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|---|---|---|---|---|---|---|
α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : CompactSpace α
inst✝¹ : MetricSpace β
inst✝ : NormedAddCommGroup E
f g : C(α, β)
C : ℝ
C0 : 0 < C
⊢ (∀ (x : α), dist ((mkOfCompact f) x) ((mkOfCompact g) x) < C) ↔ ∀ (x : α), dist (f x) (g x) < C
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compacts
#align_import topology.continuous_function.compact from "leanprover-community/mathlib"@"d3af0609f6db8691dffdc3e1fb7feb7da72698f2"
/-!
# Continuous functions on a compact space
Continuous functions `C(α, β)` from a compact space `α` to a metric space `β`
are automatically bounded, and so acquire various structures inherited from `α →ᵇ β`.
This file transfers these structures, and restates some lemmas
characterising these structures.
If you need a lemma which is proved about `α →ᵇ β` but not for `C(α, β)` when `α` is compact,
you should restate it here. You can also use
`ContinuousMap.equivBoundedOfCompact` to move functions back and forth.
-/
noncomputable section
open Topology Classical NNReal BoundedContinuousFunction BigOperators
open Set Filter Metric
open BoundedContinuousFunction
namespace ContinuousMap
variable {α β E : Type*} [TopologicalSpace α] [CompactSpace α] [MetricSpace β]
[NormedAddCommGroup E]
section
variable (α β)
/-- When `α` is compact, the bounded continuous maps `α →ᵇ β` are
equivalent to `C(α, β)`.
-/
@[simps (config := .asFn)]
def equivBoundedOfCompact : C(α, β) ≃ (α →ᵇ β) :=
⟨mkOfCompact, BoundedContinuousFunction.toContinuousMap, fun f => by
ext
rfl, fun f => by
ext
rfl⟩
#align continuous_map.equiv_bounded_of_compact ContinuousMap.equivBoundedOfCompact
theorem uniformInducing_equivBoundedOfCompact : UniformInducing (equivBoundedOfCompact α β) :=
UniformInducing.mk'
(by
simp only [hasBasis_compactConvergenceUniformity.mem_iff, uniformity_basis_dist_le.mem_iff]
exact fun s =>
⟨fun ⟨⟨a, b⟩, ⟨_, ⟨ε, hε, hb⟩⟩, hs⟩ =>
⟨{ p | ∀ x, (p.1 x, p.2 x) ∈ b }, ⟨ε, hε, fun _ h x => hb ((dist_le hε.le).mp h x)⟩,
fun f g h => hs fun x _ => h x⟩,
fun ⟨_, ⟨ε, hε, ht⟩, hs⟩ =>
⟨⟨Set.univ, { p | dist p.1 p.2 ≤ ε }⟩, ⟨isCompact_univ, ⟨ε, hε, fun _ h => h⟩⟩,
fun ⟨f, g⟩ h => hs _ _ (ht ((dist_le hε.le).mpr fun x => h x (mem_univ x)))⟩⟩)
#align continuous_map.uniform_inducing_equiv_bounded_of_compact ContinuousMap.uniformInducing_equivBoundedOfCompact
theorem uniformEmbedding_equivBoundedOfCompact : UniformEmbedding (equivBoundedOfCompact α β) :=
{ uniformInducing_equivBoundedOfCompact α β with inj := (equivBoundedOfCompact α β).injective }
#align continuous_map.uniform_embedding_equiv_bounded_of_compact ContinuousMap.uniformEmbedding_equivBoundedOfCompact
/-- When `α` is compact, the bounded continuous maps `α →ᵇ 𝕜` are
additively equivalent to `C(α, 𝕜)`.
-/
-- porting note: the following `simps` received a "maximum recursion depth" error
-- @[simps! (config := .asFn) apply symm_apply]
def addEquivBoundedOfCompact [AddMonoid β] [LipschitzAdd β] : C(α, β) ≃+ (α →ᵇ β) :=
({ toContinuousMapAddHom α β, (equivBoundedOfCompact α β).symm with } : (α →ᵇ β) ≃+ C(α, β)).symm
#align continuous_map.add_equiv_bounded_of_compact ContinuousMap.addEquivBoundedOfCompact
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_symm_apply [AddMonoid β] [LipschitzAdd β] :
⇑((addEquivBoundedOfCompact α β).symm) = toContinuousMapAddHom α β :=
rfl
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_apply [AddMonoid β] [LipschitzAdd β] :
⇑(addEquivBoundedOfCompact α β) = mkOfCompact :=
rfl
instance metricSpace : MetricSpace C(α, β) :=
(uniformEmbedding_equivBoundedOfCompact α β).comapMetricSpace _
#align continuous_map.metric_space ContinuousMap.metricSpace
/-- When `α` is compact, and `β` is a metric space, the bounded continuous maps `α →ᵇ β` are
isometric to `C(α, β)`.
-/
@[simps! (config := .asFn) toEquiv apply symm_apply]
def isometryEquivBoundedOfCompact : C(α, β) ≃ᵢ (α →ᵇ β) where
isometry_toFun _ _ := rfl
toEquiv := equivBoundedOfCompact α β
#align continuous_map.isometry_equiv_bounded_of_compact ContinuousMap.isometryEquivBoundedOfCompact
end
@[simp]
theorem _root_.BoundedContinuousFunction.dist_mkOfCompact (f g : C(α, β)) :
dist (mkOfCompact f) (mkOfCompact g) = dist f g :=
rfl
#align bounded_continuous_function.dist_mk_of_compact BoundedContinuousFunction.dist_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.dist_toContinuousMap (f g : α →ᵇ β) :
dist f.toContinuousMap g.toContinuousMap = dist f g :=
rfl
#align bounded_continuous_function.dist_to_continuous_map BoundedContinuousFunction.dist_toContinuousMap
open BoundedContinuousFunction
section
variable {f g : C(α, β)} {C : ℝ}
/-- The pointwise distance is controlled by the distance between functions, by definition. -/
theorem dist_apply_le_dist (x : α) : dist (f x) (g x) ≤ dist f g := by
simp only [← dist_mkOfCompact, dist_coe_le_dist, ← mkOfCompact_apply]
#align continuous_map.dist_apply_le_dist ContinuousMap.dist_apply_le_dist
/-- The distance between two functions is controlled by the supremum of the pointwise distances. -/
theorem dist_le (C0 : (0 : ℝ) ≤ C) : dist f g ≤ C ↔ ∀ x : α, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le C0, mkOfCompact_apply]
#align continuous_map.dist_le ContinuousMap.dist_le
theorem dist_le_iff_of_nonempty [Nonempty α] : dist f g ≤ C ↔ ∀ x, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le_iff_of_nonempty,
mkOfCompact_apply]
#align continuous_map.dist_le_iff_of_nonempty ContinuousMap.dist_le_iff_of_nonempty
theorem dist_lt_iff_of_nonempty [Nonempty α] : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
simp only [← dist_mkOfCompact, dist_lt_iff_of_nonempty_compact, mkOfCompact_apply]
#align continuous_map.dist_lt_iff_of_nonempty ContinuousMap.dist_lt_iff_of_nonempty
theorem dist_lt_of_nonempty [Nonempty α] (w : ∀ x : α, dist (f x) (g x) < C) : dist f g < C :=
dist_lt_iff_of_nonempty.2 w
#align continuous_map.dist_lt_of_nonempty ContinuousMap.dist_lt_of_nonempty
theorem dist_lt_iff (C0 : (0 : ℝ) < C) : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
rw [← dist_mkOfCompact, dist_lt_iff_of_compact C0]
|
simp only [mkOfCompact_apply]
|
theorem dist_lt_iff (C0 : (0 : ℝ) < C) : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
rw [← dist_mkOfCompact, dist_lt_iff_of_compact C0]
|
Mathlib.Topology.ContinuousFunction.Compact.152_0.Mig2jTVnn2FLKEB
|
theorem dist_lt_iff (C0 : (0 : ℝ) < C) : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C
|
Mathlib_Topology_ContinuousFunction_Compact
|
α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : CompactSpace α
inst✝¹ : MetricSpace β
inst✝ : NormedAddCommGroup E
src✝¹ : MetricSpace C(α, E) := metricSpace α E
src✝ : AddCommGroup C(α, E) := instAddCommGroupContinuousMap
x y : C(α, E)
⊢ dist x y = ‖x - y‖
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compacts
#align_import topology.continuous_function.compact from "leanprover-community/mathlib"@"d3af0609f6db8691dffdc3e1fb7feb7da72698f2"
/-!
# Continuous functions on a compact space
Continuous functions `C(α, β)` from a compact space `α` to a metric space `β`
are automatically bounded, and so acquire various structures inherited from `α →ᵇ β`.
This file transfers these structures, and restates some lemmas
characterising these structures.
If you need a lemma which is proved about `α →ᵇ β` but not for `C(α, β)` when `α` is compact,
you should restate it here. You can also use
`ContinuousMap.equivBoundedOfCompact` to move functions back and forth.
-/
noncomputable section
open Topology Classical NNReal BoundedContinuousFunction BigOperators
open Set Filter Metric
open BoundedContinuousFunction
namespace ContinuousMap
variable {α β E : Type*} [TopologicalSpace α] [CompactSpace α] [MetricSpace β]
[NormedAddCommGroup E]
section
variable (α β)
/-- When `α` is compact, the bounded continuous maps `α →ᵇ β` are
equivalent to `C(α, β)`.
-/
@[simps (config := .asFn)]
def equivBoundedOfCompact : C(α, β) ≃ (α →ᵇ β) :=
⟨mkOfCompact, BoundedContinuousFunction.toContinuousMap, fun f => by
ext
rfl, fun f => by
ext
rfl⟩
#align continuous_map.equiv_bounded_of_compact ContinuousMap.equivBoundedOfCompact
theorem uniformInducing_equivBoundedOfCompact : UniformInducing (equivBoundedOfCompact α β) :=
UniformInducing.mk'
(by
simp only [hasBasis_compactConvergenceUniformity.mem_iff, uniformity_basis_dist_le.mem_iff]
exact fun s =>
⟨fun ⟨⟨a, b⟩, ⟨_, ⟨ε, hε, hb⟩⟩, hs⟩ =>
⟨{ p | ∀ x, (p.1 x, p.2 x) ∈ b }, ⟨ε, hε, fun _ h x => hb ((dist_le hε.le).mp h x)⟩,
fun f g h => hs fun x _ => h x⟩,
fun ⟨_, ⟨ε, hε, ht⟩, hs⟩ =>
⟨⟨Set.univ, { p | dist p.1 p.2 ≤ ε }⟩, ⟨isCompact_univ, ⟨ε, hε, fun _ h => h⟩⟩,
fun ⟨f, g⟩ h => hs _ _ (ht ((dist_le hε.le).mpr fun x => h x (mem_univ x)))⟩⟩)
#align continuous_map.uniform_inducing_equiv_bounded_of_compact ContinuousMap.uniformInducing_equivBoundedOfCompact
theorem uniformEmbedding_equivBoundedOfCompact : UniformEmbedding (equivBoundedOfCompact α β) :=
{ uniformInducing_equivBoundedOfCompact α β with inj := (equivBoundedOfCompact α β).injective }
#align continuous_map.uniform_embedding_equiv_bounded_of_compact ContinuousMap.uniformEmbedding_equivBoundedOfCompact
/-- When `α` is compact, the bounded continuous maps `α →ᵇ 𝕜` are
additively equivalent to `C(α, 𝕜)`.
-/
-- porting note: the following `simps` received a "maximum recursion depth" error
-- @[simps! (config := .asFn) apply symm_apply]
def addEquivBoundedOfCompact [AddMonoid β] [LipschitzAdd β] : C(α, β) ≃+ (α →ᵇ β) :=
({ toContinuousMapAddHom α β, (equivBoundedOfCompact α β).symm with } : (α →ᵇ β) ≃+ C(α, β)).symm
#align continuous_map.add_equiv_bounded_of_compact ContinuousMap.addEquivBoundedOfCompact
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_symm_apply [AddMonoid β] [LipschitzAdd β] :
⇑((addEquivBoundedOfCompact α β).symm) = toContinuousMapAddHom α β :=
rfl
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_apply [AddMonoid β] [LipschitzAdd β] :
⇑(addEquivBoundedOfCompact α β) = mkOfCompact :=
rfl
instance metricSpace : MetricSpace C(α, β) :=
(uniformEmbedding_equivBoundedOfCompact α β).comapMetricSpace _
#align continuous_map.metric_space ContinuousMap.metricSpace
/-- When `α` is compact, and `β` is a metric space, the bounded continuous maps `α →ᵇ β` are
isometric to `C(α, β)`.
-/
@[simps! (config := .asFn) toEquiv apply symm_apply]
def isometryEquivBoundedOfCompact : C(α, β) ≃ᵢ (α →ᵇ β) where
isometry_toFun _ _ := rfl
toEquiv := equivBoundedOfCompact α β
#align continuous_map.isometry_equiv_bounded_of_compact ContinuousMap.isometryEquivBoundedOfCompact
end
@[simp]
theorem _root_.BoundedContinuousFunction.dist_mkOfCompact (f g : C(α, β)) :
dist (mkOfCompact f) (mkOfCompact g) = dist f g :=
rfl
#align bounded_continuous_function.dist_mk_of_compact BoundedContinuousFunction.dist_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.dist_toContinuousMap (f g : α →ᵇ β) :
dist f.toContinuousMap g.toContinuousMap = dist f g :=
rfl
#align bounded_continuous_function.dist_to_continuous_map BoundedContinuousFunction.dist_toContinuousMap
open BoundedContinuousFunction
section
variable {f g : C(α, β)} {C : ℝ}
/-- The pointwise distance is controlled by the distance between functions, by definition. -/
theorem dist_apply_le_dist (x : α) : dist (f x) (g x) ≤ dist f g := by
simp only [← dist_mkOfCompact, dist_coe_le_dist, ← mkOfCompact_apply]
#align continuous_map.dist_apply_le_dist ContinuousMap.dist_apply_le_dist
/-- The distance between two functions is controlled by the supremum of the pointwise distances. -/
theorem dist_le (C0 : (0 : ℝ) ≤ C) : dist f g ≤ C ↔ ∀ x : α, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le C0, mkOfCompact_apply]
#align continuous_map.dist_le ContinuousMap.dist_le
theorem dist_le_iff_of_nonempty [Nonempty α] : dist f g ≤ C ↔ ∀ x, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le_iff_of_nonempty,
mkOfCompact_apply]
#align continuous_map.dist_le_iff_of_nonempty ContinuousMap.dist_le_iff_of_nonempty
theorem dist_lt_iff_of_nonempty [Nonempty α] : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
simp only [← dist_mkOfCompact, dist_lt_iff_of_nonempty_compact, mkOfCompact_apply]
#align continuous_map.dist_lt_iff_of_nonempty ContinuousMap.dist_lt_iff_of_nonempty
theorem dist_lt_of_nonempty [Nonempty α] (w : ∀ x : α, dist (f x) (g x) < C) : dist f g < C :=
dist_lt_iff_of_nonempty.2 w
#align continuous_map.dist_lt_of_nonempty ContinuousMap.dist_lt_of_nonempty
theorem dist_lt_iff (C0 : (0 : ℝ) < C) : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
rw [← dist_mkOfCompact, dist_lt_iff_of_compact C0]
simp only [mkOfCompact_apply]
#align continuous_map.dist_lt_iff ContinuousMap.dist_lt_iff
end
instance [CompleteSpace β] : CompleteSpace C(α, β) :=
(isometryEquivBoundedOfCompact α β).completeSpace
/-- See also `ContinuousMap.continuous_eval'`. -/
@[continuity]
theorem continuous_eval : Continuous fun p : C(α, β) × α => p.1 p.2 :=
continuous_eval.comp ((isometryEquivBoundedOfCompact α β).continuous.prod_map continuous_id)
#align continuous_map.continuous_eval ContinuousMap.continuous_eval
-- TODO at some point we will need lemmas characterising this norm!
-- At the moment the only way to reason about it is to transfer `f : C(α,E)` back to `α →ᵇ E`.
instance : Norm C(α, E) where norm x := dist x 0
@[simp]
theorem _root_.BoundedContinuousFunction.norm_mkOfCompact (f : C(α, E)) : ‖mkOfCompact f‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_mk_of_compact BoundedContinuousFunction.norm_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.norm_toContinuousMap_eq (f : α →ᵇ E) :
‖f.toContinuousMap‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_to_continuous_map_eq BoundedContinuousFunction.norm_toContinuousMap_eq
open BoundedContinuousFunction
instance : NormedAddCommGroup C(α, E) :=
{ ContinuousMap.metricSpace _ _,
ContinuousMap.instAddCommGroupContinuousMap with
dist_eq := fun x y => by
|
rw [← norm_mkOfCompact, ← dist_mkOfCompact, dist_eq_norm, mkOfCompact_sub]
|
instance : NormedAddCommGroup C(α, E) :=
{ ContinuousMap.metricSpace _ _,
ContinuousMap.instAddCommGroupContinuousMap with
dist_eq := fun x y => by
|
Mathlib.Topology.ContinuousFunction.Compact.185_0.Mig2jTVnn2FLKEB
|
instance : NormedAddCommGroup C(α, E)
|
Mathlib_Topology_ContinuousFunction_Compact
|
α : Type u_1
β : Type u_2
E : Type u_3
inst✝⁶ : TopologicalSpace α
inst✝⁵ : CompactSpace α
inst✝⁴ : MetricSpace β
inst✝³ : NormedAddCommGroup E
inst✝² : Nonempty α
inst✝¹ : One E
inst✝ : NormOneClass E
⊢ ‖1‖ = 1
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compacts
#align_import topology.continuous_function.compact from "leanprover-community/mathlib"@"d3af0609f6db8691dffdc3e1fb7feb7da72698f2"
/-!
# Continuous functions on a compact space
Continuous functions `C(α, β)` from a compact space `α` to a metric space `β`
are automatically bounded, and so acquire various structures inherited from `α →ᵇ β`.
This file transfers these structures, and restates some lemmas
characterising these structures.
If you need a lemma which is proved about `α →ᵇ β` but not for `C(α, β)` when `α` is compact,
you should restate it here. You can also use
`ContinuousMap.equivBoundedOfCompact` to move functions back and forth.
-/
noncomputable section
open Topology Classical NNReal BoundedContinuousFunction BigOperators
open Set Filter Metric
open BoundedContinuousFunction
namespace ContinuousMap
variable {α β E : Type*} [TopologicalSpace α] [CompactSpace α] [MetricSpace β]
[NormedAddCommGroup E]
section
variable (α β)
/-- When `α` is compact, the bounded continuous maps `α →ᵇ β` are
equivalent to `C(α, β)`.
-/
@[simps (config := .asFn)]
def equivBoundedOfCompact : C(α, β) ≃ (α →ᵇ β) :=
⟨mkOfCompact, BoundedContinuousFunction.toContinuousMap, fun f => by
ext
rfl, fun f => by
ext
rfl⟩
#align continuous_map.equiv_bounded_of_compact ContinuousMap.equivBoundedOfCompact
theorem uniformInducing_equivBoundedOfCompact : UniformInducing (equivBoundedOfCompact α β) :=
UniformInducing.mk'
(by
simp only [hasBasis_compactConvergenceUniformity.mem_iff, uniformity_basis_dist_le.mem_iff]
exact fun s =>
⟨fun ⟨⟨a, b⟩, ⟨_, ⟨ε, hε, hb⟩⟩, hs⟩ =>
⟨{ p | ∀ x, (p.1 x, p.2 x) ∈ b }, ⟨ε, hε, fun _ h x => hb ((dist_le hε.le).mp h x)⟩,
fun f g h => hs fun x _ => h x⟩,
fun ⟨_, ⟨ε, hε, ht⟩, hs⟩ =>
⟨⟨Set.univ, { p | dist p.1 p.2 ≤ ε }⟩, ⟨isCompact_univ, ⟨ε, hε, fun _ h => h⟩⟩,
fun ⟨f, g⟩ h => hs _ _ (ht ((dist_le hε.le).mpr fun x => h x (mem_univ x)))⟩⟩)
#align continuous_map.uniform_inducing_equiv_bounded_of_compact ContinuousMap.uniformInducing_equivBoundedOfCompact
theorem uniformEmbedding_equivBoundedOfCompact : UniformEmbedding (equivBoundedOfCompact α β) :=
{ uniformInducing_equivBoundedOfCompact α β with inj := (equivBoundedOfCompact α β).injective }
#align continuous_map.uniform_embedding_equiv_bounded_of_compact ContinuousMap.uniformEmbedding_equivBoundedOfCompact
/-- When `α` is compact, the bounded continuous maps `α →ᵇ 𝕜` are
additively equivalent to `C(α, 𝕜)`.
-/
-- porting note: the following `simps` received a "maximum recursion depth" error
-- @[simps! (config := .asFn) apply symm_apply]
def addEquivBoundedOfCompact [AddMonoid β] [LipschitzAdd β] : C(α, β) ≃+ (α →ᵇ β) :=
({ toContinuousMapAddHom α β, (equivBoundedOfCompact α β).symm with } : (α →ᵇ β) ≃+ C(α, β)).symm
#align continuous_map.add_equiv_bounded_of_compact ContinuousMap.addEquivBoundedOfCompact
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_symm_apply [AddMonoid β] [LipschitzAdd β] :
⇑((addEquivBoundedOfCompact α β).symm) = toContinuousMapAddHom α β :=
rfl
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_apply [AddMonoid β] [LipschitzAdd β] :
⇑(addEquivBoundedOfCompact α β) = mkOfCompact :=
rfl
instance metricSpace : MetricSpace C(α, β) :=
(uniformEmbedding_equivBoundedOfCompact α β).comapMetricSpace _
#align continuous_map.metric_space ContinuousMap.metricSpace
/-- When `α` is compact, and `β` is a metric space, the bounded continuous maps `α →ᵇ β` are
isometric to `C(α, β)`.
-/
@[simps! (config := .asFn) toEquiv apply symm_apply]
def isometryEquivBoundedOfCompact : C(α, β) ≃ᵢ (α →ᵇ β) where
isometry_toFun _ _ := rfl
toEquiv := equivBoundedOfCompact α β
#align continuous_map.isometry_equiv_bounded_of_compact ContinuousMap.isometryEquivBoundedOfCompact
end
@[simp]
theorem _root_.BoundedContinuousFunction.dist_mkOfCompact (f g : C(α, β)) :
dist (mkOfCompact f) (mkOfCompact g) = dist f g :=
rfl
#align bounded_continuous_function.dist_mk_of_compact BoundedContinuousFunction.dist_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.dist_toContinuousMap (f g : α →ᵇ β) :
dist f.toContinuousMap g.toContinuousMap = dist f g :=
rfl
#align bounded_continuous_function.dist_to_continuous_map BoundedContinuousFunction.dist_toContinuousMap
open BoundedContinuousFunction
section
variable {f g : C(α, β)} {C : ℝ}
/-- The pointwise distance is controlled by the distance between functions, by definition. -/
theorem dist_apply_le_dist (x : α) : dist (f x) (g x) ≤ dist f g := by
simp only [← dist_mkOfCompact, dist_coe_le_dist, ← mkOfCompact_apply]
#align continuous_map.dist_apply_le_dist ContinuousMap.dist_apply_le_dist
/-- The distance between two functions is controlled by the supremum of the pointwise distances. -/
theorem dist_le (C0 : (0 : ℝ) ≤ C) : dist f g ≤ C ↔ ∀ x : α, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le C0, mkOfCompact_apply]
#align continuous_map.dist_le ContinuousMap.dist_le
theorem dist_le_iff_of_nonempty [Nonempty α] : dist f g ≤ C ↔ ∀ x, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le_iff_of_nonempty,
mkOfCompact_apply]
#align continuous_map.dist_le_iff_of_nonempty ContinuousMap.dist_le_iff_of_nonempty
theorem dist_lt_iff_of_nonempty [Nonempty α] : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
simp only [← dist_mkOfCompact, dist_lt_iff_of_nonempty_compact, mkOfCompact_apply]
#align continuous_map.dist_lt_iff_of_nonempty ContinuousMap.dist_lt_iff_of_nonempty
theorem dist_lt_of_nonempty [Nonempty α] (w : ∀ x : α, dist (f x) (g x) < C) : dist f g < C :=
dist_lt_iff_of_nonempty.2 w
#align continuous_map.dist_lt_of_nonempty ContinuousMap.dist_lt_of_nonempty
theorem dist_lt_iff (C0 : (0 : ℝ) < C) : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
rw [← dist_mkOfCompact, dist_lt_iff_of_compact C0]
simp only [mkOfCompact_apply]
#align continuous_map.dist_lt_iff ContinuousMap.dist_lt_iff
end
instance [CompleteSpace β] : CompleteSpace C(α, β) :=
(isometryEquivBoundedOfCompact α β).completeSpace
/-- See also `ContinuousMap.continuous_eval'`. -/
@[continuity]
theorem continuous_eval : Continuous fun p : C(α, β) × α => p.1 p.2 :=
continuous_eval.comp ((isometryEquivBoundedOfCompact α β).continuous.prod_map continuous_id)
#align continuous_map.continuous_eval ContinuousMap.continuous_eval
-- TODO at some point we will need lemmas characterising this norm!
-- At the moment the only way to reason about it is to transfer `f : C(α,E)` back to `α →ᵇ E`.
instance : Norm C(α, E) where norm x := dist x 0
@[simp]
theorem _root_.BoundedContinuousFunction.norm_mkOfCompact (f : C(α, E)) : ‖mkOfCompact f‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_mk_of_compact BoundedContinuousFunction.norm_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.norm_toContinuousMap_eq (f : α →ᵇ E) :
‖f.toContinuousMap‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_to_continuous_map_eq BoundedContinuousFunction.norm_toContinuousMap_eq
open BoundedContinuousFunction
instance : NormedAddCommGroup C(α, E) :=
{ ContinuousMap.metricSpace _ _,
ContinuousMap.instAddCommGroupContinuousMap with
dist_eq := fun x y => by
rw [← norm_mkOfCompact, ← dist_mkOfCompact, dist_eq_norm, mkOfCompact_sub]
dist := dist
norm := norm }
instance [Nonempty α] [One E] [NormOneClass E] : NormOneClass C(α, E) where
norm_one := by
|
simp only [← norm_mkOfCompact, mkOfCompact_one, norm_one]
|
instance [Nonempty α] [One E] [NormOneClass E] : NormOneClass C(α, E) where
norm_one := by
|
Mathlib.Topology.ContinuousFunction.Compact.193_0.Mig2jTVnn2FLKEB
|
instance [Nonempty α] [One E] [NormOneClass E] : NormOneClass C(α, E) where
norm_one
|
Mathlib_Topology_ContinuousFunction_Compact
|
α : Type u_1
β : Type u_2
E : Type u_3
inst✝⁵ : TopologicalSpace α
inst✝⁴ : CompactSpace α
inst✝³ : MetricSpace β
inst✝² : NormedAddCommGroup E
𝕜 : Type u_4
inst✝¹ : NormedField 𝕜
inst✝ : NormedSpace 𝕜 E
src✝ : C(α, E) ≃+ (α →ᵇ E) := addEquivBoundedOfCompact α E
c : 𝕜
f : C(α, E)
⊢ AddHom.toFun
{ toFun := src✝.toFun,
map_add' :=
(_ :
∀ (x y : C(α, E)),
Equiv.toFun src✝.toEquiv (x + y) = Equiv.toFun src✝.toEquiv x + Equiv.toFun src✝.toEquiv y) }
(c • f) =
(RingHom.id 𝕜) c •
AddHom.toFun
{ toFun := src✝.toFun,
map_add' :=
(_ :
∀ (x y : C(α, E)),
Equiv.toFun src✝.toEquiv (x + y) = Equiv.toFun src✝.toEquiv x + Equiv.toFun src✝.toEquiv y) }
f
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compacts
#align_import topology.continuous_function.compact from "leanprover-community/mathlib"@"d3af0609f6db8691dffdc3e1fb7feb7da72698f2"
/-!
# Continuous functions on a compact space
Continuous functions `C(α, β)` from a compact space `α` to a metric space `β`
are automatically bounded, and so acquire various structures inherited from `α →ᵇ β`.
This file transfers these structures, and restates some lemmas
characterising these structures.
If you need a lemma which is proved about `α →ᵇ β` but not for `C(α, β)` when `α` is compact,
you should restate it here. You can also use
`ContinuousMap.equivBoundedOfCompact` to move functions back and forth.
-/
noncomputable section
open Topology Classical NNReal BoundedContinuousFunction BigOperators
open Set Filter Metric
open BoundedContinuousFunction
namespace ContinuousMap
variable {α β E : Type*} [TopologicalSpace α] [CompactSpace α] [MetricSpace β]
[NormedAddCommGroup E]
section
variable (α β)
/-- When `α` is compact, the bounded continuous maps `α →ᵇ β` are
equivalent to `C(α, β)`.
-/
@[simps (config := .asFn)]
def equivBoundedOfCompact : C(α, β) ≃ (α →ᵇ β) :=
⟨mkOfCompact, BoundedContinuousFunction.toContinuousMap, fun f => by
ext
rfl, fun f => by
ext
rfl⟩
#align continuous_map.equiv_bounded_of_compact ContinuousMap.equivBoundedOfCompact
theorem uniformInducing_equivBoundedOfCompact : UniformInducing (equivBoundedOfCompact α β) :=
UniformInducing.mk'
(by
simp only [hasBasis_compactConvergenceUniformity.mem_iff, uniformity_basis_dist_le.mem_iff]
exact fun s =>
⟨fun ⟨⟨a, b⟩, ⟨_, ⟨ε, hε, hb⟩⟩, hs⟩ =>
⟨{ p | ∀ x, (p.1 x, p.2 x) ∈ b }, ⟨ε, hε, fun _ h x => hb ((dist_le hε.le).mp h x)⟩,
fun f g h => hs fun x _ => h x⟩,
fun ⟨_, ⟨ε, hε, ht⟩, hs⟩ =>
⟨⟨Set.univ, { p | dist p.1 p.2 ≤ ε }⟩, ⟨isCompact_univ, ⟨ε, hε, fun _ h => h⟩⟩,
fun ⟨f, g⟩ h => hs _ _ (ht ((dist_le hε.le).mpr fun x => h x (mem_univ x)))⟩⟩)
#align continuous_map.uniform_inducing_equiv_bounded_of_compact ContinuousMap.uniformInducing_equivBoundedOfCompact
theorem uniformEmbedding_equivBoundedOfCompact : UniformEmbedding (equivBoundedOfCompact α β) :=
{ uniformInducing_equivBoundedOfCompact α β with inj := (equivBoundedOfCompact α β).injective }
#align continuous_map.uniform_embedding_equiv_bounded_of_compact ContinuousMap.uniformEmbedding_equivBoundedOfCompact
/-- When `α` is compact, the bounded continuous maps `α →ᵇ 𝕜` are
additively equivalent to `C(α, 𝕜)`.
-/
-- porting note: the following `simps` received a "maximum recursion depth" error
-- @[simps! (config := .asFn) apply symm_apply]
def addEquivBoundedOfCompact [AddMonoid β] [LipschitzAdd β] : C(α, β) ≃+ (α →ᵇ β) :=
({ toContinuousMapAddHom α β, (equivBoundedOfCompact α β).symm with } : (α →ᵇ β) ≃+ C(α, β)).symm
#align continuous_map.add_equiv_bounded_of_compact ContinuousMap.addEquivBoundedOfCompact
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_symm_apply [AddMonoid β] [LipschitzAdd β] :
⇑((addEquivBoundedOfCompact α β).symm) = toContinuousMapAddHom α β :=
rfl
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_apply [AddMonoid β] [LipschitzAdd β] :
⇑(addEquivBoundedOfCompact α β) = mkOfCompact :=
rfl
instance metricSpace : MetricSpace C(α, β) :=
(uniformEmbedding_equivBoundedOfCompact α β).comapMetricSpace _
#align continuous_map.metric_space ContinuousMap.metricSpace
/-- When `α` is compact, and `β` is a metric space, the bounded continuous maps `α →ᵇ β` are
isometric to `C(α, β)`.
-/
@[simps! (config := .asFn) toEquiv apply symm_apply]
def isometryEquivBoundedOfCompact : C(α, β) ≃ᵢ (α →ᵇ β) where
isometry_toFun _ _ := rfl
toEquiv := equivBoundedOfCompact α β
#align continuous_map.isometry_equiv_bounded_of_compact ContinuousMap.isometryEquivBoundedOfCompact
end
@[simp]
theorem _root_.BoundedContinuousFunction.dist_mkOfCompact (f g : C(α, β)) :
dist (mkOfCompact f) (mkOfCompact g) = dist f g :=
rfl
#align bounded_continuous_function.dist_mk_of_compact BoundedContinuousFunction.dist_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.dist_toContinuousMap (f g : α →ᵇ β) :
dist f.toContinuousMap g.toContinuousMap = dist f g :=
rfl
#align bounded_continuous_function.dist_to_continuous_map BoundedContinuousFunction.dist_toContinuousMap
open BoundedContinuousFunction
section
variable {f g : C(α, β)} {C : ℝ}
/-- The pointwise distance is controlled by the distance between functions, by definition. -/
theorem dist_apply_le_dist (x : α) : dist (f x) (g x) ≤ dist f g := by
simp only [← dist_mkOfCompact, dist_coe_le_dist, ← mkOfCompact_apply]
#align continuous_map.dist_apply_le_dist ContinuousMap.dist_apply_le_dist
/-- The distance between two functions is controlled by the supremum of the pointwise distances. -/
theorem dist_le (C0 : (0 : ℝ) ≤ C) : dist f g ≤ C ↔ ∀ x : α, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le C0, mkOfCompact_apply]
#align continuous_map.dist_le ContinuousMap.dist_le
theorem dist_le_iff_of_nonempty [Nonempty α] : dist f g ≤ C ↔ ∀ x, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le_iff_of_nonempty,
mkOfCompact_apply]
#align continuous_map.dist_le_iff_of_nonempty ContinuousMap.dist_le_iff_of_nonempty
theorem dist_lt_iff_of_nonempty [Nonempty α] : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
simp only [← dist_mkOfCompact, dist_lt_iff_of_nonempty_compact, mkOfCompact_apply]
#align continuous_map.dist_lt_iff_of_nonempty ContinuousMap.dist_lt_iff_of_nonempty
theorem dist_lt_of_nonempty [Nonempty α] (w : ∀ x : α, dist (f x) (g x) < C) : dist f g < C :=
dist_lt_iff_of_nonempty.2 w
#align continuous_map.dist_lt_of_nonempty ContinuousMap.dist_lt_of_nonempty
theorem dist_lt_iff (C0 : (0 : ℝ) < C) : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
rw [← dist_mkOfCompact, dist_lt_iff_of_compact C0]
simp only [mkOfCompact_apply]
#align continuous_map.dist_lt_iff ContinuousMap.dist_lt_iff
end
instance [CompleteSpace β] : CompleteSpace C(α, β) :=
(isometryEquivBoundedOfCompact α β).completeSpace
/-- See also `ContinuousMap.continuous_eval'`. -/
@[continuity]
theorem continuous_eval : Continuous fun p : C(α, β) × α => p.1 p.2 :=
continuous_eval.comp ((isometryEquivBoundedOfCompact α β).continuous.prod_map continuous_id)
#align continuous_map.continuous_eval ContinuousMap.continuous_eval
-- TODO at some point we will need lemmas characterising this norm!
-- At the moment the only way to reason about it is to transfer `f : C(α,E)` back to `α →ᵇ E`.
instance : Norm C(α, E) where norm x := dist x 0
@[simp]
theorem _root_.BoundedContinuousFunction.norm_mkOfCompact (f : C(α, E)) : ‖mkOfCompact f‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_mk_of_compact BoundedContinuousFunction.norm_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.norm_toContinuousMap_eq (f : α →ᵇ E) :
‖f.toContinuousMap‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_to_continuous_map_eq BoundedContinuousFunction.norm_toContinuousMap_eq
open BoundedContinuousFunction
instance : NormedAddCommGroup C(α, E) :=
{ ContinuousMap.metricSpace _ _,
ContinuousMap.instAddCommGroupContinuousMap with
dist_eq := fun x y => by
rw [← norm_mkOfCompact, ← dist_mkOfCompact, dist_eq_norm, mkOfCompact_sub]
dist := dist
norm := norm }
instance [Nonempty α] [One E] [NormOneClass E] : NormOneClass C(α, E) where
norm_one := by simp only [← norm_mkOfCompact, mkOfCompact_one, norm_one]
section
variable (f : C(α, E))
-- The corresponding lemmas for `BoundedContinuousFunction` are stated with `{f}`,
-- and so can not be used in dot notation.
theorem norm_coe_le_norm (x : α) : ‖f x‖ ≤ ‖f‖ :=
(mkOfCompact f).norm_coe_le_norm x
#align continuous_map.norm_coe_le_norm ContinuousMap.norm_coe_le_norm
/-- Distance between the images of any two points is at most twice the norm of the function. -/
theorem dist_le_two_norm (x y : α) : dist (f x) (f y) ≤ 2 * ‖f‖ :=
(mkOfCompact f).dist_le_two_norm x y
#align continuous_map.dist_le_two_norm ContinuousMap.dist_le_two_norm
/-- The norm of a function is controlled by the supremum of the pointwise norms. -/
theorem norm_le {C : ℝ} (C0 : (0 : ℝ) ≤ C) : ‖f‖ ≤ C ↔ ∀ x : α, ‖f x‖ ≤ C :=
@BoundedContinuousFunction.norm_le _ _ _ _ (mkOfCompact f) _ C0
#align continuous_map.norm_le ContinuousMap.norm_le
theorem norm_le_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ ≤ M ↔ ∀ x, ‖f x‖ ≤ M :=
@BoundedContinuousFunction.norm_le_of_nonempty _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_le_of_nonempty ContinuousMap.norm_le_of_nonempty
theorem norm_lt_iff {M : ℝ} (M0 : 0 < M) : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_compact _ _ _ _ _ (mkOfCompact f) _ M0
#align continuous_map.norm_lt_iff ContinuousMap.norm_lt_iff
theorem nnnorm_lt_iff {M : ℝ≥0} (M0 : 0 < M) : ‖f‖₊ < M ↔ ∀ x : α, ‖f x‖₊ < M :=
f.norm_lt_iff M0
#align continuous_map.nnnorm_lt_iff ContinuousMap.nnnorm_lt_iff
theorem norm_lt_iff_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_nonempty_compact _ _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_lt_iff_of_nonempty ContinuousMap.norm_lt_iff_of_nonempty
theorem nnnorm_lt_iff_of_nonempty [Nonempty α] {M : ℝ≥0} : ‖f‖₊ < M ↔ ∀ x, ‖f x‖₊ < M :=
f.norm_lt_iff_of_nonempty
#align continuous_map.nnnorm_lt_iff_of_nonempty ContinuousMap.nnnorm_lt_iff_of_nonempty
theorem apply_le_norm (f : C(α, ℝ)) (x : α) : f x ≤ ‖f‖ :=
le_trans (le_abs.mpr (Or.inl (le_refl (f x)))) (f.norm_coe_le_norm x)
#align continuous_map.apply_le_norm ContinuousMap.apply_le_norm
theorem neg_norm_le_apply (f : C(α, ℝ)) (x : α) : -‖f‖ ≤ f x :=
le_trans (neg_le_neg (f.norm_coe_le_norm x)) (neg_le.mp (neg_le_abs_self (f x)))
#align continuous_map.neg_norm_le_apply ContinuousMap.neg_norm_le_apply
theorem norm_eq_iSup_norm : ‖f‖ = ⨆ x : α, ‖f x‖ :=
(mkOfCompact f).norm_eq_iSup_norm
#align continuous_map.norm_eq_supr_norm ContinuousMap.norm_eq_iSup_norm
theorem norm_restrict_mono_set {X : Type*} [TopologicalSpace X] (f : C(X, E))
{K L : TopologicalSpace.Compacts X} (hKL : K ≤ L) : ‖f.restrict K‖ ≤ ‖f.restrict L‖ :=
(norm_le _ (norm_nonneg _)).mpr fun x => norm_coe_le_norm (f.restrict L) <| Set.inclusion hKL x
#align continuous_map.norm_restrict_mono_set ContinuousMap.norm_restrict_mono_set
end
section
variable {R : Type*} [NormedRing R]
instance : NormedRing C(α, R) :=
{ (inferInstance : NormedAddCommGroup C(α, R)), ContinuousMap.instRingContinuousMap with
norm_mul := fun f g => norm_mul_le (mkOfCompact f) (mkOfCompact g) }
end
section
variable {𝕜 : Type*} [NormedField 𝕜] [NormedSpace 𝕜 E]
instance normedSpace : NormedSpace 𝕜 C(α, E) where
norm_smul_le c f := (norm_smul_le c (mkOfCompact f) : _)
#align continuous_map.normed_space ContinuousMap.normedSpace
section
variable (α 𝕜 E)
/-- When `α` is compact and `𝕜` is a normed field,
the `𝕜`-algebra of bounded continuous maps `α →ᵇ β` is
`𝕜`-linearly isometric to `C(α, β)`.
-/
def linearIsometryBoundedOfCompact : C(α, E) ≃ₗᵢ[𝕜] α →ᵇ E :=
{ addEquivBoundedOfCompact α E with
map_smul' := fun c f => by
|
ext
|
/-- When `α` is compact and `𝕜` is a normed field,
the `𝕜`-algebra of bounded continuous maps `α →ᵇ β` is
`𝕜`-linearly isometric to `C(α, β)`.
-/
def linearIsometryBoundedOfCompact : C(α, E) ≃ₗᵢ[𝕜] α →ᵇ E :=
{ addEquivBoundedOfCompact α E with
map_smul' := fun c f => by
|
Mathlib.Topology.ContinuousFunction.Compact.277_0.Mig2jTVnn2FLKEB
|
/-- When `α` is compact and `𝕜` is a normed field,
the `𝕜`-algebra of bounded continuous maps `α →ᵇ β` is
`𝕜`-linearly isometric to `C(α, β)`.
-/
def linearIsometryBoundedOfCompact : C(α, E) ≃ₗᵢ[𝕜] α →ᵇ E
|
Mathlib_Topology_ContinuousFunction_Compact
|
case h
α : Type u_1
β : Type u_2
E : Type u_3
inst✝⁵ : TopologicalSpace α
inst✝⁴ : CompactSpace α
inst✝³ : MetricSpace β
inst✝² : NormedAddCommGroup E
𝕜 : Type u_4
inst✝¹ : NormedField 𝕜
inst✝ : NormedSpace 𝕜 E
src✝ : C(α, E) ≃+ (α →ᵇ E) := addEquivBoundedOfCompact α E
c : 𝕜
f : C(α, E)
x✝ : α
⊢ (AddHom.toFun
{ toFun := src✝.toFun,
map_add' :=
(_ :
∀ (x y : C(α, E)),
Equiv.toFun src✝.toEquiv (x + y) = Equiv.toFun src✝.toEquiv x + Equiv.toFun src✝.toEquiv y) }
(c • f))
x✝ =
((RingHom.id 𝕜) c •
AddHom.toFun
{ toFun := src✝.toFun,
map_add' :=
(_ :
∀ (x y : C(α, E)),
Equiv.toFun src✝.toEquiv (x + y) = Equiv.toFun src✝.toEquiv x + Equiv.toFun src✝.toEquiv y) }
f)
x✝
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compacts
#align_import topology.continuous_function.compact from "leanprover-community/mathlib"@"d3af0609f6db8691dffdc3e1fb7feb7da72698f2"
/-!
# Continuous functions on a compact space
Continuous functions `C(α, β)` from a compact space `α` to a metric space `β`
are automatically bounded, and so acquire various structures inherited from `α →ᵇ β`.
This file transfers these structures, and restates some lemmas
characterising these structures.
If you need a lemma which is proved about `α →ᵇ β` but not for `C(α, β)` when `α` is compact,
you should restate it here. You can also use
`ContinuousMap.equivBoundedOfCompact` to move functions back and forth.
-/
noncomputable section
open Topology Classical NNReal BoundedContinuousFunction BigOperators
open Set Filter Metric
open BoundedContinuousFunction
namespace ContinuousMap
variable {α β E : Type*} [TopologicalSpace α] [CompactSpace α] [MetricSpace β]
[NormedAddCommGroup E]
section
variable (α β)
/-- When `α` is compact, the bounded continuous maps `α →ᵇ β` are
equivalent to `C(α, β)`.
-/
@[simps (config := .asFn)]
def equivBoundedOfCompact : C(α, β) ≃ (α →ᵇ β) :=
⟨mkOfCompact, BoundedContinuousFunction.toContinuousMap, fun f => by
ext
rfl, fun f => by
ext
rfl⟩
#align continuous_map.equiv_bounded_of_compact ContinuousMap.equivBoundedOfCompact
theorem uniformInducing_equivBoundedOfCompact : UniformInducing (equivBoundedOfCompact α β) :=
UniformInducing.mk'
(by
simp only [hasBasis_compactConvergenceUniformity.mem_iff, uniformity_basis_dist_le.mem_iff]
exact fun s =>
⟨fun ⟨⟨a, b⟩, ⟨_, ⟨ε, hε, hb⟩⟩, hs⟩ =>
⟨{ p | ∀ x, (p.1 x, p.2 x) ∈ b }, ⟨ε, hε, fun _ h x => hb ((dist_le hε.le).mp h x)⟩,
fun f g h => hs fun x _ => h x⟩,
fun ⟨_, ⟨ε, hε, ht⟩, hs⟩ =>
⟨⟨Set.univ, { p | dist p.1 p.2 ≤ ε }⟩, ⟨isCompact_univ, ⟨ε, hε, fun _ h => h⟩⟩,
fun ⟨f, g⟩ h => hs _ _ (ht ((dist_le hε.le).mpr fun x => h x (mem_univ x)))⟩⟩)
#align continuous_map.uniform_inducing_equiv_bounded_of_compact ContinuousMap.uniformInducing_equivBoundedOfCompact
theorem uniformEmbedding_equivBoundedOfCompact : UniformEmbedding (equivBoundedOfCompact α β) :=
{ uniformInducing_equivBoundedOfCompact α β with inj := (equivBoundedOfCompact α β).injective }
#align continuous_map.uniform_embedding_equiv_bounded_of_compact ContinuousMap.uniformEmbedding_equivBoundedOfCompact
/-- When `α` is compact, the bounded continuous maps `α →ᵇ 𝕜` are
additively equivalent to `C(α, 𝕜)`.
-/
-- porting note: the following `simps` received a "maximum recursion depth" error
-- @[simps! (config := .asFn) apply symm_apply]
def addEquivBoundedOfCompact [AddMonoid β] [LipschitzAdd β] : C(α, β) ≃+ (α →ᵇ β) :=
({ toContinuousMapAddHom α β, (equivBoundedOfCompact α β).symm with } : (α →ᵇ β) ≃+ C(α, β)).symm
#align continuous_map.add_equiv_bounded_of_compact ContinuousMap.addEquivBoundedOfCompact
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_symm_apply [AddMonoid β] [LipschitzAdd β] :
⇑((addEquivBoundedOfCompact α β).symm) = toContinuousMapAddHom α β :=
rfl
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_apply [AddMonoid β] [LipschitzAdd β] :
⇑(addEquivBoundedOfCompact α β) = mkOfCompact :=
rfl
instance metricSpace : MetricSpace C(α, β) :=
(uniformEmbedding_equivBoundedOfCompact α β).comapMetricSpace _
#align continuous_map.metric_space ContinuousMap.metricSpace
/-- When `α` is compact, and `β` is a metric space, the bounded continuous maps `α →ᵇ β` are
isometric to `C(α, β)`.
-/
@[simps! (config := .asFn) toEquiv apply symm_apply]
def isometryEquivBoundedOfCompact : C(α, β) ≃ᵢ (α →ᵇ β) where
isometry_toFun _ _ := rfl
toEquiv := equivBoundedOfCompact α β
#align continuous_map.isometry_equiv_bounded_of_compact ContinuousMap.isometryEquivBoundedOfCompact
end
@[simp]
theorem _root_.BoundedContinuousFunction.dist_mkOfCompact (f g : C(α, β)) :
dist (mkOfCompact f) (mkOfCompact g) = dist f g :=
rfl
#align bounded_continuous_function.dist_mk_of_compact BoundedContinuousFunction.dist_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.dist_toContinuousMap (f g : α →ᵇ β) :
dist f.toContinuousMap g.toContinuousMap = dist f g :=
rfl
#align bounded_continuous_function.dist_to_continuous_map BoundedContinuousFunction.dist_toContinuousMap
open BoundedContinuousFunction
section
variable {f g : C(α, β)} {C : ℝ}
/-- The pointwise distance is controlled by the distance between functions, by definition. -/
theorem dist_apply_le_dist (x : α) : dist (f x) (g x) ≤ dist f g := by
simp only [← dist_mkOfCompact, dist_coe_le_dist, ← mkOfCompact_apply]
#align continuous_map.dist_apply_le_dist ContinuousMap.dist_apply_le_dist
/-- The distance between two functions is controlled by the supremum of the pointwise distances. -/
theorem dist_le (C0 : (0 : ℝ) ≤ C) : dist f g ≤ C ↔ ∀ x : α, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le C0, mkOfCompact_apply]
#align continuous_map.dist_le ContinuousMap.dist_le
theorem dist_le_iff_of_nonempty [Nonempty α] : dist f g ≤ C ↔ ∀ x, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le_iff_of_nonempty,
mkOfCompact_apply]
#align continuous_map.dist_le_iff_of_nonempty ContinuousMap.dist_le_iff_of_nonempty
theorem dist_lt_iff_of_nonempty [Nonempty α] : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
simp only [← dist_mkOfCompact, dist_lt_iff_of_nonempty_compact, mkOfCompact_apply]
#align continuous_map.dist_lt_iff_of_nonempty ContinuousMap.dist_lt_iff_of_nonempty
theorem dist_lt_of_nonempty [Nonempty α] (w : ∀ x : α, dist (f x) (g x) < C) : dist f g < C :=
dist_lt_iff_of_nonempty.2 w
#align continuous_map.dist_lt_of_nonempty ContinuousMap.dist_lt_of_nonempty
theorem dist_lt_iff (C0 : (0 : ℝ) < C) : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
rw [← dist_mkOfCompact, dist_lt_iff_of_compact C0]
simp only [mkOfCompact_apply]
#align continuous_map.dist_lt_iff ContinuousMap.dist_lt_iff
end
instance [CompleteSpace β] : CompleteSpace C(α, β) :=
(isometryEquivBoundedOfCompact α β).completeSpace
/-- See also `ContinuousMap.continuous_eval'`. -/
@[continuity]
theorem continuous_eval : Continuous fun p : C(α, β) × α => p.1 p.2 :=
continuous_eval.comp ((isometryEquivBoundedOfCompact α β).continuous.prod_map continuous_id)
#align continuous_map.continuous_eval ContinuousMap.continuous_eval
-- TODO at some point we will need lemmas characterising this norm!
-- At the moment the only way to reason about it is to transfer `f : C(α,E)` back to `α →ᵇ E`.
instance : Norm C(α, E) where norm x := dist x 0
@[simp]
theorem _root_.BoundedContinuousFunction.norm_mkOfCompact (f : C(α, E)) : ‖mkOfCompact f‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_mk_of_compact BoundedContinuousFunction.norm_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.norm_toContinuousMap_eq (f : α →ᵇ E) :
‖f.toContinuousMap‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_to_continuous_map_eq BoundedContinuousFunction.norm_toContinuousMap_eq
open BoundedContinuousFunction
instance : NormedAddCommGroup C(α, E) :=
{ ContinuousMap.metricSpace _ _,
ContinuousMap.instAddCommGroupContinuousMap with
dist_eq := fun x y => by
rw [← norm_mkOfCompact, ← dist_mkOfCompact, dist_eq_norm, mkOfCompact_sub]
dist := dist
norm := norm }
instance [Nonempty α] [One E] [NormOneClass E] : NormOneClass C(α, E) where
norm_one := by simp only [← norm_mkOfCompact, mkOfCompact_one, norm_one]
section
variable (f : C(α, E))
-- The corresponding lemmas for `BoundedContinuousFunction` are stated with `{f}`,
-- and so can not be used in dot notation.
theorem norm_coe_le_norm (x : α) : ‖f x‖ ≤ ‖f‖ :=
(mkOfCompact f).norm_coe_le_norm x
#align continuous_map.norm_coe_le_norm ContinuousMap.norm_coe_le_norm
/-- Distance between the images of any two points is at most twice the norm of the function. -/
theorem dist_le_two_norm (x y : α) : dist (f x) (f y) ≤ 2 * ‖f‖ :=
(mkOfCompact f).dist_le_two_norm x y
#align continuous_map.dist_le_two_norm ContinuousMap.dist_le_two_norm
/-- The norm of a function is controlled by the supremum of the pointwise norms. -/
theorem norm_le {C : ℝ} (C0 : (0 : ℝ) ≤ C) : ‖f‖ ≤ C ↔ ∀ x : α, ‖f x‖ ≤ C :=
@BoundedContinuousFunction.norm_le _ _ _ _ (mkOfCompact f) _ C0
#align continuous_map.norm_le ContinuousMap.norm_le
theorem norm_le_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ ≤ M ↔ ∀ x, ‖f x‖ ≤ M :=
@BoundedContinuousFunction.norm_le_of_nonempty _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_le_of_nonempty ContinuousMap.norm_le_of_nonempty
theorem norm_lt_iff {M : ℝ} (M0 : 0 < M) : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_compact _ _ _ _ _ (mkOfCompact f) _ M0
#align continuous_map.norm_lt_iff ContinuousMap.norm_lt_iff
theorem nnnorm_lt_iff {M : ℝ≥0} (M0 : 0 < M) : ‖f‖₊ < M ↔ ∀ x : α, ‖f x‖₊ < M :=
f.norm_lt_iff M0
#align continuous_map.nnnorm_lt_iff ContinuousMap.nnnorm_lt_iff
theorem norm_lt_iff_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_nonempty_compact _ _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_lt_iff_of_nonempty ContinuousMap.norm_lt_iff_of_nonempty
theorem nnnorm_lt_iff_of_nonempty [Nonempty α] {M : ℝ≥0} : ‖f‖₊ < M ↔ ∀ x, ‖f x‖₊ < M :=
f.norm_lt_iff_of_nonempty
#align continuous_map.nnnorm_lt_iff_of_nonempty ContinuousMap.nnnorm_lt_iff_of_nonempty
theorem apply_le_norm (f : C(α, ℝ)) (x : α) : f x ≤ ‖f‖ :=
le_trans (le_abs.mpr (Or.inl (le_refl (f x)))) (f.norm_coe_le_norm x)
#align continuous_map.apply_le_norm ContinuousMap.apply_le_norm
theorem neg_norm_le_apply (f : C(α, ℝ)) (x : α) : -‖f‖ ≤ f x :=
le_trans (neg_le_neg (f.norm_coe_le_norm x)) (neg_le.mp (neg_le_abs_self (f x)))
#align continuous_map.neg_norm_le_apply ContinuousMap.neg_norm_le_apply
theorem norm_eq_iSup_norm : ‖f‖ = ⨆ x : α, ‖f x‖ :=
(mkOfCompact f).norm_eq_iSup_norm
#align continuous_map.norm_eq_supr_norm ContinuousMap.norm_eq_iSup_norm
theorem norm_restrict_mono_set {X : Type*} [TopologicalSpace X] (f : C(X, E))
{K L : TopologicalSpace.Compacts X} (hKL : K ≤ L) : ‖f.restrict K‖ ≤ ‖f.restrict L‖ :=
(norm_le _ (norm_nonneg _)).mpr fun x => norm_coe_le_norm (f.restrict L) <| Set.inclusion hKL x
#align continuous_map.norm_restrict_mono_set ContinuousMap.norm_restrict_mono_set
end
section
variable {R : Type*} [NormedRing R]
instance : NormedRing C(α, R) :=
{ (inferInstance : NormedAddCommGroup C(α, R)), ContinuousMap.instRingContinuousMap with
norm_mul := fun f g => norm_mul_le (mkOfCompact f) (mkOfCompact g) }
end
section
variable {𝕜 : Type*} [NormedField 𝕜] [NormedSpace 𝕜 E]
instance normedSpace : NormedSpace 𝕜 C(α, E) where
norm_smul_le c f := (norm_smul_le c (mkOfCompact f) : _)
#align continuous_map.normed_space ContinuousMap.normedSpace
section
variable (α 𝕜 E)
/-- When `α` is compact and `𝕜` is a normed field,
the `𝕜`-algebra of bounded continuous maps `α →ᵇ β` is
`𝕜`-linearly isometric to `C(α, β)`.
-/
def linearIsometryBoundedOfCompact : C(α, E) ≃ₗᵢ[𝕜] α →ᵇ E :=
{ addEquivBoundedOfCompact α E with
map_smul' := fun c f => by
ext
|
norm_cast
|
/-- When `α` is compact and `𝕜` is a normed field,
the `𝕜`-algebra of bounded continuous maps `α →ᵇ β` is
`𝕜`-linearly isometric to `C(α, β)`.
-/
def linearIsometryBoundedOfCompact : C(α, E) ≃ₗᵢ[𝕜] α →ᵇ E :=
{ addEquivBoundedOfCompact α E with
map_smul' := fun c f => by
ext
|
Mathlib.Topology.ContinuousFunction.Compact.277_0.Mig2jTVnn2FLKEB
|
/-- When `α` is compact and `𝕜` is a normed field,
the `𝕜`-algebra of bounded continuous maps `α →ᵇ β` is
`𝕜`-linearly isometric to `C(α, β)`.
-/
def linearIsometryBoundedOfCompact : C(α, E) ≃ₗᵢ[𝕜] α →ᵇ E
|
Mathlib_Topology_ContinuousFunction_Compact
|
X : Type u_1
𝕜 : Type u_2
β : Type u_3
γ : Type u_4
inst✝⁶ : TopologicalSpace X
inst✝⁵ : CompactSpace X
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup β
inst✝² : NormedSpace 𝕜 β
inst✝¹ : NormedAddCommGroup γ
inst✝ : NormedSpace 𝕜 γ
g : β →L[𝕜] γ
⊢ ↑(ContinuousLinearMap.compLeftContinuousCompact X g) = ContinuousLinearMap.compLeftContinuous 𝕜 X g
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compacts
#align_import topology.continuous_function.compact from "leanprover-community/mathlib"@"d3af0609f6db8691dffdc3e1fb7feb7da72698f2"
/-!
# Continuous functions on a compact space
Continuous functions `C(α, β)` from a compact space `α` to a metric space `β`
are automatically bounded, and so acquire various structures inherited from `α →ᵇ β`.
This file transfers these structures, and restates some lemmas
characterising these structures.
If you need a lemma which is proved about `α →ᵇ β` but not for `C(α, β)` when `α` is compact,
you should restate it here. You can also use
`ContinuousMap.equivBoundedOfCompact` to move functions back and forth.
-/
noncomputable section
open Topology Classical NNReal BoundedContinuousFunction BigOperators
open Set Filter Metric
open BoundedContinuousFunction
namespace ContinuousMap
variable {α β E : Type*} [TopologicalSpace α] [CompactSpace α] [MetricSpace β]
[NormedAddCommGroup E]
section
variable (α β)
/-- When `α` is compact, the bounded continuous maps `α →ᵇ β` are
equivalent to `C(α, β)`.
-/
@[simps (config := .asFn)]
def equivBoundedOfCompact : C(α, β) ≃ (α →ᵇ β) :=
⟨mkOfCompact, BoundedContinuousFunction.toContinuousMap, fun f => by
ext
rfl, fun f => by
ext
rfl⟩
#align continuous_map.equiv_bounded_of_compact ContinuousMap.equivBoundedOfCompact
theorem uniformInducing_equivBoundedOfCompact : UniformInducing (equivBoundedOfCompact α β) :=
UniformInducing.mk'
(by
simp only [hasBasis_compactConvergenceUniformity.mem_iff, uniformity_basis_dist_le.mem_iff]
exact fun s =>
⟨fun ⟨⟨a, b⟩, ⟨_, ⟨ε, hε, hb⟩⟩, hs⟩ =>
⟨{ p | ∀ x, (p.1 x, p.2 x) ∈ b }, ⟨ε, hε, fun _ h x => hb ((dist_le hε.le).mp h x)⟩,
fun f g h => hs fun x _ => h x⟩,
fun ⟨_, ⟨ε, hε, ht⟩, hs⟩ =>
⟨⟨Set.univ, { p | dist p.1 p.2 ≤ ε }⟩, ⟨isCompact_univ, ⟨ε, hε, fun _ h => h⟩⟩,
fun ⟨f, g⟩ h => hs _ _ (ht ((dist_le hε.le).mpr fun x => h x (mem_univ x)))⟩⟩)
#align continuous_map.uniform_inducing_equiv_bounded_of_compact ContinuousMap.uniformInducing_equivBoundedOfCompact
theorem uniformEmbedding_equivBoundedOfCompact : UniformEmbedding (equivBoundedOfCompact α β) :=
{ uniformInducing_equivBoundedOfCompact α β with inj := (equivBoundedOfCompact α β).injective }
#align continuous_map.uniform_embedding_equiv_bounded_of_compact ContinuousMap.uniformEmbedding_equivBoundedOfCompact
/-- When `α` is compact, the bounded continuous maps `α →ᵇ 𝕜` are
additively equivalent to `C(α, 𝕜)`.
-/
-- porting note: the following `simps` received a "maximum recursion depth" error
-- @[simps! (config := .asFn) apply symm_apply]
def addEquivBoundedOfCompact [AddMonoid β] [LipschitzAdd β] : C(α, β) ≃+ (α →ᵇ β) :=
({ toContinuousMapAddHom α β, (equivBoundedOfCompact α β).symm with } : (α →ᵇ β) ≃+ C(α, β)).symm
#align continuous_map.add_equiv_bounded_of_compact ContinuousMap.addEquivBoundedOfCompact
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_symm_apply [AddMonoid β] [LipschitzAdd β] :
⇑((addEquivBoundedOfCompact α β).symm) = toContinuousMapAddHom α β :=
rfl
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_apply [AddMonoid β] [LipschitzAdd β] :
⇑(addEquivBoundedOfCompact α β) = mkOfCompact :=
rfl
instance metricSpace : MetricSpace C(α, β) :=
(uniformEmbedding_equivBoundedOfCompact α β).comapMetricSpace _
#align continuous_map.metric_space ContinuousMap.metricSpace
/-- When `α` is compact, and `β` is a metric space, the bounded continuous maps `α →ᵇ β` are
isometric to `C(α, β)`.
