jacobson
#align ideal.jacobson_eq_self_of_is_maximal Ideal.jacobson_eq_self_of_isMaximal
instance (priority := 100) jacobson.isMaximal [H : IsMaximal I] : IsMaximal (jacobson I) :=
⟨⟨fun htop => H.1.1 (jacobson_eq_top_iff.1 htop), fun _ hJ =>
H.1.2 _ (lt_of_le_of_lt le_jacobson hJ)⟩⟩
#align ideal.jacobson.is_maximal Ideal.jacobson.isMaximal
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I :=
⟨fun hx y =>
by_cases
(fun hxy : I ⊔ span {y * x + 1} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy)
let ⟨r, hr⟩ := mem_span_singleton'.1 hq
⟨r, by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one r (y * x), hr, ← hpq, ← neg_sub, add_sub_cancel]
exact I.neg_mem hpi⟩)
fun hxy : I ⊔ span {y * x + 1} ≠ ⊤ => let ⟨M, hm1, hm2⟩ := exists_le_maximal _ hxy
suffices x ∉ M from (this <| mem_sInf.1 hx ⟨le_trans le_sup_left hm2, hm1⟩).elim
fun hxm => hm1.1.1 <| (eq_top_iff_one _).2 <| add_sub_cancel' (y * x) 1 ▸
M.sub_mem (le_sup_right.trans hm2 <| subset_span rfl) (M.mul_mem_left _ hxm),
fun hx => mem_sInf.2 fun M ⟨him, hm⟩ => by_contradiction fun hxm =>
let ⟨y, i, hi, df⟩ := hm.exists_inv hxm
let ⟨z, hz⟩ := hx (-y)
hm.1.1 <| (eq_top_iff_one _).2 <| sub_sub_cancel (z * -y * x + z) 1 ▸
M.sub_mem (by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one z, neg_mul, ← sub_eq_iff_eq_add.mpr df.symm, neg_sub,
sub_add_cancel]
exact M.mul_mem_left _ hi) <| him hz⟩
#align ideal.mem_jacobson_iff Ideal.mem_jacobson_iff
theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
∃ s, s * r - 1 ∈ I := by
cases' mem_jacobson_iff.1 h 1 with s hs
use s
simpa [mul_sub] using hs
#align ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson Ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M := by
use fun hI => ⟨{ J : Ideal R | I ≤ J ∧ J.IsMaximal }, ⟨fun _ hJ => Or.inl hJ.right, hI.symm⟩⟩
rintro ⟨M, hM, hInf⟩
refine le_antisymm (fun x hx => ?_) le_jacobson
rw [hInf, mem_sInf]
intro I hI
cases' hM I hI with is_max is_top
· exact (mem_sInf.1 hx) ⟨le_sInf_iff.1 (le_of_eq hInf) I hI, is_max⟩
· exact is_top.symm ▸ Submodule.mem_top
#align ideal.eq_jacobson_iff_Inf_maximal Ideal.eq_jacobson_iff_sInf_maximal
theorem eq_jacobson_iff_sInf_maximal' :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, ∀ (K : Ideal R), J < K → K = ⊤) ∧ I = sInf M :=
eq_jacobson_iff_sInf_maximal.trans
⟨fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ K hK =>
Or.recOn (hM.1 J hJ) (fun h => h.1.2 K hK) fun h => eq_top_iff.2 (le_of_lt (h ▸ hK)),
hM.2⟩⟩,
fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ =>
Or.recOn (Classical.em (J = ⊤)) (fun h => Or.inr h) fun h => Or.inl ⟨⟨h, hM.1 J hJ⟩⟩,
hM.2⟩⟩⟩
#align ideal.eq_jacobson_iff_Inf_maximal' Ideal.eq_jacobson_iff_sInf_maximal'
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M := by
constructor
· intro h x hx
erw [← h, mem_sInf] at hx
push_neg at hx
exact hx
· refine fun h => le_antisymm (fun x hx => ?_) le_jacobson
contrapose hx
erw [mem_sInf]
push_neg
exact h x hx
#align ideal.eq_jacobson_iff_not_mem Ideal.eq_jacobson_iff_not_mem
theorem map_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) :
RingHom.ker f ≤ I → map f I.jacobson = (map f I).jacobson := by
intro h
unfold Ideal.jacobson
-- porting note : dot notation for `RingHom.ker` does not work
have : ∀ J ∈ { J : Ideal R | I ≤ J ∧ J.IsMaximal }, RingHom.ker f ≤ J :=
fun J hJ => le_trans h hJ.left
refine Trans.trans (map_sInf hf this) (le_antisymm ?_ ?_)
·
|
refine'
sInf_le_sInf fun J hJ =>
⟨comap f J, ⟨⟨le_comap_of_map_le hJ.1, _⟩, map_comap_of_surjective f hf J⟩⟩
|
theorem map_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) :
RingHom.ker f ≤ I → map f I.jacobson = (map f I).jacobson := by
intro h
unfold Ideal.jacobson
-- porting note : dot notation for `RingHom.ker` does not work
have : ∀ J ∈ { J : Ideal R | I ≤ J ∧ J.IsMaximal }, RingHom.ker f ≤ J :=
fun J hJ => le_trans h hJ.left
refine Trans.trans (map_sInf hf this) (le_antisymm ?_ ?_)
·
|
Mathlib.RingTheory.JacobsonIdeal.178_0.Lz0MgLQMj1bGzuN
|
theorem map_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) :
RingHom.ker f ≤ I → map f I.jacobson = (map f I).jacobson
|
Mathlib_RingTheory_JacobsonIdeal
|
case refine_1
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I : Ideal R
f : R →+* S
hf : Function.Surjective ⇑f
h : RingHom.ker f ≤ I
this : ∀ J ∈ {J | I ≤ J ∧ IsMaximal J}, RingHom.ker f ≤ J
J : Ideal S
hJ : J ∈ {J | map f I ≤ J ∧ IsMaximal J}
⊢ IsMaximal (comap f J)
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
/-!
# Jacobson radical
The Jacobson radical of a ring `R` is defined to be the intersection of all maximal ideals of `R`.
This is similar to how the nilradical is equal to the intersection of all prime ideals of `R`.
We can extend the idea of the nilradical to ideals of `R`,
by letting the radical of an ideal `I` be the intersection of prime ideals containing `I`.
Under this extension, the original nilradical is the radical of the zero ideal `⊥`.
Here we define the Jacobson radical of an ideal `I` in a similar way,
as the intersection of maximal ideals containing `I`.
## Main definitions
Let `R` be a commutative ring, and `I` be an ideal of `R`
* `Ideal.jacobson I` is the jacobson radical, i.e. the infimum of all maximal ideals containing I.
* `Ideal.IsLocal I` is the proposition that the jacobson radical of `I` is itself a maximal ideal
## Main statements
* `mem_jacobson_iff` gives a characterization of members of the jacobson of I
* `Ideal.isLocal_of_isMaximal_radical`: if the radical of I is maximal then so is the jacobson
radical
## Tags
Jacobson, Jacobson radical, Local Ideal
-/
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}
open Polynomial
section Jacobson
section Ring
variable [Ring R] [Ring S] {I : Ideal R}
/-- The Jacobson radical of `I` is the infimum of all maximal (left) ideals containing `I`. -/
def jacobson (I : Ideal R) : Ideal R :=
sInf { J : Ideal R | I ≤ J ∧ IsMaximal J }
#align ideal.jacobson Ideal.jacobson
theorem le_jacobson : I ≤ jacobson I := fun _ hx => mem_sInf.mpr fun _ hJ => hJ.left hx
#align ideal.le_jacobson Ideal.le_jacobson
@[simp]
theorem jacobson_idem : jacobson (jacobson I) = jacobson I :=
le_antisymm (sInf_le_sInf fun _ hJ => ⟨sInf_le hJ, hJ.2⟩) le_jacobson
#align ideal.jacobson_idem Ideal.jacobson_idem
@[simp]
theorem jacobson_top : jacobson (⊤ : Ideal R) = ⊤ :=
eq_top_iff.2 le_jacobson
#align ideal.jacobson_top Ideal.jacobson_top
@[simp]
theorem jacobson_eq_top_iff : jacobson I = ⊤ ↔ I = ⊤ :=
⟨fun H =>
by_contradiction fun hi => let ⟨M, hm, him⟩ := exists_le_maximal I hi
lt_top_iff_ne_top.1
(lt_of_le_of_lt (show jacobson I ≤ M from sInf_le ⟨him, hm⟩) <|
lt_top_iff_ne_top.2 hm.ne_top) H,
fun H => eq_top_iff.2 <| le_sInf fun _ ⟨hij, _⟩ => H ▸ hij⟩
#align ideal.jacobson_eq_top_iff Ideal.jacobson_eq_top_iff
theorem jacobson_eq_bot : jacobson I = ⊥ → I = ⊥ := fun h => eq_bot_iff.mpr (h ▸ le_jacobson)
#align ideal.jacobson_eq_bot Ideal.jacobson_eq_bot
theorem jacobson_eq_self_of_isMaximal [H : IsMaximal I] : I.jacobson = I :=
le_antisymm (sInf_le ⟨le_of_eq rfl, H⟩) le_jacobson
#align ideal.jacobson_eq_self_of_is_maximal Ideal.jacobson_eq_self_of_isMaximal
instance (priority := 100) jacobson.isMaximal [H : IsMaximal I] : IsMaximal (jacobson I) :=
⟨⟨fun htop => H.1.1 (jacobson_eq_top_iff.1 htop), fun _ hJ =>
H.1.2 _ (lt_of_le_of_lt le_jacobson hJ)⟩⟩
#align ideal.jacobson.is_maximal Ideal.jacobson.isMaximal
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I :=
⟨fun hx y =>
by_cases
(fun hxy : I ⊔ span {y * x + 1} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy)
let ⟨r, hr⟩ := mem_span_singleton'.1 hq
⟨r, by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one r (y * x), hr, ← hpq, ← neg_sub, add_sub_cancel]
exact I.neg_mem hpi⟩)
fun hxy : I ⊔ span {y * x + 1} ≠ ⊤ => let ⟨M, hm1, hm2⟩ := exists_le_maximal _ hxy
suffices x ∉ M from (this <| mem_sInf.1 hx ⟨le_trans le_sup_left hm2, hm1⟩).elim
fun hxm => hm1.1.1 <| (eq_top_iff_one _).2 <| add_sub_cancel' (y * x) 1 ▸
M.sub_mem (le_sup_right.trans hm2 <| subset_span rfl) (M.mul_mem_left _ hxm),
fun hx => mem_sInf.2 fun M ⟨him, hm⟩ => by_contradiction fun hxm =>
let ⟨y, i, hi, df⟩ := hm.exists_inv hxm
let ⟨z, hz⟩ := hx (-y)
hm.1.1 <| (eq_top_iff_one _).2 <| sub_sub_cancel (z * -y * x + z) 1 ▸
M.sub_mem (by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one z, neg_mul, ← sub_eq_iff_eq_add.mpr df.symm, neg_sub,
sub_add_cancel]
exact M.mul_mem_left _ hi) <| him hz⟩
#align ideal.mem_jacobson_iff Ideal.mem_jacobson_iff
theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
∃ s, s * r - 1 ∈ I := by
cases' mem_jacobson_iff.1 h 1 with s hs
use s
simpa [mul_sub] using hs
#align ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson Ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M := by
use fun hI => ⟨{ J : Ideal R | I ≤ J ∧ J.IsMaximal }, ⟨fun _ hJ => Or.inl hJ.right, hI.symm⟩⟩
rintro ⟨M, hM, hInf⟩
refine le_antisymm (fun x hx => ?_) le_jacobson
rw [hInf, mem_sInf]
intro I hI
cases' hM I hI with is_max is_top
· exact (mem_sInf.1 hx) ⟨le_sInf_iff.1 (le_of_eq hInf) I hI, is_max⟩
· exact is_top.symm ▸ Submodule.mem_top
#align ideal.eq_jacobson_iff_Inf_maximal Ideal.eq_jacobson_iff_sInf_maximal
theorem eq_jacobson_iff_sInf_maximal' :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, ∀ (K : Ideal R), J < K → K = ⊤) ∧ I = sInf M :=
eq_jacobson_iff_sInf_maximal.trans
⟨fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ K hK =>
Or.recOn (hM.1 J hJ) (fun h => h.1.2 K hK) fun h => eq_top_iff.2 (le_of_lt (h ▸ hK)),
hM.2⟩⟩,
fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ =>
Or.recOn (Classical.em (J = ⊤)) (fun h => Or.inr h) fun h => Or.inl ⟨⟨h, hM.1 J hJ⟩⟩,
hM.2⟩⟩⟩
#align ideal.eq_jacobson_iff_Inf_maximal' Ideal.eq_jacobson_iff_sInf_maximal'
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M := by
constructor
· intro h x hx
erw [← h, mem_sInf] at hx
push_neg at hx
exact hx
· refine fun h => le_antisymm (fun x hx => ?_) le_jacobson
contrapose hx
erw [mem_sInf]
push_neg
exact h x hx
#align ideal.eq_jacobson_iff_not_mem Ideal.eq_jacobson_iff_not_mem
theorem map_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) :
RingHom.ker f ≤ I → map f I.jacobson = (map f I).jacobson := by
intro h
unfold Ideal.jacobson
-- porting note : dot notation for `RingHom.ker` does not work
have : ∀ J ∈ { J : Ideal R | I ≤ J ∧ J.IsMaximal }, RingHom.ker f ≤ J :=
fun J hJ => le_trans h hJ.left
refine Trans.trans (map_sInf hf this) (le_antisymm ?_ ?_)
· refine'
sInf_le_sInf fun J hJ =>
⟨comap f J, ⟨⟨le_comap_of_map_le hJ.1, _⟩, map_comap_of_surjective f hf J⟩⟩
|
haveI : J.IsMaximal := hJ.right
|
theorem map_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) :
RingHom.ker f ≤ I → map f I.jacobson = (map f I).jacobson := by
intro h
unfold Ideal.jacobson
-- porting note : dot notation for `RingHom.ker` does not work
have : ∀ J ∈ { J : Ideal R | I ≤ J ∧ J.IsMaximal }, RingHom.ker f ≤ J :=
fun J hJ => le_trans h hJ.left
refine Trans.trans (map_sInf hf this) (le_antisymm ?_ ?_)
· refine'
sInf_le_sInf fun J hJ =>
⟨comap f J, ⟨⟨le_comap_of_map_le hJ.1, _⟩, map_comap_of_surjective f hf J⟩⟩
|
Mathlib.RingTheory.JacobsonIdeal.178_0.Lz0MgLQMj1bGzuN
|
theorem map_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) :
RingHom.ker f ≤ I → map f I.jacobson = (map f I).jacobson
|
Mathlib_RingTheory_JacobsonIdeal
|
case refine_1
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I : Ideal R
f : R →+* S
hf : Function.Surjective ⇑f
h : RingHom.ker f ≤ I
this✝ : ∀ J ∈ {J | I ≤ J ∧ IsMaximal J}, RingHom.ker f ≤ J
J : Ideal S
hJ : J ∈ {J | map f I ≤ J ∧ IsMaximal J}
this : IsMaximal J
⊢ IsMaximal (comap f J)
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
/-!
# Jacobson radical
The Jacobson radical of a ring `R` is defined to be the intersection of all maximal ideals of `R`.
This is similar to how the nilradical is equal to the intersection of all prime ideals of `R`.
We can extend the idea of the nilradical to ideals of `R`,
by letting the radical of an ideal `I` be the intersection of prime ideals containing `I`.
Under this extension, the original nilradical is the radical of the zero ideal `⊥`.
Here we define the Jacobson radical of an ideal `I` in a similar way,
as the intersection of maximal ideals containing `I`.
## Main definitions
Let `R` be a commutative ring, and `I` be an ideal of `R`
* `Ideal.jacobson I` is the jacobson radical, i.e. the infimum of all maximal ideals containing I.
* `Ideal.IsLocal I` is the proposition that the jacobson radical of `I` is itself a maximal ideal
## Main statements
* `mem_jacobson_iff` gives a characterization of members of the jacobson of I
* `Ideal.isLocal_of_isMaximal_radical`: if the radical of I is maximal then so is the jacobson
radical
## Tags
Jacobson, Jacobson radical, Local Ideal
-/
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}
open Polynomial
section Jacobson
section Ring
variable [Ring R] [Ring S] {I : Ideal R}
/-- The Jacobson radical of `I` is the infimum of all maximal (left) ideals containing `I`. -/
def jacobson (I : Ideal R) : Ideal R :=
sInf { J : Ideal R | I ≤ J ∧ IsMaximal J }
#align ideal.jacobson Ideal.jacobson
theorem le_jacobson : I ≤ jacobson I := fun _ hx => mem_sInf.mpr fun _ hJ => hJ.left hx
#align ideal.le_jacobson Ideal.le_jacobson
@[simp]
theorem jacobson_idem : jacobson (jacobson I) = jacobson I :=
le_antisymm (sInf_le_sInf fun _ hJ => ⟨sInf_le hJ, hJ.2⟩) le_jacobson
#align ideal.jacobson_idem Ideal.jacobson_idem
@[simp]
theorem jacobson_top : jacobson (⊤ : Ideal R) = ⊤ :=
eq_top_iff.2 le_jacobson
#align ideal.jacobson_top Ideal.jacobson_top
@[simp]
theorem jacobson_eq_top_iff : jacobson I = ⊤ ↔ I = ⊤ :=
⟨fun H =>
by_contradiction fun hi => let ⟨M, hm, him⟩ := exists_le_maximal I hi
lt_top_iff_ne_top.1
(lt_of_le_of_lt (show jacobson I ≤ M from sInf_le ⟨him, hm⟩) <|
lt_top_iff_ne_top.2 hm.ne_top) H,
fun H => eq_top_iff.2 <| le_sInf fun _ ⟨hij, _⟩ => H ▸ hij⟩
#align ideal.jacobson_eq_top_iff Ideal.jacobson_eq_top_iff
theorem jacobson_eq_bot : jacobson I = ⊥ → I = ⊥ := fun h => eq_bot_iff.mpr (h ▸ le_jacobson)
#align ideal.jacobson_eq_bot Ideal.jacobson_eq_bot
theorem jacobson_eq_self_of_isMaximal [H : IsMaximal I] : I.jacobson = I :=
le_antisymm (sInf_le ⟨le_of_eq rfl, H⟩) le_jacobson
#align ideal.jacobson_eq_self_of_is_maximal Ideal.jacobson_eq_self_of_isMaximal
instance (priority := 100) jacobson.isMaximal [H : IsMaximal I] : IsMaximal (jacobson I) :=
⟨⟨fun htop => H.1.1 (jacobson_eq_top_iff.1 htop), fun _ hJ =>
H.1.2 _ (lt_of_le_of_lt le_jacobson hJ)⟩⟩
#align ideal.jacobson.is_maximal Ideal.jacobson.isMaximal
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I :=
⟨fun hx y =>
by_cases
(fun hxy : I ⊔ span {y * x + 1} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy)
let ⟨r, hr⟩ := mem_span_singleton'.1 hq
⟨r, by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one r (y * x), hr, ← hpq, ← neg_sub, add_sub_cancel]
exact I.neg_mem hpi⟩)
fun hxy : I ⊔ span {y * x + 1} ≠ ⊤ => let ⟨M, hm1, hm2⟩ := exists_le_maximal _ hxy
suffices x ∉ M from (this <| mem_sInf.1 hx ⟨le_trans le_sup_left hm2, hm1⟩).elim
fun hxm => hm1.1.1 <| (eq_top_iff_one _).2 <| add_sub_cancel' (y * x) 1 ▸
M.sub_mem (le_sup_right.trans hm2 <| subset_span rfl) (M.mul_mem_left _ hxm),
fun hx => mem_sInf.2 fun M ⟨him, hm⟩ => by_contradiction fun hxm =>
let ⟨y, i, hi, df⟩ := hm.exists_inv hxm
let ⟨z, hz⟩ := hx (-y)
hm.1.1 <| (eq_top_iff_one _).2 <| sub_sub_cancel (z * -y * x + z) 1 ▸
M.sub_mem (by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one z, neg_mul, ← sub_eq_iff_eq_add.mpr df.symm, neg_sub,
sub_add_cancel]
exact M.mul_mem_left _ hi) <| him hz⟩
#align ideal.mem_jacobson_iff Ideal.mem_jacobson_iff
theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
∃ s, s * r - 1 ∈ I := by
cases' mem_jacobson_iff.1 h 1 with s hs
use s
simpa [mul_sub] using hs
#align ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson Ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M := by
use fun hI => ⟨{ J : Ideal R | I ≤ J ∧ J.IsMaximal }, ⟨fun _ hJ => Or.inl hJ.right, hI.symm⟩⟩
rintro ⟨M, hM, hInf⟩
refine le_antisymm (fun x hx => ?_) le_jacobson
rw [hInf, mem_sInf]
intro I hI
cases' hM I hI with is_max is_top
· exact (mem_sInf.1 hx) ⟨le_sInf_iff.1 (le_of_eq hInf) I hI, is_max⟩
· exact is_top.symm ▸ Submodule.mem_top
#align ideal.eq_jacobson_iff_Inf_maximal Ideal.eq_jacobson_iff_sInf_maximal
theorem eq_jacobson_iff_sInf_maximal' :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, ∀ (K : Ideal R), J < K → K = ⊤) ∧ I = sInf M :=
eq_jacobson_iff_sInf_maximal.trans
⟨fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ K hK =>
Or.recOn (hM.1 J hJ) (fun h => h.1.2 K hK) fun h => eq_top_iff.2 (le_of_lt (h ▸ hK)),
hM.2⟩⟩,
fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ =>
Or.recOn (Classical.em (J = ⊤)) (fun h => Or.inr h) fun h => Or.inl ⟨⟨h, hM.1 J hJ⟩⟩,
hM.2⟩⟩⟩
#align ideal.eq_jacobson_iff_Inf_maximal' Ideal.eq_jacobson_iff_sInf_maximal'
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M := by
constructor
· intro h x hx
erw [← h, mem_sInf] at hx
push_neg at hx
exact hx
· refine fun h => le_antisymm (fun x hx => ?_) le_jacobson
contrapose hx
erw [mem_sInf]
push_neg
exact h x hx
#align ideal.eq_jacobson_iff_not_mem Ideal.eq_jacobson_iff_not_mem
theorem map_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) :
RingHom.ker f ≤ I → map f I.jacobson = (map f I).jacobson := by
intro h
unfold Ideal.jacobson
-- porting note : dot notation for `RingHom.ker` does not work
have : ∀ J ∈ { J : Ideal R | I ≤ J ∧ J.IsMaximal }, RingHom.ker f ≤ J :=
fun J hJ => le_trans h hJ.left
refine Trans.trans (map_sInf hf this) (le_antisymm ?_ ?_)
· refine'
sInf_le_sInf fun J hJ =>
⟨comap f J, ⟨⟨le_comap_of_map_le hJ.1, _⟩, map_comap_of_surjective f hf J⟩⟩
haveI : J.IsMaximal := hJ.right
|
exact comap_isMaximal_of_surjective f hf
|
theorem map_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) :
RingHom.ker f ≤ I → map f I.jacobson = (map f I).jacobson := by
intro h
unfold Ideal.jacobson
-- porting note : dot notation for `RingHom.ker` does not work
have : ∀ J ∈ { J : Ideal R | I ≤ J ∧ J.IsMaximal }, RingHom.ker f ≤ J :=
fun J hJ => le_trans h hJ.left
refine Trans.trans (map_sInf hf this) (le_antisymm ?_ ?_)
· refine'
sInf_le_sInf fun J hJ =>
⟨comap f J, ⟨⟨le_comap_of_map_le hJ.1, _⟩, map_comap_of_surjective f hf J⟩⟩
haveI : J.IsMaximal := hJ.right
|
Mathlib.RingTheory.JacobsonIdeal.178_0.Lz0MgLQMj1bGzuN
|
theorem map_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) :
RingHom.ker f ≤ I → map f I.jacobson = (map f I).jacobson
|
Mathlib_RingTheory_JacobsonIdeal
|
case refine_2
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I : Ideal R
f : R →+* S
hf : Function.Surjective ⇑f
h : RingHom.ker f ≤ I
this : ∀ J ∈ {J | I ≤ J ∧ IsMaximal J}, RingHom.ker f ≤ J
⊢ sInf {J | map f I ≤ J ∧ IsMaximal J} ≤ sInf (map f '' {J | I ≤ J ∧ IsMaximal J})
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
/-!