-/
@[simps! (config := .asFn) toEquiv apply symm_apply]
def isometryEquivBoundedOfCompact : C(α, β) ≃ᵢ (α →ᵇ β) where
isometry_toFun _ _ := rfl
toEquiv := equivBoundedOfCompact α β
#align continuous_map.isometry_equiv_bounded_of_compact ContinuousMap.isometryEquivBoundedOfCompact
end
@[simp]
theorem _root_.BoundedContinuousFunction.dist_mkOfCompact (f g : C(α, β)) :
dist (mkOfCompact f) (mkOfCompact g) = dist f g :=
rfl
#align bounded_continuous_function.dist_mk_of_compact BoundedContinuousFunction.dist_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.dist_toContinuousMap (f g : α →ᵇ β) :
dist f.toContinuousMap g.toContinuousMap = dist f g :=
rfl
#align bounded_continuous_function.dist_to_continuous_map BoundedContinuousFunction.dist_toContinuousMap
open BoundedContinuousFunction
section
variable {f g : C(α, β)} {C : ℝ}
/-- The pointwise distance is controlled by the distance between functions, by definition. -/
theorem dist_apply_le_dist (x : α) : dist (f x) (g x) ≤ dist f g := by
simp only [← dist_mkOfCompact, dist_coe_le_dist, ← mkOfCompact_apply]
#align continuous_map.dist_apply_le_dist ContinuousMap.dist_apply_le_dist
/-- The distance between two functions is controlled by the supremum of the pointwise distances. -/
theorem dist_le (C0 : (0 : ℝ) ≤ C) : dist f g ≤ C ↔ ∀ x : α, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le C0, mkOfCompact_apply]
#align continuous_map.dist_le ContinuousMap.dist_le
theorem dist_le_iff_of_nonempty [Nonempty α] : dist f g ≤ C ↔ ∀ x, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le_iff_of_nonempty,
mkOfCompact_apply]
#align continuous_map.dist_le_iff_of_nonempty ContinuousMap.dist_le_iff_of_nonempty
theorem dist_lt_iff_of_nonempty [Nonempty α] : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
simp only [← dist_mkOfCompact, dist_lt_iff_of_nonempty_compact, mkOfCompact_apply]
#align continuous_map.dist_lt_iff_of_nonempty ContinuousMap.dist_lt_iff_of_nonempty
theorem dist_lt_of_nonempty [Nonempty α] (w : ∀ x : α, dist (f x) (g x) < C) : dist f g < C :=
dist_lt_iff_of_nonempty.2 w
#align continuous_map.dist_lt_of_nonempty ContinuousMap.dist_lt_of_nonempty
theorem dist_lt_iff (C0 : (0 : ℝ) < C) : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
rw [← dist_mkOfCompact, dist_lt_iff_of_compact C0]
simp only [mkOfCompact_apply]
#align continuous_map.dist_lt_iff ContinuousMap.dist_lt_iff
end
instance [CompleteSpace β] : CompleteSpace C(α, β) :=
(isometryEquivBoundedOfCompact α β).completeSpace
/-- See also `ContinuousMap.continuous_eval'`. -/
@[continuity]
theorem continuous_eval : Continuous fun p : C(α, β) × α => p.1 p.2 :=
continuous_eval.comp ((isometryEquivBoundedOfCompact α β).continuous.prod_map continuous_id)
#align continuous_map.continuous_eval ContinuousMap.continuous_eval
-- TODO at some point we will need lemmas characterising this norm!
-- At the moment the only way to reason about it is to transfer `f : C(α,E)` back to `α →ᵇ E`.
instance : Norm C(α, E) where norm x := dist x 0
@[simp]
theorem _root_.BoundedContinuousFunction.norm_mkOfCompact (f : C(α, E)) : ‖mkOfCompact f‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_mk_of_compact BoundedContinuousFunction.norm_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.norm_toContinuousMap_eq (f : α →ᵇ E) :
‖f.toContinuousMap‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_to_continuous_map_eq BoundedContinuousFunction.norm_toContinuousMap_eq
open BoundedContinuousFunction
instance : NormedAddCommGroup C(α, E) :=
{ ContinuousMap.metricSpace _ _,
ContinuousMap.instAddCommGroupContinuousMap with
dist_eq := fun x y => by
rw [← norm_mkOfCompact, ← dist_mkOfCompact, dist_eq_norm, mkOfCompact_sub]
dist := dist
norm := norm }
instance [Nonempty α] [One E] [NormOneClass E] : NormOneClass C(α, E) where
norm_one := by simp only [← norm_mkOfCompact, mkOfCompact_one, norm_one]
section
variable (f : C(α, E))
-- The corresponding lemmas for `BoundedContinuousFunction` are stated with `{f}`,
-- and so can not be used in dot notation.
theorem norm_coe_le_norm (x : α) : ‖f x‖ ≤ ‖f‖ :=
(mkOfCompact f).norm_coe_le_norm x
#align continuous_map.norm_coe_le_norm ContinuousMap.norm_coe_le_norm
/-- Distance between the images of any two points is at most twice the norm of the function. -/
theorem dist_le_two_norm (x y : α) : dist (f x) (f y) ≤ 2 * ‖f‖ :=
(mkOfCompact f).dist_le_two_norm x y
#align continuous_map.dist_le_two_norm ContinuousMap.dist_le_two_norm
/-- The norm of a function is controlled by the supremum of the pointwise norms. -/
theorem norm_le {C : ℝ} (C0 : (0 : ℝ) ≤ C) : ‖f‖ ≤ C ↔ ∀ x : α, ‖f x‖ ≤ C :=
@BoundedContinuousFunction.norm_le _ _ _ _ (mkOfCompact f) _ C0
#align continuous_map.norm_le ContinuousMap.norm_le
theorem norm_le_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ ≤ M ↔ ∀ x, ‖f x‖ ≤ M :=
@BoundedContinuousFunction.norm_le_of_nonempty _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_le_of_nonempty ContinuousMap.norm_le_of_nonempty
theorem norm_lt_iff {M : ℝ} (M0 : 0 < M) : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_compact _ _ _ _ _ (mkOfCompact f) _ M0
#align continuous_map.norm_lt_iff ContinuousMap.norm_lt_iff
theorem nnnorm_lt_iff {M : ℝ≥0} (M0 : 0 < M) : ‖f‖₊ < M ↔ ∀ x : α, ‖f x‖₊ < M :=
f.norm_lt_iff M0
#align continuous_map.nnnorm_lt_iff ContinuousMap.nnnorm_lt_iff
theorem norm_lt_iff_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_nonempty_compact _ _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_lt_iff_of_nonempty ContinuousMap.norm_lt_iff_of_nonempty
theorem nnnorm_lt_iff_of_nonempty [Nonempty α] {M : ℝ≥0} : ‖f‖₊ < M ↔ ∀ x, ‖f x‖₊ < M :=
f.norm_lt_iff_of_nonempty
#align continuous_map.nnnorm_lt_iff_of_nonempty ContinuousMap.nnnorm_lt_iff_of_nonempty
theorem apply_le_norm (f : C(α, ℝ)) (x : α) : f x ≤ ‖f‖ :=
le_trans (le_abs.mpr (Or.inl (le_refl (f x)))) (f.norm_coe_le_norm x)
#align continuous_map.apply_le_norm ContinuousMap.apply_le_norm
theorem neg_norm_le_apply (f : C(α, ℝ)) (x : α) : -‖f‖ ≤ f x :=
le_trans (neg_le_neg (f.norm_coe_le_norm x)) (neg_le.mp (neg_le_abs_self (f x)))
#align continuous_map.neg_norm_le_apply ContinuousMap.neg_norm_le_apply
theorem norm_eq_iSup_norm : ‖f‖ = ⨆ x : α, ‖f x‖ :=
(mkOfCompact f).norm_eq_iSup_norm
#align continuous_map.norm_eq_supr_norm ContinuousMap.norm_eq_iSup_norm
theorem norm_restrict_mono_set {X : Type*} [TopologicalSpace X] (f : C(X, E))
{K L : TopologicalSpace.Compacts X} (hKL : K ≤ L) : ‖f.restrict K‖ ≤ ‖f.restrict L‖ :=
(norm_le _ (norm_nonneg _)).mpr fun x => norm_coe_le_norm (f.restrict L) <| Set.inclusion hKL x
#align continuous_map.norm_restrict_mono_set ContinuousMap.norm_restrict_mono_set
end
section
variable {R : Type*} [NormedRing R]
instance : NormedRing C(α, R) :=
{ (inferInstance : NormedAddCommGroup C(α, R)), ContinuousMap.instRingContinuousMap with
norm_mul := fun f g => norm_mul_le (mkOfCompact f) (mkOfCompact g) }
end
section
variable {𝕜 : Type*} [NormedField 𝕜] [NormedSpace 𝕜 E]
instance normedSpace : NormedSpace 𝕜 C(α, E) where
norm_smul_le c f := (norm_smul_le c (mkOfCompact f) : _)
#align continuous_map.normed_space ContinuousMap.normedSpace
section
variable (α 𝕜 E)
/-- When `α` is compact and `𝕜` is a normed field,
the `𝕜`-algebra of bounded continuous maps `α →ᵇ β` is
`𝕜`-linearly isometric to `C(α, β)`.
-/
def linearIsometryBoundedOfCompact : C(α, E) ≃ₗᵢ[𝕜] α →ᵇ E :=
{ addEquivBoundedOfCompact α E with
map_smul' := fun c f => by
ext
norm_cast
norm_map' := fun f => rfl }
#align continuous_map.linear_isometry_bounded_of_compact ContinuousMap.linearIsometryBoundedOfCompact
variable {α E}
-- to match `BoundedContinuousFunction.evalClm`
/-- The evaluation at a point, as a continuous linear map from `C(α, 𝕜)` to `𝕜`. -/
def evalClm (x : α) : C(α, E) →L[𝕜] E :=
(BoundedContinuousFunction.evalClm 𝕜 x).comp
(linearIsometryBoundedOfCompact α E 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_map.eval_clm ContinuousMap.evalClm
end
-- this lemma and the next are the analogues of those autogenerated by `@[simps]` for
-- `equivBoundedOfCompact`, `addEquivBoundedOfCompact`
@[simp]
theorem linearIsometryBoundedOfCompact_symm_apply (f : α →ᵇ E) :
(linearIsometryBoundedOfCompact α E 𝕜).symm f = f.toContinuousMap :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_symm_apply ContinuousMap.linearIsometryBoundedOfCompact_symm_apply
@[simp]
theorem linearIsometryBoundedOfCompact_apply_apply (f : C(α, E)) (a : α) :
(linearIsometryBoundedOfCompact α E 𝕜 f) a = f a :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_apply_apply ContinuousMap.linearIsometryBoundedOfCompact_apply_apply
@[simp]
theorem linearIsometryBoundedOfCompact_toIsometryEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toIsometryEquiv = isometryEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_isometry_equiv ContinuousMap.linearIsometryBoundedOfCompact_toIsometryEquiv
@[simp] -- porting note: adjusted LHS because `simpNF` complained it simplified.
theorem linearIsometryBoundedOfCompact_toAddEquiv :
((linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv : C(α, E) ≃+ (α →ᵇ E)) =
addEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_add_equiv ContinuousMap.linearIsometryBoundedOfCompact_toAddEquiv
@[simp]
theorem linearIsometryBoundedOfCompact_of_compact_toEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv.toEquiv = equivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_of_compact_to_equiv ContinuousMap.linearIsometryBoundedOfCompact_of_compact_toEquiv
end
section
variable {𝕜 : Type*} {γ : Type*} [NormedField 𝕜] [NormedRing γ] [NormedAlgebra 𝕜 γ]
instance : NormedAlgebra 𝕜 C(α, γ) :=
{ ContinuousMap.normedSpace, ContinuousMap.algebra with }
end
end ContinuousMap
namespace ContinuousMap
section UniformContinuity
variable {α β : Type*}
variable [MetricSpace α] [CompactSpace α] [MetricSpace β]
/-!
We now set up some declarations making it convenient to use uniform continuity.
-/
theorem uniform_continuity (f : C(α, β)) (ε : ℝ) (h : 0 < ε) :
∃ δ > 0, ∀ {x y}, dist x y < δ → dist (f x) (f y) < ε :=
Metric.uniformContinuous_iff.mp (CompactSpace.uniformContinuous_of_continuous f.continuous) ε h
#align continuous_map.uniform_continuity ContinuousMap.uniform_continuity
-- This definition allows us to separate the choice of some `δ`,
-- and the corresponding use of `dist a b < δ → dist (f a) (f b) < ε`,
-- even across different declarations.
/-- An arbitrarily chosen modulus of uniform continuity for a given function `f` and `ε > 0`. -/
def modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) : ℝ :=
Classical.choose (uniform_continuity f ε h)
#align continuous_map.modulus ContinuousMap.modulus
theorem modulus_pos (f : C(α, β)) {ε : ℝ} {h : 0 < ε} : 0 < f.modulus ε h :=
(Classical.choose_spec (uniform_continuity f ε h)).1
#align continuous_map.modulus_pos ContinuousMap.modulus_pos
theorem dist_lt_of_dist_lt_modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) {a b : α}
(w : dist a b < f.modulus ε h) : dist (f a) (f b) < ε :=
(Classical.choose_spec (uniform_continuity f ε h)).2 w
#align continuous_map.dist_lt_of_dist_lt_modulus ContinuousMap.dist_lt_of_dist_lt_modulus
end UniformContinuity
end ContinuousMap
section CompLeft
variable (X : Type*) {𝕜 β γ : Type*} [TopologicalSpace X] [CompactSpace X]
[NontriviallyNormedField 𝕜]
variable [NormedAddCommGroup β] [NormedSpace 𝕜 β] [NormedAddCommGroup γ] [NormedSpace 𝕜 γ]
open ContinuousMap
/-- Postcomposition of continuous functions into a normed module by a continuous linear map is a
continuous linear map.
Transferred version of `ContinuousLinearMap.compLeftContinuousBounded`,
upgraded version of `ContinuousLinearMap.compLeftContinuous`,
similar to `LinearMap.compLeft`. -/
protected def ContinuousLinearMap.compLeftContinuousCompact (g : β →L[𝕜] γ) :
C(X, β) →L[𝕜] C(X, γ) :=
(linearIsometryBoundedOfCompact X γ 𝕜).symm.toLinearIsometry.toContinuousLinearMap.comp <|
(g.compLeftContinuousBounded X).comp <|
(linearIsometryBoundedOfCompact X β 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_linear_map.comp_left_continuous_compact ContinuousLinearMap.compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.toLinear_compLeftContinuousCompact (g : β →L[𝕜] γ) :
(g.compLeftContinuousCompact X : C(X, β) →ₗ[𝕜] C(X, γ)) = g.compLeftContinuous 𝕜 X := by
|
ext f
|
@[simp]
theorem ContinuousLinearMap.toLinear_compLeftContinuousCompact (g : β →L[𝕜] γ) :
(g.compLeftContinuousCompact X : C(X, β) →ₗ[𝕜] C(X, γ)) = g.compLeftContinuous 𝕜 X := by
|
Mathlib.Topology.ContinuousFunction.Compact.406_0.Mig2jTVnn2FLKEB
|
@[simp]
theorem ContinuousLinearMap.toLinear_compLeftContinuousCompact (g : β →L[𝕜] γ) :
(g.compLeftContinuousCompact X : C(X, β) →ₗ[𝕜] C(X, γ)) = g.compLeftContinuous 𝕜 X
|
Mathlib_Topology_ContinuousFunction_Compact
|
case h.h
X : Type u_1
𝕜 : Type u_2
β : Type u_3
γ : Type u_4
inst✝⁶ : TopologicalSpace X
inst✝⁵ : CompactSpace X
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup β
inst✝² : NormedSpace 𝕜 β
inst✝¹ : NormedAddCommGroup γ
inst✝ : NormedSpace 𝕜 γ
g : β →L[𝕜] γ
f : C(X, β)
a✝ : X
⊢ (↑(ContinuousLinearMap.compLeftContinuousCompact X g) f) a✝ = ((ContinuousLinearMap.compLeftContinuous 𝕜 X g) f) a✝
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compacts
#align_import topology.continuous_function.compact from "leanprover-community/mathlib"@"d3af0609f6db8691dffdc3e1fb7feb7da72698f2"
/-!
# Continuous functions on a compact space
Continuous functions `C(α, β)` from a compact space `α` to a metric space `β`
are automatically bounded, and so acquire various structures inherited from `α →ᵇ β`.
This file transfers these structures, and restates some lemmas
characterising these structures.
If you need a lemma which is proved about `α →ᵇ β` but not for `C(α, β)` when `α` is compact,
you should restate it here. You can also use
`ContinuousMap.equivBoundedOfCompact` to move functions back and forth.
-/
noncomputable section
open Topology Classical NNReal BoundedContinuousFunction BigOperators
open Set Filter Metric
open BoundedContinuousFunction
namespace ContinuousMap
variable {α β E : Type*} [TopologicalSpace α] [CompactSpace α] [MetricSpace β]
[NormedAddCommGroup E]
section
variable (α β)
/-- When `α` is compact, the bounded continuous maps `α →ᵇ β` are
equivalent to `C(α, β)`.
-/
@[simps (config := .asFn)]
def equivBoundedOfCompact : C(α, β) ≃ (α →ᵇ β) :=
⟨mkOfCompact, BoundedContinuousFunction.toContinuousMap, fun f => by
ext
rfl, fun f => by
ext
rfl⟩
#align continuous_map.equiv_bounded_of_compact ContinuousMap.equivBoundedOfCompact
theorem uniformInducing_equivBoundedOfCompact : UniformInducing (equivBoundedOfCompact α β) :=
UniformInducing.mk'
(by
simp only [hasBasis_compactConvergenceUniformity.mem_iff, uniformity_basis_dist_le.mem_iff]
exact fun s =>
⟨fun ⟨⟨a, b⟩, ⟨_, ⟨ε, hε, hb⟩⟩, hs⟩ =>
⟨{ p | ∀ x, (p.1 x, p.2 x) ∈ b }, ⟨ε, hε, fun _ h x => hb ((dist_le hε.le).mp h x)⟩,
fun f g h => hs fun x _ => h x⟩,
fun ⟨_, ⟨ε, hε, ht⟩, hs⟩ =>
⟨⟨Set.univ, { p | dist p.1 p.2 ≤ ε }⟩, ⟨isCompact_univ, ⟨ε, hε, fun _ h => h⟩⟩,
fun ⟨f, g⟩ h => hs _ _ (ht ((dist_le hε.le).mpr fun x => h x (mem_univ x)))⟩⟩)
#align continuous_map.uniform_inducing_equiv_bounded_of_compact ContinuousMap.uniformInducing_equivBoundedOfCompact
theorem uniformEmbedding_equivBoundedOfCompact : UniformEmbedding (equivBoundedOfCompact α β) :=
{ uniformInducing_equivBoundedOfCompact α β with inj := (equivBoundedOfCompact α β).injective }
#align continuous_map.uniform_embedding_equiv_bounded_of_compact ContinuousMap.uniformEmbedding_equivBoundedOfCompact
/-- When `α` is compact, the bounded continuous maps `α →ᵇ 𝕜` are
additively equivalent to `C(α, 𝕜)`.
-/
-- porting note: the following `simps` received a "maximum recursion depth" error
-- @[simps! (config := .asFn) apply symm_apply]
def addEquivBoundedOfCompact [AddMonoid β] [LipschitzAdd β] : C(α, β) ≃+ (α →ᵇ β) :=
({ toContinuousMapAddHom α β, (equivBoundedOfCompact α β).symm with } : (α →ᵇ β) ≃+ C(α, β)).symm
#align continuous_map.add_equiv_bounded_of_compact ContinuousMap.addEquivBoundedOfCompact
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_symm_apply [AddMonoid β] [LipschitzAdd β] :
⇑((addEquivBoundedOfCompact α β).symm) = toContinuousMapAddHom α β :=
rfl
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_apply [AddMonoid β] [LipschitzAdd β] :
⇑(addEquivBoundedOfCompact α β) = mkOfCompact :=
rfl
instance metricSpace : MetricSpace C(α, β) :=
(uniformEmbedding_equivBoundedOfCompact α β).comapMetricSpace _
#align continuous_map.metric_space ContinuousMap.metricSpace
/-- When `α` is compact, and `β` is a metric space, the bounded continuous maps `α →ᵇ β` are
isometric to `C(α, β)`.
-/
@[simps! (config := .asFn) toEquiv apply symm_apply]
def isometryEquivBoundedOfCompact : C(α, β) ≃ᵢ (α →ᵇ β) where
isometry_toFun _ _ := rfl
toEquiv := equivBoundedOfCompact α β
#align continuous_map.isometry_equiv_bounded_of_compact ContinuousMap.isometryEquivBoundedOfCompact
end
@[simp]
theorem _root_.BoundedContinuousFunction.dist_mkOfCompact (f g : C(α, β)) :
dist (mkOfCompact f) (mkOfCompact g) = dist f g :=
rfl
#align bounded_continuous_function.dist_mk_of_compact BoundedContinuousFunction.dist_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.dist_toContinuousMap (f g : α →ᵇ β) :
dist f.toContinuousMap g.toContinuousMap = dist f g :=
rfl
#align bounded_continuous_function.dist_to_continuous_map BoundedContinuousFunction.dist_toContinuousMap
open BoundedContinuousFunction
section
variable {f g : C(α, β)} {C : ℝ}
/-- The pointwise distance is controlled by the distance between functions, by definition. -/
theorem dist_apply_le_dist (x : α) : dist (f x) (g x) ≤ dist f g := by
simp only [← dist_mkOfCompact, dist_coe_le_dist, ← mkOfCompact_apply]
#align continuous_map.dist_apply_le_dist ContinuousMap.dist_apply_le_dist
/-- The distance between two functions is controlled by the supremum of the pointwise distances. -/
theorem dist_le (C0 : (0 : ℝ) ≤ C) : dist f g ≤ C ↔ ∀ x : α, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le C0, mkOfCompact_apply]
#align continuous_map.dist_le ContinuousMap.dist_le
theorem dist_le_iff_of_nonempty [Nonempty α] : dist f g ≤ C ↔ ∀ x, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le_iff_of_nonempty,
mkOfCompact_apply]
#align continuous_map.dist_le_iff_of_nonempty ContinuousMap.dist_le_iff_of_nonempty
theorem dist_lt_iff_of_nonempty [Nonempty α] : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
simp only [← dist_mkOfCompact, dist_lt_iff_of_nonempty_compact, mkOfCompact_apply]
#align continuous_map.dist_lt_iff_of_nonempty ContinuousMap.dist_lt_iff_of_nonempty
theorem dist_lt_of_nonempty [Nonempty α] (w : ∀ x : α, dist (f x) (g x) < C) : dist f g < C :=
dist_lt_iff_of_nonempty.2 w
#align continuous_map.dist_lt_of_nonempty ContinuousMap.dist_lt_of_nonempty
theorem dist_lt_iff (C0 : (0 : ℝ) < C) : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
rw [← dist_mkOfCompact, dist_lt_iff_of_compact C0]
simp only [mkOfCompact_apply]
#align continuous_map.dist_lt_iff ContinuousMap.dist_lt_iff
end
instance [CompleteSpace β] : CompleteSpace C(α, β) :=
(isometryEquivBoundedOfCompact α β).completeSpace
/-- See also `ContinuousMap.continuous_eval'`. -/
@[continuity]
theorem continuous_eval : Continuous fun p : C(α, β) × α => p.1 p.2 :=
continuous_eval.comp ((isometryEquivBoundedOfCompact α β).continuous.prod_map continuous_id)
#align continuous_map.continuous_eval ContinuousMap.continuous_eval
-- TODO at some point we will need lemmas characterising this norm!
-- At the moment the only way to reason about it is to transfer `f : C(α,E)` back to `α →ᵇ E`.
instance : Norm C(α, E) where norm x := dist x 0
@[simp]
theorem _root_.BoundedContinuousFunction.norm_mkOfCompact (f : C(α, E)) : ‖mkOfCompact f‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_mk_of_compact BoundedContinuousFunction.norm_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.norm_toContinuousMap_eq (f : α →ᵇ E) :
‖f.toContinuousMap‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_to_continuous_map_eq BoundedContinuousFunction.norm_toContinuousMap_eq
open BoundedContinuousFunction
instance : NormedAddCommGroup C(α, E) :=
{ ContinuousMap.metricSpace _ _,
ContinuousMap.instAddCommGroupContinuousMap with
dist_eq := fun x y => by
rw [← norm_mkOfCompact, ← dist_mkOfCompact, dist_eq_norm, mkOfCompact_sub]
dist := dist
norm := norm }
instance [Nonempty α] [One E] [NormOneClass E] : NormOneClass C(α, E) where
norm_one := by simp only [← norm_mkOfCompact, mkOfCompact_one, norm_one]
section
variable (f : C(α, E))
-- The corresponding lemmas for `BoundedContinuousFunction` are stated with `{f}`,
-- and so can not be used in dot notation.
theorem norm_coe_le_norm (x : α) : ‖f x‖ ≤ ‖f‖ :=
(mkOfCompact f).norm_coe_le_norm x
#align continuous_map.norm_coe_le_norm ContinuousMap.norm_coe_le_norm
/-- Distance between the images of any two points is at most twice the norm of the function. -/
theorem dist_le_two_norm (x y : α) : dist (f x) (f y) ≤ 2 * ‖f‖ :=
(mkOfCompact f).dist_le_two_norm x y
#align continuous_map.dist_le_two_norm ContinuousMap.dist_le_two_norm
/-- The norm of a function is controlled by the supremum of the pointwise norms. -/
theorem norm_le {C : ℝ} (C0 : (0 : ℝ) ≤ C) : ‖f‖ ≤ C ↔ ∀ x : α, ‖f x‖ ≤ C :=
@BoundedContinuousFunction.norm_le _ _ _ _ (mkOfCompact f) _ C0
#align continuous_map.norm_le ContinuousMap.norm_le
theorem norm_le_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ ≤ M ↔ ∀ x, ‖f x‖ ≤ M :=
@BoundedContinuousFunction.norm_le_of_nonempty _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_le_of_nonempty ContinuousMap.norm_le_of_nonempty
theorem norm_lt_iff {M : ℝ} (M0 : 0 < M) : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_compact _ _ _ _ _ (mkOfCompact f) _ M0
#align continuous_map.norm_lt_iff ContinuousMap.norm_lt_iff
theorem nnnorm_lt_iff {M : ℝ≥0} (M0 : 0 < M) : ‖f‖₊ < M ↔ ∀ x : α, ‖f x‖₊ < M :=
f.norm_lt_iff M0
#align continuous_map.nnnorm_lt_iff ContinuousMap.nnnorm_lt_iff
theorem norm_lt_iff_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_nonempty_compact _ _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_lt_iff_of_nonempty ContinuousMap.norm_lt_iff_of_nonempty
theorem nnnorm_lt_iff_of_nonempty [Nonempty α] {M : ℝ≥0} : ‖f‖₊ < M ↔ ∀ x, ‖f x‖₊ < M :=
f.norm_lt_iff_of_nonempty
#align continuous_map.nnnorm_lt_iff_of_nonempty ContinuousMap.nnnorm_lt_iff_of_nonempty
theorem apply_le_norm (f : C(α, ℝ)) (x : α) : f x ≤ ‖f‖ :=
le_trans (le_abs.mpr (Or.inl (le_refl (f x)))) (f.norm_coe_le_norm x)
#align continuous_map.apply_le_norm ContinuousMap.apply_le_norm
theorem neg_norm_le_apply (f : C(α, ℝ)) (x : α) : -‖f‖ ≤ f x :=
le_trans (neg_le_neg (f.norm_coe_le_norm x)) (neg_le.mp (neg_le_abs_self (f x)))
#align continuous_map.neg_norm_le_apply ContinuousMap.neg_norm_le_apply
theorem norm_eq_iSup_norm : ‖f‖ = ⨆ x : α, ‖f x‖ :=
(mkOfCompact f).norm_eq_iSup_norm
#align continuous_map.norm_eq_supr_norm ContinuousMap.norm_eq_iSup_norm
theorem norm_restrict_mono_set {X : Type*} [TopologicalSpace X] (f : C(X, E))
{K L : TopologicalSpace.Compacts X} (hKL : K ≤ L) : ‖f.restrict K‖ ≤ ‖f.restrict L‖ :=
(norm_le _ (norm_nonneg _)).mpr fun x => norm_coe_le_norm (f.restrict L) <| Set.inclusion hKL x
#align continuous_map.norm_restrict_mono_set ContinuousMap.norm_restrict_mono_set
end
section
variable {R : Type*} [NormedRing R]
instance : NormedRing C(α, R) :=
{ (inferInstance : NormedAddCommGroup C(α, R)), ContinuousMap.instRingContinuousMap with
norm_mul := fun f g => norm_mul_le (mkOfCompact f) (mkOfCompact g) }
end
section
variable {𝕜 : Type*} [NormedField 𝕜] [NormedSpace 𝕜 E]
instance normedSpace : NormedSpace 𝕜 C(α, E) where
norm_smul_le c f := (norm_smul_le c (mkOfCompact f) : _)
#align continuous_map.normed_space ContinuousMap.normedSpace
section
variable (α 𝕜 E)
/-- When `α` is compact and `𝕜` is a normed field,
the `𝕜`-algebra of bounded continuous maps `α →ᵇ β` is
`𝕜`-linearly isometric to `C(α, β)`.
-/
def linearIsometryBoundedOfCompact : C(α, E) ≃ₗᵢ[𝕜] α →ᵇ E :=
{ addEquivBoundedOfCompact α E with
map_smul' := fun c f => by
ext
norm_cast
norm_map' := fun f => rfl }
#align continuous_map.linear_isometry_bounded_of_compact ContinuousMap.linearIsometryBoundedOfCompact
variable {α E}
-- to match `BoundedContinuousFunction.evalClm`
/-- The evaluation at a point, as a continuous linear map from `C(α, 𝕜)` to `𝕜`. -/
def evalClm (x : α) : C(α, E) →L[𝕜] E :=
(BoundedContinuousFunction.evalClm 𝕜 x).comp
(linearIsometryBoundedOfCompact α E 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_map.eval_clm ContinuousMap.evalClm
end
-- this lemma and the next are the analogues of those autogenerated by `@[simps]` for
-- `equivBoundedOfCompact`, `addEquivBoundedOfCompact`
@[simp]
theorem linearIsometryBoundedOfCompact_symm_apply (f : α →ᵇ E) :
(linearIsometryBoundedOfCompact α E 𝕜).symm f = f.toContinuousMap :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_symm_apply ContinuousMap.linearIsometryBoundedOfCompact_symm_apply
@[simp]
theorem linearIsometryBoundedOfCompact_apply_apply (f : C(α, E)) (a : α) :
(linearIsometryBoundedOfCompact α E 𝕜 f) a = f a :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_apply_apply ContinuousMap.linearIsometryBoundedOfCompact_apply_apply
@[simp]
theorem linearIsometryBoundedOfCompact_toIsometryEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toIsometryEquiv = isometryEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_isometry_equiv ContinuousMap.linearIsometryBoundedOfCompact_toIsometryEquiv
@[simp] -- porting note: adjusted LHS because `simpNF` complained it simplified.
theorem linearIsometryBoundedOfCompact_toAddEquiv :
((linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv : C(α, E) ≃+ (α →ᵇ E)) =
addEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_add_equiv ContinuousMap.linearIsometryBoundedOfCompact_toAddEquiv
@[simp]
theorem linearIsometryBoundedOfCompact_of_compact_toEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv.toEquiv = equivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_of_compact_to_equiv ContinuousMap.linearIsometryBoundedOfCompact_of_compact_toEquiv
end
section
variable {𝕜 : Type*} {γ : Type*} [NormedField 𝕜] [NormedRing γ] [NormedAlgebra 𝕜 γ]
instance : NormedAlgebra 𝕜 C(α, γ) :=
{ ContinuousMap.normedSpace, ContinuousMap.algebra with }
end
end ContinuousMap
namespace ContinuousMap
section UniformContinuity
variable {α β : Type*}
variable [MetricSpace α] [CompactSpace α] [MetricSpace β]
/-!
We now set up some declarations making it convenient to use uniform continuity.
-/
theorem uniform_continuity (f : C(α, β)) (ε : ℝ) (h : 0 < ε) :
∃ δ > 0, ∀ {x y}, dist x y < δ → dist (f x) (f y) < ε :=
Metric.uniformContinuous_iff.mp (CompactSpace.uniformContinuous_of_continuous f.continuous) ε h
#align continuous_map.uniform_continuity ContinuousMap.uniform_continuity
-- This definition allows us to separate the choice of some `δ`,
-- and the corresponding use of `dist a b < δ → dist (f a) (f b) < ε`,
-- even across different declarations.
/-- An arbitrarily chosen modulus of uniform continuity for a given function `f` and `ε > 0`. -/
def modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) : ℝ :=
Classical.choose (uniform_continuity f ε h)
#align continuous_map.modulus ContinuousMap.modulus
theorem modulus_pos (f : C(α, β)) {ε : ℝ} {h : 0 < ε} : 0 < f.modulus ε h :=
(Classical.choose_spec (uniform_continuity f ε h)).1
#align continuous_map.modulus_pos ContinuousMap.modulus_pos
theorem dist_lt_of_dist_lt_modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) {a b : α}
(w : dist a b < f.modulus ε h) : dist (f a) (f b) < ε :=
(Classical.choose_spec (uniform_continuity f ε h)).2 w
#align continuous_map.dist_lt_of_dist_lt_modulus ContinuousMap.dist_lt_of_dist_lt_modulus
end UniformContinuity
end ContinuousMap
section CompLeft
variable (X : Type*) {𝕜 β γ : Type*} [TopologicalSpace X] [CompactSpace X]
[NontriviallyNormedField 𝕜]
variable [NormedAddCommGroup β] [NormedSpace 𝕜 β] [NormedAddCommGroup γ] [NormedSpace 𝕜 γ]
open ContinuousMap
/-- Postcomposition of continuous functions into a normed module by a continuous linear map is a
continuous linear map.
Transferred version of `ContinuousLinearMap.compLeftContinuousBounded`,
upgraded version of `ContinuousLinearMap.compLeftContinuous`,
similar to `LinearMap.compLeft`. -/
protected def ContinuousLinearMap.compLeftContinuousCompact (g : β →L[𝕜] γ) :
C(X, β) →L[𝕜] C(X, γ) :=
(linearIsometryBoundedOfCompact X γ 𝕜).symm.toLinearIsometry.toContinuousLinearMap.comp <|
(g.compLeftContinuousBounded X).comp <|
(linearIsometryBoundedOfCompact X β 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_linear_map.comp_left_continuous_compact ContinuousLinearMap.compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.toLinear_compLeftContinuousCompact (g : β →L[𝕜] γ) :
(g.compLeftContinuousCompact X : C(X, β) →ₗ[𝕜] C(X, γ)) = g.compLeftContinuous 𝕜 X := by
ext f
|
rfl
|
@[simp]
theorem ContinuousLinearMap.toLinear_compLeftContinuousCompact (g : β →L[𝕜] γ) :
(g.compLeftContinuousCompact X : C(X, β) →ₗ[𝕜] C(X, γ)) = g.compLeftContinuous 𝕜 X := by
ext f
|
Mathlib.Topology.ContinuousFunction.Compact.406_0.Mig2jTVnn2FLKEB
|
@[simp]
theorem ContinuousLinearMap.toLinear_compLeftContinuousCompact (g : β →L[𝕜] γ) :
(g.compLeftContinuousCompact X : C(X, β) →ₗ[𝕜] C(X, γ)) = g.compLeftContinuous 𝕜 X
|
Mathlib_Topology_ContinuousFunction_Compact
|
X : Type u_1
Y : Type u_2
T : Type u_3
inst✝⁴ : TopologicalSpace X
inst✝³ : CompactSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : CompactSpace Y
inst✝ : MetricSpace T
f : C(X, Y)
⊢ Continuous fun g => comp g f
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compacts
#align_import topology.continuous_function.compact from "leanprover-community/mathlib"@"d3af0609f6db8691dffdc3e1fb7feb7da72698f2"
/-!
# Continuous functions on a compact space
Continuous functions `C(α, β)` from a compact space `α` to a metric space `β`
are automatically bounded, and so acquire various structures inherited from `α →ᵇ β`.
This file transfers these structures, and restates some lemmas
characterising these structures.
If you need a lemma which is proved about `α →ᵇ β` but not for `C(α, β)` when `α` is compact,
you should restate it here. You can also use
`ContinuousMap.equivBoundedOfCompact` to move functions back and forth.
-/
noncomputable section
open Topology Classical NNReal BoundedContinuousFunction BigOperators
open Set Filter Metric
open BoundedContinuousFunction
namespace ContinuousMap
variable {α β E : Type*} [TopologicalSpace α] [CompactSpace α] [MetricSpace β]
[NormedAddCommGroup E]
section
variable (α β)
/-- When `α` is compact, the bounded continuous maps `α →ᵇ β` are
equivalent to `C(α, β)`.
-/
@[simps (config := .asFn)]
def equivBoundedOfCompact : C(α, β) ≃ (α →ᵇ β) :=
⟨mkOfCompact, BoundedContinuousFunction.toContinuousMap, fun f => by
ext
rfl, fun f => by
ext
rfl⟩
#align continuous_map.equiv_bounded_of_compact ContinuousMap.equivBoundedOfCompact
theorem uniformInducing_equivBoundedOfCompact : UniformInducing (equivBoundedOfCompact α β) :=
UniformInducing.mk'
(by
simp only [hasBasis_compactConvergenceUniformity.mem_iff, uniformity_basis_dist_le.mem_iff]
exact fun s =>
⟨fun ⟨⟨a, b⟩, ⟨_, ⟨ε, hε, hb⟩⟩, hs⟩ =>
⟨{ p | ∀ x, (p.1 x, p.2 x) ∈ b }, ⟨ε, hε, fun _ h x => hb ((dist_le hε.le).mp h x)⟩,
fun f g h => hs fun x _ => h x⟩,
fun ⟨_, ⟨ε, hε, ht⟩, hs⟩ =>
⟨⟨Set.univ, { p | dist p.1 p.2 ≤ ε }⟩, ⟨isCompact_univ, ⟨ε, hε, fun _ h => h⟩⟩,
fun ⟨f, g⟩ h => hs _ _ (ht ((dist_le hε.le).mpr fun x => h x (mem_univ x)))⟩⟩)
#align continuous_map.uniform_inducing_equiv_bounded_of_compact ContinuousMap.uniformInducing_equivBoundedOfCompact
theorem uniformEmbedding_equivBoundedOfCompact : UniformEmbedding (equivBoundedOfCompact α β) :=
{ uniformInducing_equivBoundedOfCompact α β with inj := (equivBoundedOfCompact α β).injective }
#align continuous_map.uniform_embedding_equiv_bounded_of_compact ContinuousMap.uniformEmbedding_equivBoundedOfCompact
/-- When `α` is compact, the bounded continuous maps `α →ᵇ 𝕜` are
additively equivalent to `C(α, 𝕜)`.
-/
-- porting note: the following `simps` received a "maximum recursion depth" error
-- @[simps! (config := .asFn) apply symm_apply]
def addEquivBoundedOfCompact [AddMonoid β] [LipschitzAdd β] : C(α, β) ≃+ (α →ᵇ β) :=
({ toContinuousMapAddHom α β, (equivBoundedOfCompact α β).symm with } : (α →ᵇ β) ≃+ C(α, β)).symm
#align continuous_map.add_equiv_bounded_of_compact ContinuousMap.addEquivBoundedOfCompact
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_symm_apply [AddMonoid β] [LipschitzAdd β] :
⇑((addEquivBoundedOfCompact α β).symm) = toContinuousMapAddHom α β :=
rfl
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_apply [AddMonoid β] [LipschitzAdd β] :
⇑(addEquivBoundedOfCompact α β) = mkOfCompact :=
rfl
instance metricSpace : MetricSpace C(α, β) :=
(uniformEmbedding_equivBoundedOfCompact α β).comapMetricSpace _
#align continuous_map.metric_space ContinuousMap.metricSpace
/-- When `α` is compact, and `β` is a metric space, the bounded continuous maps `α →ᵇ β` are
isometric to `C(α, β)`.
-/
@[simps! (config := .asFn) toEquiv apply symm_apply]
def isometryEquivBoundedOfCompact : C(α, β) ≃ᵢ (α →ᵇ β) where
isometry_toFun _ _ := rfl
toEquiv := equivBoundedOfCompact α β
#align continuous_map.isometry_equiv_bounded_of_compact ContinuousMap.isometryEquivBoundedOfCompact
end
@[simp]
theorem _root_.BoundedContinuousFunction.dist_mkOfCompact (f g : C(α, β)) :
dist (mkOfCompact f) (mkOfCompact g) = dist f g :=
rfl
#align bounded_continuous_function.dist_mk_of_compact BoundedContinuousFunction.dist_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.dist_toContinuousMap (f g : α →ᵇ β) :
dist f.toContinuousMap g.toContinuousMap = dist f g :=
rfl
#align bounded_continuous_function.dist_to_continuous_map BoundedContinuousFunction.dist_toContinuousMap
open BoundedContinuousFunction
section
variable {f g : C(α, β)} {C : ℝ}
/-- The pointwise distance is controlled by the distance between functions, by definition. -/
theorem dist_apply_le_dist (x : α) : dist (f x) (g x) ≤ dist f g := by
simp only [← dist_mkOfCompact, dist_coe_le_dist, ← mkOfCompact_apply]
#align continuous_map.dist_apply_le_dist ContinuousMap.dist_apply_le_dist
/-- The distance between two functions is controlled by the supremum of the pointwise distances. -/
theorem dist_le (C0 : (0 : ℝ) ≤ C) : dist f g ≤ C ↔ ∀ x : α, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le C0, mkOfCompact_apply]
#align continuous_map.dist_le ContinuousMap.dist_le
theorem dist_le_iff_of_nonempty [Nonempty α] : dist f g ≤ C ↔ ∀ x, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le_iff_of_nonempty,
mkOfCompact_apply]
#align continuous_map.dist_le_iff_of_nonempty ContinuousMap.dist_le_iff_of_nonempty
theorem dist_lt_iff_of_nonempty [Nonempty α] : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
simp only [← dist_mkOfCompact, dist_lt_iff_of_nonempty_compact, mkOfCompact_apply]
#align continuous_map.dist_lt_iff_of_nonempty ContinuousMap.dist_lt_iff_of_nonempty
theorem dist_lt_of_nonempty [Nonempty α] (w : ∀ x : α, dist (f x) (g x) < C) : dist f g < C :=
dist_lt_iff_of_nonempty.2 w
#align continuous_map.dist_lt_of_nonempty ContinuousMap.dist_lt_of_nonempty
theorem dist_lt_iff (C0 : (0 : ℝ) < C) : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
rw [← dist_mkOfCompact, dist_lt_iff_of_compact C0]
simp only [mkOfCompact_apply]
#align continuous_map.dist_lt_iff ContinuousMap.dist_lt_iff
end
instance [CompleteSpace β] : CompleteSpace C(α, β) :=
(isometryEquivBoundedOfCompact α β).completeSpace
/-- See also `ContinuousMap.continuous_eval'`. -/
@[continuity]
theorem continuous_eval : Continuous fun p : C(α, β) × α => p.1 p.2 :=
continuous_eval.comp ((isometryEquivBoundedOfCompact α β).continuous.prod_map continuous_id)
#align continuous_map.continuous_eval ContinuousMap.continuous_eval
-- TODO at some point we will need lemmas characterising this norm!
-- At the moment the only way to reason about it is to transfer `f : C(α,E)` back to `α →ᵇ E`.
instance : Norm C(α, E) where norm x := dist x 0
@[simp]
theorem _root_.BoundedContinuousFunction.norm_mkOfCompact (f : C(α, E)) : ‖mkOfCompact f‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_mk_of_compact BoundedContinuousFunction.norm_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.norm_toContinuousMap_eq (f : α →ᵇ E) :
‖f.toContinuousMap‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_to_continuous_map_eq BoundedContinuousFunction.norm_toContinuousMap_eq
open BoundedContinuousFunction
instance : NormedAddCommGroup C(α, E) :=
{ ContinuousMap.metricSpace _ _,
ContinuousMap.instAddCommGroupContinuousMap with
dist_eq := fun x y => by
rw [← norm_mkOfCompact, ← dist_mkOfCompact, dist_eq_norm, mkOfCompact_sub]
dist := dist
norm := norm }
instance [Nonempty α] [One E] [NormOneClass E] : NormOneClass C(α, E) where
norm_one := by simp only [← norm_mkOfCompact, mkOfCompact_one, norm_one]
section
variable (f : C(α, E))
-- The corresponding lemmas for `BoundedContinuousFunction` are stated with `{f}`,
-- and so can not be used in dot notation.
theorem norm_coe_le_norm (x : α) : ‖f x‖ ≤ ‖f‖ :=
(mkOfCompact f).norm_coe_le_norm x
#align continuous_map.norm_coe_le_norm ContinuousMap.norm_coe_le_norm
/-- Distance between the images of any two points is at most twice the norm of the function. -/
theorem dist_le_two_norm (x y : α) : dist (f x) (f y) ≤ 2 * ‖f‖ :=
(mkOfCompact f).dist_le_two_norm x y
#align continuous_map.dist_le_two_norm ContinuousMap.dist_le_two_norm
/-- The norm of a function is controlled by the supremum of the pointwise norms. -/
theorem norm_le {C : ℝ} (C0 : (0 : ℝ) ≤ C) : ‖f‖ ≤ C ↔ ∀ x : α, ‖f x‖ ≤ C :=
@BoundedContinuousFunction.norm_le _ _ _ _ (mkOfCompact f) _ C0
#align continuous_map.norm_le ContinuousMap.norm_le
theorem norm_le_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ ≤ M ↔ ∀ x, ‖f x‖ ≤ M :=
@BoundedContinuousFunction.norm_le_of_nonempty _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_le_of_nonempty ContinuousMap.norm_le_of_nonempty
theorem norm_lt_iff {M : ℝ} (M0 : 0 < M) : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_compact _ _ _ _ _ (mkOfCompact f) _ M0
#align continuous_map.norm_lt_iff ContinuousMap.norm_lt_iff
theorem nnnorm_lt_iff {M : ℝ≥0} (M0 : 0 < M) : ‖f‖₊ < M ↔ ∀ x : α, ‖f x‖₊ < M :=
f.norm_lt_iff M0
#align continuous_map.nnnorm_lt_iff ContinuousMap.nnnorm_lt_iff
theorem norm_lt_iff_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_nonempty_compact _ _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_lt_iff_of_nonempty ContinuousMap.norm_lt_iff_of_nonempty
theorem nnnorm_lt_iff_of_nonempty [Nonempty α] {M : ℝ≥0} : ‖f‖₊ < M ↔ ∀ x, ‖f x‖₊ < M :=
f.norm_lt_iff_of_nonempty
#align continuous_map.nnnorm_lt_iff_of_nonempty ContinuousMap.nnnorm_lt_iff_of_nonempty
theorem apply_le_norm (f : C(α, ℝ)) (x : α) : f x ≤ ‖f‖ :=
le_trans (le_abs.mpr (Or.inl (le_refl (f x)))) (f.norm_coe_le_norm x)
#align continuous_map.apply_le_norm ContinuousMap.apply_le_norm
theorem neg_norm_le_apply (f : C(α, ℝ)) (x : α) : -‖f‖ ≤ f x :=
le_trans (neg_le_neg (f.norm_coe_le_norm x)) (neg_le.mp (neg_le_abs_self (f x)))
#align continuous_map.neg_norm_le_apply ContinuousMap.neg_norm_le_apply
theorem norm_eq_iSup_norm : ‖f‖ = ⨆ x : α, ‖f x‖ :=
(mkOfCompact f).norm_eq_iSup_norm
#align continuous_map.norm_eq_supr_norm ContinuousMap.norm_eq_iSup_norm
theorem norm_restrict_mono_set {X : Type*} [TopologicalSpace X] (f : C(X, E))
{K L : TopologicalSpace.Compacts X} (hKL : K ≤ L) : ‖f.restrict K‖ ≤ ‖f.restrict L‖ :=
(norm_le _ (norm_nonneg _)).mpr fun x => norm_coe_le_norm (f.restrict L) <| Set.inclusion hKL x
#align continuous_map.norm_restrict_mono_set ContinuousMap.norm_restrict_mono_set
end
section
variable {R : Type*} [NormedRing R]
instance : NormedRing C(α, R) :=
{ (inferInstance : NormedAddCommGroup C(α, R)), ContinuousMap.instRingContinuousMap with
norm_mul := fun f g => norm_mul_le (mkOfCompact f) (mkOfCompact g) }
end
section
variable {𝕜 : Type*} [NormedField 𝕜] [NormedSpace 𝕜 E]
instance normedSpace : NormedSpace 𝕜 C(α, E) where
norm_smul_le c f := (norm_smul_le c (mkOfCompact f) : _)
#align continuous_map.normed_space ContinuousMap.normedSpace
section
variable (α 𝕜 E)
/-- When `α` is compact and `𝕜` is a normed field,
the `𝕜`-algebra of bounded continuous maps `α →ᵇ β` is
`𝕜`-linearly isometric to `C(α, β)`.
-/
def linearIsometryBoundedOfCompact : C(α, E) ≃ₗᵢ[𝕜] α →ᵇ E :=
{ addEquivBoundedOfCompact α E with
map_smul' := fun c f => by
ext
norm_cast
norm_map' := fun f => rfl }
#align continuous_map.linear_isometry_bounded_of_compact ContinuousMap.linearIsometryBoundedOfCompact
variable {α E}
-- to match `BoundedContinuousFunction.evalClm`
/-- The evaluation at a point, as a continuous linear map from `C(α, 𝕜)` to `𝕜`. -/
def evalClm (x : α) : C(α, E) →L[𝕜] E :=
(BoundedContinuousFunction.evalClm 𝕜 x).comp
(linearIsometryBoundedOfCompact α E 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_map.eval_clm ContinuousMap.evalClm
end
-- this lemma and the next are the analogues of those autogenerated by `@[simps]` for
-- `equivBoundedOfCompact`, `addEquivBoundedOfCompact`
@[simp]
theorem linearIsometryBoundedOfCompact_symm_apply (f : α →ᵇ E) :
(linearIsometryBoundedOfCompact α E 𝕜).symm f = f.toContinuousMap :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_symm_apply ContinuousMap.linearIsometryBoundedOfCompact_symm_apply
@[simp]
theorem linearIsometryBoundedOfCompact_apply_apply (f : C(α, E)) (a : α) :
(linearIsometryBoundedOfCompact α E 𝕜 f) a = f a :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_apply_apply ContinuousMap.linearIsometryBoundedOfCompact_apply_apply
@[simp]
theorem linearIsometryBoundedOfCompact_toIsometryEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toIsometryEquiv = isometryEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_isometry_equiv ContinuousMap.linearIsometryBoundedOfCompact_toIsometryEquiv
@[simp] -- porting note: adjusted LHS because `simpNF` complained it simplified.
theorem linearIsometryBoundedOfCompact_toAddEquiv :
((linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv : C(α, E) ≃+ (α →ᵇ E)) =
addEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_add_equiv ContinuousMap.linearIsometryBoundedOfCompact_toAddEquiv
@[simp]
theorem linearIsometryBoundedOfCompact_of_compact_toEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv.toEquiv = equivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_of_compact_to_equiv ContinuousMap.linearIsometryBoundedOfCompact_of_compact_toEquiv
end
section
variable {𝕜 : Type*} {γ : Type*} [NormedField 𝕜] [NormedRing γ] [NormedAlgebra 𝕜 γ]
instance : NormedAlgebra 𝕜 C(α, γ) :=
{ ContinuousMap.normedSpace, ContinuousMap.algebra with }
end
end ContinuousMap
namespace ContinuousMap
section UniformContinuity
variable {α β : Type*}
variable [MetricSpace α] [CompactSpace α] [MetricSpace β]
/-!
We now set up some declarations making it convenient to use uniform continuity.
-/
theorem uniform_continuity (f : C(α, β)) (ε : ℝ) (h : 0 < ε) :
∃ δ > 0, ∀ {x y}, dist x y < δ → dist (f x) (f y) < ε :=
Metric.uniformContinuous_iff.mp (CompactSpace.uniformContinuous_of_continuous f.continuous) ε h
#align continuous_map.uniform_continuity ContinuousMap.uniform_continuity
-- This definition allows us to separate the choice of some `δ`,
-- and the corresponding use of `dist a b < δ → dist (f a) (f b) < ε`,
-- even across different declarations.
/-- An arbitrarily chosen modulus of uniform continuity for a given function `f` and `ε > 0`. -/
def modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) : ℝ :=
Classical.choose (uniform_continuity f ε h)
#align continuous_map.modulus ContinuousMap.modulus
theorem modulus_pos (f : C(α, β)) {ε : ℝ} {h : 0 < ε} : 0 < f.modulus ε h :=
(Classical.choose_spec (uniform_continuity f ε h)).1
#align continuous_map.modulus_pos ContinuousMap.modulus_pos
theorem dist_lt_of_dist_lt_modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) {a b : α}
(w : dist a b < f.modulus ε h) : dist (f a) (f b) < ε :=
(Classical.choose_spec (uniform_continuity f ε h)).2 w
#align continuous_map.dist_lt_of_dist_lt_modulus ContinuousMap.dist_lt_of_dist_lt_modulus
end UniformContinuity
end ContinuousMap
section CompLeft
variable (X : Type*) {𝕜 β γ : Type*} [TopologicalSpace X] [CompactSpace X]
[NontriviallyNormedField 𝕜]
variable [NormedAddCommGroup β] [NormedSpace 𝕜 β] [NormedAddCommGroup γ] [NormedSpace 𝕜 γ]
open ContinuousMap
/-- Postcomposition of continuous functions into a normed module by a continuous linear map is a
continuous linear map.
Transferred version of `ContinuousLinearMap.compLeftContinuousBounded`,
upgraded version of `ContinuousLinearMap.compLeftContinuous`,
similar to `LinearMap.compLeft`. -/
protected def ContinuousLinearMap.compLeftContinuousCompact (g : β →L[𝕜] γ) :
C(X, β) →L[𝕜] C(X, γ) :=
(linearIsometryBoundedOfCompact X γ 𝕜).symm.toLinearIsometry.toContinuousLinearMap.comp <|
(g.compLeftContinuousBounded X).comp <|
(linearIsometryBoundedOfCompact X β 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_linear_map.comp_left_continuous_compact ContinuousLinearMap.compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.toLinear_compLeftContinuousCompact (g : β →L[𝕜] γ) :
(g.compLeftContinuousCompact X : C(X, β) →ₗ[𝕜] C(X, γ)) = g.compLeftContinuous 𝕜 X := by
ext f
rfl
#align continuous_linear_map.to_linear_comp_left_continuous_compact ContinuousLinearMap.toLinear_compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.compLeftContinuousCompact_apply (g : β →L[𝕜] γ) (f : C(X, β)) (x : X) :
g.compLeftContinuousCompact X f x = g (f x) :=
rfl
#align continuous_linear_map.comp_left_continuous_compact_apply ContinuousLinearMap.compLeftContinuousCompact_apply
end CompLeft
namespace ContinuousMap
/-!
We now setup variations on `compRight* f`, where `f : C(X, Y)`
(that is, precomposition by a continuous map),
as a morphism `C(Y, T) → C(X, T)`, respecting various types of structure.
In particular:
* `compRightContinuousMap`, the bundled continuous map (for this we need `X Y` compact).
* `compRightHomeomorph`, when we precompose by a homeomorphism.
* `compRightAlgHom`, when `T = R` is a topological ring.
-/
section CompRight
/-- Precomposition by a continuous map is itself a continuous map between spaces of continuous maps.
-/
def compRightContinuousMap {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y)) : C(C(Y, T), C(X, T)) where
toFun g := g.comp f
continuous_toFun := by
|
refine' Metric.continuous_iff.mpr _
|
/-- Precomposition by a continuous map is itself a continuous map between spaces of continuous maps.
-/
def compRightContinuousMap {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y)) : C(C(Y, T), C(X, T)) where
toFun g := g.comp f
continuous_toFun := by
|
Mathlib.Topology.ContinuousFunction.Compact.437_0.Mig2jTVnn2FLKEB
|
/-- Precomposition by a continuous map is itself a continuous map between spaces of continuous maps.
-/
def compRightContinuousMap {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y)) : C(C(Y, T), C(X, T)) where
toFun g
|
Mathlib_Topology_ContinuousFunction_Compact
|
X : Type u_1
Y : Type u_2
T : Type u_3
inst✝⁴ : TopologicalSpace X
inst✝³ : CompactSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : CompactSpace Y
inst✝ : MetricSpace T
f : C(X, Y)
⊢ ∀ (b : C(Y, T)), ∀ ε > 0, ∃ δ > 0, ∀ (a : C(Y, T)), dist a b < δ → dist (comp a f) (comp b f) < ε
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compacts
#align_import topology.continuous_function.compact from "leanprover-community/mathlib"@"d3af0609f6db8691dffdc3e1fb7feb7da72698f2"
/-!
# Continuous functions on a compact space
Continuous functions `C(α, β)` from a compact space `α` to a metric space `β`
are automatically bounded, and so acquire various structures inherited from `α →ᵇ β`.
This file transfers these structures, and restates some lemmas
characterising these structures.
If you need a lemma which is proved about `α →ᵇ β` but not for `C(α, β)` when `α` is compact,
you should restate it here. You can also use
`ContinuousMap.equivBoundedOfCompact` to move functions back and forth.
-/
noncomputable section
open Topology Classical NNReal BoundedContinuousFunction BigOperators
open Set Filter Metric
open BoundedContinuousFunction
namespace ContinuousMap
variable {α β E : Type*} [TopologicalSpace α] [CompactSpace α] [MetricSpace β]
[NormedAddCommGroup E]
section
variable (α β)
/-- When `α` is compact, the bounded continuous maps `α →ᵇ β` are
equivalent to `C(α, β)`.
-/
@[simps (config := .asFn)]
def equivBoundedOfCompact : C(α, β) ≃ (α →ᵇ β) :=
⟨mkOfCompact, BoundedContinuousFunction.toContinuousMap, fun f => by
ext
rfl, fun f => by
ext
rfl⟩
#align continuous_map.equiv_bounded_of_compact ContinuousMap.equivBoundedOfCompact
theorem uniformInducing_equivBoundedOfCompact : UniformInducing (equivBoundedOfCompact α β) :=
UniformInducing.mk'
(by
simp only [hasBasis_compactConvergenceUniformity.mem_iff, uniformity_basis_dist_le.mem_iff]
exact fun s =>
⟨fun ⟨⟨a, b⟩, ⟨_, ⟨ε, hε, hb⟩⟩, hs⟩ =>
⟨{ p | ∀ x, (p.1 x, p.2 x) ∈ b }, ⟨ε, hε, fun _ h x => hb ((dist_le hε.le).mp h x)⟩,
fun f g h => hs fun x _ => h x⟩,
fun ⟨_, ⟨ε, hε, ht⟩, hs⟩ =>
⟨⟨Set.univ, { p | dist p.1 p.2 ≤ ε }⟩, ⟨isCompact_univ, ⟨ε, hε, fun _ h => h⟩⟩,
fun ⟨f, g⟩ h => hs _ _ (ht ((dist_le hε.le).mpr fun x => h x (mem_univ x)))⟩⟩)
#align continuous_map.uniform_inducing_equiv_bounded_of_compact ContinuousMap.uniformInducing_equivBoundedOfCompact
theorem uniformEmbedding_equivBoundedOfCompact : UniformEmbedding (equivBoundedOfCompact α β) :=
{ uniformInducing_equivBoundedOfCompact α β with inj := (equivBoundedOfCompact α β).injective }
#align continuous_map.uniform_embedding_equiv_bounded_of_compact ContinuousMap.uniformEmbedding_equivBoundedOfCompact
/-- When `α` is compact, the bounded continuous maps `α →ᵇ 𝕜` are
additively equivalent to `C(α, 𝕜)`.
-/
-- porting note: the following `simps` received a "maximum recursion depth" error
-- @[simps! (config := .asFn) apply symm_apply]
def addEquivBoundedOfCompact [AddMonoid β] [LipschitzAdd β] : C(α, β) ≃+ (α →ᵇ β) :=
({ toContinuousMapAddHom α β, (equivBoundedOfCompact α β).symm with } : (α →ᵇ β) ≃+ C(α, β)).symm
#align continuous_map.add_equiv_bounded_of_compact ContinuousMap.addEquivBoundedOfCompact
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_symm_apply [AddMonoid β] [LipschitzAdd β] :
⇑((addEquivBoundedOfCompact α β).symm) = toContinuousMapAddHom α β :=
rfl
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_apply [AddMonoid β] [LipschitzAdd β] :
⇑(addEquivBoundedOfCompact α β) = mkOfCompact :=
rfl
instance metricSpace : MetricSpace C(α, β) :=
(uniformEmbedding_equivBoundedOfCompact α β).comapMetricSpace _
#align continuous_map.metric_space ContinuousMap.metricSpace
/-- When `α` is compact, and `β` is a metric space, the bounded continuous maps `α →ᵇ β` are
isometric to `C(α, β)`.
-/
@[simps! (config := .asFn) toEquiv apply symm_apply]
def isometryEquivBoundedOfCompact : C(α, β) ≃ᵢ (α →ᵇ β) where
isometry_toFun _ _ := rfl
toEquiv := equivBoundedOfCompact α β
#align continuous_map.isometry_equiv_bounded_of_compact ContinuousMap.isometryEquivBoundedOfCompact
end
@[simp]
theorem _root_.BoundedContinuousFunction.dist_mkOfCompact (f g : C(α, β)) :
dist (mkOfCompact f) (mkOfCompact g) = dist f g :=
rfl
#align bounded_continuous_function.dist_mk_of_compact BoundedContinuousFunction.dist_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.dist_toContinuousMap (f g : α →ᵇ β) :
dist f.toContinuousMap g.toContinuousMap = dist f g :=
rfl
#align bounded_continuous_function.dist_to_continuous_map BoundedContinuousFunction.dist_toContinuousMap
open BoundedContinuousFunction
section
variable {f g : C(α, β)} {C : ℝ}
/-- The pointwise distance is controlled by the distance between functions, by definition. -/
theorem dist_apply_le_dist (x : α) : dist (f x) (g x) ≤ dist f g := by
simp only [← dist_mkOfCompact, dist_coe_le_dist, ← mkOfCompact_apply]
#align continuous_map.dist_apply_le_dist ContinuousMap.dist_apply_le_dist
/-- The distance between two functions is controlled by the supremum of the pointwise distances. -/
theorem dist_le (C0 : (0 : ℝ) ≤ C) : dist f g ≤ C ↔ ∀ x : α, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le C0, mkOfCompact_apply]
#align continuous_map.dist_le ContinuousMap.dist_le
theorem dist_le_iff_of_nonempty [Nonempty α] : dist f g ≤ C ↔ ∀ x, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le_iff_of_nonempty,
mkOfCompact_apply]
#align continuous_map.dist_le_iff_of_nonempty ContinuousMap.dist_le_iff_of_nonempty
theorem dist_lt_iff_of_nonempty [Nonempty α] : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
simp only [← dist_mkOfCompact, dist_lt_iff_of_nonempty_compact, mkOfCompact_apply]
#align continuous_map.dist_lt_iff_of_nonempty ContinuousMap.dist_lt_iff_of_nonempty
theorem dist_lt_of_nonempty [Nonempty α] (w : ∀ x : α, dist (f x) (g x) < C) : dist f g < C :=
dist_lt_iff_of_nonempty.2 w
#align continuous_map.dist_lt_of_nonempty ContinuousMap.dist_lt_of_nonempty
theorem dist_lt_iff (C0 : (0 : ℝ) < C) : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
rw [← dist_mkOfCompact, dist_lt_iff_of_compact C0]
simp only [mkOfCompact_apply]
#align continuous_map.dist_lt_iff ContinuousMap.dist_lt_iff
end
instance [CompleteSpace β] : CompleteSpace C(α, β) :=
(isometryEquivBoundedOfCompact α β).completeSpace
/-- See also `ContinuousMap.continuous_eval'`. -/
@[continuity]
theorem continuous_eval : Continuous fun p : C(α, β) × α => p.1 p.2 :=
continuous_eval.comp ((isometryEquivBoundedOfCompact α β).continuous.prod_map continuous_id)
#align continuous_map.continuous_eval ContinuousMap.continuous_eval
-- TODO at some point we will need lemmas characterising this norm!