# Jacobson radical
The Jacobson radical of a ring `R` is defined to be the intersection of all maximal ideals of `R`.
This is similar to how the nilradical is equal to the intersection of all prime ideals of `R`.
We can extend the idea of the nilradical to ideals of `R`,
by letting the radical of an ideal `I` be the intersection of prime ideals containing `I`.
Under this extension, the original nilradical is the radical of the zero ideal `⊥`.
Here we define the Jacobson radical of an ideal `I` in a similar way,
as the intersection of maximal ideals containing `I`.
## Main definitions
Let `R` be a commutative ring, and `I` be an ideal of `R`
* `Ideal.jacobson I` is the jacobson radical, i.e. the infimum of all maximal ideals containing I.
* `Ideal.IsLocal I` is the proposition that the jacobson radical of `I` is itself a maximal ideal
## Main statements
* `mem_jacobson_iff` gives a characterization of members of the jacobson of I
* `Ideal.isLocal_of_isMaximal_radical`: if the radical of I is maximal then so is the jacobson
radical
## Tags
Jacobson, Jacobson radical, Local Ideal
-/
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}
open Polynomial
section Jacobson
section Ring
variable [Ring R] [Ring S] {I : Ideal R}
/-- The Jacobson radical of `I` is the infimum of all maximal (left) ideals containing `I`. -/
def jacobson (I : Ideal R) : Ideal R :=
sInf { J : Ideal R | I ≤ J ∧ IsMaximal J }
#align ideal.jacobson Ideal.jacobson
theorem le_jacobson : I ≤ jacobson I := fun _ hx => mem_sInf.mpr fun _ hJ => hJ.left hx
#align ideal.le_jacobson Ideal.le_jacobson
@[simp]
theorem jacobson_idem : jacobson (jacobson I) = jacobson I :=
le_antisymm (sInf_le_sInf fun _ hJ => ⟨sInf_le hJ, hJ.2⟩) le_jacobson
#align ideal.jacobson_idem Ideal.jacobson_idem
@[simp]
theorem jacobson_top : jacobson (⊤ : Ideal R) = ⊤ :=
eq_top_iff.2 le_jacobson
#align ideal.jacobson_top Ideal.jacobson_top
@[simp]
theorem jacobson_eq_top_iff : jacobson I = ⊤ ↔ I = ⊤ :=
⟨fun H =>
by_contradiction fun hi => let ⟨M, hm, him⟩ := exists_le_maximal I hi
lt_top_iff_ne_top.1
(lt_of_le_of_lt (show jacobson I ≤ M from sInf_le ⟨him, hm⟩) <|
lt_top_iff_ne_top.2 hm.ne_top) H,
fun H => eq_top_iff.2 <| le_sInf fun _ ⟨hij, _⟩ => H ▸ hij⟩
#align ideal.jacobson_eq_top_iff Ideal.jacobson_eq_top_iff
theorem jacobson_eq_bot : jacobson I = ⊥ → I = ⊥ := fun h => eq_bot_iff.mpr (h ▸ le_jacobson)
#align ideal.jacobson_eq_bot Ideal.jacobson_eq_bot
theorem jacobson_eq_self_of_isMaximal [H : IsMaximal I] : I.jacobson = I :=
le_antisymm (sInf_le ⟨le_of_eq rfl, H⟩) le_jacobson
#align ideal.jacobson_eq_self_of_is_maximal Ideal.jacobson_eq_self_of_isMaximal
instance (priority := 100) jacobson.isMaximal [H : IsMaximal I] : IsMaximal (jacobson I) :=
⟨⟨fun htop => H.1.1 (jacobson_eq_top_iff.1 htop), fun _ hJ =>
H.1.2 _ (lt_of_le_of_lt le_jacobson hJ)⟩⟩
#align ideal.jacobson.is_maximal Ideal.jacobson.isMaximal
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I :=
⟨fun hx y =>
by_cases
(fun hxy : I ⊔ span {y * x + 1} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy)
let ⟨r, hr⟩ := mem_span_singleton'.1 hq
⟨r, by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one r (y * x), hr, ← hpq, ← neg_sub, add_sub_cancel]
exact I.neg_mem hpi⟩)
fun hxy : I ⊔ span {y * x + 1} ≠ ⊤ => let ⟨M, hm1, hm2⟩ := exists_le_maximal _ hxy
suffices x ∉ M from (this <| mem_sInf.1 hx ⟨le_trans le_sup_left hm2, hm1⟩).elim
fun hxm => hm1.1.1 <| (eq_top_iff_one _).2 <| add_sub_cancel' (y * x) 1 ▸
M.sub_mem (le_sup_right.trans hm2 <| subset_span rfl) (M.mul_mem_left _ hxm),
fun hx => mem_sInf.2 fun M ⟨him, hm⟩ => by_contradiction fun hxm =>
let ⟨y, i, hi, df⟩ := hm.exists_inv hxm
let ⟨z, hz⟩ := hx (-y)
hm.1.1 <| (eq_top_iff_one _).2 <| sub_sub_cancel (z * -y * x + z) 1 ▸
M.sub_mem (by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one z, neg_mul, ← sub_eq_iff_eq_add.mpr df.symm, neg_sub,
sub_add_cancel]
exact M.mul_mem_left _ hi) <| him hz⟩
#align ideal.mem_jacobson_iff Ideal.mem_jacobson_iff
theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
∃ s, s * r - 1 ∈ I := by
cases' mem_jacobson_iff.1 h 1 with s hs
use s
simpa [mul_sub] using hs
#align ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson Ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M := by
use fun hI => ⟨{ J : Ideal R | I ≤ J ∧ J.IsMaximal }, ⟨fun _ hJ => Or.inl hJ.right, hI.symm⟩⟩
rintro ⟨M, hM, hInf⟩
refine le_antisymm (fun x hx => ?_) le_jacobson
rw [hInf, mem_sInf]
intro I hI
cases' hM I hI with is_max is_top
· exact (mem_sInf.1 hx) ⟨le_sInf_iff.1 (le_of_eq hInf) I hI, is_max⟩
· exact is_top.symm ▸ Submodule.mem_top
#align ideal.eq_jacobson_iff_Inf_maximal Ideal.eq_jacobson_iff_sInf_maximal
theorem eq_jacobson_iff_sInf_maximal' :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, ∀ (K : Ideal R), J < K → K = ⊤) ∧ I = sInf M :=
eq_jacobson_iff_sInf_maximal.trans
⟨fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ K hK =>
Or.recOn (hM.1 J hJ) (fun h => h.1.2 K hK) fun h => eq_top_iff.2 (le_of_lt (h ▸ hK)),
hM.2⟩⟩,
fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ =>
Or.recOn (Classical.em (J = ⊤)) (fun h => Or.inr h) fun h => Or.inl ⟨⟨h, hM.1 J hJ⟩⟩,
hM.2⟩⟩⟩
#align ideal.eq_jacobson_iff_Inf_maximal' Ideal.eq_jacobson_iff_sInf_maximal'
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M := by
constructor
· intro h x hx
erw [← h, mem_sInf] at hx
push_neg at hx
exact hx
· refine fun h => le_antisymm (fun x hx => ?_) le_jacobson
contrapose hx
erw [mem_sInf]
push_neg
exact h x hx
#align ideal.eq_jacobson_iff_not_mem Ideal.eq_jacobson_iff_not_mem
theorem map_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) :
RingHom.ker f ≤ I → map f I.jacobson = (map f I).jacobson := by
intro h
unfold Ideal.jacobson
-- porting note : dot notation for `RingHom.ker` does not work
have : ∀ J ∈ { J : Ideal R | I ≤ J ∧ J.IsMaximal }, RingHom.ker f ≤ J :=
fun J hJ => le_trans h hJ.left
refine Trans.trans (map_sInf hf this) (le_antisymm ?_ ?_)
· refine'
sInf_le_sInf fun J hJ =>
⟨comap f J, ⟨⟨le_comap_of_map_le hJ.1, _⟩, map_comap_of_surjective f hf J⟩⟩
haveI : J.IsMaximal := hJ.right
exact comap_isMaximal_of_surjective f hf
·
|
refine' sInf_le_sInf_of_subset_insert_top fun j hj => hj.recOn fun J hJ => _
|
theorem map_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) :
RingHom.ker f ≤ I → map f I.jacobson = (map f I).jacobson := by
intro h
unfold Ideal.jacobson
-- porting note : dot notation for `RingHom.ker` does not work
have : ∀ J ∈ { J : Ideal R | I ≤ J ∧ J.IsMaximal }, RingHom.ker f ≤ J :=
fun J hJ => le_trans h hJ.left
refine Trans.trans (map_sInf hf this) (le_antisymm ?_ ?_)
· refine'
sInf_le_sInf fun J hJ =>
⟨comap f J, ⟨⟨le_comap_of_map_le hJ.1, _⟩, map_comap_of_surjective f hf J⟩⟩
haveI : J.IsMaximal := hJ.right
exact comap_isMaximal_of_surjective f hf
·
|
Mathlib.RingTheory.JacobsonIdeal.178_0.Lz0MgLQMj1bGzuN
|
theorem map_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) :
RingHom.ker f ≤ I → map f I.jacobson = (map f I).jacobson
|
Mathlib_RingTheory_JacobsonIdeal
|
case refine_2
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I : Ideal R
f : R →+* S
hf : Function.Surjective ⇑f
h : RingHom.ker f ≤ I
this : ∀ J ∈ {J | I ≤ J ∧ IsMaximal J}, RingHom.ker f ≤ J
j : Ideal S
hj : j ∈ map f '' {J | I ≤ J ∧ IsMaximal J}
J : Ideal R
hJ : J ∈ {J | I ≤ J ∧ IsMaximal J} ∧ map f J = j
⊢ j ∈ insert ⊤ {J | map f I ≤ J ∧ IsMaximal J}
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
/-!
# Jacobson radical
The Jacobson radical of a ring `R` is defined to be the intersection of all maximal ideals of `R`.
This is similar to how the nilradical is equal to the intersection of all prime ideals of `R`.
We can extend the idea of the nilradical to ideals of `R`,
by letting the radical of an ideal `I` be the intersection of prime ideals containing `I`.
Under this extension, the original nilradical is the radical of the zero ideal `⊥`.
Here we define the Jacobson radical of an ideal `I` in a similar way,
as the intersection of maximal ideals containing `I`.
## Main definitions
Let `R` be a commutative ring, and `I` be an ideal of `R`
* `Ideal.jacobson I` is the jacobson radical, i.e. the infimum of all maximal ideals containing I.
* `Ideal.IsLocal I` is the proposition that the jacobson radical of `I` is itself a maximal ideal
## Main statements
* `mem_jacobson_iff` gives a characterization of members of the jacobson of I
* `Ideal.isLocal_of_isMaximal_radical`: if the radical of I is maximal then so is the jacobson
radical
## Tags
Jacobson, Jacobson radical, Local Ideal
-/
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}
open Polynomial
section Jacobson
section Ring
variable [Ring R] [Ring S] {I : Ideal R}
/-- The Jacobson radical of `I` is the infimum of all maximal (left) ideals containing `I`. -/
def jacobson (I : Ideal R) : Ideal R :=
sInf { J : Ideal R | I ≤ J ∧ IsMaximal J }
#align ideal.jacobson Ideal.jacobson
theorem le_jacobson : I ≤ jacobson I := fun _ hx => mem_sInf.mpr fun _ hJ => hJ.left hx
#align ideal.le_jacobson Ideal.le_jacobson
@[simp]
theorem jacobson_idem : jacobson (jacobson I) = jacobson I :=
le_antisymm (sInf_le_sInf fun _ hJ => ⟨sInf_le hJ, hJ.2⟩) le_jacobson
#align ideal.jacobson_idem Ideal.jacobson_idem
@[simp]
theorem jacobson_top : jacobson (⊤ : Ideal R) = ⊤ :=
eq_top_iff.2 le_jacobson
#align ideal.jacobson_top Ideal.jacobson_top
@[simp]
theorem jacobson_eq_top_iff : jacobson I = ⊤ ↔ I = ⊤ :=
⟨fun H =>
by_contradiction fun hi => let ⟨M, hm, him⟩ := exists_le_maximal I hi
lt_top_iff_ne_top.1
(lt_of_le_of_lt (show jacobson I ≤ M from sInf_le ⟨him, hm⟩) <|
lt_top_iff_ne_top.2 hm.ne_top) H,
fun H => eq_top_iff.2 <| le_sInf fun _ ⟨hij, _⟩ => H ▸ hij⟩
#align ideal.jacobson_eq_top_iff Ideal.jacobson_eq_top_iff
theorem jacobson_eq_bot : jacobson I = ⊥ → I = ⊥ := fun h => eq_bot_iff.mpr (h ▸ le_jacobson)
#align ideal.jacobson_eq_bot Ideal.jacobson_eq_bot
theorem jacobson_eq_self_of_isMaximal [H : IsMaximal I] : I.jacobson = I :=
le_antisymm (sInf_le ⟨le_of_eq rfl, H⟩) le_jacobson
#align ideal.jacobson_eq_self_of_is_maximal Ideal.jacobson_eq_self_of_isMaximal
instance (priority := 100) jacobson.isMaximal [H : IsMaximal I] : IsMaximal (jacobson I) :=
⟨⟨fun htop => H.1.1 (jacobson_eq_top_iff.1 htop), fun _ hJ =>
H.1.2 _ (lt_of_le_of_lt le_jacobson hJ)⟩⟩
#align ideal.jacobson.is_maximal Ideal.jacobson.isMaximal
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I :=
⟨fun hx y =>
by_cases
(fun hxy : I ⊔ span {y * x + 1} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy)
let ⟨r, hr⟩ := mem_span_singleton'.1 hq
⟨r, by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one r (y * x), hr, ← hpq, ← neg_sub, add_sub_cancel]
exact I.neg_mem hpi⟩)
fun hxy : I ⊔ span {y * x + 1} ≠ ⊤ => let ⟨M, hm1, hm2⟩ := exists_le_maximal _ hxy
suffices x ∉ M from (this <| mem_sInf.1 hx ⟨le_trans le_sup_left hm2, hm1⟩).elim
fun hxm => hm1.1.1 <| (eq_top_iff_one _).2 <| add_sub_cancel' (y * x) 1 ▸
M.sub_mem (le_sup_right.trans hm2 <| subset_span rfl) (M.mul_mem_left _ hxm),
fun hx => mem_sInf.2 fun M ⟨him, hm⟩ => by_contradiction fun hxm =>
let ⟨y, i, hi, df⟩ := hm.exists_inv hxm
let ⟨z, hz⟩ := hx (-y)
hm.1.1 <| (eq_top_iff_one _).2 <| sub_sub_cancel (z * -y * x + z) 1 ▸
M.sub_mem (by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one z, neg_mul, ← sub_eq_iff_eq_add.mpr df.symm, neg_sub,
sub_add_cancel]
exact M.mul_mem_left _ hi) <| him hz⟩
#align ideal.mem_jacobson_iff Ideal.mem_jacobson_iff
theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
∃ s, s * r - 1 ∈ I := by
cases' mem_jacobson_iff.1 h 1 with s hs
use s
simpa [mul_sub] using hs
#align ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson Ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M := by
use fun hI => ⟨{ J : Ideal R | I ≤ J ∧ J.IsMaximal }, ⟨fun _ hJ => Or.inl hJ.right, hI.symm⟩⟩
rintro ⟨M, hM, hInf⟩
refine le_antisymm (fun x hx => ?_) le_jacobson
rw [hInf, mem_sInf]
intro I hI
cases' hM I hI with is_max is_top
· exact (mem_sInf.1 hx) ⟨le_sInf_iff.1 (le_of_eq hInf) I hI, is_max⟩
· exact is_top.symm ▸ Submodule.mem_top
#align ideal.eq_jacobson_iff_Inf_maximal Ideal.eq_jacobson_iff_sInf_maximal
theorem eq_jacobson_iff_sInf_maximal' :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, ∀ (K : Ideal R), J < K → K = ⊤) ∧ I = sInf M :=
eq_jacobson_iff_sInf_maximal.trans
⟨fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ K hK =>
Or.recOn (hM.1 J hJ) (fun h => h.1.2 K hK) fun h => eq_top_iff.2 (le_of_lt (h ▸ hK)),
hM.2⟩⟩,
fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ =>
Or.recOn (Classical.em (J = ⊤)) (fun h => Or.inr h) fun h => Or.inl ⟨⟨h, hM.1 J hJ⟩⟩,
hM.2⟩⟩⟩
#align ideal.eq_jacobson_iff_Inf_maximal' Ideal.eq_jacobson_iff_sInf_maximal'
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M := by
constructor
· intro h x hx
erw [← h, mem_sInf] at hx
push_neg at hx
exact hx
· refine fun h => le_antisymm (fun x hx => ?_) le_jacobson
contrapose hx
erw [mem_sInf]
push_neg
exact h x hx
#align ideal.eq_jacobson_iff_not_mem Ideal.eq_jacobson_iff_not_mem
theorem map_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) :
RingHom.ker f ≤ I → map f I.jacobson = (map f I).jacobson := by
intro h
unfold Ideal.jacobson
-- porting note : dot notation for `RingHom.ker` does not work
have : ∀ J ∈ { J : Ideal R | I ≤ J ∧ J.IsMaximal }, RingHom.ker f ≤ J :=
fun J hJ => le_trans h hJ.left
refine Trans.trans (map_sInf hf this) (le_antisymm ?_ ?_)
· refine'
sInf_le_sInf fun J hJ =>
⟨comap f J, ⟨⟨le_comap_of_map_le hJ.1, _⟩, map_comap_of_surjective f hf J⟩⟩
haveI : J.IsMaximal := hJ.right
exact comap_isMaximal_of_surjective f hf
· refine' sInf_le_sInf_of_subset_insert_top fun j hj => hj.recOn fun J hJ => _
|
rw [← hJ.2]
|
theorem map_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) :
RingHom.ker f ≤ I → map f I.jacobson = (map f I).jacobson := by
intro h
unfold Ideal.jacobson
-- porting note : dot notation for `RingHom.ker` does not work
have : ∀ J ∈ { J : Ideal R | I ≤ J ∧ J.IsMaximal }, RingHom.ker f ≤ J :=
fun J hJ => le_trans h hJ.left
refine Trans.trans (map_sInf hf this) (le_antisymm ?_ ?_)
· refine'
sInf_le_sInf fun J hJ =>
⟨comap f J, ⟨⟨le_comap_of_map_le hJ.1, _⟩, map_comap_of_surjective f hf J⟩⟩
haveI : J.IsMaximal := hJ.right
exact comap_isMaximal_of_surjective f hf
· refine' sInf_le_sInf_of_subset_insert_top fun j hj => hj.recOn fun J hJ => _
|
Mathlib.RingTheory.JacobsonIdeal.178_0.Lz0MgLQMj1bGzuN
|
theorem map_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) :
RingHom.ker f ≤ I → map f I.jacobson = (map f I).jacobson
|
Mathlib_RingTheory_JacobsonIdeal
|
case refine_2
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I : Ideal R
f : R →+* S
hf : Function.Surjective ⇑f
h : RingHom.ker f ≤ I
this : ∀ J ∈ {J | I ≤ J ∧ IsMaximal J}, RingHom.ker f ≤ J
j : Ideal S
hj : j ∈ map f '' {J | I ≤ J ∧ IsMaximal J}
J : Ideal R
hJ : J ∈ {J | I ≤ J ∧ IsMaximal J} ∧ map f J = j
⊢ map f J ∈ insert ⊤ {J | map f I ≤ J ∧ IsMaximal J}
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
/-!
# Jacobson radical
The Jacobson radical of a ring `R` is defined to be the intersection of all maximal ideals of `R`.
This is similar to how the nilradical is equal to the intersection of all prime ideals of `R`.
We can extend the idea of the nilradical to ideals of `R`,
by letting the radical of an ideal `I` be the intersection of prime ideals containing `I`.
Under this extension, the original nilradical is the radical of the zero ideal `⊥`.
Here we define the Jacobson radical of an ideal `I` in a similar way,
as the intersection of maximal ideals containing `I`.
## Main definitions
Let `R` be a commutative ring, and `I` be an ideal of `R`
* `Ideal.jacobson I` is the jacobson radical, i.e. the infimum of all maximal ideals containing I.