-- At the moment the only way to reason about it is to transfer `f : C(α,E)` back to `α →ᵇ E`.
instance : Norm C(α, E) where norm x := dist x 0
@[simp]
theorem _root_.BoundedContinuousFunction.norm_mkOfCompact (f : C(α, E)) : ‖mkOfCompact f‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_mk_of_compact BoundedContinuousFunction.norm_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.norm_toContinuousMap_eq (f : α →ᵇ E) :
‖f.toContinuousMap‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_to_continuous_map_eq BoundedContinuousFunction.norm_toContinuousMap_eq
open BoundedContinuousFunction
instance : NormedAddCommGroup C(α, E) :=
{ ContinuousMap.metricSpace _ _,
ContinuousMap.instAddCommGroupContinuousMap with
dist_eq := fun x y => by
rw [← norm_mkOfCompact, ← dist_mkOfCompact, dist_eq_norm, mkOfCompact_sub]
dist := dist
norm := norm }
instance [Nonempty α] [One E] [NormOneClass E] : NormOneClass C(α, E) where
norm_one := by simp only [← norm_mkOfCompact, mkOfCompact_one, norm_one]
section
variable (f : C(α, E))
-- The corresponding lemmas for `BoundedContinuousFunction` are stated with `{f}`,
-- and so can not be used in dot notation.
theorem norm_coe_le_norm (x : α) : ‖f x‖ ≤ ‖f‖ :=
(mkOfCompact f).norm_coe_le_norm x
#align continuous_map.norm_coe_le_norm ContinuousMap.norm_coe_le_norm
/-- Distance between the images of any two points is at most twice the norm of the function. -/
theorem dist_le_two_norm (x y : α) : dist (f x) (f y) ≤ 2 * ‖f‖ :=
(mkOfCompact f).dist_le_two_norm x y
#align continuous_map.dist_le_two_norm ContinuousMap.dist_le_two_norm
/-- The norm of a function is controlled by the supremum of the pointwise norms. -/
theorem norm_le {C : ℝ} (C0 : (0 : ℝ) ≤ C) : ‖f‖ ≤ C ↔ ∀ x : α, ‖f x‖ ≤ C :=
@BoundedContinuousFunction.norm_le _ _ _ _ (mkOfCompact f) _ C0
#align continuous_map.norm_le ContinuousMap.norm_le
theorem norm_le_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ ≤ M ↔ ∀ x, ‖f x‖ ≤ M :=
@BoundedContinuousFunction.norm_le_of_nonempty _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_le_of_nonempty ContinuousMap.norm_le_of_nonempty
theorem norm_lt_iff {M : ℝ} (M0 : 0 < M) : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_compact _ _ _ _ _ (mkOfCompact f) _ M0
#align continuous_map.norm_lt_iff ContinuousMap.norm_lt_iff
theorem nnnorm_lt_iff {M : ℝ≥0} (M0 : 0 < M) : ‖f‖₊ < M ↔ ∀ x : α, ‖f x‖₊ < M :=
f.norm_lt_iff M0
#align continuous_map.nnnorm_lt_iff ContinuousMap.nnnorm_lt_iff
theorem norm_lt_iff_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_nonempty_compact _ _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_lt_iff_of_nonempty ContinuousMap.norm_lt_iff_of_nonempty
theorem nnnorm_lt_iff_of_nonempty [Nonempty α] {M : ℝ≥0} : ‖f‖₊ < M ↔ ∀ x, ‖f x‖₊ < M :=
f.norm_lt_iff_of_nonempty
#align continuous_map.nnnorm_lt_iff_of_nonempty ContinuousMap.nnnorm_lt_iff_of_nonempty
theorem apply_le_norm (f : C(α, ℝ)) (x : α) : f x ≤ ‖f‖ :=
le_trans (le_abs.mpr (Or.inl (le_refl (f x)))) (f.norm_coe_le_norm x)
#align continuous_map.apply_le_norm ContinuousMap.apply_le_norm
theorem neg_norm_le_apply (f : C(α, ℝ)) (x : α) : -‖f‖ ≤ f x :=
le_trans (neg_le_neg (f.norm_coe_le_norm x)) (neg_le.mp (neg_le_abs_self (f x)))
#align continuous_map.neg_norm_le_apply ContinuousMap.neg_norm_le_apply
theorem norm_eq_iSup_norm : ‖f‖ = ⨆ x : α, ‖f x‖ :=
(mkOfCompact f).norm_eq_iSup_norm
#align continuous_map.norm_eq_supr_norm ContinuousMap.norm_eq_iSup_norm
theorem norm_restrict_mono_set {X : Type*} [TopologicalSpace X] (f : C(X, E))
{K L : TopologicalSpace.Compacts X} (hKL : K ≤ L) : ‖f.restrict K‖ ≤ ‖f.restrict L‖ :=
(norm_le _ (norm_nonneg _)).mpr fun x => norm_coe_le_norm (f.restrict L) <| Set.inclusion hKL x
#align continuous_map.norm_restrict_mono_set ContinuousMap.norm_restrict_mono_set
end
section
variable {R : Type*} [NormedRing R]
instance : NormedRing C(α, R) :=
{ (inferInstance : NormedAddCommGroup C(α, R)), ContinuousMap.instRingContinuousMap with
norm_mul := fun f g => norm_mul_le (mkOfCompact f) (mkOfCompact g) }
end
section
variable {𝕜 : Type*} [NormedField 𝕜] [NormedSpace 𝕜 E]
instance normedSpace : NormedSpace 𝕜 C(α, E) where
norm_smul_le c f := (norm_smul_le c (mkOfCompact f) : _)
#align continuous_map.normed_space ContinuousMap.normedSpace
section
variable (α 𝕜 E)
/-- When `α` is compact and `𝕜` is a normed field,
the `𝕜`-algebra of bounded continuous maps `α →ᵇ β` is
`𝕜`-linearly isometric to `C(α, β)`.
-/
def linearIsometryBoundedOfCompact : C(α, E) ≃ₗᵢ[𝕜] α →ᵇ E :=
{ addEquivBoundedOfCompact α E with
map_smul' := fun c f => by
ext
norm_cast
norm_map' := fun f => rfl }
#align continuous_map.linear_isometry_bounded_of_compact ContinuousMap.linearIsometryBoundedOfCompact
variable {α E}
-- to match `BoundedContinuousFunction.evalClm`
/-- The evaluation at a point, as a continuous linear map from `C(α, 𝕜)` to `𝕜`. -/
def evalClm (x : α) : C(α, E) →L[𝕜] E :=
(BoundedContinuousFunction.evalClm 𝕜 x).comp
(linearIsometryBoundedOfCompact α E 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_map.eval_clm ContinuousMap.evalClm
end
-- this lemma and the next are the analogues of those autogenerated by `@[simps]` for
-- `equivBoundedOfCompact`, `addEquivBoundedOfCompact`
@[simp]
theorem linearIsometryBoundedOfCompact_symm_apply (f : α →ᵇ E) :
(linearIsometryBoundedOfCompact α E 𝕜).symm f = f.toContinuousMap :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_symm_apply ContinuousMap.linearIsometryBoundedOfCompact_symm_apply
@[simp]
theorem linearIsometryBoundedOfCompact_apply_apply (f : C(α, E)) (a : α) :
(linearIsometryBoundedOfCompact α E 𝕜 f) a = f a :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_apply_apply ContinuousMap.linearIsometryBoundedOfCompact_apply_apply
@[simp]
theorem linearIsometryBoundedOfCompact_toIsometryEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toIsometryEquiv = isometryEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_isometry_equiv ContinuousMap.linearIsometryBoundedOfCompact_toIsometryEquiv
@[simp] -- porting note: adjusted LHS because `simpNF` complained it simplified.
theorem linearIsometryBoundedOfCompact_toAddEquiv :
((linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv : C(α, E) ≃+ (α →ᵇ E)) =
addEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_add_equiv ContinuousMap.linearIsometryBoundedOfCompact_toAddEquiv
@[simp]
theorem linearIsometryBoundedOfCompact_of_compact_toEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv.toEquiv = equivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_of_compact_to_equiv ContinuousMap.linearIsometryBoundedOfCompact_of_compact_toEquiv
end
section
variable {𝕜 : Type*} {γ : Type*} [NormedField 𝕜] [NormedRing γ] [NormedAlgebra 𝕜 γ]
instance : NormedAlgebra 𝕜 C(α, γ) :=
{ ContinuousMap.normedSpace, ContinuousMap.algebra with }
end
end ContinuousMap
namespace ContinuousMap
section UniformContinuity
variable {α β : Type*}
variable [MetricSpace α] [CompactSpace α] [MetricSpace β]
/-!
We now set up some declarations making it convenient to use uniform continuity.
-/
theorem uniform_continuity (f : C(α, β)) (ε : ℝ) (h : 0 < ε) :
∃ δ > 0, ∀ {x y}, dist x y < δ → dist (f x) (f y) < ε :=
Metric.uniformContinuous_iff.mp (CompactSpace.uniformContinuous_of_continuous f.continuous) ε h
#align continuous_map.uniform_continuity ContinuousMap.uniform_continuity
-- This definition allows us to separate the choice of some `δ`,
-- and the corresponding use of `dist a b < δ → dist (f a) (f b) < ε`,
-- even across different declarations.
/-- An arbitrarily chosen modulus of uniform continuity for a given function `f` and `ε > 0`. -/
def modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) : ℝ :=
Classical.choose (uniform_continuity f ε h)
#align continuous_map.modulus ContinuousMap.modulus
theorem modulus_pos (f : C(α, β)) {ε : ℝ} {h : 0 < ε} : 0 < f.modulus ε h :=
(Classical.choose_spec (uniform_continuity f ε h)).1
#align continuous_map.modulus_pos ContinuousMap.modulus_pos
theorem dist_lt_of_dist_lt_modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) {a b : α}
(w : dist a b < f.modulus ε h) : dist (f a) (f b) < ε :=
(Classical.choose_spec (uniform_continuity f ε h)).2 w
#align continuous_map.dist_lt_of_dist_lt_modulus ContinuousMap.dist_lt_of_dist_lt_modulus
end UniformContinuity
end ContinuousMap
section CompLeft
variable (X : Type*) {𝕜 β γ : Type*} [TopologicalSpace X] [CompactSpace X]
[NontriviallyNormedField 𝕜]
variable [NormedAddCommGroup β] [NormedSpace 𝕜 β] [NormedAddCommGroup γ] [NormedSpace 𝕜 γ]
open ContinuousMap
/-- Postcomposition of continuous functions into a normed module by a continuous linear map is a
continuous linear map.
Transferred version of `ContinuousLinearMap.compLeftContinuousBounded`,
upgraded version of `ContinuousLinearMap.compLeftContinuous`,
similar to `LinearMap.compLeft`. -/
protected def ContinuousLinearMap.compLeftContinuousCompact (g : β →L[𝕜] γ) :
C(X, β) →L[𝕜] C(X, γ) :=
(linearIsometryBoundedOfCompact X γ 𝕜).symm.toLinearIsometry.toContinuousLinearMap.comp <|
(g.compLeftContinuousBounded X).comp <|
(linearIsometryBoundedOfCompact X β 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_linear_map.comp_left_continuous_compact ContinuousLinearMap.compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.toLinear_compLeftContinuousCompact (g : β →L[𝕜] γ) :
(g.compLeftContinuousCompact X : C(X, β) →ₗ[𝕜] C(X, γ)) = g.compLeftContinuous 𝕜 X := by
ext f
rfl
#align continuous_linear_map.to_linear_comp_left_continuous_compact ContinuousLinearMap.toLinear_compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.compLeftContinuousCompact_apply (g : β →L[𝕜] γ) (f : C(X, β)) (x : X) :
g.compLeftContinuousCompact X f x = g (f x) :=
rfl
#align continuous_linear_map.comp_left_continuous_compact_apply ContinuousLinearMap.compLeftContinuousCompact_apply
end CompLeft
namespace ContinuousMap
/-!
We now setup variations on `compRight* f`, where `f : C(X, Y)`
(that is, precomposition by a continuous map),
as a morphism `C(Y, T) → C(X, T)`, respecting various types of structure.
In particular:
* `compRightContinuousMap`, the bundled continuous map (for this we need `X Y` compact).
* `compRightHomeomorph`, when we precompose by a homeomorphism.
* `compRightAlgHom`, when `T = R` is a topological ring.
-/
section CompRight
/-- Precomposition by a continuous map is itself a continuous map between spaces of continuous maps.
-/
def compRightContinuousMap {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y)) : C(C(Y, T), C(X, T)) where
toFun g := g.comp f
continuous_toFun := by
refine' Metric.continuous_iff.mpr _
|
intro g ε ε_pos
|
/-- Precomposition by a continuous map is itself a continuous map between spaces of continuous maps.
-/
def compRightContinuousMap {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y)) : C(C(Y, T), C(X, T)) where
toFun g := g.comp f
continuous_toFun := by
refine' Metric.continuous_iff.mpr _
|
Mathlib.Topology.ContinuousFunction.Compact.437_0.Mig2jTVnn2FLKEB
|
/-- Precomposition by a continuous map is itself a continuous map between spaces of continuous maps.
-/
def compRightContinuousMap {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y)) : C(C(Y, T), C(X, T)) where
toFun g
|
Mathlib_Topology_ContinuousFunction_Compact
|
X : Type u_1
Y : Type u_2
T : Type u_3
inst✝⁴ : TopologicalSpace X
inst✝³ : CompactSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : CompactSpace Y
inst✝ : MetricSpace T
f : C(X, Y)
g : C(Y, T)
ε : ℝ
ε_pos : ε > 0
⊢ ∃ δ > 0, ∀ (a : C(Y, T)), dist a g < δ → dist (comp a f) (comp g f) < ε
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compacts
#align_import topology.continuous_function.compact from "leanprover-community/mathlib"@"d3af0609f6db8691dffdc3e1fb7feb7da72698f2"
/-!
# Continuous functions on a compact space
Continuous functions `C(α, β)` from a compact space `α` to a metric space `β`
are automatically bounded, and so acquire various structures inherited from `α →ᵇ β`.
This file transfers these structures, and restates some lemmas
characterising these structures.
If you need a lemma which is proved about `α →ᵇ β` but not for `C(α, β)` when `α` is compact,
you should restate it here. You can also use
`ContinuousMap.equivBoundedOfCompact` to move functions back and forth.
-/
noncomputable section
open Topology Classical NNReal BoundedContinuousFunction BigOperators
open Set Filter Metric
open BoundedContinuousFunction
namespace ContinuousMap
variable {α β E : Type*} [TopologicalSpace α] [CompactSpace α] [MetricSpace β]
[NormedAddCommGroup E]
section
variable (α β)
/-- When `α` is compact, the bounded continuous maps `α →ᵇ β` are
equivalent to `C(α, β)`.
-/
@[simps (config := .asFn)]
def equivBoundedOfCompact : C(α, β) ≃ (α →ᵇ β) :=
⟨mkOfCompact, BoundedContinuousFunction.toContinuousMap, fun f => by
ext
rfl, fun f => by
ext
rfl⟩
#align continuous_map.equiv_bounded_of_compact ContinuousMap.equivBoundedOfCompact
theorem uniformInducing_equivBoundedOfCompact : UniformInducing (equivBoundedOfCompact α β) :=
UniformInducing.mk'
(by
simp only [hasBasis_compactConvergenceUniformity.mem_iff, uniformity_basis_dist_le.mem_iff]
exact fun s =>
⟨fun ⟨⟨a, b⟩, ⟨_, ⟨ε, hε, hb⟩⟩, hs⟩ =>
⟨{ p | ∀ x, (p.1 x, p.2 x) ∈ b }, ⟨ε, hε, fun _ h x => hb ((dist_le hε.le).mp h x)⟩,
fun f g h => hs fun x _ => h x⟩,
fun ⟨_, ⟨ε, hε, ht⟩, hs⟩ =>
⟨⟨Set.univ, { p | dist p.1 p.2 ≤ ε }⟩, ⟨isCompact_univ, ⟨ε, hε, fun _ h => h⟩⟩,
fun ⟨f, g⟩ h => hs _ _ (ht ((dist_le hε.le).mpr fun x => h x (mem_univ x)))⟩⟩)
#align continuous_map.uniform_inducing_equiv_bounded_of_compact ContinuousMap.uniformInducing_equivBoundedOfCompact
theorem uniformEmbedding_equivBoundedOfCompact : UniformEmbedding (equivBoundedOfCompact α β) :=
{ uniformInducing_equivBoundedOfCompact α β with inj := (equivBoundedOfCompact α β).injective }
#align continuous_map.uniform_embedding_equiv_bounded_of_compact ContinuousMap.uniformEmbedding_equivBoundedOfCompact
/-- When `α` is compact, the bounded continuous maps `α →ᵇ 𝕜` are
additively equivalent to `C(α, 𝕜)`.
-/
-- porting note: the following `simps` received a "maximum recursion depth" error
-- @[simps! (config := .asFn) apply symm_apply]
def addEquivBoundedOfCompact [AddMonoid β] [LipschitzAdd β] : C(α, β) ≃+ (α →ᵇ β) :=
({ toContinuousMapAddHom α β, (equivBoundedOfCompact α β).symm with } : (α →ᵇ β) ≃+ C(α, β)).symm
#align continuous_map.add_equiv_bounded_of_compact ContinuousMap.addEquivBoundedOfCompact
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_symm_apply [AddMonoid β] [LipschitzAdd β] :
⇑((addEquivBoundedOfCompact α β).symm) = toContinuousMapAddHom α β :=
rfl
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_apply [AddMonoid β] [LipschitzAdd β] :
⇑(addEquivBoundedOfCompact α β) = mkOfCompact :=
rfl
instance metricSpace : MetricSpace C(α, β) :=
(uniformEmbedding_equivBoundedOfCompact α β).comapMetricSpace _
#align continuous_map.metric_space ContinuousMap.metricSpace
/-- When `α` is compact, and `β` is a metric space, the bounded continuous maps `α →ᵇ β` are
isometric to `C(α, β)`.
-/
@[simps! (config := .asFn) toEquiv apply symm_apply]
def isometryEquivBoundedOfCompact : C(α, β) ≃ᵢ (α →ᵇ β) where
isometry_toFun _ _ := rfl
toEquiv := equivBoundedOfCompact α β
#align continuous_map.isometry_equiv_bounded_of_compact ContinuousMap.isometryEquivBoundedOfCompact
end
@[simp]
theorem _root_.BoundedContinuousFunction.dist_mkOfCompact (f g : C(α, β)) :
dist (mkOfCompact f) (mkOfCompact g) = dist f g :=
rfl
#align bounded_continuous_function.dist_mk_of_compact BoundedContinuousFunction.dist_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.dist_toContinuousMap (f g : α →ᵇ β) :
dist f.toContinuousMap g.toContinuousMap = dist f g :=
rfl
#align bounded_continuous_function.dist_to_continuous_map BoundedContinuousFunction.dist_toContinuousMap
open BoundedContinuousFunction
section
variable {f g : C(α, β)} {C : ℝ}
/-- The pointwise distance is controlled by the distance between functions, by definition. -/
theorem dist_apply_le_dist (x : α) : dist (f x) (g x) ≤ dist f g := by
simp only [← dist_mkOfCompact, dist_coe_le_dist, ← mkOfCompact_apply]
#align continuous_map.dist_apply_le_dist ContinuousMap.dist_apply_le_dist
/-- The distance between two functions is controlled by the supremum of the pointwise distances. -/
theorem dist_le (C0 : (0 : ℝ) ≤ C) : dist f g ≤ C ↔ ∀ x : α, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le C0, mkOfCompact_apply]
#align continuous_map.dist_le ContinuousMap.dist_le
theorem dist_le_iff_of_nonempty [Nonempty α] : dist f g ≤ C ↔ ∀ x, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le_iff_of_nonempty,
mkOfCompact_apply]
#align continuous_map.dist_le_iff_of_nonempty ContinuousMap.dist_le_iff_of_nonempty
theorem dist_lt_iff_of_nonempty [Nonempty α] : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
simp only [← dist_mkOfCompact, dist_lt_iff_of_nonempty_compact, mkOfCompact_apply]
#align continuous_map.dist_lt_iff_of_nonempty ContinuousMap.dist_lt_iff_of_nonempty
theorem dist_lt_of_nonempty [Nonempty α] (w : ∀ x : α, dist (f x) (g x) < C) : dist f g < C :=
dist_lt_iff_of_nonempty.2 w
#align continuous_map.dist_lt_of_nonempty ContinuousMap.dist_lt_of_nonempty
theorem dist_lt_iff (C0 : (0 : ℝ) < C) : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
rw [← dist_mkOfCompact, dist_lt_iff_of_compact C0]
simp only [mkOfCompact_apply]
#align continuous_map.dist_lt_iff ContinuousMap.dist_lt_iff
end
instance [CompleteSpace β] : CompleteSpace C(α, β) :=
(isometryEquivBoundedOfCompact α β).completeSpace
/-- See also `ContinuousMap.continuous_eval'`. -/
@[continuity]
theorem continuous_eval : Continuous fun p : C(α, β) × α => p.1 p.2 :=
continuous_eval.comp ((isometryEquivBoundedOfCompact α β).continuous.prod_map continuous_id)
#align continuous_map.continuous_eval ContinuousMap.continuous_eval
-- TODO at some point we will need lemmas characterising this norm!
-- At the moment the only way to reason about it is to transfer `f : C(α,E)` back to `α →ᵇ E`.
instance : Norm C(α, E) where norm x := dist x 0
@[simp]
theorem _root_.BoundedContinuousFunction.norm_mkOfCompact (f : C(α, E)) : ‖mkOfCompact f‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_mk_of_compact BoundedContinuousFunction.norm_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.norm_toContinuousMap_eq (f : α →ᵇ E) :
‖f.toContinuousMap‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_to_continuous_map_eq BoundedContinuousFunction.norm_toContinuousMap_eq
open BoundedContinuousFunction
instance : NormedAddCommGroup C(α, E) :=
{ ContinuousMap.metricSpace _ _,
ContinuousMap.instAddCommGroupContinuousMap with
dist_eq := fun x y => by
rw [← norm_mkOfCompact, ← dist_mkOfCompact, dist_eq_norm, mkOfCompact_sub]
dist := dist
norm := norm }
instance [Nonempty α] [One E] [NormOneClass E] : NormOneClass C(α, E) where
norm_one := by simp only [← norm_mkOfCompact, mkOfCompact_one, norm_one]
section
variable (f : C(α, E))
-- The corresponding lemmas for `BoundedContinuousFunction` are stated with `{f}`,
-- and so can not be used in dot notation.
theorem norm_coe_le_norm (x : α) : ‖f x‖ ≤ ‖f‖ :=
(mkOfCompact f).norm_coe_le_norm x
#align continuous_map.norm_coe_le_norm ContinuousMap.norm_coe_le_norm
/-- Distance between the images of any two points is at most twice the norm of the function. -/
theorem dist_le_two_norm (x y : α) : dist (f x) (f y) ≤ 2 * ‖f‖ :=
(mkOfCompact f).dist_le_two_norm x y
#align continuous_map.dist_le_two_norm ContinuousMap.dist_le_two_norm
/-- The norm of a function is controlled by the supremum of the pointwise norms. -/
theorem norm_le {C : ℝ} (C0 : (0 : ℝ) ≤ C) : ‖f‖ ≤ C ↔ ∀ x : α, ‖f x‖ ≤ C :=
@BoundedContinuousFunction.norm_le _ _ _ _ (mkOfCompact f) _ C0
#align continuous_map.norm_le ContinuousMap.norm_le
theorem norm_le_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ ≤ M ↔ ∀ x, ‖f x‖ ≤ M :=
@BoundedContinuousFunction.norm_le_of_nonempty _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_le_of_nonempty ContinuousMap.norm_le_of_nonempty
theorem norm_lt_iff {M : ℝ} (M0 : 0 < M) : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_compact _ _ _ _ _ (mkOfCompact f) _ M0
#align continuous_map.norm_lt_iff ContinuousMap.norm_lt_iff
theorem nnnorm_lt_iff {M : ℝ≥0} (M0 : 0 < M) : ‖f‖₊ < M ↔ ∀ x : α, ‖f x‖₊ < M :=
f.norm_lt_iff M0
#align continuous_map.nnnorm_lt_iff ContinuousMap.nnnorm_lt_iff
theorem norm_lt_iff_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_nonempty_compact _ _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_lt_iff_of_nonempty ContinuousMap.norm_lt_iff_of_nonempty
theorem nnnorm_lt_iff_of_nonempty [Nonempty α] {M : ℝ≥0} : ‖f‖₊ < M ↔ ∀ x, ‖f x‖₊ < M :=
f.norm_lt_iff_of_nonempty
#align continuous_map.nnnorm_lt_iff_of_nonempty ContinuousMap.nnnorm_lt_iff_of_nonempty
theorem apply_le_norm (f : C(α, ℝ)) (x : α) : f x ≤ ‖f‖ :=
le_trans (le_abs.mpr (Or.inl (le_refl (f x)))) (f.norm_coe_le_norm x)
#align continuous_map.apply_le_norm ContinuousMap.apply_le_norm
theorem neg_norm_le_apply (f : C(α, ℝ)) (x : α) : -‖f‖ ≤ f x :=
le_trans (neg_le_neg (f.norm_coe_le_norm x)) (neg_le.mp (neg_le_abs_self (f x)))
#align continuous_map.neg_norm_le_apply ContinuousMap.neg_norm_le_apply
theorem norm_eq_iSup_norm : ‖f‖ = ⨆ x : α, ‖f x‖ :=
(mkOfCompact f).norm_eq_iSup_norm
#align continuous_map.norm_eq_supr_norm ContinuousMap.norm_eq_iSup_norm
theorem norm_restrict_mono_set {X : Type*} [TopologicalSpace X] (f : C(X, E))
{K L : TopologicalSpace.Compacts X} (hKL : K ≤ L) : ‖f.restrict K‖ ≤ ‖f.restrict L‖ :=
(norm_le _ (norm_nonneg _)).mpr fun x => norm_coe_le_norm (f.restrict L) <| Set.inclusion hKL x
#align continuous_map.norm_restrict_mono_set ContinuousMap.norm_restrict_mono_set
end
section
variable {R : Type*} [NormedRing R]
instance : NormedRing C(α, R) :=
{ (inferInstance : NormedAddCommGroup C(α, R)), ContinuousMap.instRingContinuousMap with
norm_mul := fun f g => norm_mul_le (mkOfCompact f) (mkOfCompact g) }
end
section
variable {𝕜 : Type*} [NormedField 𝕜] [NormedSpace 𝕜 E]
instance normedSpace : NormedSpace 𝕜 C(α, E) where
norm_smul_le c f := (norm_smul_le c (mkOfCompact f) : _)
#align continuous_map.normed_space ContinuousMap.normedSpace
section
variable (α 𝕜 E)
/-- When `α` is compact and `𝕜` is a normed field,
the `𝕜`-algebra of bounded continuous maps `α →ᵇ β` is
`𝕜`-linearly isometric to `C(α, β)`.
-/
def linearIsometryBoundedOfCompact : C(α, E) ≃ₗᵢ[𝕜] α →ᵇ E :=
{ addEquivBoundedOfCompact α E with
map_smul' := fun c f => by
ext
norm_cast
norm_map' := fun f => rfl }
#align continuous_map.linear_isometry_bounded_of_compact ContinuousMap.linearIsometryBoundedOfCompact
variable {α E}
-- to match `BoundedContinuousFunction.evalClm`
/-- The evaluation at a point, as a continuous linear map from `C(α, 𝕜)` to `𝕜`. -/
def evalClm (x : α) : C(α, E) →L[𝕜] E :=
(BoundedContinuousFunction.evalClm 𝕜 x).comp
(linearIsometryBoundedOfCompact α E 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_map.eval_clm ContinuousMap.evalClm
end
-- this lemma and the next are the analogues of those autogenerated by `@[simps]` for
-- `equivBoundedOfCompact`, `addEquivBoundedOfCompact`
@[simp]
theorem linearIsometryBoundedOfCompact_symm_apply (f : α →ᵇ E) :
(linearIsometryBoundedOfCompact α E 𝕜).symm f = f.toContinuousMap :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_symm_apply ContinuousMap.linearIsometryBoundedOfCompact_symm_apply
@[simp]
theorem linearIsometryBoundedOfCompact_apply_apply (f : C(α, E)) (a : α) :
(linearIsometryBoundedOfCompact α E 𝕜 f) a = f a :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_apply_apply ContinuousMap.linearIsometryBoundedOfCompact_apply_apply
@[simp]
theorem linearIsometryBoundedOfCompact_toIsometryEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toIsometryEquiv = isometryEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_isometry_equiv ContinuousMap.linearIsometryBoundedOfCompact_toIsometryEquiv
@[simp] -- porting note: adjusted LHS because `simpNF` complained it simplified.
theorem linearIsometryBoundedOfCompact_toAddEquiv :
((linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv : C(α, E) ≃+ (α →ᵇ E)) =
addEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_add_equiv ContinuousMap.linearIsometryBoundedOfCompact_toAddEquiv
@[simp]
theorem linearIsometryBoundedOfCompact_of_compact_toEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv.toEquiv = equivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_of_compact_to_equiv ContinuousMap.linearIsometryBoundedOfCompact_of_compact_toEquiv
end
section
variable {𝕜 : Type*} {γ : Type*} [NormedField 𝕜] [NormedRing γ] [NormedAlgebra 𝕜 γ]
instance : NormedAlgebra 𝕜 C(α, γ) :=
{ ContinuousMap.normedSpace, ContinuousMap.algebra with }
end
end ContinuousMap
namespace ContinuousMap
section UniformContinuity
variable {α β : Type*}
variable [MetricSpace α] [CompactSpace α] [MetricSpace β]
/-!
We now set up some declarations making it convenient to use uniform continuity.
-/
theorem uniform_continuity (f : C(α, β)) (ε : ℝ) (h : 0 < ε) :
∃ δ > 0, ∀ {x y}, dist x y < δ → dist (f x) (f y) < ε :=
Metric.uniformContinuous_iff.mp (CompactSpace.uniformContinuous_of_continuous f.continuous) ε h
#align continuous_map.uniform_continuity ContinuousMap.uniform_continuity
-- This definition allows us to separate the choice of some `δ`,
-- and the corresponding use of `dist a b < δ → dist (f a) (f b) < ε`,
-- even across different declarations.
/-- An arbitrarily chosen modulus of uniform continuity for a given function `f` and `ε > 0`. -/
def modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) : ℝ :=
Classical.choose (uniform_continuity f ε h)
#align continuous_map.modulus ContinuousMap.modulus
theorem modulus_pos (f : C(α, β)) {ε : ℝ} {h : 0 < ε} : 0 < f.modulus ε h :=
(Classical.choose_spec (uniform_continuity f ε h)).1
#align continuous_map.modulus_pos ContinuousMap.modulus_pos
theorem dist_lt_of_dist_lt_modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) {a b : α}
(w : dist a b < f.modulus ε h) : dist (f a) (f b) < ε :=
(Classical.choose_spec (uniform_continuity f ε h)).2 w
#align continuous_map.dist_lt_of_dist_lt_modulus ContinuousMap.dist_lt_of_dist_lt_modulus
end UniformContinuity
end ContinuousMap
section CompLeft
variable (X : Type*) {𝕜 β γ : Type*} [TopologicalSpace X] [CompactSpace X]
[NontriviallyNormedField 𝕜]
variable [NormedAddCommGroup β] [NormedSpace 𝕜 β] [NormedAddCommGroup γ] [NormedSpace 𝕜 γ]
open ContinuousMap
/-- Postcomposition of continuous functions into a normed module by a continuous linear map is a
continuous linear map.
Transferred version of `ContinuousLinearMap.compLeftContinuousBounded`,
upgraded version of `ContinuousLinearMap.compLeftContinuous`,
similar to `LinearMap.compLeft`. -/
protected def ContinuousLinearMap.compLeftContinuousCompact (g : β →L[𝕜] γ) :
C(X, β) →L[𝕜] C(X, γ) :=
(linearIsometryBoundedOfCompact X γ 𝕜).symm.toLinearIsometry.toContinuousLinearMap.comp <|
(g.compLeftContinuousBounded X).comp <|
(linearIsometryBoundedOfCompact X β 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_linear_map.comp_left_continuous_compact ContinuousLinearMap.compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.toLinear_compLeftContinuousCompact (g : β →L[𝕜] γ) :
(g.compLeftContinuousCompact X : C(X, β) →ₗ[𝕜] C(X, γ)) = g.compLeftContinuous 𝕜 X := by
ext f
rfl
#align continuous_linear_map.to_linear_comp_left_continuous_compact ContinuousLinearMap.toLinear_compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.compLeftContinuousCompact_apply (g : β →L[𝕜] γ) (f : C(X, β)) (x : X) :
g.compLeftContinuousCompact X f x = g (f x) :=
rfl
#align continuous_linear_map.comp_left_continuous_compact_apply ContinuousLinearMap.compLeftContinuousCompact_apply
end CompLeft
namespace ContinuousMap
/-!
We now setup variations on `compRight* f`, where `f : C(X, Y)`
(that is, precomposition by a continuous map),
as a morphism `C(Y, T) → C(X, T)`, respecting various types of structure.
In particular:
* `compRightContinuousMap`, the bundled continuous map (for this we need `X Y` compact).
* `compRightHomeomorph`, when we precompose by a homeomorphism.
* `compRightAlgHom`, when `T = R` is a topological ring.
-/
section CompRight
/-- Precomposition by a continuous map is itself a continuous map between spaces of continuous maps.
-/
def compRightContinuousMap {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y)) : C(C(Y, T), C(X, T)) where
toFun g := g.comp f
continuous_toFun := by
refine' Metric.continuous_iff.mpr _
intro g ε ε_pos
|
refine' ⟨ε, ε_pos, fun g' h => _⟩
|
/-- Precomposition by a continuous map is itself a continuous map between spaces of continuous maps.
-/
def compRightContinuousMap {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y)) : C(C(Y, T), C(X, T)) where
toFun g := g.comp f
continuous_toFun := by
refine' Metric.continuous_iff.mpr _
intro g ε ε_pos
|
Mathlib.Topology.ContinuousFunction.Compact.437_0.Mig2jTVnn2FLKEB
|
/-- Precomposition by a continuous map is itself a continuous map between spaces of continuous maps.
-/
def compRightContinuousMap {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y)) : C(C(Y, T), C(X, T)) where
toFun g
|
Mathlib_Topology_ContinuousFunction_Compact
|
X : Type u_1
Y : Type u_2
T : Type u_3
inst✝⁴ : TopologicalSpace X
inst✝³ : CompactSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : CompactSpace Y
inst✝ : MetricSpace T
f : C(X, Y)
g : C(Y, T)
ε : ℝ
ε_pos : ε > 0
g' : C(Y, T)
h : dist g' g < ε
⊢ dist (comp g' f) (comp g f) < ε
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compacts
#align_import topology.continuous_function.compact from "leanprover-community/mathlib"@"d3af0609f6db8691dffdc3e1fb7feb7da72698f2"
/-!
# Continuous functions on a compact space
Continuous functions `C(α, β)` from a compact space `α` to a metric space `β`
are automatically bounded, and so acquire various structures inherited from `α →ᵇ β`.
This file transfers these structures, and restates some lemmas
characterising these structures.
If you need a lemma which is proved about `α →ᵇ β` but not for `C(α, β)` when `α` is compact,
you should restate it here. You can also use
`ContinuousMap.equivBoundedOfCompact` to move functions back and forth.
-/
noncomputable section
open Topology Classical NNReal BoundedContinuousFunction BigOperators
open Set Filter Metric
open BoundedContinuousFunction
namespace ContinuousMap
variable {α β E : Type*} [TopologicalSpace α] [CompactSpace α] [MetricSpace β]
[NormedAddCommGroup E]
section
variable (α β)
/-- When `α` is compact, the bounded continuous maps `α →ᵇ β` are
equivalent to `C(α, β)`.
-/
@[simps (config := .asFn)]
def equivBoundedOfCompact : C(α, β) ≃ (α →ᵇ β) :=
⟨mkOfCompact, BoundedContinuousFunction.toContinuousMap, fun f => by
ext
rfl, fun f => by
ext
rfl⟩
#align continuous_map.equiv_bounded_of_compact ContinuousMap.equivBoundedOfCompact
theorem uniformInducing_equivBoundedOfCompact : UniformInducing (equivBoundedOfCompact α β) :=
UniformInducing.mk'
(by
simp only [hasBasis_compactConvergenceUniformity.mem_iff, uniformity_basis_dist_le.mem_iff]
exact fun s =>
⟨fun ⟨⟨a, b⟩, ⟨_, ⟨ε, hε, hb⟩⟩, hs⟩ =>
⟨{ p | ∀ x, (p.1 x, p.2 x) ∈ b }, ⟨ε, hε, fun _ h x => hb ((dist_le hε.le).mp h x)⟩,
fun f g h => hs fun x _ => h x⟩,
fun ⟨_, ⟨ε, hε, ht⟩, hs⟩ =>
⟨⟨Set.univ, { p | dist p.1 p.2 ≤ ε }⟩, ⟨isCompact_univ, ⟨ε, hε, fun _ h => h⟩⟩,
fun ⟨f, g⟩ h => hs _ _ (ht ((dist_le hε.le).mpr fun x => h x (mem_univ x)))⟩⟩)
#align continuous_map.uniform_inducing_equiv_bounded_of_compact ContinuousMap.uniformInducing_equivBoundedOfCompact
theorem uniformEmbedding_equivBoundedOfCompact : UniformEmbedding (equivBoundedOfCompact α β) :=
{ uniformInducing_equivBoundedOfCompact α β with inj := (equivBoundedOfCompact α β).injective }
#align continuous_map.uniform_embedding_equiv_bounded_of_compact ContinuousMap.uniformEmbedding_equivBoundedOfCompact
/-- When `α` is compact, the bounded continuous maps `α →ᵇ 𝕜` are
additively equivalent to `C(α, 𝕜)`.
-/
-- porting note: the following `simps` received a "maximum recursion depth" error
-- @[simps! (config := .asFn) apply symm_apply]
def addEquivBoundedOfCompact [AddMonoid β] [LipschitzAdd β] : C(α, β) ≃+ (α →ᵇ β) :=
({ toContinuousMapAddHom α β, (equivBoundedOfCompact α β).symm with } : (α →ᵇ β) ≃+ C(α, β)).symm
#align continuous_map.add_equiv_bounded_of_compact ContinuousMap.addEquivBoundedOfCompact
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_symm_apply [AddMonoid β] [LipschitzAdd β] :
⇑((addEquivBoundedOfCompact α β).symm) = toContinuousMapAddHom α β :=
rfl
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_apply [AddMonoid β] [LipschitzAdd β] :
⇑(addEquivBoundedOfCompact α β) = mkOfCompact :=
rfl
instance metricSpace : MetricSpace C(α, β) :=
(uniformEmbedding_equivBoundedOfCompact α β).comapMetricSpace _
#align continuous_map.metric_space ContinuousMap.metricSpace
/-- When `α` is compact, and `β` is a metric space, the bounded continuous maps `α →ᵇ β` are
isometric to `C(α, β)`.
-/
@[simps! (config := .asFn) toEquiv apply symm_apply]
def isometryEquivBoundedOfCompact : C(α, β) ≃ᵢ (α →ᵇ β) where
isometry_toFun _ _ := rfl
toEquiv := equivBoundedOfCompact α β
#align continuous_map.isometry_equiv_bounded_of_compact ContinuousMap.isometryEquivBoundedOfCompact
end
@[simp]
theorem _root_.BoundedContinuousFunction.dist_mkOfCompact (f g : C(α, β)) :
dist (mkOfCompact f) (mkOfCompact g) = dist f g :=
rfl
#align bounded_continuous_function.dist_mk_of_compact BoundedContinuousFunction.dist_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.dist_toContinuousMap (f g : α →ᵇ β) :
dist f.toContinuousMap g.toContinuousMap = dist f g :=
rfl
#align bounded_continuous_function.dist_to_continuous_map BoundedContinuousFunction.dist_toContinuousMap
open BoundedContinuousFunction
section
variable {f g : C(α, β)} {C : ℝ}
/-- The pointwise distance is controlled by the distance between functions, by definition. -/
theorem dist_apply_le_dist (x : α) : dist (f x) (g x) ≤ dist f g := by
simp only [← dist_mkOfCompact, dist_coe_le_dist, ← mkOfCompact_apply]
#align continuous_map.dist_apply_le_dist ContinuousMap.dist_apply_le_dist
/-- The distance between two functions is controlled by the supremum of the pointwise distances. -/
theorem dist_le (C0 : (0 : ℝ) ≤ C) : dist f g ≤ C ↔ ∀ x : α, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le C0, mkOfCompact_apply]
#align continuous_map.dist_le ContinuousMap.dist_le
theorem dist_le_iff_of_nonempty [Nonempty α] : dist f g ≤ C ↔ ∀ x, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le_iff_of_nonempty,
mkOfCompact_apply]
#align continuous_map.dist_le_iff_of_nonempty ContinuousMap.dist_le_iff_of_nonempty
theorem dist_lt_iff_of_nonempty [Nonempty α] : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
simp only [← dist_mkOfCompact, dist_lt_iff_of_nonempty_compact, mkOfCompact_apply]
#align continuous_map.dist_lt_iff_of_nonempty ContinuousMap.dist_lt_iff_of_nonempty
theorem dist_lt_of_nonempty [Nonempty α] (w : ∀ x : α, dist (f x) (g x) < C) : dist f g < C :=
dist_lt_iff_of_nonempty.2 w
#align continuous_map.dist_lt_of_nonempty ContinuousMap.dist_lt_of_nonempty
theorem dist_lt_iff (C0 : (0 : ℝ) < C) : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
rw [← dist_mkOfCompact, dist_lt_iff_of_compact C0]
simp only [mkOfCompact_apply]
#align continuous_map.dist_lt_iff ContinuousMap.dist_lt_iff
end
instance [CompleteSpace β] : CompleteSpace C(α, β) :=
(isometryEquivBoundedOfCompact α β).completeSpace
/-- See also `ContinuousMap.continuous_eval'`. -/
@[continuity]
theorem continuous_eval : Continuous fun p : C(α, β) × α => p.1 p.2 :=
continuous_eval.comp ((isometryEquivBoundedOfCompact α β).continuous.prod_map continuous_id)
#align continuous_map.continuous_eval ContinuousMap.continuous_eval
-- TODO at some point we will need lemmas characterising this norm!
-- At the moment the only way to reason about it is to transfer `f : C(α,E)` back to `α →ᵇ E`.
instance : Norm C(α, E) where norm x := dist x 0
@[simp]
theorem _root_.BoundedContinuousFunction.norm_mkOfCompact (f : C(α, E)) : ‖mkOfCompact f‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_mk_of_compact BoundedContinuousFunction.norm_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.norm_toContinuousMap_eq (f : α →ᵇ E) :
‖f.toContinuousMap‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_to_continuous_map_eq BoundedContinuousFunction.norm_toContinuousMap_eq
open BoundedContinuousFunction
instance : NormedAddCommGroup C(α, E) :=
{ ContinuousMap.metricSpace _ _,
ContinuousMap.instAddCommGroupContinuousMap with
dist_eq := fun x y => by
rw [← norm_mkOfCompact, ← dist_mkOfCompact, dist_eq_norm, mkOfCompact_sub]
dist := dist
norm := norm }
instance [Nonempty α] [One E] [NormOneClass E] : NormOneClass C(α, E) where
norm_one := by simp only [← norm_mkOfCompact, mkOfCompact_one, norm_one]
section
variable (f : C(α, E))
-- The corresponding lemmas for `BoundedContinuousFunction` are stated with `{f}`,
-- and so can not be used in dot notation.
theorem norm_coe_le_norm (x : α) : ‖f x‖ ≤ ‖f‖ :=
(mkOfCompact f).norm_coe_le_norm x
#align continuous_map.norm_coe_le_norm ContinuousMap.norm_coe_le_norm
/-- Distance between the images of any two points is at most twice the norm of the function. -/
theorem dist_le_two_norm (x y : α) : dist (f x) (f y) ≤ 2 * ‖f‖ :=
(mkOfCompact f).dist_le_two_norm x y
#align continuous_map.dist_le_two_norm ContinuousMap.dist_le_two_norm
/-- The norm of a function is controlled by the supremum of the pointwise norms. -/
theorem norm_le {C : ℝ} (C0 : (0 : ℝ) ≤ C) : ‖f‖ ≤ C ↔ ∀ x : α, ‖f x‖ ≤ C :=
@BoundedContinuousFunction.norm_le _ _ _ _ (mkOfCompact f) _ C0
#align continuous_map.norm_le ContinuousMap.norm_le
theorem norm_le_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ ≤ M ↔ ∀ x, ‖f x‖ ≤ M :=
@BoundedContinuousFunction.norm_le_of_nonempty _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_le_of_nonempty ContinuousMap.norm_le_of_nonempty
theorem norm_lt_iff {M : ℝ} (M0 : 0 < M) : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_compact _ _ _ _ _ (mkOfCompact f) _ M0
#align continuous_map.norm_lt_iff ContinuousMap.norm_lt_iff
theorem nnnorm_lt_iff {M : ℝ≥0} (M0 : 0 < M) : ‖f‖₊ < M ↔ ∀ x : α, ‖f x‖₊ < M :=
f.norm_lt_iff M0
#align continuous_map.nnnorm_lt_iff ContinuousMap.nnnorm_lt_iff
theorem norm_lt_iff_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_nonempty_compact _ _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_lt_iff_of_nonempty ContinuousMap.norm_lt_iff_of_nonempty
theorem nnnorm_lt_iff_of_nonempty [Nonempty α] {M : ℝ≥0} : ‖f‖₊ < M ↔ ∀ x, ‖f x‖₊ < M :=
f.norm_lt_iff_of_nonempty
#align continuous_map.nnnorm_lt_iff_of_nonempty ContinuousMap.nnnorm_lt_iff_of_nonempty
theorem apply_le_norm (f : C(α, ℝ)) (x : α) : f x ≤ ‖f‖ :=
le_trans (le_abs.mpr (Or.inl (le_refl (f x)))) (f.norm_coe_le_norm x)
#align continuous_map.apply_le_norm ContinuousMap.apply_le_norm
theorem neg_norm_le_apply (f : C(α, ℝ)) (x : α) : -‖f‖ ≤ f x :=
le_trans (neg_le_neg (f.norm_coe_le_norm x)) (neg_le.mp (neg_le_abs_self (f x)))
#align continuous_map.neg_norm_le_apply ContinuousMap.neg_norm_le_apply
theorem norm_eq_iSup_norm : ‖f‖ = ⨆ x : α, ‖f x‖ :=
(mkOfCompact f).norm_eq_iSup_norm
#align continuous_map.norm_eq_supr_norm ContinuousMap.norm_eq_iSup_norm
theorem norm_restrict_mono_set {X : Type*} [TopologicalSpace X] (f : C(X, E))
{K L : TopologicalSpace.Compacts X} (hKL : K ≤ L) : ‖f.restrict K‖ ≤ ‖f.restrict L‖ :=
(norm_le _ (norm_nonneg _)).mpr fun x => norm_coe_le_norm (f.restrict L) <| Set.inclusion hKL x
#align continuous_map.norm_restrict_mono_set ContinuousMap.norm_restrict_mono_set
end
section
variable {R : Type*} [NormedRing R]
instance : NormedRing C(α, R) :=
{ (inferInstance : NormedAddCommGroup C(α, R)), ContinuousMap.instRingContinuousMap with
norm_mul := fun f g => norm_mul_le (mkOfCompact f) (mkOfCompact g) }
end
section
variable {𝕜 : Type*} [NormedField 𝕜] [NormedSpace 𝕜 E]
instance normedSpace : NormedSpace 𝕜 C(α, E) where
norm_smul_le c f := (norm_smul_le c (mkOfCompact f) : _)
#align continuous_map.normed_space ContinuousMap.normedSpace
section
variable (α 𝕜 E)
/-- When `α` is compact and `𝕜` is a normed field,
the `𝕜`-algebra of bounded continuous maps `α →ᵇ β` is
`𝕜`-linearly isometric to `C(α, β)`.
-/
def linearIsometryBoundedOfCompact : C(α, E) ≃ₗᵢ[𝕜] α →ᵇ E :=
{ addEquivBoundedOfCompact α E with
map_smul' := fun c f => by
ext
norm_cast
norm_map' := fun f => rfl }
#align continuous_map.linear_isometry_bounded_of_compact ContinuousMap.linearIsometryBoundedOfCompact
variable {α E}
-- to match `BoundedContinuousFunction.evalClm`
/-- The evaluation at a point, as a continuous linear map from `C(α, 𝕜)` to `𝕜`. -/
def evalClm (x : α) : C(α, E) →L[𝕜] E :=
(BoundedContinuousFunction.evalClm 𝕜 x).comp
(linearIsometryBoundedOfCompact α E 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_map.eval_clm ContinuousMap.evalClm
end
-- this lemma and the next are the analogues of those autogenerated by `@[simps]` for
-- `equivBoundedOfCompact`, `addEquivBoundedOfCompact`
@[simp]
theorem linearIsometryBoundedOfCompact_symm_apply (f : α →ᵇ E) :
(linearIsometryBoundedOfCompact α E 𝕜).symm f = f.toContinuousMap :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_symm_apply ContinuousMap.linearIsometryBoundedOfCompact_symm_apply
@[simp]
theorem linearIsometryBoundedOfCompact_apply_apply (f : C(α, E)) (a : α) :
(linearIsometryBoundedOfCompact α E 𝕜 f) a = f a :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_apply_apply ContinuousMap.linearIsometryBoundedOfCompact_apply_apply
@[simp]
theorem linearIsometryBoundedOfCompact_toIsometryEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toIsometryEquiv = isometryEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_isometry_equiv ContinuousMap.linearIsometryBoundedOfCompact_toIsometryEquiv
@[simp] -- porting note: adjusted LHS because `simpNF` complained it simplified.
theorem linearIsometryBoundedOfCompact_toAddEquiv :
((linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv : C(α, E) ≃+ (α →ᵇ E)) =
addEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_add_equiv ContinuousMap.linearIsometryBoundedOfCompact_toAddEquiv
@[simp]
theorem linearIsometryBoundedOfCompact_of_compact_toEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv.toEquiv = equivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_of_compact_to_equiv ContinuousMap.linearIsometryBoundedOfCompact_of_compact_toEquiv
end
section
variable {𝕜 : Type*} {γ : Type*} [NormedField 𝕜] [NormedRing γ] [NormedAlgebra 𝕜 γ]
instance : NormedAlgebra 𝕜 C(α, γ) :=
{ ContinuousMap.normedSpace, ContinuousMap.algebra with }
end
end ContinuousMap
namespace ContinuousMap
section UniformContinuity
variable {α β : Type*}
variable [MetricSpace α] [CompactSpace α] [MetricSpace β]
/-!
We now set up some declarations making it convenient to use uniform continuity.
-/
theorem uniform_continuity (f : C(α, β)) (ε : ℝ) (h : 0 < ε) :
∃ δ > 0, ∀ {x y}, dist x y < δ → dist (f x) (f y) < ε :=
Metric.uniformContinuous_iff.mp (CompactSpace.uniformContinuous_of_continuous f.continuous) ε h
#align continuous_map.uniform_continuity ContinuousMap.uniform_continuity
-- This definition allows us to separate the choice of some `δ`,
-- and the corresponding use of `dist a b < δ → dist (f a) (f b) < ε`,
-- even across different declarations.
/-- An arbitrarily chosen modulus of uniform continuity for a given function `f` and `ε > 0`. -/
def modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) : ℝ :=
Classical.choose (uniform_continuity f ε h)
#align continuous_map.modulus ContinuousMap.modulus
theorem modulus_pos (f : C(α, β)) {ε : ℝ} {h : 0 < ε} : 0 < f.modulus ε h :=
(Classical.choose_spec (uniform_continuity f ε h)).1
#align continuous_map.modulus_pos ContinuousMap.modulus_pos
theorem dist_lt_of_dist_lt_modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) {a b : α}
(w : dist a b < f.modulus ε h) : dist (f a) (f b) < ε :=
(Classical.choose_spec (uniform_continuity f ε h)).2 w
#align continuous_map.dist_lt_of_dist_lt_modulus ContinuousMap.dist_lt_of_dist_lt_modulus
end UniformContinuity
end ContinuousMap
section CompLeft
variable (X : Type*) {𝕜 β γ : Type*} [TopologicalSpace X] [CompactSpace X]
[NontriviallyNormedField 𝕜]
variable [NormedAddCommGroup β] [NormedSpace 𝕜 β] [NormedAddCommGroup γ] [NormedSpace 𝕜 γ]
open ContinuousMap
/-- Postcomposition of continuous functions into a normed module by a continuous linear map is a
continuous linear map.
Transferred version of `ContinuousLinearMap.compLeftContinuousBounded`,
upgraded version of `ContinuousLinearMap.compLeftContinuous`,
similar to `LinearMap.compLeft`. -/
protected def ContinuousLinearMap.compLeftContinuousCompact (g : β →L[𝕜] γ) :
C(X, β) →L[𝕜] C(X, γ) :=
(linearIsometryBoundedOfCompact X γ 𝕜).symm.toLinearIsometry.toContinuousLinearMap.comp <|
(g.compLeftContinuousBounded X).comp <|
(linearIsometryBoundedOfCompact X β 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_linear_map.comp_left_continuous_compact ContinuousLinearMap.compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.toLinear_compLeftContinuousCompact (g : β →L[𝕜] γ) :
(g.compLeftContinuousCompact X : C(X, β) →ₗ[𝕜] C(X, γ)) = g.compLeftContinuous 𝕜 X := by
ext f
rfl
#align continuous_linear_map.to_linear_comp_left_continuous_compact ContinuousLinearMap.toLinear_compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.compLeftContinuousCompact_apply (g : β →L[𝕜] γ) (f : C(X, β)) (x : X) :
g.compLeftContinuousCompact X f x = g (f x) :=
rfl
#align continuous_linear_map.comp_left_continuous_compact_apply ContinuousLinearMap.compLeftContinuousCompact_apply
end CompLeft
namespace ContinuousMap
/-!
We now setup variations on `compRight* f`, where `f : C(X, Y)`
(that is, precomposition by a continuous map),
as a morphism `C(Y, T) → C(X, T)`, respecting various types of structure.
In particular:
* `compRightContinuousMap`, the bundled continuous map (for this we need `X Y` compact).
* `compRightHomeomorph`, when we precompose by a homeomorphism.
* `compRightAlgHom`, when `T = R` is a topological ring.
-/
section CompRight
/-- Precomposition by a continuous map is itself a continuous map between spaces of continuous maps.
-/
def compRightContinuousMap {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y)) : C(C(Y, T), C(X, T)) where
toFun g := g.comp f
continuous_toFun := by
refine' Metric.continuous_iff.mpr _
intro g ε ε_pos
refine' ⟨ε, ε_pos, fun g' h => _⟩
|
rw [ContinuousMap.dist_lt_iff ε_pos] at h ⊢
|
/-- Precomposition by a continuous map is itself a continuous map between spaces of continuous maps.
-/
def compRightContinuousMap {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y)) : C(C(Y, T), C(X, T)) where
toFun g := g.comp f
continuous_toFun := by
refine' Metric.continuous_iff.mpr _
intro g ε ε_pos
refine' ⟨ε, ε_pos, fun g' h => _⟩
|
Mathlib.Topology.ContinuousFunction.Compact.437_0.Mig2jTVnn2FLKEB
|
/-- Precomposition by a continuous map is itself a continuous map between spaces of continuous maps.
-/
def compRightContinuousMap {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y)) : C(C(Y, T), C(X, T)) where
toFun g
|
Mathlib_Topology_ContinuousFunction_Compact
|
X : Type u_1
Y : Type u_2
T : Type u_3
inst✝⁴ : TopologicalSpace X
inst✝³ : CompactSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : CompactSpace Y
inst✝ : MetricSpace T
f : C(X, Y)
g : C(Y, T)
ε : ℝ
ε_pos : ε > 0
g' : C(Y, T)
h : ∀ (x : Y), dist (g' x) (g x) < ε
⊢ ∀ (x : X), dist ((comp g' f) x) ((comp g f) x) < ε
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compacts
#align_import topology.continuous_function.compact from "leanprover-community/mathlib"@"d3af0609f6db8691dffdc3e1fb7feb7da72698f2"
/-!
# Continuous functions on a compact space
Continuous functions `C(α, β)` from a compact space `α` to a metric space `β`
are automatically bounded, and so acquire various structures inherited from `α →ᵇ β`.
This file transfers these structures, and restates some lemmas
characterising these structures.
If you need a lemma which is proved about `α →ᵇ β` but not for `C(α, β)` when `α` is compact,
you should restate it here. You can also use
`ContinuousMap.equivBoundedOfCompact` to move functions back and forth.
-/
noncomputable section
open Topology Classical NNReal BoundedContinuousFunction BigOperators
open Set Filter Metric
open BoundedContinuousFunction
namespace ContinuousMap
variable {α β E : Type*} [TopologicalSpace α] [CompactSpace α] [MetricSpace β]
[NormedAddCommGroup E]
section
variable (α β)
/-- When `α` is compact, the bounded continuous maps `α →ᵇ β` are
equivalent to `C(α, β)`.
-/
@[simps (config := .asFn)]
def equivBoundedOfCompact : C(α, β) ≃ (α →ᵇ β) :=
⟨mkOfCompact, BoundedContinuousFunction.toContinuousMap, fun f => by
ext
rfl, fun f => by
ext
rfl⟩
#align continuous_map.equiv_bounded_of_compact ContinuousMap.equivBoundedOfCompact
theorem uniformInducing_equivBoundedOfCompact : UniformInducing (equivBoundedOfCompact α β) :=
UniformInducing.mk'
(by
simp only [hasBasis_compactConvergenceUniformity.mem_iff, uniformity_basis_dist_le.mem_iff]
exact fun s =>
⟨fun ⟨⟨a, b⟩, ⟨_, ⟨ε, hε, hb⟩⟩, hs⟩ =>
⟨{ p | ∀ x, (p.1 x, p.2 x) ∈ b }, ⟨ε, hε, fun _ h x => hb ((dist_le hε.le).mp h x)⟩,
fun f g h => hs fun x _ => h x⟩,
fun ⟨_, ⟨ε, hε, ht⟩, hs⟩ =>
⟨⟨Set.univ, { p | dist p.1 p.2 ≤ ε }⟩, ⟨isCompact_univ, ⟨ε, hε, fun _ h => h⟩⟩,
fun ⟨f, g⟩ h => hs _ _ (ht ((dist_le hε.le).mpr fun x => h x (mem_univ x)))⟩⟩)
#align continuous_map.uniform_inducing_equiv_bounded_of_compact ContinuousMap.uniformInducing_equivBoundedOfCompact
theorem uniformEmbedding_equivBoundedOfCompact : UniformEmbedding (equivBoundedOfCompact α β) :=
{ uniformInducing_equivBoundedOfCompact α β with inj := (equivBoundedOfCompact α β).injective }
#align continuous_map.uniform_embedding_equiv_bounded_of_compact ContinuousMap.uniformEmbedding_equivBoundedOfCompact
/-- When `α` is compact, the bounded continuous maps `α →ᵇ 𝕜` are
additively equivalent to `C(α, 𝕜)`.
-/
-- porting note: the following `simps` received a "maximum recursion depth" error
-- @[simps! (config := .asFn) apply symm_apply]
def addEquivBoundedOfCompact [AddMonoid β] [LipschitzAdd β] : C(α, β) ≃+ (α →ᵇ β) :=
({ toContinuousMapAddHom α β, (equivBoundedOfCompact α β).symm with } : (α →ᵇ β) ≃+ C(α, β)).symm
#align continuous_map.add_equiv_bounded_of_compact ContinuousMap.addEquivBoundedOfCompact
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_symm_apply [AddMonoid β] [LipschitzAdd β] :
⇑((addEquivBoundedOfCompact α β).symm) = toContinuousMapAddHom α β :=
rfl
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_apply [AddMonoid β] [LipschitzAdd β] :
⇑(addEquivBoundedOfCompact α β) = mkOfCompact :=
rfl
instance metricSpace : MetricSpace C(α, β) :=
(uniformEmbedding_equivBoundedOfCompact α β).comapMetricSpace _
#align continuous_map.metric_space ContinuousMap.metricSpace
/-- When `α` is compact, and `β` is a metric space, the bounded continuous maps `α →ᵇ β` are
isometric to `C(α, β)`.
-/
@[simps! (config := .asFn) toEquiv apply symm_apply]
def isometryEquivBoundedOfCompact : C(α, β) ≃ᵢ (α →ᵇ β) where
isometry_toFun _ _ := rfl
toEquiv := equivBoundedOfCompact α β
#align continuous_map.isometry_equiv_bounded_of_compact ContinuousMap.isometryEquivBoundedOfCompact
end
@[simp]
theorem _root_.BoundedContinuousFunction.dist_mkOfCompact (f g : C(α, β)) :
dist (mkOfCompact f) (mkOfCompact g) = dist f g :=
rfl
#align bounded_continuous_function.dist_mk_of_compact BoundedContinuousFunction.dist_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.dist_toContinuousMap (f g : α →ᵇ β) :
dist f.toContinuousMap g.toContinuousMap = dist f g :=
rfl
#align bounded_continuous_function.dist_to_continuous_map BoundedContinuousFunction.dist_toContinuousMap
open BoundedContinuousFunction
section
variable {f g : C(α, β)} {C : ℝ}
/-- The pointwise distance is controlled by the distance between functions, by definition. -/
theorem dist_apply_le_dist (x : α) : dist (f x) (g x) ≤ dist f g := by
simp only [← dist_mkOfCompact, dist_coe_le_dist, ← mkOfCompact_apply]
#align continuous_map.dist_apply_le_dist ContinuousMap.dist_apply_le_dist
/-- The distance between two functions is controlled by the supremum of the pointwise distances. -/
theorem dist_le (C0 : (0 : ℝ) ≤ C) : dist f g ≤ C ↔ ∀ x : α, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le C0, mkOfCompact_apply]
#align continuous_map.dist_le ContinuousMap.dist_le
theorem dist_le_iff_of_nonempty [Nonempty α] : dist f g ≤ C ↔ ∀ x, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le_iff_of_nonempty,
mkOfCompact_apply]
#align continuous_map.dist_le_iff_of_nonempty ContinuousMap.dist_le_iff_of_nonempty
theorem dist_lt_iff_of_nonempty [Nonempty α] : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
simp only [← dist_mkOfCompact, dist_lt_iff_of_nonempty_compact, mkOfCompact_apply]
#align continuous_map.dist_lt_iff_of_nonempty ContinuousMap.dist_lt_iff_of_nonempty
theorem dist_lt_of_nonempty [Nonempty α] (w : ∀ x : α, dist (f x) (g x) < C) : dist f g < C :=
dist_lt_iff_of_nonempty.2 w
#align continuous_map.dist_lt_of_nonempty ContinuousMap.dist_lt_of_nonempty
theorem dist_lt_iff (C0 : (0 : ℝ) < C) : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
rw [← dist_mkOfCompact, dist_lt_iff_of_compact C0]
simp only [mkOfCompact_apply]
#align continuous_map.dist_lt_iff ContinuousMap.dist_lt_iff
end
instance [CompleteSpace β] : CompleteSpace C(α, β) :=
(isometryEquivBoundedOfCompact α β).completeSpace
/-- See also `ContinuousMap.continuous_eval'`. -/
@[continuity]
theorem continuous_eval : Continuous fun p : C(α, β) × α => p.1 p.2 :=
continuous_eval.comp ((isometryEquivBoundedOfCompact α β).continuous.prod_map continuous_id)
#align continuous_map.continuous_eval ContinuousMap.continuous_eval
-- TODO at some point we will need lemmas characterising this norm!