* `Ideal.IsLocal I` is the proposition that the jacobson radical of `I` is itself a maximal ideal
## Main statements
* `mem_jacobson_iff` gives a characterization of members of the jacobson of I
* `Ideal.isLocal_of_isMaximal_radical`: if the radical of I is maximal then so is the jacobson
radical
## Tags
Jacobson, Jacobson radical, Local Ideal
-/
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}
open Polynomial
section Jacobson
section Ring
variable [Ring R] [Ring S] {I : Ideal R}
/-- The Jacobson radical of `I` is the infimum of all maximal (left) ideals containing `I`. -/
def jacobson (I : Ideal R) : Ideal R :=
sInf { J : Ideal R | I ≤ J ∧ IsMaximal J }
#align ideal.jacobson Ideal.jacobson
theorem le_jacobson : I ≤ jacobson I := fun _ hx => mem_sInf.mpr fun _ hJ => hJ.left hx
#align ideal.le_jacobson Ideal.le_jacobson
@[simp]
theorem jacobson_idem : jacobson (jacobson I) = jacobson I :=
le_antisymm (sInf_le_sInf fun _ hJ => ⟨sInf_le hJ, hJ.2⟩) le_jacobson
#align ideal.jacobson_idem Ideal.jacobson_idem
@[simp]
theorem jacobson_top : jacobson (⊤ : Ideal R) = ⊤ :=
eq_top_iff.2 le_jacobson
#align ideal.jacobson_top Ideal.jacobson_top
@[simp]
theorem jacobson_eq_top_iff : jacobson I = ⊤ ↔ I = ⊤ :=
⟨fun H =>
by_contradiction fun hi => let ⟨M, hm, him⟩ := exists_le_maximal I hi
lt_top_iff_ne_top.1
(lt_of_le_of_lt (show jacobson I ≤ M from sInf_le ⟨him, hm⟩) <|
lt_top_iff_ne_top.2 hm.ne_top) H,
fun H => eq_top_iff.2 <| le_sInf fun _ ⟨hij, _⟩ => H ▸ hij⟩
#align ideal.jacobson_eq_top_iff Ideal.jacobson_eq_top_iff
theorem jacobson_eq_bot : jacobson I = ⊥ → I = ⊥ := fun h => eq_bot_iff.mpr (h ▸ le_jacobson)
#align ideal.jacobson_eq_bot Ideal.jacobson_eq_bot
theorem jacobson_eq_self_of_isMaximal [H : IsMaximal I] : I.jacobson = I :=
le_antisymm (sInf_le ⟨le_of_eq rfl, H⟩) le_jacobson
#align ideal.jacobson_eq_self_of_is_maximal Ideal.jacobson_eq_self_of_isMaximal
instance (priority := 100) jacobson.isMaximal [H : IsMaximal I] : IsMaximal (jacobson I) :=
⟨⟨fun htop => H.1.1 (jacobson_eq_top_iff.1 htop), fun _ hJ =>
H.1.2 _ (lt_of_le_of_lt le_jacobson hJ)⟩⟩
#align ideal.jacobson.is_maximal Ideal.jacobson.isMaximal
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I :=
⟨fun hx y =>
by_cases
(fun hxy : I ⊔ span {y * x + 1} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy)
let ⟨r, hr⟩ := mem_span_singleton'.1 hq
⟨r, by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one r (y * x), hr, ← hpq, ← neg_sub, add_sub_cancel]
exact I.neg_mem hpi⟩)
fun hxy : I ⊔ span {y * x + 1} ≠ ⊤ => let ⟨M, hm1, hm2⟩ := exists_le_maximal _ hxy
suffices x ∉ M from (this <| mem_sInf.1 hx ⟨le_trans le_sup_left hm2, hm1⟩).elim
fun hxm => hm1.1.1 <| (eq_top_iff_one _).2 <| add_sub_cancel' (y * x) 1 ▸
M.sub_mem (le_sup_right.trans hm2 <| subset_span rfl) (M.mul_mem_left _ hxm),
fun hx => mem_sInf.2 fun M ⟨him, hm⟩ => by_contradiction fun hxm =>
let ⟨y, i, hi, df⟩ := hm.exists_inv hxm
let ⟨z, hz⟩ := hx (-y)
hm.1.1 <| (eq_top_iff_one _).2 <| sub_sub_cancel (z * -y * x + z) 1 ▸
M.sub_mem (by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one z, neg_mul, ← sub_eq_iff_eq_add.mpr df.symm, neg_sub,
sub_add_cancel]
exact M.mul_mem_left _ hi) <| him hz⟩
#align ideal.mem_jacobson_iff Ideal.mem_jacobson_iff
theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
∃ s, s * r - 1 ∈ I := by
cases' mem_jacobson_iff.1 h 1 with s hs
use s
simpa [mul_sub] using hs
#align ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson Ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M := by
use fun hI => ⟨{ J : Ideal R | I ≤ J ∧ J.IsMaximal }, ⟨fun _ hJ => Or.inl hJ.right, hI.symm⟩⟩
rintro ⟨M, hM, hInf⟩
refine le_antisymm (fun x hx => ?_) le_jacobson
rw [hInf, mem_sInf]
intro I hI
cases' hM I hI with is_max is_top
· exact (mem_sInf.1 hx) ⟨le_sInf_iff.1 (le_of_eq hInf) I hI, is_max⟩
· exact is_top.symm ▸ Submodule.mem_top
#align ideal.eq_jacobson_iff_Inf_maximal Ideal.eq_jacobson_iff_sInf_maximal
theorem eq_jacobson_iff_sInf_maximal' :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, ∀ (K : Ideal R), J < K → K = ⊤) ∧ I = sInf M :=
eq_jacobson_iff_sInf_maximal.trans
⟨fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ K hK =>
Or.recOn (hM.1 J hJ) (fun h => h.1.2 K hK) fun h => eq_top_iff.2 (le_of_lt (h ▸ hK)),
hM.2⟩⟩,
fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ =>
Or.recOn (Classical.em (J = ⊤)) (fun h => Or.inr h) fun h => Or.inl ⟨⟨h, hM.1 J hJ⟩⟩,
hM.2⟩⟩⟩
#align ideal.eq_jacobson_iff_Inf_maximal' Ideal.eq_jacobson_iff_sInf_maximal'
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M := by
constructor
· intro h x hx
erw [← h, mem_sInf] at hx
push_neg at hx
exact hx
· refine fun h => le_antisymm (fun x hx => ?_) le_jacobson
contrapose hx
erw [mem_sInf]
push_neg
exact h x hx
#align ideal.eq_jacobson_iff_not_mem Ideal.eq_jacobson_iff_not_mem
theorem map_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) :
RingHom.ker f ≤ I → map f I.jacobson = (map f I).jacobson := by
intro h
unfold Ideal.jacobson
-- porting note : dot notation for `RingHom.ker` does not work
have : ∀ J ∈ { J : Ideal R | I ≤ J ∧ J.IsMaximal }, RingHom.ker f ≤ J :=
fun J hJ => le_trans h hJ.left
refine Trans.trans (map_sInf hf this) (le_antisymm ?_ ?_)
· refine'
sInf_le_sInf fun J hJ =>
⟨comap f J, ⟨⟨le_comap_of_map_le hJ.1, _⟩, map_comap_of_surjective f hf J⟩⟩
haveI : J.IsMaximal := hJ.right
exact comap_isMaximal_of_surjective f hf
· refine' sInf_le_sInf_of_subset_insert_top fun j hj => hj.recOn fun J hJ => _
rw [← hJ.2]
|
cases' map_eq_top_or_isMaximal_of_surjective f hf hJ.left.right with htop hmax
|
theorem map_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) :
RingHom.ker f ≤ I → map f I.jacobson = (map f I).jacobson := by
intro h
unfold Ideal.jacobson
-- porting note : dot notation for `RingHom.ker` does not work
have : ∀ J ∈ { J : Ideal R | I ≤ J ∧ J.IsMaximal }, RingHom.ker f ≤ J :=
fun J hJ => le_trans h hJ.left
refine Trans.trans (map_sInf hf this) (le_antisymm ?_ ?_)
· refine'
sInf_le_sInf fun J hJ =>
⟨comap f J, ⟨⟨le_comap_of_map_le hJ.1, _⟩, map_comap_of_surjective f hf J⟩⟩
haveI : J.IsMaximal := hJ.right
exact comap_isMaximal_of_surjective f hf
· refine' sInf_le_sInf_of_subset_insert_top fun j hj => hj.recOn fun J hJ => _
rw [← hJ.2]
|
Mathlib.RingTheory.JacobsonIdeal.178_0.Lz0MgLQMj1bGzuN
|
theorem map_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) :
RingHom.ker f ≤ I → map f I.jacobson = (map f I).jacobson
|
Mathlib_RingTheory_JacobsonIdeal
|
case refine_2.inl
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I : Ideal R
f : R →+* S
hf : Function.Surjective ⇑f
h : RingHom.ker f ≤ I
this : ∀ J ∈ {J | I ≤ J ∧ IsMaximal J}, RingHom.ker f ≤ J
j : Ideal S
hj : j ∈ map f '' {J | I ≤ J ∧ IsMaximal J}
J : Ideal R
hJ : J ∈ {J | I ≤ J ∧ IsMaximal J} ∧ map f J = j
htop : map f J = ⊤
⊢ map f J ∈ insert ⊤ {J | map f I ≤ J ∧ IsMaximal J}
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
/-!
# Jacobson radical
The Jacobson radical of a ring `R` is defined to be the intersection of all maximal ideals of `R`.
This is similar to how the nilradical is equal to the intersection of all prime ideals of `R`.
We can extend the idea of the nilradical to ideals of `R`,
by letting the radical of an ideal `I` be the intersection of prime ideals containing `I`.
Under this extension, the original nilradical is the radical of the zero ideal `⊥`.
Here we define the Jacobson radical of an ideal `I` in a similar way,
as the intersection of maximal ideals containing `I`.
## Main definitions
Let `R` be a commutative ring, and `I` be an ideal of `R`
* `Ideal.jacobson I` is the jacobson radical, i.e. the infimum of all maximal ideals containing I.
* `Ideal.IsLocal I` is the proposition that the jacobson radical of `I` is itself a maximal ideal
## Main statements
* `mem_jacobson_iff` gives a characterization of members of the jacobson of I
* `Ideal.isLocal_of_isMaximal_radical`: if the radical of I is maximal then so is the jacobson
radical
## Tags
Jacobson, Jacobson radical, Local Ideal
-/
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}
open Polynomial
section Jacobson
section Ring
variable [Ring R] [Ring S] {I : Ideal R}
/-- The Jacobson radical of `I` is the infimum of all maximal (left) ideals containing `I`. -/
def jacobson (I : Ideal R) : Ideal R :=
sInf { J : Ideal R | I ≤ J ∧ IsMaximal J }
#align ideal.jacobson Ideal.jacobson
theorem le_jacobson : I ≤ jacobson I := fun _ hx => mem_sInf.mpr fun _ hJ => hJ.left hx
#align ideal.le_jacobson Ideal.le_jacobson
@[simp]
theorem jacobson_idem : jacobson (jacobson I) = jacobson I :=
le_antisymm (sInf_le_sInf fun _ hJ => ⟨sInf_le hJ, hJ.2⟩) le_jacobson
#align ideal.jacobson_idem Ideal.jacobson_idem
@[simp]
theorem jacobson_top : jacobson (⊤ : Ideal R) = ⊤ :=
eq_top_iff.2 le_jacobson
#align ideal.jacobson_top Ideal.jacobson_top
@[simp]
theorem jacobson_eq_top_iff : jacobson I = ⊤ ↔ I = ⊤ :=
⟨fun H =>
by_contradiction fun hi => let ⟨M, hm, him⟩ := exists_le_maximal I hi
lt_top_iff_ne_top.1
(lt_of_le_of_lt (show jacobson I ≤ M from sInf_le ⟨him, hm⟩) <|
lt_top_iff_ne_top.2 hm.ne_top) H,
fun H => eq_top_iff.2 <| le_sInf fun _ ⟨hij, _⟩ => H ▸ hij⟩
#align ideal.jacobson_eq_top_iff Ideal.jacobson_eq_top_iff
theorem jacobson_eq_bot : jacobson I = ⊥ → I = ⊥ := fun h => eq_bot_iff.mpr (h ▸ le_jacobson)
#align ideal.jacobson_eq_bot Ideal.jacobson_eq_bot
theorem jacobson_eq_self_of_isMaximal [H : IsMaximal I] : I.jacobson = I :=
le_antisymm (sInf_le ⟨le_of_eq rfl, H⟩) le_jacobson
#align ideal.jacobson_eq_self_of_is_maximal Ideal.jacobson_eq_self_of_isMaximal
instance (priority := 100) jacobson.isMaximal [H : IsMaximal I] : IsMaximal (jacobson I) :=
⟨⟨fun htop => H.1.1 (jacobson_eq_top_iff.1 htop), fun _ hJ =>
H.1.2 _ (lt_of_le_of_lt le_jacobson hJ)⟩⟩
#align ideal.jacobson.is_maximal Ideal.jacobson.isMaximal
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I :=
⟨fun hx y =>
by_cases
(fun hxy : I ⊔ span {y * x + 1} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy)
let ⟨r, hr⟩ := mem_span_singleton'.1 hq
⟨r, by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one r (y * x), hr, ← hpq, ← neg_sub, add_sub_cancel]
exact I.neg_mem hpi⟩)
fun hxy : I ⊔ span {y * x + 1} ≠ ⊤ => let ⟨M, hm1, hm2⟩ := exists_le_maximal _ hxy
suffices x ∉ M from (this <| mem_sInf.1 hx ⟨le_trans le_sup_left hm2, hm1⟩).elim
fun hxm => hm1.1.1 <| (eq_top_iff_one _).2 <| add_sub_cancel' (y * x) 1 ▸
M.sub_mem (le_sup_right.trans hm2 <| subset_span rfl) (M.mul_mem_left _ hxm),
fun hx => mem_sInf.2 fun M ⟨him, hm⟩ => by_contradiction fun hxm =>
let ⟨y, i, hi, df⟩ := hm.exists_inv hxm
let ⟨z, hz⟩ := hx (-y)
hm.1.1 <| (eq_top_iff_one _).2 <| sub_sub_cancel (z * -y * x + z) 1 ▸
M.sub_mem (by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one z, neg_mul, ← sub_eq_iff_eq_add.mpr df.symm, neg_sub,
sub_add_cancel]
exact M.mul_mem_left _ hi) <| him hz⟩
#align ideal.mem_jacobson_iff Ideal.mem_jacobson_iff
theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
∃ s, s * r - 1 ∈ I := by
cases' mem_jacobson_iff.1 h 1 with s hs
use s
simpa [mul_sub] using hs
#align ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson Ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M := by
use fun hI => ⟨{ J : Ideal R | I ≤ J ∧ J.IsMaximal }, ⟨fun _ hJ => Or.inl hJ.right, hI.symm⟩⟩
rintro ⟨M, hM, hInf⟩
refine le_antisymm (fun x hx => ?_) le_jacobson
rw [hInf, mem_sInf]
intro I hI
cases' hM I hI with is_max is_top
· exact (mem_sInf.1 hx) ⟨le_sInf_iff.1 (le_of_eq hInf) I hI, is_max⟩
· exact is_top.symm ▸ Submodule.mem_top
#align ideal.eq_jacobson_iff_Inf_maximal Ideal.eq_jacobson_iff_sInf_maximal
theorem eq_jacobson_iff_sInf_maximal' :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, ∀ (K : Ideal R), J < K → K = ⊤) ∧ I = sInf M :=
eq_jacobson_iff_sInf_maximal.trans
⟨fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ K hK =>
Or.recOn (hM.1 J hJ) (fun h => h.1.2 K hK) fun h => eq_top_iff.2 (le_of_lt (h ▸ hK)),
hM.2⟩⟩,
fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ =>
Or.recOn (Classical.em (J = ⊤)) (fun h => Or.inr h) fun h => Or.inl ⟨⟨h, hM.1 J hJ⟩⟩,
hM.2⟩⟩⟩
#align ideal.eq_jacobson_iff_Inf_maximal' Ideal.eq_jacobson_iff_sInf_maximal'
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M := by
constructor
· intro h x hx
erw [← h, mem_sInf] at hx
push_neg at hx
exact hx
· refine fun h => le_antisymm (fun x hx => ?_) le_jacobson
contrapose hx
erw [mem_sInf]
push_neg
exact h x hx
#align ideal.eq_jacobson_iff_not_mem Ideal.eq_jacobson_iff_not_mem
theorem map_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) :
RingHom.ker f ≤ I → map f I.jacobson = (map f I).jacobson := by
intro h
unfold Ideal.jacobson
-- porting note : dot notation for `RingHom.ker` does not work
have : ∀ J ∈ { J : Ideal R | I ≤ J ∧ J.IsMaximal }, RingHom.ker f ≤ J :=
fun J hJ => le_trans h hJ.left
refine Trans.trans (map_sInf hf this) (le_antisymm ?_ ?_)
· refine'
sInf_le_sInf fun J hJ =>
⟨comap f J, ⟨⟨le_comap_of_map_le hJ.1, _⟩, map_comap_of_surjective f hf J⟩⟩
haveI : J.IsMaximal := hJ.right
exact comap_isMaximal_of_surjective f hf
· refine' sInf_le_sInf_of_subset_insert_top fun j hj => hj.recOn fun J hJ => _
rw [← hJ.2]
cases' map_eq_top_or_isMaximal_of_surjective f hf hJ.left.right with htop hmax
·
|
exact htop.symm ▸ Set.mem_insert ⊤ _
|
theorem map_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) :
RingHom.ker f ≤ I → map f I.jacobson = (map f I).jacobson := by
intro h
unfold Ideal.jacobson
-- porting note : dot notation for `RingHom.ker` does not work
have : ∀ J ∈ { J : Ideal R | I ≤ J ∧ J.IsMaximal }, RingHom.ker f ≤ J :=
fun J hJ => le_trans h hJ.left
refine Trans.trans (map_sInf hf this) (le_antisymm ?_ ?_)
· refine'
sInf_le_sInf fun J hJ =>
⟨comap f J, ⟨⟨le_comap_of_map_le hJ.1, _⟩, map_comap_of_surjective f hf J⟩⟩
haveI : J.IsMaximal := hJ.right
exact comap_isMaximal_of_surjective f hf
· refine' sInf_le_sInf_of_subset_insert_top fun j hj => hj.recOn fun J hJ => _
rw [← hJ.2]
cases' map_eq_top_or_isMaximal_of_surjective f hf hJ.left.right with htop hmax
·
|
Mathlib.RingTheory.JacobsonIdeal.178_0.Lz0MgLQMj1bGzuN
|
theorem map_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) :
RingHom.ker f ≤ I → map f I.jacobson = (map f I).jacobson
|
Mathlib_RingTheory_JacobsonIdeal
|
case refine_2.inr
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I : Ideal R
f : R →+* S
hf : Function.Surjective ⇑f
h : RingHom.ker f ≤ I
this : ∀ J ∈ {J | I ≤ J ∧ IsMaximal J}, RingHom.ker f ≤ J
j : Ideal S
hj : j ∈ map f '' {J | I ≤ J ∧ IsMaximal J}
J : Ideal R
hJ : J ∈ {J | I ≤ J ∧ IsMaximal J} ∧ map f J = j
hmax : IsMaximal (map f J)
⊢ map f J ∈ insert ⊤ {J | map f I ≤ J ∧ IsMaximal J}
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
/-!
# Jacobson radical
The Jacobson radical of a ring `R` is defined to be the intersection of all maximal ideals of `R`.
This is similar to how the nilradical is equal to the intersection of all prime ideals of `R`.
We can extend the idea of the nilradical to ideals of `R`,
by letting the radical of an ideal `I` be the intersection of prime ideals containing `I`.
Under this extension, the original nilradical is the radical of the zero ideal `⊥`.
Here we define the Jacobson radical of an ideal `I` in a similar way,
as the intersection of maximal ideals containing `I`.
## Main definitions
Let `R` be a commutative ring, and `I` be an ideal of `R`
* `Ideal.jacobson I` is the jacobson radical, i.e. the infimum of all maximal ideals containing I.
* `Ideal.IsLocal I` is the proposition that the jacobson radical of `I` is itself a maximal ideal
## Main statements
* `mem_jacobson_iff` gives a characterization of members of the jacobson of I
* `Ideal.isLocal_of_isMaximal_radical`: if the radical of I is maximal then so is the jacobson
radical
## Tags
Jacobson, Jacobson radical, Local Ideal
-/
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}
open Polynomial
section Jacobson
section Ring
variable [Ring R] [Ring S] {I : Ideal R}
/-- The Jacobson radical of `I` is the infimum of all maximal (left) ideals containing `I`. -/
def jacobson (I : Ideal R) : Ideal R :=
sInf { J : Ideal R | I ≤ J ∧ IsMaximal J }
#align ideal.jacobson Ideal.jacobson
theorem le_jacobson : I ≤ jacobson I := fun _ hx => mem_sInf.mpr fun _ hJ => hJ.left hx
#align ideal.le_jacobson Ideal.le_jacobson
@[simp]
theorem jacobson_idem : jacobson (jacobson I) = jacobson I :=
le_antisymm (sInf_le_sInf fun _ hJ => ⟨sInf_le hJ, hJ.2⟩) le_jacobson
#align ideal.jacobson_idem Ideal.jacobson_idem
@[simp]
theorem jacobson_top : jacobson (⊤ : Ideal R) = ⊤ :=
eq_top_iff.2 le_jacobson
#align ideal.jacobson_top Ideal.jacobson_top
@[simp]
theorem jacobson_eq_top_iff : jacobson I = ⊤ ↔ I = ⊤ :=
⟨fun H =>
by_contradiction fun hi => let ⟨M, hm, him⟩ := exists_le_maximal I hi
lt_top_iff_ne_top.1
(lt_of_le_of_lt (show jacobson I ≤ M from sInf_le ⟨him, hm⟩) <|
lt_top_iff_ne_top.2 hm.ne_top) H,
fun H => eq_top_iff.2 <| le_sInf fun _ ⟨hij, _⟩ => H ▸ hij⟩
#align ideal.jacobson_eq_top_iff Ideal.jacobson_eq_top_iff
theorem jacobson_eq_bot : jacobson I = ⊥ → I = ⊥ := fun h => eq_bot_iff.mpr (h ▸ le_jacobson)
#align ideal.jacobson_eq_bot Ideal.jacobson_eq_bot
theorem jacobson_eq_self_of_isMaximal [H : IsMaximal I] : I.jacobson = I :=
le_antisymm (sInf_le ⟨le_of_eq rfl, H⟩) le_jacobson
#align ideal.jacobson_eq_self_of_is_maximal Ideal.jacobson_eq_self_of_isMaximal
instance (priority := 100) jacobson.isMaximal [H : IsMaximal I] : IsMaximal (jacobson I) :=
⟨⟨fun htop => H.1.1 (jacobson_eq_top_iff.1 htop), fun _ hJ =>
H.1.2 _ (lt_of_le_of_lt le_jacobson hJ)⟩⟩
#align ideal.jacobson.is_maximal Ideal.jacobson.isMaximal
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I :=
⟨fun hx y =>
by_cases
(fun hxy : I ⊔ span {y * x + 1} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy)
let ⟨r, hr⟩ := mem_span_singleton'.1 hq
⟨r, by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one r (y * x), hr, ← hpq, ← neg_sub, add_sub_cancel]
exact I.neg_mem hpi⟩)
fun hxy : I ⊔ span {y * x + 1} ≠ ⊤ => let ⟨M, hm1, hm2⟩ := exists_le_maximal _ hxy
suffices x ∉ M from (this <| mem_sInf.1 hx ⟨le_trans le_sup_left hm2, hm1⟩).elim
fun hxm => hm1.1.1 <| (eq_top_iff_one _).2 <| add_sub_cancel' (y * x) 1 ▸
M.sub_mem (le_sup_right.trans hm2 <| subset_span rfl) (M.mul_mem_left _ hxm),
fun hx => mem_sInf.2 fun M ⟨him, hm⟩ => by_contradiction fun hxm =>
let ⟨y, i, hi, df⟩ := hm.exists_inv hxm
let ⟨z, hz⟩ := hx (-y)
hm.1.1 <| (eq_top_iff_one _).2 <| sub_sub_cancel (z * -y * x + z) 1 ▸
M.sub_mem (by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one z, neg_mul, ← sub_eq_iff_eq_add.mpr df.symm, neg_sub,
sub_add_cancel]
exact M.mul_mem_left _ hi) <| him hz⟩
#align ideal.mem_jacobson_iff Ideal.mem_jacobson_iff
theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
∃ s, s * r - 1 ∈ I := by
cases' mem_jacobson_iff.1 h 1 with s hs
use s
simpa [mul_sub] using hs
#align ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson Ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M := by
use fun hI => ⟨{ J : Ideal R | I ≤ J ∧ J.IsMaximal }, ⟨fun _ hJ => Or.inl hJ.right, hI.symm⟩⟩
rintro ⟨M, hM, hInf⟩
refine le_antisymm (fun x hx => ?_) le_jacobson
rw [hInf, mem_sInf]
intro I hI
cases' hM I hI with is_max is_top
· exact (mem_sInf.1 hx) ⟨le_sInf_iff.1 (le_of_eq hInf) I hI, is_max⟩
· exact is_top.symm ▸ Submodule.mem_top
#align ideal.eq_jacobson_iff_Inf_maximal Ideal.eq_jacobson_iff_sInf_maximal
theorem eq_jacobson_iff_sInf_maximal' :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, ∀ (K : Ideal R), J < K → K = ⊤) ∧ I = sInf M :=
eq_jacobson_iff_sInf_maximal.trans
⟨fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ K hK =>
Or.recOn (hM.1 J hJ) (fun h => h.1.2 K hK) fun h => eq_top_iff.2 (le_of_lt (h ▸ hK)),
hM.2⟩⟩,
fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ =>
Or.recOn (Classical.em (J = ⊤)) (fun h => Or.inr h) fun h => Or.inl ⟨⟨h, hM.1 J hJ⟩⟩,
hM.2⟩⟩⟩
#align ideal.eq_jacobson_iff_Inf_maximal' Ideal.eq_jacobson_iff_sInf_maximal'
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M := by
constructor
· intro h x hx
erw [← h, mem_sInf] at hx
push_neg at hx
exact hx
· refine fun h => le_antisymm (fun x hx => ?_) le_jacobson
contrapose hx
erw [mem_sInf]
push_neg
exact h x hx
#align ideal.eq_jacobson_iff_not_mem Ideal.eq_jacobson_iff_not_mem
theorem map_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) :
RingHom.ker f ≤ I → map f I.jacobson = (map f I).jacobson := by
intro h
unfold Ideal.jacobson
-- porting note : dot notation for `RingHom.ker` does not work
have : ∀ J ∈ { J : Ideal R | I ≤ J ∧ J.IsMaximal }, RingHom.ker f ≤ J :=
fun J hJ => le_trans h hJ.left
refine Trans.trans (map_sInf hf this) (le_antisymm ?_ ?_)
· refine'
sInf_le_sInf fun J hJ =>
⟨comap f J, ⟨⟨le_comap_of_map_le hJ.1, _⟩, map_comap_of_surjective f hf J⟩⟩
haveI : J.IsMaximal := hJ.right
exact comap_isMaximal_of_surjective f hf
· refine' sInf_le_sInf_of_subset_insert_top fun j hj => hj.recOn fun J hJ => _
rw [← hJ.2]
cases' map_eq_top_or_isMaximal_of_surjective f hf hJ.left.right with htop hmax
· exact htop.symm ▸ Set.mem_insert ⊤ _
·
|
exact Set.mem_insert_of_mem ⊤ ⟨map_mono hJ.1.1, hmax⟩
|
theorem map_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) :
RingHom.ker f ≤ I → map f I.jacobson = (map f I).jacobson := by
intro h
unfold Ideal.jacobson
-- porting note : dot notation for `RingHom.ker` does not work
have : ∀ J ∈ { J : Ideal R | I ≤ J ∧ J.IsMaximal }, RingHom.ker f ≤ J :=
fun J hJ => le_trans h hJ.left
refine Trans.trans (map_sInf hf this) (le_antisymm ?_ ?_)
· refine'
sInf_le_sInf fun J hJ =>
⟨comap f J, ⟨⟨le_comap_of_map_le hJ.1, _⟩, map_comap_of_surjective f hf J⟩⟩
haveI : J.IsMaximal := hJ.right
exact comap_isMaximal_of_surjective f hf
· refine' sInf_le_sInf_of_subset_insert_top fun j hj => hj.recOn fun J hJ => _
rw [← hJ.2]
cases' map_eq_top_or_isMaximal_of_surjective f hf hJ.left.right with htop hmax
· exact htop.symm ▸ Set.mem_insert ⊤ _
·
|
Mathlib.RingTheory.JacobsonIdeal.178_0.Lz0MgLQMj1bGzuN
|
theorem map_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) :
RingHom.ker f ≤ I → map f I.jacobson = (map f I).jacobson
|
Mathlib_RingTheory_JacobsonIdeal
|
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I : Ideal R
f : R →+* S
hf : Function.Surjective ⇑f
K : Ideal S
⊢ comap f (jacobson K) = jacobson (comap f K)
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
/-!
# Jacobson radical
The Jacobson radical of a ring `R` is defined to be the intersection of all maximal ideals of `R`.
This is similar to how the nilradical is equal to the intersection of all prime ideals of `R`.
We can extend the idea of the nilradical to ideals of `R`,
by letting the radical of an ideal `I` be the intersection of prime ideals containing `I`.
Under this extension, the original nilradical is the radical of the zero ideal `⊥`.
Here we define the Jacobson radical of an ideal `I` in a similar way,
as the intersection of maximal ideals containing `I`.
## Main definitions
Let `R` be a commutative ring, and `I` be an ideal of `R`
* `Ideal.jacobson I` is the jacobson radical, i.e. the infimum of all maximal ideals containing I.
* `Ideal.IsLocal I` is the proposition that the jacobson radical of `I` is itself a maximal ideal
## Main statements
* `mem_jacobson_iff` gives a characterization of members of the jacobson of I
* `Ideal.isLocal_of_isMaximal_radical`: if the radical of I is maximal then so is the jacobson
radical
## Tags
Jacobson, Jacobson radical, Local Ideal
-/
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}
open Polynomial
section Jacobson
section Ring
variable [Ring R] [Ring S] {I : Ideal R}
/-- The Jacobson radical of `I` is the infimum of all maximal (left) ideals containing `I`. -/
def jacobson (I : Ideal R) : Ideal R :=
sInf { J : Ideal R | I ≤ J ∧ IsMaximal J }
#align ideal.jacobson Ideal.jacobson
theorem le_jacobson : I ≤ jacobson I := fun _ hx => mem_sInf.mpr fun _ hJ => hJ.left hx
#align ideal.le_jacobson Ideal.le_jacobson
@[simp]
theorem jacobson_idem : jacobson (jacobson I) = jacobson I :=
le_antisymm (sInf_le_sInf fun _ hJ => ⟨sInf_le hJ, hJ.2⟩) le_jacobson
#align ideal.jacobson_idem Ideal.jacobson_idem
@[simp]
theorem jacobson_top : jacobson (⊤ : Ideal R) = ⊤ :=
eq_top_iff.2 le_jacobson
#align ideal.jacobson_top Ideal.jacobson_top
@[simp]
theorem jacobson_eq_top_iff : jacobson I = ⊤ ↔ I = ⊤ :=
⟨fun H =>
by_contradiction fun hi => let ⟨M, hm, him⟩ := exists_le_maximal I hi
lt_top_iff_ne_top.1
(lt_of_le_of_lt (show jacobson I ≤ M from sInf_le ⟨him, hm⟩) <|
lt_top_iff_ne_top.2 hm.ne_top) H,
fun H => eq_top_iff.2 <| le_sInf fun _ ⟨hij, _⟩ => H ▸ hij⟩
#align ideal.jacobson_eq_top_iff Ideal.jacobson_eq_top_iff
theorem jacobson_eq_bot : jacobson I = ⊥ → I = ⊥ := fun h => eq_bot_iff.mpr (h ▸ le_jacobson)
#align ideal.jacobson_eq_bot Ideal.jacobson_eq_bot
theorem jacobson_eq_self_of_isMaximal [H : IsMaximal I] : I.jacobson = I :=
le_antisymm (sInf_le ⟨le_of_eq rfl, H⟩) le_jacobson
#align ideal.jacobson_eq_self_of_is_maximal Ideal.jacobson_eq_self_of_isMaximal
instance (priority := 100) jacobson.isMaximal [H : IsMaximal I] : IsMaximal (jacobson I) :=
⟨⟨fun htop => H.1.1 (jacobson_eq_top_iff.1 htop), fun _ hJ =>
H.1.2 _ (lt_of_le_of_lt le_jacobson hJ)⟩⟩
#align ideal.jacobson.is_maximal Ideal.jacobson.isMaximal
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I :=
⟨fun hx y =>
by_cases
(fun hxy : I ⊔ span {y * x + 1} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy)
let ⟨r, hr⟩ := mem_span_singleton'.1 hq
⟨r, by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one r (y * x), hr, ← hpq, ← neg_sub, add_sub_cancel]
exact I.neg_mem hpi⟩)
fun hxy : I ⊔ span {y * x + 1} ≠ ⊤ => let ⟨M, hm1, hm2⟩ := exists_le_maximal _ hxy
suffices x ∉ M from (this <| mem_sInf.1 hx ⟨le_trans le_sup_left hm2, hm1⟩).elim
fun hxm => hm1.1.1 <| (eq_top_iff_one _).2 <| add_sub_cancel' (y * x) 1 ▸
M.sub_mem (le_sup_right.trans hm2 <| subset_span rfl) (M.mul_mem_left _ hxm),
fun hx => mem_sInf.2 fun M ⟨him, hm⟩ => by_contradiction fun hxm =>
let ⟨y, i, hi, df⟩ := hm.exists_inv hxm
let ⟨z, hz⟩ := hx (-y)
hm.1.1 <| (eq_top_iff_one _).2 <| sub_sub_cancel (z * -y * x + z) 1 ▸
M.sub_mem (by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one z, neg_mul, ← sub_eq_iff_eq_add.mpr df.symm, neg_sub,
sub_add_cancel]
exact M.mul_mem_left _ hi) <| him hz⟩
#align ideal.mem_jacobson_iff Ideal.mem_jacobson_iff
theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
∃ s, s * r - 1 ∈ I := by
cases' mem_jacobson_iff.1 h 1 with s hs
use s
simpa [mul_sub] using hs
#align ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson Ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M := by
use fun hI => ⟨{ J : Ideal R | I ≤ J ∧ J.IsMaximal }, ⟨fun _ hJ => Or.inl hJ.right, hI.symm⟩⟩
rintro ⟨M, hM, hInf⟩
refine le_antisymm (fun x hx => ?_) le_jacobson
rw [hInf, mem_sInf]
intro I hI
cases' hM I hI with is_max is_top
· exact (mem_sInf.1 hx) ⟨le_sInf_iff.1 (le_of_eq hInf) I hI, is_max⟩
· exact is_top.symm ▸ Submodule.mem_top
#align ideal.eq_jacobson_iff_Inf_maximal Ideal.eq_jacobson_iff_sInf_maximal
theorem eq_jacobson_iff_sInf_maximal' :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, ∀ (K : Ideal R), J < K → K = ⊤) ∧ I = sInf M :=
eq_jacobson_iff_sInf_maximal.trans
⟨fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ K hK =>
Or.recOn (hM.1 J hJ) (fun h => h.1.2 K hK) fun h => eq_top_iff.2 (le_of_lt (h ▸ hK)),
hM.2⟩⟩,
fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ =>
Or.recOn (Classical.em (J = ⊤)) (fun h => Or.inr h) fun h => Or.inl ⟨⟨h, hM.1 J hJ⟩⟩,
hM.2⟩⟩⟩
#align ideal.eq_jacobson_iff_Inf_maximal' Ideal.eq_jacobson_iff_sInf_maximal'
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M := by
constructor
· intro h x hx
erw [← h, mem_sInf] at hx
push_neg at hx
exact hx
· refine fun h => le_antisymm (fun x hx => ?_) le_jacobson
contrapose hx
erw [mem_sInf]
push_neg
exact h x hx
#align ideal.eq_jacobson_iff_not_mem Ideal.eq_jacobson_iff_not_mem
theorem map_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) :
RingHom.ker f ≤ I → map f I.jacobson = (map f I).jacobson := by
intro h
unfold Ideal.jacobson
-- porting note : dot notation for `RingHom.ker` does not work
have : ∀ J ∈ { J : Ideal R | I ≤ J ∧ J.IsMaximal }, RingHom.ker f ≤ J :=
fun J hJ => le_trans h hJ.left
refine Trans.trans (map_sInf hf this) (le_antisymm ?_ ?_)
· refine'
sInf_le_sInf fun J hJ =>
⟨comap f J, ⟨⟨le_comap_of_map_le hJ.1, _⟩, map_comap_of_surjective f hf J⟩⟩
haveI : J.IsMaximal := hJ.right
exact comap_isMaximal_of_surjective f hf
· refine' sInf_le_sInf_of_subset_insert_top fun j hj => hj.recOn fun J hJ => _
rw [← hJ.2]
cases' map_eq_top_or_isMaximal_of_surjective f hf hJ.left.right with htop hmax
· exact htop.symm ▸ Set.mem_insert ⊤ _
· exact Set.mem_insert_of_mem ⊤ ⟨map_mono hJ.1.1, hmax⟩
#align ideal.map_jacobson_of_surjective Ideal.map_jacobson_of_surjective
theorem map_jacobson_of_bijective {f : R →+* S} (hf : Function.Bijective f) :
map f I.jacobson = (map f I).jacobson :=
map_jacobson_of_surjective hf.right
(le_trans (le_of_eq (f.injective_iff_ker_eq_bot.1 hf.left)) bot_le)
#align ideal.map_jacobson_of_bijective Ideal.map_jacobson_of_bijective
theorem comap_jacobson {f : R →+* S} {K : Ideal S} :
comap f K.jacobson = sInf (comap f '' { J : Ideal S | K ≤ J ∧ J.IsMaximal }) :=
Trans.trans (comap_sInf' f _) sInf_eq_iInf.symm
#align ideal.comap_jacobson Ideal.comap_jacobson
theorem comap_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) {K : Ideal S} :
comap f K.jacobson = (comap f K).jacobson := by
|
unfold Ideal.jacobson
|
theorem comap_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) {K : Ideal S} :
comap f K.jacobson = (comap f K).jacobson := by
|
Mathlib.RingTheory.JacobsonIdeal.209_0.Lz0MgLQMj1bGzuN
|
theorem comap_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) {K : Ideal S} :
comap f K.jacobson = (comap f K).jacobson
|
Mathlib_RingTheory_JacobsonIdeal
|
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I : Ideal R
f : R →+* S
hf : Function.Surjective ⇑f
K : Ideal S
⊢ comap f (sInf {J | K ≤ J ∧ IsMaximal J}) = sInf {J | comap f K ≤ J ∧ IsMaximal J}
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
/-!
# Jacobson radical
The Jacobson radical of a ring `R` is defined to be the intersection of all maximal ideals of `R`.
This is similar to how the nilradical is equal to the intersection of all prime ideals of `R`.
We can extend the idea of the nilradical to ideals of `R`,
by letting the radical of an ideal `I` be the intersection of prime ideals containing `I`.
Under this extension, the original nilradical is the radical of the zero ideal `⊥`.
Here we define the Jacobson radical of an ideal `I` in a similar way,
as the intersection of maximal ideals containing `I`.
## Main definitions
Let `R` be a commutative ring, and `I` be an ideal of `R`
* `Ideal.jacobson I` is the jacobson radical, i.e. the infimum of all maximal ideals containing I.
* `Ideal.IsLocal I` is the proposition that the jacobson radical of `I` is itself a maximal ideal
## Main statements
* `mem_jacobson_iff` gives a characterization of members of the jacobson of I
* `Ideal.isLocal_of_isMaximal_radical`: if the radical of I is maximal then so is the jacobson
radical
## Tags
Jacobson, Jacobson radical, Local Ideal
-/
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}
open Polynomial
section Jacobson
section Ring
variable [Ring R] [Ring S] {I : Ideal R}
/-- The Jacobson radical of `I` is the infimum of all maximal (left) ideals containing `I`. -/
def jacobson (I : Ideal R) : Ideal R :=
sInf { J : Ideal R | I ≤ J ∧ IsMaximal J }
#align ideal.jacobson Ideal.jacobson
theorem le_jacobson : I ≤ jacobson I := fun _ hx => mem_sInf.mpr fun _ hJ => hJ.left hx
#align ideal.le_jacobson Ideal.le_jacobson
@[simp]
theorem jacobson_idem : jacobson (jacobson I) = jacobson I :=
le_antisymm (sInf_le_sInf fun _ hJ => ⟨sInf_le hJ, hJ.2⟩) le_jacobson
#align ideal.jacobson_idem Ideal.jacobson_idem
@[simp]
theorem jacobson_top : jacobson (⊤ : Ideal R) = ⊤ :=
eq_top_iff.2 le_jacobson
#align ideal.jacobson_top Ideal.jacobson_top
@[simp]
theorem jacobson_eq_top_iff : jacobson I = ⊤ ↔ I = ⊤ :=
⟨fun H =>
by_contradiction fun hi => let ⟨M, hm, him⟩ := exists_le_maximal I hi
lt_top_iff_ne_top.1
(lt_of_le_of_lt (show jacobson I ≤ M from sInf_le ⟨him, hm⟩) <|
lt_top_iff_ne_top.2 hm.ne_top) H,
fun H => eq_top_iff.2 <| le_sInf fun _ ⟨hij, _⟩ => H ▸ hij⟩
#align ideal.jacobson_eq_top_iff Ideal.jacobson_eq_top_iff
theorem jacobson_eq_bot : jacobson I = ⊥ → I = ⊥ := fun h => eq_bot_iff.mpr (h ▸ le_jacobson)
#align ideal.jacobson_eq_bot Ideal.jacobson_eq_bot
theorem jacobson_eq_self_of_isMaximal [H : IsMaximal I] : I.jacobson = I :=
le_antisymm (sInf_le ⟨le_of_eq rfl, H⟩) le_jacobson
#align ideal.jacobson_eq_self_of_is_maximal Ideal.jacobson_eq_self_of_isMaximal
instance (priority := 100) jacobson.isMaximal [H : IsMaximal I] : IsMaximal (jacobson I) :=
⟨⟨fun htop => H.1.1 (jacobson_eq_top_iff.1 htop), fun _ hJ =>
H.1.2 _ (lt_of_le_of_lt le_jacobson hJ)⟩⟩
#align ideal.jacobson.is_maximal Ideal.jacobson.isMaximal
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I :=
⟨fun hx y =>
by_cases
(fun hxy : I ⊔ span {y * x + 1} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy)
let ⟨r, hr⟩ := mem_span_singleton'.1 hq
⟨r, by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one r (y * x), hr, ← hpq, ← neg_sub, add_sub_cancel]
exact I.neg_mem hpi⟩)
fun hxy : I ⊔ span {y * x + 1} ≠ ⊤ => let ⟨M, hm1, hm2⟩ := exists_le_maximal _ hxy
suffices x ∉ M from (this <| mem_sInf.1 hx ⟨le_trans le_sup_left hm2, hm1⟩).elim
fun hxm => hm1.1.1 <| (eq_top_iff_one _).2 <| add_sub_cancel' (y * x) 1 ▸
M.sub_mem (le_sup_right.trans hm2 <| subset_span rfl) (M.mul_mem_left _ hxm),
fun hx => mem_sInf.2 fun M ⟨him, hm⟩ => by_contradiction fun hxm =>
let ⟨y, i, hi, df⟩ := hm.exists_inv hxm
let ⟨z, hz⟩ := hx (-y)
hm.1.1 <| (eq_top_iff_one _).2 <| sub_sub_cancel (z * -y * x + z) 1 ▸
M.sub_mem (by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one z, neg_mul, ← sub_eq_iff_eq_add.mpr df.symm, neg_sub,
sub_add_cancel]
exact M.mul_mem_left _ hi) <| him hz⟩
#align ideal.mem_jacobson_iff Ideal.mem_jacobson_iff
theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
∃ s, s * r - 1 ∈ I := by
cases' mem_jacobson_iff.1 h 1 with s hs
use s
simpa [mul_sub] using hs
#align ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson Ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M := by
use fun hI => ⟨{ J : Ideal R | I ≤ J ∧ J.IsMaximal }, ⟨fun _ hJ => Or.inl hJ.right, hI.symm⟩⟩
rintro ⟨M, hM, hInf⟩
refine le_antisymm (fun x hx => ?_) le_jacobson
rw [hInf, mem_sInf]
intro I hI
cases' hM I hI with is_max is_top
· exact (mem_sInf.1 hx) ⟨le_sInf_iff.1 (le_of_eq hInf) I hI, is_max⟩
· exact is_top.symm ▸ Submodule.mem_top
#align ideal.eq_jacobson_iff_Inf_maximal Ideal.eq_jacobson_iff_sInf_maximal
theorem eq_jacobson_iff_sInf_maximal' :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, ∀ (K : Ideal R), J < K → K = ⊤) ∧ I = sInf M :=
eq_jacobson_iff_sInf_maximal.trans
⟨fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ K hK =>
Or.recOn (hM.1 J hJ) (fun h => h.1.2 K hK) fun h => eq_top_iff.2 (le_of_lt (h ▸ hK)),
hM.2⟩⟩,
fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ =>
Or.recOn (Classical.em (J = ⊤)) (fun h => Or.inr h) fun h => Or.inl ⟨⟨h, hM.1 J hJ⟩⟩,
hM.2⟩⟩⟩
#align ideal.eq_jacobson_iff_Inf_maximal' Ideal.eq_jacobson_iff_sInf_maximal'
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M := by
constructor
· intro h x hx
erw [← h, mem_sInf] at hx
push_neg at hx
exact hx
· refine fun h => le_antisymm (fun x hx => ?_) le_jacobson
contrapose hx
erw [mem_sInf]
push_neg
exact h x hx
#align ideal.eq_jacobson_iff_not_mem Ideal.eq_jacobson_iff_not_mem
theorem map_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) :
RingHom.ker f ≤ I → map f I.jacobson = (map f I).jacobson := by
intro h
unfold Ideal.jacobson
-- porting note : dot notation for `RingHom.ker` does not work
have : ∀ J ∈ { J : Ideal R | I ≤ J ∧ J.IsMaximal }, RingHom.ker f ≤ J :=
fun J hJ => le_trans h hJ.left
refine Trans.trans (map_sInf hf this) (le_antisymm ?_ ?_)
· refine'
sInf_le_sInf fun J hJ =>
⟨comap f J, ⟨⟨le_comap_of_map_le hJ.1, _⟩, map_comap_of_surjective f hf J⟩⟩
haveI : J.IsMaximal := hJ.right
exact comap_isMaximal_of_surjective f hf
· refine' sInf_le_sInf_of_subset_insert_top fun j hj => hj.recOn fun J hJ => _
rw [← hJ.2]
cases' map_eq_top_or_isMaximal_of_surjective f hf hJ.left.right with htop hmax
· exact htop.symm ▸ Set.mem_insert ⊤ _
· exact Set.mem_insert_of_mem ⊤ ⟨map_mono hJ.1.1, hmax⟩
#align ideal.map_jacobson_of_surjective Ideal.map_jacobson_of_surjective
theorem map_jacobson_of_bijective {f : R →+* S} (hf : Function.Bijective f) :
map f I.jacobson = (map f I).jacobson :=
map_jacobson_of_surjective hf.right
(le_trans (le_of_eq (f.injective_iff_ker_eq_bot.1 hf.left)) bot_le)
#align ideal.map_jacobson_of_bijective Ideal.map_jacobson_of_bijective
theorem comap_jacobson {f : R →+* S} {K : Ideal S} :
comap f K.jacobson = sInf (comap f '' { J : Ideal S | K ≤ J ∧ J.IsMaximal }) :=
Trans.trans (comap_sInf' f _) sInf_eq_iInf.symm
#align ideal.comap_jacobson Ideal.comap_jacobson
theorem comap_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) {K : Ideal S} :
comap f K.jacobson = (comap f K).jacobson := by
unfold Ideal.jacobson
|
refine' le_antisymm _ _
|
theorem comap_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) {K : Ideal S} :
comap f K.jacobson = (comap f K).jacobson := by
unfold Ideal.jacobson
|
Mathlib.RingTheory.JacobsonIdeal.209_0.Lz0MgLQMj1bGzuN
|
theorem comap_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) {K : Ideal S} :
comap f K.jacobson = (comap f K).jacobson
|
Mathlib_RingTheory_JacobsonIdeal
|
case refine'_1
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I : Ideal R
f : R →+* S
hf : Function.Surjective ⇑f
K : Ideal S
⊢ comap f (sInf {J | K ≤ J ∧ IsMaximal J}) ≤ sInf {J | comap f K ≤ J ∧ IsMaximal J}
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
/-!
# Jacobson radical
The Jacobson radical of a ring `R` is defined to be the intersection of all maximal ideals of `R`.
This is similar to how the nilradical is equal to the intersection of all prime ideals of `R`.
We can extend the idea of the nilradical to ideals of `R`,
by letting the radical of an ideal `I` be the intersection of prime ideals containing `I`.
Under this extension, the original nilradical is the radical of the zero ideal `⊥`.
Here we define the Jacobson radical of an ideal `I` in a similar way,
as the intersection of maximal ideals containing `I`.
## Main definitions
Let `R` be a commutative ring, and `I` be an ideal of `R`
* `Ideal.jacobson I` is the jacobson radical, i.e. the infimum of all maximal ideals containing I.