-- At the moment the only way to reason about it is to transfer `f : C(α,E)` back to `α →ᵇ E`.
instance : Norm C(α, E) where norm x := dist x 0
@[simp]
theorem _root_.BoundedContinuousFunction.norm_mkOfCompact (f : C(α, E)) : ‖mkOfCompact f‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_mk_of_compact BoundedContinuousFunction.norm_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.norm_toContinuousMap_eq (f : α →ᵇ E) :
‖f.toContinuousMap‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_to_continuous_map_eq BoundedContinuousFunction.norm_toContinuousMap_eq
open BoundedContinuousFunction
instance : NormedAddCommGroup C(α, E) :=
{ ContinuousMap.metricSpace _ _,
ContinuousMap.instAddCommGroupContinuousMap with
dist_eq := fun x y => by
rw [← norm_mkOfCompact, ← dist_mkOfCompact, dist_eq_norm, mkOfCompact_sub]
dist := dist
norm := norm }
instance [Nonempty α] [One E] [NormOneClass E] : NormOneClass C(α, E) where
norm_one := by simp only [← norm_mkOfCompact, mkOfCompact_one, norm_one]
section
variable (f : C(α, E))
-- The corresponding lemmas for `BoundedContinuousFunction` are stated with `{f}`,
-- and so can not be used in dot notation.
theorem norm_coe_le_norm (x : α) : ‖f x‖ ≤ ‖f‖ :=
(mkOfCompact f).norm_coe_le_norm x
#align continuous_map.norm_coe_le_norm ContinuousMap.norm_coe_le_norm
/-- Distance between the images of any two points is at most twice the norm of the function. -/
theorem dist_le_two_norm (x y : α) : dist (f x) (f y) ≤ 2 * ‖f‖ :=
(mkOfCompact f).dist_le_two_norm x y
#align continuous_map.dist_le_two_norm ContinuousMap.dist_le_two_norm
/-- The norm of a function is controlled by the supremum of the pointwise norms. -/
theorem norm_le {C : ℝ} (C0 : (0 : ℝ) ≤ C) : ‖f‖ ≤ C ↔ ∀ x : α, ‖f x‖ ≤ C :=
@BoundedContinuousFunction.norm_le _ _ _ _ (mkOfCompact f) _ C0
#align continuous_map.norm_le ContinuousMap.norm_le
theorem norm_le_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ ≤ M ↔ ∀ x, ‖f x‖ ≤ M :=
@BoundedContinuousFunction.norm_le_of_nonempty _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_le_of_nonempty ContinuousMap.norm_le_of_nonempty
theorem norm_lt_iff {M : ℝ} (M0 : 0 < M) : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_compact _ _ _ _ _ (mkOfCompact f) _ M0
#align continuous_map.norm_lt_iff ContinuousMap.norm_lt_iff
theorem nnnorm_lt_iff {M : ℝ≥0} (M0 : 0 < M) : ‖f‖₊ < M ↔ ∀ x : α, ‖f x‖₊ < M :=
f.norm_lt_iff M0
#align continuous_map.nnnorm_lt_iff ContinuousMap.nnnorm_lt_iff
theorem norm_lt_iff_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_nonempty_compact _ _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_lt_iff_of_nonempty ContinuousMap.norm_lt_iff_of_nonempty
theorem nnnorm_lt_iff_of_nonempty [Nonempty α] {M : ℝ≥0} : ‖f‖₊ < M ↔ ∀ x, ‖f x‖₊ < M :=
f.norm_lt_iff_of_nonempty
#align continuous_map.nnnorm_lt_iff_of_nonempty ContinuousMap.nnnorm_lt_iff_of_nonempty
theorem apply_le_norm (f : C(α, ℝ)) (x : α) : f x ≤ ‖f‖ :=
le_trans (le_abs.mpr (Or.inl (le_refl (f x)))) (f.norm_coe_le_norm x)
#align continuous_map.apply_le_norm ContinuousMap.apply_le_norm
theorem neg_norm_le_apply (f : C(α, ℝ)) (x : α) : -‖f‖ ≤ f x :=
le_trans (neg_le_neg (f.norm_coe_le_norm x)) (neg_le.mp (neg_le_abs_self (f x)))
#align continuous_map.neg_norm_le_apply ContinuousMap.neg_norm_le_apply
theorem norm_eq_iSup_norm : ‖f‖ = ⨆ x : α, ‖f x‖ :=
(mkOfCompact f).norm_eq_iSup_norm
#align continuous_map.norm_eq_supr_norm ContinuousMap.norm_eq_iSup_norm
theorem norm_restrict_mono_set {X : Type*} [TopologicalSpace X] (f : C(X, E))
{K L : TopologicalSpace.Compacts X} (hKL : K ≤ L) : ‖f.restrict K‖ ≤ ‖f.restrict L‖ :=
(norm_le _ (norm_nonneg _)).mpr fun x => norm_coe_le_norm (f.restrict L) <| Set.inclusion hKL x
#align continuous_map.norm_restrict_mono_set ContinuousMap.norm_restrict_mono_set
end
section
variable {R : Type*} [NormedRing R]
instance : NormedRing C(α, R) :=
{ (inferInstance : NormedAddCommGroup C(α, R)), ContinuousMap.instRingContinuousMap with
norm_mul := fun f g => norm_mul_le (mkOfCompact f) (mkOfCompact g) }
end
section
variable {𝕜 : Type*} [NormedField 𝕜] [NormedSpace 𝕜 E]
instance normedSpace : NormedSpace 𝕜 C(α, E) where
norm_smul_le c f := (norm_smul_le c (mkOfCompact f) : _)
#align continuous_map.normed_space ContinuousMap.normedSpace
section
variable (α 𝕜 E)
/-- When `α` is compact and `𝕜` is a normed field,
the `𝕜`-algebra of bounded continuous maps `α →ᵇ β` is
`𝕜`-linearly isometric to `C(α, β)`.
-/
def linearIsometryBoundedOfCompact : C(α, E) ≃ₗᵢ[𝕜] α →ᵇ E :=
{ addEquivBoundedOfCompact α E with
map_smul' := fun c f => by
ext
norm_cast
norm_map' := fun f => rfl }
#align continuous_map.linear_isometry_bounded_of_compact ContinuousMap.linearIsometryBoundedOfCompact
variable {α E}
-- to match `BoundedContinuousFunction.evalClm`
/-- The evaluation at a point, as a continuous linear map from `C(α, 𝕜)` to `𝕜`. -/
def evalClm (x : α) : C(α, E) →L[𝕜] E :=
(BoundedContinuousFunction.evalClm 𝕜 x).comp
(linearIsometryBoundedOfCompact α E 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_map.eval_clm ContinuousMap.evalClm
end
-- this lemma and the next are the analogues of those autogenerated by `@[simps]` for
-- `equivBoundedOfCompact`, `addEquivBoundedOfCompact`
@[simp]
theorem linearIsometryBoundedOfCompact_symm_apply (f : α →ᵇ E) :
(linearIsometryBoundedOfCompact α E 𝕜).symm f = f.toContinuousMap :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_symm_apply ContinuousMap.linearIsometryBoundedOfCompact_symm_apply
@[simp]
theorem linearIsometryBoundedOfCompact_apply_apply (f : C(α, E)) (a : α) :
(linearIsometryBoundedOfCompact α E 𝕜 f) a = f a :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_apply_apply ContinuousMap.linearIsometryBoundedOfCompact_apply_apply
@[simp]
theorem linearIsometryBoundedOfCompact_toIsometryEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toIsometryEquiv = isometryEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_isometry_equiv ContinuousMap.linearIsometryBoundedOfCompact_toIsometryEquiv
@[simp] -- porting note: adjusted LHS because `simpNF` complained it simplified.
theorem linearIsometryBoundedOfCompact_toAddEquiv :
((linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv : C(α, E) ≃+ (α →ᵇ E)) =
addEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_add_equiv ContinuousMap.linearIsometryBoundedOfCompact_toAddEquiv
@[simp]
theorem linearIsometryBoundedOfCompact_of_compact_toEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv.toEquiv = equivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_of_compact_to_equiv ContinuousMap.linearIsometryBoundedOfCompact_of_compact_toEquiv
end
section
variable {𝕜 : Type*} {γ : Type*} [NormedField 𝕜] [NormedRing γ] [NormedAlgebra 𝕜 γ]
instance : NormedAlgebra 𝕜 C(α, γ) :=
{ ContinuousMap.normedSpace, ContinuousMap.algebra with }
end
end ContinuousMap
namespace ContinuousMap
section UniformContinuity
variable {α β : Type*}
variable [MetricSpace α] [CompactSpace α] [MetricSpace β]
/-!
We now set up some declarations making it convenient to use uniform continuity.
-/
theorem uniform_continuity (f : C(α, β)) (ε : ℝ) (h : 0 < ε) :
∃ δ > 0, ∀ {x y}, dist x y < δ → dist (f x) (f y) < ε :=
Metric.uniformContinuous_iff.mp (CompactSpace.uniformContinuous_of_continuous f.continuous) ε h
#align continuous_map.uniform_continuity ContinuousMap.uniform_continuity
-- This definition allows us to separate the choice of some `δ`,
-- and the corresponding use of `dist a b < δ → dist (f a) (f b) < ε`,
-- even across different declarations.
/-- An arbitrarily chosen modulus of uniform continuity for a given function `f` and `ε > 0`. -/
def modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) : ℝ :=
Classical.choose (uniform_continuity f ε h)
#align continuous_map.modulus ContinuousMap.modulus
theorem modulus_pos (f : C(α, β)) {ε : ℝ} {h : 0 < ε} : 0 < f.modulus ε h :=
(Classical.choose_spec (uniform_continuity f ε h)).1
#align continuous_map.modulus_pos ContinuousMap.modulus_pos
theorem dist_lt_of_dist_lt_modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) {a b : α}
(w : dist a b < f.modulus ε h) : dist (f a) (f b) < ε :=
(Classical.choose_spec (uniform_continuity f ε h)).2 w
#align continuous_map.dist_lt_of_dist_lt_modulus ContinuousMap.dist_lt_of_dist_lt_modulus
end UniformContinuity
end ContinuousMap
section CompLeft
variable (X : Type*) {𝕜 β γ : Type*} [TopologicalSpace X] [CompactSpace X]
[NontriviallyNormedField 𝕜]
variable [NormedAddCommGroup β] [NormedSpace 𝕜 β] [NormedAddCommGroup γ] [NormedSpace 𝕜 γ]
open ContinuousMap
/-- Postcomposition of continuous functions into a normed module by a continuous linear map is a
continuous linear map.
Transferred version of `ContinuousLinearMap.compLeftContinuousBounded`,
upgraded version of `ContinuousLinearMap.compLeftContinuous`,
similar to `LinearMap.compLeft`. -/
protected def ContinuousLinearMap.compLeftContinuousCompact (g : β →L[𝕜] γ) :
C(X, β) →L[𝕜] C(X, γ) :=
(linearIsometryBoundedOfCompact X γ 𝕜).symm.toLinearIsometry.toContinuousLinearMap.comp <|
(g.compLeftContinuousBounded X).comp <|
(linearIsometryBoundedOfCompact X β 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_linear_map.comp_left_continuous_compact ContinuousLinearMap.compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.toLinear_compLeftContinuousCompact (g : β →L[𝕜] γ) :
(g.compLeftContinuousCompact X : C(X, β) →ₗ[𝕜] C(X, γ)) = g.compLeftContinuous 𝕜 X := by
ext f
rfl
#align continuous_linear_map.to_linear_comp_left_continuous_compact ContinuousLinearMap.toLinear_compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.compLeftContinuousCompact_apply (g : β →L[𝕜] γ) (f : C(X, β)) (x : X) :
g.compLeftContinuousCompact X f x = g (f x) :=
rfl
#align continuous_linear_map.comp_left_continuous_compact_apply ContinuousLinearMap.compLeftContinuousCompact_apply
end CompLeft
namespace ContinuousMap
/-!
We now setup variations on `compRight* f`, where `f : C(X, Y)`
(that is, precomposition by a continuous map),
as a morphism `C(Y, T) → C(X, T)`, respecting various types of structure.
In particular:
* `compRightContinuousMap`, the bundled continuous map (for this we need `X Y` compact).
* `compRightHomeomorph`, when we precompose by a homeomorphism.
* `compRightAlgHom`, when `T = R` is a topological ring.
-/
section CompRight
/-- Precomposition by a continuous map is itself a continuous map between spaces of continuous maps.
-/
def compRightContinuousMap {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y)) : C(C(Y, T), C(X, T)) where
toFun g := g.comp f
continuous_toFun := by
refine' Metric.continuous_iff.mpr _
intro g ε ε_pos
refine' ⟨ε, ε_pos, fun g' h => _⟩
rw [ContinuousMap.dist_lt_iff ε_pos] at h ⊢
|
exact fun x => h (f x)
|
/-- Precomposition by a continuous map is itself a continuous map between spaces of continuous maps.
-/
def compRightContinuousMap {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y)) : C(C(Y, T), C(X, T)) where
toFun g := g.comp f
continuous_toFun := by
refine' Metric.continuous_iff.mpr _
intro g ε ε_pos
refine' ⟨ε, ε_pos, fun g' h => _⟩
rw [ContinuousMap.dist_lt_iff ε_pos] at h ⊢
|
Mathlib.Topology.ContinuousFunction.Compact.437_0.Mig2jTVnn2FLKEB
|
/-- Precomposition by a continuous map is itself a continuous map between spaces of continuous maps.
-/
def compRightContinuousMap {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y)) : C(C(Y, T), C(X, T)) where
toFun g
|
Mathlib_Topology_ContinuousFunction_Compact
|
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : LocallyCompactSpace X
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
ι : Type u_3
F : ι → C(X, E)
hF : ∀ (K : Compacts X), Summable fun i => ‖restrict (↑K) (F i)‖
⊢ Summable F
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compacts
#align_import topology.continuous_function.compact from "leanprover-community/mathlib"@"d3af0609f6db8691dffdc3e1fb7feb7da72698f2"
/-!
# Continuous functions on a compact space
Continuous functions `C(α, β)` from a compact space `α` to a metric space `β`
are automatically bounded, and so acquire various structures inherited from `α →ᵇ β`.
This file transfers these structures, and restates some lemmas
characterising these structures.
If you need a lemma which is proved about `α →ᵇ β` but not for `C(α, β)` when `α` is compact,
you should restate it here. You can also use
`ContinuousMap.equivBoundedOfCompact` to move functions back and forth.
-/
noncomputable section
open Topology Classical NNReal BoundedContinuousFunction BigOperators
open Set Filter Metric
open BoundedContinuousFunction
namespace ContinuousMap
variable {α β E : Type*} [TopologicalSpace α] [CompactSpace α] [MetricSpace β]
[NormedAddCommGroup E]
section
variable (α β)
/-- When `α` is compact, the bounded continuous maps `α →ᵇ β` are
equivalent to `C(α, β)`.
-/
@[simps (config := .asFn)]
def equivBoundedOfCompact : C(α, β) ≃ (α →ᵇ β) :=
⟨mkOfCompact, BoundedContinuousFunction.toContinuousMap, fun f => by
ext
rfl, fun f => by
ext
rfl⟩
#align continuous_map.equiv_bounded_of_compact ContinuousMap.equivBoundedOfCompact
theorem uniformInducing_equivBoundedOfCompact : UniformInducing (equivBoundedOfCompact α β) :=
UniformInducing.mk'
(by
simp only [hasBasis_compactConvergenceUniformity.mem_iff, uniformity_basis_dist_le.mem_iff]
exact fun s =>
⟨fun ⟨⟨a, b⟩, ⟨_, ⟨ε, hε, hb⟩⟩, hs⟩ =>
⟨{ p | ∀ x, (p.1 x, p.2 x) ∈ b }, ⟨ε, hε, fun _ h x => hb ((dist_le hε.le).mp h x)⟩,
fun f g h => hs fun x _ => h x⟩,
fun ⟨_, ⟨ε, hε, ht⟩, hs⟩ =>
⟨⟨Set.univ, { p | dist p.1 p.2 ≤ ε }⟩, ⟨isCompact_univ, ⟨ε, hε, fun _ h => h⟩⟩,
fun ⟨f, g⟩ h => hs _ _ (ht ((dist_le hε.le).mpr fun x => h x (mem_univ x)))⟩⟩)
#align continuous_map.uniform_inducing_equiv_bounded_of_compact ContinuousMap.uniformInducing_equivBoundedOfCompact
theorem uniformEmbedding_equivBoundedOfCompact : UniformEmbedding (equivBoundedOfCompact α β) :=
{ uniformInducing_equivBoundedOfCompact α β with inj := (equivBoundedOfCompact α β).injective }
#align continuous_map.uniform_embedding_equiv_bounded_of_compact ContinuousMap.uniformEmbedding_equivBoundedOfCompact
/-- When `α` is compact, the bounded continuous maps `α →ᵇ 𝕜` are
additively equivalent to `C(α, 𝕜)`.
-/
-- porting note: the following `simps` received a "maximum recursion depth" error
-- @[simps! (config := .asFn) apply symm_apply]
def addEquivBoundedOfCompact [AddMonoid β] [LipschitzAdd β] : C(α, β) ≃+ (α →ᵇ β) :=
({ toContinuousMapAddHom α β, (equivBoundedOfCompact α β).symm with } : (α →ᵇ β) ≃+ C(α, β)).symm
#align continuous_map.add_equiv_bounded_of_compact ContinuousMap.addEquivBoundedOfCompact
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_symm_apply [AddMonoid β] [LipschitzAdd β] :
⇑((addEquivBoundedOfCompact α β).symm) = toContinuousMapAddHom α β :=
rfl
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_apply [AddMonoid β] [LipschitzAdd β] :
⇑(addEquivBoundedOfCompact α β) = mkOfCompact :=
rfl
instance metricSpace : MetricSpace C(α, β) :=
(uniformEmbedding_equivBoundedOfCompact α β).comapMetricSpace _
#align continuous_map.metric_space ContinuousMap.metricSpace
/-- When `α` is compact, and `β` is a metric space, the bounded continuous maps `α →ᵇ β` are
isometric to `C(α, β)`.
-/
@[simps! (config := .asFn) toEquiv apply symm_apply]
def isometryEquivBoundedOfCompact : C(α, β) ≃ᵢ (α →ᵇ β) where
isometry_toFun _ _ := rfl
toEquiv := equivBoundedOfCompact α β
#align continuous_map.isometry_equiv_bounded_of_compact ContinuousMap.isometryEquivBoundedOfCompact
end
@[simp]
theorem _root_.BoundedContinuousFunction.dist_mkOfCompact (f g : C(α, β)) :
dist (mkOfCompact f) (mkOfCompact g) = dist f g :=
rfl
#align bounded_continuous_function.dist_mk_of_compact BoundedContinuousFunction.dist_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.dist_toContinuousMap (f g : α →ᵇ β) :
dist f.toContinuousMap g.toContinuousMap = dist f g :=
rfl
#align bounded_continuous_function.dist_to_continuous_map BoundedContinuousFunction.dist_toContinuousMap
open BoundedContinuousFunction
section
variable {f g : C(α, β)} {C : ℝ}
/-- The pointwise distance is controlled by the distance between functions, by definition. -/
theorem dist_apply_le_dist (x : α) : dist (f x) (g x) ≤ dist f g := by
simp only [← dist_mkOfCompact, dist_coe_le_dist, ← mkOfCompact_apply]
#align continuous_map.dist_apply_le_dist ContinuousMap.dist_apply_le_dist
/-- The distance between two functions is controlled by the supremum of the pointwise distances. -/
theorem dist_le (C0 : (0 : ℝ) ≤ C) : dist f g ≤ C ↔ ∀ x : α, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le C0, mkOfCompact_apply]
#align continuous_map.dist_le ContinuousMap.dist_le
theorem dist_le_iff_of_nonempty [Nonempty α] : dist f g ≤ C ↔ ∀ x, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le_iff_of_nonempty,
mkOfCompact_apply]
#align continuous_map.dist_le_iff_of_nonempty ContinuousMap.dist_le_iff_of_nonempty
theorem dist_lt_iff_of_nonempty [Nonempty α] : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
simp only [← dist_mkOfCompact, dist_lt_iff_of_nonempty_compact, mkOfCompact_apply]
#align continuous_map.dist_lt_iff_of_nonempty ContinuousMap.dist_lt_iff_of_nonempty
theorem dist_lt_of_nonempty [Nonempty α] (w : ∀ x : α, dist (f x) (g x) < C) : dist f g < C :=
dist_lt_iff_of_nonempty.2 w
#align continuous_map.dist_lt_of_nonempty ContinuousMap.dist_lt_of_nonempty
theorem dist_lt_iff (C0 : (0 : ℝ) < C) : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
rw [← dist_mkOfCompact, dist_lt_iff_of_compact C0]
simp only [mkOfCompact_apply]
#align continuous_map.dist_lt_iff ContinuousMap.dist_lt_iff
end
instance [CompleteSpace β] : CompleteSpace C(α, β) :=
(isometryEquivBoundedOfCompact α β).completeSpace
/-- See also `ContinuousMap.continuous_eval'`. -/
@[continuity]
theorem continuous_eval : Continuous fun p : C(α, β) × α => p.1 p.2 :=
continuous_eval.comp ((isometryEquivBoundedOfCompact α β).continuous.prod_map continuous_id)
#align continuous_map.continuous_eval ContinuousMap.continuous_eval
-- TODO at some point we will need lemmas characterising this norm!
-- At the moment the only way to reason about it is to transfer `f : C(α,E)` back to `α →ᵇ E`.
instance : Norm C(α, E) where norm x := dist x 0
@[simp]
theorem _root_.BoundedContinuousFunction.norm_mkOfCompact (f : C(α, E)) : ‖mkOfCompact f‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_mk_of_compact BoundedContinuousFunction.norm_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.norm_toContinuousMap_eq (f : α →ᵇ E) :
‖f.toContinuousMap‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_to_continuous_map_eq BoundedContinuousFunction.norm_toContinuousMap_eq
open BoundedContinuousFunction
instance : NormedAddCommGroup C(α, E) :=
{ ContinuousMap.metricSpace _ _,
ContinuousMap.instAddCommGroupContinuousMap with
dist_eq := fun x y => by
rw [← norm_mkOfCompact, ← dist_mkOfCompact, dist_eq_norm, mkOfCompact_sub]
dist := dist
norm := norm }
instance [Nonempty α] [One E] [NormOneClass E] : NormOneClass C(α, E) where
norm_one := by simp only [← norm_mkOfCompact, mkOfCompact_one, norm_one]
section
variable (f : C(α, E))
-- The corresponding lemmas for `BoundedContinuousFunction` are stated with `{f}`,
-- and so can not be used in dot notation.
theorem norm_coe_le_norm (x : α) : ‖f x‖ ≤ ‖f‖ :=
(mkOfCompact f).norm_coe_le_norm x
#align continuous_map.norm_coe_le_norm ContinuousMap.norm_coe_le_norm
/-- Distance between the images of any two points is at most twice the norm of the function. -/
theorem dist_le_two_norm (x y : α) : dist (f x) (f y) ≤ 2 * ‖f‖ :=
(mkOfCompact f).dist_le_two_norm x y
#align continuous_map.dist_le_two_norm ContinuousMap.dist_le_two_norm
/-- The norm of a function is controlled by the supremum of the pointwise norms. -/
theorem norm_le {C : ℝ} (C0 : (0 : ℝ) ≤ C) : ‖f‖ ≤ C ↔ ∀ x : α, ‖f x‖ ≤ C :=
@BoundedContinuousFunction.norm_le _ _ _ _ (mkOfCompact f) _ C0
#align continuous_map.norm_le ContinuousMap.norm_le
theorem norm_le_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ ≤ M ↔ ∀ x, ‖f x‖ ≤ M :=
@BoundedContinuousFunction.norm_le_of_nonempty _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_le_of_nonempty ContinuousMap.norm_le_of_nonempty
theorem norm_lt_iff {M : ℝ} (M0 : 0 < M) : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_compact _ _ _ _ _ (mkOfCompact f) _ M0
#align continuous_map.norm_lt_iff ContinuousMap.norm_lt_iff
theorem nnnorm_lt_iff {M : ℝ≥0} (M0 : 0 < M) : ‖f‖₊ < M ↔ ∀ x : α, ‖f x‖₊ < M :=
f.norm_lt_iff M0
#align continuous_map.nnnorm_lt_iff ContinuousMap.nnnorm_lt_iff
theorem norm_lt_iff_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_nonempty_compact _ _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_lt_iff_of_nonempty ContinuousMap.norm_lt_iff_of_nonempty
theorem nnnorm_lt_iff_of_nonempty [Nonempty α] {M : ℝ≥0} : ‖f‖₊ < M ↔ ∀ x, ‖f x‖₊ < M :=
f.norm_lt_iff_of_nonempty
#align continuous_map.nnnorm_lt_iff_of_nonempty ContinuousMap.nnnorm_lt_iff_of_nonempty
theorem apply_le_norm (f : C(α, ℝ)) (x : α) : f x ≤ ‖f‖ :=
le_trans (le_abs.mpr (Or.inl (le_refl (f x)))) (f.norm_coe_le_norm x)
#align continuous_map.apply_le_norm ContinuousMap.apply_le_norm
theorem neg_norm_le_apply (f : C(α, ℝ)) (x : α) : -‖f‖ ≤ f x :=
le_trans (neg_le_neg (f.norm_coe_le_norm x)) (neg_le.mp (neg_le_abs_self (f x)))
#align continuous_map.neg_norm_le_apply ContinuousMap.neg_norm_le_apply
theorem norm_eq_iSup_norm : ‖f‖ = ⨆ x : α, ‖f x‖ :=
(mkOfCompact f).norm_eq_iSup_norm
#align continuous_map.norm_eq_supr_norm ContinuousMap.norm_eq_iSup_norm
theorem norm_restrict_mono_set {X : Type*} [TopologicalSpace X] (f : C(X, E))
{K L : TopologicalSpace.Compacts X} (hKL : K ≤ L) : ‖f.restrict K‖ ≤ ‖f.restrict L‖ :=
(norm_le _ (norm_nonneg _)).mpr fun x => norm_coe_le_norm (f.restrict L) <| Set.inclusion hKL x
#align continuous_map.norm_restrict_mono_set ContinuousMap.norm_restrict_mono_set
end
section
variable {R : Type*} [NormedRing R]
instance : NormedRing C(α, R) :=
{ (inferInstance : NormedAddCommGroup C(α, R)), ContinuousMap.instRingContinuousMap with
norm_mul := fun f g => norm_mul_le (mkOfCompact f) (mkOfCompact g) }
end
section
variable {𝕜 : Type*} [NormedField 𝕜] [NormedSpace 𝕜 E]
instance normedSpace : NormedSpace 𝕜 C(α, E) where
norm_smul_le c f := (norm_smul_le c (mkOfCompact f) : _)
#align continuous_map.normed_space ContinuousMap.normedSpace
section
variable (α 𝕜 E)
/-- When `α` is compact and `𝕜` is a normed field,
the `𝕜`-algebra of bounded continuous maps `α →ᵇ β` is
`𝕜`-linearly isometric to `C(α, β)`.
-/
def linearIsometryBoundedOfCompact : C(α, E) ≃ₗᵢ[𝕜] α →ᵇ E :=
{ addEquivBoundedOfCompact α E with
map_smul' := fun c f => by
ext
norm_cast
norm_map' := fun f => rfl }
#align continuous_map.linear_isometry_bounded_of_compact ContinuousMap.linearIsometryBoundedOfCompact
variable {α E}
-- to match `BoundedContinuousFunction.evalClm`
/-- The evaluation at a point, as a continuous linear map from `C(α, 𝕜)` to `𝕜`. -/
def evalClm (x : α) : C(α, E) →L[𝕜] E :=
(BoundedContinuousFunction.evalClm 𝕜 x).comp
(linearIsometryBoundedOfCompact α E 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_map.eval_clm ContinuousMap.evalClm
end
-- this lemma and the next are the analogues of those autogenerated by `@[simps]` for
-- `equivBoundedOfCompact`, `addEquivBoundedOfCompact`
@[simp]
theorem linearIsometryBoundedOfCompact_symm_apply (f : α →ᵇ E) :
(linearIsometryBoundedOfCompact α E 𝕜).symm f = f.toContinuousMap :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_symm_apply ContinuousMap.linearIsometryBoundedOfCompact_symm_apply
@[simp]
theorem linearIsometryBoundedOfCompact_apply_apply (f : C(α, E)) (a : α) :
(linearIsometryBoundedOfCompact α E 𝕜 f) a = f a :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_apply_apply ContinuousMap.linearIsometryBoundedOfCompact_apply_apply
@[simp]
theorem linearIsometryBoundedOfCompact_toIsometryEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toIsometryEquiv = isometryEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_isometry_equiv ContinuousMap.linearIsometryBoundedOfCompact_toIsometryEquiv
@[simp] -- porting note: adjusted LHS because `simpNF` complained it simplified.
theorem linearIsometryBoundedOfCompact_toAddEquiv :
((linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv : C(α, E) ≃+ (α →ᵇ E)) =
addEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_add_equiv ContinuousMap.linearIsometryBoundedOfCompact_toAddEquiv
@[simp]
theorem linearIsometryBoundedOfCompact_of_compact_toEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv.toEquiv = equivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_of_compact_to_equiv ContinuousMap.linearIsometryBoundedOfCompact_of_compact_toEquiv
end
section
variable {𝕜 : Type*} {γ : Type*} [NormedField 𝕜] [NormedRing γ] [NormedAlgebra 𝕜 γ]
instance : NormedAlgebra 𝕜 C(α, γ) :=
{ ContinuousMap.normedSpace, ContinuousMap.algebra with }
end
end ContinuousMap
namespace ContinuousMap
section UniformContinuity
variable {α β : Type*}
variable [MetricSpace α] [CompactSpace α] [MetricSpace β]
/-!
We now set up some declarations making it convenient to use uniform continuity.
-/
theorem uniform_continuity (f : C(α, β)) (ε : ℝ) (h : 0 < ε) :
∃ δ > 0, ∀ {x y}, dist x y < δ → dist (f x) (f y) < ε :=
Metric.uniformContinuous_iff.mp (CompactSpace.uniformContinuous_of_continuous f.continuous) ε h
#align continuous_map.uniform_continuity ContinuousMap.uniform_continuity
-- This definition allows us to separate the choice of some `δ`,
-- and the corresponding use of `dist a b < δ → dist (f a) (f b) < ε`,
-- even across different declarations.
/-- An arbitrarily chosen modulus of uniform continuity for a given function `f` and `ε > 0`. -/
def modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) : ℝ :=
Classical.choose (uniform_continuity f ε h)
#align continuous_map.modulus ContinuousMap.modulus
theorem modulus_pos (f : C(α, β)) {ε : ℝ} {h : 0 < ε} : 0 < f.modulus ε h :=
(Classical.choose_spec (uniform_continuity f ε h)).1
#align continuous_map.modulus_pos ContinuousMap.modulus_pos
theorem dist_lt_of_dist_lt_modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) {a b : α}
(w : dist a b < f.modulus ε h) : dist (f a) (f b) < ε :=
(Classical.choose_spec (uniform_continuity f ε h)).2 w
#align continuous_map.dist_lt_of_dist_lt_modulus ContinuousMap.dist_lt_of_dist_lt_modulus
end UniformContinuity
end ContinuousMap
section CompLeft
variable (X : Type*) {𝕜 β γ : Type*} [TopologicalSpace X] [CompactSpace X]
[NontriviallyNormedField 𝕜]
variable [NormedAddCommGroup β] [NormedSpace 𝕜 β] [NormedAddCommGroup γ] [NormedSpace 𝕜 γ]
open ContinuousMap
/-- Postcomposition of continuous functions into a normed module by a continuous linear map is a
continuous linear map.
Transferred version of `ContinuousLinearMap.compLeftContinuousBounded`,
upgraded version of `ContinuousLinearMap.compLeftContinuous`,
similar to `LinearMap.compLeft`. -/
protected def ContinuousLinearMap.compLeftContinuousCompact (g : β →L[𝕜] γ) :
C(X, β) →L[𝕜] C(X, γ) :=
(linearIsometryBoundedOfCompact X γ 𝕜).symm.toLinearIsometry.toContinuousLinearMap.comp <|
(g.compLeftContinuousBounded X).comp <|
(linearIsometryBoundedOfCompact X β 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_linear_map.comp_left_continuous_compact ContinuousLinearMap.compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.toLinear_compLeftContinuousCompact (g : β →L[𝕜] γ) :
(g.compLeftContinuousCompact X : C(X, β) →ₗ[𝕜] C(X, γ)) = g.compLeftContinuous 𝕜 X := by
ext f
rfl
#align continuous_linear_map.to_linear_comp_left_continuous_compact ContinuousLinearMap.toLinear_compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.compLeftContinuousCompact_apply (g : β →L[𝕜] γ) (f : C(X, β)) (x : X) :
g.compLeftContinuousCompact X f x = g (f x) :=
rfl
#align continuous_linear_map.comp_left_continuous_compact_apply ContinuousLinearMap.compLeftContinuousCompact_apply
end CompLeft
namespace ContinuousMap
/-!
We now setup variations on `compRight* f`, where `f : C(X, Y)`
(that is, precomposition by a continuous map),
as a morphism `C(Y, T) → C(X, T)`, respecting various types of structure.
In particular:
* `compRightContinuousMap`, the bundled continuous map (for this we need `X Y` compact).
* `compRightHomeomorph`, when we precompose by a homeomorphism.
* `compRightAlgHom`, when `T = R` is a topological ring.
-/
section CompRight
/-- Precomposition by a continuous map is itself a continuous map between spaces of continuous maps.
-/
def compRightContinuousMap {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y)) : C(C(Y, T), C(X, T)) where
toFun g := g.comp f
continuous_toFun := by
refine' Metric.continuous_iff.mpr _
intro g ε ε_pos
refine' ⟨ε, ε_pos, fun g' h => _⟩
rw [ContinuousMap.dist_lt_iff ε_pos] at h ⊢
exact fun x => h (f x)
#align continuous_map.comp_right_continuous_map ContinuousMap.compRightContinuousMap
@[simp]
theorem compRightContinuousMap_apply {X Y : Type*} (T : Type*) [TopologicalSpace X]
[CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y))
(g : C(Y, T)) : (compRightContinuousMap T f) g = g.comp f :=
rfl
#align continuous_map.comp_right_continuous_map_apply ContinuousMap.compRightContinuousMap_apply
/-- Precomposition by a homeomorphism is itself a homeomorphism between spaces of continuous maps.
-/
def compRightHomeomorph {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : X ≃ₜ Y) : C(Y, T) ≃ₜ C(X, T) where
toFun := compRightContinuousMap T f.toContinuousMap
invFun := compRightContinuousMap T f.symm.toContinuousMap
left_inv g := ext fun _ => congr_arg g (f.apply_symm_apply _)
right_inv g := ext fun _ => congr_arg g (f.symm_apply_apply _)
#align continuous_map.comp_right_homeomorph ContinuousMap.compRightHomeomorph
theorem compRightAlgHom_continuous {X Y : Type*} (R A : Type*) [TopologicalSpace X]
[CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [CommSemiring R] [Semiring A]
[MetricSpace A] [TopologicalSemiring A] [Algebra R A] (f : C(X, Y)) :
Continuous (compRightAlgHom R A f) :=
map_continuous (compRightContinuousMap A f)
#align continuous_map.comp_right_alg_hom_continuous ContinuousMap.compRightAlgHom_continuous
end CompRight
section LocalNormalConvergence
/-! ### Local normal convergence
A sum of continuous functions (on a locally compact space) is "locally normally convergent" if the
sum of its sup-norms on any compact subset is summable. This implies convergence in the topology
of `C(X, E)` (i.e. locally uniform convergence). -/
open TopologicalSpace
variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
variable {E : Type*} [NormedAddCommGroup E] [CompleteSpace E]
theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F := by
|
refine' (ContinuousMap.exists_tendsto_compactOpen_iff_forall _).2 fun K hK => _
|
theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F := by
|
Mathlib.Topology.ContinuousFunction.Compact.491_0.Mig2jTVnn2FLKEB
|
theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F
|
Mathlib_Topology_ContinuousFunction_Compact
|
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : LocallyCompactSpace X
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
ι : Type u_3
F : ι → C(X, E)
hF : ∀ (K : Compacts X), Summable fun i => ‖restrict (↑K) (F i)‖
K : Set X
hK : IsCompact K
⊢ ∃ f, Tendsto (fun i => restrict K (∑ b in i, F b)) atTop (𝓝 f)
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compacts
#align_import topology.continuous_function.compact from "leanprover-community/mathlib"@"d3af0609f6db8691dffdc3e1fb7feb7da72698f2"
/-!
# Continuous functions on a compact space
Continuous functions `C(α, β)` from a compact space `α` to a metric space `β`
are automatically bounded, and so acquire various structures inherited from `α →ᵇ β`.
This file transfers these structures, and restates some lemmas
characterising these structures.
If you need a lemma which is proved about `α →ᵇ β` but not for `C(α, β)` when `α` is compact,
you should restate it here. You can also use
`ContinuousMap.equivBoundedOfCompact` to move functions back and forth.
-/
noncomputable section
open Topology Classical NNReal BoundedContinuousFunction BigOperators
open Set Filter Metric
open BoundedContinuousFunction
namespace ContinuousMap
variable {α β E : Type*} [TopologicalSpace α] [CompactSpace α] [MetricSpace β]
[NormedAddCommGroup E]
section
variable (α β)
/-- When `α` is compact, the bounded continuous maps `α →ᵇ β` are
equivalent to `C(α, β)`.
-/
@[simps (config := .asFn)]
def equivBoundedOfCompact : C(α, β) ≃ (α →ᵇ β) :=
⟨mkOfCompact, BoundedContinuousFunction.toContinuousMap, fun f => by
ext
rfl, fun f => by
ext
rfl⟩
#align continuous_map.equiv_bounded_of_compact ContinuousMap.equivBoundedOfCompact
theorem uniformInducing_equivBoundedOfCompact : UniformInducing (equivBoundedOfCompact α β) :=
UniformInducing.mk'
(by
simp only [hasBasis_compactConvergenceUniformity.mem_iff, uniformity_basis_dist_le.mem_iff]
exact fun s =>
⟨fun ⟨⟨a, b⟩, ⟨_, ⟨ε, hε, hb⟩⟩, hs⟩ =>
⟨{ p | ∀ x, (p.1 x, p.2 x) ∈ b }, ⟨ε, hε, fun _ h x => hb ((dist_le hε.le).mp h x)⟩,
fun f g h => hs fun x _ => h x⟩,
fun ⟨_, ⟨ε, hε, ht⟩, hs⟩ =>
⟨⟨Set.univ, { p | dist p.1 p.2 ≤ ε }⟩, ⟨isCompact_univ, ⟨ε, hε, fun _ h => h⟩⟩,
fun ⟨f, g⟩ h => hs _ _ (ht ((dist_le hε.le).mpr fun x => h x (mem_univ x)))⟩⟩)
#align continuous_map.uniform_inducing_equiv_bounded_of_compact ContinuousMap.uniformInducing_equivBoundedOfCompact
theorem uniformEmbedding_equivBoundedOfCompact : UniformEmbedding (equivBoundedOfCompact α β) :=
{ uniformInducing_equivBoundedOfCompact α β with inj := (equivBoundedOfCompact α β).injective }
#align continuous_map.uniform_embedding_equiv_bounded_of_compact ContinuousMap.uniformEmbedding_equivBoundedOfCompact
/-- When `α` is compact, the bounded continuous maps `α →ᵇ 𝕜` are
additively equivalent to `C(α, 𝕜)`.
-/
-- porting note: the following `simps` received a "maximum recursion depth" error
-- @[simps! (config := .asFn) apply symm_apply]
def addEquivBoundedOfCompact [AddMonoid β] [LipschitzAdd β] : C(α, β) ≃+ (α →ᵇ β) :=
({ toContinuousMapAddHom α β, (equivBoundedOfCompact α β).symm with } : (α →ᵇ β) ≃+ C(α, β)).symm
#align continuous_map.add_equiv_bounded_of_compact ContinuousMap.addEquivBoundedOfCompact
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_symm_apply [AddMonoid β] [LipschitzAdd β] :
⇑((addEquivBoundedOfCompact α β).symm) = toContinuousMapAddHom α β :=
rfl
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_apply [AddMonoid β] [LipschitzAdd β] :
⇑(addEquivBoundedOfCompact α β) = mkOfCompact :=
rfl
instance metricSpace : MetricSpace C(α, β) :=
(uniformEmbedding_equivBoundedOfCompact α β).comapMetricSpace _
#align continuous_map.metric_space ContinuousMap.metricSpace
/-- When `α` is compact, and `β` is a metric space, the bounded continuous maps `α →ᵇ β` are
isometric to `C(α, β)`.
-/
@[simps! (config := .asFn) toEquiv apply symm_apply]
def isometryEquivBoundedOfCompact : C(α, β) ≃ᵢ (α →ᵇ β) where
isometry_toFun _ _ := rfl
toEquiv := equivBoundedOfCompact α β
#align continuous_map.isometry_equiv_bounded_of_compact ContinuousMap.isometryEquivBoundedOfCompact
end
@[simp]
theorem _root_.BoundedContinuousFunction.dist_mkOfCompact (f g : C(α, β)) :
dist (mkOfCompact f) (mkOfCompact g) = dist f g :=
rfl
#align bounded_continuous_function.dist_mk_of_compact BoundedContinuousFunction.dist_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.dist_toContinuousMap (f g : α →ᵇ β) :
dist f.toContinuousMap g.toContinuousMap = dist f g :=
rfl
#align bounded_continuous_function.dist_to_continuous_map BoundedContinuousFunction.dist_toContinuousMap
open BoundedContinuousFunction
section
variable {f g : C(α, β)} {C : ℝ}
/-- The pointwise distance is controlled by the distance between functions, by definition. -/
theorem dist_apply_le_dist (x : α) : dist (f x) (g x) ≤ dist f g := by
simp only [← dist_mkOfCompact, dist_coe_le_dist, ← mkOfCompact_apply]
#align continuous_map.dist_apply_le_dist ContinuousMap.dist_apply_le_dist
/-- The distance between two functions is controlled by the supremum of the pointwise distances. -/
theorem dist_le (C0 : (0 : ℝ) ≤ C) : dist f g ≤ C ↔ ∀ x : α, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le C0, mkOfCompact_apply]
#align continuous_map.dist_le ContinuousMap.dist_le
theorem dist_le_iff_of_nonempty [Nonempty α] : dist f g ≤ C ↔ ∀ x, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le_iff_of_nonempty,
mkOfCompact_apply]
#align continuous_map.dist_le_iff_of_nonempty ContinuousMap.dist_le_iff_of_nonempty
theorem dist_lt_iff_of_nonempty [Nonempty α] : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
simp only [← dist_mkOfCompact, dist_lt_iff_of_nonempty_compact, mkOfCompact_apply]
#align continuous_map.dist_lt_iff_of_nonempty ContinuousMap.dist_lt_iff_of_nonempty
theorem dist_lt_of_nonempty [Nonempty α] (w : ∀ x : α, dist (f x) (g x) < C) : dist f g < C :=
dist_lt_iff_of_nonempty.2 w
#align continuous_map.dist_lt_of_nonempty ContinuousMap.dist_lt_of_nonempty
theorem dist_lt_iff (C0 : (0 : ℝ) < C) : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
rw [← dist_mkOfCompact, dist_lt_iff_of_compact C0]
simp only [mkOfCompact_apply]
#align continuous_map.dist_lt_iff ContinuousMap.dist_lt_iff
end
instance [CompleteSpace β] : CompleteSpace C(α, β) :=
(isometryEquivBoundedOfCompact α β).completeSpace
/-- See also `ContinuousMap.continuous_eval'`. -/
@[continuity]
theorem continuous_eval : Continuous fun p : C(α, β) × α => p.1 p.2 :=
continuous_eval.comp ((isometryEquivBoundedOfCompact α β).continuous.prod_map continuous_id)
#align continuous_map.continuous_eval ContinuousMap.continuous_eval
-- TODO at some point we will need lemmas characterising this norm!
-- At the moment the only way to reason about it is to transfer `f : C(α,E)` back to `α →ᵇ E`.
instance : Norm C(α, E) where norm x := dist x 0
@[simp]
theorem _root_.BoundedContinuousFunction.norm_mkOfCompact (f : C(α, E)) : ‖mkOfCompact f‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_mk_of_compact BoundedContinuousFunction.norm_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.norm_toContinuousMap_eq (f : α →ᵇ E) :
‖f.toContinuousMap‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_to_continuous_map_eq BoundedContinuousFunction.norm_toContinuousMap_eq
open BoundedContinuousFunction
instance : NormedAddCommGroup C(α, E) :=
{ ContinuousMap.metricSpace _ _,
ContinuousMap.instAddCommGroupContinuousMap with
dist_eq := fun x y => by
rw [← norm_mkOfCompact, ← dist_mkOfCompact, dist_eq_norm, mkOfCompact_sub]
dist := dist
norm := norm }
instance [Nonempty α] [One E] [NormOneClass E] : NormOneClass C(α, E) where
norm_one := by simp only [← norm_mkOfCompact, mkOfCompact_one, norm_one]
section
variable (f : C(α, E))
-- The corresponding lemmas for `BoundedContinuousFunction` are stated with `{f}`,
-- and so can not be used in dot notation.
theorem norm_coe_le_norm (x : α) : ‖f x‖ ≤ ‖f‖ :=
(mkOfCompact f).norm_coe_le_norm x
#align continuous_map.norm_coe_le_norm ContinuousMap.norm_coe_le_norm
/-- Distance between the images of any two points is at most twice the norm of the function. -/
theorem dist_le_two_norm (x y : α) : dist (f x) (f y) ≤ 2 * ‖f‖ :=
(mkOfCompact f).dist_le_two_norm x y
#align continuous_map.dist_le_two_norm ContinuousMap.dist_le_two_norm
/-- The norm of a function is controlled by the supremum of the pointwise norms. -/
theorem norm_le {C : ℝ} (C0 : (0 : ℝ) ≤ C) : ‖f‖ ≤ C ↔ ∀ x : α, ‖f x‖ ≤ C :=
@BoundedContinuousFunction.norm_le _ _ _ _ (mkOfCompact f) _ C0
#align continuous_map.norm_le ContinuousMap.norm_le
theorem norm_le_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ ≤ M ↔ ∀ x, ‖f x‖ ≤ M :=
@BoundedContinuousFunction.norm_le_of_nonempty _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_le_of_nonempty ContinuousMap.norm_le_of_nonempty
theorem norm_lt_iff {M : ℝ} (M0 : 0 < M) : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_compact _ _ _ _ _ (mkOfCompact f) _ M0
#align continuous_map.norm_lt_iff ContinuousMap.norm_lt_iff
theorem nnnorm_lt_iff {M : ℝ≥0} (M0 : 0 < M) : ‖f‖₊ < M ↔ ∀ x : α, ‖f x‖₊ < M :=
f.norm_lt_iff M0
#align continuous_map.nnnorm_lt_iff ContinuousMap.nnnorm_lt_iff
theorem norm_lt_iff_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_nonempty_compact _ _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_lt_iff_of_nonempty ContinuousMap.norm_lt_iff_of_nonempty
theorem nnnorm_lt_iff_of_nonempty [Nonempty α] {M : ℝ≥0} : ‖f‖₊ < M ↔ ∀ x, ‖f x‖₊ < M :=
f.norm_lt_iff_of_nonempty
#align continuous_map.nnnorm_lt_iff_of_nonempty ContinuousMap.nnnorm_lt_iff_of_nonempty
theorem apply_le_norm (f : C(α, ℝ)) (x : α) : f x ≤ ‖f‖ :=
le_trans (le_abs.mpr (Or.inl (le_refl (f x)))) (f.norm_coe_le_norm x)
#align continuous_map.apply_le_norm ContinuousMap.apply_le_norm
theorem neg_norm_le_apply (f : C(α, ℝ)) (x : α) : -‖f‖ ≤ f x :=
le_trans (neg_le_neg (f.norm_coe_le_norm x)) (neg_le.mp (neg_le_abs_self (f x)))
#align continuous_map.neg_norm_le_apply ContinuousMap.neg_norm_le_apply
theorem norm_eq_iSup_norm : ‖f‖ = ⨆ x : α, ‖f x‖ :=
(mkOfCompact f).norm_eq_iSup_norm
#align continuous_map.norm_eq_supr_norm ContinuousMap.norm_eq_iSup_norm
theorem norm_restrict_mono_set {X : Type*} [TopologicalSpace X] (f : C(X, E))
{K L : TopologicalSpace.Compacts X} (hKL : K ≤ L) : ‖f.restrict K‖ ≤ ‖f.restrict L‖ :=
(norm_le _ (norm_nonneg _)).mpr fun x => norm_coe_le_norm (f.restrict L) <| Set.inclusion hKL x
#align continuous_map.norm_restrict_mono_set ContinuousMap.norm_restrict_mono_set
end
section
variable {R : Type*} [NormedRing R]
instance : NormedRing C(α, R) :=
{ (inferInstance : NormedAddCommGroup C(α, R)), ContinuousMap.instRingContinuousMap with
norm_mul := fun f g => norm_mul_le (mkOfCompact f) (mkOfCompact g) }
end
section
variable {𝕜 : Type*} [NormedField 𝕜] [NormedSpace 𝕜 E]
instance normedSpace : NormedSpace 𝕜 C(α, E) where
norm_smul_le c f := (norm_smul_le c (mkOfCompact f) : _)
#align continuous_map.normed_space ContinuousMap.normedSpace
section
variable (α 𝕜 E)
/-- When `α` is compact and `𝕜` is a normed field,
the `𝕜`-algebra of bounded continuous maps `α →ᵇ β` is
`𝕜`-linearly isometric to `C(α, β)`.
-/
def linearIsometryBoundedOfCompact : C(α, E) ≃ₗᵢ[𝕜] α →ᵇ E :=
{ addEquivBoundedOfCompact α E with
map_smul' := fun c f => by
ext
norm_cast
norm_map' := fun f => rfl }
#align continuous_map.linear_isometry_bounded_of_compact ContinuousMap.linearIsometryBoundedOfCompact
variable {α E}
-- to match `BoundedContinuousFunction.evalClm`
/-- The evaluation at a point, as a continuous linear map from `C(α, 𝕜)` to `𝕜`. -/
def evalClm (x : α) : C(α, E) →L[𝕜] E :=
(BoundedContinuousFunction.evalClm 𝕜 x).comp
(linearIsometryBoundedOfCompact α E 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_map.eval_clm ContinuousMap.evalClm
end
-- this lemma and the next are the analogues of those autogenerated by `@[simps]` for
-- `equivBoundedOfCompact`, `addEquivBoundedOfCompact`
@[simp]
theorem linearIsometryBoundedOfCompact_symm_apply (f : α →ᵇ E) :
(linearIsometryBoundedOfCompact α E 𝕜).symm f = f.toContinuousMap :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_symm_apply ContinuousMap.linearIsometryBoundedOfCompact_symm_apply
@[simp]
theorem linearIsometryBoundedOfCompact_apply_apply (f : C(α, E)) (a : α) :
(linearIsometryBoundedOfCompact α E 𝕜 f) a = f a :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_apply_apply ContinuousMap.linearIsometryBoundedOfCompact_apply_apply
@[simp]
theorem linearIsometryBoundedOfCompact_toIsometryEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toIsometryEquiv = isometryEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_isometry_equiv ContinuousMap.linearIsometryBoundedOfCompact_toIsometryEquiv
@[simp] -- porting note: adjusted LHS because `simpNF` complained it simplified.
theorem linearIsometryBoundedOfCompact_toAddEquiv :
((linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv : C(α, E) ≃+ (α →ᵇ E)) =
addEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_add_equiv ContinuousMap.linearIsometryBoundedOfCompact_toAddEquiv
@[simp]
theorem linearIsometryBoundedOfCompact_of_compact_toEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv.toEquiv = equivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_of_compact_to_equiv ContinuousMap.linearIsometryBoundedOfCompact_of_compact_toEquiv
end
section
variable {𝕜 : Type*} {γ : Type*} [NormedField 𝕜] [NormedRing γ] [NormedAlgebra 𝕜 γ]
instance : NormedAlgebra 𝕜 C(α, γ) :=
{ ContinuousMap.normedSpace, ContinuousMap.algebra with }
end
end ContinuousMap
namespace ContinuousMap
section UniformContinuity
variable {α β : Type*}
variable [MetricSpace α] [CompactSpace α] [MetricSpace β]
/-!
We now set up some declarations making it convenient to use uniform continuity.
-/
theorem uniform_continuity (f : C(α, β)) (ε : ℝ) (h : 0 < ε) :
∃ δ > 0, ∀ {x y}, dist x y < δ → dist (f x) (f y) < ε :=
Metric.uniformContinuous_iff.mp (CompactSpace.uniformContinuous_of_continuous f.continuous) ε h
#align continuous_map.uniform_continuity ContinuousMap.uniform_continuity
-- This definition allows us to separate the choice of some `δ`,
-- and the corresponding use of `dist a b < δ → dist (f a) (f b) < ε`,
-- even across different declarations.
/-- An arbitrarily chosen modulus of uniform continuity for a given function `f` and `ε > 0`. -/
def modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) : ℝ :=
Classical.choose (uniform_continuity f ε h)
#align continuous_map.modulus ContinuousMap.modulus
theorem modulus_pos (f : C(α, β)) {ε : ℝ} {h : 0 < ε} : 0 < f.modulus ε h :=
(Classical.choose_spec (uniform_continuity f ε h)).1
#align continuous_map.modulus_pos ContinuousMap.modulus_pos
theorem dist_lt_of_dist_lt_modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) {a b : α}
(w : dist a b < f.modulus ε h) : dist (f a) (f b) < ε :=
(Classical.choose_spec (uniform_continuity f ε h)).2 w
#align continuous_map.dist_lt_of_dist_lt_modulus ContinuousMap.dist_lt_of_dist_lt_modulus
end UniformContinuity
end ContinuousMap
section CompLeft
variable (X : Type*) {𝕜 β γ : Type*} [TopologicalSpace X] [CompactSpace X]
[NontriviallyNormedField 𝕜]
variable [NormedAddCommGroup β] [NormedSpace 𝕜 β] [NormedAddCommGroup γ] [NormedSpace 𝕜 γ]
open ContinuousMap
/-- Postcomposition of continuous functions into a normed module by a continuous linear map is a
continuous linear map.
Transferred version of `ContinuousLinearMap.compLeftContinuousBounded`,
upgraded version of `ContinuousLinearMap.compLeftContinuous`,
similar to `LinearMap.compLeft`. -/
protected def ContinuousLinearMap.compLeftContinuousCompact (g : β →L[𝕜] γ) :
C(X, β) →L[𝕜] C(X, γ) :=
(linearIsometryBoundedOfCompact X γ 𝕜).symm.toLinearIsometry.toContinuousLinearMap.comp <|
(g.compLeftContinuousBounded X).comp <|
(linearIsometryBoundedOfCompact X β 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_linear_map.comp_left_continuous_compact ContinuousLinearMap.compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.toLinear_compLeftContinuousCompact (g : β →L[𝕜] γ) :
(g.compLeftContinuousCompact X : C(X, β) →ₗ[𝕜] C(X, γ)) = g.compLeftContinuous 𝕜 X := by
ext f
rfl
#align continuous_linear_map.to_linear_comp_left_continuous_compact ContinuousLinearMap.toLinear_compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.compLeftContinuousCompact_apply (g : β →L[𝕜] γ) (f : C(X, β)) (x : X) :
g.compLeftContinuousCompact X f x = g (f x) :=
rfl
#align continuous_linear_map.comp_left_continuous_compact_apply ContinuousLinearMap.compLeftContinuousCompact_apply
end CompLeft
namespace ContinuousMap
/-!
We now setup variations on `compRight* f`, where `f : C(X, Y)`
(that is, precomposition by a continuous map),
as a morphism `C(Y, T) → C(X, T)`, respecting various types of structure.
In particular:
* `compRightContinuousMap`, the bundled continuous map (for this we need `X Y` compact).
* `compRightHomeomorph`, when we precompose by a homeomorphism.
* `compRightAlgHom`, when `T = R` is a topological ring.
-/
section CompRight
/-- Precomposition by a continuous map is itself a continuous map between spaces of continuous maps.
-/
def compRightContinuousMap {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y)) : C(C(Y, T), C(X, T)) where
toFun g := g.comp f
continuous_toFun := by
refine' Metric.continuous_iff.mpr _
intro g ε ε_pos
refine' ⟨ε, ε_pos, fun g' h => _⟩
rw [ContinuousMap.dist_lt_iff ε_pos] at h ⊢
exact fun x => h (f x)
#align continuous_map.comp_right_continuous_map ContinuousMap.compRightContinuousMap
@[simp]
theorem compRightContinuousMap_apply {X Y : Type*} (T : Type*) [TopologicalSpace X]
[CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y))
(g : C(Y, T)) : (compRightContinuousMap T f) g = g.comp f :=
rfl
#align continuous_map.comp_right_continuous_map_apply ContinuousMap.compRightContinuousMap_apply
/-- Precomposition by a homeomorphism is itself a homeomorphism between spaces of continuous maps.
-/
def compRightHomeomorph {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : X ≃ₜ Y) : C(Y, T) ≃ₜ C(X, T) where
toFun := compRightContinuousMap T f.toContinuousMap
invFun := compRightContinuousMap T f.symm.toContinuousMap
left_inv g := ext fun _ => congr_arg g (f.apply_symm_apply _)
right_inv g := ext fun _ => congr_arg g (f.symm_apply_apply _)
#align continuous_map.comp_right_homeomorph ContinuousMap.compRightHomeomorph
theorem compRightAlgHom_continuous {X Y : Type*} (R A : Type*) [TopologicalSpace X]
[CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [CommSemiring R] [Semiring A]
[MetricSpace A] [TopologicalSemiring A] [Algebra R A] (f : C(X, Y)) :
Continuous (compRightAlgHom R A f) :=
map_continuous (compRightContinuousMap A f)
#align continuous_map.comp_right_alg_hom_continuous ContinuousMap.compRightAlgHom_continuous
end CompRight
section LocalNormalConvergence
/-! ### Local normal convergence
A sum of continuous functions (on a locally compact space) is "locally normally convergent" if the
sum of its sup-norms on any compact subset is summable. This implies convergence in the topology
of `C(X, E)` (i.e. locally uniform convergence). -/
open TopologicalSpace
variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
variable {E : Type*} [NormedAddCommGroup E] [CompleteSpace E]
theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F := by
refine' (ContinuousMap.exists_tendsto_compactOpen_iff_forall _).2 fun K hK => _
|
lift K to Compacts X using hK
|
theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F := by
refine' (ContinuousMap.exists_tendsto_compactOpen_iff_forall _).2 fun K hK => _
|
Mathlib.Topology.ContinuousFunction.Compact.491_0.Mig2jTVnn2FLKEB
|
theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F
|
Mathlib_Topology_ContinuousFunction_Compact
|
case intro
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : LocallyCompactSpace X
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
ι : Type u_3
F : ι → C(X, E)
hF : ∀ (K : Compacts X), Summable fun i => ‖restrict (↑K) (F i)‖
K : Compacts X
⊢ ∃ f, Tendsto (fun i => restrict (↑K) (∑ b in i, F b)) atTop (𝓝 f)
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compacts
#align_import topology.continuous_function.compact from "leanprover-community/mathlib"@"d3af0609f6db8691dffdc3e1fb7feb7da72698f2"
/-!
# Continuous functions on a compact space
Continuous functions `C(α, β)` from a compact space `α` to a metric space `β`
are automatically bounded, and so acquire various structures inherited from `α →ᵇ β`.
This file transfers these structures, and restates some lemmas
characterising these structures.
If you need a lemma which is proved about `α →ᵇ β` but not for `C(α, β)` when `α` is compact,
you should restate it here. You can also use
`ContinuousMap.equivBoundedOfCompact` to move functions back and forth.
-/
noncomputable section
open Topology Classical NNReal BoundedContinuousFunction BigOperators
open Set Filter Metric
open BoundedContinuousFunction
namespace ContinuousMap
variable {α β E : Type*} [TopologicalSpace α] [CompactSpace α] [MetricSpace β]
[NormedAddCommGroup E]
section
variable (α β)
/-- When `α` is compact, the bounded continuous maps `α →ᵇ β` are
equivalent to `C(α, β)`.
-/
@[simps (config := .asFn)]
def equivBoundedOfCompact : C(α, β) ≃ (α →ᵇ β) :=
⟨mkOfCompact, BoundedContinuousFunction.toContinuousMap, fun f => by
ext
rfl, fun f => by
ext
rfl⟩
#align continuous_map.equiv_bounded_of_compact ContinuousMap.equivBoundedOfCompact
theorem uniformInducing_equivBoundedOfCompact : UniformInducing (equivBoundedOfCompact α β) :=
UniformInducing.mk'
(by
simp only [hasBasis_compactConvergenceUniformity.mem_iff, uniformity_basis_dist_le.mem_iff]
exact fun s =>
⟨fun ⟨⟨a, b⟩, ⟨_, ⟨ε, hε, hb⟩⟩, hs⟩ =>
⟨{ p | ∀ x, (p.1 x, p.2 x) ∈ b }, ⟨ε, hε, fun _ h x => hb ((dist_le hε.le).mp h x)⟩,
fun f g h => hs fun x _ => h x⟩,
fun ⟨_, ⟨ε, hε, ht⟩, hs⟩ =>
⟨⟨Set.univ, { p | dist p.1 p.2 ≤ ε }⟩, ⟨isCompact_univ, ⟨ε, hε, fun _ h => h⟩⟩,
fun ⟨f, g⟩ h => hs _ _ (ht ((dist_le hε.le).mpr fun x => h x (mem_univ x)))⟩⟩)
#align continuous_map.uniform_inducing_equiv_bounded_of_compact ContinuousMap.uniformInducing_equivBoundedOfCompact
theorem uniformEmbedding_equivBoundedOfCompact : UniformEmbedding (equivBoundedOfCompact α β) :=
{ uniformInducing_equivBoundedOfCompact α β with inj := (equivBoundedOfCompact α β).injective }
#align continuous_map.uniform_embedding_equiv_bounded_of_compact ContinuousMap.uniformEmbedding_equivBoundedOfCompact
/-- When `α` is compact, the bounded continuous maps `α →ᵇ 𝕜` are
additively equivalent to `C(α, 𝕜)`.
-/
-- porting note: the following `simps` received a "maximum recursion depth" error
-- @[simps! (config := .asFn) apply symm_apply]
def addEquivBoundedOfCompact [AddMonoid β] [LipschitzAdd β] : C(α, β) ≃+ (α →ᵇ β) :=
({ toContinuousMapAddHom α β, (equivBoundedOfCompact α β).symm with } : (α →ᵇ β) ≃+ C(α, β)).symm
#align continuous_map.add_equiv_bounded_of_compact ContinuousMap.addEquivBoundedOfCompact
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_symm_apply [AddMonoid β] [LipschitzAdd β] :
⇑((addEquivBoundedOfCompact α β).symm) = toContinuousMapAddHom α β :=
rfl
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_apply [AddMonoid β] [LipschitzAdd β] :
⇑(addEquivBoundedOfCompact α β) = mkOfCompact :=
rfl
instance metricSpace : MetricSpace C(α, β) :=
(uniformEmbedding_equivBoundedOfCompact α β).comapMetricSpace _
#align continuous_map.metric_space ContinuousMap.metricSpace
/-- When `α` is compact, and `β` is a metric space, the bounded continuous maps `α →ᵇ β` are
isometric to `C(α, β)`.
-/
@[simps! (config := .asFn) toEquiv apply symm_apply]
def isometryEquivBoundedOfCompact : C(α, β) ≃ᵢ (α →ᵇ β) where
isometry_toFun _ _ := rfl
toEquiv := equivBoundedOfCompact α β
#align continuous_map.isometry_equiv_bounded_of_compact ContinuousMap.isometryEquivBoundedOfCompact
end
@[simp]
theorem _root_.BoundedContinuousFunction.dist_mkOfCompact (f g : C(α, β)) :
dist (mkOfCompact f) (mkOfCompact g) = dist f g :=
rfl
#align bounded_continuous_function.dist_mk_of_compact BoundedContinuousFunction.dist_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.dist_toContinuousMap (f g : α →ᵇ β) :
dist f.toContinuousMap g.toContinuousMap = dist f g :=
rfl
#align bounded_continuous_function.dist_to_continuous_map BoundedContinuousFunction.dist_toContinuousMap
open BoundedContinuousFunction
section
variable {f g : C(α, β)} {C : ℝ}
/-- The pointwise distance is controlled by the distance between functions, by definition. -/
theorem dist_apply_le_dist (x : α) : dist (f x) (g x) ≤ dist f g := by
simp only [← dist_mkOfCompact, dist_coe_le_dist, ← mkOfCompact_apply]
#align continuous_map.dist_apply_le_dist ContinuousMap.dist_apply_le_dist
/-- The distance between two functions is controlled by the supremum of the pointwise distances. -/
theorem dist_le (C0 : (0 : ℝ) ≤ C) : dist f g ≤ C ↔ ∀ x : α, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le C0, mkOfCompact_apply]
#align continuous_map.dist_le ContinuousMap.dist_le
theorem dist_le_iff_of_nonempty [Nonempty α] : dist f g ≤ C ↔ ∀ x, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le_iff_of_nonempty,
mkOfCompact_apply]
#align continuous_map.dist_le_iff_of_nonempty ContinuousMap.dist_le_iff_of_nonempty
theorem dist_lt_iff_of_nonempty [Nonempty α] : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
simp only [← dist_mkOfCompact, dist_lt_iff_of_nonempty_compact, mkOfCompact_apply]
#align continuous_map.dist_lt_iff_of_nonempty ContinuousMap.dist_lt_iff_of_nonempty
theorem dist_lt_of_nonempty [Nonempty α] (w : ∀ x : α, dist (f x) (g x) < C) : dist f g < C :=
dist_lt_iff_of_nonempty.2 w
#align continuous_map.dist_lt_of_nonempty ContinuousMap.dist_lt_of_nonempty
theorem dist_lt_iff (C0 : (0 : ℝ) < C) : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
rw [← dist_mkOfCompact, dist_lt_iff_of_compact C0]
simp only [mkOfCompact_apply]
#align continuous_map.dist_lt_iff ContinuousMap.dist_lt_iff
end
instance [CompleteSpace β] : CompleteSpace C(α, β) :=
(isometryEquivBoundedOfCompact α β).completeSpace
/-- See also `ContinuousMap.continuous_eval'`. -/
@[continuity]
theorem continuous_eval : Continuous fun p : C(α, β) × α => p.1 p.2 :=
continuous_eval.comp ((isometryEquivBoundedOfCompact α β).continuous.prod_map continuous_id)
#align continuous_map.continuous_eval ContinuousMap.continuous_eval
-- TODO at some point we will need lemmas characterising this norm!
-- At the moment the only way to reason about it is to transfer `f : C(α,E)` back to `α →ᵇ E`.
instance : Norm C(α, E) where norm x := dist x 0
@[simp]
theorem _root_.BoundedContinuousFunction.norm_mkOfCompact (f : C(α, E)) : ‖mkOfCompact f‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_mk_of_compact BoundedContinuousFunction.norm_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.norm_toContinuousMap_eq (f : α →ᵇ E) :
‖f.toContinuousMap‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_to_continuous_map_eq BoundedContinuousFunction.norm_toContinuousMap_eq
open BoundedContinuousFunction
instance : NormedAddCommGroup C(α, E) :=
{ ContinuousMap.metricSpace _ _,
ContinuousMap.instAddCommGroupContinuousMap with
dist_eq := fun x y => by
rw [← norm_mkOfCompact, ← dist_mkOfCompact, dist_eq_norm, mkOfCompact_sub]
dist := dist
norm := norm }
instance [Nonempty α] [One E] [NormOneClass E] : NormOneClass C(α, E) where
norm_one := by simp only [← norm_mkOfCompact, mkOfCompact_one, norm_one]
section
variable (f : C(α, E))
-- The corresponding lemmas for `BoundedContinuousFunction` are stated with `{f}`,
-- and so can not be used in dot notation.
theorem norm_coe_le_norm (x : α) : ‖f x‖ ≤ ‖f‖ :=
(mkOfCompact f).norm_coe_le_norm x
#align continuous_map.norm_coe_le_norm ContinuousMap.norm_coe_le_norm
/-- Distance between the images of any two points is at most twice the norm of the function. -/
theorem dist_le_two_norm (x y : α) : dist (f x) (f y) ≤ 2 * ‖f‖ :=
(mkOfCompact f).dist_le_two_norm x y
#align continuous_map.dist_le_two_norm ContinuousMap.dist_le_two_norm
/-- The norm of a function is controlled by the supremum of the pointwise norms. -/
theorem norm_le {C : ℝ} (C0 : (0 : ℝ) ≤ C) : ‖f‖ ≤ C ↔ ∀ x : α, ‖f x‖ ≤ C :=
@BoundedContinuousFunction.norm_le _ _ _ _ (mkOfCompact f) _ C0
#align continuous_map.norm_le ContinuousMap.norm_le
theorem norm_le_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ ≤ M ↔ ∀ x, ‖f x‖ ≤ M :=
@BoundedContinuousFunction.norm_le_of_nonempty _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_le_of_nonempty ContinuousMap.norm_le_of_nonempty
theorem norm_lt_iff {M : ℝ} (M0 : 0 < M) : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_compact _ _ _ _ _ (mkOfCompact f) _ M0
#align continuous_map.norm_lt_iff ContinuousMap.norm_lt_iff
theorem nnnorm_lt_iff {M : ℝ≥0} (M0 : 0 < M) : ‖f‖₊ < M ↔ ∀ x : α, ‖f x‖₊ < M :=
f.norm_lt_iff M0
#align continuous_map.nnnorm_lt_iff ContinuousMap.nnnorm_lt_iff
theorem norm_lt_iff_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_nonempty_compact _ _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_lt_iff_of_nonempty ContinuousMap.norm_lt_iff_of_nonempty
theorem nnnorm_lt_iff_of_nonempty [Nonempty α] {M : ℝ≥0} : ‖f‖₊ < M ↔ ∀ x, ‖f x‖₊ < M :=
f.norm_lt_iff_of_nonempty
#align continuous_map.nnnorm_lt_iff_of_nonempty ContinuousMap.nnnorm_lt_iff_of_nonempty
theorem apply_le_norm (f : C(α, ℝ)) (x : α) : f x ≤ ‖f‖ :=
le_trans (le_abs.mpr (Or.inl (le_refl (f x)))) (f.norm_coe_le_norm x)
#align continuous_map.apply_le_norm ContinuousMap.apply_le_norm
theorem neg_norm_le_apply (f : C(α, ℝ)) (x : α) : -‖f‖ ≤ f x :=
le_trans (neg_le_neg (f.norm_coe_le_norm x)) (neg_le.mp (neg_le_abs_self (f x)))
#align continuous_map.neg_norm_le_apply ContinuousMap.neg_norm_le_apply
theorem norm_eq_iSup_norm : ‖f‖ = ⨆ x : α, ‖f x‖ :=
(mkOfCompact f).norm_eq_iSup_norm
#align continuous_map.norm_eq_supr_norm ContinuousMap.norm_eq_iSup_norm
theorem norm_restrict_mono_set {X : Type*} [TopologicalSpace X] (f : C(X, E))
{K L : TopologicalSpace.Compacts X} (hKL : K ≤ L) : ‖f.restrict K‖ ≤ ‖f.restrict L‖ :=
(norm_le _ (norm_nonneg _)).mpr fun x => norm_coe_le_norm (f.restrict L) <| Set.inclusion hKL x
#align continuous_map.norm_restrict_mono_set ContinuousMap.norm_restrict_mono_set
end
section
variable {R : Type*} [NormedRing R]
instance : NormedRing C(α, R) :=
{ (inferInstance : NormedAddCommGroup C(α, R)), ContinuousMap.instRingContinuousMap with
norm_mul := fun f g => norm_mul_le (mkOfCompact f) (mkOfCompact g) }
end
section
variable {𝕜 : Type*} [NormedField 𝕜] [NormedSpace 𝕜 E]
instance normedSpace : NormedSpace 𝕜 C(α, E) where
norm_smul_le c f := (norm_smul_le c (mkOfCompact f) : _)
#align continuous_map.normed_space ContinuousMap.normedSpace
section
variable (α 𝕜 E)
/-- When `α` is compact and `𝕜` is a normed field,
the `𝕜`-algebra of bounded continuous maps `α →ᵇ β` is
`𝕜`-linearly isometric to `C(α, β)`.
-/
def linearIsometryBoundedOfCompact : C(α, E) ≃ₗᵢ[𝕜] α →ᵇ E :=
{ addEquivBoundedOfCompact α E with
map_smul' := fun c f => by
ext
norm_cast
norm_map' := fun f => rfl }
#align continuous_map.linear_isometry_bounded_of_compact ContinuousMap.linearIsometryBoundedOfCompact
variable {α E}
-- to match `BoundedContinuousFunction.evalClm`
/-- The evaluation at a point, as a continuous linear map from `C(α, 𝕜)` to `𝕜`. -/
def evalClm (x : α) : C(α, E) →L[𝕜] E :=
(BoundedContinuousFunction.evalClm 𝕜 x).comp
(linearIsometryBoundedOfCompact α E 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_map.eval_clm ContinuousMap.evalClm
end
-- this lemma and the next are the analogues of those autogenerated by `@[simps]` for
-- `equivBoundedOfCompact`, `addEquivBoundedOfCompact`
@[simp]
theorem linearIsometryBoundedOfCompact_symm_apply (f : α →ᵇ E) :
(linearIsometryBoundedOfCompact α E 𝕜).symm f = f.toContinuousMap :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_symm_apply ContinuousMap.linearIsometryBoundedOfCompact_symm_apply
@[simp]
theorem linearIsometryBoundedOfCompact_apply_apply (f : C(α, E)) (a : α) :
(linearIsometryBoundedOfCompact α E 𝕜 f) a = f a :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_apply_apply ContinuousMap.linearIsometryBoundedOfCompact_apply_apply
@[simp]
theorem linearIsometryBoundedOfCompact_toIsometryEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toIsometryEquiv = isometryEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_isometry_equiv ContinuousMap.linearIsometryBoundedOfCompact_toIsometryEquiv
@[simp] -- porting note: adjusted LHS because `simpNF` complained it simplified.
theorem linearIsometryBoundedOfCompact_toAddEquiv :
((linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv : C(α, E) ≃+ (α →ᵇ E)) =
addEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_add_equiv ContinuousMap.linearIsometryBoundedOfCompact_toAddEquiv
@[simp]
theorem linearIsometryBoundedOfCompact_of_compact_toEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv.toEquiv = equivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_of_compact_to_equiv ContinuousMap.linearIsometryBoundedOfCompact_of_compact_toEquiv
end
section
variable {𝕜 : Type*} {γ : Type*} [NormedField 𝕜] [NormedRing γ] [NormedAlgebra 𝕜 γ]
instance : NormedAlgebra 𝕜 C(α, γ) :=
{ ContinuousMap.normedSpace, ContinuousMap.algebra with }
end
end ContinuousMap
namespace ContinuousMap
section UniformContinuity
variable {α β : Type*}
variable [MetricSpace α] [CompactSpace α] [MetricSpace β]
/-!
We now set up some declarations making it convenient to use uniform continuity.
-/
theorem uniform_continuity (f : C(α, β)) (ε : ℝ) (h : 0 < ε) :
∃ δ > 0, ∀ {x y}, dist x y < δ → dist (f x) (f y) < ε :=
Metric.uniformContinuous_iff.mp (CompactSpace.uniformContinuous_of_continuous f.continuous) ε h
#align continuous_map.uniform_continuity ContinuousMap.uniform_continuity
-- This definition allows us to separate the choice of some `δ`,
-- and the corresponding use of `dist a b < δ → dist (f a) (f b) < ε`,
-- even across different declarations.
/-- An arbitrarily chosen modulus of uniform continuity for a given function `f` and `ε > 0`. -/
def modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) : ℝ :=
Classical.choose (uniform_continuity f ε h)
#align continuous_map.modulus ContinuousMap.modulus
theorem modulus_pos (f : C(α, β)) {ε : ℝ} {h : 0 < ε} : 0 < f.modulus ε h :=
(Classical.choose_spec (uniform_continuity f ε h)).1
#align continuous_map.modulus_pos ContinuousMap.modulus_pos
theorem dist_lt_of_dist_lt_modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) {a b : α}
(w : dist a b < f.modulus ε h) : dist (f a) (f b) < ε :=
(Classical.choose_spec (uniform_continuity f ε h)).2 w
#align continuous_map.dist_lt_of_dist_lt_modulus ContinuousMap.dist_lt_of_dist_lt_modulus
end UniformContinuity
end ContinuousMap
section CompLeft
variable (X : Type*) {𝕜 β γ : Type*} [TopologicalSpace X] [CompactSpace X]
[NontriviallyNormedField 𝕜]
variable [NormedAddCommGroup β] [NormedSpace 𝕜 β] [NormedAddCommGroup γ] [NormedSpace 𝕜 γ]
open ContinuousMap
/-- Postcomposition of continuous functions into a normed module by a continuous linear map is a
continuous linear map.
Transferred version of `ContinuousLinearMap.compLeftContinuousBounded`,
upgraded version of `ContinuousLinearMap.compLeftContinuous`,
similar to `LinearMap.compLeft`. -/
protected def ContinuousLinearMap.compLeftContinuousCompact (g : β →L[𝕜] γ) :
C(X, β) →L[𝕜] C(X, γ) :=
(linearIsometryBoundedOfCompact X γ 𝕜).symm.toLinearIsometry.toContinuousLinearMap.comp <|
(g.compLeftContinuousBounded X).comp <|
(linearIsometryBoundedOfCompact X β 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_linear_map.comp_left_continuous_compact ContinuousLinearMap.compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.toLinear_compLeftContinuousCompact (g : β →L[𝕜] γ) :
(g.compLeftContinuousCompact X : C(X, β) →ₗ[𝕜] C(X, γ)) = g.compLeftContinuous 𝕜 X := by
ext f
rfl
#align continuous_linear_map.to_linear_comp_left_continuous_compact ContinuousLinearMap.toLinear_compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.compLeftContinuousCompact_apply (g : β →L[𝕜] γ) (f : C(X, β)) (x : X) :
g.compLeftContinuousCompact X f x = g (f x) :=
rfl
#align continuous_linear_map.comp_left_continuous_compact_apply ContinuousLinearMap.compLeftContinuousCompact_apply
end CompLeft
namespace ContinuousMap
/-!
We now setup variations on `compRight* f`, where `f : C(X, Y)`
(that is, precomposition by a continuous map),
as a morphism `C(Y, T) → C(X, T)`, respecting various types of structure.
In particular:
* `compRightContinuousMap`, the bundled continuous map (for this we need `X Y` compact).
* `compRightHomeomorph`, when we precompose by a homeomorphism.
* `compRightAlgHom`, when `T = R` is a topological ring.
-/
section CompRight
/-- Precomposition by a continuous map is itself a continuous map between spaces of continuous maps.
-/
def compRightContinuousMap {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y)) : C(C(Y, T), C(X, T)) where
toFun g := g.comp f
continuous_toFun := by
refine' Metric.continuous_iff.mpr _
intro g ε ε_pos
refine' ⟨ε, ε_pos, fun g' h => _⟩
rw [ContinuousMap.dist_lt_iff ε_pos] at h ⊢
exact fun x => h (f x)
#align continuous_map.comp_right_continuous_map ContinuousMap.compRightContinuousMap
@[simp]
theorem compRightContinuousMap_apply {X Y : Type*} (T : Type*) [TopologicalSpace X]
[CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y))
(g : C(Y, T)) : (compRightContinuousMap T f) g = g.comp f :=
rfl
#align continuous_map.comp_right_continuous_map_apply ContinuousMap.compRightContinuousMap_apply
/-- Precomposition by a homeomorphism is itself a homeomorphism between spaces of continuous maps.
-/
def compRightHomeomorph {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : X ≃ₜ Y) : C(Y, T) ≃ₜ C(X, T) where
toFun := compRightContinuousMap T f.toContinuousMap
invFun := compRightContinuousMap T f.symm.toContinuousMap
left_inv g := ext fun _ => congr_arg g (f.apply_symm_apply _)
right_inv g := ext fun _ => congr_arg g (f.symm_apply_apply _)
#align continuous_map.comp_right_homeomorph ContinuousMap.compRightHomeomorph
theorem compRightAlgHom_continuous {X Y : Type*} (R A : Type*) [TopologicalSpace X]
[CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [CommSemiring R] [Semiring A]
[MetricSpace A] [TopologicalSemiring A] [Algebra R A] (f : C(X, Y)) :
Continuous (compRightAlgHom R A f) :=
map_continuous (compRightContinuousMap A f)
#align continuous_map.comp_right_alg_hom_continuous ContinuousMap.compRightAlgHom_continuous
end CompRight
section LocalNormalConvergence
/-! ### Local normal convergence
A sum of continuous functions (on a locally compact space) is "locally normally convergent" if the
sum of its sup-norms on any compact subset is summable. This implies convergence in the topology
of `C(X, E)` (i.e. locally uniform convergence). -/
open TopologicalSpace
variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
variable {E : Type*} [NormedAddCommGroup E] [CompleteSpace E]
theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F := by
refine' (ContinuousMap.exists_tendsto_compactOpen_iff_forall _).2 fun K hK => _
lift K to Compacts X using hK
|
have A : ∀ s : Finset ι, restrict (↑K) (∑ i in s, F i) = ∑ i in s, restrict K (F i) := by
intro s
ext1 x
simp
-- This used to be the end of the proof before leanprover/lean4#2644
erw [restrict_apply, restrict_apply, restrict_apply, restrict_apply]
simp? says simp only [coe_sum, Finset.sum_apply]
congr!
|
theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F := by
refine' (ContinuousMap.exists_tendsto_compactOpen_iff_forall _).2 fun K hK => _
lift K to Compacts X using hK
|
Mathlib.Topology.ContinuousFunction.Compact.491_0.Mig2jTVnn2FLKEB
|
theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F
|
Mathlib_Topology_ContinuousFunction_Compact
|
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : LocallyCompactSpace X
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
ι : Type u_3
F : ι → C(X, E)
hF : ∀ (K : Compacts X), Summable fun i => ‖restrict (↑K) (F i)‖
K : Compacts X
⊢ ∀ (s : Finset ι), restrict (↑K) (∑ i in s, F i) = ∑ i in s, restrict (↑K) (F i)
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compacts
#align_import topology.continuous_function.compact from "leanprover-community/mathlib"@"d3af0609f6db8691dffdc3e1fb7feb7da72698f2"
/-!
# Continuous functions on a compact space
Continuous functions `C(α, β)` from a compact space `α` to a metric space `β`
are automatically bounded, and so acquire various structures inherited from `α →ᵇ β`.
This file transfers these structures, and restates some lemmas
characterising these structures.
If you need a lemma which is proved about `α →ᵇ β` but not for `C(α, β)` when `α` is compact,
you should restate it here. You can also use
`ContinuousMap.equivBoundedOfCompact` to move functions back and forth.
-/
noncomputable section
open Topology Classical NNReal BoundedContinuousFunction BigOperators
open Set Filter Metric
open BoundedContinuousFunction
namespace ContinuousMap
variable {α β E : Type*} [TopologicalSpace α] [CompactSpace α] [MetricSpace β]
[NormedAddCommGroup E]
section
variable (α β)
/-- When `α` is compact, the bounded continuous maps `α →ᵇ β` are
equivalent to `C(α, β)`.
-/
@[simps (config := .asFn)]
def equivBoundedOfCompact : C(α, β) ≃ (α →ᵇ β) :=
⟨mkOfCompact, BoundedContinuousFunction.toContinuousMap, fun f => by
ext
rfl, fun f => by
ext
rfl⟩
#align continuous_map.equiv_bounded_of_compact ContinuousMap.equivBoundedOfCompact
theorem uniformInducing_equivBoundedOfCompact : UniformInducing (equivBoundedOfCompact α β) :=
UniformInducing.mk'
(by
simp only [hasBasis_compactConvergenceUniformity.mem_iff, uniformity_basis_dist_le.mem_iff]
exact fun s =>
⟨fun ⟨⟨a, b⟩, ⟨_, ⟨ε, hε, hb⟩⟩, hs⟩ =>
⟨{ p | ∀ x, (p.1 x, p.2 x) ∈ b }, ⟨ε, hε, fun _ h x => hb ((dist_le hε.le).mp h x)⟩,
fun f g h => hs fun x _ => h x⟩,
fun ⟨_, ⟨ε, hε, ht⟩, hs⟩ =>
⟨⟨Set.univ, { p | dist p.1 p.2 ≤ ε }⟩, ⟨isCompact_univ, ⟨ε, hε, fun _ h => h⟩⟩,
fun ⟨f, g⟩ h => hs _ _ (ht ((dist_le hε.le).mpr fun x => h x (mem_univ x)))⟩⟩)
#align continuous_map.uniform_inducing_equiv_bounded_of_compact ContinuousMap.uniformInducing_equivBoundedOfCompact
theorem uniformEmbedding_equivBoundedOfCompact : UniformEmbedding (equivBoundedOfCompact α β) :=
{ uniformInducing_equivBoundedOfCompact α β with inj := (equivBoundedOfCompact α β).injective }
#align continuous_map.uniform_embedding_equiv_bounded_of_compact ContinuousMap.uniformEmbedding_equivBoundedOfCompact
/-- When `α` is compact, the bounded continuous maps `α →ᵇ 𝕜` are
additively equivalent to `C(α, 𝕜)`.
-/
-- porting note: the following `simps` received a "maximum recursion depth" error
-- @[simps! (config := .asFn) apply symm_apply]
def addEquivBoundedOfCompact [AddMonoid β] [LipschitzAdd β] : C(α, β) ≃+ (α →ᵇ β) :=
({ toContinuousMapAddHom α β, (equivBoundedOfCompact α β).symm with } : (α →ᵇ β) ≃+ C(α, β)).symm
#align continuous_map.add_equiv_bounded_of_compact ContinuousMap.addEquivBoundedOfCompact
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_symm_apply [AddMonoid β] [LipschitzAdd β] :
⇑((addEquivBoundedOfCompact α β).symm) = toContinuousMapAddHom α β :=
rfl
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_apply [AddMonoid β] [LipschitzAdd β] :
⇑(addEquivBoundedOfCompact α β) = mkOfCompact :=
rfl
instance metricSpace : MetricSpace C(α, β) :=
(uniformEmbedding_equivBoundedOfCompact α β).comapMetricSpace _
#align continuous_map.metric_space ContinuousMap.metricSpace
/-- When `α` is compact, and `β` is a metric space, the bounded continuous maps `α →ᵇ β` are
isometric to `C(α, β)`.
-/
@[simps! (config := .asFn) toEquiv apply symm_apply]
def isometryEquivBoundedOfCompact : C(α, β) ≃ᵢ (α →ᵇ β) where
isometry_toFun _ _ := rfl
toEquiv := equivBoundedOfCompact α β
#align continuous_map.isometry_equiv_bounded_of_compact ContinuousMap.isometryEquivBoundedOfCompact
end
@[simp]
theorem _root_.BoundedContinuousFunction.dist_mkOfCompact (f g : C(α, β)) :
dist (mkOfCompact f) (mkOfCompact g) = dist f g :=
rfl
#align bounded_continuous_function.dist_mk_of_compact BoundedContinuousFunction.dist_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.dist_toContinuousMap (f g : α →ᵇ β) :
dist f.toContinuousMap g.toContinuousMap = dist f g :=
rfl
#align bounded_continuous_function.dist_to_continuous_map BoundedContinuousFunction.dist_toContinuousMap
open BoundedContinuousFunction
section
variable {f g : C(α, β)} {C : ℝ}
/-- The pointwise distance is controlled by the distance between functions, by definition. -/
theorem dist_apply_le_dist (x : α) : dist (f x) (g x) ≤ dist f g := by
simp only [← dist_mkOfCompact, dist_coe_le_dist, ← mkOfCompact_apply]
#align continuous_map.dist_apply_le_dist ContinuousMap.dist_apply_le_dist
/-- The distance between two functions is controlled by the supremum of the pointwise distances. -/
theorem dist_le (C0 : (0 : ℝ) ≤ C) : dist f g ≤ C ↔ ∀ x : α, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le C0, mkOfCompact_apply]
#align continuous_map.dist_le ContinuousMap.dist_le
theorem dist_le_iff_of_nonempty [Nonempty α] : dist f g ≤ C ↔ ∀ x, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le_iff_of_nonempty,
mkOfCompact_apply]
#align continuous_map.dist_le_iff_of_nonempty ContinuousMap.dist_le_iff_of_nonempty
theorem dist_lt_iff_of_nonempty [Nonempty α] : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
simp only [← dist_mkOfCompact, dist_lt_iff_of_nonempty_compact, mkOfCompact_apply]
#align continuous_map.dist_lt_iff_of_nonempty ContinuousMap.dist_lt_iff_of_nonempty
theorem dist_lt_of_nonempty [Nonempty α] (w : ∀ x : α, dist (f x) (g x) < C) : dist f g < C :=
dist_lt_iff_of_nonempty.2 w
#align continuous_map.dist_lt_of_nonempty ContinuousMap.dist_lt_of_nonempty
theorem dist_lt_iff (C0 : (0 : ℝ) < C) : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
rw [← dist_mkOfCompact, dist_lt_iff_of_compact C0]
simp only [mkOfCompact_apply]
#align continuous_map.dist_lt_iff ContinuousMap.dist_lt_iff
end
instance [CompleteSpace β] : CompleteSpace C(α, β) :=
(isometryEquivBoundedOfCompact α β).completeSpace
/-- See also `ContinuousMap.continuous_eval'`. -/
@[continuity]
theorem continuous_eval : Continuous fun p : C(α, β) × α => p.1 p.2 :=
continuous_eval.comp ((isometryEquivBoundedOfCompact α β).continuous.prod_map continuous_id)
#align continuous_map.continuous_eval ContinuousMap.continuous_eval
-- TODO at some point we will need lemmas characterising this norm!
-- At the moment the only way to reason about it is to transfer `f : C(α,E)` back to `α →ᵇ E`.
instance : Norm C(α, E) where norm x := dist x 0
@[simp]
theorem _root_.BoundedContinuousFunction.norm_mkOfCompact (f : C(α, E)) : ‖mkOfCompact f‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_mk_of_compact BoundedContinuousFunction.norm_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.norm_toContinuousMap_eq (f : α →ᵇ E) :
‖f.toContinuousMap‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_to_continuous_map_eq BoundedContinuousFunction.norm_toContinuousMap_eq
open BoundedContinuousFunction
instance : NormedAddCommGroup C(α, E) :=
{ ContinuousMap.metricSpace _ _,
ContinuousMap.instAddCommGroupContinuousMap with
dist_eq := fun x y => by
rw [← norm_mkOfCompact, ← dist_mkOfCompact, dist_eq_norm, mkOfCompact_sub]
dist := dist
norm := norm }
instance [Nonempty α] [One E] [NormOneClass E] : NormOneClass C(α, E) where
norm_one := by simp only [← norm_mkOfCompact, mkOfCompact_one, norm_one]
section
variable (f : C(α, E))
-- The corresponding lemmas for `BoundedContinuousFunction` are stated with `{f}`,
-- and so can not be used in dot notation.
theorem norm_coe_le_norm (x : α) : ‖f x‖ ≤ ‖f‖ :=
(mkOfCompact f).norm_coe_le_norm x
#align continuous_map.norm_coe_le_norm ContinuousMap.norm_coe_le_norm
/-- Distance between the images of any two points is at most twice the norm of the function. -/
theorem dist_le_two_norm (x y : α) : dist (f x) (f y) ≤ 2 * ‖f‖ :=
(mkOfCompact f).dist_le_two_norm x y
#align continuous_map.dist_le_two_norm ContinuousMap.dist_le_two_norm
/-- The norm of a function is controlled by the supremum of the pointwise norms. -/
theorem norm_le {C : ℝ} (C0 : (0 : ℝ) ≤ C) : ‖f‖ ≤ C ↔ ∀ x : α, ‖f x‖ ≤ C :=
@BoundedContinuousFunction.norm_le _ _ _ _ (mkOfCompact f) _ C0
#align continuous_map.norm_le ContinuousMap.norm_le
theorem norm_le_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ ≤ M ↔ ∀ x, ‖f x‖ ≤ M :=
@BoundedContinuousFunction.norm_le_of_nonempty _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_le_of_nonempty ContinuousMap.norm_le_of_nonempty
theorem norm_lt_iff {M : ℝ} (M0 : 0 < M) : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_compact _ _ _ _ _ (mkOfCompact f) _ M0
#align continuous_map.norm_lt_iff ContinuousMap.norm_lt_iff
theorem nnnorm_lt_iff {M : ℝ≥0} (M0 : 0 < M) : ‖f‖₊ < M ↔ ∀ x : α, ‖f x‖₊ < M :=
f.norm_lt_iff M0
#align continuous_map.nnnorm_lt_iff ContinuousMap.nnnorm_lt_iff
theorem norm_lt_iff_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_nonempty_compact _ _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_lt_iff_of_nonempty ContinuousMap.norm_lt_iff_of_nonempty
theorem nnnorm_lt_iff_of_nonempty [Nonempty α] {M : ℝ≥0} : ‖f‖₊ < M ↔ ∀ x, ‖f x‖₊ < M :=
f.norm_lt_iff_of_nonempty
#align continuous_map.nnnorm_lt_iff_of_nonempty ContinuousMap.nnnorm_lt_iff_of_nonempty
theorem apply_le_norm (f : C(α, ℝ)) (x : α) : f x ≤ ‖f‖ :=
le_trans (le_abs.mpr (Or.inl (le_refl (f x)))) (f.norm_coe_le_norm x)
#align continuous_map.apply_le_norm ContinuousMap.apply_le_norm
theorem neg_norm_le_apply (f : C(α, ℝ)) (x : α) : -‖f‖ ≤ f x :=
le_trans (neg_le_neg (f.norm_coe_le_norm x)) (neg_le.mp (neg_le_abs_self (f x)))
#align continuous_map.neg_norm_le_apply ContinuousMap.neg_norm_le_apply
theorem norm_eq_iSup_norm : ‖f‖ = ⨆ x : α, ‖f x‖ :=
(mkOfCompact f).norm_eq_iSup_norm
#align continuous_map.norm_eq_supr_norm ContinuousMap.norm_eq_iSup_norm
theorem norm_restrict_mono_set {X : Type*} [TopologicalSpace X] (f : C(X, E))
{K L : TopologicalSpace.Compacts X} (hKL : K ≤ L) : ‖f.restrict K‖ ≤ ‖f.restrict L‖ :=
(norm_le _ (norm_nonneg _)).mpr fun x => norm_coe_le_norm (f.restrict L) <| Set.inclusion hKL x
#align continuous_map.norm_restrict_mono_set ContinuousMap.norm_restrict_mono_set
end
section
variable {R : Type*} [NormedRing R]
instance : NormedRing C(α, R) :=
{ (inferInstance : NormedAddCommGroup C(α, R)), ContinuousMap.instRingContinuousMap with
norm_mul := fun f g => norm_mul_le (mkOfCompact f) (mkOfCompact g) }
end
section
variable {𝕜 : Type*} [NormedField 𝕜] [NormedSpace 𝕜 E]
instance normedSpace : NormedSpace 𝕜 C(α, E) where
norm_smul_le c f := (norm_smul_le c (mkOfCompact f) : _)
#align continuous_map.normed_space ContinuousMap.normedSpace
section
variable (α 𝕜 E)
/-- When `α` is compact and `𝕜` is a normed field,
the `𝕜`-algebra of bounded continuous maps `α →ᵇ β` is
`𝕜`-linearly isometric to `C(α, β)`.
-/
def linearIsometryBoundedOfCompact : C(α, E) ≃ₗᵢ[𝕜] α →ᵇ E :=
{ addEquivBoundedOfCompact α E with
map_smul' := fun c f => by
ext
norm_cast
norm_map' := fun f => rfl }
#align continuous_map.linear_isometry_bounded_of_compact ContinuousMap.linearIsometryBoundedOfCompact
variable {α E}
-- to match `BoundedContinuousFunction.evalClm`
/-- The evaluation at a point, as a continuous linear map from `C(α, 𝕜)` to `𝕜`. -/
def evalClm (x : α) : C(α, E) →L[𝕜] E :=
(BoundedContinuousFunction.evalClm 𝕜 x).comp
(linearIsometryBoundedOfCompact α E 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_map.eval_clm ContinuousMap.evalClm
end
-- this lemma and the next are the analogues of those autogenerated by `@[simps]` for
-- `equivBoundedOfCompact`, `addEquivBoundedOfCompact`
@[simp]
theorem linearIsometryBoundedOfCompact_symm_apply (f : α →ᵇ E) :
(linearIsometryBoundedOfCompact α E 𝕜).symm f = f.toContinuousMap :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_symm_apply ContinuousMap.linearIsometryBoundedOfCompact_symm_apply
@[simp]
theorem linearIsometryBoundedOfCompact_apply_apply (f : C(α, E)) (a : α) :
(linearIsometryBoundedOfCompact α E 𝕜 f) a = f a :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_apply_apply ContinuousMap.linearIsometryBoundedOfCompact_apply_apply
@[simp]
theorem linearIsometryBoundedOfCompact_toIsometryEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toIsometryEquiv = isometryEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_isometry_equiv ContinuousMap.linearIsometryBoundedOfCompact_toIsometryEquiv
@[simp] -- porting note: adjusted LHS because `simpNF` complained it simplified.
theorem linearIsometryBoundedOfCompact_toAddEquiv :
((linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv : C(α, E) ≃+ (α →ᵇ E)) =
addEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_add_equiv ContinuousMap.linearIsometryBoundedOfCompact_toAddEquiv
@[simp]
theorem linearIsometryBoundedOfCompact_of_compact_toEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv.toEquiv = equivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_of_compact_to_equiv ContinuousMap.linearIsometryBoundedOfCompact_of_compact_toEquiv
end
section
variable {𝕜 : Type*} {γ : Type*} [NormedField 𝕜] [NormedRing γ] [NormedAlgebra 𝕜 γ]
instance : NormedAlgebra 𝕜 C(α, γ) :=
{ ContinuousMap.normedSpace, ContinuousMap.algebra with }
end
end ContinuousMap
namespace ContinuousMap
section UniformContinuity
variable {α β : Type*}
variable [MetricSpace α] [CompactSpace α] [MetricSpace β]
/-!
We now set up some declarations making it convenient to use uniform continuity.
-/
theorem uniform_continuity (f : C(α, β)) (ε : ℝ) (h : 0 < ε) :
∃ δ > 0, ∀ {x y}, dist x y < δ → dist (f x) (f y) < ε :=
Metric.uniformContinuous_iff.mp (CompactSpace.uniformContinuous_of_continuous f.continuous) ε h
#align continuous_map.uniform_continuity ContinuousMap.uniform_continuity
-- This definition allows us to separate the choice of some `δ`,
-- and the corresponding use of `dist a b < δ → dist (f a) (f b) < ε`,
-- even across different declarations.
/-- An arbitrarily chosen modulus of uniform continuity for a given function `f` and `ε > 0`. -/
def modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) : ℝ :=
Classical.choose (uniform_continuity f ε h)
#align continuous_map.modulus ContinuousMap.modulus
theorem modulus_pos (f : C(α, β)) {ε : ℝ} {h : 0 < ε} : 0 < f.modulus ε h :=
(Classical.choose_spec (uniform_continuity f ε h)).1
#align continuous_map.modulus_pos ContinuousMap.modulus_pos
theorem dist_lt_of_dist_lt_modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) {a b : α}
(w : dist a b < f.modulus ε h) : dist (f a) (f b) < ε :=
(Classical.choose_spec (uniform_continuity f ε h)).2 w
#align continuous_map.dist_lt_of_dist_lt_modulus ContinuousMap.dist_lt_of_dist_lt_modulus
end UniformContinuity
end ContinuousMap
section CompLeft
variable (X : Type*) {𝕜 β γ : Type*} [TopologicalSpace X] [CompactSpace X]
[NontriviallyNormedField 𝕜]
variable [NormedAddCommGroup β] [NormedSpace 𝕜 β] [NormedAddCommGroup γ] [NormedSpace 𝕜 γ]
open ContinuousMap
/-- Postcomposition of continuous functions into a normed module by a continuous linear map is a
continuous linear map.
Transferred version of `ContinuousLinearMap.compLeftContinuousBounded`,
upgraded version of `ContinuousLinearMap.compLeftContinuous`,
similar to `LinearMap.compLeft`. -/
protected def ContinuousLinearMap.compLeftContinuousCompact (g : β →L[𝕜] γ) :
C(X, β) →L[𝕜] C(X, γ) :=
(linearIsometryBoundedOfCompact X γ 𝕜).symm.toLinearIsometry.toContinuousLinearMap.comp <|
(g.compLeftContinuousBounded X).comp <|
(linearIsometryBoundedOfCompact X β 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_linear_map.comp_left_continuous_compact ContinuousLinearMap.compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.toLinear_compLeftContinuousCompact (g : β →L[𝕜] γ) :
(g.compLeftContinuousCompact X : C(X, β) →ₗ[𝕜] C(X, γ)) = g.compLeftContinuous 𝕜 X := by
ext f
rfl
#align continuous_linear_map.to_linear_comp_left_continuous_compact ContinuousLinearMap.toLinear_compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.compLeftContinuousCompact_apply (g : β →L[𝕜] γ) (f : C(X, β)) (x : X) :
g.compLeftContinuousCompact X f x = g (f x) :=
rfl
#align continuous_linear_map.comp_left_continuous_compact_apply ContinuousLinearMap.compLeftContinuousCompact_apply
end CompLeft
namespace ContinuousMap
/-!
We now setup variations on `compRight* f`, where `f : C(X, Y)`
(that is, precomposition by a continuous map),
as a morphism `C(Y, T) → C(X, T)`, respecting various types of structure.
In particular:
* `compRightContinuousMap`, the bundled continuous map (for this we need `X Y` compact).
* `compRightHomeomorph`, when we precompose by a homeomorphism.
* `compRightAlgHom`, when `T = R` is a topological ring.
-/
section CompRight
/-- Precomposition by a continuous map is itself a continuous map between spaces of continuous maps.
-/
def compRightContinuousMap {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y)) : C(C(Y, T), C(X, T)) where
toFun g := g.comp f
continuous_toFun := by
refine' Metric.continuous_iff.mpr _
intro g ε ε_pos
refine' ⟨ε, ε_pos, fun g' h => _⟩
rw [ContinuousMap.dist_lt_iff ε_pos] at h ⊢
exact fun x => h (f x)
#align continuous_map.comp_right_continuous_map ContinuousMap.compRightContinuousMap
@[simp]
theorem compRightContinuousMap_apply {X Y : Type*} (T : Type*) [TopologicalSpace X]
[CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y))
(g : C(Y, T)) : (compRightContinuousMap T f) g = g.comp f :=
rfl
#align continuous_map.comp_right_continuous_map_apply ContinuousMap.compRightContinuousMap_apply
/-- Precomposition by a homeomorphism is itself a homeomorphism between spaces of continuous maps.
-/
def compRightHomeomorph {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : X ≃ₜ Y) : C(Y, T) ≃ₜ C(X, T) where
toFun := compRightContinuousMap T f.toContinuousMap
invFun := compRightContinuousMap T f.symm.toContinuousMap
left_inv g := ext fun _ => congr_arg g (f.apply_symm_apply _)
right_inv g := ext fun _ => congr_arg g (f.symm_apply_apply _)
#align continuous_map.comp_right_homeomorph ContinuousMap.compRightHomeomorph
theorem compRightAlgHom_continuous {X Y : Type*} (R A : Type*) [TopologicalSpace X]
[CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [CommSemiring R] [Semiring A]
[MetricSpace A] [TopologicalSemiring A] [Algebra R A] (f : C(X, Y)) :
Continuous (compRightAlgHom R A f) :=
map_continuous (compRightContinuousMap A f)
#align continuous_map.comp_right_alg_hom_continuous ContinuousMap.compRightAlgHom_continuous
end CompRight
section LocalNormalConvergence
/-! ### Local normal convergence
A sum of continuous functions (on a locally compact space) is "locally normally convergent" if the
sum of its sup-norms on any compact subset is summable. This implies convergence in the topology
of `C(X, E)` (i.e. locally uniform convergence). -/
open TopologicalSpace
variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
variable {E : Type*} [NormedAddCommGroup E] [CompleteSpace E]
theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F := by
refine' (ContinuousMap.exists_tendsto_compactOpen_iff_forall _).2 fun K hK => _
lift K to Compacts X using hK
have A : ∀ s : Finset ι, restrict (↑K) (∑ i in s, F i) = ∑ i in s, restrict K (F i) := by
|
intro s
|
theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F := by
refine' (ContinuousMap.exists_tendsto_compactOpen_iff_forall _).2 fun K hK => _
lift K to Compacts X using hK
have A : ∀ s : Finset ι, restrict (↑K) (∑ i in s, F i) = ∑ i in s, restrict K (F i) := by
|
Mathlib.Topology.ContinuousFunction.Compact.491_0.Mig2jTVnn2FLKEB
|
theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F
|
Mathlib_Topology_ContinuousFunction_Compact
|
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : LocallyCompactSpace X
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
ι : Type u_3
F : ι → C(X, E)
hF : ∀ (K : Compacts X), Summable fun i => ‖restrict (↑K) (F i)‖
K : Compacts X
s : Finset ι
⊢ restrict (↑K) (∑ i in s, F i) = ∑ i in s, restrict (↑K) (F i)
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compacts
#align_import topology.continuous_function.compact from "leanprover-community/mathlib"@"d3af0609f6db8691dffdc3e1fb7feb7da72698f2"
/-!
# Continuous functions on a compact space
Continuous functions `C(α, β)` from a compact space `α` to a metric space `β`
are automatically bounded, and so acquire various structures inherited from `α →ᵇ β`.
This file transfers these structures, and restates some lemmas
characterising these structures.
If you need a lemma which is proved about `α →ᵇ β` but not for `C(α, β)` when `α` is compact,
you should restate it here. You can also use
`ContinuousMap.equivBoundedOfCompact` to move functions back and forth.
-/
noncomputable section
open Topology Classical NNReal BoundedContinuousFunction BigOperators
open Set Filter Metric
open BoundedContinuousFunction
namespace ContinuousMap
variable {α β E : Type*} [TopologicalSpace α] [CompactSpace α] [MetricSpace β]
[NormedAddCommGroup E]
section
variable (α β)
/-- When `α` is compact, the bounded continuous maps `α →ᵇ β` are
equivalent to `C(α, β)`.
-/
@[simps (config := .asFn)]
def equivBoundedOfCompact : C(α, β) ≃ (α →ᵇ β) :=
⟨mkOfCompact, BoundedContinuousFunction.toContinuousMap, fun f => by
ext
rfl, fun f => by
ext
rfl⟩
#align continuous_map.equiv_bounded_of_compact ContinuousMap.equivBoundedOfCompact
theorem uniformInducing_equivBoundedOfCompact : UniformInducing (equivBoundedOfCompact α β) :=
UniformInducing.mk'
(by
simp only [hasBasis_compactConvergenceUniformity.mem_iff, uniformity_basis_dist_le.mem_iff]
exact fun s =>
⟨fun ⟨⟨a, b⟩, ⟨_, ⟨ε, hε, hb⟩⟩, hs⟩ =>
⟨{ p | ∀ x, (p.1 x, p.2 x) ∈ b }, ⟨ε, hε, fun _ h x => hb ((dist_le hε.le).mp h x)⟩,
fun f g h => hs fun x _ => h x⟩,
fun ⟨_, ⟨ε, hε, ht⟩, hs⟩ =>
⟨⟨Set.univ, { p | dist p.1 p.2 ≤ ε }⟩, ⟨isCompact_univ, ⟨ε, hε, fun _ h => h⟩⟩,
fun ⟨f, g⟩ h => hs _ _ (ht ((dist_le hε.le).mpr fun x => h x (mem_univ x)))⟩⟩)
#align continuous_map.uniform_inducing_equiv_bounded_of_compact ContinuousMap.uniformInducing_equivBoundedOfCompact
theorem uniformEmbedding_equivBoundedOfCompact : UniformEmbedding (equivBoundedOfCompact α β) :=
{ uniformInducing_equivBoundedOfCompact α β with inj := (equivBoundedOfCompact α β).injective }
#align continuous_map.uniform_embedding_equiv_bounded_of_compact ContinuousMap.uniformEmbedding_equivBoundedOfCompact
/-- When `α` is compact, the bounded continuous maps `α →ᵇ 𝕜` are
additively equivalent to `C(α, 𝕜)`.
-/
-- porting note: the following `simps` received a "maximum recursion depth" error
-- @[simps! (config := .asFn) apply symm_apply]
def addEquivBoundedOfCompact [AddMonoid β] [LipschitzAdd β] : C(α, β) ≃+ (α →ᵇ β) :=
({ toContinuousMapAddHom α β, (equivBoundedOfCompact α β).symm with } : (α →ᵇ β) ≃+ C(α, β)).symm
#align continuous_map.add_equiv_bounded_of_compact ContinuousMap.addEquivBoundedOfCompact
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_symm_apply [AddMonoid β] [LipschitzAdd β] :
⇑((addEquivBoundedOfCompact α β).symm) = toContinuousMapAddHom α β :=
rfl
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_apply [AddMonoid β] [LipschitzAdd β] :
⇑(addEquivBoundedOfCompact α β) = mkOfCompact :=
rfl
instance metricSpace : MetricSpace C(α, β) :=
(uniformEmbedding_equivBoundedOfCompact α β).comapMetricSpace _
#align continuous_map.metric_space ContinuousMap.metricSpace
/-- When `α` is compact, and `β` is a metric space, the bounded continuous maps `α →ᵇ β` are
isometric to `C(α, β)`.
-/
@[simps! (config := .asFn) toEquiv apply symm_apply]
def isometryEquivBoundedOfCompact : C(α, β) ≃ᵢ (α →ᵇ β) where
isometry_toFun _ _ := rfl
toEquiv := equivBoundedOfCompact α β
#align continuous_map.isometry_equiv_bounded_of_compact ContinuousMap.isometryEquivBoundedOfCompact
end
@[simp]
theorem _root_.BoundedContinuousFunction.dist_mkOfCompact (f g : C(α, β)) :
dist (mkOfCompact f) (mkOfCompact g) = dist f g :=
rfl
#align bounded_continuous_function.dist_mk_of_compact BoundedContinuousFunction.dist_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.dist_toContinuousMap (f g : α →ᵇ β) :
dist f.toContinuousMap g.toContinuousMap = dist f g :=
rfl
#align bounded_continuous_function.dist_to_continuous_map BoundedContinuousFunction.dist_toContinuousMap
open BoundedContinuousFunction
section
variable {f g : C(α, β)} {C : ℝ}
/-- The pointwise distance is controlled by the distance between functions, by definition. -/
theorem dist_apply_le_dist (x : α) : dist (f x) (g x) ≤ dist f g := by
simp only [← dist_mkOfCompact, dist_coe_le_dist, ← mkOfCompact_apply]
#align continuous_map.dist_apply_le_dist ContinuousMap.dist_apply_le_dist
/-- The distance between two functions is controlled by the supremum of the pointwise distances. -/
theorem dist_le (C0 : (0 : ℝ) ≤ C) : dist f g ≤ C ↔ ∀ x : α, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le C0, mkOfCompact_apply]
#align continuous_map.dist_le ContinuousMap.dist_le
theorem dist_le_iff_of_nonempty [Nonempty α] : dist f g ≤ C ↔ ∀ x, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le_iff_of_nonempty,
mkOfCompact_apply]
#align continuous_map.dist_le_iff_of_nonempty ContinuousMap.dist_le_iff_of_nonempty
theorem dist_lt_iff_of_nonempty [Nonempty α] : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
simp only [← dist_mkOfCompact, dist_lt_iff_of_nonempty_compact, mkOfCompact_apply]
#align continuous_map.dist_lt_iff_of_nonempty ContinuousMap.dist_lt_iff_of_nonempty
theorem dist_lt_of_nonempty [Nonempty α] (w : ∀ x : α, dist (f x) (g x) < C) : dist f g < C :=
dist_lt_iff_of_nonempty.2 w
#align continuous_map.dist_lt_of_nonempty ContinuousMap.dist_lt_of_nonempty
theorem dist_lt_iff (C0 : (0 : ℝ) < C) : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
rw [← dist_mkOfCompact, dist_lt_iff_of_compact C0]
simp only [mkOfCompact_apply]
#align continuous_map.dist_lt_iff ContinuousMap.dist_lt_iff
end
instance [CompleteSpace β] : CompleteSpace C(α, β) :=
(isometryEquivBoundedOfCompact α β).completeSpace
/-- See also `ContinuousMap.continuous_eval'`. -/
@[continuity]
theorem continuous_eval : Continuous fun p : C(α, β) × α => p.1 p.2 :=
continuous_eval.comp ((isometryEquivBoundedOfCompact α β).continuous.prod_map continuous_id)
#align continuous_map.continuous_eval ContinuousMap.continuous_eval
-- TODO at some point we will need lemmas characterising this norm!
-- At the moment the only way to reason about it is to transfer `f : C(α,E)` back to `α →ᵇ E`.
instance : Norm C(α, E) where norm x := dist x 0
@[simp]
theorem _root_.BoundedContinuousFunction.norm_mkOfCompact (f : C(α, E)) : ‖mkOfCompact f‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_mk_of_compact BoundedContinuousFunction.norm_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.norm_toContinuousMap_eq (f : α →ᵇ E) :
‖f.toContinuousMap‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_to_continuous_map_eq BoundedContinuousFunction.norm_toContinuousMap_eq
open BoundedContinuousFunction
instance : NormedAddCommGroup C(α, E) :=
{ ContinuousMap.metricSpace _ _,
ContinuousMap.instAddCommGroupContinuousMap with
dist_eq := fun x y => by
rw [← norm_mkOfCompact, ← dist_mkOfCompact, dist_eq_norm, mkOfCompact_sub]
dist := dist
norm := norm }
instance [Nonempty α] [One E] [NormOneClass E] : NormOneClass C(α, E) where
norm_one := by simp only [← norm_mkOfCompact, mkOfCompact_one, norm_one]
section
variable (f : C(α, E))
-- The corresponding lemmas for `BoundedContinuousFunction` are stated with `{f}`,
-- and so can not be used in dot notation.
theorem norm_coe_le_norm (x : α) : ‖f x‖ ≤ ‖f‖ :=
(mkOfCompact f).norm_coe_le_norm x
#align continuous_map.norm_coe_le_norm ContinuousMap.norm_coe_le_norm
/-- Distance between the images of any two points is at most twice the norm of the function. -/
theorem dist_le_two_norm (x y : α) : dist (f x) (f y) ≤ 2 * ‖f‖ :=
(mkOfCompact f).dist_le_two_norm x y
#align continuous_map.dist_le_two_norm ContinuousMap.dist_le_two_norm
/-- The norm of a function is controlled by the supremum of the pointwise norms. -/
theorem norm_le {C : ℝ} (C0 : (0 : ℝ) ≤ C) : ‖f‖ ≤ C ↔ ∀ x : α, ‖f x‖ ≤ C :=
@BoundedContinuousFunction.norm_le _ _ _ _ (mkOfCompact f) _ C0
#align continuous_map.norm_le ContinuousMap.norm_le
theorem norm_le_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ ≤ M ↔ ∀ x, ‖f x‖ ≤ M :=
@BoundedContinuousFunction.norm_le_of_nonempty _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_le_of_nonempty ContinuousMap.norm_le_of_nonempty
theorem norm_lt_iff {M : ℝ} (M0 : 0 < M) : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_compact _ _ _ _ _ (mkOfCompact f) _ M0
#align continuous_map.norm_lt_iff ContinuousMap.norm_lt_iff
theorem nnnorm_lt_iff {M : ℝ≥0} (M0 : 0 < M) : ‖f‖₊ < M ↔ ∀ x : α, ‖f x‖₊ < M :=
f.norm_lt_iff M0
#align continuous_map.nnnorm_lt_iff ContinuousMap.nnnorm_lt_iff
theorem norm_lt_iff_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_nonempty_compact _ _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_lt_iff_of_nonempty ContinuousMap.norm_lt_iff_of_nonempty
theorem nnnorm_lt_iff_of_nonempty [Nonempty α] {M : ℝ≥0} : ‖f‖₊ < M ↔ ∀ x, ‖f x‖₊ < M :=
f.norm_lt_iff_of_nonempty
#align continuous_map.nnnorm_lt_iff_of_nonempty ContinuousMap.nnnorm_lt_iff_of_nonempty
theorem apply_le_norm (f : C(α, ℝ)) (x : α) : f x ≤ ‖f‖ :=
le_trans (le_abs.mpr (Or.inl (le_refl (f x)))) (f.norm_coe_le_norm x)
#align continuous_map.apply_le_norm ContinuousMap.apply_le_norm
theorem neg_norm_le_apply (f : C(α, ℝ)) (x : α) : -‖f‖ ≤ f x :=
le_trans (neg_le_neg (f.norm_coe_le_norm x)) (neg_le.mp (neg_le_abs_self (f x)))
#align continuous_map.neg_norm_le_apply ContinuousMap.neg_norm_le_apply
theorem norm_eq_iSup_norm : ‖f‖ = ⨆ x : α, ‖f x‖ :=
(mkOfCompact f).norm_eq_iSup_norm
#align continuous_map.norm_eq_supr_norm ContinuousMap.norm_eq_iSup_norm
theorem norm_restrict_mono_set {X : Type*} [TopologicalSpace X] (f : C(X, E))
{K L : TopologicalSpace.Compacts X} (hKL : K ≤ L) : ‖f.restrict K‖ ≤ ‖f.restrict L‖ :=
(norm_le _ (norm_nonneg _)).mpr fun x => norm_coe_le_norm (f.restrict L) <| Set.inclusion hKL x
#align continuous_map.norm_restrict_mono_set ContinuousMap.norm_restrict_mono_set
end
section
variable {R : Type*} [NormedRing R]
instance : NormedRing C(α, R) :=
{ (inferInstance : NormedAddCommGroup C(α, R)), ContinuousMap.instRingContinuousMap with
norm_mul := fun f g => norm_mul_le (mkOfCompact f) (mkOfCompact g) }
end
section
variable {𝕜 : Type*} [NormedField 𝕜] [NormedSpace 𝕜 E]
instance normedSpace : NormedSpace 𝕜 C(α, E) where
norm_smul_le c f := (norm_smul_le c (mkOfCompact f) : _)
#align continuous_map.normed_space ContinuousMap.normedSpace
section
variable (α 𝕜 E)
/-- When `α` is compact and `𝕜` is a normed field,
the `𝕜`-algebra of bounded continuous maps `α →ᵇ β` is
`𝕜`-linearly isometric to `C(α, β)`.
-/
def linearIsometryBoundedOfCompact : C(α, E) ≃ₗᵢ[𝕜] α →ᵇ E :=
{ addEquivBoundedOfCompact α E with
map_smul' := fun c f => by
ext
norm_cast
norm_map' := fun f => rfl }
#align continuous_map.linear_isometry_bounded_of_compact ContinuousMap.linearIsometryBoundedOfCompact
variable {α E}
-- to match `BoundedContinuousFunction.evalClm`
/-- The evaluation at a point, as a continuous linear map from `C(α, 𝕜)` to `𝕜`. -/
def evalClm (x : α) : C(α, E) →L[𝕜] E :=
(BoundedContinuousFunction.evalClm 𝕜 x).comp
(linearIsometryBoundedOfCompact α E 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_map.eval_clm ContinuousMap.evalClm
end
-- this lemma and the next are the analogues of those autogenerated by `@[simps]` for
-- `equivBoundedOfCompact`, `addEquivBoundedOfCompact`
@[simp]
theorem linearIsometryBoundedOfCompact_symm_apply (f : α →ᵇ E) :
(linearIsometryBoundedOfCompact α E 𝕜).symm f = f.toContinuousMap :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_symm_apply ContinuousMap.linearIsometryBoundedOfCompact_symm_apply
@[simp]
theorem linearIsometryBoundedOfCompact_apply_apply (f : C(α, E)) (a : α) :
(linearIsometryBoundedOfCompact α E 𝕜 f) a = f a :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_apply_apply ContinuousMap.linearIsometryBoundedOfCompact_apply_apply
@[simp]
theorem linearIsometryBoundedOfCompact_toIsometryEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toIsometryEquiv = isometryEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_isometry_equiv ContinuousMap.linearIsometryBoundedOfCompact_toIsometryEquiv
@[simp] -- porting note: adjusted LHS because `simpNF` complained it simplified.
theorem linearIsometryBoundedOfCompact_toAddEquiv :
((linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv : C(α, E) ≃+ (α →ᵇ E)) =
addEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_add_equiv ContinuousMap.linearIsometryBoundedOfCompact_toAddEquiv
@[simp]
theorem linearIsometryBoundedOfCompact_of_compact_toEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv.toEquiv = equivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_of_compact_to_equiv ContinuousMap.linearIsometryBoundedOfCompact_of_compact_toEquiv
end
section
variable {𝕜 : Type*} {γ : Type*} [NormedField 𝕜] [NormedRing γ] [NormedAlgebra 𝕜 γ]
instance : NormedAlgebra 𝕜 C(α, γ) :=
{ ContinuousMap.normedSpace, ContinuousMap.algebra with }
end
end ContinuousMap
namespace ContinuousMap
section UniformContinuity
variable {α β : Type*}
variable [MetricSpace α] [CompactSpace α] [MetricSpace β]
/-!
We now set up some declarations making it convenient to use uniform continuity.
-/
theorem uniform_continuity (f : C(α, β)) (ε : ℝ) (h : 0 < ε) :
∃ δ > 0, ∀ {x y}, dist x y < δ → dist (f x) (f y) < ε :=
Metric.uniformContinuous_iff.mp (CompactSpace.uniformContinuous_of_continuous f.continuous) ε h
#align continuous_map.uniform_continuity ContinuousMap.uniform_continuity
-- This definition allows us to separate the choice of some `δ`,
-- and the corresponding use of `dist a b < δ → dist (f a) (f b) < ε`,
-- even across different declarations.
/-- An arbitrarily chosen modulus of uniform continuity for a given function `f` and `ε > 0`. -/
def modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) : ℝ :=
Classical.choose (uniform_continuity f ε h)
#align continuous_map.modulus ContinuousMap.modulus
theorem modulus_pos (f : C(α, β)) {ε : ℝ} {h : 0 < ε} : 0 < f.modulus ε h :=
(Classical.choose_spec (uniform_continuity f ε h)).1
#align continuous_map.modulus_pos ContinuousMap.modulus_pos
theorem dist_lt_of_dist_lt_modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) {a b : α}
(w : dist a b < f.modulus ε h) : dist (f a) (f b) < ε :=
(Classical.choose_spec (uniform_continuity f ε h)).2 w
#align continuous_map.dist_lt_of_dist_lt_modulus ContinuousMap.dist_lt_of_dist_lt_modulus
end UniformContinuity
end ContinuousMap
section CompLeft
variable (X : Type*) {𝕜 β γ : Type*} [TopologicalSpace X] [CompactSpace X]
[NontriviallyNormedField 𝕜]
variable [NormedAddCommGroup β] [NormedSpace 𝕜 β] [NormedAddCommGroup γ] [NormedSpace 𝕜 γ]
open ContinuousMap
/-- Postcomposition of continuous functions into a normed module by a continuous linear map is a
continuous linear map.
Transferred version of `ContinuousLinearMap.compLeftContinuousBounded`,
upgraded version of `ContinuousLinearMap.compLeftContinuous`,
similar to `LinearMap.compLeft`. -/
protected def ContinuousLinearMap.compLeftContinuousCompact (g : β →L[𝕜] γ) :
C(X, β) →L[𝕜] C(X, γ) :=
(linearIsometryBoundedOfCompact X γ 𝕜).symm.toLinearIsometry.toContinuousLinearMap.comp <|
(g.compLeftContinuousBounded X).comp <|
(linearIsometryBoundedOfCompact X β 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_linear_map.comp_left_continuous_compact ContinuousLinearMap.compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.toLinear_compLeftContinuousCompact (g : β →L[𝕜] γ) :
(g.compLeftContinuousCompact X : C(X, β) →ₗ[𝕜] C(X, γ)) = g.compLeftContinuous 𝕜 X := by
ext f
rfl
#align continuous_linear_map.to_linear_comp_left_continuous_compact ContinuousLinearMap.toLinear_compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.compLeftContinuousCompact_apply (g : β →L[𝕜] γ) (f : C(X, β)) (x : X) :
g.compLeftContinuousCompact X f x = g (f x) :=
rfl
#align continuous_linear_map.comp_left_continuous_compact_apply ContinuousLinearMap.compLeftContinuousCompact_apply
end CompLeft
namespace ContinuousMap
/-!
We now setup variations on `compRight* f`, where `f : C(X, Y)`
(that is, precomposition by a continuous map),
as a morphism `C(Y, T) → C(X, T)`, respecting various types of structure.
In particular:
* `compRightContinuousMap`, the bundled continuous map (for this we need `X Y` compact).
* `compRightHomeomorph`, when we precompose by a homeomorphism.