* `Ideal.IsLocal I` is the proposition that the jacobson radical of `I` is itself a maximal ideal
## Main statements
* `mem_jacobson_iff` gives a characterization of members of the jacobson of I
* `Ideal.isLocal_of_isMaximal_radical`: if the radical of I is maximal then so is the jacobson
radical
## Tags
Jacobson, Jacobson radical, Local Ideal
-/
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}
open Polynomial
section Jacobson
section Ring
variable [Ring R] [Ring S] {I : Ideal R}
/-- The Jacobson radical of `I` is the infimum of all maximal (left) ideals containing `I`. -/
def jacobson (I : Ideal R) : Ideal R :=
sInf { J : Ideal R | I ≤ J ∧ IsMaximal J }
#align ideal.jacobson Ideal.jacobson
theorem le_jacobson : I ≤ jacobson I := fun _ hx => mem_sInf.mpr fun _ hJ => hJ.left hx
#align ideal.le_jacobson Ideal.le_jacobson
@[simp]
theorem jacobson_idem : jacobson (jacobson I) = jacobson I :=
le_antisymm (sInf_le_sInf fun _ hJ => ⟨sInf_le hJ, hJ.2⟩) le_jacobson
#align ideal.jacobson_idem Ideal.jacobson_idem
@[simp]
theorem jacobson_top : jacobson (⊤ : Ideal R) = ⊤ :=
eq_top_iff.2 le_jacobson
#align ideal.jacobson_top Ideal.jacobson_top
@[simp]
theorem jacobson_eq_top_iff : jacobson I = ⊤ ↔ I = ⊤ :=
⟨fun H =>
by_contradiction fun hi => let ⟨M, hm, him⟩ := exists_le_maximal I hi
lt_top_iff_ne_top.1
(lt_of_le_of_lt (show jacobson I ≤ M from sInf_le ⟨him, hm⟩) <|
lt_top_iff_ne_top.2 hm.ne_top) H,
fun H => eq_top_iff.2 <| le_sInf fun _ ⟨hij, _⟩ => H ▸ hij⟩
#align ideal.jacobson_eq_top_iff Ideal.jacobson_eq_top_iff
theorem jacobson_eq_bot : jacobson I = ⊥ → I = ⊥ := fun h => eq_bot_iff.mpr (h ▸ le_jacobson)
#align ideal.jacobson_eq_bot Ideal.jacobson_eq_bot
theorem jacobson_eq_self_of_isMaximal [H : IsMaximal I] : I.jacobson = I :=
le_antisymm (sInf_le ⟨le_of_eq rfl, H⟩) le_jacobson
#align ideal.jacobson_eq_self_of_is_maximal Ideal.jacobson_eq_self_of_isMaximal
instance (priority := 100) jacobson.isMaximal [H : IsMaximal I] : IsMaximal (jacobson I) :=
⟨⟨fun htop => H.1.1 (jacobson_eq_top_iff.1 htop), fun _ hJ =>
H.1.2 _ (lt_of_le_of_lt le_jacobson hJ)⟩⟩
#align ideal.jacobson.is_maximal Ideal.jacobson.isMaximal
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I :=
⟨fun hx y =>
by_cases
(fun hxy : I ⊔ span {y * x + 1} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy)
let ⟨r, hr⟩ := mem_span_singleton'.1 hq
⟨r, by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one r (y * x), hr, ← hpq, ← neg_sub, add_sub_cancel]
exact I.neg_mem hpi⟩)
fun hxy : I ⊔ span {y * x + 1} ≠ ⊤ => let ⟨M, hm1, hm2⟩ := exists_le_maximal _ hxy
suffices x ∉ M from (this <| mem_sInf.1 hx ⟨le_trans le_sup_left hm2, hm1⟩).elim
fun hxm => hm1.1.1 <| (eq_top_iff_one _).2 <| add_sub_cancel' (y * x) 1 ▸
M.sub_mem (le_sup_right.trans hm2 <| subset_span rfl) (M.mul_mem_left _ hxm),
fun hx => mem_sInf.2 fun M ⟨him, hm⟩ => by_contradiction fun hxm =>
let ⟨y, i, hi, df⟩ := hm.exists_inv hxm
let ⟨z, hz⟩ := hx (-y)
hm.1.1 <| (eq_top_iff_one _).2 <| sub_sub_cancel (z * -y * x + z) 1 ▸
M.sub_mem (by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one z, neg_mul, ← sub_eq_iff_eq_add.mpr df.symm, neg_sub,
sub_add_cancel]
exact M.mul_mem_left _ hi) <| him hz⟩
#align ideal.mem_jacobson_iff Ideal.mem_jacobson_iff
theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
∃ s, s * r - 1 ∈ I := by
cases' mem_jacobson_iff.1 h 1 with s hs
use s
simpa [mul_sub] using hs
#align ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson Ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M := by
use fun hI => ⟨{ J : Ideal R | I ≤ J ∧ J.IsMaximal }, ⟨fun _ hJ => Or.inl hJ.right, hI.symm⟩⟩
rintro ⟨M, hM, hInf⟩
refine le_antisymm (fun x hx => ?_) le_jacobson
rw [hInf, mem_sInf]
intro I hI
cases' hM I hI with is_max is_top
· exact (mem_sInf.1 hx) ⟨le_sInf_iff.1 (le_of_eq hInf) I hI, is_max⟩
· exact is_top.symm ▸ Submodule.mem_top
#align ideal.eq_jacobson_iff_Inf_maximal Ideal.eq_jacobson_iff_sInf_maximal
theorem eq_jacobson_iff_sInf_maximal' :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, ∀ (K : Ideal R), J < K → K = ⊤) ∧ I = sInf M :=
eq_jacobson_iff_sInf_maximal.trans
⟨fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ K hK =>
Or.recOn (hM.1 J hJ) (fun h => h.1.2 K hK) fun h => eq_top_iff.2 (le_of_lt (h ▸ hK)),
hM.2⟩⟩,
fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ =>
Or.recOn (Classical.em (J = ⊤)) (fun h => Or.inr h) fun h => Or.inl ⟨⟨h, hM.1 J hJ⟩⟩,
hM.2⟩⟩⟩
#align ideal.eq_jacobson_iff_Inf_maximal' Ideal.eq_jacobson_iff_sInf_maximal'
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M := by
constructor
· intro h x hx
erw [← h, mem_sInf] at hx
push_neg at hx
exact hx
· refine fun h => le_antisymm (fun x hx => ?_) le_jacobson
contrapose hx
erw [mem_sInf]
push_neg
exact h x hx
#align ideal.eq_jacobson_iff_not_mem Ideal.eq_jacobson_iff_not_mem
theorem map_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) :
RingHom.ker f ≤ I → map f I.jacobson = (map f I).jacobson := by
intro h
unfold Ideal.jacobson
-- porting note : dot notation for `RingHom.ker` does not work
have : ∀ J ∈ { J : Ideal R | I ≤ J ∧ J.IsMaximal }, RingHom.ker f ≤ J :=
fun J hJ => le_trans h hJ.left
refine Trans.trans (map_sInf hf this) (le_antisymm ?_ ?_)
· refine'
sInf_le_sInf fun J hJ =>
⟨comap f J, ⟨⟨le_comap_of_map_le hJ.1, _⟩, map_comap_of_surjective f hf J⟩⟩
haveI : J.IsMaximal := hJ.right
exact comap_isMaximal_of_surjective f hf
· refine' sInf_le_sInf_of_subset_insert_top fun j hj => hj.recOn fun J hJ => _
rw [← hJ.2]
cases' map_eq_top_or_isMaximal_of_surjective f hf hJ.left.right with htop hmax
· exact htop.symm ▸ Set.mem_insert ⊤ _
· exact Set.mem_insert_of_mem ⊤ ⟨map_mono hJ.1.1, hmax⟩
#align ideal.map_jacobson_of_surjective Ideal.map_jacobson_of_surjective
theorem map_jacobson_of_bijective {f : R →+* S} (hf : Function.Bijective f) :
map f I.jacobson = (map f I).jacobson :=
map_jacobson_of_surjective hf.right
(le_trans (le_of_eq (f.injective_iff_ker_eq_bot.1 hf.left)) bot_le)
#align ideal.map_jacobson_of_bijective Ideal.map_jacobson_of_bijective
theorem comap_jacobson {f : R →+* S} {K : Ideal S} :
comap f K.jacobson = sInf (comap f '' { J : Ideal S | K ≤ J ∧ J.IsMaximal }) :=
Trans.trans (comap_sInf' f _) sInf_eq_iInf.symm
#align ideal.comap_jacobson Ideal.comap_jacobson
theorem comap_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) {K : Ideal S} :
comap f K.jacobson = (comap f K).jacobson := by
unfold Ideal.jacobson
refine' le_antisymm _ _
·
|
refine le_trans (comap_mono (le_of_eq (Trans.trans top_inf_eq.symm sInf_insert.symm))) ?_
|
theorem comap_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) {K : Ideal S} :
comap f K.jacobson = (comap f K).jacobson := by
unfold Ideal.jacobson
refine' le_antisymm _ _
·
|
Mathlib.RingTheory.JacobsonIdeal.209_0.Lz0MgLQMj1bGzuN
|
theorem comap_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) {K : Ideal S} :
comap f K.jacobson = (comap f K).jacobson
|
Mathlib_RingTheory_JacobsonIdeal
|
case refine'_1
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I : Ideal R
f : R →+* S
hf : Function.Surjective ⇑f
K : Ideal S
⊢ comap f (sInf (insert ⊤ {J | K ≤ J ∧ IsMaximal J})) ≤ sInf {J | comap f K ≤ J ∧ IsMaximal J}
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
/-!
# Jacobson radical
The Jacobson radical of a ring `R` is defined to be the intersection of all maximal ideals of `R`.
This is similar to how the nilradical is equal to the intersection of all prime ideals of `R`.
We can extend the idea of the nilradical to ideals of `R`,
by letting the radical of an ideal `I` be the intersection of prime ideals containing `I`.
Under this extension, the original nilradical is the radical of the zero ideal `⊥`.
Here we define the Jacobson radical of an ideal `I` in a similar way,
as the intersection of maximal ideals containing `I`.
## Main definitions
Let `R` be a commutative ring, and `I` be an ideal of `R`
* `Ideal.jacobson I` is the jacobson radical, i.e. the infimum of all maximal ideals containing I.
* `Ideal.IsLocal I` is the proposition that the jacobson radical of `I` is itself a maximal ideal
## Main statements
* `mem_jacobson_iff` gives a characterization of members of the jacobson of I
* `Ideal.isLocal_of_isMaximal_radical`: if the radical of I is maximal then so is the jacobson
radical
## Tags
Jacobson, Jacobson radical, Local Ideal
-/
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}
open Polynomial
section Jacobson
section Ring
variable [Ring R] [Ring S] {I : Ideal R}
/-- The Jacobson radical of `I` is the infimum of all maximal (left) ideals containing `I`. -/
def jacobson (I : Ideal R) : Ideal R :=
sInf { J : Ideal R | I ≤ J ∧ IsMaximal J }
#align ideal.jacobson Ideal.jacobson
theorem le_jacobson : I ≤ jacobson I := fun _ hx => mem_sInf.mpr fun _ hJ => hJ.left hx
#align ideal.le_jacobson Ideal.le_jacobson
@[simp]
theorem jacobson_idem : jacobson (jacobson I) = jacobson I :=
le_antisymm (sInf_le_sInf fun _ hJ => ⟨sInf_le hJ, hJ.2⟩) le_jacobson
#align ideal.jacobson_idem Ideal.jacobson_idem
@[simp]
theorem jacobson_top : jacobson (⊤ : Ideal R) = ⊤ :=
eq_top_iff.2 le_jacobson
#align ideal.jacobson_top Ideal.jacobson_top
@[simp]
theorem jacobson_eq_top_iff : jacobson I = ⊤ ↔ I = ⊤ :=
⟨fun H =>
by_contradiction fun hi => let ⟨M, hm, him⟩ := exists_le_maximal I hi
lt_top_iff_ne_top.1
(lt_of_le_of_lt (show jacobson I ≤ M from sInf_le ⟨him, hm⟩) <|
lt_top_iff_ne_top.2 hm.ne_top) H,
fun H => eq_top_iff.2 <| le_sInf fun _ ⟨hij, _⟩ => H ▸ hij⟩
#align ideal.jacobson_eq_top_iff Ideal.jacobson_eq_top_iff
theorem jacobson_eq_bot : jacobson I = ⊥ → I = ⊥ := fun h => eq_bot_iff.mpr (h ▸ le_jacobson)
#align ideal.jacobson_eq_bot Ideal.jacobson_eq_bot
theorem jacobson_eq_self_of_isMaximal [H : IsMaximal I] : I.jacobson = I :=
le_antisymm (sInf_le ⟨le_of_eq rfl, H⟩) le_jacobson
#align ideal.jacobson_eq_self_of_is_maximal Ideal.jacobson_eq_self_of_isMaximal
instance (priority := 100) jacobson.isMaximal [H : IsMaximal I] : IsMaximal (jacobson I) :=
⟨⟨fun htop => H.1.1 (jacobson_eq_top_iff.1 htop), fun _ hJ =>
H.1.2 _ (lt_of_le_of_lt le_jacobson hJ)⟩⟩
#align ideal.jacobson.is_maximal Ideal.jacobson.isMaximal
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I :=
⟨fun hx y =>
by_cases
(fun hxy : I ⊔ span {y * x + 1} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy)
let ⟨r, hr⟩ := mem_span_singleton'.1 hq
⟨r, by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one r (y * x), hr, ← hpq, ← neg_sub, add_sub_cancel]
exact I.neg_mem hpi⟩)
fun hxy : I ⊔ span {y * x + 1} ≠ ⊤ => let ⟨M, hm1, hm2⟩ := exists_le_maximal _ hxy
suffices x ∉ M from (this <| mem_sInf.1 hx ⟨le_trans le_sup_left hm2, hm1⟩).elim
fun hxm => hm1.1.1 <| (eq_top_iff_one _).2 <| add_sub_cancel' (y * x) 1 ▸
M.sub_mem (le_sup_right.trans hm2 <| subset_span rfl) (M.mul_mem_left _ hxm),
fun hx => mem_sInf.2 fun M ⟨him, hm⟩ => by_contradiction fun hxm =>
let ⟨y, i, hi, df⟩ := hm.exists_inv hxm
let ⟨z, hz⟩ := hx (-y)
hm.1.1 <| (eq_top_iff_one _).2 <| sub_sub_cancel (z * -y * x + z) 1 ▸
M.sub_mem (by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one z, neg_mul, ← sub_eq_iff_eq_add.mpr df.symm, neg_sub,
sub_add_cancel]
exact M.mul_mem_left _ hi) <| him hz⟩
#align ideal.mem_jacobson_iff Ideal.mem_jacobson_iff
theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
∃ s, s * r - 1 ∈ I := by
cases' mem_jacobson_iff.1 h 1 with s hs
use s
simpa [mul_sub] using hs
#align ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson Ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M := by
use fun hI => ⟨{ J : Ideal R | I ≤ J ∧ J.IsMaximal }, ⟨fun _ hJ => Or.inl hJ.right, hI.symm⟩⟩
rintro ⟨M, hM, hInf⟩
refine le_antisymm (fun x hx => ?_) le_jacobson
rw [hInf, mem_sInf]
intro I hI
cases' hM I hI with is_max is_top
· exact (mem_sInf.1 hx) ⟨le_sInf_iff.1 (le_of_eq hInf) I hI, is_max⟩
· exact is_top.symm ▸ Submodule.mem_top
#align ideal.eq_jacobson_iff_Inf_maximal Ideal.eq_jacobson_iff_sInf_maximal
theorem eq_jacobson_iff_sInf_maximal' :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, ∀ (K : Ideal R), J < K → K = ⊤) ∧ I = sInf M :=
eq_jacobson_iff_sInf_maximal.trans
⟨fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ K hK =>
Or.recOn (hM.1 J hJ) (fun h => h.1.2 K hK) fun h => eq_top_iff.2 (le_of_lt (h ▸ hK)),
hM.2⟩⟩,
fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ =>
Or.recOn (Classical.em (J = ⊤)) (fun h => Or.inr h) fun h => Or.inl ⟨⟨h, hM.1 J hJ⟩⟩,
hM.2⟩⟩⟩
#align ideal.eq_jacobson_iff_Inf_maximal' Ideal.eq_jacobson_iff_sInf_maximal'
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M := by
constructor
· intro h x hx
erw [← h, mem_sInf] at hx
push_neg at hx
exact hx
· refine fun h => le_antisymm (fun x hx => ?_) le_jacobson
contrapose hx
erw [mem_sInf]
push_neg
exact h x hx
#align ideal.eq_jacobson_iff_not_mem Ideal.eq_jacobson_iff_not_mem
theorem map_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) :
RingHom.ker f ≤ I → map f I.jacobson = (map f I).jacobson := by
intro h
unfold Ideal.jacobson
-- porting note : dot notation for `RingHom.ker` does not work
have : ∀ J ∈ { J : Ideal R | I ≤ J ∧ J.IsMaximal }, RingHom.ker f ≤ J :=
fun J hJ => le_trans h hJ.left
refine Trans.trans (map_sInf hf this) (le_antisymm ?_ ?_)
· refine'
sInf_le_sInf fun J hJ =>
⟨comap f J, ⟨⟨le_comap_of_map_le hJ.1, _⟩, map_comap_of_surjective f hf J⟩⟩
haveI : J.IsMaximal := hJ.right
exact comap_isMaximal_of_surjective f hf
· refine' sInf_le_sInf_of_subset_insert_top fun j hj => hj.recOn fun J hJ => _
rw [← hJ.2]
cases' map_eq_top_or_isMaximal_of_surjective f hf hJ.left.right with htop hmax
· exact htop.symm ▸ Set.mem_insert ⊤ _
· exact Set.mem_insert_of_mem ⊤ ⟨map_mono hJ.1.1, hmax⟩
#align ideal.map_jacobson_of_surjective Ideal.map_jacobson_of_surjective
theorem map_jacobson_of_bijective {f : R →+* S} (hf : Function.Bijective f) :
map f I.jacobson = (map f I).jacobson :=
map_jacobson_of_surjective hf.right
(le_trans (le_of_eq (f.injective_iff_ker_eq_bot.1 hf.left)) bot_le)
#align ideal.map_jacobson_of_bijective Ideal.map_jacobson_of_bijective
theorem comap_jacobson {f : R →+* S} {K : Ideal S} :
comap f K.jacobson = sInf (comap f '' { J : Ideal S | K ≤ J ∧ J.IsMaximal }) :=
Trans.trans (comap_sInf' f _) sInf_eq_iInf.symm
#align ideal.comap_jacobson Ideal.comap_jacobson
theorem comap_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) {K : Ideal S} :
comap f K.jacobson = (comap f K).jacobson := by
unfold Ideal.jacobson
refine' le_antisymm _ _
· refine le_trans (comap_mono (le_of_eq (Trans.trans top_inf_eq.symm sInf_insert.symm))) ?_
|
rw [comap_sInf', sInf_eq_iInf]
|
theorem comap_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) {K : Ideal S} :
comap f K.jacobson = (comap f K).jacobson := by
unfold Ideal.jacobson
refine' le_antisymm _ _
· refine le_trans (comap_mono (le_of_eq (Trans.trans top_inf_eq.symm sInf_insert.symm))) ?_
|
Mathlib.RingTheory.JacobsonIdeal.209_0.Lz0MgLQMj1bGzuN
|
theorem comap_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) {K : Ideal S} :
comap f K.jacobson = (comap f K).jacobson
|
Mathlib_RingTheory_JacobsonIdeal
|
case refine'_1
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I : Ideal R
f : R →+* S
hf : Function.Surjective ⇑f
K : Ideal S
⊢ ⨅ I ∈ comap f '' insert ⊤ {J | K ≤ J ∧ IsMaximal J}, I ≤ ⨅ a ∈ {J | comap f K ≤ J ∧ IsMaximal J}, a
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
/-!
# Jacobson radical
The Jacobson radical of a ring `R` is defined to be the intersection of all maximal ideals of `R`.
This is similar to how the nilradical is equal to the intersection of all prime ideals of `R`.
We can extend the idea of the nilradical to ideals of `R`,
by letting the radical of an ideal `I` be the intersection of prime ideals containing `I`.
Under this extension, the original nilradical is the radical of the zero ideal `⊥`.
Here we define the Jacobson radical of an ideal `I` in a similar way,
as the intersection of maximal ideals containing `I`.
## Main definitions
Let `R` be a commutative ring, and `I` be an ideal of `R`
* `Ideal.jacobson I` is the jacobson radical, i.e. the infimum of all maximal ideals containing I.
* `Ideal.IsLocal I` is the proposition that the jacobson radical of `I` is itself a maximal ideal
## Main statements
* `mem_jacobson_iff` gives a characterization of members of the jacobson of I
* `Ideal.isLocal_of_isMaximal_radical`: if the radical of I is maximal then so is the jacobson
radical
## Tags
Jacobson, Jacobson radical, Local Ideal
-/
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}
open Polynomial
section Jacobson
section Ring
variable [Ring R] [Ring S] {I : Ideal R}
/-- The Jacobson radical of `I` is the infimum of all maximal (left) ideals containing `I`. -/
def jacobson (I : Ideal R) : Ideal R :=
sInf { J : Ideal R | I ≤ J ∧ IsMaximal J }
#align ideal.jacobson Ideal.jacobson
theorem le_jacobson : I ≤ jacobson I := fun _ hx => mem_sInf.mpr fun _ hJ => hJ.left hx
#align ideal.le_jacobson Ideal.le_jacobson
@[simp]
theorem jacobson_idem : jacobson (jacobson I) = jacobson I :=
le_antisymm (sInf_le_sInf fun _ hJ => ⟨sInf_le hJ, hJ.2⟩) le_jacobson
#align ideal.jacobson_idem Ideal.jacobson_idem
@[simp]
theorem jacobson_top : jacobson (⊤ : Ideal R) = ⊤ :=
eq_top_iff.2 le_jacobson
#align ideal.jacobson_top Ideal.jacobson_top
@[simp]
theorem jacobson_eq_top_iff : jacobson I = ⊤ ↔ I = ⊤ :=
⟨fun H =>
by_contradiction fun hi => let ⟨M, hm, him⟩ := exists_le_maximal I hi
lt_top_iff_ne_top.1
(lt_of_le_of_lt (show jacobson I ≤ M from sInf_le ⟨him, hm⟩) <|
lt_top_iff_ne_top.2 hm.ne_top) H,
fun H => eq_top_iff.2 <| le_sInf fun _ ⟨hij, _⟩ => H ▸ hij⟩
#align ideal.jacobson_eq_top_iff Ideal.jacobson_eq_top_iff
theorem jacobson_eq_bot : jacobson I = ⊥ → I = ⊥ := fun h => eq_bot_iff.mpr (h ▸ le_jacobson)
#align ideal.jacobson_eq_bot Ideal.jacobson_eq_bot
theorem jacobson_eq_self_of_isMaximal [H : IsMaximal I] : I.jacobson = I :=
le_antisymm (sInf_le ⟨le_of_eq rfl, H⟩) le_jacobson
#align ideal.jacobson_eq_self_of_is_maximal Ideal.jacobson_eq_self_of_isMaximal
instance (priority := 100) jacobson.isMaximal [H : IsMaximal I] : IsMaximal (jacobson I) :=
⟨⟨fun htop => H.1.1 (jacobson_eq_top_iff.1 htop), fun _ hJ =>
H.1.2 _ (lt_of_le_of_lt le_jacobson hJ)⟩⟩
#align ideal.jacobson.is_maximal Ideal.jacobson.isMaximal
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I :=
⟨fun hx y =>
by_cases
(fun hxy : I ⊔ span {y * x + 1} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy)
let ⟨r, hr⟩ := mem_span_singleton'.1 hq
⟨r, by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one r (y * x), hr, ← hpq, ← neg_sub, add_sub_cancel]
exact I.neg_mem hpi⟩)
fun hxy : I ⊔ span {y * x + 1} ≠ ⊤ => let ⟨M, hm1, hm2⟩ := exists_le_maximal _ hxy
suffices x ∉ M from (this <| mem_sInf.1 hx ⟨le_trans le_sup_left hm2, hm1⟩).elim
fun hxm => hm1.1.1 <| (eq_top_iff_one _).2 <| add_sub_cancel' (y * x) 1 ▸
M.sub_mem (le_sup_right.trans hm2 <| subset_span rfl) (M.mul_mem_left _ hxm),
fun hx => mem_sInf.2 fun M ⟨him, hm⟩ => by_contradiction fun hxm =>
let ⟨y, i, hi, df⟩ := hm.exists_inv hxm
let ⟨z, hz⟩ := hx (-y)
hm.1.1 <| (eq_top_iff_one _).2 <| sub_sub_cancel (z * -y * x + z) 1 ▸
M.sub_mem (by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one z, neg_mul, ← sub_eq_iff_eq_add.mpr df.symm, neg_sub,
sub_add_cancel]
exact M.mul_mem_left _ hi) <| him hz⟩
#align ideal.mem_jacobson_iff Ideal.mem_jacobson_iff
theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
∃ s, s * r - 1 ∈ I := by
cases' mem_jacobson_iff.1 h 1 with s hs
use s
simpa [mul_sub] using hs
#align ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson Ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M := by
use fun hI => ⟨{ J : Ideal R | I ≤ J ∧ J.IsMaximal }, ⟨fun _ hJ => Or.inl hJ.right, hI.symm⟩⟩
rintro ⟨M, hM, hInf⟩
refine le_antisymm (fun x hx => ?_) le_jacobson
rw [hInf, mem_sInf]
intro I hI
cases' hM I hI with is_max is_top
· exact (mem_sInf.1 hx) ⟨le_sInf_iff.1 (le_of_eq hInf) I hI, is_max⟩
· exact is_top.symm ▸ Submodule.mem_top
#align ideal.eq_jacobson_iff_Inf_maximal Ideal.eq_jacobson_iff_sInf_maximal
theorem eq_jacobson_iff_sInf_maximal' :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, ∀ (K : Ideal R), J < K → K = ⊤) ∧ I = sInf M :=
eq_jacobson_iff_sInf_maximal.trans
⟨fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ K hK =>
Or.recOn (hM.1 J hJ) (fun h => h.1.2 K hK) fun h => eq_top_iff.2 (le_of_lt (h ▸ hK)),
hM.2⟩⟩,
fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ =>
Or.recOn (Classical.em (J = ⊤)) (fun h => Or.inr h) fun h => Or.inl ⟨⟨h, hM.1 J hJ⟩⟩,
hM.2⟩⟩⟩
#align ideal.eq_jacobson_iff_Inf_maximal' Ideal.eq_jacobson_iff_sInf_maximal'
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M := by
constructor
· intro h x hx
erw [← h, mem_sInf] at hx
push_neg at hx
exact hx
· refine fun h => le_antisymm (fun x hx => ?_) le_jacobson
contrapose hx
erw [mem_sInf]
push_neg
exact h x hx
#align ideal.eq_jacobson_iff_not_mem Ideal.eq_jacobson_iff_not_mem
theorem map_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) :
RingHom.ker f ≤ I → map f I.jacobson = (map f I).jacobson := by
intro h
unfold Ideal.jacobson
-- porting note : dot notation for `RingHom.ker` does not work
have : ∀ J ∈ { J : Ideal R | I ≤ J ∧ J.IsMaximal }, RingHom.ker f ≤ J :=
fun J hJ => le_trans h hJ.left
refine Trans.trans (map_sInf hf this) (le_antisymm ?_ ?_)
· refine'
sInf_le_sInf fun J hJ =>
⟨comap f J, ⟨⟨le_comap_of_map_le hJ.1, _⟩, map_comap_of_surjective f hf J⟩⟩
haveI : J.IsMaximal := hJ.right
exact comap_isMaximal_of_surjective f hf
· refine' sInf_le_sInf_of_subset_insert_top fun j hj => hj.recOn fun J hJ => _
rw [← hJ.2]
cases' map_eq_top_or_isMaximal_of_surjective f hf hJ.left.right with htop hmax
· exact htop.symm ▸ Set.mem_insert ⊤ _
· exact Set.mem_insert_of_mem ⊤ ⟨map_mono hJ.1.1, hmax⟩
#align ideal.map_jacobson_of_surjective Ideal.map_jacobson_of_surjective
theorem map_jacobson_of_bijective {f : R →+* S} (hf : Function.Bijective f) :
map f I.jacobson = (map f I).jacobson :=
map_jacobson_of_surjective hf.right
(le_trans (le_of_eq (f.injective_iff_ker_eq_bot.1 hf.left)) bot_le)
#align ideal.map_jacobson_of_bijective Ideal.map_jacobson_of_bijective
theorem comap_jacobson {f : R →+* S} {K : Ideal S} :
comap f K.jacobson = sInf (comap f '' { J : Ideal S | K ≤ J ∧ J.IsMaximal }) :=
Trans.trans (comap_sInf' f _) sInf_eq_iInf.symm
#align ideal.comap_jacobson Ideal.comap_jacobson
theorem comap_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) {K : Ideal S} :
comap f K.jacobson = (comap f K).jacobson := by
unfold Ideal.jacobson
refine' le_antisymm _ _
· refine le_trans (comap_mono (le_of_eq (Trans.trans top_inf_eq.symm sInf_insert.symm))) ?_
rw [comap_sInf', sInf_eq_iInf]
|
refine' iInf_le_iInf_of_subset fun J hJ => _
|
theorem comap_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) {K : Ideal S} :
comap f K.jacobson = (comap f K).jacobson := by
unfold Ideal.jacobson
refine' le_antisymm _ _
· refine le_trans (comap_mono (le_of_eq (Trans.trans top_inf_eq.symm sInf_insert.symm))) ?_
rw [comap_sInf', sInf_eq_iInf]
|
Mathlib.RingTheory.JacobsonIdeal.209_0.Lz0MgLQMj1bGzuN
|
theorem comap_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) {K : Ideal S} :
comap f K.jacobson = (comap f K).jacobson
|
Mathlib_RingTheory_JacobsonIdeal
|
case refine'_1
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I : Ideal R
f : R →+* S
hf : Function.Surjective ⇑f
K : Ideal S
J : Ideal R
hJ : J ∈ {J | comap f K ≤ J ∧ IsMaximal J}
⊢ J ∈ comap f '' insert ⊤ {J | K ≤ J ∧ IsMaximal J}
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
/-!
# Jacobson radical
The Jacobson radical of a ring `R` is defined to be the intersection of all maximal ideals of `R`.
This is similar to how the nilradical is equal to the intersection of all prime ideals of `R`.
We can extend the idea of the nilradical to ideals of `R`,
by letting the radical of an ideal `I` be the intersection of prime ideals containing `I`.
Under this extension, the original nilradical is the radical of the zero ideal `⊥`.
Here we define the Jacobson radical of an ideal `I` in a similar way,
as the intersection of maximal ideals containing `I`.
## Main definitions
Let `R` be a commutative ring, and `I` be an ideal of `R`
* `Ideal.jacobson I` is the jacobson radical, i.e. the infimum of all maximal ideals containing I.
* `Ideal.IsLocal I` is the proposition that the jacobson radical of `I` is itself a maximal ideal
## Main statements
* `mem_jacobson_iff` gives a characterization of members of the jacobson of I
* `Ideal.isLocal_of_isMaximal_radical`: if the radical of I is maximal then so is the jacobson
radical
## Tags
Jacobson, Jacobson radical, Local Ideal
-/
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}
open Polynomial
section Jacobson
section Ring
variable [Ring R] [Ring S] {I : Ideal R}
/-- The Jacobson radical of `I` is the infimum of all maximal (left) ideals containing `I`. -/
def jacobson (I : Ideal R) : Ideal R :=
sInf { J : Ideal R | I ≤ J ∧ IsMaximal J }
#align ideal.jacobson Ideal.jacobson
theorem le_jacobson : I ≤ jacobson I := fun _ hx => mem_sInf.mpr fun _ hJ => hJ.left hx
#align ideal.le_jacobson Ideal.le_jacobson
@[simp]
theorem jacobson_idem : jacobson (jacobson I) = jacobson I :=
le_antisymm (sInf_le_sInf fun _ hJ => ⟨sInf_le hJ, hJ.2⟩) le_jacobson
#align ideal.jacobson_idem Ideal.jacobson_idem
@[simp]
theorem jacobson_top : jacobson (⊤ : Ideal R) = ⊤ :=
eq_top_iff.2 le_jacobson
#align ideal.jacobson_top Ideal.jacobson_top
@[simp]
theorem jacobson_eq_top_iff : jacobson I = ⊤ ↔ I = ⊤ :=
⟨fun H =>
by_contradiction fun hi => let ⟨M, hm, him⟩ := exists_le_maximal I hi
lt_top_iff_ne_top.1
(lt_of_le_of_lt (show jacobson I ≤ M from sInf_le ⟨him, hm⟩) <|
lt_top_iff_ne_top.2 hm.ne_top) H,
fun H => eq_top_iff.2 <| le_sInf fun _ ⟨hij, _⟩ => H ▸ hij⟩
#align ideal.jacobson_eq_top_iff Ideal.jacobson_eq_top_iff
theorem jacobson_eq_bot : jacobson I = ⊥ → I = ⊥ := fun h => eq_bot_iff.mpr (h ▸ le_jacobson)
#align ideal.jacobson_eq_bot Ideal.jacobson_eq_bot
theorem jacobson_eq_self_of_isMaximal [H : IsMaximal I] : I.jacobson = I :=
le_antisymm (sInf_le ⟨le_of_eq rfl, H⟩) le_jacobson
#align ideal.jacobson_eq_self_of_is_maximal Ideal.jacobson_eq_self_of_isMaximal
instance (priority := 100) jacobson.isMaximal [H : IsMaximal I] : IsMaximal (jacobson I) :=
⟨⟨fun htop => H.1.1 (jacobson_eq_top_iff.1 htop), fun _ hJ =>
H.1.2 _ (lt_of_le_of_lt le_jacobson hJ)⟩⟩
#align ideal.jacobson.is_maximal Ideal.jacobson.isMaximal
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I :=
⟨fun hx y =>
by_cases
(fun hxy : I ⊔ span {y * x + 1} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy)
let ⟨r, hr⟩ := mem_span_singleton'.1 hq
⟨r, by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one r (y * x), hr, ← hpq, ← neg_sub, add_sub_cancel]
exact I.neg_mem hpi⟩)
fun hxy : I ⊔ span {y * x + 1} ≠ ⊤ => let ⟨M, hm1, hm2⟩ := exists_le_maximal _ hxy
suffices x ∉ M from (this <| mem_sInf.1 hx ⟨le_trans le_sup_left hm2, hm1⟩).elim
fun hxm => hm1.1.1 <| (eq_top_iff_one _).2 <| add_sub_cancel' (y * x) 1 ▸
M.sub_mem (le_sup_right.trans hm2 <| subset_span rfl) (M.mul_mem_left _ hxm),
fun hx => mem_sInf.2 fun M ⟨him, hm⟩ => by_contradiction fun hxm =>
let ⟨y, i, hi, df⟩ := hm.exists_inv hxm
let ⟨z, hz⟩ := hx (-y)
hm.1.1 <| (eq_top_iff_one _).2 <| sub_sub_cancel (z * -y * x + z) 1 ▸
M.sub_mem (by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one z, neg_mul, ← sub_eq_iff_eq_add.mpr df.symm, neg_sub,
sub_add_cancel]
exact M.mul_mem_left _ hi) <| him hz⟩
#align ideal.mem_jacobson_iff Ideal.mem_jacobson_iff
theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
∃ s, s * r - 1 ∈ I := by
cases' mem_jacobson_iff.1 h 1 with s hs
use s
simpa [mul_sub] using hs
#align ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson Ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M := by
use fun hI => ⟨{ J : Ideal R | I ≤ J ∧ J.IsMaximal }, ⟨fun _ hJ => Or.inl hJ.right, hI.symm⟩⟩
rintro ⟨M, hM, hInf⟩
refine le_antisymm (fun x hx => ?_) le_jacobson
rw [hInf, mem_sInf]
intro I hI
cases' hM I hI with is_max is_top
· exact (mem_sInf.1 hx) ⟨le_sInf_iff.1 (le_of_eq hInf) I hI, is_max⟩
· exact is_top.symm ▸ Submodule.mem_top
#align ideal.eq_jacobson_iff_Inf_maximal Ideal.eq_jacobson_iff_sInf_maximal
theorem eq_jacobson_iff_sInf_maximal' :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, ∀ (K : Ideal R), J < K → K = ⊤) ∧ I = sInf M :=
eq_jacobson_iff_sInf_maximal.trans
⟨fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ K hK =>
Or.recOn (hM.1 J hJ) (fun h => h.1.2 K hK) fun h => eq_top_iff.2 (le_of_lt (h ▸ hK)),
hM.2⟩⟩,
fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ =>
Or.recOn (Classical.em (J = ⊤)) (fun h => Or.inr h) fun h => Or.inl ⟨⟨h, hM.1 J hJ⟩⟩,
hM.2⟩⟩⟩
#align ideal.eq_jacobson_iff_Inf_maximal' Ideal.eq_jacobson_iff_sInf_maximal'
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M := by
constructor
· intro h x hx
erw [← h, mem_sInf] at hx
push_neg at hx
exact hx
· refine fun h => le_antisymm (fun x hx => ?_) le_jacobson
contrapose hx
erw [mem_sInf]
push_neg
exact h x hx
#align ideal.eq_jacobson_iff_not_mem Ideal.eq_jacobson_iff_not_mem
theorem map_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) :
RingHom.ker f ≤ I → map f I.jacobson = (map f I).jacobson := by
intro h
unfold Ideal.jacobson
-- porting note : dot notation for `RingHom.ker` does not work
have : ∀ J ∈ { J : Ideal R | I ≤ J ∧ J.IsMaximal }, RingHom.ker f ≤ J :=
fun J hJ => le_trans h hJ.left
refine Trans.trans (map_sInf hf this) (le_antisymm ?_ ?_)
· refine'
sInf_le_sInf fun J hJ =>
⟨comap f J, ⟨⟨le_comap_of_map_le hJ.1, _⟩, map_comap_of_surjective f hf J⟩⟩
haveI : J.IsMaximal := hJ.right
exact comap_isMaximal_of_surjective f hf
· refine' sInf_le_sInf_of_subset_insert_top fun j hj => hj.recOn fun J hJ => _
rw [← hJ.2]
cases' map_eq_top_or_isMaximal_of_surjective f hf hJ.left.right with htop hmax
· exact htop.symm ▸ Set.mem_insert ⊤ _
· exact Set.mem_insert_of_mem ⊤ ⟨map_mono hJ.1.1, hmax⟩
#align ideal.map_jacobson_of_surjective Ideal.map_jacobson_of_surjective
theorem map_jacobson_of_bijective {f : R →+* S} (hf : Function.Bijective f) :
map f I.jacobson = (map f I).jacobson :=
map_jacobson_of_surjective hf.right
(le_trans (le_of_eq (f.injective_iff_ker_eq_bot.1 hf.left)) bot_le)
#align ideal.map_jacobson_of_bijective Ideal.map_jacobson_of_bijective
theorem comap_jacobson {f : R →+* S} {K : Ideal S} :
comap f K.jacobson = sInf (comap f '' { J : Ideal S | K ≤ J ∧ J.IsMaximal }) :=
Trans.trans (comap_sInf' f _) sInf_eq_iInf.symm
#align ideal.comap_jacobson Ideal.comap_jacobson
theorem comap_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) {K : Ideal S} :
comap f K.jacobson = (comap f K).jacobson := by
unfold Ideal.jacobson
refine' le_antisymm _ _
· refine le_trans (comap_mono (le_of_eq (Trans.trans top_inf_eq.symm sInf_insert.symm))) ?_
rw [comap_sInf', sInf_eq_iInf]
refine' iInf_le_iInf_of_subset fun J hJ => _
|
have : comap f (map f J) = J :=
Trans.trans (comap_map_of_surjective f hf J)
(le_antisymm (sup_le_iff.2 ⟨le_of_eq rfl, le_trans (comap_mono bot_le) hJ.left⟩)
le_sup_left)
|
theorem comap_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) {K : Ideal S} :
comap f K.jacobson = (comap f K).jacobson := by
unfold Ideal.jacobson
refine' le_antisymm _ _
· refine le_trans (comap_mono (le_of_eq (Trans.trans top_inf_eq.symm sInf_insert.symm))) ?_
rw [comap_sInf', sInf_eq_iInf]
refine' iInf_le_iInf_of_subset fun J hJ => _
|
Mathlib.RingTheory.JacobsonIdeal.209_0.Lz0MgLQMj1bGzuN
|
theorem comap_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) {K : Ideal S} :
comap f K.jacobson = (comap f K).jacobson
|
Mathlib_RingTheory_JacobsonIdeal
|
case refine'_1
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I : Ideal R
f : R →+* S
hf : Function.Surjective ⇑f
K : Ideal S
J : Ideal R
hJ : J ∈ {J | comap f K ≤ J ∧ IsMaximal J}
this : comap f (map f J) = J
⊢ J ∈ comap f '' insert ⊤ {J | K ≤ J ∧ IsMaximal J}
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
/-!
# Jacobson radical
The Jacobson radical of a ring `R` is defined to be the intersection of all maximal ideals of `R`.
This is similar to how the nilradical is equal to the intersection of all prime ideals of `R`.
We can extend the idea of the nilradical to ideals of `R`,
by letting the radical of an ideal `I` be the intersection of prime ideals containing `I`.
Under this extension, the original nilradical is the radical of the zero ideal `⊥`.
Here we define the Jacobson radical of an ideal `I` in a similar way,
as the intersection of maximal ideals containing `I`.
## Main definitions
Let `R` be a commutative ring, and `I` be an ideal of `R`
* `Ideal.jacobson I` is the jacobson radical, i.e. the infimum of all maximal ideals containing I.
* `Ideal.IsLocal I` is the proposition that the jacobson radical of `I` is itself a maximal ideal
## Main statements
* `mem_jacobson_iff` gives a characterization of members of the jacobson of I
* `Ideal.isLocal_of_isMaximal_radical`: if the radical of I is maximal then so is the jacobson
radical
## Tags
Jacobson, Jacobson radical, Local Ideal
-/
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}
open Polynomial
section Jacobson
section Ring
variable [Ring R] [Ring S] {I : Ideal R}
/-- The Jacobson radical of `I` is the infimum of all maximal (left) ideals containing `I`. -/
def jacobson (I : Ideal R) : Ideal R :=
sInf { J : Ideal R | I ≤ J ∧ IsMaximal J }
#align ideal.jacobson Ideal.jacobson
theorem le_jacobson : I ≤ jacobson I := fun _ hx => mem_sInf.mpr fun _ hJ => hJ.left hx
#align ideal.le_jacobson Ideal.le_jacobson
@[simp]
theorem jacobson_idem : jacobson (jacobson I) = jacobson I :=
le_antisymm (sInf_le_sInf fun _ hJ => ⟨sInf_le hJ, hJ.2⟩) le_jacobson
#align ideal.jacobson_idem Ideal.jacobson_idem
@[simp]
theorem jacobson_top : jacobson (⊤ : Ideal R) = ⊤ :=
eq_top_iff.2 le_jacobson
#align ideal.jacobson_top Ideal.jacobson_top
@[simp]
theorem jacobson_eq_top_iff : jacobson I = ⊤ ↔ I = ⊤ :=
⟨fun H =>
by_contradiction fun hi => let ⟨M, hm, him⟩ := exists_le_maximal I hi
lt_top_iff_ne_top.1
(lt_of_le_of_lt (show jacobson I ≤ M from sInf_le ⟨him, hm⟩) <|
lt_top_iff_ne_top.2 hm.ne_top) H,
fun H => eq_top_iff.2 <| le_sInf fun _ ⟨hij, _⟩ => H ▸ hij⟩
#align ideal.jacobson_eq_top_iff Ideal.jacobson_eq_top_iff
theorem jacobson_eq_bot : jacobson I = ⊥ → I = ⊥ := fun h => eq_bot_iff.mpr (h ▸ le_jacobson)
#align ideal.jacobson_eq_bot Ideal.jacobson_eq_bot
theorem jacobson_eq_self_of_isMaximal [H : IsMaximal I] : I.jacobson = I :=
le_antisymm (sInf_le ⟨le_of_eq rfl, H⟩) le_jacobson
#align ideal.jacobson_eq_self_of_is_maximal Ideal.jacobson_eq_self_of_isMaximal
instance (priority := 100) jacobson.isMaximal [H : IsMaximal I] : IsMaximal (jacobson I) :=
⟨⟨fun htop => H.1.1 (jacobson_eq_top_iff.1 htop), fun _ hJ =>
H.1.2 _ (lt_of_le_of_lt le_jacobson hJ)⟩⟩
#align ideal.jacobson.is_maximal Ideal.jacobson.isMaximal
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I :=
⟨fun hx y =>
by_cases
(fun hxy : I ⊔ span {y * x + 1} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy)
let ⟨r, hr⟩ := mem_span_singleton'.1 hq
⟨r, by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one r (y * x), hr, ← hpq, ← neg_sub, add_sub_cancel]
exact I.neg_mem hpi⟩)
fun hxy : I ⊔ span {y * x + 1} ≠ ⊤ => let ⟨M, hm1, hm2⟩ := exists_le_maximal _ hxy
suffices x ∉ M from (this <| mem_sInf.1 hx ⟨le_trans le_sup_left hm2, hm1⟩).elim
fun hxm => hm1.1.1 <| (eq_top_iff_one _).2 <| add_sub_cancel' (y * x) 1 ▸
M.sub_mem (le_sup_right.trans hm2 <| subset_span rfl) (M.mul_mem_left _ hxm),
fun hx => mem_sInf.2 fun M ⟨him, hm⟩ => by_contradiction fun hxm =>
let ⟨y, i, hi, df⟩ := hm.exists_inv hxm
let ⟨z, hz⟩ := hx (-y)
hm.1.1 <| (eq_top_iff_one _).2 <| sub_sub_cancel (z * -y * x + z) 1 ▸
M.sub_mem (by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one z, neg_mul, ← sub_eq_iff_eq_add.mpr df.symm, neg_sub,
sub_add_cancel]
exact M.mul_mem_left _ hi) <| him hz⟩
#align ideal.mem_jacobson_iff Ideal.mem_jacobson_iff
theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
∃ s, s * r - 1 ∈ I := by
cases' mem_jacobson_iff.1 h 1 with s hs
use s
simpa [mul_sub] using hs
#align ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson Ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M := by
use fun hI => ⟨{ J : Ideal R | I ≤ J ∧ J.IsMaximal }, ⟨fun _ hJ => Or.inl hJ.right, hI.symm⟩⟩
rintro ⟨M, hM, hInf⟩
refine le_antisymm (fun x hx => ?_) le_jacobson
rw [hInf, mem_sInf]
intro I hI
cases' hM I hI with is_max is_top
· exact (mem_sInf.1 hx) ⟨le_sInf_iff.1 (le_of_eq hInf) I hI, is_max⟩
· exact is_top.symm ▸ Submodule.mem_top
#align ideal.eq_jacobson_iff_Inf_maximal Ideal.eq_jacobson_iff_sInf_maximal
theorem eq_jacobson_iff_sInf_maximal' :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, ∀ (K : Ideal R), J < K → K = ⊤) ∧ I = sInf M :=
eq_jacobson_iff_sInf_maximal.trans
⟨fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ K hK =>
Or.recOn (hM.1 J hJ) (fun h => h.1.2 K hK) fun h => eq_top_iff.2 (le_of_lt (h ▸ hK)),
hM.2⟩⟩,
fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ =>
Or.recOn (Classical.em (J = ⊤)) (fun h => Or.inr h) fun h => Or.inl ⟨⟨h, hM.1 J hJ⟩⟩,
hM.2⟩⟩⟩
#align ideal.eq_jacobson_iff_Inf_maximal' Ideal.eq_jacobson_iff_sInf_maximal'
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M := by
constructor
· intro h x hx
erw [← h, mem_sInf] at hx
push_neg at hx
exact hx
· refine fun h => le_antisymm (fun x hx => ?_) le_jacobson
contrapose hx
erw [mem_sInf]
push_neg
exact h x hx
#align ideal.eq_jacobson_iff_not_mem Ideal.eq_jacobson_iff_not_mem
theorem map_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) :
RingHom.ker f ≤ I → map f I.jacobson = (map f I).jacobson := by
intro h
unfold Ideal.jacobson
-- porting note : dot notation for `RingHom.ker` does not work
have : ∀ J ∈ { J : Ideal R | I ≤ J ∧ J.IsMaximal }, RingHom.ker f ≤ J :=
fun J hJ => le_trans h hJ.left
refine Trans.trans (map_sInf hf this) (le_antisymm ?_ ?_)
· refine'
sInf_le_sInf fun J hJ =>
⟨comap f J, ⟨⟨le_comap_of_map_le hJ.1, _⟩, map_comap_of_surjective f hf J⟩⟩
haveI : J.IsMaximal := hJ.right
exact comap_isMaximal_of_surjective f hf
· refine' sInf_le_sInf_of_subset_insert_top fun j hj => hj.recOn fun J hJ => _
rw [← hJ.2]
cases' map_eq_top_or_isMaximal_of_surjective f hf hJ.left.right with htop hmax
· exact htop.symm ▸ Set.mem_insert ⊤ _
· exact Set.mem_insert_of_mem ⊤ ⟨map_mono hJ.1.1, hmax⟩
#align ideal.map_jacobson_of_surjective Ideal.map_jacobson_of_surjective
theorem map_jacobson_of_bijective {f : R →+* S} (hf : Function.Bijective f) :
map f I.jacobson = (map f I).jacobson :=
map_jacobson_of_surjective hf.right
(le_trans (le_of_eq (f.injective_iff_ker_eq_bot.1 hf.left)) bot_le)
#align ideal.map_jacobson_of_bijective Ideal.map_jacobson_of_bijective
theorem comap_jacobson {f : R →+* S} {K : Ideal S} :
comap f K.jacobson = sInf (comap f '' { J : Ideal S | K ≤ J ∧ J.IsMaximal }) :=
Trans.trans (comap_sInf' f _) sInf_eq_iInf.symm
#align ideal.comap_jacobson Ideal.comap_jacobson
theorem comap_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) {K : Ideal S} :
comap f K.jacobson = (comap f K).jacobson := by
unfold Ideal.jacobson
refine' le_antisymm _ _
· refine le_trans (comap_mono (le_of_eq (Trans.trans top_inf_eq.symm sInf_insert.symm))) ?_
rw [comap_sInf', sInf_eq_iInf]
refine' iInf_le_iInf_of_subset fun J hJ => _
have : comap f (map f J) = J :=
Trans.trans (comap_map_of_surjective f hf J)
(le_antisymm (sup_le_iff.2 ⟨le_of_eq rfl, le_trans (comap_mono bot_le) hJ.left⟩)
le_sup_left)
|
cases' map_eq_top_or_isMaximal_of_surjective _ hf hJ.right with htop hmax
|
theorem comap_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) {K : Ideal S} :
comap f K.jacobson = (comap f K).jacobson := by
unfold Ideal.jacobson
refine' le_antisymm _ _
· refine le_trans (comap_mono (le_of_eq (Trans.trans top_inf_eq.symm sInf_insert.symm))) ?_
rw [comap_sInf', sInf_eq_iInf]
refine' iInf_le_iInf_of_subset fun J hJ => _
have : comap f (map f J) = J :=
Trans.trans (comap_map_of_surjective f hf J)
(le_antisymm (sup_le_iff.2 ⟨le_of_eq rfl, le_trans (comap_mono bot_le) hJ.left⟩)
le_sup_left)
|
Mathlib.RingTheory.JacobsonIdeal.209_0.Lz0MgLQMj1bGzuN
|
theorem comap_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) {K : Ideal S} :
comap f K.jacobson = (comap f K).jacobson
|
Mathlib_RingTheory_JacobsonIdeal
|
case refine'_1.inl
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I : Ideal R
f : R →+* S
hf : Function.Surjective ⇑f
K : Ideal S
J : Ideal R
hJ : J ∈ {J | comap f K ≤ J ∧ IsMaximal J}
this : comap f (map f J) = J
htop : map f J = ⊤
⊢ J ∈ comap f '' insert ⊤ {J | K ≤ J ∧ IsMaximal J}
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
/-!