* `compRightAlgHom`, when `T = R` is a topological ring.
-/
section CompRight
/-- Precomposition by a continuous map is itself a continuous map between spaces of continuous maps.
-/
def compRightContinuousMap {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y)) : C(C(Y, T), C(X, T)) where
toFun g := g.comp f
continuous_toFun := by
refine' Metric.continuous_iff.mpr _
intro g ε ε_pos
refine' ⟨ε, ε_pos, fun g' h => _⟩
rw [ContinuousMap.dist_lt_iff ε_pos] at h ⊢
exact fun x => h (f x)
#align continuous_map.comp_right_continuous_map ContinuousMap.compRightContinuousMap
@[simp]
theorem compRightContinuousMap_apply {X Y : Type*} (T : Type*) [TopologicalSpace X]
[CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y))
(g : C(Y, T)) : (compRightContinuousMap T f) g = g.comp f :=
rfl
#align continuous_map.comp_right_continuous_map_apply ContinuousMap.compRightContinuousMap_apply
/-- Precomposition by a homeomorphism is itself a homeomorphism between spaces of continuous maps.
-/
def compRightHomeomorph {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : X ≃ₜ Y) : C(Y, T) ≃ₜ C(X, T) where
toFun := compRightContinuousMap T f.toContinuousMap
invFun := compRightContinuousMap T f.symm.toContinuousMap
left_inv g := ext fun _ => congr_arg g (f.apply_symm_apply _)
right_inv g := ext fun _ => congr_arg g (f.symm_apply_apply _)
#align continuous_map.comp_right_homeomorph ContinuousMap.compRightHomeomorph
theorem compRightAlgHom_continuous {X Y : Type*} (R A : Type*) [TopologicalSpace X]
[CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [CommSemiring R] [Semiring A]
[MetricSpace A] [TopologicalSemiring A] [Algebra R A] (f : C(X, Y)) :
Continuous (compRightAlgHom R A f) :=
map_continuous (compRightContinuousMap A f)
#align continuous_map.comp_right_alg_hom_continuous ContinuousMap.compRightAlgHom_continuous
end CompRight
section LocalNormalConvergence
/-! ### Local normal convergence
A sum of continuous functions (on a locally compact space) is "locally normally convergent" if the
sum of its sup-norms on any compact subset is summable. This implies convergence in the topology
of `C(X, E)` (i.e. locally uniform convergence). -/
open TopologicalSpace
variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
variable {E : Type*} [NormedAddCommGroup E] [CompleteSpace E]
theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F := by
refine' (ContinuousMap.exists_tendsto_compactOpen_iff_forall _).2 fun K hK => _
lift K to Compacts X using hK
have A : ∀ s : Finset ι, restrict (↑K) (∑ i in s, F i) = ∑ i in s, restrict K (F i) := by
intro s
|
ext1 x
|
theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F := by
refine' (ContinuousMap.exists_tendsto_compactOpen_iff_forall _).2 fun K hK => _
lift K to Compacts X using hK
have A : ∀ s : Finset ι, restrict (↑K) (∑ i in s, F i) = ∑ i in s, restrict K (F i) := by
intro s
|
Mathlib.Topology.ContinuousFunction.Compact.491_0.Mig2jTVnn2FLKEB
|
theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F
|
Mathlib_Topology_ContinuousFunction_Compact
|
case h
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : LocallyCompactSpace X
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
ι : Type u_3
F : ι → C(X, E)
hF : ∀ (K : Compacts X), Summable fun i => ‖restrict (↑K) (F i)‖
K : Compacts X
s : Finset ι
x : ↑↑K
⊢ (restrict (↑K) (∑ i in s, F i)) x = (∑ i in s, restrict (↑K) (F i)) x
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compacts
#align_import topology.continuous_function.compact from "leanprover-community/mathlib"@"d3af0609f6db8691dffdc3e1fb7feb7da72698f2"
/-!
# Continuous functions on a compact space
Continuous functions `C(α, β)` from a compact space `α` to a metric space `β`
are automatically bounded, and so acquire various structures inherited from `α →ᵇ β`.
This file transfers these structures, and restates some lemmas
characterising these structures.
If you need a lemma which is proved about `α →ᵇ β` but not for `C(α, β)` when `α` is compact,
you should restate it here. You can also use
`ContinuousMap.equivBoundedOfCompact` to move functions back and forth.
-/
noncomputable section
open Topology Classical NNReal BoundedContinuousFunction BigOperators
open Set Filter Metric
open BoundedContinuousFunction
namespace ContinuousMap
variable {α β E : Type*} [TopologicalSpace α] [CompactSpace α] [MetricSpace β]
[NormedAddCommGroup E]
section
variable (α β)
/-- When `α` is compact, the bounded continuous maps `α →ᵇ β` are
equivalent to `C(α, β)`.
-/
@[simps (config := .asFn)]
def equivBoundedOfCompact : C(α, β) ≃ (α →ᵇ β) :=
⟨mkOfCompact, BoundedContinuousFunction.toContinuousMap, fun f => by
ext
rfl, fun f => by
ext
rfl⟩
#align continuous_map.equiv_bounded_of_compact ContinuousMap.equivBoundedOfCompact
theorem uniformInducing_equivBoundedOfCompact : UniformInducing (equivBoundedOfCompact α β) :=
UniformInducing.mk'
(by
simp only [hasBasis_compactConvergenceUniformity.mem_iff, uniformity_basis_dist_le.mem_iff]
exact fun s =>
⟨fun ⟨⟨a, b⟩, ⟨_, ⟨ε, hε, hb⟩⟩, hs⟩ =>
⟨{ p | ∀ x, (p.1 x, p.2 x) ∈ b }, ⟨ε, hε, fun _ h x => hb ((dist_le hε.le).mp h x)⟩,
fun f g h => hs fun x _ => h x⟩,
fun ⟨_, ⟨ε, hε, ht⟩, hs⟩ =>
⟨⟨Set.univ, { p | dist p.1 p.2 ≤ ε }⟩, ⟨isCompact_univ, ⟨ε, hε, fun _ h => h⟩⟩,
fun ⟨f, g⟩ h => hs _ _ (ht ((dist_le hε.le).mpr fun x => h x (mem_univ x)))⟩⟩)
#align continuous_map.uniform_inducing_equiv_bounded_of_compact ContinuousMap.uniformInducing_equivBoundedOfCompact
theorem uniformEmbedding_equivBoundedOfCompact : UniformEmbedding (equivBoundedOfCompact α β) :=
{ uniformInducing_equivBoundedOfCompact α β with inj := (equivBoundedOfCompact α β).injective }
#align continuous_map.uniform_embedding_equiv_bounded_of_compact ContinuousMap.uniformEmbedding_equivBoundedOfCompact
/-- When `α` is compact, the bounded continuous maps `α →ᵇ 𝕜` are
additively equivalent to `C(α, 𝕜)`.
-/
-- porting note: the following `simps` received a "maximum recursion depth" error
-- @[simps! (config := .asFn) apply symm_apply]
def addEquivBoundedOfCompact [AddMonoid β] [LipschitzAdd β] : C(α, β) ≃+ (α →ᵇ β) :=
({ toContinuousMapAddHom α β, (equivBoundedOfCompact α β).symm with } : (α →ᵇ β) ≃+ C(α, β)).symm
#align continuous_map.add_equiv_bounded_of_compact ContinuousMap.addEquivBoundedOfCompact
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_symm_apply [AddMonoid β] [LipschitzAdd β] :
⇑((addEquivBoundedOfCompact α β).symm) = toContinuousMapAddHom α β :=
rfl
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_apply [AddMonoid β] [LipschitzAdd β] :
⇑(addEquivBoundedOfCompact α β) = mkOfCompact :=
rfl
instance metricSpace : MetricSpace C(α, β) :=
(uniformEmbedding_equivBoundedOfCompact α β).comapMetricSpace _
#align continuous_map.metric_space ContinuousMap.metricSpace
/-- When `α` is compact, and `β` is a metric space, the bounded continuous maps `α →ᵇ β` are
isometric to `C(α, β)`.
-/
@[simps! (config := .asFn) toEquiv apply symm_apply]
def isometryEquivBoundedOfCompact : C(α, β) ≃ᵢ (α →ᵇ β) where
isometry_toFun _ _ := rfl
toEquiv := equivBoundedOfCompact α β
#align continuous_map.isometry_equiv_bounded_of_compact ContinuousMap.isometryEquivBoundedOfCompact
end
@[simp]
theorem _root_.BoundedContinuousFunction.dist_mkOfCompact (f g : C(α, β)) :
dist (mkOfCompact f) (mkOfCompact g) = dist f g :=
rfl
#align bounded_continuous_function.dist_mk_of_compact BoundedContinuousFunction.dist_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.dist_toContinuousMap (f g : α →ᵇ β) :
dist f.toContinuousMap g.toContinuousMap = dist f g :=
rfl
#align bounded_continuous_function.dist_to_continuous_map BoundedContinuousFunction.dist_toContinuousMap
open BoundedContinuousFunction
section
variable {f g : C(α, β)} {C : ℝ}
/-- The pointwise distance is controlled by the distance between functions, by definition. -/
theorem dist_apply_le_dist (x : α) : dist (f x) (g x) ≤ dist f g := by
simp only [← dist_mkOfCompact, dist_coe_le_dist, ← mkOfCompact_apply]
#align continuous_map.dist_apply_le_dist ContinuousMap.dist_apply_le_dist
/-- The distance between two functions is controlled by the supremum of the pointwise distances. -/
theorem dist_le (C0 : (0 : ℝ) ≤ C) : dist f g ≤ C ↔ ∀ x : α, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le C0, mkOfCompact_apply]
#align continuous_map.dist_le ContinuousMap.dist_le
theorem dist_le_iff_of_nonempty [Nonempty α] : dist f g ≤ C ↔ ∀ x, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le_iff_of_nonempty,
mkOfCompact_apply]
#align continuous_map.dist_le_iff_of_nonempty ContinuousMap.dist_le_iff_of_nonempty
theorem dist_lt_iff_of_nonempty [Nonempty α] : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
simp only [← dist_mkOfCompact, dist_lt_iff_of_nonempty_compact, mkOfCompact_apply]
#align continuous_map.dist_lt_iff_of_nonempty ContinuousMap.dist_lt_iff_of_nonempty
theorem dist_lt_of_nonempty [Nonempty α] (w : ∀ x : α, dist (f x) (g x) < C) : dist f g < C :=
dist_lt_iff_of_nonempty.2 w
#align continuous_map.dist_lt_of_nonempty ContinuousMap.dist_lt_of_nonempty
theorem dist_lt_iff (C0 : (0 : ℝ) < C) : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
rw [← dist_mkOfCompact, dist_lt_iff_of_compact C0]
simp only [mkOfCompact_apply]
#align continuous_map.dist_lt_iff ContinuousMap.dist_lt_iff
end
instance [CompleteSpace β] : CompleteSpace C(α, β) :=
(isometryEquivBoundedOfCompact α β).completeSpace
/-- See also `ContinuousMap.continuous_eval'`. -/
@[continuity]
theorem continuous_eval : Continuous fun p : C(α, β) × α => p.1 p.2 :=
continuous_eval.comp ((isometryEquivBoundedOfCompact α β).continuous.prod_map continuous_id)
#align continuous_map.continuous_eval ContinuousMap.continuous_eval
-- TODO at some point we will need lemmas characterising this norm!
-- At the moment the only way to reason about it is to transfer `f : C(α,E)` back to `α →ᵇ E`.
instance : Norm C(α, E) where norm x := dist x 0
@[simp]
theorem _root_.BoundedContinuousFunction.norm_mkOfCompact (f : C(α, E)) : ‖mkOfCompact f‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_mk_of_compact BoundedContinuousFunction.norm_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.norm_toContinuousMap_eq (f : α →ᵇ E) :
‖f.toContinuousMap‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_to_continuous_map_eq BoundedContinuousFunction.norm_toContinuousMap_eq
open BoundedContinuousFunction
instance : NormedAddCommGroup C(α, E) :=
{ ContinuousMap.metricSpace _ _,
ContinuousMap.instAddCommGroupContinuousMap with
dist_eq := fun x y => by
rw [← norm_mkOfCompact, ← dist_mkOfCompact, dist_eq_norm, mkOfCompact_sub]
dist := dist
norm := norm }
instance [Nonempty α] [One E] [NormOneClass E] : NormOneClass C(α, E) where
norm_one := by simp only [← norm_mkOfCompact, mkOfCompact_one, norm_one]
section
variable (f : C(α, E))
-- The corresponding lemmas for `BoundedContinuousFunction` are stated with `{f}`,
-- and so can not be used in dot notation.
theorem norm_coe_le_norm (x : α) : ‖f x‖ ≤ ‖f‖ :=
(mkOfCompact f).norm_coe_le_norm x
#align continuous_map.norm_coe_le_norm ContinuousMap.norm_coe_le_norm
/-- Distance between the images of any two points is at most twice the norm of the function. -/
theorem dist_le_two_norm (x y : α) : dist (f x) (f y) ≤ 2 * ‖f‖ :=
(mkOfCompact f).dist_le_two_norm x y
#align continuous_map.dist_le_two_norm ContinuousMap.dist_le_two_norm
/-- The norm of a function is controlled by the supremum of the pointwise norms. -/
theorem norm_le {C : ℝ} (C0 : (0 : ℝ) ≤ C) : ‖f‖ ≤ C ↔ ∀ x : α, ‖f x‖ ≤ C :=
@BoundedContinuousFunction.norm_le _ _ _ _ (mkOfCompact f) _ C0
#align continuous_map.norm_le ContinuousMap.norm_le
theorem norm_le_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ ≤ M ↔ ∀ x, ‖f x‖ ≤ M :=
@BoundedContinuousFunction.norm_le_of_nonempty _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_le_of_nonempty ContinuousMap.norm_le_of_nonempty
theorem norm_lt_iff {M : ℝ} (M0 : 0 < M) : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_compact _ _ _ _ _ (mkOfCompact f) _ M0
#align continuous_map.norm_lt_iff ContinuousMap.norm_lt_iff
theorem nnnorm_lt_iff {M : ℝ≥0} (M0 : 0 < M) : ‖f‖₊ < M ↔ ∀ x : α, ‖f x‖₊ < M :=
f.norm_lt_iff M0
#align continuous_map.nnnorm_lt_iff ContinuousMap.nnnorm_lt_iff
theorem norm_lt_iff_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_nonempty_compact _ _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_lt_iff_of_nonempty ContinuousMap.norm_lt_iff_of_nonempty
theorem nnnorm_lt_iff_of_nonempty [Nonempty α] {M : ℝ≥0} : ‖f‖₊ < M ↔ ∀ x, ‖f x‖₊ < M :=
f.norm_lt_iff_of_nonempty
#align continuous_map.nnnorm_lt_iff_of_nonempty ContinuousMap.nnnorm_lt_iff_of_nonempty
theorem apply_le_norm (f : C(α, ℝ)) (x : α) : f x ≤ ‖f‖ :=
le_trans (le_abs.mpr (Or.inl (le_refl (f x)))) (f.norm_coe_le_norm x)
#align continuous_map.apply_le_norm ContinuousMap.apply_le_norm
theorem neg_norm_le_apply (f : C(α, ℝ)) (x : α) : -‖f‖ ≤ f x :=
le_trans (neg_le_neg (f.norm_coe_le_norm x)) (neg_le.mp (neg_le_abs_self (f x)))
#align continuous_map.neg_norm_le_apply ContinuousMap.neg_norm_le_apply
theorem norm_eq_iSup_norm : ‖f‖ = ⨆ x : α, ‖f x‖ :=
(mkOfCompact f).norm_eq_iSup_norm
#align continuous_map.norm_eq_supr_norm ContinuousMap.norm_eq_iSup_norm
theorem norm_restrict_mono_set {X : Type*} [TopologicalSpace X] (f : C(X, E))
{K L : TopologicalSpace.Compacts X} (hKL : K ≤ L) : ‖f.restrict K‖ ≤ ‖f.restrict L‖ :=
(norm_le _ (norm_nonneg _)).mpr fun x => norm_coe_le_norm (f.restrict L) <| Set.inclusion hKL x
#align continuous_map.norm_restrict_mono_set ContinuousMap.norm_restrict_mono_set
end
section
variable {R : Type*} [NormedRing R]
instance : NormedRing C(α, R) :=
{ (inferInstance : NormedAddCommGroup C(α, R)), ContinuousMap.instRingContinuousMap with
norm_mul := fun f g => norm_mul_le (mkOfCompact f) (mkOfCompact g) }
end
section
variable {𝕜 : Type*} [NormedField 𝕜] [NormedSpace 𝕜 E]
instance normedSpace : NormedSpace 𝕜 C(α, E) where
norm_smul_le c f := (norm_smul_le c (mkOfCompact f) : _)
#align continuous_map.normed_space ContinuousMap.normedSpace
section
variable (α 𝕜 E)
/-- When `α` is compact and `𝕜` is a normed field,
the `𝕜`-algebra of bounded continuous maps `α →ᵇ β` is
`𝕜`-linearly isometric to `C(α, β)`.
-/
def linearIsometryBoundedOfCompact : C(α, E) ≃ₗᵢ[𝕜] α →ᵇ E :=
{ addEquivBoundedOfCompact α E with
map_smul' := fun c f => by
ext
norm_cast
norm_map' := fun f => rfl }
#align continuous_map.linear_isometry_bounded_of_compact ContinuousMap.linearIsometryBoundedOfCompact
variable {α E}
-- to match `BoundedContinuousFunction.evalClm`
/-- The evaluation at a point, as a continuous linear map from `C(α, 𝕜)` to `𝕜`. -/
def evalClm (x : α) : C(α, E) →L[𝕜] E :=
(BoundedContinuousFunction.evalClm 𝕜 x).comp
(linearIsometryBoundedOfCompact α E 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_map.eval_clm ContinuousMap.evalClm
end
-- this lemma and the next are the analogues of those autogenerated by `@[simps]` for
-- `equivBoundedOfCompact`, `addEquivBoundedOfCompact`
@[simp]
theorem linearIsometryBoundedOfCompact_symm_apply (f : α →ᵇ E) :
(linearIsometryBoundedOfCompact α E 𝕜).symm f = f.toContinuousMap :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_symm_apply ContinuousMap.linearIsometryBoundedOfCompact_symm_apply
@[simp]
theorem linearIsometryBoundedOfCompact_apply_apply (f : C(α, E)) (a : α) :
(linearIsometryBoundedOfCompact α E 𝕜 f) a = f a :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_apply_apply ContinuousMap.linearIsometryBoundedOfCompact_apply_apply
@[simp]
theorem linearIsometryBoundedOfCompact_toIsometryEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toIsometryEquiv = isometryEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_isometry_equiv ContinuousMap.linearIsometryBoundedOfCompact_toIsometryEquiv
@[simp] -- porting note: adjusted LHS because `simpNF` complained it simplified.
theorem linearIsometryBoundedOfCompact_toAddEquiv :
((linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv : C(α, E) ≃+ (α →ᵇ E)) =
addEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_add_equiv ContinuousMap.linearIsometryBoundedOfCompact_toAddEquiv
@[simp]
theorem linearIsometryBoundedOfCompact_of_compact_toEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv.toEquiv = equivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_of_compact_to_equiv ContinuousMap.linearIsometryBoundedOfCompact_of_compact_toEquiv
end
section
variable {𝕜 : Type*} {γ : Type*} [NormedField 𝕜] [NormedRing γ] [NormedAlgebra 𝕜 γ]
instance : NormedAlgebra 𝕜 C(α, γ) :=
{ ContinuousMap.normedSpace, ContinuousMap.algebra with }
end
end ContinuousMap
namespace ContinuousMap
section UniformContinuity
variable {α β : Type*}
variable [MetricSpace α] [CompactSpace α] [MetricSpace β]
/-!
We now set up some declarations making it convenient to use uniform continuity.
-/
theorem uniform_continuity (f : C(α, β)) (ε : ℝ) (h : 0 < ε) :
∃ δ > 0, ∀ {x y}, dist x y < δ → dist (f x) (f y) < ε :=
Metric.uniformContinuous_iff.mp (CompactSpace.uniformContinuous_of_continuous f.continuous) ε h
#align continuous_map.uniform_continuity ContinuousMap.uniform_continuity
-- This definition allows us to separate the choice of some `δ`,
-- and the corresponding use of `dist a b < δ → dist (f a) (f b) < ε`,
-- even across different declarations.
/-- An arbitrarily chosen modulus of uniform continuity for a given function `f` and `ε > 0`. -/
def modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) : ℝ :=
Classical.choose (uniform_continuity f ε h)
#align continuous_map.modulus ContinuousMap.modulus
theorem modulus_pos (f : C(α, β)) {ε : ℝ} {h : 0 < ε} : 0 < f.modulus ε h :=
(Classical.choose_spec (uniform_continuity f ε h)).1
#align continuous_map.modulus_pos ContinuousMap.modulus_pos
theorem dist_lt_of_dist_lt_modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) {a b : α}
(w : dist a b < f.modulus ε h) : dist (f a) (f b) < ε :=
(Classical.choose_spec (uniform_continuity f ε h)).2 w
#align continuous_map.dist_lt_of_dist_lt_modulus ContinuousMap.dist_lt_of_dist_lt_modulus
end UniformContinuity
end ContinuousMap
section CompLeft
variable (X : Type*) {𝕜 β γ : Type*} [TopologicalSpace X] [CompactSpace X]
[NontriviallyNormedField 𝕜]
variable [NormedAddCommGroup β] [NormedSpace 𝕜 β] [NormedAddCommGroup γ] [NormedSpace 𝕜 γ]
open ContinuousMap
/-- Postcomposition of continuous functions into a normed module by a continuous linear map is a
continuous linear map.
Transferred version of `ContinuousLinearMap.compLeftContinuousBounded`,
upgraded version of `ContinuousLinearMap.compLeftContinuous`,
similar to `LinearMap.compLeft`. -/
protected def ContinuousLinearMap.compLeftContinuousCompact (g : β →L[𝕜] γ) :
C(X, β) →L[𝕜] C(X, γ) :=
(linearIsometryBoundedOfCompact X γ 𝕜).symm.toLinearIsometry.toContinuousLinearMap.comp <|
(g.compLeftContinuousBounded X).comp <|
(linearIsometryBoundedOfCompact X β 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_linear_map.comp_left_continuous_compact ContinuousLinearMap.compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.toLinear_compLeftContinuousCompact (g : β →L[𝕜] γ) :
(g.compLeftContinuousCompact X : C(X, β) →ₗ[𝕜] C(X, γ)) = g.compLeftContinuous 𝕜 X := by
ext f
rfl
#align continuous_linear_map.to_linear_comp_left_continuous_compact ContinuousLinearMap.toLinear_compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.compLeftContinuousCompact_apply (g : β →L[𝕜] γ) (f : C(X, β)) (x : X) :
g.compLeftContinuousCompact X f x = g (f x) :=
rfl
#align continuous_linear_map.comp_left_continuous_compact_apply ContinuousLinearMap.compLeftContinuousCompact_apply
end CompLeft
namespace ContinuousMap
/-!
We now setup variations on `compRight* f`, where `f : C(X, Y)`
(that is, precomposition by a continuous map),
as a morphism `C(Y, T) → C(X, T)`, respecting various types of structure.
In particular:
* `compRightContinuousMap`, the bundled continuous map (for this we need `X Y` compact).
* `compRightHomeomorph`, when we precompose by a homeomorphism.
* `compRightAlgHom`, when `T = R` is a topological ring.
-/
section CompRight
/-- Precomposition by a continuous map is itself a continuous map between spaces of continuous maps.
-/
def compRightContinuousMap {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y)) : C(C(Y, T), C(X, T)) where
toFun g := g.comp f
continuous_toFun := by
refine' Metric.continuous_iff.mpr _
intro g ε ε_pos
refine' ⟨ε, ε_pos, fun g' h => _⟩
rw [ContinuousMap.dist_lt_iff ε_pos] at h ⊢
exact fun x => h (f x)
#align continuous_map.comp_right_continuous_map ContinuousMap.compRightContinuousMap
@[simp]
theorem compRightContinuousMap_apply {X Y : Type*} (T : Type*) [TopologicalSpace X]
[CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y))
(g : C(Y, T)) : (compRightContinuousMap T f) g = g.comp f :=
rfl
#align continuous_map.comp_right_continuous_map_apply ContinuousMap.compRightContinuousMap_apply
/-- Precomposition by a homeomorphism is itself a homeomorphism between spaces of continuous maps.
-/
def compRightHomeomorph {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : X ≃ₜ Y) : C(Y, T) ≃ₜ C(X, T) where
toFun := compRightContinuousMap T f.toContinuousMap
invFun := compRightContinuousMap T f.symm.toContinuousMap
left_inv g := ext fun _ => congr_arg g (f.apply_symm_apply _)
right_inv g := ext fun _ => congr_arg g (f.symm_apply_apply _)
#align continuous_map.comp_right_homeomorph ContinuousMap.compRightHomeomorph
theorem compRightAlgHom_continuous {X Y : Type*} (R A : Type*) [TopologicalSpace X]
[CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [CommSemiring R] [Semiring A]
[MetricSpace A] [TopologicalSemiring A] [Algebra R A] (f : C(X, Y)) :
Continuous (compRightAlgHom R A f) :=
map_continuous (compRightContinuousMap A f)
#align continuous_map.comp_right_alg_hom_continuous ContinuousMap.compRightAlgHom_continuous
end CompRight
section LocalNormalConvergence
/-! ### Local normal convergence
A sum of continuous functions (on a locally compact space) is "locally normally convergent" if the
sum of its sup-norms on any compact subset is summable. This implies convergence in the topology
of `C(X, E)` (i.e. locally uniform convergence). -/
open TopologicalSpace
variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
variable {E : Type*} [NormedAddCommGroup E] [CompleteSpace E]
theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F := by
refine' (ContinuousMap.exists_tendsto_compactOpen_iff_forall _).2 fun K hK => _
lift K to Compacts X using hK
have A : ∀ s : Finset ι, restrict (↑K) (∑ i in s, F i) = ∑ i in s, restrict K (F i) := by
intro s
ext1 x
|
simp
|
theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F := by
refine' (ContinuousMap.exists_tendsto_compactOpen_iff_forall _).2 fun K hK => _
lift K to Compacts X using hK
have A : ∀ s : Finset ι, restrict (↑K) (∑ i in s, F i) = ∑ i in s, restrict K (F i) := by
intro s
ext1 x
|
Mathlib.Topology.ContinuousFunction.Compact.491_0.Mig2jTVnn2FLKEB
|
theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F
|
Mathlib_Topology_ContinuousFunction_Compact
|
case h
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : LocallyCompactSpace X
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
ι : Type u_3
F : ι → C(X, E)
hF : ∀ (K : Compacts X), Summable fun i => ‖restrict (↑K) (F i)‖
K : Compacts X
s : Finset ι
x : ↑↑K
⊢ (restrict (↑K) (∑ i in s, F i)) x = ∑ c in s, (restrict (↑K) (F c)) x
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compacts
#align_import topology.continuous_function.compact from "leanprover-community/mathlib"@"d3af0609f6db8691dffdc3e1fb7feb7da72698f2"
/-!
# Continuous functions on a compact space
Continuous functions `C(α, β)` from a compact space `α` to a metric space `β`
are automatically bounded, and so acquire various structures inherited from `α →ᵇ β`.
This file transfers these structures, and restates some lemmas
characterising these structures.
If you need a lemma which is proved about `α →ᵇ β` but not for `C(α, β)` when `α` is compact,
you should restate it here. You can also use
`ContinuousMap.equivBoundedOfCompact` to move functions back and forth.
-/
noncomputable section
open Topology Classical NNReal BoundedContinuousFunction BigOperators
open Set Filter Metric
open BoundedContinuousFunction
namespace ContinuousMap
variable {α β E : Type*} [TopologicalSpace α] [CompactSpace α] [MetricSpace β]
[NormedAddCommGroup E]
section
variable (α β)
/-- When `α` is compact, the bounded continuous maps `α →ᵇ β` are
equivalent to `C(α, β)`.
-/
@[simps (config := .asFn)]
def equivBoundedOfCompact : C(α, β) ≃ (α →ᵇ β) :=
⟨mkOfCompact, BoundedContinuousFunction.toContinuousMap, fun f => by
ext
rfl, fun f => by
ext
rfl⟩
#align continuous_map.equiv_bounded_of_compact ContinuousMap.equivBoundedOfCompact
theorem uniformInducing_equivBoundedOfCompact : UniformInducing (equivBoundedOfCompact α β) :=
UniformInducing.mk'
(by
simp only [hasBasis_compactConvergenceUniformity.mem_iff, uniformity_basis_dist_le.mem_iff]
exact fun s =>
⟨fun ⟨⟨a, b⟩, ⟨_, ⟨ε, hε, hb⟩⟩, hs⟩ =>
⟨{ p | ∀ x, (p.1 x, p.2 x) ∈ b }, ⟨ε, hε, fun _ h x => hb ((dist_le hε.le).mp h x)⟩,
fun f g h => hs fun x _ => h x⟩,
fun ⟨_, ⟨ε, hε, ht⟩, hs⟩ =>
⟨⟨Set.univ, { p | dist p.1 p.2 ≤ ε }⟩, ⟨isCompact_univ, ⟨ε, hε, fun _ h => h⟩⟩,
fun ⟨f, g⟩ h => hs _ _ (ht ((dist_le hε.le).mpr fun x => h x (mem_univ x)))⟩⟩)
#align continuous_map.uniform_inducing_equiv_bounded_of_compact ContinuousMap.uniformInducing_equivBoundedOfCompact
theorem uniformEmbedding_equivBoundedOfCompact : UniformEmbedding (equivBoundedOfCompact α β) :=
{ uniformInducing_equivBoundedOfCompact α β with inj := (equivBoundedOfCompact α β).injective }
#align continuous_map.uniform_embedding_equiv_bounded_of_compact ContinuousMap.uniformEmbedding_equivBoundedOfCompact
/-- When `α` is compact, the bounded continuous maps `α →ᵇ 𝕜` are
additively equivalent to `C(α, 𝕜)`.
-/
-- porting note: the following `simps` received a "maximum recursion depth" error
-- @[simps! (config := .asFn) apply symm_apply]
def addEquivBoundedOfCompact [AddMonoid β] [LipschitzAdd β] : C(α, β) ≃+ (α →ᵇ β) :=
({ toContinuousMapAddHom α β, (equivBoundedOfCompact α β).symm with } : (α →ᵇ β) ≃+ C(α, β)).symm
#align continuous_map.add_equiv_bounded_of_compact ContinuousMap.addEquivBoundedOfCompact
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_symm_apply [AddMonoid β] [LipschitzAdd β] :
⇑((addEquivBoundedOfCompact α β).symm) = toContinuousMapAddHom α β :=
rfl
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_apply [AddMonoid β] [LipschitzAdd β] :
⇑(addEquivBoundedOfCompact α β) = mkOfCompact :=
rfl
instance metricSpace : MetricSpace C(α, β) :=
(uniformEmbedding_equivBoundedOfCompact α β).comapMetricSpace _
#align continuous_map.metric_space ContinuousMap.metricSpace
/-- When `α` is compact, and `β` is a metric space, the bounded continuous maps `α →ᵇ β` are
isometric to `C(α, β)`.
-/
@[simps! (config := .asFn) toEquiv apply symm_apply]
def isometryEquivBoundedOfCompact : C(α, β) ≃ᵢ (α →ᵇ β) where
isometry_toFun _ _ := rfl
toEquiv := equivBoundedOfCompact α β
#align continuous_map.isometry_equiv_bounded_of_compact ContinuousMap.isometryEquivBoundedOfCompact
end
@[simp]
theorem _root_.BoundedContinuousFunction.dist_mkOfCompact (f g : C(α, β)) :
dist (mkOfCompact f) (mkOfCompact g) = dist f g :=
rfl
#align bounded_continuous_function.dist_mk_of_compact BoundedContinuousFunction.dist_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.dist_toContinuousMap (f g : α →ᵇ β) :
dist f.toContinuousMap g.toContinuousMap = dist f g :=
rfl
#align bounded_continuous_function.dist_to_continuous_map BoundedContinuousFunction.dist_toContinuousMap
open BoundedContinuousFunction
section
variable {f g : C(α, β)} {C : ℝ}
/-- The pointwise distance is controlled by the distance between functions, by definition. -/
theorem dist_apply_le_dist (x : α) : dist (f x) (g x) ≤ dist f g := by
simp only [← dist_mkOfCompact, dist_coe_le_dist, ← mkOfCompact_apply]
#align continuous_map.dist_apply_le_dist ContinuousMap.dist_apply_le_dist
/-- The distance between two functions is controlled by the supremum of the pointwise distances. -/
theorem dist_le (C0 : (0 : ℝ) ≤ C) : dist f g ≤ C ↔ ∀ x : α, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le C0, mkOfCompact_apply]
#align continuous_map.dist_le ContinuousMap.dist_le
theorem dist_le_iff_of_nonempty [Nonempty α] : dist f g ≤ C ↔ ∀ x, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le_iff_of_nonempty,
mkOfCompact_apply]
#align continuous_map.dist_le_iff_of_nonempty ContinuousMap.dist_le_iff_of_nonempty
theorem dist_lt_iff_of_nonempty [Nonempty α] : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
simp only [← dist_mkOfCompact, dist_lt_iff_of_nonempty_compact, mkOfCompact_apply]
#align continuous_map.dist_lt_iff_of_nonempty ContinuousMap.dist_lt_iff_of_nonempty
theorem dist_lt_of_nonempty [Nonempty α] (w : ∀ x : α, dist (f x) (g x) < C) : dist f g < C :=
dist_lt_iff_of_nonempty.2 w
#align continuous_map.dist_lt_of_nonempty ContinuousMap.dist_lt_of_nonempty
theorem dist_lt_iff (C0 : (0 : ℝ) < C) : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
rw [← dist_mkOfCompact, dist_lt_iff_of_compact C0]
simp only [mkOfCompact_apply]
#align continuous_map.dist_lt_iff ContinuousMap.dist_lt_iff
end
instance [CompleteSpace β] : CompleteSpace C(α, β) :=
(isometryEquivBoundedOfCompact α β).completeSpace
/-- See also `ContinuousMap.continuous_eval'`. -/
@[continuity]
theorem continuous_eval : Continuous fun p : C(α, β) × α => p.1 p.2 :=
continuous_eval.comp ((isometryEquivBoundedOfCompact α β).continuous.prod_map continuous_id)
#align continuous_map.continuous_eval ContinuousMap.continuous_eval
-- TODO at some point we will need lemmas characterising this norm!
-- At the moment the only way to reason about it is to transfer `f : C(α,E)` back to `α →ᵇ E`.
instance : Norm C(α, E) where norm x := dist x 0
@[simp]
theorem _root_.BoundedContinuousFunction.norm_mkOfCompact (f : C(α, E)) : ‖mkOfCompact f‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_mk_of_compact BoundedContinuousFunction.norm_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.norm_toContinuousMap_eq (f : α →ᵇ E) :
‖f.toContinuousMap‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_to_continuous_map_eq BoundedContinuousFunction.norm_toContinuousMap_eq
open BoundedContinuousFunction
instance : NormedAddCommGroup C(α, E) :=
{ ContinuousMap.metricSpace _ _,
ContinuousMap.instAddCommGroupContinuousMap with
dist_eq := fun x y => by
rw [← norm_mkOfCompact, ← dist_mkOfCompact, dist_eq_norm, mkOfCompact_sub]
dist := dist
norm := norm }
instance [Nonempty α] [One E] [NormOneClass E] : NormOneClass C(α, E) where
norm_one := by simp only [← norm_mkOfCompact, mkOfCompact_one, norm_one]
section
variable (f : C(α, E))
-- The corresponding lemmas for `BoundedContinuousFunction` are stated with `{f}`,
-- and so can not be used in dot notation.
theorem norm_coe_le_norm (x : α) : ‖f x‖ ≤ ‖f‖ :=
(mkOfCompact f).norm_coe_le_norm x
#align continuous_map.norm_coe_le_norm ContinuousMap.norm_coe_le_norm
/-- Distance between the images of any two points is at most twice the norm of the function. -/
theorem dist_le_two_norm (x y : α) : dist (f x) (f y) ≤ 2 * ‖f‖ :=
(mkOfCompact f).dist_le_two_norm x y
#align continuous_map.dist_le_two_norm ContinuousMap.dist_le_two_norm
/-- The norm of a function is controlled by the supremum of the pointwise norms. -/
theorem norm_le {C : ℝ} (C0 : (0 : ℝ) ≤ C) : ‖f‖ ≤ C ↔ ∀ x : α, ‖f x‖ ≤ C :=
@BoundedContinuousFunction.norm_le _ _ _ _ (mkOfCompact f) _ C0
#align continuous_map.norm_le ContinuousMap.norm_le
theorem norm_le_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ ≤ M ↔ ∀ x, ‖f x‖ ≤ M :=
@BoundedContinuousFunction.norm_le_of_nonempty _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_le_of_nonempty ContinuousMap.norm_le_of_nonempty
theorem norm_lt_iff {M : ℝ} (M0 : 0 < M) : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_compact _ _ _ _ _ (mkOfCompact f) _ M0
#align continuous_map.norm_lt_iff ContinuousMap.norm_lt_iff
theorem nnnorm_lt_iff {M : ℝ≥0} (M0 : 0 < M) : ‖f‖₊ < M ↔ ∀ x : α, ‖f x‖₊ < M :=
f.norm_lt_iff M0
#align continuous_map.nnnorm_lt_iff ContinuousMap.nnnorm_lt_iff
theorem norm_lt_iff_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_nonempty_compact _ _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_lt_iff_of_nonempty ContinuousMap.norm_lt_iff_of_nonempty
theorem nnnorm_lt_iff_of_nonempty [Nonempty α] {M : ℝ≥0} : ‖f‖₊ < M ↔ ∀ x, ‖f x‖₊ < M :=
f.norm_lt_iff_of_nonempty
#align continuous_map.nnnorm_lt_iff_of_nonempty ContinuousMap.nnnorm_lt_iff_of_nonempty
theorem apply_le_norm (f : C(α, ℝ)) (x : α) : f x ≤ ‖f‖ :=
le_trans (le_abs.mpr (Or.inl (le_refl (f x)))) (f.norm_coe_le_norm x)
#align continuous_map.apply_le_norm ContinuousMap.apply_le_norm
theorem neg_norm_le_apply (f : C(α, ℝ)) (x : α) : -‖f‖ ≤ f x :=
le_trans (neg_le_neg (f.norm_coe_le_norm x)) (neg_le.mp (neg_le_abs_self (f x)))
#align continuous_map.neg_norm_le_apply ContinuousMap.neg_norm_le_apply
theorem norm_eq_iSup_norm : ‖f‖ = ⨆ x : α, ‖f x‖ :=
(mkOfCompact f).norm_eq_iSup_norm
#align continuous_map.norm_eq_supr_norm ContinuousMap.norm_eq_iSup_norm
theorem norm_restrict_mono_set {X : Type*} [TopologicalSpace X] (f : C(X, E))
{K L : TopologicalSpace.Compacts X} (hKL : K ≤ L) : ‖f.restrict K‖ ≤ ‖f.restrict L‖ :=
(norm_le _ (norm_nonneg _)).mpr fun x => norm_coe_le_norm (f.restrict L) <| Set.inclusion hKL x
#align continuous_map.norm_restrict_mono_set ContinuousMap.norm_restrict_mono_set
end
section
variable {R : Type*} [NormedRing R]
instance : NormedRing C(α, R) :=
{ (inferInstance : NormedAddCommGroup C(α, R)), ContinuousMap.instRingContinuousMap with
norm_mul := fun f g => norm_mul_le (mkOfCompact f) (mkOfCompact g) }
end
section
variable {𝕜 : Type*} [NormedField 𝕜] [NormedSpace 𝕜 E]
instance normedSpace : NormedSpace 𝕜 C(α, E) where
norm_smul_le c f := (norm_smul_le c (mkOfCompact f) : _)
#align continuous_map.normed_space ContinuousMap.normedSpace
section
variable (α 𝕜 E)
/-- When `α` is compact and `𝕜` is a normed field,
the `𝕜`-algebra of bounded continuous maps `α →ᵇ β` is
`𝕜`-linearly isometric to `C(α, β)`.
-/
def linearIsometryBoundedOfCompact : C(α, E) ≃ₗᵢ[𝕜] α →ᵇ E :=
{ addEquivBoundedOfCompact α E with
map_smul' := fun c f => by
ext
norm_cast
norm_map' := fun f => rfl }
#align continuous_map.linear_isometry_bounded_of_compact ContinuousMap.linearIsometryBoundedOfCompact
variable {α E}
-- to match `BoundedContinuousFunction.evalClm`
/-- The evaluation at a point, as a continuous linear map from `C(α, 𝕜)` to `𝕜`. -/
def evalClm (x : α) : C(α, E) →L[𝕜] E :=
(BoundedContinuousFunction.evalClm 𝕜 x).comp
(linearIsometryBoundedOfCompact α E 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_map.eval_clm ContinuousMap.evalClm
end
-- this lemma and the next are the analogues of those autogenerated by `@[simps]` for
-- `equivBoundedOfCompact`, `addEquivBoundedOfCompact`
@[simp]
theorem linearIsometryBoundedOfCompact_symm_apply (f : α →ᵇ E) :
(linearIsometryBoundedOfCompact α E 𝕜).symm f = f.toContinuousMap :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_symm_apply ContinuousMap.linearIsometryBoundedOfCompact_symm_apply
@[simp]
theorem linearIsometryBoundedOfCompact_apply_apply (f : C(α, E)) (a : α) :
(linearIsometryBoundedOfCompact α E 𝕜 f) a = f a :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_apply_apply ContinuousMap.linearIsometryBoundedOfCompact_apply_apply
@[simp]
theorem linearIsometryBoundedOfCompact_toIsometryEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toIsometryEquiv = isometryEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_isometry_equiv ContinuousMap.linearIsometryBoundedOfCompact_toIsometryEquiv
@[simp] -- porting note: adjusted LHS because `simpNF` complained it simplified.
theorem linearIsometryBoundedOfCompact_toAddEquiv :
((linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv : C(α, E) ≃+ (α →ᵇ E)) =
addEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_add_equiv ContinuousMap.linearIsometryBoundedOfCompact_toAddEquiv
@[simp]
theorem linearIsometryBoundedOfCompact_of_compact_toEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv.toEquiv = equivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_of_compact_to_equiv ContinuousMap.linearIsometryBoundedOfCompact_of_compact_toEquiv
end
section
variable {𝕜 : Type*} {γ : Type*} [NormedField 𝕜] [NormedRing γ] [NormedAlgebra 𝕜 γ]
instance : NormedAlgebra 𝕜 C(α, γ) :=
{ ContinuousMap.normedSpace, ContinuousMap.algebra with }
end
end ContinuousMap
namespace ContinuousMap
section UniformContinuity
variable {α β : Type*}
variable [MetricSpace α] [CompactSpace α] [MetricSpace β]
/-!
We now set up some declarations making it convenient to use uniform continuity.
-/
theorem uniform_continuity (f : C(α, β)) (ε : ℝ) (h : 0 < ε) :
∃ δ > 0, ∀ {x y}, dist x y < δ → dist (f x) (f y) < ε :=
Metric.uniformContinuous_iff.mp (CompactSpace.uniformContinuous_of_continuous f.continuous) ε h
#align continuous_map.uniform_continuity ContinuousMap.uniform_continuity
-- This definition allows us to separate the choice of some `δ`,
-- and the corresponding use of `dist a b < δ → dist (f a) (f b) < ε`,
-- even across different declarations.
/-- An arbitrarily chosen modulus of uniform continuity for a given function `f` and `ε > 0`. -/
def modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) : ℝ :=
Classical.choose (uniform_continuity f ε h)
#align continuous_map.modulus ContinuousMap.modulus
theorem modulus_pos (f : C(α, β)) {ε : ℝ} {h : 0 < ε} : 0 < f.modulus ε h :=
(Classical.choose_spec (uniform_continuity f ε h)).1
#align continuous_map.modulus_pos ContinuousMap.modulus_pos
theorem dist_lt_of_dist_lt_modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) {a b : α}
(w : dist a b < f.modulus ε h) : dist (f a) (f b) < ε :=
(Classical.choose_spec (uniform_continuity f ε h)).2 w
#align continuous_map.dist_lt_of_dist_lt_modulus ContinuousMap.dist_lt_of_dist_lt_modulus
end UniformContinuity
end ContinuousMap
section CompLeft
variable (X : Type*) {𝕜 β γ : Type*} [TopologicalSpace X] [CompactSpace X]
[NontriviallyNormedField 𝕜]
variable [NormedAddCommGroup β] [NormedSpace 𝕜 β] [NormedAddCommGroup γ] [NormedSpace 𝕜 γ]
open ContinuousMap
/-- Postcomposition of continuous functions into a normed module by a continuous linear map is a
continuous linear map.
Transferred version of `ContinuousLinearMap.compLeftContinuousBounded`,
upgraded version of `ContinuousLinearMap.compLeftContinuous`,
similar to `LinearMap.compLeft`. -/
protected def ContinuousLinearMap.compLeftContinuousCompact (g : β →L[𝕜] γ) :
C(X, β) →L[𝕜] C(X, γ) :=
(linearIsometryBoundedOfCompact X γ 𝕜).symm.toLinearIsometry.toContinuousLinearMap.comp <|
(g.compLeftContinuousBounded X).comp <|
(linearIsometryBoundedOfCompact X β 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_linear_map.comp_left_continuous_compact ContinuousLinearMap.compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.toLinear_compLeftContinuousCompact (g : β →L[𝕜] γ) :
(g.compLeftContinuousCompact X : C(X, β) →ₗ[𝕜] C(X, γ)) = g.compLeftContinuous 𝕜 X := by
ext f
rfl
#align continuous_linear_map.to_linear_comp_left_continuous_compact ContinuousLinearMap.toLinear_compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.compLeftContinuousCompact_apply (g : β →L[𝕜] γ) (f : C(X, β)) (x : X) :
g.compLeftContinuousCompact X f x = g (f x) :=
rfl
#align continuous_linear_map.comp_left_continuous_compact_apply ContinuousLinearMap.compLeftContinuousCompact_apply
end CompLeft
namespace ContinuousMap
/-!
We now setup variations on `compRight* f`, where `f : C(X, Y)`
(that is, precomposition by a continuous map),
as a morphism `C(Y, T) → C(X, T)`, respecting various types of structure.
In particular:
* `compRightContinuousMap`, the bundled continuous map (for this we need `X Y` compact).
* `compRightHomeomorph`, when we precompose by a homeomorphism.
* `compRightAlgHom`, when `T = R` is a topological ring.
-/
section CompRight
/-- Precomposition by a continuous map is itself a continuous map between spaces of continuous maps.
-/
def compRightContinuousMap {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y)) : C(C(Y, T), C(X, T)) where
toFun g := g.comp f
continuous_toFun := by
refine' Metric.continuous_iff.mpr _
intro g ε ε_pos
refine' ⟨ε, ε_pos, fun g' h => _⟩
rw [ContinuousMap.dist_lt_iff ε_pos] at h ⊢
exact fun x => h (f x)
#align continuous_map.comp_right_continuous_map ContinuousMap.compRightContinuousMap
@[simp]
theorem compRightContinuousMap_apply {X Y : Type*} (T : Type*) [TopologicalSpace X]
[CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y))
(g : C(Y, T)) : (compRightContinuousMap T f) g = g.comp f :=
rfl
#align continuous_map.comp_right_continuous_map_apply ContinuousMap.compRightContinuousMap_apply
/-- Precomposition by a homeomorphism is itself a homeomorphism between spaces of continuous maps.
-/
def compRightHomeomorph {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : X ≃ₜ Y) : C(Y, T) ≃ₜ C(X, T) where
toFun := compRightContinuousMap T f.toContinuousMap
invFun := compRightContinuousMap T f.symm.toContinuousMap
left_inv g := ext fun _ => congr_arg g (f.apply_symm_apply _)
right_inv g := ext fun _ => congr_arg g (f.symm_apply_apply _)
#align continuous_map.comp_right_homeomorph ContinuousMap.compRightHomeomorph
theorem compRightAlgHom_continuous {X Y : Type*} (R A : Type*) [TopologicalSpace X]
[CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [CommSemiring R] [Semiring A]
[MetricSpace A] [TopologicalSemiring A] [Algebra R A] (f : C(X, Y)) :
Continuous (compRightAlgHom R A f) :=
map_continuous (compRightContinuousMap A f)
#align continuous_map.comp_right_alg_hom_continuous ContinuousMap.compRightAlgHom_continuous
end CompRight
section LocalNormalConvergence
/-! ### Local normal convergence
A sum of continuous functions (on a locally compact space) is "locally normally convergent" if the
sum of its sup-norms on any compact subset is summable. This implies convergence in the topology
of `C(X, E)` (i.e. locally uniform convergence). -/
open TopologicalSpace
variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
variable {E : Type*} [NormedAddCommGroup E] [CompleteSpace E]
theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F := by
refine' (ContinuousMap.exists_tendsto_compactOpen_iff_forall _).2 fun K hK => _
lift K to Compacts X using hK
have A : ∀ s : Finset ι, restrict (↑K) (∑ i in s, F i) = ∑ i in s, restrict K (F i) := by
intro s
ext1 x
simp
-- This used to be the end of the proof before leanprover/lean4#2644
|
erw [restrict_apply, restrict_apply, restrict_apply, restrict_apply]
|
theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F := by
refine' (ContinuousMap.exists_tendsto_compactOpen_iff_forall _).2 fun K hK => _
lift K to Compacts X using hK
have A : ∀ s : Finset ι, restrict (↑K) (∑ i in s, F i) = ∑ i in s, restrict K (F i) := by
intro s
ext1 x
simp
-- This used to be the end of the proof before leanprover/lean4#2644
|
Mathlib.Topology.ContinuousFunction.Compact.491_0.Mig2jTVnn2FLKEB
|
theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F
|
Mathlib_Topology_ContinuousFunction_Compact
|
case h
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : LocallyCompactSpace X
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
ι : Type u_3
F : ι → C(X, E)
hF : ∀ (K : Compacts X), Summable fun i => ‖restrict (↑K) (F i)‖
K : Compacts X
s : Finset ι
x : ↑↑K
⊢ (∑ i in s, F i) ↑x = ∑ c in s, (restrict (↑K) (F c)) x
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compacts
#align_import topology.continuous_function.compact from "leanprover-community/mathlib"@"d3af0609f6db8691dffdc3e1fb7feb7da72698f2"
/-!
# Continuous functions on a compact space
Continuous functions `C(α, β)` from a compact space `α` to a metric space `β`
are automatically bounded, and so acquire various structures inherited from `α →ᵇ β`.
This file transfers these structures, and restates some lemmas
characterising these structures.
If you need a lemma which is proved about `α →ᵇ β` but not for `C(α, β)` when `α` is compact,
you should restate it here. You can also use
`ContinuousMap.equivBoundedOfCompact` to move functions back and forth.
-/
noncomputable section
open Topology Classical NNReal BoundedContinuousFunction BigOperators
open Set Filter Metric
open BoundedContinuousFunction
namespace ContinuousMap
variable {α β E : Type*} [TopologicalSpace α] [CompactSpace α] [MetricSpace β]
[NormedAddCommGroup E]
section
variable (α β)
/-- When `α` is compact, the bounded continuous maps `α →ᵇ β` are
equivalent to `C(α, β)`.
-/
@[simps (config := .asFn)]
def equivBoundedOfCompact : C(α, β) ≃ (α →ᵇ β) :=
⟨mkOfCompact, BoundedContinuousFunction.toContinuousMap, fun f => by
ext
rfl, fun f => by
ext
rfl⟩
#align continuous_map.equiv_bounded_of_compact ContinuousMap.equivBoundedOfCompact
theorem uniformInducing_equivBoundedOfCompact : UniformInducing (equivBoundedOfCompact α β) :=
UniformInducing.mk'
(by
simp only [hasBasis_compactConvergenceUniformity.mem_iff, uniformity_basis_dist_le.mem_iff]
exact fun s =>
⟨fun ⟨⟨a, b⟩, ⟨_, ⟨ε, hε, hb⟩⟩, hs⟩ =>
⟨{ p | ∀ x, (p.1 x, p.2 x) ∈ b }, ⟨ε, hε, fun _ h x => hb ((dist_le hε.le).mp h x)⟩,
fun f g h => hs fun x _ => h x⟩,
fun ⟨_, ⟨ε, hε, ht⟩, hs⟩ =>
⟨⟨Set.univ, { p | dist p.1 p.2 ≤ ε }⟩, ⟨isCompact_univ, ⟨ε, hε, fun _ h => h⟩⟩,
fun ⟨f, g⟩ h => hs _ _ (ht ((dist_le hε.le).mpr fun x => h x (mem_univ x)))⟩⟩)
#align continuous_map.uniform_inducing_equiv_bounded_of_compact ContinuousMap.uniformInducing_equivBoundedOfCompact
theorem uniformEmbedding_equivBoundedOfCompact : UniformEmbedding (equivBoundedOfCompact α β) :=
{ uniformInducing_equivBoundedOfCompact α β with inj := (equivBoundedOfCompact α β).injective }
#align continuous_map.uniform_embedding_equiv_bounded_of_compact ContinuousMap.uniformEmbedding_equivBoundedOfCompact
/-- When `α` is compact, the bounded continuous maps `α →ᵇ 𝕜` are
additively equivalent to `C(α, 𝕜)`.
-/
-- porting note: the following `simps` received a "maximum recursion depth" error
-- @[simps! (config := .asFn) apply symm_apply]
def addEquivBoundedOfCompact [AddMonoid β] [LipschitzAdd β] : C(α, β) ≃+ (α →ᵇ β) :=
({ toContinuousMapAddHom α β, (equivBoundedOfCompact α β).symm with } : (α →ᵇ β) ≃+ C(α, β)).symm
#align continuous_map.add_equiv_bounded_of_compact ContinuousMap.addEquivBoundedOfCompact
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_symm_apply [AddMonoid β] [LipschitzAdd β] :
⇑((addEquivBoundedOfCompact α β).symm) = toContinuousMapAddHom α β :=
rfl
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_apply [AddMonoid β] [LipschitzAdd β] :
⇑(addEquivBoundedOfCompact α β) = mkOfCompact :=
rfl
instance metricSpace : MetricSpace C(α, β) :=
(uniformEmbedding_equivBoundedOfCompact α β).comapMetricSpace _
#align continuous_map.metric_space ContinuousMap.metricSpace
/-- When `α` is compact, and `β` is a metric space, the bounded continuous maps `α →ᵇ β` are
isometric to `C(α, β)`.
-/
@[simps! (config := .asFn) toEquiv apply symm_apply]
def isometryEquivBoundedOfCompact : C(α, β) ≃ᵢ (α →ᵇ β) where
isometry_toFun _ _ := rfl
toEquiv := equivBoundedOfCompact α β
#align continuous_map.isometry_equiv_bounded_of_compact ContinuousMap.isometryEquivBoundedOfCompact
end
@[simp]
theorem _root_.BoundedContinuousFunction.dist_mkOfCompact (f g : C(α, β)) :
dist (mkOfCompact f) (mkOfCompact g) = dist f g :=
rfl
#align bounded_continuous_function.dist_mk_of_compact BoundedContinuousFunction.dist_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.dist_toContinuousMap (f g : α →ᵇ β) :
dist f.toContinuousMap g.toContinuousMap = dist f g :=
rfl
#align bounded_continuous_function.dist_to_continuous_map BoundedContinuousFunction.dist_toContinuousMap
open BoundedContinuousFunction
section
variable {f g : C(α, β)} {C : ℝ}
/-- The pointwise distance is controlled by the distance between functions, by definition. -/
theorem dist_apply_le_dist (x : α) : dist (f x) (g x) ≤ dist f g := by
simp only [← dist_mkOfCompact, dist_coe_le_dist, ← mkOfCompact_apply]
#align continuous_map.dist_apply_le_dist ContinuousMap.dist_apply_le_dist
/-- The distance between two functions is controlled by the supremum of the pointwise distances. -/
theorem dist_le (C0 : (0 : ℝ) ≤ C) : dist f g ≤ C ↔ ∀ x : α, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le C0, mkOfCompact_apply]
#align continuous_map.dist_le ContinuousMap.dist_le
theorem dist_le_iff_of_nonempty [Nonempty α] : dist f g ≤ C ↔ ∀ x, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le_iff_of_nonempty,
mkOfCompact_apply]
#align continuous_map.dist_le_iff_of_nonempty ContinuousMap.dist_le_iff_of_nonempty
theorem dist_lt_iff_of_nonempty [Nonempty α] : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
simp only [← dist_mkOfCompact, dist_lt_iff_of_nonempty_compact, mkOfCompact_apply]
#align continuous_map.dist_lt_iff_of_nonempty ContinuousMap.dist_lt_iff_of_nonempty
theorem dist_lt_of_nonempty [Nonempty α] (w : ∀ x : α, dist (f x) (g x) < C) : dist f g < C :=
dist_lt_iff_of_nonempty.2 w
#align continuous_map.dist_lt_of_nonempty ContinuousMap.dist_lt_of_nonempty
theorem dist_lt_iff (C0 : (0 : ℝ) < C) : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
rw [← dist_mkOfCompact, dist_lt_iff_of_compact C0]
simp only [mkOfCompact_apply]
#align continuous_map.dist_lt_iff ContinuousMap.dist_lt_iff
end
instance [CompleteSpace β] : CompleteSpace C(α, β) :=
(isometryEquivBoundedOfCompact α β).completeSpace
/-- See also `ContinuousMap.continuous_eval'`. -/
@[continuity]
theorem continuous_eval : Continuous fun p : C(α, β) × α => p.1 p.2 :=
continuous_eval.comp ((isometryEquivBoundedOfCompact α β).continuous.prod_map continuous_id)
#align continuous_map.continuous_eval ContinuousMap.continuous_eval
-- TODO at some point we will need lemmas characterising this norm!
-- At the moment the only way to reason about it is to transfer `f : C(α,E)` back to `α →ᵇ E`.
instance : Norm C(α, E) where norm x := dist x 0
@[simp]
theorem _root_.BoundedContinuousFunction.norm_mkOfCompact (f : C(α, E)) : ‖mkOfCompact f‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_mk_of_compact BoundedContinuousFunction.norm_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.norm_toContinuousMap_eq (f : α →ᵇ E) :
‖f.toContinuousMap‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_to_continuous_map_eq BoundedContinuousFunction.norm_toContinuousMap_eq
open BoundedContinuousFunction
instance : NormedAddCommGroup C(α, E) :=
{ ContinuousMap.metricSpace _ _,
ContinuousMap.instAddCommGroupContinuousMap with
dist_eq := fun x y => by
rw [← norm_mkOfCompact, ← dist_mkOfCompact, dist_eq_norm, mkOfCompact_sub]
dist := dist
norm := norm }
instance [Nonempty α] [One E] [NormOneClass E] : NormOneClass C(α, E) where
norm_one := by simp only [← norm_mkOfCompact, mkOfCompact_one, norm_one]
section
variable (f : C(α, E))
-- The corresponding lemmas for `BoundedContinuousFunction` are stated with `{f}`,
-- and so can not be used in dot notation.
theorem norm_coe_le_norm (x : α) : ‖f x‖ ≤ ‖f‖ :=
(mkOfCompact f).norm_coe_le_norm x
#align continuous_map.norm_coe_le_norm ContinuousMap.norm_coe_le_norm
/-- Distance between the images of any two points is at most twice the norm of the function. -/
theorem dist_le_two_norm (x y : α) : dist (f x) (f y) ≤ 2 * ‖f‖ :=
(mkOfCompact f).dist_le_two_norm x y
#align continuous_map.dist_le_two_norm ContinuousMap.dist_le_two_norm
/-- The norm of a function is controlled by the supremum of the pointwise norms. -/
theorem norm_le {C : ℝ} (C0 : (0 : ℝ) ≤ C) : ‖f‖ ≤ C ↔ ∀ x : α, ‖f x‖ ≤ C :=
@BoundedContinuousFunction.norm_le _ _ _ _ (mkOfCompact f) _ C0
#align continuous_map.norm_le ContinuousMap.norm_le
theorem norm_le_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ ≤ M ↔ ∀ x, ‖f x‖ ≤ M :=
@BoundedContinuousFunction.norm_le_of_nonempty _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_le_of_nonempty ContinuousMap.norm_le_of_nonempty
theorem norm_lt_iff {M : ℝ} (M0 : 0 < M) : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_compact _ _ _ _ _ (mkOfCompact f) _ M0
#align continuous_map.norm_lt_iff ContinuousMap.norm_lt_iff
theorem nnnorm_lt_iff {M : ℝ≥0} (M0 : 0 < M) : ‖f‖₊ < M ↔ ∀ x : α, ‖f x‖₊ < M :=
f.norm_lt_iff M0
#align continuous_map.nnnorm_lt_iff ContinuousMap.nnnorm_lt_iff
theorem norm_lt_iff_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_nonempty_compact _ _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_lt_iff_of_nonempty ContinuousMap.norm_lt_iff_of_nonempty
theorem nnnorm_lt_iff_of_nonempty [Nonempty α] {M : ℝ≥0} : ‖f‖₊ < M ↔ ∀ x, ‖f x‖₊ < M :=
f.norm_lt_iff_of_nonempty
#align continuous_map.nnnorm_lt_iff_of_nonempty ContinuousMap.nnnorm_lt_iff_of_nonempty
theorem apply_le_norm (f : C(α, ℝ)) (x : α) : f x ≤ ‖f‖ :=
le_trans (le_abs.mpr (Or.inl (le_refl (f x)))) (f.norm_coe_le_norm x)
#align continuous_map.apply_le_norm ContinuousMap.apply_le_norm
theorem neg_norm_le_apply (f : C(α, ℝ)) (x : α) : -‖f‖ ≤ f x :=
le_trans (neg_le_neg (f.norm_coe_le_norm x)) (neg_le.mp (neg_le_abs_self (f x)))
#align continuous_map.neg_norm_le_apply ContinuousMap.neg_norm_le_apply
theorem norm_eq_iSup_norm : ‖f‖ = ⨆ x : α, ‖f x‖ :=
(mkOfCompact f).norm_eq_iSup_norm
#align continuous_map.norm_eq_supr_norm ContinuousMap.norm_eq_iSup_norm
theorem norm_restrict_mono_set {X : Type*} [TopologicalSpace X] (f : C(X, E))
{K L : TopologicalSpace.Compacts X} (hKL : K ≤ L) : ‖f.restrict K‖ ≤ ‖f.restrict L‖ :=
(norm_le _ (norm_nonneg _)).mpr fun x => norm_coe_le_norm (f.restrict L) <| Set.inclusion hKL x
#align continuous_map.norm_restrict_mono_set ContinuousMap.norm_restrict_mono_set
end
section
variable {R : Type*} [NormedRing R]
instance : NormedRing C(α, R) :=
{ (inferInstance : NormedAddCommGroup C(α, R)), ContinuousMap.instRingContinuousMap with
norm_mul := fun f g => norm_mul_le (mkOfCompact f) (mkOfCompact g) }
end
section
variable {𝕜 : Type*} [NormedField 𝕜] [NormedSpace 𝕜 E]
instance normedSpace : NormedSpace 𝕜 C(α, E) where
norm_smul_le c f := (norm_smul_le c (mkOfCompact f) : _)
#align continuous_map.normed_space ContinuousMap.normedSpace
section
variable (α 𝕜 E)
/-- When `α` is compact and `𝕜` is a normed field,
the `𝕜`-algebra of bounded continuous maps `α →ᵇ β` is
`𝕜`-linearly isometric to `C(α, β)`.