# Jacobson radical
The Jacobson radical of a ring `R` is defined to be the intersection of all maximal ideals of `R`.
This is similar to how the nilradical is equal to the intersection of all prime ideals of `R`.
We can extend the idea of the nilradical to ideals of `R`,
by letting the radical of an ideal `I` be the intersection of prime ideals containing `I`.
Under this extension, the original nilradical is the radical of the zero ideal `⊥`.
Here we define the Jacobson radical of an ideal `I` in a similar way,
as the intersection of maximal ideals containing `I`.
## Main definitions
Let `R` be a commutative ring, and `I` be an ideal of `R`
* `Ideal.jacobson I` is the jacobson radical, i.e. the infimum of all maximal ideals containing I.
* `Ideal.IsLocal I` is the proposition that the jacobson radical of `I` is itself a maximal ideal
## Main statements
* `mem_jacobson_iff` gives a characterization of members of the jacobson of I
* `Ideal.isLocal_of_isMaximal_radical`: if the radical of I is maximal then so is the jacobson
radical
## Tags
Jacobson, Jacobson radical, Local Ideal
-/
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}
open Polynomial
section Jacobson
section Ring
variable [Ring R] [Ring S] {I : Ideal R}
/-- The Jacobson radical of `I` is the infimum of all maximal (left) ideals containing `I`. -/
def jacobson (I : Ideal R) : Ideal R :=
sInf { J : Ideal R | I ≤ J ∧ IsMaximal J }
#align ideal.jacobson Ideal.jacobson
theorem le_jacobson : I ≤ jacobson I := fun _ hx => mem_sInf.mpr fun _ hJ => hJ.left hx
#align ideal.le_jacobson Ideal.le_jacobson
@[simp]
theorem jacobson_idem : jacobson (jacobson I) = jacobson I :=
le_antisymm (sInf_le_sInf fun _ hJ => ⟨sInf_le hJ, hJ.2⟩) le_jacobson
#align ideal.jacobson_idem Ideal.jacobson_idem
@[simp]
theorem jacobson_top : jacobson (⊤ : Ideal R) = ⊤ :=
eq_top_iff.2 le_jacobson
#align ideal.jacobson_top Ideal.jacobson_top
@[simp]
theorem jacobson_eq_top_iff : jacobson I = ⊤ ↔ I = ⊤ :=
⟨fun H =>
by_contradiction fun hi => let ⟨M, hm, him⟩ := exists_le_maximal I hi
lt_top_iff_ne_top.1
(lt_of_le_of_lt (show jacobson I ≤ M from sInf_le ⟨him, hm⟩) <|
lt_top_iff_ne_top.2 hm.ne_top) H,
fun H => eq_top_iff.2 <| le_sInf fun _ ⟨hij, _⟩ => H ▸ hij⟩
#align ideal.jacobson_eq_top_iff Ideal.jacobson_eq_top_iff
theorem jacobson_eq_bot : jacobson I = ⊥ → I = ⊥ := fun h => eq_bot_iff.mpr (h ▸ le_jacobson)
#align ideal.jacobson_eq_bot Ideal.jacobson_eq_bot
theorem jacobson_eq_self_of_isMaximal [H : IsMaximal I] : I.jacobson = I :=
le_antisymm (sInf_le ⟨le_of_eq rfl, H⟩) le_jacobson
#align ideal.jacobson_eq_self_of_is_maximal Ideal.jacobson_eq_self_of_isMaximal
instance (priority := 100) jacobson.isMaximal [H : IsMaximal I] : IsMaximal (jacobson I) :=
⟨⟨fun htop => H.1.1 (jacobson_eq_top_iff.1 htop), fun _ hJ =>
H.1.2 _ (lt_of_le_of_lt le_jacobson hJ)⟩⟩
#align ideal.jacobson.is_maximal Ideal.jacobson.isMaximal
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I :=
⟨fun hx y =>
by_cases
(fun hxy : I ⊔ span {y * x + 1} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy)
let ⟨r, hr⟩ := mem_span_singleton'.1 hq
⟨r, by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one r (y * x), hr, ← hpq, ← neg_sub, add_sub_cancel]
exact I.neg_mem hpi⟩)
fun hxy : I ⊔ span {y * x + 1} ≠ ⊤ => let ⟨M, hm1, hm2⟩ := exists_le_maximal _ hxy
suffices x ∉ M from (this <| mem_sInf.1 hx ⟨le_trans le_sup_left hm2, hm1⟩).elim
fun hxm => hm1.1.1 <| (eq_top_iff_one _).2 <| add_sub_cancel' (y * x) 1 ▸
M.sub_mem (le_sup_right.trans hm2 <| subset_span rfl) (M.mul_mem_left _ hxm),
fun hx => mem_sInf.2 fun M ⟨him, hm⟩ => by_contradiction fun hxm =>
let ⟨y, i, hi, df⟩ := hm.exists_inv hxm
let ⟨z, hz⟩ := hx (-y)
hm.1.1 <| (eq_top_iff_one _).2 <| sub_sub_cancel (z * -y * x + z) 1 ▸
M.sub_mem (by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one z, neg_mul, ← sub_eq_iff_eq_add.mpr df.symm, neg_sub,
sub_add_cancel]
exact M.mul_mem_left _ hi) <| him hz⟩
#align ideal.mem_jacobson_iff Ideal.mem_jacobson_iff
theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
∃ s, s * r - 1 ∈ I := by
cases' mem_jacobson_iff.1 h 1 with s hs
use s
simpa [mul_sub] using hs
#align ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson Ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M := by
use fun hI => ⟨{ J : Ideal R | I ≤ J ∧ J.IsMaximal }, ⟨fun _ hJ => Or.inl hJ.right, hI.symm⟩⟩
rintro ⟨M, hM, hInf⟩
refine le_antisymm (fun x hx => ?_) le_jacobson
rw [hInf, mem_sInf]
intro I hI
cases' hM I hI with is_max is_top
· exact (mem_sInf.1 hx) ⟨le_sInf_iff.1 (le_of_eq hInf) I hI, is_max⟩
· exact is_top.symm ▸ Submodule.mem_top
#align ideal.eq_jacobson_iff_Inf_maximal Ideal.eq_jacobson_iff_sInf_maximal
theorem eq_jacobson_iff_sInf_maximal' :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, ∀ (K : Ideal R), J < K → K = ⊤) ∧ I = sInf M :=
eq_jacobson_iff_sInf_maximal.trans
⟨fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ K hK =>
Or.recOn (hM.1 J hJ) (fun h => h.1.2 K hK) fun h => eq_top_iff.2 (le_of_lt (h ▸ hK)),
hM.2⟩⟩,
fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ =>
Or.recOn (Classical.em (J = ⊤)) (fun h => Or.inr h) fun h => Or.inl ⟨⟨h, hM.1 J hJ⟩⟩,
hM.2⟩⟩⟩
#align ideal.eq_jacobson_iff_Inf_maximal' Ideal.eq_jacobson_iff_sInf_maximal'
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M := by
constructor
· intro h x hx
erw [← h, mem_sInf] at hx
push_neg at hx
exact hx
· refine fun h => le_antisymm (fun x hx => ?_) le_jacobson
contrapose hx
erw [mem_sInf]
push_neg
exact h x hx
#align ideal.eq_jacobson_iff_not_mem Ideal.eq_jacobson_iff_not_mem
theorem map_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) :
RingHom.ker f ≤ I → map f I.jacobson = (map f I).jacobson := by
intro h
unfold Ideal.jacobson
-- porting note : dot notation for `RingHom.ker` does not work
have : ∀ J ∈ { J : Ideal R | I ≤ J ∧ J.IsMaximal }, RingHom.ker f ≤ J :=
fun J hJ => le_trans h hJ.left
refine Trans.trans (map_sInf hf this) (le_antisymm ?_ ?_)
· refine'
sInf_le_sInf fun J hJ =>
⟨comap f J, ⟨⟨le_comap_of_map_le hJ.1, _⟩, map_comap_of_surjective f hf J⟩⟩
haveI : J.IsMaximal := hJ.right
exact comap_isMaximal_of_surjective f hf
· refine' sInf_le_sInf_of_subset_insert_top fun j hj => hj.recOn fun J hJ => _
rw [← hJ.2]
cases' map_eq_top_or_isMaximal_of_surjective f hf hJ.left.right with htop hmax
· exact htop.symm ▸ Set.mem_insert ⊤ _
· exact Set.mem_insert_of_mem ⊤ ⟨map_mono hJ.1.1, hmax⟩
#align ideal.map_jacobson_of_surjective Ideal.map_jacobson_of_surjective
theorem map_jacobson_of_bijective {f : R →+* S} (hf : Function.Bijective f) :
map f I.jacobson = (map f I).jacobson :=
map_jacobson_of_surjective hf.right
(le_trans (le_of_eq (f.injective_iff_ker_eq_bot.1 hf.left)) bot_le)
#align ideal.map_jacobson_of_bijective Ideal.map_jacobson_of_bijective
theorem comap_jacobson {f : R →+* S} {K : Ideal S} :
comap f K.jacobson = sInf (comap f '' { J : Ideal S | K ≤ J ∧ J.IsMaximal }) :=
Trans.trans (comap_sInf' f _) sInf_eq_iInf.symm
#align ideal.comap_jacobson Ideal.comap_jacobson
theorem comap_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) {K : Ideal S} :
comap f K.jacobson = (comap f K).jacobson := by
unfold Ideal.jacobson
refine' le_antisymm _ _
· refine le_trans (comap_mono (le_of_eq (Trans.trans top_inf_eq.symm sInf_insert.symm))) ?_
rw [comap_sInf', sInf_eq_iInf]
refine' iInf_le_iInf_of_subset fun J hJ => _
have : comap f (map f J) = J :=
Trans.trans (comap_map_of_surjective f hf J)
(le_antisymm (sup_le_iff.2 ⟨le_of_eq rfl, le_trans (comap_mono bot_le) hJ.left⟩)
le_sup_left)
cases' map_eq_top_or_isMaximal_of_surjective _ hf hJ.right with htop hmax
·
|
exact ⟨⊤, ⟨Set.mem_insert ⊤ _, htop ▸ this⟩⟩
|
theorem comap_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) {K : Ideal S} :
comap f K.jacobson = (comap f K).jacobson := by
unfold Ideal.jacobson
refine' le_antisymm _ _
· refine le_trans (comap_mono (le_of_eq (Trans.trans top_inf_eq.symm sInf_insert.symm))) ?_
rw [comap_sInf', sInf_eq_iInf]
refine' iInf_le_iInf_of_subset fun J hJ => _
have : comap f (map f J) = J :=
Trans.trans (comap_map_of_surjective f hf J)
(le_antisymm (sup_le_iff.2 ⟨le_of_eq rfl, le_trans (comap_mono bot_le) hJ.left⟩)
le_sup_left)
cases' map_eq_top_or_isMaximal_of_surjective _ hf hJ.right with htop hmax
·
|
Mathlib.RingTheory.JacobsonIdeal.209_0.Lz0MgLQMj1bGzuN
|
theorem comap_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) {K : Ideal S} :
comap f K.jacobson = (comap f K).jacobson
|
Mathlib_RingTheory_JacobsonIdeal
|
case refine'_1.inr
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I : Ideal R
f : R →+* S
hf : Function.Surjective ⇑f
K : Ideal S
J : Ideal R
hJ : J ∈ {J | comap f K ≤ J ∧ IsMaximal J}
this : comap f (map f J) = J
hmax : IsMaximal (map f J)
⊢ J ∈ comap f '' insert ⊤ {J | K ≤ J ∧ IsMaximal J}
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
/-!
# Jacobson radical
The Jacobson radical of a ring `R` is defined to be the intersection of all maximal ideals of `R`.
This is similar to how the nilradical is equal to the intersection of all prime ideals of `R`.
We can extend the idea of the nilradical to ideals of `R`,
by letting the radical of an ideal `I` be the intersection of prime ideals containing `I`.
Under this extension, the original nilradical is the radical of the zero ideal `⊥`.
Here we define the Jacobson radical of an ideal `I` in a similar way,
as the intersection of maximal ideals containing `I`.
## Main definitions
Let `R` be a commutative ring, and `I` be an ideal of `R`
* `Ideal.jacobson I` is the jacobson radical, i.e. the infimum of all maximal ideals containing I.
* `Ideal.IsLocal I` is the proposition that the jacobson radical of `I` is itself a maximal ideal
## Main statements
* `mem_jacobson_iff` gives a characterization of members of the jacobson of I
* `Ideal.isLocal_of_isMaximal_radical`: if the radical of I is maximal then so is the jacobson
radical
## Tags
Jacobson, Jacobson radical, Local Ideal
-/
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}
open Polynomial
section Jacobson
section Ring
variable [Ring R] [Ring S] {I : Ideal R}
/-- The Jacobson radical of `I` is the infimum of all maximal (left) ideals containing `I`. -/
def jacobson (I : Ideal R) : Ideal R :=
sInf { J : Ideal R | I ≤ J ∧ IsMaximal J }
#align ideal.jacobson Ideal.jacobson
theorem le_jacobson : I ≤ jacobson I := fun _ hx => mem_sInf.mpr fun _ hJ => hJ.left hx
#align ideal.le_jacobson Ideal.le_jacobson
@[simp]
theorem jacobson_idem : jacobson (jacobson I) = jacobson I :=
le_antisymm (sInf_le_sInf fun _ hJ => ⟨sInf_le hJ, hJ.2⟩) le_jacobson
#align ideal.jacobson_idem Ideal.jacobson_idem
@[simp]
theorem jacobson_top : jacobson (⊤ : Ideal R) = ⊤ :=
eq_top_iff.2 le_jacobson
#align ideal.jacobson_top Ideal.jacobson_top
@[simp]
theorem jacobson_eq_top_iff : jacobson I = ⊤ ↔ I = ⊤ :=
⟨fun H =>
by_contradiction fun hi => let ⟨M, hm, him⟩ := exists_le_maximal I hi
lt_top_iff_ne_top.1
(lt_of_le_of_lt (show jacobson I ≤ M from sInf_le ⟨him, hm⟩) <|
lt_top_iff_ne_top.2 hm.ne_top) H,
fun H => eq_top_iff.2 <| le_sInf fun _ ⟨hij, _⟩ => H ▸ hij⟩
#align ideal.jacobson_eq_top_iff Ideal.jacobson_eq_top_iff
theorem jacobson_eq_bot : jacobson I = ⊥ → I = ⊥ := fun h => eq_bot_iff.mpr (h ▸ le_jacobson)
#align ideal.jacobson_eq_bot Ideal.jacobson_eq_bot
theorem jacobson_eq_self_of_isMaximal [H : IsMaximal I] : I.jacobson = I :=
le_antisymm (sInf_le ⟨le_of_eq rfl, H⟩) le_jacobson
#align ideal.jacobson_eq_self_of_is_maximal Ideal.jacobson_eq_self_of_isMaximal
instance (priority := 100) jacobson.isMaximal [H : IsMaximal I] : IsMaximal (jacobson I) :=
⟨⟨fun htop => H.1.1 (jacobson_eq_top_iff.1 htop), fun _ hJ =>
H.1.2 _ (lt_of_le_of_lt le_jacobson hJ)⟩⟩
#align ideal.jacobson.is_maximal Ideal.jacobson.isMaximal
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I :=
⟨fun hx y =>
by_cases
(fun hxy : I ⊔ span {y * x + 1} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy)
let ⟨r, hr⟩ := mem_span_singleton'.1 hq
⟨r, by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one r (y * x), hr, ← hpq, ← neg_sub, add_sub_cancel]
exact I.neg_mem hpi⟩)
fun hxy : I ⊔ span {y * x + 1} ≠ ⊤ => let ⟨M, hm1, hm2⟩ := exists_le_maximal _ hxy
suffices x ∉ M from (this <| mem_sInf.1 hx ⟨le_trans le_sup_left hm2, hm1⟩).elim
fun hxm => hm1.1.1 <| (eq_top_iff_one _).2 <| add_sub_cancel' (y * x) 1 ▸
M.sub_mem (le_sup_right.trans hm2 <| subset_span rfl) (M.mul_mem_left _ hxm),
fun hx => mem_sInf.2 fun M ⟨him, hm⟩ => by_contradiction fun hxm =>
let ⟨y, i, hi, df⟩ := hm.exists_inv hxm
let ⟨z, hz⟩ := hx (-y)
hm.1.1 <| (eq_top_iff_one _).2 <| sub_sub_cancel (z * -y * x + z) 1 ▸
M.sub_mem (by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one z, neg_mul, ← sub_eq_iff_eq_add.mpr df.symm, neg_sub,
sub_add_cancel]
exact M.mul_mem_left _ hi) <| him hz⟩
#align ideal.mem_jacobson_iff Ideal.mem_jacobson_iff
theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
∃ s, s * r - 1 ∈ I := by
cases' mem_jacobson_iff.1 h 1 with s hs
use s
simpa [mul_sub] using hs
#align ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson Ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M := by
use fun hI => ⟨{ J : Ideal R | I ≤ J ∧ J.IsMaximal }, ⟨fun _ hJ => Or.inl hJ.right, hI.symm⟩⟩
rintro ⟨M, hM, hInf⟩
refine le_antisymm (fun x hx => ?_) le_jacobson
rw [hInf, mem_sInf]
intro I hI
cases' hM I hI with is_max is_top
· exact (mem_sInf.1 hx) ⟨le_sInf_iff.1 (le_of_eq hInf) I hI, is_max⟩
· exact is_top.symm ▸ Submodule.mem_top
#align ideal.eq_jacobson_iff_Inf_maximal Ideal.eq_jacobson_iff_sInf_maximal
theorem eq_jacobson_iff_sInf_maximal' :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, ∀ (K : Ideal R), J < K → K = ⊤) ∧ I = sInf M :=
eq_jacobson_iff_sInf_maximal.trans
⟨fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ K hK =>
Or.recOn (hM.1 J hJ) (fun h => h.1.2 K hK) fun h => eq_top_iff.2 (le_of_lt (h ▸ hK)),
hM.2⟩⟩,
fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ =>
Or.recOn (Classical.em (J = ⊤)) (fun h => Or.inr h) fun h => Or.inl ⟨⟨h, hM.1 J hJ⟩⟩,
hM.2⟩⟩⟩
#align ideal.eq_jacobson_iff_Inf_maximal' Ideal.eq_jacobson_iff_sInf_maximal'
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M := by
constructor
· intro h x hx
erw [← h, mem_sInf] at hx
push_neg at hx
exact hx
· refine fun h => le_antisymm (fun x hx => ?_) le_jacobson
contrapose hx
erw [mem_sInf]
push_neg
exact h x hx
#align ideal.eq_jacobson_iff_not_mem Ideal.eq_jacobson_iff_not_mem
theorem map_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) :
RingHom.ker f ≤ I → map f I.jacobson = (map f I).jacobson := by
intro h
unfold Ideal.jacobson
-- porting note : dot notation for `RingHom.ker` does not work
have : ∀ J ∈ { J : Ideal R | I ≤ J ∧ J.IsMaximal }, RingHom.ker f ≤ J :=
fun J hJ => le_trans h hJ.left
refine Trans.trans (map_sInf hf this) (le_antisymm ?_ ?_)
· refine'
sInf_le_sInf fun J hJ =>
⟨comap f J, ⟨⟨le_comap_of_map_le hJ.1, _⟩, map_comap_of_surjective f hf J⟩⟩
haveI : J.IsMaximal := hJ.right
exact comap_isMaximal_of_surjective f hf
· refine' sInf_le_sInf_of_subset_insert_top fun j hj => hj.recOn fun J hJ => _
rw [← hJ.2]
cases' map_eq_top_or_isMaximal_of_surjective f hf hJ.left.right with htop hmax
· exact htop.symm ▸ Set.mem_insert ⊤ _
· exact Set.mem_insert_of_mem ⊤ ⟨map_mono hJ.1.1, hmax⟩
#align ideal.map_jacobson_of_surjective Ideal.map_jacobson_of_surjective
theorem map_jacobson_of_bijective {f : R →+* S} (hf : Function.Bijective f) :
map f I.jacobson = (map f I).jacobson :=
map_jacobson_of_surjective hf.right
(le_trans (le_of_eq (f.injective_iff_ker_eq_bot.1 hf.left)) bot_le)
#align ideal.map_jacobson_of_bijective Ideal.map_jacobson_of_bijective
theorem comap_jacobson {f : R →+* S} {K : Ideal S} :
comap f K.jacobson = sInf (comap f '' { J : Ideal S | K ≤ J ∧ J.IsMaximal }) :=
Trans.trans (comap_sInf' f _) sInf_eq_iInf.symm
#align ideal.comap_jacobson Ideal.comap_jacobson
theorem comap_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) {K : Ideal S} :
comap f K.jacobson = (comap f K).jacobson := by
unfold Ideal.jacobson
refine' le_antisymm _ _
· refine le_trans (comap_mono (le_of_eq (Trans.trans top_inf_eq.symm sInf_insert.symm))) ?_
rw [comap_sInf', sInf_eq_iInf]
refine' iInf_le_iInf_of_subset fun J hJ => _
have : comap f (map f J) = J :=
Trans.trans (comap_map_of_surjective f hf J)
(le_antisymm (sup_le_iff.2 ⟨le_of_eq rfl, le_trans (comap_mono bot_le) hJ.left⟩)
le_sup_left)
cases' map_eq_top_or_isMaximal_of_surjective _ hf hJ.right with htop hmax
· exact ⟨⊤, ⟨Set.mem_insert ⊤ _, htop ▸ this⟩⟩
·
|
exact ⟨map f J, ⟨Set.mem_insert_of_mem _ ⟨le_map_of_comap_le_of_surjective f hf hJ.1, hmax⟩,
this⟩⟩
|
theorem comap_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) {K : Ideal S} :
comap f K.jacobson = (comap f K).jacobson := by
unfold Ideal.jacobson
refine' le_antisymm _ _
· refine le_trans (comap_mono (le_of_eq (Trans.trans top_inf_eq.symm sInf_insert.symm))) ?_
rw [comap_sInf', sInf_eq_iInf]
refine' iInf_le_iInf_of_subset fun J hJ => _
have : comap f (map f J) = J :=
Trans.trans (comap_map_of_surjective f hf J)
(le_antisymm (sup_le_iff.2 ⟨le_of_eq rfl, le_trans (comap_mono bot_le) hJ.left⟩)
le_sup_left)
cases' map_eq_top_or_isMaximal_of_surjective _ hf hJ.right with htop hmax
· exact ⟨⊤, ⟨Set.mem_insert ⊤ _, htop ▸ this⟩⟩
·
|
Mathlib.RingTheory.JacobsonIdeal.209_0.Lz0MgLQMj1bGzuN
|
theorem comap_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) {K : Ideal S} :
comap f K.jacobson = (comap f K).jacobson
|
Mathlib_RingTheory_JacobsonIdeal
|
case refine'_2
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I : Ideal R
f : R →+* S
hf : Function.Surjective ⇑f
K : Ideal S
⊢ sInf {J | comap f K ≤ J ∧ IsMaximal J} ≤ comap f (sInf {J | K ≤ J ∧ IsMaximal J})
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
/-!