-/
def linearIsometryBoundedOfCompact : C(α, E) ≃ₗᵢ[𝕜] α →ᵇ E :=
{ addEquivBoundedOfCompact α E with
map_smul' := fun c f => by
ext
norm_cast
norm_map' := fun f => rfl }
#align continuous_map.linear_isometry_bounded_of_compact ContinuousMap.linearIsometryBoundedOfCompact
variable {α E}
-- to match `BoundedContinuousFunction.evalClm`
/-- The evaluation at a point, as a continuous linear map from `C(α, 𝕜)` to `𝕜`. -/
def evalClm (x : α) : C(α, E) →L[𝕜] E :=
(BoundedContinuousFunction.evalClm 𝕜 x).comp
(linearIsometryBoundedOfCompact α E 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_map.eval_clm ContinuousMap.evalClm
end
-- this lemma and the next are the analogues of those autogenerated by `@[simps]` for
-- `equivBoundedOfCompact`, `addEquivBoundedOfCompact`
@[simp]
theorem linearIsometryBoundedOfCompact_symm_apply (f : α →ᵇ E) :
(linearIsometryBoundedOfCompact α E 𝕜).symm f = f.toContinuousMap :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_symm_apply ContinuousMap.linearIsometryBoundedOfCompact_symm_apply
@[simp]
theorem linearIsometryBoundedOfCompact_apply_apply (f : C(α, E)) (a : α) :
(linearIsometryBoundedOfCompact α E 𝕜 f) a = f a :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_apply_apply ContinuousMap.linearIsometryBoundedOfCompact_apply_apply
@[simp]
theorem linearIsometryBoundedOfCompact_toIsometryEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toIsometryEquiv = isometryEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_isometry_equiv ContinuousMap.linearIsometryBoundedOfCompact_toIsometryEquiv
@[simp] -- porting note: adjusted LHS because `simpNF` complained it simplified.
theorem linearIsometryBoundedOfCompact_toAddEquiv :
((linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv : C(α, E) ≃+ (α →ᵇ E)) =
addEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_add_equiv ContinuousMap.linearIsometryBoundedOfCompact_toAddEquiv
@[simp]
theorem linearIsometryBoundedOfCompact_of_compact_toEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv.toEquiv = equivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_of_compact_to_equiv ContinuousMap.linearIsometryBoundedOfCompact_of_compact_toEquiv
end
section
variable {𝕜 : Type*} {γ : Type*} [NormedField 𝕜] [NormedRing γ] [NormedAlgebra 𝕜 γ]
instance : NormedAlgebra 𝕜 C(α, γ) :=
{ ContinuousMap.normedSpace, ContinuousMap.algebra with }
end
end ContinuousMap
namespace ContinuousMap
section UniformContinuity
variable {α β : Type*}
variable [MetricSpace α] [CompactSpace α] [MetricSpace β]
/-!
We now set up some declarations making it convenient to use uniform continuity.
-/
theorem uniform_continuity (f : C(α, β)) (ε : ℝ) (h : 0 < ε) :
∃ δ > 0, ∀ {x y}, dist x y < δ → dist (f x) (f y) < ε :=
Metric.uniformContinuous_iff.mp (CompactSpace.uniformContinuous_of_continuous f.continuous) ε h
#align continuous_map.uniform_continuity ContinuousMap.uniform_continuity
-- This definition allows us to separate the choice of some `δ`,
-- and the corresponding use of `dist a b < δ → dist (f a) (f b) < ε`,
-- even across different declarations.
/-- An arbitrarily chosen modulus of uniform continuity for a given function `f` and `ε > 0`. -/
def modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) : ℝ :=
Classical.choose (uniform_continuity f ε h)
#align continuous_map.modulus ContinuousMap.modulus
theorem modulus_pos (f : C(α, β)) {ε : ℝ} {h : 0 < ε} : 0 < f.modulus ε h :=
(Classical.choose_spec (uniform_continuity f ε h)).1
#align continuous_map.modulus_pos ContinuousMap.modulus_pos
theorem dist_lt_of_dist_lt_modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) {a b : α}
(w : dist a b < f.modulus ε h) : dist (f a) (f b) < ε :=
(Classical.choose_spec (uniform_continuity f ε h)).2 w
#align continuous_map.dist_lt_of_dist_lt_modulus ContinuousMap.dist_lt_of_dist_lt_modulus
end UniformContinuity
end ContinuousMap
section CompLeft
variable (X : Type*) {𝕜 β γ : Type*} [TopologicalSpace X] [CompactSpace X]
[NontriviallyNormedField 𝕜]
variable [NormedAddCommGroup β] [NormedSpace 𝕜 β] [NormedAddCommGroup γ] [NormedSpace 𝕜 γ]
open ContinuousMap
/-- Postcomposition of continuous functions into a normed module by a continuous linear map is a
continuous linear map.
Transferred version of `ContinuousLinearMap.compLeftContinuousBounded`,
upgraded version of `ContinuousLinearMap.compLeftContinuous`,
similar to `LinearMap.compLeft`. -/
protected def ContinuousLinearMap.compLeftContinuousCompact (g : β →L[𝕜] γ) :
C(X, β) →L[𝕜] C(X, γ) :=
(linearIsometryBoundedOfCompact X γ 𝕜).symm.toLinearIsometry.toContinuousLinearMap.comp <|
(g.compLeftContinuousBounded X).comp <|
(linearIsometryBoundedOfCompact X β 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_linear_map.comp_left_continuous_compact ContinuousLinearMap.compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.toLinear_compLeftContinuousCompact (g : β →L[𝕜] γ) :
(g.compLeftContinuousCompact X : C(X, β) →ₗ[𝕜] C(X, γ)) = g.compLeftContinuous 𝕜 X := by
ext f
rfl
#align continuous_linear_map.to_linear_comp_left_continuous_compact ContinuousLinearMap.toLinear_compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.compLeftContinuousCompact_apply (g : β →L[𝕜] γ) (f : C(X, β)) (x : X) :
g.compLeftContinuousCompact X f x = g (f x) :=
rfl
#align continuous_linear_map.comp_left_continuous_compact_apply ContinuousLinearMap.compLeftContinuousCompact_apply
end CompLeft
namespace ContinuousMap
/-!
We now setup variations on `compRight* f`, where `f : C(X, Y)`
(that is, precomposition by a continuous map),
as a morphism `C(Y, T) → C(X, T)`, respecting various types of structure.
In particular:
* `compRightContinuousMap`, the bundled continuous map (for this we need `X Y` compact).
* `compRightHomeomorph`, when we precompose by a homeomorphism.
* `compRightAlgHom`, when `T = R` is a topological ring.
-/
section CompRight
/-- Precomposition by a continuous map is itself a continuous map between spaces of continuous maps.
-/
def compRightContinuousMap {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y)) : C(C(Y, T), C(X, T)) where
toFun g := g.comp f
continuous_toFun := by
refine' Metric.continuous_iff.mpr _
intro g ε ε_pos
refine' ⟨ε, ε_pos, fun g' h => _⟩
rw [ContinuousMap.dist_lt_iff ε_pos] at h ⊢
exact fun x => h (f x)
#align continuous_map.comp_right_continuous_map ContinuousMap.compRightContinuousMap
@[simp]
theorem compRightContinuousMap_apply {X Y : Type*} (T : Type*) [TopologicalSpace X]
[CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y))
(g : C(Y, T)) : (compRightContinuousMap T f) g = g.comp f :=
rfl
#align continuous_map.comp_right_continuous_map_apply ContinuousMap.compRightContinuousMap_apply
/-- Precomposition by a homeomorphism is itself a homeomorphism between spaces of continuous maps.
-/
def compRightHomeomorph {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : X ≃ₜ Y) : C(Y, T) ≃ₜ C(X, T) where
toFun := compRightContinuousMap T f.toContinuousMap
invFun := compRightContinuousMap T f.symm.toContinuousMap
left_inv g := ext fun _ => congr_arg g (f.apply_symm_apply _)
right_inv g := ext fun _ => congr_arg g (f.symm_apply_apply _)
#align continuous_map.comp_right_homeomorph ContinuousMap.compRightHomeomorph
theorem compRightAlgHom_continuous {X Y : Type*} (R A : Type*) [TopologicalSpace X]
[CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [CommSemiring R] [Semiring A]
[MetricSpace A] [TopologicalSemiring A] [Algebra R A] (f : C(X, Y)) :
Continuous (compRightAlgHom R A f) :=
map_continuous (compRightContinuousMap A f)
#align continuous_map.comp_right_alg_hom_continuous ContinuousMap.compRightAlgHom_continuous
end CompRight
section LocalNormalConvergence
/-! ### Local normal convergence
A sum of continuous functions (on a locally compact space) is "locally normally convergent" if the
sum of its sup-norms on any compact subset is summable. This implies convergence in the topology
of `C(X, E)` (i.e. locally uniform convergence). -/
open TopologicalSpace
variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
variable {E : Type*} [NormedAddCommGroup E] [CompleteSpace E]
theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F := by
refine' (ContinuousMap.exists_tendsto_compactOpen_iff_forall _).2 fun K hK => _
lift K to Compacts X using hK
have A : ∀ s : Finset ι, restrict (↑K) (∑ i in s, F i) = ∑ i in s, restrict K (F i) := by
intro s
ext1 x
simp
-- This used to be the end of the proof before leanprover/lean4#2644
erw [restrict_apply, restrict_apply, restrict_apply, restrict_apply]
|
simp? says simp only [coe_sum, Finset.sum_apply]
|
theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F := by
refine' (ContinuousMap.exists_tendsto_compactOpen_iff_forall _).2 fun K hK => _
lift K to Compacts X using hK
have A : ∀ s : Finset ι, restrict (↑K) (∑ i in s, F i) = ∑ i in s, restrict K (F i) := by
intro s
ext1 x
simp
-- This used to be the end of the proof before leanprover/lean4#2644
erw [restrict_apply, restrict_apply, restrict_apply, restrict_apply]
|
Mathlib.Topology.ContinuousFunction.Compact.491_0.Mig2jTVnn2FLKEB
|
theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F
|
Mathlib_Topology_ContinuousFunction_Compact
|
case h
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : LocallyCompactSpace X
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
ι : Type u_3
F : ι → C(X, E)
hF : ∀ (K : Compacts X), Summable fun i => ‖restrict (↑K) (F i)‖
K : Compacts X
s : Finset ι
x : ↑↑K
⊢ (∑ i in s, F i) ↑x = ∑ c in s, (restrict (↑K) (F c)) x
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compacts
#align_import topology.continuous_function.compact from "leanprover-community/mathlib"@"d3af0609f6db8691dffdc3e1fb7feb7da72698f2"
/-!
# Continuous functions on a compact space
Continuous functions `C(α, β)` from a compact space `α` to a metric space `β`
are automatically bounded, and so acquire various structures inherited from `α →ᵇ β`.
This file transfers these structures, and restates some lemmas
characterising these structures.
If you need a lemma which is proved about `α →ᵇ β` but not for `C(α, β)` when `α` is compact,
you should restate it here. You can also use
`ContinuousMap.equivBoundedOfCompact` to move functions back and forth.
-/
noncomputable section
open Topology Classical NNReal BoundedContinuousFunction BigOperators
open Set Filter Metric
open BoundedContinuousFunction
namespace ContinuousMap
variable {α β E : Type*} [TopologicalSpace α] [CompactSpace α] [MetricSpace β]
[NormedAddCommGroup E]
section
variable (α β)
/-- When `α` is compact, the bounded continuous maps `α →ᵇ β` are
equivalent to `C(α, β)`.
-/
@[simps (config := .asFn)]
def equivBoundedOfCompact : C(α, β) ≃ (α →ᵇ β) :=
⟨mkOfCompact, BoundedContinuousFunction.toContinuousMap, fun f => by
ext
rfl, fun f => by
ext
rfl⟩
#align continuous_map.equiv_bounded_of_compact ContinuousMap.equivBoundedOfCompact
theorem uniformInducing_equivBoundedOfCompact : UniformInducing (equivBoundedOfCompact α β) :=
UniformInducing.mk'
(by
simp only [hasBasis_compactConvergenceUniformity.mem_iff, uniformity_basis_dist_le.mem_iff]
exact fun s =>
⟨fun ⟨⟨a, b⟩, ⟨_, ⟨ε, hε, hb⟩⟩, hs⟩ =>
⟨{ p | ∀ x, (p.1 x, p.2 x) ∈ b }, ⟨ε, hε, fun _ h x => hb ((dist_le hε.le).mp h x)⟩,
fun f g h => hs fun x _ => h x⟩,
fun ⟨_, ⟨ε, hε, ht⟩, hs⟩ =>
⟨⟨Set.univ, { p | dist p.1 p.2 ≤ ε }⟩, ⟨isCompact_univ, ⟨ε, hε, fun _ h => h⟩⟩,
fun ⟨f, g⟩ h => hs _ _ (ht ((dist_le hε.le).mpr fun x => h x (mem_univ x)))⟩⟩)
#align continuous_map.uniform_inducing_equiv_bounded_of_compact ContinuousMap.uniformInducing_equivBoundedOfCompact
theorem uniformEmbedding_equivBoundedOfCompact : UniformEmbedding (equivBoundedOfCompact α β) :=
{ uniformInducing_equivBoundedOfCompact α β with inj := (equivBoundedOfCompact α β).injective }
#align continuous_map.uniform_embedding_equiv_bounded_of_compact ContinuousMap.uniformEmbedding_equivBoundedOfCompact
/-- When `α` is compact, the bounded continuous maps `α →ᵇ 𝕜` are
additively equivalent to `C(α, 𝕜)`.
-/
-- porting note: the following `simps` received a "maximum recursion depth" error
-- @[simps! (config := .asFn) apply symm_apply]
def addEquivBoundedOfCompact [AddMonoid β] [LipschitzAdd β] : C(α, β) ≃+ (α →ᵇ β) :=
({ toContinuousMapAddHom α β, (equivBoundedOfCompact α β).symm with } : (α →ᵇ β) ≃+ C(α, β)).symm
#align continuous_map.add_equiv_bounded_of_compact ContinuousMap.addEquivBoundedOfCompact
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_symm_apply [AddMonoid β] [LipschitzAdd β] :
⇑((addEquivBoundedOfCompact α β).symm) = toContinuousMapAddHom α β :=
rfl
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_apply [AddMonoid β] [LipschitzAdd β] :
⇑(addEquivBoundedOfCompact α β) = mkOfCompact :=
rfl
instance metricSpace : MetricSpace C(α, β) :=
(uniformEmbedding_equivBoundedOfCompact α β).comapMetricSpace _
#align continuous_map.metric_space ContinuousMap.metricSpace
/-- When `α` is compact, and `β` is a metric space, the bounded continuous maps `α →ᵇ β` are
isometric to `C(α, β)`.
-/
@[simps! (config := .asFn) toEquiv apply symm_apply]
def isometryEquivBoundedOfCompact : C(α, β) ≃ᵢ (α →ᵇ β) where
isometry_toFun _ _ := rfl
toEquiv := equivBoundedOfCompact α β
#align continuous_map.isometry_equiv_bounded_of_compact ContinuousMap.isometryEquivBoundedOfCompact
end
@[simp]
theorem _root_.BoundedContinuousFunction.dist_mkOfCompact (f g : C(α, β)) :
dist (mkOfCompact f) (mkOfCompact g) = dist f g :=
rfl
#align bounded_continuous_function.dist_mk_of_compact BoundedContinuousFunction.dist_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.dist_toContinuousMap (f g : α →ᵇ β) :
dist f.toContinuousMap g.toContinuousMap = dist f g :=
rfl
#align bounded_continuous_function.dist_to_continuous_map BoundedContinuousFunction.dist_toContinuousMap
open BoundedContinuousFunction
section
variable {f g : C(α, β)} {C : ℝ}
/-- The pointwise distance is controlled by the distance between functions, by definition. -/
theorem dist_apply_le_dist (x : α) : dist (f x) (g x) ≤ dist f g := by
simp only [← dist_mkOfCompact, dist_coe_le_dist, ← mkOfCompact_apply]
#align continuous_map.dist_apply_le_dist ContinuousMap.dist_apply_le_dist
/-- The distance between two functions is controlled by the supremum of the pointwise distances. -/
theorem dist_le (C0 : (0 : ℝ) ≤ C) : dist f g ≤ C ↔ ∀ x : α, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le C0, mkOfCompact_apply]
#align continuous_map.dist_le ContinuousMap.dist_le
theorem dist_le_iff_of_nonempty [Nonempty α] : dist f g ≤ C ↔ ∀ x, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le_iff_of_nonempty,
mkOfCompact_apply]
#align continuous_map.dist_le_iff_of_nonempty ContinuousMap.dist_le_iff_of_nonempty
theorem dist_lt_iff_of_nonempty [Nonempty α] : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
simp only [← dist_mkOfCompact, dist_lt_iff_of_nonempty_compact, mkOfCompact_apply]
#align continuous_map.dist_lt_iff_of_nonempty ContinuousMap.dist_lt_iff_of_nonempty
theorem dist_lt_of_nonempty [Nonempty α] (w : ∀ x : α, dist (f x) (g x) < C) : dist f g < C :=
dist_lt_iff_of_nonempty.2 w
#align continuous_map.dist_lt_of_nonempty ContinuousMap.dist_lt_of_nonempty
theorem dist_lt_iff (C0 : (0 : ℝ) < C) : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
rw [← dist_mkOfCompact, dist_lt_iff_of_compact C0]
simp only [mkOfCompact_apply]
#align continuous_map.dist_lt_iff ContinuousMap.dist_lt_iff
end
instance [CompleteSpace β] : CompleteSpace C(α, β) :=
(isometryEquivBoundedOfCompact α β).completeSpace
/-- See also `ContinuousMap.continuous_eval'`. -/
@[continuity]
theorem continuous_eval : Continuous fun p : C(α, β) × α => p.1 p.2 :=
continuous_eval.comp ((isometryEquivBoundedOfCompact α β).continuous.prod_map continuous_id)
#align continuous_map.continuous_eval ContinuousMap.continuous_eval
-- TODO at some point we will need lemmas characterising this norm!
-- At the moment the only way to reason about it is to transfer `f : C(α,E)` back to `α →ᵇ E`.
instance : Norm C(α, E) where norm x := dist x 0
@[simp]
theorem _root_.BoundedContinuousFunction.norm_mkOfCompact (f : C(α, E)) : ‖mkOfCompact f‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_mk_of_compact BoundedContinuousFunction.norm_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.norm_toContinuousMap_eq (f : α →ᵇ E) :
‖f.toContinuousMap‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_to_continuous_map_eq BoundedContinuousFunction.norm_toContinuousMap_eq
open BoundedContinuousFunction
instance : NormedAddCommGroup C(α, E) :=
{ ContinuousMap.metricSpace _ _,
ContinuousMap.instAddCommGroupContinuousMap with
dist_eq := fun x y => by
rw [← norm_mkOfCompact, ← dist_mkOfCompact, dist_eq_norm, mkOfCompact_sub]
dist := dist
norm := norm }
instance [Nonempty α] [One E] [NormOneClass E] : NormOneClass C(α, E) where
norm_one := by simp only [← norm_mkOfCompact, mkOfCompact_one, norm_one]
section
variable (f : C(α, E))
-- The corresponding lemmas for `BoundedContinuousFunction` are stated with `{f}`,
-- and so can not be used in dot notation.
theorem norm_coe_le_norm (x : α) : ‖f x‖ ≤ ‖f‖ :=
(mkOfCompact f).norm_coe_le_norm x
#align continuous_map.norm_coe_le_norm ContinuousMap.norm_coe_le_norm
/-- Distance between the images of any two points is at most twice the norm of the function. -/
theorem dist_le_two_norm (x y : α) : dist (f x) (f y) ≤ 2 * ‖f‖ :=
(mkOfCompact f).dist_le_two_norm x y
#align continuous_map.dist_le_two_norm ContinuousMap.dist_le_two_norm
/-- The norm of a function is controlled by the supremum of the pointwise norms. -/
theorem norm_le {C : ℝ} (C0 : (0 : ℝ) ≤ C) : ‖f‖ ≤ C ↔ ∀ x : α, ‖f x‖ ≤ C :=
@BoundedContinuousFunction.norm_le _ _ _ _ (mkOfCompact f) _ C0
#align continuous_map.norm_le ContinuousMap.norm_le
theorem norm_le_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ ≤ M ↔ ∀ x, ‖f x‖ ≤ M :=
@BoundedContinuousFunction.norm_le_of_nonempty _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_le_of_nonempty ContinuousMap.norm_le_of_nonempty
theorem norm_lt_iff {M : ℝ} (M0 : 0 < M) : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_compact _ _ _ _ _ (mkOfCompact f) _ M0
#align continuous_map.norm_lt_iff ContinuousMap.norm_lt_iff
theorem nnnorm_lt_iff {M : ℝ≥0} (M0 : 0 < M) : ‖f‖₊ < M ↔ ∀ x : α, ‖f x‖₊ < M :=
f.norm_lt_iff M0
#align continuous_map.nnnorm_lt_iff ContinuousMap.nnnorm_lt_iff
theorem norm_lt_iff_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_nonempty_compact _ _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_lt_iff_of_nonempty ContinuousMap.norm_lt_iff_of_nonempty
theorem nnnorm_lt_iff_of_nonempty [Nonempty α] {M : ℝ≥0} : ‖f‖₊ < M ↔ ∀ x, ‖f x‖₊ < M :=
f.norm_lt_iff_of_nonempty
#align continuous_map.nnnorm_lt_iff_of_nonempty ContinuousMap.nnnorm_lt_iff_of_nonempty
theorem apply_le_norm (f : C(α, ℝ)) (x : α) : f x ≤ ‖f‖ :=
le_trans (le_abs.mpr (Or.inl (le_refl (f x)))) (f.norm_coe_le_norm x)
#align continuous_map.apply_le_norm ContinuousMap.apply_le_norm
theorem neg_norm_le_apply (f : C(α, ℝ)) (x : α) : -‖f‖ ≤ f x :=
le_trans (neg_le_neg (f.norm_coe_le_norm x)) (neg_le.mp (neg_le_abs_self (f x)))
#align continuous_map.neg_norm_le_apply ContinuousMap.neg_norm_le_apply
theorem norm_eq_iSup_norm : ‖f‖ = ⨆ x : α, ‖f x‖ :=
(mkOfCompact f).norm_eq_iSup_norm
#align continuous_map.norm_eq_supr_norm ContinuousMap.norm_eq_iSup_norm
theorem norm_restrict_mono_set {X : Type*} [TopologicalSpace X] (f : C(X, E))
{K L : TopologicalSpace.Compacts X} (hKL : K ≤ L) : ‖f.restrict K‖ ≤ ‖f.restrict L‖ :=
(norm_le _ (norm_nonneg _)).mpr fun x => norm_coe_le_norm (f.restrict L) <| Set.inclusion hKL x
#align continuous_map.norm_restrict_mono_set ContinuousMap.norm_restrict_mono_set
end
section
variable {R : Type*} [NormedRing R]
instance : NormedRing C(α, R) :=
{ (inferInstance : NormedAddCommGroup C(α, R)), ContinuousMap.instRingContinuousMap with
norm_mul := fun f g => norm_mul_le (mkOfCompact f) (mkOfCompact g) }
end
section
variable {𝕜 : Type*} [NormedField 𝕜] [NormedSpace 𝕜 E]
instance normedSpace : NormedSpace 𝕜 C(α, E) where
norm_smul_le c f := (norm_smul_le c (mkOfCompact f) : _)
#align continuous_map.normed_space ContinuousMap.normedSpace
section
variable (α 𝕜 E)
/-- When `α` is compact and `𝕜` is a normed field,
the `𝕜`-algebra of bounded continuous maps `α →ᵇ β` is
`𝕜`-linearly isometric to `C(α, β)`.
-/
def linearIsometryBoundedOfCompact : C(α, E) ≃ₗᵢ[𝕜] α →ᵇ E :=
{ addEquivBoundedOfCompact α E with
map_smul' := fun c f => by
ext
norm_cast
norm_map' := fun f => rfl }
#align continuous_map.linear_isometry_bounded_of_compact ContinuousMap.linearIsometryBoundedOfCompact
variable {α E}
-- to match `BoundedContinuousFunction.evalClm`
/-- The evaluation at a point, as a continuous linear map from `C(α, 𝕜)` to `𝕜`. -/
def evalClm (x : α) : C(α, E) →L[𝕜] E :=
(BoundedContinuousFunction.evalClm 𝕜 x).comp
(linearIsometryBoundedOfCompact α E 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_map.eval_clm ContinuousMap.evalClm
end
-- this lemma and the next are the analogues of those autogenerated by `@[simps]` for
-- `equivBoundedOfCompact`, `addEquivBoundedOfCompact`
@[simp]
theorem linearIsometryBoundedOfCompact_symm_apply (f : α →ᵇ E) :
(linearIsometryBoundedOfCompact α E 𝕜).symm f = f.toContinuousMap :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_symm_apply ContinuousMap.linearIsometryBoundedOfCompact_symm_apply
@[simp]
theorem linearIsometryBoundedOfCompact_apply_apply (f : C(α, E)) (a : α) :
(linearIsometryBoundedOfCompact α E 𝕜 f) a = f a :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_apply_apply ContinuousMap.linearIsometryBoundedOfCompact_apply_apply
@[simp]
theorem linearIsometryBoundedOfCompact_toIsometryEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toIsometryEquiv = isometryEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_isometry_equiv ContinuousMap.linearIsometryBoundedOfCompact_toIsometryEquiv
@[simp] -- porting note: adjusted LHS because `simpNF` complained it simplified.
theorem linearIsometryBoundedOfCompact_toAddEquiv :
((linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv : C(α, E) ≃+ (α →ᵇ E)) =
addEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_add_equiv ContinuousMap.linearIsometryBoundedOfCompact_toAddEquiv
@[simp]
theorem linearIsometryBoundedOfCompact_of_compact_toEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv.toEquiv = equivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_of_compact_to_equiv ContinuousMap.linearIsometryBoundedOfCompact_of_compact_toEquiv
end
section
variable {𝕜 : Type*} {γ : Type*} [NormedField 𝕜] [NormedRing γ] [NormedAlgebra 𝕜 γ]
instance : NormedAlgebra 𝕜 C(α, γ) :=
{ ContinuousMap.normedSpace, ContinuousMap.algebra with }
end
end ContinuousMap
namespace ContinuousMap
section UniformContinuity
variable {α β : Type*}
variable [MetricSpace α] [CompactSpace α] [MetricSpace β]
/-!
We now set up some declarations making it convenient to use uniform continuity.
-/
theorem uniform_continuity (f : C(α, β)) (ε : ℝ) (h : 0 < ε) :
∃ δ > 0, ∀ {x y}, dist x y < δ → dist (f x) (f y) < ε :=
Metric.uniformContinuous_iff.mp (CompactSpace.uniformContinuous_of_continuous f.continuous) ε h
#align continuous_map.uniform_continuity ContinuousMap.uniform_continuity
-- This definition allows us to separate the choice of some `δ`,
-- and the corresponding use of `dist a b < δ → dist (f a) (f b) < ε`,
-- even across different declarations.
/-- An arbitrarily chosen modulus of uniform continuity for a given function `f` and `ε > 0`. -/
def modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) : ℝ :=
Classical.choose (uniform_continuity f ε h)
#align continuous_map.modulus ContinuousMap.modulus
theorem modulus_pos (f : C(α, β)) {ε : ℝ} {h : 0 < ε} : 0 < f.modulus ε h :=
(Classical.choose_spec (uniform_continuity f ε h)).1
#align continuous_map.modulus_pos ContinuousMap.modulus_pos
theorem dist_lt_of_dist_lt_modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) {a b : α}
(w : dist a b < f.modulus ε h) : dist (f a) (f b) < ε :=
(Classical.choose_spec (uniform_continuity f ε h)).2 w
#align continuous_map.dist_lt_of_dist_lt_modulus ContinuousMap.dist_lt_of_dist_lt_modulus
end UniformContinuity
end ContinuousMap
section CompLeft
variable (X : Type*) {𝕜 β γ : Type*} [TopologicalSpace X] [CompactSpace X]
[NontriviallyNormedField 𝕜]
variable [NormedAddCommGroup β] [NormedSpace 𝕜 β] [NormedAddCommGroup γ] [NormedSpace 𝕜 γ]
open ContinuousMap
/-- Postcomposition of continuous functions into a normed module by a continuous linear map is a
continuous linear map.
Transferred version of `ContinuousLinearMap.compLeftContinuousBounded`,
upgraded version of `ContinuousLinearMap.compLeftContinuous`,
similar to `LinearMap.compLeft`. -/
protected def ContinuousLinearMap.compLeftContinuousCompact (g : β →L[𝕜] γ) :
C(X, β) →L[𝕜] C(X, γ) :=
(linearIsometryBoundedOfCompact X γ 𝕜).symm.toLinearIsometry.toContinuousLinearMap.comp <|
(g.compLeftContinuousBounded X).comp <|
(linearIsometryBoundedOfCompact X β 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_linear_map.comp_left_continuous_compact ContinuousLinearMap.compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.toLinear_compLeftContinuousCompact (g : β →L[𝕜] γ) :
(g.compLeftContinuousCompact X : C(X, β) →ₗ[𝕜] C(X, γ)) = g.compLeftContinuous 𝕜 X := by
ext f
rfl
#align continuous_linear_map.to_linear_comp_left_continuous_compact ContinuousLinearMap.toLinear_compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.compLeftContinuousCompact_apply (g : β →L[𝕜] γ) (f : C(X, β)) (x : X) :
g.compLeftContinuousCompact X f x = g (f x) :=
rfl
#align continuous_linear_map.comp_left_continuous_compact_apply ContinuousLinearMap.compLeftContinuousCompact_apply
end CompLeft
namespace ContinuousMap
/-!
We now setup variations on `compRight* f`, where `f : C(X, Y)`
(that is, precomposition by a continuous map),
as a morphism `C(Y, T) → C(X, T)`, respecting various types of structure.
In particular:
* `compRightContinuousMap`, the bundled continuous map (for this we need `X Y` compact).
* `compRightHomeomorph`, when we precompose by a homeomorphism.
* `compRightAlgHom`, when `T = R` is a topological ring.
-/
section CompRight
/-- Precomposition by a continuous map is itself a continuous map between spaces of continuous maps.
-/
def compRightContinuousMap {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y)) : C(C(Y, T), C(X, T)) where
toFun g := g.comp f
continuous_toFun := by
refine' Metric.continuous_iff.mpr _
intro g ε ε_pos
refine' ⟨ε, ε_pos, fun g' h => _⟩
rw [ContinuousMap.dist_lt_iff ε_pos] at h ⊢
exact fun x => h (f x)
#align continuous_map.comp_right_continuous_map ContinuousMap.compRightContinuousMap
@[simp]
theorem compRightContinuousMap_apply {X Y : Type*} (T : Type*) [TopologicalSpace X]
[CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y))
(g : C(Y, T)) : (compRightContinuousMap T f) g = g.comp f :=
rfl
#align continuous_map.comp_right_continuous_map_apply ContinuousMap.compRightContinuousMap_apply
/-- Precomposition by a homeomorphism is itself a homeomorphism between spaces of continuous maps.
-/
def compRightHomeomorph {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : X ≃ₜ Y) : C(Y, T) ≃ₜ C(X, T) where
toFun := compRightContinuousMap T f.toContinuousMap
invFun := compRightContinuousMap T f.symm.toContinuousMap
left_inv g := ext fun _ => congr_arg g (f.apply_symm_apply _)
right_inv g := ext fun _ => congr_arg g (f.symm_apply_apply _)
#align continuous_map.comp_right_homeomorph ContinuousMap.compRightHomeomorph
theorem compRightAlgHom_continuous {X Y : Type*} (R A : Type*) [TopologicalSpace X]
[CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [CommSemiring R] [Semiring A]
[MetricSpace A] [TopologicalSemiring A] [Algebra R A] (f : C(X, Y)) :
Continuous (compRightAlgHom R A f) :=
map_continuous (compRightContinuousMap A f)
#align continuous_map.comp_right_alg_hom_continuous ContinuousMap.compRightAlgHom_continuous
end CompRight
section LocalNormalConvergence
/-! ### Local normal convergence
A sum of continuous functions (on a locally compact space) is "locally normally convergent" if the
sum of its sup-norms on any compact subset is summable. This implies convergence in the topology
of `C(X, E)` (i.e. locally uniform convergence). -/
open TopologicalSpace
variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
variable {E : Type*} [NormedAddCommGroup E] [CompleteSpace E]
theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F := by
refine' (ContinuousMap.exists_tendsto_compactOpen_iff_forall _).2 fun K hK => _
lift K to Compacts X using hK
have A : ∀ s : Finset ι, restrict (↑K) (∑ i in s, F i) = ∑ i in s, restrict K (F i) := by
intro s
ext1 x
simp
-- This used to be the end of the proof before leanprover/lean4#2644
erw [restrict_apply, restrict_apply, restrict_apply, restrict_apply]
simp? says
|
simp only [coe_sum, Finset.sum_apply]
|
theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F := by
refine' (ContinuousMap.exists_tendsto_compactOpen_iff_forall _).2 fun K hK => _
lift K to Compacts X using hK
have A : ∀ s : Finset ι, restrict (↑K) (∑ i in s, F i) = ∑ i in s, restrict K (F i) := by
intro s
ext1 x
simp
-- This used to be the end of the proof before leanprover/lean4#2644
erw [restrict_apply, restrict_apply, restrict_apply, restrict_apply]
simp? says
|
Mathlib.Topology.ContinuousFunction.Compact.491_0.Mig2jTVnn2FLKEB
|
theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F
|
Mathlib_Topology_ContinuousFunction_Compact
|
case h
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : LocallyCompactSpace X
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
ι : Type u_3
F : ι → C(X, E)
hF : ∀ (K : Compacts X), Summable fun i => ‖restrict (↑K) (F i)‖
K : Compacts X
s : Finset ι
x : ↑↑K
⊢ ∑ c in s, (F c) ↑x = ∑ c in s, (restrict (↑K) (F c)) x
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compacts
#align_import topology.continuous_function.compact from "leanprover-community/mathlib"@"d3af0609f6db8691dffdc3e1fb7feb7da72698f2"
/-!
# Continuous functions on a compact space
Continuous functions `C(α, β)` from a compact space `α` to a metric space `β`
are automatically bounded, and so acquire various structures inherited from `α →ᵇ β`.
This file transfers these structures, and restates some lemmas
characterising these structures.
If you need a lemma which is proved about `α →ᵇ β` but not for `C(α, β)` when `α` is compact,
you should restate it here. You can also use
`ContinuousMap.equivBoundedOfCompact` to move functions back and forth.
-/
noncomputable section
open Topology Classical NNReal BoundedContinuousFunction BigOperators
open Set Filter Metric
open BoundedContinuousFunction
namespace ContinuousMap
variable {α β E : Type*} [TopologicalSpace α] [CompactSpace α] [MetricSpace β]
[NormedAddCommGroup E]
section
variable (α β)
/-- When `α` is compact, the bounded continuous maps `α →ᵇ β` are
equivalent to `C(α, β)`.
-/
@[simps (config := .asFn)]
def equivBoundedOfCompact : C(α, β) ≃ (α →ᵇ β) :=
⟨mkOfCompact, BoundedContinuousFunction.toContinuousMap, fun f => by
ext
rfl, fun f => by
ext
rfl⟩
#align continuous_map.equiv_bounded_of_compact ContinuousMap.equivBoundedOfCompact
theorem uniformInducing_equivBoundedOfCompact : UniformInducing (equivBoundedOfCompact α β) :=
UniformInducing.mk'
(by
simp only [hasBasis_compactConvergenceUniformity.mem_iff, uniformity_basis_dist_le.mem_iff]
exact fun s =>
⟨fun ⟨⟨a, b⟩, ⟨_, ⟨ε, hε, hb⟩⟩, hs⟩ =>
⟨{ p | ∀ x, (p.1 x, p.2 x) ∈ b }, ⟨ε, hε, fun _ h x => hb ((dist_le hε.le).mp h x)⟩,
fun f g h => hs fun x _ => h x⟩,
fun ⟨_, ⟨ε, hε, ht⟩, hs⟩ =>
⟨⟨Set.univ, { p | dist p.1 p.2 ≤ ε }⟩, ⟨isCompact_univ, ⟨ε, hε, fun _ h => h⟩⟩,
fun ⟨f, g⟩ h => hs _ _ (ht ((dist_le hε.le).mpr fun x => h x (mem_univ x)))⟩⟩)
#align continuous_map.uniform_inducing_equiv_bounded_of_compact ContinuousMap.uniformInducing_equivBoundedOfCompact
theorem uniformEmbedding_equivBoundedOfCompact : UniformEmbedding (equivBoundedOfCompact α β) :=
{ uniformInducing_equivBoundedOfCompact α β with inj := (equivBoundedOfCompact α β).injective }
#align continuous_map.uniform_embedding_equiv_bounded_of_compact ContinuousMap.uniformEmbedding_equivBoundedOfCompact
/-- When `α` is compact, the bounded continuous maps `α →ᵇ 𝕜` are
additively equivalent to `C(α, 𝕜)`.
-/
-- porting note: the following `simps` received a "maximum recursion depth" error
-- @[simps! (config := .asFn) apply symm_apply]
def addEquivBoundedOfCompact [AddMonoid β] [LipschitzAdd β] : C(α, β) ≃+ (α →ᵇ β) :=
({ toContinuousMapAddHom α β, (equivBoundedOfCompact α β).symm with } : (α →ᵇ β) ≃+ C(α, β)).symm
#align continuous_map.add_equiv_bounded_of_compact ContinuousMap.addEquivBoundedOfCompact
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_symm_apply [AddMonoid β] [LipschitzAdd β] :
⇑((addEquivBoundedOfCompact α β).symm) = toContinuousMapAddHom α β :=
rfl
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_apply [AddMonoid β] [LipschitzAdd β] :
⇑(addEquivBoundedOfCompact α β) = mkOfCompact :=
rfl
instance metricSpace : MetricSpace C(α, β) :=
(uniformEmbedding_equivBoundedOfCompact α β).comapMetricSpace _
#align continuous_map.metric_space ContinuousMap.metricSpace
/-- When `α` is compact, and `β` is a metric space, the bounded continuous maps `α →ᵇ β` are
isometric to `C(α, β)`.
-/
@[simps! (config := .asFn) toEquiv apply symm_apply]
def isometryEquivBoundedOfCompact : C(α, β) ≃ᵢ (α →ᵇ β) where
isometry_toFun _ _ := rfl
toEquiv := equivBoundedOfCompact α β
#align continuous_map.isometry_equiv_bounded_of_compact ContinuousMap.isometryEquivBoundedOfCompact
end
@[simp]
theorem _root_.BoundedContinuousFunction.dist_mkOfCompact (f g : C(α, β)) :
dist (mkOfCompact f) (mkOfCompact g) = dist f g :=
rfl
#align bounded_continuous_function.dist_mk_of_compact BoundedContinuousFunction.dist_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.dist_toContinuousMap (f g : α →ᵇ β) :
dist f.toContinuousMap g.toContinuousMap = dist f g :=
rfl
#align bounded_continuous_function.dist_to_continuous_map BoundedContinuousFunction.dist_toContinuousMap
open BoundedContinuousFunction
section
variable {f g : C(α, β)} {C : ℝ}
/-- The pointwise distance is controlled by the distance between functions, by definition. -/
theorem dist_apply_le_dist (x : α) : dist (f x) (g x) ≤ dist f g := by
simp only [← dist_mkOfCompact, dist_coe_le_dist, ← mkOfCompact_apply]
#align continuous_map.dist_apply_le_dist ContinuousMap.dist_apply_le_dist
/-- The distance between two functions is controlled by the supremum of the pointwise distances. -/
theorem dist_le (C0 : (0 : ℝ) ≤ C) : dist f g ≤ C ↔ ∀ x : α, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le C0, mkOfCompact_apply]
#align continuous_map.dist_le ContinuousMap.dist_le
theorem dist_le_iff_of_nonempty [Nonempty α] : dist f g ≤ C ↔ ∀ x, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le_iff_of_nonempty,
mkOfCompact_apply]
#align continuous_map.dist_le_iff_of_nonempty ContinuousMap.dist_le_iff_of_nonempty
theorem dist_lt_iff_of_nonempty [Nonempty α] : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
simp only [← dist_mkOfCompact, dist_lt_iff_of_nonempty_compact, mkOfCompact_apply]
#align continuous_map.dist_lt_iff_of_nonempty ContinuousMap.dist_lt_iff_of_nonempty
theorem dist_lt_of_nonempty [Nonempty α] (w : ∀ x : α, dist (f x) (g x) < C) : dist f g < C :=
dist_lt_iff_of_nonempty.2 w
#align continuous_map.dist_lt_of_nonempty ContinuousMap.dist_lt_of_nonempty
theorem dist_lt_iff (C0 : (0 : ℝ) < C) : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
rw [← dist_mkOfCompact, dist_lt_iff_of_compact C0]
simp only [mkOfCompact_apply]
#align continuous_map.dist_lt_iff ContinuousMap.dist_lt_iff
end
instance [CompleteSpace β] : CompleteSpace C(α, β) :=
(isometryEquivBoundedOfCompact α β).completeSpace
/-- See also `ContinuousMap.continuous_eval'`. -/
@[continuity]
theorem continuous_eval : Continuous fun p : C(α, β) × α => p.1 p.2 :=
continuous_eval.comp ((isometryEquivBoundedOfCompact α β).continuous.prod_map continuous_id)
#align continuous_map.continuous_eval ContinuousMap.continuous_eval
-- TODO at some point we will need lemmas characterising this norm!
-- At the moment the only way to reason about it is to transfer `f : C(α,E)` back to `α →ᵇ E`.
instance : Norm C(α, E) where norm x := dist x 0
@[simp]
theorem _root_.BoundedContinuousFunction.norm_mkOfCompact (f : C(α, E)) : ‖mkOfCompact f‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_mk_of_compact BoundedContinuousFunction.norm_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.norm_toContinuousMap_eq (f : α →ᵇ E) :
‖f.toContinuousMap‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_to_continuous_map_eq BoundedContinuousFunction.norm_toContinuousMap_eq
open BoundedContinuousFunction
instance : NormedAddCommGroup C(α, E) :=
{ ContinuousMap.metricSpace _ _,
ContinuousMap.instAddCommGroupContinuousMap with
dist_eq := fun x y => by
rw [← norm_mkOfCompact, ← dist_mkOfCompact, dist_eq_norm, mkOfCompact_sub]
dist := dist
norm := norm }
instance [Nonempty α] [One E] [NormOneClass E] : NormOneClass C(α, E) where
norm_one := by simp only [← norm_mkOfCompact, mkOfCompact_one, norm_one]
section
variable (f : C(α, E))
-- The corresponding lemmas for `BoundedContinuousFunction` are stated with `{f}`,
-- and so can not be used in dot notation.
theorem norm_coe_le_norm (x : α) : ‖f x‖ ≤ ‖f‖ :=
(mkOfCompact f).norm_coe_le_norm x
#align continuous_map.norm_coe_le_norm ContinuousMap.norm_coe_le_norm
/-- Distance between the images of any two points is at most twice the norm of the function. -/
theorem dist_le_two_norm (x y : α) : dist (f x) (f y) ≤ 2 * ‖f‖ :=
(mkOfCompact f).dist_le_two_norm x y
#align continuous_map.dist_le_two_norm ContinuousMap.dist_le_two_norm
/-- The norm of a function is controlled by the supremum of the pointwise norms. -/
theorem norm_le {C : ℝ} (C0 : (0 : ℝ) ≤ C) : ‖f‖ ≤ C ↔ ∀ x : α, ‖f x‖ ≤ C :=
@BoundedContinuousFunction.norm_le _ _ _ _ (mkOfCompact f) _ C0
#align continuous_map.norm_le ContinuousMap.norm_le
theorem norm_le_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ ≤ M ↔ ∀ x, ‖f x‖ ≤ M :=
@BoundedContinuousFunction.norm_le_of_nonempty _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_le_of_nonempty ContinuousMap.norm_le_of_nonempty
theorem norm_lt_iff {M : ℝ} (M0 : 0 < M) : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_compact _ _ _ _ _ (mkOfCompact f) _ M0
#align continuous_map.norm_lt_iff ContinuousMap.norm_lt_iff
theorem nnnorm_lt_iff {M : ℝ≥0} (M0 : 0 < M) : ‖f‖₊ < M ↔ ∀ x : α, ‖f x‖₊ < M :=
f.norm_lt_iff M0
#align continuous_map.nnnorm_lt_iff ContinuousMap.nnnorm_lt_iff
theorem norm_lt_iff_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_nonempty_compact _ _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_lt_iff_of_nonempty ContinuousMap.norm_lt_iff_of_nonempty
theorem nnnorm_lt_iff_of_nonempty [Nonempty α] {M : ℝ≥0} : ‖f‖₊ < M ↔ ∀ x, ‖f x‖₊ < M :=
f.norm_lt_iff_of_nonempty
#align continuous_map.nnnorm_lt_iff_of_nonempty ContinuousMap.nnnorm_lt_iff_of_nonempty
theorem apply_le_norm (f : C(α, ℝ)) (x : α) : f x ≤ ‖f‖ :=
le_trans (le_abs.mpr (Or.inl (le_refl (f x)))) (f.norm_coe_le_norm x)
#align continuous_map.apply_le_norm ContinuousMap.apply_le_norm
theorem neg_norm_le_apply (f : C(α, ℝ)) (x : α) : -‖f‖ ≤ f x :=
le_trans (neg_le_neg (f.norm_coe_le_norm x)) (neg_le.mp (neg_le_abs_self (f x)))
#align continuous_map.neg_norm_le_apply ContinuousMap.neg_norm_le_apply
theorem norm_eq_iSup_norm : ‖f‖ = ⨆ x : α, ‖f x‖ :=
(mkOfCompact f).norm_eq_iSup_norm
#align continuous_map.norm_eq_supr_norm ContinuousMap.norm_eq_iSup_norm
theorem norm_restrict_mono_set {X : Type*} [TopologicalSpace X] (f : C(X, E))
{K L : TopologicalSpace.Compacts X} (hKL : K ≤ L) : ‖f.restrict K‖ ≤ ‖f.restrict L‖ :=
(norm_le _ (norm_nonneg _)).mpr fun x => norm_coe_le_norm (f.restrict L) <| Set.inclusion hKL x
#align continuous_map.norm_restrict_mono_set ContinuousMap.norm_restrict_mono_set
end
section
variable {R : Type*} [NormedRing R]
instance : NormedRing C(α, R) :=
{ (inferInstance : NormedAddCommGroup C(α, R)), ContinuousMap.instRingContinuousMap with
norm_mul := fun f g => norm_mul_le (mkOfCompact f) (mkOfCompact g) }
end
section
variable {𝕜 : Type*} [NormedField 𝕜] [NormedSpace 𝕜 E]
instance normedSpace : NormedSpace 𝕜 C(α, E) where
norm_smul_le c f := (norm_smul_le c (mkOfCompact f) : _)
#align continuous_map.normed_space ContinuousMap.normedSpace
section
variable (α 𝕜 E)
/-- When `α` is compact and `𝕜` is a normed field,
the `𝕜`-algebra of bounded continuous maps `α →ᵇ β` is
`𝕜`-linearly isometric to `C(α, β)`.
-/
def linearIsometryBoundedOfCompact : C(α, E) ≃ₗᵢ[𝕜] α →ᵇ E :=
{ addEquivBoundedOfCompact α E with
map_smul' := fun c f => by
ext
norm_cast
norm_map' := fun f => rfl }
#align continuous_map.linear_isometry_bounded_of_compact ContinuousMap.linearIsometryBoundedOfCompact
variable {α E}
-- to match `BoundedContinuousFunction.evalClm`
/-- The evaluation at a point, as a continuous linear map from `C(α, 𝕜)` to `𝕜`. -/
def evalClm (x : α) : C(α, E) →L[𝕜] E :=
(BoundedContinuousFunction.evalClm 𝕜 x).comp
(linearIsometryBoundedOfCompact α E 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_map.eval_clm ContinuousMap.evalClm
end
-- this lemma and the next are the analogues of those autogenerated by `@[simps]` for
-- `equivBoundedOfCompact`, `addEquivBoundedOfCompact`
@[simp]
theorem linearIsometryBoundedOfCompact_symm_apply (f : α →ᵇ E) :
(linearIsometryBoundedOfCompact α E 𝕜).symm f = f.toContinuousMap :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_symm_apply ContinuousMap.linearIsometryBoundedOfCompact_symm_apply
@[simp]
theorem linearIsometryBoundedOfCompact_apply_apply (f : C(α, E)) (a : α) :
(linearIsometryBoundedOfCompact α E 𝕜 f) a = f a :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_apply_apply ContinuousMap.linearIsometryBoundedOfCompact_apply_apply
@[simp]
theorem linearIsometryBoundedOfCompact_toIsometryEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toIsometryEquiv = isometryEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_isometry_equiv ContinuousMap.linearIsometryBoundedOfCompact_toIsometryEquiv
@[simp] -- porting note: adjusted LHS because `simpNF` complained it simplified.
theorem linearIsometryBoundedOfCompact_toAddEquiv :
((linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv : C(α, E) ≃+ (α →ᵇ E)) =
addEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_add_equiv ContinuousMap.linearIsometryBoundedOfCompact_toAddEquiv
@[simp]
theorem linearIsometryBoundedOfCompact_of_compact_toEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv.toEquiv = equivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_of_compact_to_equiv ContinuousMap.linearIsometryBoundedOfCompact_of_compact_toEquiv
end
section
variable {𝕜 : Type*} {γ : Type*} [NormedField 𝕜] [NormedRing γ] [NormedAlgebra 𝕜 γ]
instance : NormedAlgebra 𝕜 C(α, γ) :=
{ ContinuousMap.normedSpace, ContinuousMap.algebra with }
end
end ContinuousMap
namespace ContinuousMap
section UniformContinuity
variable {α β : Type*}
variable [MetricSpace α] [CompactSpace α] [MetricSpace β]
/-!
We now set up some declarations making it convenient to use uniform continuity.
-/
theorem uniform_continuity (f : C(α, β)) (ε : ℝ) (h : 0 < ε) :
∃ δ > 0, ∀ {x y}, dist x y < δ → dist (f x) (f y) < ε :=
Metric.uniformContinuous_iff.mp (CompactSpace.uniformContinuous_of_continuous f.continuous) ε h
#align continuous_map.uniform_continuity ContinuousMap.uniform_continuity
-- This definition allows us to separate the choice of some `δ`,
-- and the corresponding use of `dist a b < δ → dist (f a) (f b) < ε`,
-- even across different declarations.
/-- An arbitrarily chosen modulus of uniform continuity for a given function `f` and `ε > 0`. -/
def modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) : ℝ :=
Classical.choose (uniform_continuity f ε h)
#align continuous_map.modulus ContinuousMap.modulus
theorem modulus_pos (f : C(α, β)) {ε : ℝ} {h : 0 < ε} : 0 < f.modulus ε h :=
(Classical.choose_spec (uniform_continuity f ε h)).1
#align continuous_map.modulus_pos ContinuousMap.modulus_pos
theorem dist_lt_of_dist_lt_modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) {a b : α}
(w : dist a b < f.modulus ε h) : dist (f a) (f b) < ε :=
(Classical.choose_spec (uniform_continuity f ε h)).2 w
#align continuous_map.dist_lt_of_dist_lt_modulus ContinuousMap.dist_lt_of_dist_lt_modulus
end UniformContinuity
end ContinuousMap
section CompLeft
variable (X : Type*) {𝕜 β γ : Type*} [TopologicalSpace X] [CompactSpace X]
[NontriviallyNormedField 𝕜]
variable [NormedAddCommGroup β] [NormedSpace 𝕜 β] [NormedAddCommGroup γ] [NormedSpace 𝕜 γ]
open ContinuousMap
/-- Postcomposition of continuous functions into a normed module by a continuous linear map is a
continuous linear map.
Transferred version of `ContinuousLinearMap.compLeftContinuousBounded`,
upgraded version of `ContinuousLinearMap.compLeftContinuous`,
similar to `LinearMap.compLeft`. -/
protected def ContinuousLinearMap.compLeftContinuousCompact (g : β →L[𝕜] γ) :
C(X, β) →L[𝕜] C(X, γ) :=
(linearIsometryBoundedOfCompact X γ 𝕜).symm.toLinearIsometry.toContinuousLinearMap.comp <|
(g.compLeftContinuousBounded X).comp <|
(linearIsometryBoundedOfCompact X β 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_linear_map.comp_left_continuous_compact ContinuousLinearMap.compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.toLinear_compLeftContinuousCompact (g : β →L[𝕜] γ) :
(g.compLeftContinuousCompact X : C(X, β) →ₗ[𝕜] C(X, γ)) = g.compLeftContinuous 𝕜 X := by
ext f
rfl
#align continuous_linear_map.to_linear_comp_left_continuous_compact ContinuousLinearMap.toLinear_compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.compLeftContinuousCompact_apply (g : β →L[𝕜] γ) (f : C(X, β)) (x : X) :
g.compLeftContinuousCompact X f x = g (f x) :=
rfl
#align continuous_linear_map.comp_left_continuous_compact_apply ContinuousLinearMap.compLeftContinuousCompact_apply
end CompLeft
namespace ContinuousMap
/-!
We now setup variations on `compRight* f`, where `f : C(X, Y)`
(that is, precomposition by a continuous map),
as a morphism `C(Y, T) → C(X, T)`, respecting various types of structure.
In particular:
* `compRightContinuousMap`, the bundled continuous map (for this we need `X Y` compact).
* `compRightHomeomorph`, when we precompose by a homeomorphism.
* `compRightAlgHom`, when `T = R` is a topological ring.
-/
section CompRight
/-- Precomposition by a continuous map is itself a continuous map between spaces of continuous maps.
-/
def compRightContinuousMap {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y)) : C(C(Y, T), C(X, T)) where
toFun g := g.comp f
continuous_toFun := by
refine' Metric.continuous_iff.mpr _
intro g ε ε_pos
refine' ⟨ε, ε_pos, fun g' h => _⟩
rw [ContinuousMap.dist_lt_iff ε_pos] at h ⊢
exact fun x => h (f x)
#align continuous_map.comp_right_continuous_map ContinuousMap.compRightContinuousMap
@[simp]
theorem compRightContinuousMap_apply {X Y : Type*} (T : Type*) [TopologicalSpace X]
[CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y))
(g : C(Y, T)) : (compRightContinuousMap T f) g = g.comp f :=
rfl
#align continuous_map.comp_right_continuous_map_apply ContinuousMap.compRightContinuousMap_apply
/-- Precomposition by a homeomorphism is itself a homeomorphism between spaces of continuous maps.
-/
def compRightHomeomorph {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : X ≃ₜ Y) : C(Y, T) ≃ₜ C(X, T) where
toFun := compRightContinuousMap T f.toContinuousMap
invFun := compRightContinuousMap T f.symm.toContinuousMap
left_inv g := ext fun _ => congr_arg g (f.apply_symm_apply _)
right_inv g := ext fun _ => congr_arg g (f.symm_apply_apply _)
#align continuous_map.comp_right_homeomorph ContinuousMap.compRightHomeomorph
theorem compRightAlgHom_continuous {X Y : Type*} (R A : Type*) [TopologicalSpace X]
[CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [CommSemiring R] [Semiring A]
[MetricSpace A] [TopologicalSemiring A] [Algebra R A] (f : C(X, Y)) :
Continuous (compRightAlgHom R A f) :=
map_continuous (compRightContinuousMap A f)
#align continuous_map.comp_right_alg_hom_continuous ContinuousMap.compRightAlgHom_continuous
end CompRight
section LocalNormalConvergence
/-! ### Local normal convergence
A sum of continuous functions (on a locally compact space) is "locally normally convergent" if the
sum of its sup-norms on any compact subset is summable. This implies convergence in the topology
of `C(X, E)` (i.e. locally uniform convergence). -/
open TopologicalSpace
variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
variable {E : Type*} [NormedAddCommGroup E] [CompleteSpace E]
theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F := by
refine' (ContinuousMap.exists_tendsto_compactOpen_iff_forall _).2 fun K hK => _
lift K to Compacts X using hK
have A : ∀ s : Finset ι, restrict (↑K) (∑ i in s, F i) = ∑ i in s, restrict K (F i) := by
intro s
ext1 x
simp
-- This used to be the end of the proof before leanprover/lean4#2644
erw [restrict_apply, restrict_apply, restrict_apply, restrict_apply]
simp? says simp only [coe_sum, Finset.sum_apply]
|
congr!
|
theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F := by
refine' (ContinuousMap.exists_tendsto_compactOpen_iff_forall _).2 fun K hK => _
lift K to Compacts X using hK
have A : ∀ s : Finset ι, restrict (↑K) (∑ i in s, F i) = ∑ i in s, restrict K (F i) := by
intro s
ext1 x
simp
-- This used to be the end of the proof before leanprover/lean4#2644
erw [restrict_apply, restrict_apply, restrict_apply, restrict_apply]
simp? says simp only [coe_sum, Finset.sum_apply]
|
Mathlib.Topology.ContinuousFunction.Compact.491_0.Mig2jTVnn2FLKEB
|
theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F
|
Mathlib_Topology_ContinuousFunction_Compact
|
case intro
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : LocallyCompactSpace X
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
ι : Type u_3
F : ι → C(X, E)
hF : ∀ (K : Compacts X), Summable fun i => ‖restrict (↑K) (F i)‖
K : Compacts X
A : ∀ (s : Finset ι), restrict (↑K) (∑ i in s, F i) = ∑ i in s, restrict (↑K) (F i)
⊢ ∃ f, Tendsto (fun i => restrict (↑K) (∑ b in i, F b)) atTop (𝓝 f)
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compacts
#align_import topology.continuous_function.compact from "leanprover-community/mathlib"@"d3af0609f6db8691dffdc3e1fb7feb7da72698f2"
/-!
# Continuous functions on a compact space
Continuous functions `C(α, β)` from a compact space `α` to a metric space `β`
are automatically bounded, and so acquire various structures inherited from `α →ᵇ β`.
This file transfers these structures, and restates some lemmas
characterising these structures.
If you need a lemma which is proved about `α →ᵇ β` but not for `C(α, β)` when `α` is compact,
you should restate it here. You can also use
`ContinuousMap.equivBoundedOfCompact` to move functions back and forth.
-/
noncomputable section
open Topology Classical NNReal BoundedContinuousFunction BigOperators
open Set Filter Metric
open BoundedContinuousFunction
namespace ContinuousMap
variable {α β E : Type*} [TopologicalSpace α] [CompactSpace α] [MetricSpace β]
[NormedAddCommGroup E]
section
variable (α β)
/-- When `α` is compact, the bounded continuous maps `α →ᵇ β` are
equivalent to `C(α, β)`.
-/
@[simps (config := .asFn)]
def equivBoundedOfCompact : C(α, β) ≃ (α →ᵇ β) :=
⟨mkOfCompact, BoundedContinuousFunction.toContinuousMap, fun f => by
ext
rfl, fun f => by
ext
rfl⟩
#align continuous_map.equiv_bounded_of_compact ContinuousMap.equivBoundedOfCompact
theorem uniformInducing_equivBoundedOfCompact : UniformInducing (equivBoundedOfCompact α β) :=
UniformInducing.mk'
(by
simp only [hasBasis_compactConvergenceUniformity.mem_iff, uniformity_basis_dist_le.mem_iff]
exact fun s =>
⟨fun ⟨⟨a, b⟩, ⟨_, ⟨ε, hε, hb⟩⟩, hs⟩ =>
⟨{ p | ∀ x, (p.1 x, p.2 x) ∈ b }, ⟨ε, hε, fun _ h x => hb ((dist_le hε.le).mp h x)⟩,
fun f g h => hs fun x _ => h x⟩,
fun ⟨_, ⟨ε, hε, ht⟩, hs⟩ =>
⟨⟨Set.univ, { p | dist p.1 p.2 ≤ ε }⟩, ⟨isCompact_univ, ⟨ε, hε, fun _ h => h⟩⟩,
fun ⟨f, g⟩ h => hs _ _ (ht ((dist_le hε.le).mpr fun x => h x (mem_univ x)))⟩⟩)
#align continuous_map.uniform_inducing_equiv_bounded_of_compact ContinuousMap.uniformInducing_equivBoundedOfCompact
theorem uniformEmbedding_equivBoundedOfCompact : UniformEmbedding (equivBoundedOfCompact α β) :=
{ uniformInducing_equivBoundedOfCompact α β with inj := (equivBoundedOfCompact α β).injective }
#align continuous_map.uniform_embedding_equiv_bounded_of_compact ContinuousMap.uniformEmbedding_equivBoundedOfCompact
/-- When `α` is compact, the bounded continuous maps `α →ᵇ 𝕜` are
additively equivalent to `C(α, 𝕜)`.
-/
-- porting note: the following `simps` received a "maximum recursion depth" error
-- @[simps! (config := .asFn) apply symm_apply]
def addEquivBoundedOfCompact [AddMonoid β] [LipschitzAdd β] : C(α, β) ≃+ (α →ᵇ β) :=
({ toContinuousMapAddHom α β, (equivBoundedOfCompact α β).symm with } : (α →ᵇ β) ≃+ C(α, β)).symm
#align continuous_map.add_equiv_bounded_of_compact ContinuousMap.addEquivBoundedOfCompact
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_symm_apply [AddMonoid β] [LipschitzAdd β] :
⇑((addEquivBoundedOfCompact α β).symm) = toContinuousMapAddHom α β :=
rfl
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_apply [AddMonoid β] [LipschitzAdd β] :
⇑(addEquivBoundedOfCompact α β) = mkOfCompact :=
rfl
instance metricSpace : MetricSpace C(α, β) :=
(uniformEmbedding_equivBoundedOfCompact α β).comapMetricSpace _
#align continuous_map.metric_space ContinuousMap.metricSpace
/-- When `α` is compact, and `β` is a metric space, the bounded continuous maps `α →ᵇ β` are
isometric to `C(α, β)`.