# Jacobson radical
The Jacobson radical of a ring `R` is defined to be the intersection of all maximal ideals of `R`.
This is similar to how the nilradical is equal to the intersection of all prime ideals of `R`.
We can extend the idea of the nilradical to ideals of `R`,
by letting the radical of an ideal `I` be the intersection of prime ideals containing `I`.
Under this extension, the original nilradical is the radical of the zero ideal `⊥`.
Here we define the Jacobson radical of an ideal `I` in a similar way,
as the intersection of maximal ideals containing `I`.
## Main definitions
Let `R` be a commutative ring, and `I` be an ideal of `R`
* `Ideal.jacobson I` is the jacobson radical, i.e. the infimum of all maximal ideals containing I.
* `Ideal.IsLocal I` is the proposition that the jacobson radical of `I` is itself a maximal ideal
## Main statements
* `mem_jacobson_iff` gives a characterization of members of the jacobson of I
* `Ideal.isLocal_of_isMaximal_radical`: if the radical of I is maximal then so is the jacobson
radical
## Tags
Jacobson, Jacobson radical, Local Ideal
-/
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}
open Polynomial
section Jacobson
section Ring
variable [Ring R] [Ring S] {I : Ideal R}
/-- The Jacobson radical of `I` is the infimum of all maximal (left) ideals containing `I`. -/
def jacobson (I : Ideal R) : Ideal R :=
sInf { J : Ideal R | I ≤ J ∧ IsMaximal J }
#align ideal.jacobson Ideal.jacobson
theorem le_jacobson : I ≤ jacobson I := fun _ hx => mem_sInf.mpr fun _ hJ => hJ.left hx
#align ideal.le_jacobson Ideal.le_jacobson
@[simp]
theorem jacobson_idem : jacobson (jacobson I) = jacobson I :=
le_antisymm (sInf_le_sInf fun _ hJ => ⟨sInf_le hJ, hJ.2⟩) le_jacobson
#align ideal.jacobson_idem Ideal.jacobson_idem
@[simp]
theorem jacobson_top : jacobson (⊤ : Ideal R) = ⊤ :=
eq_top_iff.2 le_jacobson
#align ideal.jacobson_top Ideal.jacobson_top
@[simp]
theorem jacobson_eq_top_iff : jacobson I = ⊤ ↔ I = ⊤ :=
⟨fun H =>
by_contradiction fun hi => let ⟨M, hm, him⟩ := exists_le_maximal I hi
lt_top_iff_ne_top.1
(lt_of_le_of_lt (show jacobson I ≤ M from sInf_le ⟨him, hm⟩) <|
lt_top_iff_ne_top.2 hm.ne_top) H,
fun H => eq_top_iff.2 <| le_sInf fun _ ⟨hij, _⟩ => H ▸ hij⟩
#align ideal.jacobson_eq_top_iff Ideal.jacobson_eq_top_iff
theorem jacobson_eq_bot : jacobson I = ⊥ → I = ⊥ := fun h => eq_bot_iff.mpr (h ▸ le_jacobson)
#align ideal.jacobson_eq_bot Ideal.jacobson_eq_bot
theorem jacobson_eq_self_of_isMaximal [H : IsMaximal I] : I.jacobson = I :=
le_antisymm (sInf_le ⟨le_of_eq rfl, H⟩) le_jacobson
#align ideal.jacobson_eq_self_of_is_maximal Ideal.jacobson_eq_self_of_isMaximal
instance (priority := 100) jacobson.isMaximal [H : IsMaximal I] : IsMaximal (jacobson I) :=
⟨⟨fun htop => H.1.1 (jacobson_eq_top_iff.1 htop), fun _ hJ =>
H.1.2 _ (lt_of_le_of_lt le_jacobson hJ)⟩⟩
#align ideal.jacobson.is_maximal Ideal.jacobson.isMaximal
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I :=
⟨fun hx y =>
by_cases
(fun hxy : I ⊔ span {y * x + 1} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy)
let ⟨r, hr⟩ := mem_span_singleton'.1 hq
⟨r, by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one r (y * x), hr, ← hpq, ← neg_sub, add_sub_cancel]
exact I.neg_mem hpi⟩)
fun hxy : I ⊔ span {y * x + 1} ≠ ⊤ => let ⟨M, hm1, hm2⟩ := exists_le_maximal _ hxy
suffices x ∉ M from (this <| mem_sInf.1 hx ⟨le_trans le_sup_left hm2, hm1⟩).elim
fun hxm => hm1.1.1 <| (eq_top_iff_one _).2 <| add_sub_cancel' (y * x) 1 ▸
M.sub_mem (le_sup_right.trans hm2 <| subset_span rfl) (M.mul_mem_left _ hxm),
fun hx => mem_sInf.2 fun M ⟨him, hm⟩ => by_contradiction fun hxm =>
let ⟨y, i, hi, df⟩ := hm.exists_inv hxm
let ⟨z, hz⟩ := hx (-y)
hm.1.1 <| (eq_top_iff_one _).2 <| sub_sub_cancel (z * -y * x + z) 1 ▸
M.sub_mem (by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one z, neg_mul, ← sub_eq_iff_eq_add.mpr df.symm, neg_sub,
sub_add_cancel]
exact M.mul_mem_left _ hi) <| him hz⟩
#align ideal.mem_jacobson_iff Ideal.mem_jacobson_iff
theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
∃ s, s * r - 1 ∈ I := by
cases' mem_jacobson_iff.1 h 1 with s hs
use s
simpa [mul_sub] using hs
#align ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson Ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M := by
use fun hI => ⟨{ J : Ideal R | I ≤ J ∧ J.IsMaximal }, ⟨fun _ hJ => Or.inl hJ.right, hI.symm⟩⟩
rintro ⟨M, hM, hInf⟩
refine le_antisymm (fun x hx => ?_) le_jacobson
rw [hInf, mem_sInf]
intro I hI
cases' hM I hI with is_max is_top
· exact (mem_sInf.1 hx) ⟨le_sInf_iff.1 (le_of_eq hInf) I hI, is_max⟩
· exact is_top.symm ▸ Submodule.mem_top
#align ideal.eq_jacobson_iff_Inf_maximal Ideal.eq_jacobson_iff_sInf_maximal
theorem eq_jacobson_iff_sInf_maximal' :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, ∀ (K : Ideal R), J < K → K = ⊤) ∧ I = sInf M :=
eq_jacobson_iff_sInf_maximal.trans
⟨fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ K hK =>
Or.recOn (hM.1 J hJ) (fun h => h.1.2 K hK) fun h => eq_top_iff.2 (le_of_lt (h ▸ hK)),
hM.2⟩⟩,
fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ =>
Or.recOn (Classical.em (J = ⊤)) (fun h => Or.inr h) fun h => Or.inl ⟨⟨h, hM.1 J hJ⟩⟩,
hM.2⟩⟩⟩
#align ideal.eq_jacobson_iff_Inf_maximal' Ideal.eq_jacobson_iff_sInf_maximal'
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M := by
constructor
· intro h x hx
erw [← h, mem_sInf] at hx
push_neg at hx
exact hx
· refine fun h => le_antisymm (fun x hx => ?_) le_jacobson
contrapose hx
erw [mem_sInf]
push_neg
exact h x hx
#align ideal.eq_jacobson_iff_not_mem Ideal.eq_jacobson_iff_not_mem
theorem map_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) :
RingHom.ker f ≤ I → map f I.jacobson = (map f I).jacobson := by
intro h
unfold Ideal.jacobson
-- porting note : dot notation for `RingHom.ker` does not work
have : ∀ J ∈ { J : Ideal R | I ≤ J ∧ J.IsMaximal }, RingHom.ker f ≤ J :=
fun J hJ => le_trans h hJ.left
refine Trans.trans (map_sInf hf this) (le_antisymm ?_ ?_)
· refine'
sInf_le_sInf fun J hJ =>
⟨comap f J, ⟨⟨le_comap_of_map_le hJ.1, _⟩, map_comap_of_surjective f hf J⟩⟩
haveI : J.IsMaximal := hJ.right
exact comap_isMaximal_of_surjective f hf
· refine' sInf_le_sInf_of_subset_insert_top fun j hj => hj.recOn fun J hJ => _
rw [← hJ.2]
cases' map_eq_top_or_isMaximal_of_surjective f hf hJ.left.right with htop hmax
· exact htop.symm ▸ Set.mem_insert ⊤ _
· exact Set.mem_insert_of_mem ⊤ ⟨map_mono hJ.1.1, hmax⟩
#align ideal.map_jacobson_of_surjective Ideal.map_jacobson_of_surjective
theorem map_jacobson_of_bijective {f : R →+* S} (hf : Function.Bijective f) :
map f I.jacobson = (map f I).jacobson :=
map_jacobson_of_surjective hf.right
(le_trans (le_of_eq (f.injective_iff_ker_eq_bot.1 hf.left)) bot_le)
#align ideal.map_jacobson_of_bijective Ideal.map_jacobson_of_bijective
theorem comap_jacobson {f : R →+* S} {K : Ideal S} :
comap f K.jacobson = sInf (comap f '' { J : Ideal S | K ≤ J ∧ J.IsMaximal }) :=
Trans.trans (comap_sInf' f _) sInf_eq_iInf.symm
#align ideal.comap_jacobson Ideal.comap_jacobson
theorem comap_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) {K : Ideal S} :
comap f K.jacobson = (comap f K).jacobson := by
unfold Ideal.jacobson
refine' le_antisymm _ _
· refine le_trans (comap_mono (le_of_eq (Trans.trans top_inf_eq.symm sInf_insert.symm))) ?_
rw [comap_sInf', sInf_eq_iInf]
refine' iInf_le_iInf_of_subset fun J hJ => _
have : comap f (map f J) = J :=
Trans.trans (comap_map_of_surjective f hf J)
(le_antisymm (sup_le_iff.2 ⟨le_of_eq rfl, le_trans (comap_mono bot_le) hJ.left⟩)
le_sup_left)
cases' map_eq_top_or_isMaximal_of_surjective _ hf hJ.right with htop hmax
· exact ⟨⊤, ⟨Set.mem_insert ⊤ _, htop ▸ this⟩⟩
· exact ⟨map f J, ⟨Set.mem_insert_of_mem _ ⟨le_map_of_comap_le_of_surjective f hf hJ.1, hmax⟩,
this⟩⟩
·
|
rw [comap_sInf]
|
theorem comap_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) {K : Ideal S} :
comap f K.jacobson = (comap f K).jacobson := by
unfold Ideal.jacobson
refine' le_antisymm _ _
· refine le_trans (comap_mono (le_of_eq (Trans.trans top_inf_eq.symm sInf_insert.symm))) ?_
rw [comap_sInf', sInf_eq_iInf]
refine' iInf_le_iInf_of_subset fun J hJ => _
have : comap f (map f J) = J :=
Trans.trans (comap_map_of_surjective f hf J)
(le_antisymm (sup_le_iff.2 ⟨le_of_eq rfl, le_trans (comap_mono bot_le) hJ.left⟩)
le_sup_left)
cases' map_eq_top_or_isMaximal_of_surjective _ hf hJ.right with htop hmax
· exact ⟨⊤, ⟨Set.mem_insert ⊤ _, htop ▸ this⟩⟩
· exact ⟨map f J, ⟨Set.mem_insert_of_mem _ ⟨le_map_of_comap_le_of_surjective f hf hJ.1, hmax⟩,
this⟩⟩
·
|
Mathlib.RingTheory.JacobsonIdeal.209_0.Lz0MgLQMj1bGzuN
|
theorem comap_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) {K : Ideal S} :
comap f K.jacobson = (comap f K).jacobson
|
Mathlib_RingTheory_JacobsonIdeal
|
case refine'_2
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
I : Ideal R
f : R →+* S
hf : Function.Surjective ⇑f
K : Ideal S
⊢ sInf {J | comap f K ≤ J ∧ IsMaximal J} ≤ ⨅ I ∈ {J | K ≤ J ∧ IsMaximal J}, comap f I
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
/-!
# Jacobson radical
The Jacobson radical of a ring `R` is defined to be the intersection of all maximal ideals of `R`.
This is similar to how the nilradical is equal to the intersection of all prime ideals of `R`.
We can extend the idea of the nilradical to ideals of `R`,
by letting the radical of an ideal `I` be the intersection of prime ideals containing `I`.
Under this extension, the original nilradical is the radical of the zero ideal `⊥`.
Here we define the Jacobson radical of an ideal `I` in a similar way,
as the intersection of maximal ideals containing `I`.
## Main definitions
Let `R` be a commutative ring, and `I` be an ideal of `R`
* `Ideal.jacobson I` is the jacobson radical, i.e. the infimum of all maximal ideals containing I.
* `Ideal.IsLocal I` is the proposition that the jacobson radical of `I` is itself a maximal ideal
## Main statements
* `mem_jacobson_iff` gives a characterization of members of the jacobson of I
* `Ideal.isLocal_of_isMaximal_radical`: if the radical of I is maximal then so is the jacobson
radical
## Tags
Jacobson, Jacobson radical, Local Ideal
-/
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}
open Polynomial
section Jacobson
section Ring
variable [Ring R] [Ring S] {I : Ideal R}
/-- The Jacobson radical of `I` is the infimum of all maximal (left) ideals containing `I`. -/
def jacobson (I : Ideal R) : Ideal R :=
sInf { J : Ideal R | I ≤ J ∧ IsMaximal J }
#align ideal.jacobson Ideal.jacobson
theorem le_jacobson : I ≤ jacobson I := fun _ hx => mem_sInf.mpr fun _ hJ => hJ.left hx
#align ideal.le_jacobson Ideal.le_jacobson
@[simp]
theorem jacobson_idem : jacobson (jacobson I) = jacobson I :=
le_antisymm (sInf_le_sInf fun _ hJ => ⟨sInf_le hJ, hJ.2⟩) le_jacobson
#align ideal.jacobson_idem Ideal.jacobson_idem
@[simp]
theorem jacobson_top : jacobson (⊤ : Ideal R) = ⊤ :=
eq_top_iff.2 le_jacobson
#align ideal.jacobson_top Ideal.jacobson_top
@[simp]
theorem jacobson_eq_top_iff : jacobson I = ⊤ ↔ I = ⊤ :=
⟨fun H =>
by_contradiction fun hi => let ⟨M, hm, him⟩ := exists_le_maximal I hi
lt_top_iff_ne_top.1
(lt_of_le_of_lt (show jacobson I ≤ M from sInf_le ⟨him, hm⟩) <|
lt_top_iff_ne_top.2 hm.ne_top) H,
fun H => eq_top_iff.2 <| le_sInf fun _ ⟨hij, _⟩ => H ▸ hij⟩
#align ideal.jacobson_eq_top_iff Ideal.jacobson_eq_top_iff
theorem jacobson_eq_bot : jacobson I = ⊥ → I = ⊥ := fun h => eq_bot_iff.mpr (h ▸ le_jacobson)
#align ideal.jacobson_eq_bot Ideal.jacobson_eq_bot
theorem jacobson_eq_self_of_isMaximal [H : IsMaximal I] : I.jacobson = I :=
le_antisymm (sInf_le ⟨le_of_eq rfl, H⟩) le_jacobson
#align ideal.jacobson_eq_self_of_is_maximal Ideal.jacobson_eq_self_of_isMaximal
instance (priority := 100) jacobson.isMaximal [H : IsMaximal I] : IsMaximal (jacobson I) :=
⟨⟨fun htop => H.1.1 (jacobson_eq_top_iff.1 htop), fun _ hJ =>
H.1.2 _ (lt_of_le_of_lt le_jacobson hJ)⟩⟩
#align ideal.jacobson.is_maximal Ideal.jacobson.isMaximal
theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I :=
⟨fun hx y =>
by_cases
(fun hxy : I ⊔ span {y * x + 1} = ⊤ =>
let ⟨p, hpi, q, hq, hpq⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy)
let ⟨r, hr⟩ := mem_span_singleton'.1 hq
⟨r, by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one r (y * x), hr, ← hpq, ← neg_sub, add_sub_cancel]
exact I.neg_mem hpi⟩)
fun hxy : I ⊔ span {y * x + 1} ≠ ⊤ => let ⟨M, hm1, hm2⟩ := exists_le_maximal _ hxy
suffices x ∉ M from (this <| mem_sInf.1 hx ⟨le_trans le_sup_left hm2, hm1⟩).elim
fun hxm => hm1.1.1 <| (eq_top_iff_one _).2 <| add_sub_cancel' (y * x) 1 ▸
M.sub_mem (le_sup_right.trans hm2 <| subset_span rfl) (M.mul_mem_left _ hxm),
fun hx => mem_sInf.2 fun M ⟨him, hm⟩ => by_contradiction fun hxm =>
let ⟨y, i, hi, df⟩ := hm.exists_inv hxm
let ⟨z, hz⟩ := hx (-y)
hm.1.1 <| (eq_top_iff_one _).2 <| sub_sub_cancel (z * -y * x + z) 1 ▸
M.sub_mem (by
-- Porting note : supply `mul_add_one` with explicit variables
rw [mul_assoc, ← mul_add_one z, neg_mul, ← sub_eq_iff_eq_add.mpr df.symm, neg_sub,
sub_add_cancel]
exact M.mul_mem_left _ hi) <| him hz⟩
#align ideal.mem_jacobson_iff Ideal.mem_jacobson_iff
theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
∃ s, s * r - 1 ∈ I := by
cases' mem_jacobson_iff.1 h 1 with s hs
use s
simpa [mul_sub] using hs
#align ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson Ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson
/-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M := by
use fun hI => ⟨{ J : Ideal R | I ≤ J ∧ J.IsMaximal }, ⟨fun _ hJ => Or.inl hJ.right, hI.symm⟩⟩
rintro ⟨M, hM, hInf⟩
refine le_antisymm (fun x hx => ?_) le_jacobson
rw [hInf, mem_sInf]
intro I hI
cases' hM I hI with is_max is_top
· exact (mem_sInf.1 hx) ⟨le_sInf_iff.1 (le_of_eq hInf) I hI, is_max⟩
· exact is_top.symm ▸ Submodule.mem_top
#align ideal.eq_jacobson_iff_Inf_maximal Ideal.eq_jacobson_iff_sInf_maximal
theorem eq_jacobson_iff_sInf_maximal' :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, ∀ (K : Ideal R), J < K → K = ⊤) ∧ I = sInf M :=
eq_jacobson_iff_sInf_maximal.trans
⟨fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ K hK =>
Or.recOn (hM.1 J hJ) (fun h => h.1.2 K hK) fun h => eq_top_iff.2 (le_of_lt (h ▸ hK)),
hM.2⟩⟩,
fun h =>
let ⟨M, hM⟩ := h
⟨M,
⟨fun J hJ =>
Or.recOn (Classical.em (J = ⊤)) (fun h => Or.inr h) fun h => Or.inl ⟨⟨h, hM.1 J hJ⟩⟩,
hM.2⟩⟩⟩
#align ideal.eq_jacobson_iff_Inf_maximal' Ideal.eq_jacobson_iff_sInf_maximal'
/-- An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. -/
theorem eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ (x) (_ : x ∉ I), ∃ M : Ideal R, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M := by
constructor
· intro h x hx
erw [← h, mem_sInf] at hx
push_neg at hx
exact hx
· refine fun h => le_antisymm (fun x hx => ?_) le_jacobson
contrapose hx
erw [mem_sInf]
push_neg
exact h x hx
#align ideal.eq_jacobson_iff_not_mem Ideal.eq_jacobson_iff_not_mem
theorem map_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) :
RingHom.ker f ≤ I → map f I.jacobson = (map f I).jacobson := by
intro h
unfold Ideal.jacobson
-- porting note : dot notation for `RingHom.ker` does not work
have : ∀ J ∈ { J : Ideal R | I ≤ J ∧ J.IsMaximal }, RingHom.ker f ≤ J :=
fun J hJ => le_trans h hJ.left
refine Trans.trans (map_sInf hf this) (le_antisymm ?_ ?_)
· refine'
sInf_le_sInf fun J hJ =>
⟨comap f J, ⟨⟨le_comap_of_map_le hJ.1, _⟩, map_comap_of_surjective f hf J⟩⟩
haveI : J.IsMaximal := hJ.right
exact comap_isMaximal_of_surjective f hf
· refine' sInf_le_sInf_of_subset_insert_top fun j hj => hj.recOn fun J hJ => _
rw [← hJ.2]
cases' map_eq_top_or_isMaximal_of_surjective f hf hJ.left.right with htop hmax
· exact htop.symm ▸ Set.mem_insert ⊤ _
· exact Set.mem_insert_of_mem ⊤ ⟨map_mono hJ.1.1, hmax⟩
#align ideal.map_jacobson_of_surjective Ideal.map_jacobson_of_surjective
theorem map_jacobson_of_bijective {f : R →+* S} (hf : Function.Bijective f) :
map f I.jacobson = (map f I).jacobson :=
map_jacobson_of_surjective hf.right
(le_trans (le_of_eq (f.injective_iff_ker_eq_bot.1 hf.left)) bot_le)
#align ideal.map_jacobson_of_bijective Ideal.map_jacobson_of_bijective
theorem comap_jacobson {f : R →+* S} {K : Ideal S} :
comap f K.jacobson = sInf (comap f '' { J : Ideal S | K ≤ J ∧ J.IsMaximal }) :=
Trans.trans (comap_sInf' f _) sInf_eq_iInf.symm
#align ideal.comap_jacobson Ideal.comap_jacobson
theorem comap_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) {K : Ideal S} :
comap f K.jacobson = (comap f K).jacobson := by
unfold Ideal.jacobson
refine' le_antisymm _ _
· refine le_trans (comap_mono (le_of_eq (Trans.trans top_inf_eq.symm sInf_insert.symm))) ?_
rw [comap_sInf', sInf_eq_iInf]
refine' iInf_le_iInf_of_subset fun J hJ => _
have : comap f (map f J) = J :=
Trans.trans (comap_map_of_surjective f hf J)
(le_antisymm (sup_le_iff.2 ⟨le_of_eq rfl, le_trans (comap_mono bot_le) hJ.left⟩)
le_sup_left)
cases' map_eq_top_or_isMaximal_of_surjective _ hf hJ.right with htop hmax
· exact ⟨⊤, ⟨Set.mem_insert ⊤ _, htop ▸ this⟩⟩
· exact ⟨map f J, ⟨Set.mem_insert_of_mem _ ⟨le_map_of_comap_le_of_surjective f hf hJ.1, hmax⟩,
this⟩⟩
· rw [comap_sInf]
|
refine' le_iInf_iff.2 fun J => le_iInf_iff.2 fun hJ => _
|
theorem comap_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) {K : Ideal S} :
comap f K.jacobson = (comap f K).jacobson := by
unfold Ideal.jacobson
refine' le_antisymm _ _
· refine le_trans (comap_mono (le_of_eq (Trans.trans top_inf_eq.symm sInf_insert.symm))) ?_
rw [comap_sInf', sInf_eq_iInf]
refine' iInf_le_iInf_of_subset fun J hJ => _
have : comap f (map f J) = J :=
Trans.trans (comap_map_of_surjective f hf J)
(le_antisymm (sup_le_iff.2 ⟨le_of_eq rfl, le_trans (comap_mono bot_le) hJ.left⟩)
le_sup_left)
cases' map_eq_top_or_isMaximal_of_surjective _ hf hJ.right with htop hmax
· exact ⟨⊤, ⟨Set.mem_insert ⊤ _, htop ▸ this⟩⟩
· exact ⟨map f J, ⟨Set.mem_insert_of_mem _ ⟨le_map_of_comap_le_of_surjective f hf hJ.1, hmax⟩,
this⟩⟩
· rw [comap_sInf]
|
Mathlib.RingTheory.JacobsonIdeal.209_0.Lz0MgLQMj1bGzuN
|
theorem comap_jacobson_of_surjective {f : R →+* S} (hf : Function.Surjective f) {K : Ideal S} :
comap f K.jacobson = (comap f K).jacobson
|
Mathlib_RingTheory_JacobsonIdeal
|
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