-/
@[simps! (config := .asFn) toEquiv apply symm_apply]
def isometryEquivBoundedOfCompact : C(α, β) ≃ᵢ (α →ᵇ β) where
isometry_toFun _ _ := rfl
toEquiv := equivBoundedOfCompact α β
#align continuous_map.isometry_equiv_bounded_of_compact ContinuousMap.isometryEquivBoundedOfCompact
end
@[simp]
theorem _root_.BoundedContinuousFunction.dist_mkOfCompact (f g : C(α, β)) :
dist (mkOfCompact f) (mkOfCompact g) = dist f g :=
rfl
#align bounded_continuous_function.dist_mk_of_compact BoundedContinuousFunction.dist_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.dist_toContinuousMap (f g : α →ᵇ β) :
dist f.toContinuousMap g.toContinuousMap = dist f g :=
rfl
#align bounded_continuous_function.dist_to_continuous_map BoundedContinuousFunction.dist_toContinuousMap
open BoundedContinuousFunction
section
variable {f g : C(α, β)} {C : ℝ}
/-- The pointwise distance is controlled by the distance between functions, by definition. -/
theorem dist_apply_le_dist (x : α) : dist (f x) (g x) ≤ dist f g := by
simp only [← dist_mkOfCompact, dist_coe_le_dist, ← mkOfCompact_apply]
#align continuous_map.dist_apply_le_dist ContinuousMap.dist_apply_le_dist
/-- The distance between two functions is controlled by the supremum of the pointwise distances. -/
theorem dist_le (C0 : (0 : ℝ) ≤ C) : dist f g ≤ C ↔ ∀ x : α, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le C0, mkOfCompact_apply]
#align continuous_map.dist_le ContinuousMap.dist_le
theorem dist_le_iff_of_nonempty [Nonempty α] : dist f g ≤ C ↔ ∀ x, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le_iff_of_nonempty,
mkOfCompact_apply]
#align continuous_map.dist_le_iff_of_nonempty ContinuousMap.dist_le_iff_of_nonempty
theorem dist_lt_iff_of_nonempty [Nonempty α] : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
simp only [← dist_mkOfCompact, dist_lt_iff_of_nonempty_compact, mkOfCompact_apply]
#align continuous_map.dist_lt_iff_of_nonempty ContinuousMap.dist_lt_iff_of_nonempty
theorem dist_lt_of_nonempty [Nonempty α] (w : ∀ x : α, dist (f x) (g x) < C) : dist f g < C :=
dist_lt_iff_of_nonempty.2 w
#align continuous_map.dist_lt_of_nonempty ContinuousMap.dist_lt_of_nonempty
theorem dist_lt_iff (C0 : (0 : ℝ) < C) : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
rw [← dist_mkOfCompact, dist_lt_iff_of_compact C0]
simp only [mkOfCompact_apply]
#align continuous_map.dist_lt_iff ContinuousMap.dist_lt_iff
end
instance [CompleteSpace β] : CompleteSpace C(α, β) :=
(isometryEquivBoundedOfCompact α β).completeSpace
/-- See also `ContinuousMap.continuous_eval'`. -/
@[continuity]
theorem continuous_eval : Continuous fun p : C(α, β) × α => p.1 p.2 :=
continuous_eval.comp ((isometryEquivBoundedOfCompact α β).continuous.prod_map continuous_id)
#align continuous_map.continuous_eval ContinuousMap.continuous_eval
-- TODO at some point we will need lemmas characterising this norm!
-- At the moment the only way to reason about it is to transfer `f : C(α,E)` back to `α →ᵇ E`.
instance : Norm C(α, E) where norm x := dist x 0
@[simp]
theorem _root_.BoundedContinuousFunction.norm_mkOfCompact (f : C(α, E)) : ‖mkOfCompact f‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_mk_of_compact BoundedContinuousFunction.norm_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.norm_toContinuousMap_eq (f : α →ᵇ E) :
‖f.toContinuousMap‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_to_continuous_map_eq BoundedContinuousFunction.norm_toContinuousMap_eq
open BoundedContinuousFunction
instance : NormedAddCommGroup C(α, E) :=
{ ContinuousMap.metricSpace _ _,
ContinuousMap.instAddCommGroupContinuousMap with
dist_eq := fun x y => by
rw [← norm_mkOfCompact, ← dist_mkOfCompact, dist_eq_norm, mkOfCompact_sub]
dist := dist
norm := norm }
instance [Nonempty α] [One E] [NormOneClass E] : NormOneClass C(α, E) where
norm_one := by simp only [← norm_mkOfCompact, mkOfCompact_one, norm_one]
section
variable (f : C(α, E))
-- The corresponding lemmas for `BoundedContinuousFunction` are stated with `{f}`,
-- and so can not be used in dot notation.
theorem norm_coe_le_norm (x : α) : ‖f x‖ ≤ ‖f‖ :=
(mkOfCompact f).norm_coe_le_norm x
#align continuous_map.norm_coe_le_norm ContinuousMap.norm_coe_le_norm
/-- Distance between the images of any two points is at most twice the norm of the function. -/
theorem dist_le_two_norm (x y : α) : dist (f x) (f y) ≤ 2 * ‖f‖ :=
(mkOfCompact f).dist_le_two_norm x y
#align continuous_map.dist_le_two_norm ContinuousMap.dist_le_two_norm
/-- The norm of a function is controlled by the supremum of the pointwise norms. -/
theorem norm_le {C : ℝ} (C0 : (0 : ℝ) ≤ C) : ‖f‖ ≤ C ↔ ∀ x : α, ‖f x‖ ≤ C :=
@BoundedContinuousFunction.norm_le _ _ _ _ (mkOfCompact f) _ C0
#align continuous_map.norm_le ContinuousMap.norm_le
theorem norm_le_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ ≤ M ↔ ∀ x, ‖f x‖ ≤ M :=
@BoundedContinuousFunction.norm_le_of_nonempty _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_le_of_nonempty ContinuousMap.norm_le_of_nonempty
theorem norm_lt_iff {M : ℝ} (M0 : 0 < M) : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_compact _ _ _ _ _ (mkOfCompact f) _ M0
#align continuous_map.norm_lt_iff ContinuousMap.norm_lt_iff
theorem nnnorm_lt_iff {M : ℝ≥0} (M0 : 0 < M) : ‖f‖₊ < M ↔ ∀ x : α, ‖f x‖₊ < M :=
f.norm_lt_iff M0
#align continuous_map.nnnorm_lt_iff ContinuousMap.nnnorm_lt_iff
theorem norm_lt_iff_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_nonempty_compact _ _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_lt_iff_of_nonempty ContinuousMap.norm_lt_iff_of_nonempty
theorem nnnorm_lt_iff_of_nonempty [Nonempty α] {M : ℝ≥0} : ‖f‖₊ < M ↔ ∀ x, ‖f x‖₊ < M :=
f.norm_lt_iff_of_nonempty
#align continuous_map.nnnorm_lt_iff_of_nonempty ContinuousMap.nnnorm_lt_iff_of_nonempty
theorem apply_le_norm (f : C(α, ℝ)) (x : α) : f x ≤ ‖f‖ :=
le_trans (le_abs.mpr (Or.inl (le_refl (f x)))) (f.norm_coe_le_norm x)
#align continuous_map.apply_le_norm ContinuousMap.apply_le_norm
theorem neg_norm_le_apply (f : C(α, ℝ)) (x : α) : -‖f‖ ≤ f x :=
le_trans (neg_le_neg (f.norm_coe_le_norm x)) (neg_le.mp (neg_le_abs_self (f x)))
#align continuous_map.neg_norm_le_apply ContinuousMap.neg_norm_le_apply
theorem norm_eq_iSup_norm : ‖f‖ = ⨆ x : α, ‖f x‖ :=
(mkOfCompact f).norm_eq_iSup_norm
#align continuous_map.norm_eq_supr_norm ContinuousMap.norm_eq_iSup_norm
theorem norm_restrict_mono_set {X : Type*} [TopologicalSpace X] (f : C(X, E))
{K L : TopologicalSpace.Compacts X} (hKL : K ≤ L) : ‖f.restrict K‖ ≤ ‖f.restrict L‖ :=
(norm_le _ (norm_nonneg _)).mpr fun x => norm_coe_le_norm (f.restrict L) <| Set.inclusion hKL x
#align continuous_map.norm_restrict_mono_set ContinuousMap.norm_restrict_mono_set
end
section
variable {R : Type*} [NormedRing R]
instance : NormedRing C(α, R) :=
{ (inferInstance : NormedAddCommGroup C(α, R)), ContinuousMap.instRingContinuousMap with
norm_mul := fun f g => norm_mul_le (mkOfCompact f) (mkOfCompact g) }
end
section
variable {𝕜 : Type*} [NormedField 𝕜] [NormedSpace 𝕜 E]
instance normedSpace : NormedSpace 𝕜 C(α, E) where
norm_smul_le c f := (norm_smul_le c (mkOfCompact f) : _)
#align continuous_map.normed_space ContinuousMap.normedSpace
section
variable (α 𝕜 E)
/-- When `α` is compact and `𝕜` is a normed field,
the `𝕜`-algebra of bounded continuous maps `α →ᵇ β` is
`𝕜`-linearly isometric to `C(α, β)`.
-/
def linearIsometryBoundedOfCompact : C(α, E) ≃ₗᵢ[𝕜] α →ᵇ E :=
{ addEquivBoundedOfCompact α E with
map_smul' := fun c f => by
ext
norm_cast
norm_map' := fun f => rfl }
#align continuous_map.linear_isometry_bounded_of_compact ContinuousMap.linearIsometryBoundedOfCompact
variable {α E}
-- to match `BoundedContinuousFunction.evalClm`
/-- The evaluation at a point, as a continuous linear map from `C(α, 𝕜)` to `𝕜`. -/
def evalClm (x : α) : C(α, E) →L[𝕜] E :=
(BoundedContinuousFunction.evalClm 𝕜 x).comp
(linearIsometryBoundedOfCompact α E 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_map.eval_clm ContinuousMap.evalClm
end
-- this lemma and the next are the analogues of those autogenerated by `@[simps]` for
-- `equivBoundedOfCompact`, `addEquivBoundedOfCompact`
@[simp]
theorem linearIsometryBoundedOfCompact_symm_apply (f : α →ᵇ E) :
(linearIsometryBoundedOfCompact α E 𝕜).symm f = f.toContinuousMap :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_symm_apply ContinuousMap.linearIsometryBoundedOfCompact_symm_apply
@[simp]
theorem linearIsometryBoundedOfCompact_apply_apply (f : C(α, E)) (a : α) :
(linearIsometryBoundedOfCompact α E 𝕜 f) a = f a :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_apply_apply ContinuousMap.linearIsometryBoundedOfCompact_apply_apply
@[simp]
theorem linearIsometryBoundedOfCompact_toIsometryEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toIsometryEquiv = isometryEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_isometry_equiv ContinuousMap.linearIsometryBoundedOfCompact_toIsometryEquiv
@[simp] -- porting note: adjusted LHS because `simpNF` complained it simplified.
theorem linearIsometryBoundedOfCompact_toAddEquiv :
((linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv : C(α, E) ≃+ (α →ᵇ E)) =
addEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_add_equiv ContinuousMap.linearIsometryBoundedOfCompact_toAddEquiv
@[simp]
theorem linearIsometryBoundedOfCompact_of_compact_toEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv.toEquiv = equivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_of_compact_to_equiv ContinuousMap.linearIsometryBoundedOfCompact_of_compact_toEquiv
end
section
variable {𝕜 : Type*} {γ : Type*} [NormedField 𝕜] [NormedRing γ] [NormedAlgebra 𝕜 γ]
instance : NormedAlgebra 𝕜 C(α, γ) :=
{ ContinuousMap.normedSpace, ContinuousMap.algebra with }
end
end ContinuousMap
namespace ContinuousMap
section UniformContinuity
variable {α β : Type*}
variable [MetricSpace α] [CompactSpace α] [MetricSpace β]
/-!
We now set up some declarations making it convenient to use uniform continuity.
-/
theorem uniform_continuity (f : C(α, β)) (ε : ℝ) (h : 0 < ε) :
∃ δ > 0, ∀ {x y}, dist x y < δ → dist (f x) (f y) < ε :=
Metric.uniformContinuous_iff.mp (CompactSpace.uniformContinuous_of_continuous f.continuous) ε h
#align continuous_map.uniform_continuity ContinuousMap.uniform_continuity
-- This definition allows us to separate the choice of some `δ`,
-- and the corresponding use of `dist a b < δ → dist (f a) (f b) < ε`,
-- even across different declarations.
/-- An arbitrarily chosen modulus of uniform continuity for a given function `f` and `ε > 0`. -/
def modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) : ℝ :=
Classical.choose (uniform_continuity f ε h)
#align continuous_map.modulus ContinuousMap.modulus
theorem modulus_pos (f : C(α, β)) {ε : ℝ} {h : 0 < ε} : 0 < f.modulus ε h :=
(Classical.choose_spec (uniform_continuity f ε h)).1
#align continuous_map.modulus_pos ContinuousMap.modulus_pos
theorem dist_lt_of_dist_lt_modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) {a b : α}
(w : dist a b < f.modulus ε h) : dist (f a) (f b) < ε :=
(Classical.choose_spec (uniform_continuity f ε h)).2 w
#align continuous_map.dist_lt_of_dist_lt_modulus ContinuousMap.dist_lt_of_dist_lt_modulus
end UniformContinuity
end ContinuousMap
section CompLeft
variable (X : Type*) {𝕜 β γ : Type*} [TopologicalSpace X] [CompactSpace X]
[NontriviallyNormedField 𝕜]
variable [NormedAddCommGroup β] [NormedSpace 𝕜 β] [NormedAddCommGroup γ] [NormedSpace 𝕜 γ]
open ContinuousMap
/-- Postcomposition of continuous functions into a normed module by a continuous linear map is a
continuous linear map.
Transferred version of `ContinuousLinearMap.compLeftContinuousBounded`,
upgraded version of `ContinuousLinearMap.compLeftContinuous`,
similar to `LinearMap.compLeft`. -/
protected def ContinuousLinearMap.compLeftContinuousCompact (g : β →L[𝕜] γ) :
C(X, β) →L[𝕜] C(X, γ) :=
(linearIsometryBoundedOfCompact X γ 𝕜).symm.toLinearIsometry.toContinuousLinearMap.comp <|
(g.compLeftContinuousBounded X).comp <|
(linearIsometryBoundedOfCompact X β 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_linear_map.comp_left_continuous_compact ContinuousLinearMap.compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.toLinear_compLeftContinuousCompact (g : β →L[𝕜] γ) :
(g.compLeftContinuousCompact X : C(X, β) →ₗ[𝕜] C(X, γ)) = g.compLeftContinuous 𝕜 X := by
ext f
rfl
#align continuous_linear_map.to_linear_comp_left_continuous_compact ContinuousLinearMap.toLinear_compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.compLeftContinuousCompact_apply (g : β →L[𝕜] γ) (f : C(X, β)) (x : X) :
g.compLeftContinuousCompact X f x = g (f x) :=
rfl
#align continuous_linear_map.comp_left_continuous_compact_apply ContinuousLinearMap.compLeftContinuousCompact_apply
end CompLeft
namespace ContinuousMap
/-!
We now setup variations on `compRight* f`, where `f : C(X, Y)`
(that is, precomposition by a continuous map),
as a morphism `C(Y, T) → C(X, T)`, respecting various types of structure.
In particular:
* `compRightContinuousMap`, the bundled continuous map (for this we need `X Y` compact).
* `compRightHomeomorph`, when we precompose by a homeomorphism.
* `compRightAlgHom`, when `T = R` is a topological ring.
-/
section CompRight
/-- Precomposition by a continuous map is itself a continuous map between spaces of continuous maps.
-/
def compRightContinuousMap {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y)) : C(C(Y, T), C(X, T)) where
toFun g := g.comp f
continuous_toFun := by
refine' Metric.continuous_iff.mpr _
intro g ε ε_pos
refine' ⟨ε, ε_pos, fun g' h => _⟩
rw [ContinuousMap.dist_lt_iff ε_pos] at h ⊢
exact fun x => h (f x)
#align continuous_map.comp_right_continuous_map ContinuousMap.compRightContinuousMap
@[simp]
theorem compRightContinuousMap_apply {X Y : Type*} (T : Type*) [TopologicalSpace X]
[CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y))
(g : C(Y, T)) : (compRightContinuousMap T f) g = g.comp f :=
rfl
#align continuous_map.comp_right_continuous_map_apply ContinuousMap.compRightContinuousMap_apply
/-- Precomposition by a homeomorphism is itself a homeomorphism between spaces of continuous maps.
-/
def compRightHomeomorph {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : X ≃ₜ Y) : C(Y, T) ≃ₜ C(X, T) where
toFun := compRightContinuousMap T f.toContinuousMap
invFun := compRightContinuousMap T f.symm.toContinuousMap
left_inv g := ext fun _ => congr_arg g (f.apply_symm_apply _)
right_inv g := ext fun _ => congr_arg g (f.symm_apply_apply _)
#align continuous_map.comp_right_homeomorph ContinuousMap.compRightHomeomorph
theorem compRightAlgHom_continuous {X Y : Type*} (R A : Type*) [TopologicalSpace X]
[CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [CommSemiring R] [Semiring A]
[MetricSpace A] [TopologicalSemiring A] [Algebra R A] (f : C(X, Y)) :
Continuous (compRightAlgHom R A f) :=
map_continuous (compRightContinuousMap A f)
#align continuous_map.comp_right_alg_hom_continuous ContinuousMap.compRightAlgHom_continuous
end CompRight
section LocalNormalConvergence
/-! ### Local normal convergence
A sum of continuous functions (on a locally compact space) is "locally normally convergent" if the
sum of its sup-norms on any compact subset is summable. This implies convergence in the topology
of `C(X, E)` (i.e. locally uniform convergence). -/
open TopologicalSpace
variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
variable {E : Type*} [NormedAddCommGroup E] [CompleteSpace E]
theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F := by
refine' (ContinuousMap.exists_tendsto_compactOpen_iff_forall _).2 fun K hK => _
lift K to Compacts X using hK
have A : ∀ s : Finset ι, restrict (↑K) (∑ i in s, F i) = ∑ i in s, restrict K (F i) := by
intro s
ext1 x
simp
-- This used to be the end of the proof before leanprover/lean4#2644
erw [restrict_apply, restrict_apply, restrict_apply, restrict_apply]
simp? says simp only [coe_sum, Finset.sum_apply]
congr!
|
simpa only [HasSum, A] using (hF K).of_norm
|
theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F := by
refine' (ContinuousMap.exists_tendsto_compactOpen_iff_forall _).2 fun K hK => _
lift K to Compacts X using hK
have A : ∀ s : Finset ι, restrict (↑K) (∑ i in s, F i) = ∑ i in s, restrict K (F i) := by
intro s
ext1 x
simp
-- This used to be the end of the proof before leanprover/lean4#2644
erw [restrict_apply, restrict_apply, restrict_apply, restrict_apply]
simp? says simp only [coe_sum, Finset.sum_apply]
congr!
|
Mathlib.Topology.ContinuousFunction.Compact.491_0.Mig2jTVnn2FLKEB
|
theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F
|
Mathlib_Topology_ContinuousFunction_Compact
|
α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : NormedAddCommGroup β
inst✝² : StarAddMonoid β
inst✝¹ : NormedStarGroup β
inst✝ : CompactSpace α
f : C(α, β)
⊢ ‖star f‖ = ‖f‖
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compacts
#align_import topology.continuous_function.compact from "leanprover-community/mathlib"@"d3af0609f6db8691dffdc3e1fb7feb7da72698f2"
/-!
# Continuous functions on a compact space
Continuous functions `C(α, β)` from a compact space `α` to a metric space `β`
are automatically bounded, and so acquire various structures inherited from `α →ᵇ β`.
This file transfers these structures, and restates some lemmas
characterising these structures.
If you need a lemma which is proved about `α →ᵇ β` but not for `C(α, β)` when `α` is compact,
you should restate it here. You can also use
`ContinuousMap.equivBoundedOfCompact` to move functions back and forth.
-/
noncomputable section
open Topology Classical NNReal BoundedContinuousFunction BigOperators
open Set Filter Metric
open BoundedContinuousFunction
namespace ContinuousMap
variable {α β E : Type*} [TopologicalSpace α] [CompactSpace α] [MetricSpace β]
[NormedAddCommGroup E]
section
variable (α β)
/-- When `α` is compact, the bounded continuous maps `α →ᵇ β` are
equivalent to `C(α, β)`.
-/
@[simps (config := .asFn)]
def equivBoundedOfCompact : C(α, β) ≃ (α →ᵇ β) :=
⟨mkOfCompact, BoundedContinuousFunction.toContinuousMap, fun f => by
ext
rfl, fun f => by
ext
rfl⟩
#align continuous_map.equiv_bounded_of_compact ContinuousMap.equivBoundedOfCompact
theorem uniformInducing_equivBoundedOfCompact : UniformInducing (equivBoundedOfCompact α β) :=
UniformInducing.mk'
(by
simp only [hasBasis_compactConvergenceUniformity.mem_iff, uniformity_basis_dist_le.mem_iff]
exact fun s =>
⟨fun ⟨⟨a, b⟩, ⟨_, ⟨ε, hε, hb⟩⟩, hs⟩ =>
⟨{ p | ∀ x, (p.1 x, p.2 x) ∈ b }, ⟨ε, hε, fun _ h x => hb ((dist_le hε.le).mp h x)⟩,
fun f g h => hs fun x _ => h x⟩,
fun ⟨_, ⟨ε, hε, ht⟩, hs⟩ =>
⟨⟨Set.univ, { p | dist p.1 p.2 ≤ ε }⟩, ⟨isCompact_univ, ⟨ε, hε, fun _ h => h⟩⟩,
fun ⟨f, g⟩ h => hs _ _ (ht ((dist_le hε.le).mpr fun x => h x (mem_univ x)))⟩⟩)
#align continuous_map.uniform_inducing_equiv_bounded_of_compact ContinuousMap.uniformInducing_equivBoundedOfCompact
theorem uniformEmbedding_equivBoundedOfCompact : UniformEmbedding (equivBoundedOfCompact α β) :=
{ uniformInducing_equivBoundedOfCompact α β with inj := (equivBoundedOfCompact α β).injective }
#align continuous_map.uniform_embedding_equiv_bounded_of_compact ContinuousMap.uniformEmbedding_equivBoundedOfCompact
/-- When `α` is compact, the bounded continuous maps `α →ᵇ 𝕜` are
additively equivalent to `C(α, 𝕜)`.
-/
-- porting note: the following `simps` received a "maximum recursion depth" error
-- @[simps! (config := .asFn) apply symm_apply]
def addEquivBoundedOfCompact [AddMonoid β] [LipschitzAdd β] : C(α, β) ≃+ (α →ᵇ β) :=
({ toContinuousMapAddHom α β, (equivBoundedOfCompact α β).symm with } : (α →ᵇ β) ≃+ C(α, β)).symm
#align continuous_map.add_equiv_bounded_of_compact ContinuousMap.addEquivBoundedOfCompact
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_symm_apply [AddMonoid β] [LipschitzAdd β] :
⇑((addEquivBoundedOfCompact α β).symm) = toContinuousMapAddHom α β :=
rfl
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_apply [AddMonoid β] [LipschitzAdd β] :
⇑(addEquivBoundedOfCompact α β) = mkOfCompact :=
rfl
instance metricSpace : MetricSpace C(α, β) :=
(uniformEmbedding_equivBoundedOfCompact α β).comapMetricSpace _
#align continuous_map.metric_space ContinuousMap.metricSpace
/-- When `α` is compact, and `β` is a metric space, the bounded continuous maps `α →ᵇ β` are
isometric to `C(α, β)`.
-/
@[simps! (config := .asFn) toEquiv apply symm_apply]
def isometryEquivBoundedOfCompact : C(α, β) ≃ᵢ (α →ᵇ β) where
isometry_toFun _ _ := rfl
toEquiv := equivBoundedOfCompact α β
#align continuous_map.isometry_equiv_bounded_of_compact ContinuousMap.isometryEquivBoundedOfCompact
end
@[simp]
theorem _root_.BoundedContinuousFunction.dist_mkOfCompact (f g : C(α, β)) :
dist (mkOfCompact f) (mkOfCompact g) = dist f g :=
rfl
#align bounded_continuous_function.dist_mk_of_compact BoundedContinuousFunction.dist_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.dist_toContinuousMap (f g : α →ᵇ β) :
dist f.toContinuousMap g.toContinuousMap = dist f g :=
rfl
#align bounded_continuous_function.dist_to_continuous_map BoundedContinuousFunction.dist_toContinuousMap
open BoundedContinuousFunction
section
variable {f g : C(α, β)} {C : ℝ}
/-- The pointwise distance is controlled by the distance between functions, by definition. -/
theorem dist_apply_le_dist (x : α) : dist (f x) (g x) ≤ dist f g := by
simp only [← dist_mkOfCompact, dist_coe_le_dist, ← mkOfCompact_apply]
#align continuous_map.dist_apply_le_dist ContinuousMap.dist_apply_le_dist
/-- The distance between two functions is controlled by the supremum of the pointwise distances. -/
theorem dist_le (C0 : (0 : ℝ) ≤ C) : dist f g ≤ C ↔ ∀ x : α, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le C0, mkOfCompact_apply]
#align continuous_map.dist_le ContinuousMap.dist_le
theorem dist_le_iff_of_nonempty [Nonempty α] : dist f g ≤ C ↔ ∀ x, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le_iff_of_nonempty,
mkOfCompact_apply]
#align continuous_map.dist_le_iff_of_nonempty ContinuousMap.dist_le_iff_of_nonempty
theorem dist_lt_iff_of_nonempty [Nonempty α] : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
simp only [← dist_mkOfCompact, dist_lt_iff_of_nonempty_compact, mkOfCompact_apply]
#align continuous_map.dist_lt_iff_of_nonempty ContinuousMap.dist_lt_iff_of_nonempty
theorem dist_lt_of_nonempty [Nonempty α] (w : ∀ x : α, dist (f x) (g x) < C) : dist f g < C :=
dist_lt_iff_of_nonempty.2 w
#align continuous_map.dist_lt_of_nonempty ContinuousMap.dist_lt_of_nonempty
theorem dist_lt_iff (C0 : (0 : ℝ) < C) : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
rw [← dist_mkOfCompact, dist_lt_iff_of_compact C0]
simp only [mkOfCompact_apply]
#align continuous_map.dist_lt_iff ContinuousMap.dist_lt_iff
end
instance [CompleteSpace β] : CompleteSpace C(α, β) :=
(isometryEquivBoundedOfCompact α β).completeSpace
/-- See also `ContinuousMap.continuous_eval'`. -/
@[continuity]
theorem continuous_eval : Continuous fun p : C(α, β) × α => p.1 p.2 :=
continuous_eval.comp ((isometryEquivBoundedOfCompact α β).continuous.prod_map continuous_id)
#align continuous_map.continuous_eval ContinuousMap.continuous_eval
-- TODO at some point we will need lemmas characterising this norm!
-- At the moment the only way to reason about it is to transfer `f : C(α,E)` back to `α →ᵇ E`.
instance : Norm C(α, E) where norm x := dist x 0
@[simp]
theorem _root_.BoundedContinuousFunction.norm_mkOfCompact (f : C(α, E)) : ‖mkOfCompact f‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_mk_of_compact BoundedContinuousFunction.norm_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.norm_toContinuousMap_eq (f : α →ᵇ E) :
‖f.toContinuousMap‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_to_continuous_map_eq BoundedContinuousFunction.norm_toContinuousMap_eq
open BoundedContinuousFunction
instance : NormedAddCommGroup C(α, E) :=
{ ContinuousMap.metricSpace _ _,
ContinuousMap.instAddCommGroupContinuousMap with
dist_eq := fun x y => by
rw [← norm_mkOfCompact, ← dist_mkOfCompact, dist_eq_norm, mkOfCompact_sub]
dist := dist
norm := norm }
instance [Nonempty α] [One E] [NormOneClass E] : NormOneClass C(α, E) where
norm_one := by simp only [← norm_mkOfCompact, mkOfCompact_one, norm_one]
section
variable (f : C(α, E))
-- The corresponding lemmas for `BoundedContinuousFunction` are stated with `{f}`,
-- and so can not be used in dot notation.
theorem norm_coe_le_norm (x : α) : ‖f x‖ ≤ ‖f‖ :=
(mkOfCompact f).norm_coe_le_norm x
#align continuous_map.norm_coe_le_norm ContinuousMap.norm_coe_le_norm
/-- Distance between the images of any two points is at most twice the norm of the function. -/
theorem dist_le_two_norm (x y : α) : dist (f x) (f y) ≤ 2 * ‖f‖ :=
(mkOfCompact f).dist_le_two_norm x y
#align continuous_map.dist_le_two_norm ContinuousMap.dist_le_two_norm
/-- The norm of a function is controlled by the supremum of the pointwise norms. -/
theorem norm_le {C : ℝ} (C0 : (0 : ℝ) ≤ C) : ‖f‖ ≤ C ↔ ∀ x : α, ‖f x‖ ≤ C :=
@BoundedContinuousFunction.norm_le _ _ _ _ (mkOfCompact f) _ C0
#align continuous_map.norm_le ContinuousMap.norm_le
theorem norm_le_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ ≤ M ↔ ∀ x, ‖f x‖ ≤ M :=
@BoundedContinuousFunction.norm_le_of_nonempty _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_le_of_nonempty ContinuousMap.norm_le_of_nonempty
theorem norm_lt_iff {M : ℝ} (M0 : 0 < M) : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_compact _ _ _ _ _ (mkOfCompact f) _ M0
#align continuous_map.norm_lt_iff ContinuousMap.norm_lt_iff
theorem nnnorm_lt_iff {M : ℝ≥0} (M0 : 0 < M) : ‖f‖₊ < M ↔ ∀ x : α, ‖f x‖₊ < M :=
f.norm_lt_iff M0
#align continuous_map.nnnorm_lt_iff ContinuousMap.nnnorm_lt_iff
theorem norm_lt_iff_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_nonempty_compact _ _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_lt_iff_of_nonempty ContinuousMap.norm_lt_iff_of_nonempty
theorem nnnorm_lt_iff_of_nonempty [Nonempty α] {M : ℝ≥0} : ‖f‖₊ < M ↔ ∀ x, ‖f x‖₊ < M :=
f.norm_lt_iff_of_nonempty
#align continuous_map.nnnorm_lt_iff_of_nonempty ContinuousMap.nnnorm_lt_iff_of_nonempty
theorem apply_le_norm (f : C(α, ℝ)) (x : α) : f x ≤ ‖f‖ :=
le_trans (le_abs.mpr (Or.inl (le_refl (f x)))) (f.norm_coe_le_norm x)
#align continuous_map.apply_le_norm ContinuousMap.apply_le_norm
theorem neg_norm_le_apply (f : C(α, ℝ)) (x : α) : -‖f‖ ≤ f x :=
le_trans (neg_le_neg (f.norm_coe_le_norm x)) (neg_le.mp (neg_le_abs_self (f x)))
#align continuous_map.neg_norm_le_apply ContinuousMap.neg_norm_le_apply
theorem norm_eq_iSup_norm : ‖f‖ = ⨆ x : α, ‖f x‖ :=
(mkOfCompact f).norm_eq_iSup_norm
#align continuous_map.norm_eq_supr_norm ContinuousMap.norm_eq_iSup_norm
theorem norm_restrict_mono_set {X : Type*} [TopologicalSpace X] (f : C(X, E))
{K L : TopologicalSpace.Compacts X} (hKL : K ≤ L) : ‖f.restrict K‖ ≤ ‖f.restrict L‖ :=
(norm_le _ (norm_nonneg _)).mpr fun x => norm_coe_le_norm (f.restrict L) <| Set.inclusion hKL x
#align continuous_map.norm_restrict_mono_set ContinuousMap.norm_restrict_mono_set
end
section
variable {R : Type*} [NormedRing R]
instance : NormedRing C(α, R) :=
{ (inferInstance : NormedAddCommGroup C(α, R)), ContinuousMap.instRingContinuousMap with
norm_mul := fun f g => norm_mul_le (mkOfCompact f) (mkOfCompact g) }
end
section
variable {𝕜 : Type*} [NormedField 𝕜] [NormedSpace 𝕜 E]
instance normedSpace : NormedSpace 𝕜 C(α, E) where
norm_smul_le c f := (norm_smul_le c (mkOfCompact f) : _)
#align continuous_map.normed_space ContinuousMap.normedSpace
section
variable (α 𝕜 E)
/-- When `α` is compact and `𝕜` is a normed field,
the `𝕜`-algebra of bounded continuous maps `α →ᵇ β` is
`𝕜`-linearly isometric to `C(α, β)`.
-/
def linearIsometryBoundedOfCompact : C(α, E) ≃ₗᵢ[𝕜] α →ᵇ E :=
{ addEquivBoundedOfCompact α E with
map_smul' := fun c f => by
ext
norm_cast
norm_map' := fun f => rfl }
#align continuous_map.linear_isometry_bounded_of_compact ContinuousMap.linearIsometryBoundedOfCompact
variable {α E}
-- to match `BoundedContinuousFunction.evalClm`
/-- The evaluation at a point, as a continuous linear map from `C(α, 𝕜)` to `𝕜`. -/
def evalClm (x : α) : C(α, E) →L[𝕜] E :=
(BoundedContinuousFunction.evalClm 𝕜 x).comp
(linearIsometryBoundedOfCompact α E 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_map.eval_clm ContinuousMap.evalClm
end
-- this lemma and the next are the analogues of those autogenerated by `@[simps]` for
-- `equivBoundedOfCompact`, `addEquivBoundedOfCompact`
@[simp]
theorem linearIsometryBoundedOfCompact_symm_apply (f : α →ᵇ E) :
(linearIsometryBoundedOfCompact α E 𝕜).symm f = f.toContinuousMap :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_symm_apply ContinuousMap.linearIsometryBoundedOfCompact_symm_apply
@[simp]
theorem linearIsometryBoundedOfCompact_apply_apply (f : C(α, E)) (a : α) :
(linearIsometryBoundedOfCompact α E 𝕜 f) a = f a :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_apply_apply ContinuousMap.linearIsometryBoundedOfCompact_apply_apply
@[simp]
theorem linearIsometryBoundedOfCompact_toIsometryEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toIsometryEquiv = isometryEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_isometry_equiv ContinuousMap.linearIsometryBoundedOfCompact_toIsometryEquiv
@[simp] -- porting note: adjusted LHS because `simpNF` complained it simplified.
theorem linearIsometryBoundedOfCompact_toAddEquiv :
((linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv : C(α, E) ≃+ (α →ᵇ E)) =
addEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_add_equiv ContinuousMap.linearIsometryBoundedOfCompact_toAddEquiv
@[simp]
theorem linearIsometryBoundedOfCompact_of_compact_toEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv.toEquiv = equivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_of_compact_to_equiv ContinuousMap.linearIsometryBoundedOfCompact_of_compact_toEquiv
end
section
variable {𝕜 : Type*} {γ : Type*} [NormedField 𝕜] [NormedRing γ] [NormedAlgebra 𝕜 γ]
instance : NormedAlgebra 𝕜 C(α, γ) :=
{ ContinuousMap.normedSpace, ContinuousMap.algebra with }
end
end ContinuousMap
namespace ContinuousMap
section UniformContinuity
variable {α β : Type*}
variable [MetricSpace α] [CompactSpace α] [MetricSpace β]
/-!
We now set up some declarations making it convenient to use uniform continuity.
-/
theorem uniform_continuity (f : C(α, β)) (ε : ℝ) (h : 0 < ε) :
∃ δ > 0, ∀ {x y}, dist x y < δ → dist (f x) (f y) < ε :=
Metric.uniformContinuous_iff.mp (CompactSpace.uniformContinuous_of_continuous f.continuous) ε h
#align continuous_map.uniform_continuity ContinuousMap.uniform_continuity
-- This definition allows us to separate the choice of some `δ`,
-- and the corresponding use of `dist a b < δ → dist (f a) (f b) < ε`,
-- even across different declarations.
/-- An arbitrarily chosen modulus of uniform continuity for a given function `f` and `ε > 0`. -/
def modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) : ℝ :=
Classical.choose (uniform_continuity f ε h)
#align continuous_map.modulus ContinuousMap.modulus
theorem modulus_pos (f : C(α, β)) {ε : ℝ} {h : 0 < ε} : 0 < f.modulus ε h :=
(Classical.choose_spec (uniform_continuity f ε h)).1
#align continuous_map.modulus_pos ContinuousMap.modulus_pos
theorem dist_lt_of_dist_lt_modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) {a b : α}
(w : dist a b < f.modulus ε h) : dist (f a) (f b) < ε :=
(Classical.choose_spec (uniform_continuity f ε h)).2 w
#align continuous_map.dist_lt_of_dist_lt_modulus ContinuousMap.dist_lt_of_dist_lt_modulus
end UniformContinuity
end ContinuousMap
section CompLeft
variable (X : Type*) {𝕜 β γ : Type*} [TopologicalSpace X] [CompactSpace X]
[NontriviallyNormedField 𝕜]
variable [NormedAddCommGroup β] [NormedSpace 𝕜 β] [NormedAddCommGroup γ] [NormedSpace 𝕜 γ]
open ContinuousMap
/-- Postcomposition of continuous functions into a normed module by a continuous linear map is a
continuous linear map.
Transferred version of `ContinuousLinearMap.compLeftContinuousBounded`,
upgraded version of `ContinuousLinearMap.compLeftContinuous`,
similar to `LinearMap.compLeft`. -/
protected def ContinuousLinearMap.compLeftContinuousCompact (g : β →L[𝕜] γ) :
C(X, β) →L[𝕜] C(X, γ) :=
(linearIsometryBoundedOfCompact X γ 𝕜).symm.toLinearIsometry.toContinuousLinearMap.comp <|
(g.compLeftContinuousBounded X).comp <|
(linearIsometryBoundedOfCompact X β 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_linear_map.comp_left_continuous_compact ContinuousLinearMap.compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.toLinear_compLeftContinuousCompact (g : β →L[𝕜] γ) :
(g.compLeftContinuousCompact X : C(X, β) →ₗ[𝕜] C(X, γ)) = g.compLeftContinuous 𝕜 X := by
ext f
rfl
#align continuous_linear_map.to_linear_comp_left_continuous_compact ContinuousLinearMap.toLinear_compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.compLeftContinuousCompact_apply (g : β →L[𝕜] γ) (f : C(X, β)) (x : X) :
g.compLeftContinuousCompact X f x = g (f x) :=
rfl
#align continuous_linear_map.comp_left_continuous_compact_apply ContinuousLinearMap.compLeftContinuousCompact_apply
end CompLeft
namespace ContinuousMap
/-!
We now setup variations on `compRight* f`, where `f : C(X, Y)`
(that is, precomposition by a continuous map),
as a morphism `C(Y, T) → C(X, T)`, respecting various types of structure.
In particular:
* `compRightContinuousMap`, the bundled continuous map (for this we need `X Y` compact).
* `compRightHomeomorph`, when we precompose by a homeomorphism.
* `compRightAlgHom`, when `T = R` is a topological ring.
-/
section CompRight
/-- Precomposition by a continuous map is itself a continuous map between spaces of continuous maps.
-/
def compRightContinuousMap {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y)) : C(C(Y, T), C(X, T)) where
toFun g := g.comp f
continuous_toFun := by
refine' Metric.continuous_iff.mpr _
intro g ε ε_pos
refine' ⟨ε, ε_pos, fun g' h => _⟩
rw [ContinuousMap.dist_lt_iff ε_pos] at h ⊢
exact fun x => h (f x)
#align continuous_map.comp_right_continuous_map ContinuousMap.compRightContinuousMap
@[simp]
theorem compRightContinuousMap_apply {X Y : Type*} (T : Type*) [TopologicalSpace X]
[CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y))
(g : C(Y, T)) : (compRightContinuousMap T f) g = g.comp f :=
rfl
#align continuous_map.comp_right_continuous_map_apply ContinuousMap.compRightContinuousMap_apply
/-- Precomposition by a homeomorphism is itself a homeomorphism between spaces of continuous maps.
-/
def compRightHomeomorph {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : X ≃ₜ Y) : C(Y, T) ≃ₜ C(X, T) where
toFun := compRightContinuousMap T f.toContinuousMap
invFun := compRightContinuousMap T f.symm.toContinuousMap
left_inv g := ext fun _ => congr_arg g (f.apply_symm_apply _)
right_inv g := ext fun _ => congr_arg g (f.symm_apply_apply _)
#align continuous_map.comp_right_homeomorph ContinuousMap.compRightHomeomorph
theorem compRightAlgHom_continuous {X Y : Type*} (R A : Type*) [TopologicalSpace X]
[CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [CommSemiring R] [Semiring A]
[MetricSpace A] [TopologicalSemiring A] [Algebra R A] (f : C(X, Y)) :
Continuous (compRightAlgHom R A f) :=
map_continuous (compRightContinuousMap A f)
#align continuous_map.comp_right_alg_hom_continuous ContinuousMap.compRightAlgHom_continuous
end CompRight
section LocalNormalConvergence
/-! ### Local normal convergence
A sum of continuous functions (on a locally compact space) is "locally normally convergent" if the
sum of its sup-norms on any compact subset is summable. This implies convergence in the topology
of `C(X, E)` (i.e. locally uniform convergence). -/
open TopologicalSpace
variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
variable {E : Type*} [NormedAddCommGroup E] [CompleteSpace E]
theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F := by
refine' (ContinuousMap.exists_tendsto_compactOpen_iff_forall _).2 fun K hK => _
lift K to Compacts X using hK
have A : ∀ s : Finset ι, restrict (↑K) (∑ i in s, F i) = ∑ i in s, restrict K (F i) := by
intro s
ext1 x
simp
-- This used to be the end of the proof before leanprover/lean4#2644
erw [restrict_apply, restrict_apply, restrict_apply, restrict_apply]
simp? says simp only [coe_sum, Finset.sum_apply]
congr!
simpa only [HasSum, A] using (hF K).of_norm
#align continuous_map.summable_of_locally_summable_norm ContinuousMap.summable_of_locally_summable_norm
end LocalNormalConvergence
/-!
### Star structures
In this section, if `β` is a normed ⋆-group, then so is the space of
continuous functions from `α` to `β`, by using the star operation pointwise.
Furthermore, if `α` is compact and `β` is a C⋆-ring, then `C(α, β)` is a C⋆-ring. -/
section NormedSpace
variable {α : Type*} {β : Type*}
variable [TopologicalSpace α] [NormedAddCommGroup β] [StarAddMonoid β] [NormedStarGroup β]
theorem _root_.BoundedContinuousFunction.mkOfCompact_star [CompactSpace α] (f : C(α, β)) :
mkOfCompact (star f) = star (mkOfCompact f) :=
rfl
#align bounded_continuous_function.mk_of_compact_star BoundedContinuousFunction.mkOfCompact_star
instance [CompactSpace α] : NormedStarGroup C(α, β) where
norm_star f := by
|
rw [← BoundedContinuousFunction.norm_mkOfCompact, BoundedContinuousFunction.mkOfCompact_star,
norm_star, BoundedContinuousFunction.norm_mkOfCompact]
|
instance [CompactSpace α] : NormedStarGroup C(α, β) where
norm_star f := by
|
Mathlib.Topology.ContinuousFunction.Compact.528_0.Mig2jTVnn2FLKEB
|
instance [CompactSpace α] : NormedStarGroup C(α, β) where
norm_star f
|
Mathlib_Topology_ContinuousFunction_Compact
|
α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : NormedRing β
inst✝² : StarRing β
inst✝¹ : CompactSpace α
inst✝ : CstarRing β
f : C(α, β)
⊢ ‖star f * f‖ = ‖f‖ * ‖f‖
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compacts
#align_import topology.continuous_function.compact from "leanprover-community/mathlib"@"d3af0609f6db8691dffdc3e1fb7feb7da72698f2"
/-!
# Continuous functions on a compact space
Continuous functions `C(α, β)` from a compact space `α` to a metric space `β`
are automatically bounded, and so acquire various structures inherited from `α →ᵇ β`.
This file transfers these structures, and restates some lemmas
characterising these structures.
If you need a lemma which is proved about `α →ᵇ β` but not for `C(α, β)` when `α` is compact,
you should restate it here. You can also use
`ContinuousMap.equivBoundedOfCompact` to move functions back and forth.
-/
noncomputable section
open Topology Classical NNReal BoundedContinuousFunction BigOperators
open Set Filter Metric
open BoundedContinuousFunction
namespace ContinuousMap
variable {α β E : Type*} [TopologicalSpace α] [CompactSpace α] [MetricSpace β]
[NormedAddCommGroup E]
section
variable (α β)
/-- When `α` is compact, the bounded continuous maps `α →ᵇ β` are
equivalent to `C(α, β)`.
-/
@[simps (config := .asFn)]
def equivBoundedOfCompact : C(α, β) ≃ (α →ᵇ β) :=
⟨mkOfCompact, BoundedContinuousFunction.toContinuousMap, fun f => by
ext
rfl, fun f => by
ext
rfl⟩
#align continuous_map.equiv_bounded_of_compact ContinuousMap.equivBoundedOfCompact
theorem uniformInducing_equivBoundedOfCompact : UniformInducing (equivBoundedOfCompact α β) :=
UniformInducing.mk'
(by
simp only [hasBasis_compactConvergenceUniformity.mem_iff, uniformity_basis_dist_le.mem_iff]
exact fun s =>
⟨fun ⟨⟨a, b⟩, ⟨_, ⟨ε, hε, hb⟩⟩, hs⟩ =>
⟨{ p | ∀ x, (p.1 x, p.2 x) ∈ b }, ⟨ε, hε, fun _ h x => hb ((dist_le hε.le).mp h x)⟩,
fun f g h => hs fun x _ => h x⟩,
fun ⟨_, ⟨ε, hε, ht⟩, hs⟩ =>
⟨⟨Set.univ, { p | dist p.1 p.2 ≤ ε }⟩, ⟨isCompact_univ, ⟨ε, hε, fun _ h => h⟩⟩,
fun ⟨f, g⟩ h => hs _ _ (ht ((dist_le hε.le).mpr fun x => h x (mem_univ x)))⟩⟩)
#align continuous_map.uniform_inducing_equiv_bounded_of_compact ContinuousMap.uniformInducing_equivBoundedOfCompact
theorem uniformEmbedding_equivBoundedOfCompact : UniformEmbedding (equivBoundedOfCompact α β) :=
{ uniformInducing_equivBoundedOfCompact α β with inj := (equivBoundedOfCompact α β).injective }
#align continuous_map.uniform_embedding_equiv_bounded_of_compact ContinuousMap.uniformEmbedding_equivBoundedOfCompact
/-- When `α` is compact, the bounded continuous maps `α →ᵇ 𝕜` are
additively equivalent to `C(α, 𝕜)`.
-/
-- porting note: the following `simps` received a "maximum recursion depth" error
-- @[simps! (config := .asFn) apply symm_apply]
def addEquivBoundedOfCompact [AddMonoid β] [LipschitzAdd β] : C(α, β) ≃+ (α →ᵇ β) :=
({ toContinuousMapAddHom α β, (equivBoundedOfCompact α β).symm with } : (α →ᵇ β) ≃+ C(α, β)).symm
#align continuous_map.add_equiv_bounded_of_compact ContinuousMap.addEquivBoundedOfCompact
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_symm_apply [AddMonoid β] [LipschitzAdd β] :
⇑((addEquivBoundedOfCompact α β).symm) = toContinuousMapAddHom α β :=
rfl
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_apply [AddMonoid β] [LipschitzAdd β] :
⇑(addEquivBoundedOfCompact α β) = mkOfCompact :=
rfl
instance metricSpace : MetricSpace C(α, β) :=
(uniformEmbedding_equivBoundedOfCompact α β).comapMetricSpace _
#align continuous_map.metric_space ContinuousMap.metricSpace
/-- When `α` is compact, and `β` is a metric space, the bounded continuous maps `α →ᵇ β` are
isometric to `C(α, β)`.
-/
@[simps! (config := .asFn) toEquiv apply symm_apply]
def isometryEquivBoundedOfCompact : C(α, β) ≃ᵢ (α →ᵇ β) where
isometry_toFun _ _ := rfl
toEquiv := equivBoundedOfCompact α β
#align continuous_map.isometry_equiv_bounded_of_compact ContinuousMap.isometryEquivBoundedOfCompact
end
@[simp]
theorem _root_.BoundedContinuousFunction.dist_mkOfCompact (f g : C(α, β)) :
dist (mkOfCompact f) (mkOfCompact g) = dist f g :=
rfl
#align bounded_continuous_function.dist_mk_of_compact BoundedContinuousFunction.dist_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.dist_toContinuousMap (f g : α →ᵇ β) :
dist f.toContinuousMap g.toContinuousMap = dist f g :=
rfl
#align bounded_continuous_function.dist_to_continuous_map BoundedContinuousFunction.dist_toContinuousMap
open BoundedContinuousFunction
section
variable {f g : C(α, β)} {C : ℝ}
/-- The pointwise distance is controlled by the distance between functions, by definition. -/
theorem dist_apply_le_dist (x : α) : dist (f x) (g x) ≤ dist f g := by
simp only [← dist_mkOfCompact, dist_coe_le_dist, ← mkOfCompact_apply]
#align continuous_map.dist_apply_le_dist ContinuousMap.dist_apply_le_dist
/-- The distance between two functions is controlled by the supremum of the pointwise distances. -/
theorem dist_le (C0 : (0 : ℝ) ≤ C) : dist f g ≤ C ↔ ∀ x : α, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le C0, mkOfCompact_apply]
#align continuous_map.dist_le ContinuousMap.dist_le
theorem dist_le_iff_of_nonempty [Nonempty α] : dist f g ≤ C ↔ ∀ x, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le_iff_of_nonempty,
mkOfCompact_apply]
#align continuous_map.dist_le_iff_of_nonempty ContinuousMap.dist_le_iff_of_nonempty
theorem dist_lt_iff_of_nonempty [Nonempty α] : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
simp only [← dist_mkOfCompact, dist_lt_iff_of_nonempty_compact, mkOfCompact_apply]
#align continuous_map.dist_lt_iff_of_nonempty ContinuousMap.dist_lt_iff_of_nonempty
theorem dist_lt_of_nonempty [Nonempty α] (w : ∀ x : α, dist (f x) (g x) < C) : dist f g < C :=
dist_lt_iff_of_nonempty.2 w
#align continuous_map.dist_lt_of_nonempty ContinuousMap.dist_lt_of_nonempty
theorem dist_lt_iff (C0 : (0 : ℝ) < C) : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
rw [← dist_mkOfCompact, dist_lt_iff_of_compact C0]
simp only [mkOfCompact_apply]
#align continuous_map.dist_lt_iff ContinuousMap.dist_lt_iff
end
instance [CompleteSpace β] : CompleteSpace C(α, β) :=
(isometryEquivBoundedOfCompact α β).completeSpace
/-- See also `ContinuousMap.continuous_eval'`. -/
@[continuity]
theorem continuous_eval : Continuous fun p : C(α, β) × α => p.1 p.2 :=
continuous_eval.comp ((isometryEquivBoundedOfCompact α β).continuous.prod_map continuous_id)
#align continuous_map.continuous_eval ContinuousMap.continuous_eval
-- TODO at some point we will need lemmas characterising this norm!
-- At the moment the only way to reason about it is to transfer `f : C(α,E)` back to `α →ᵇ E`.
instance : Norm C(α, E) where norm x := dist x 0
@[simp]
theorem _root_.BoundedContinuousFunction.norm_mkOfCompact (f : C(α, E)) : ‖mkOfCompact f‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_mk_of_compact BoundedContinuousFunction.norm_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.norm_toContinuousMap_eq (f : α →ᵇ E) :
‖f.toContinuousMap‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_to_continuous_map_eq BoundedContinuousFunction.norm_toContinuousMap_eq
open BoundedContinuousFunction
instance : NormedAddCommGroup C(α, E) :=
{ ContinuousMap.metricSpace _ _,
ContinuousMap.instAddCommGroupContinuousMap with
dist_eq := fun x y => by
rw [← norm_mkOfCompact, ← dist_mkOfCompact, dist_eq_norm, mkOfCompact_sub]
dist := dist
norm := norm }
instance [Nonempty α] [One E] [NormOneClass E] : NormOneClass C(α, E) where
norm_one := by simp only [← norm_mkOfCompact, mkOfCompact_one, norm_one]
section
variable (f : C(α, E))
-- The corresponding lemmas for `BoundedContinuousFunction` are stated with `{f}`,
-- and so can not be used in dot notation.
theorem norm_coe_le_norm (x : α) : ‖f x‖ ≤ ‖f‖ :=
(mkOfCompact f).norm_coe_le_norm x
#align continuous_map.norm_coe_le_norm ContinuousMap.norm_coe_le_norm
/-- Distance between the images of any two points is at most twice the norm of the function. -/
theorem dist_le_two_norm (x y : α) : dist (f x) (f y) ≤ 2 * ‖f‖ :=
(mkOfCompact f).dist_le_two_norm x y
#align continuous_map.dist_le_two_norm ContinuousMap.dist_le_two_norm
/-- The norm of a function is controlled by the supremum of the pointwise norms. -/
theorem norm_le {C : ℝ} (C0 : (0 : ℝ) ≤ C) : ‖f‖ ≤ C ↔ ∀ x : α, ‖f x‖ ≤ C :=
@BoundedContinuousFunction.norm_le _ _ _ _ (mkOfCompact f) _ C0
#align continuous_map.norm_le ContinuousMap.norm_le
theorem norm_le_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ ≤ M ↔ ∀ x, ‖f x‖ ≤ M :=
@BoundedContinuousFunction.norm_le_of_nonempty _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_le_of_nonempty ContinuousMap.norm_le_of_nonempty
theorem norm_lt_iff {M : ℝ} (M0 : 0 < M) : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_compact _ _ _ _ _ (mkOfCompact f) _ M0
#align continuous_map.norm_lt_iff ContinuousMap.norm_lt_iff
theorem nnnorm_lt_iff {M : ℝ≥0} (M0 : 0 < M) : ‖f‖₊ < M ↔ ∀ x : α, ‖f x‖₊ < M :=
f.norm_lt_iff M0
#align continuous_map.nnnorm_lt_iff ContinuousMap.nnnorm_lt_iff
theorem norm_lt_iff_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_nonempty_compact _ _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_lt_iff_of_nonempty ContinuousMap.norm_lt_iff_of_nonempty
theorem nnnorm_lt_iff_of_nonempty [Nonempty α] {M : ℝ≥0} : ‖f‖₊ < M ↔ ∀ x, ‖f x‖₊ < M :=
f.norm_lt_iff_of_nonempty
#align continuous_map.nnnorm_lt_iff_of_nonempty ContinuousMap.nnnorm_lt_iff_of_nonempty
theorem apply_le_norm (f : C(α, ℝ)) (x : α) : f x ≤ ‖f‖ :=
le_trans (le_abs.mpr (Or.inl (le_refl (f x)))) (f.norm_coe_le_norm x)
#align continuous_map.apply_le_norm ContinuousMap.apply_le_norm
theorem neg_norm_le_apply (f : C(α, ℝ)) (x : α) : -‖f‖ ≤ f x :=
le_trans (neg_le_neg (f.norm_coe_le_norm x)) (neg_le.mp (neg_le_abs_self (f x)))
#align continuous_map.neg_norm_le_apply ContinuousMap.neg_norm_le_apply
theorem norm_eq_iSup_norm : ‖f‖ = ⨆ x : α, ‖f x‖ :=
(mkOfCompact f).norm_eq_iSup_norm
#align continuous_map.norm_eq_supr_norm ContinuousMap.norm_eq_iSup_norm
theorem norm_restrict_mono_set {X : Type*} [TopologicalSpace X] (f : C(X, E))
{K L : TopologicalSpace.Compacts X} (hKL : K ≤ L) : ‖f.restrict K‖ ≤ ‖f.restrict L‖ :=
(norm_le _ (norm_nonneg _)).mpr fun x => norm_coe_le_norm (f.restrict L) <| Set.inclusion hKL x
#align continuous_map.norm_restrict_mono_set ContinuousMap.norm_restrict_mono_set
end
section
variable {R : Type*} [NormedRing R]
instance : NormedRing C(α, R) :=
{ (inferInstance : NormedAddCommGroup C(α, R)), ContinuousMap.instRingContinuousMap with
norm_mul := fun f g => norm_mul_le (mkOfCompact f) (mkOfCompact g) }
end
section
variable {𝕜 : Type*} [NormedField 𝕜] [NormedSpace 𝕜 E]
instance normedSpace : NormedSpace 𝕜 C(α, E) where
norm_smul_le c f := (norm_smul_le c (mkOfCompact f) : _)
#align continuous_map.normed_space ContinuousMap.normedSpace
section
variable (α 𝕜 E)
/-- When `α` is compact and `𝕜` is a normed field,
the `𝕜`-algebra of bounded continuous maps `α →ᵇ β` is
`𝕜`-linearly isometric to `C(α, β)`.
-/
def linearIsometryBoundedOfCompact : C(α, E) ≃ₗᵢ[𝕜] α →ᵇ E :=
{ addEquivBoundedOfCompact α E with
map_smul' := fun c f => by
ext
norm_cast
norm_map' := fun f => rfl }
#align continuous_map.linear_isometry_bounded_of_compact ContinuousMap.linearIsometryBoundedOfCompact
variable {α E}
-- to match `BoundedContinuousFunction.evalClm`
/-- The evaluation at a point, as a continuous linear map from `C(α, 𝕜)` to `𝕜`. -/
def evalClm (x : α) : C(α, E) →L[𝕜] E :=
(BoundedContinuousFunction.evalClm 𝕜 x).comp
(linearIsometryBoundedOfCompact α E 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_map.eval_clm ContinuousMap.evalClm
end
-- this lemma and the next are the analogues of those autogenerated by `@[simps]` for
-- `equivBoundedOfCompact`, `addEquivBoundedOfCompact`
@[simp]
theorem linearIsometryBoundedOfCompact_symm_apply (f : α →ᵇ E) :
(linearIsometryBoundedOfCompact α E 𝕜).symm f = f.toContinuousMap :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_symm_apply ContinuousMap.linearIsometryBoundedOfCompact_symm_apply
@[simp]
theorem linearIsometryBoundedOfCompact_apply_apply (f : C(α, E)) (a : α) :
(linearIsometryBoundedOfCompact α E 𝕜 f) a = f a :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_apply_apply ContinuousMap.linearIsometryBoundedOfCompact_apply_apply
@[simp]
theorem linearIsometryBoundedOfCompact_toIsometryEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toIsometryEquiv = isometryEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_isometry_equiv ContinuousMap.linearIsometryBoundedOfCompact_toIsometryEquiv
@[simp] -- porting note: adjusted LHS because `simpNF` complained it simplified.
theorem linearIsometryBoundedOfCompact_toAddEquiv :
((linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv : C(α, E) ≃+ (α →ᵇ E)) =
addEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_add_equiv ContinuousMap.linearIsometryBoundedOfCompact_toAddEquiv
@[simp]
theorem linearIsometryBoundedOfCompact_of_compact_toEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv.toEquiv = equivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_of_compact_to_equiv ContinuousMap.linearIsometryBoundedOfCompact_of_compact_toEquiv
end
section
variable {𝕜 : Type*} {γ : Type*} [NormedField 𝕜] [NormedRing γ] [NormedAlgebra 𝕜 γ]
instance : NormedAlgebra 𝕜 C(α, γ) :=
{ ContinuousMap.normedSpace, ContinuousMap.algebra with }
end
end ContinuousMap
namespace ContinuousMap
section UniformContinuity
variable {α β : Type*}
variable [MetricSpace α] [CompactSpace α] [MetricSpace β]
/-!
We now set up some declarations making it convenient to use uniform continuity.
-/
theorem uniform_continuity (f : C(α, β)) (ε : ℝ) (h : 0 < ε) :
∃ δ > 0, ∀ {x y}, dist x y < δ → dist (f x) (f y) < ε :=
Metric.uniformContinuous_iff.mp (CompactSpace.uniformContinuous_of_continuous f.continuous) ε h
#align continuous_map.uniform_continuity ContinuousMap.uniform_continuity
-- This definition allows us to separate the choice of some `δ`,
-- and the corresponding use of `dist a b < δ → dist (f a) (f b) < ε`,
-- even across different declarations.
/-- An arbitrarily chosen modulus of uniform continuity for a given function `f` and `ε > 0`. -/
def modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) : ℝ :=
Classical.choose (uniform_continuity f ε h)
#align continuous_map.modulus ContinuousMap.modulus
theorem modulus_pos (f : C(α, β)) {ε : ℝ} {h : 0 < ε} : 0 < f.modulus ε h :=
(Classical.choose_spec (uniform_continuity f ε h)).1
#align continuous_map.modulus_pos ContinuousMap.modulus_pos
theorem dist_lt_of_dist_lt_modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) {a b : α}
(w : dist a b < f.modulus ε h) : dist (f a) (f b) < ε :=
(Classical.choose_spec (uniform_continuity f ε h)).2 w
#align continuous_map.dist_lt_of_dist_lt_modulus ContinuousMap.dist_lt_of_dist_lt_modulus
end UniformContinuity
end ContinuousMap
section CompLeft
variable (X : Type*) {𝕜 β γ : Type*} [TopologicalSpace X] [CompactSpace X]
[NontriviallyNormedField 𝕜]
variable [NormedAddCommGroup β] [NormedSpace 𝕜 β] [NormedAddCommGroup γ] [NormedSpace 𝕜 γ]
open ContinuousMap
/-- Postcomposition of continuous functions into a normed module by a continuous linear map is a
continuous linear map.
Transferred version of `ContinuousLinearMap.compLeftContinuousBounded`,
upgraded version of `ContinuousLinearMap.compLeftContinuous`,
similar to `LinearMap.compLeft`. -/
protected def ContinuousLinearMap.compLeftContinuousCompact (g : β →L[𝕜] γ) :
C(X, β) →L[𝕜] C(X, γ) :=
(linearIsometryBoundedOfCompact X γ 𝕜).symm.toLinearIsometry.toContinuousLinearMap.comp <|
(g.compLeftContinuousBounded X).comp <|
(linearIsometryBoundedOfCompact X β 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_linear_map.comp_left_continuous_compact ContinuousLinearMap.compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.toLinear_compLeftContinuousCompact (g : β →L[𝕜] γ) :
(g.compLeftContinuousCompact X : C(X, β) →ₗ[𝕜] C(X, γ)) = g.compLeftContinuous 𝕜 X := by
ext f
rfl
#align continuous_linear_map.to_linear_comp_left_continuous_compact ContinuousLinearMap.toLinear_compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.compLeftContinuousCompact_apply (g : β →L[𝕜] γ) (f : C(X, β)) (x : X) :
g.compLeftContinuousCompact X f x = g (f x) :=
rfl
#align continuous_linear_map.comp_left_continuous_compact_apply ContinuousLinearMap.compLeftContinuousCompact_apply
end CompLeft
namespace ContinuousMap
/-!
We now setup variations on `compRight* f`, where `f : C(X, Y)`
(that is, precomposition by a continuous map),
as a morphism `C(Y, T) → C(X, T)`, respecting various types of structure.
In particular:
* `compRightContinuousMap`, the bundled continuous map (for this we need `X Y` compact).
* `compRightHomeomorph`, when we precompose by a homeomorphism.
* `compRightAlgHom`, when `T = R` is a topological ring.
-/
section CompRight
/-- Precomposition by a continuous map is itself a continuous map between spaces of continuous maps.
-/
def compRightContinuousMap {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y)) : C(C(Y, T), C(X, T)) where
toFun g := g.comp f
continuous_toFun := by
refine' Metric.continuous_iff.mpr _
intro g ε ε_pos
refine' ⟨ε, ε_pos, fun g' h => _⟩
rw [ContinuousMap.dist_lt_iff ε_pos] at h ⊢
exact fun x => h (f x)
#align continuous_map.comp_right_continuous_map ContinuousMap.compRightContinuousMap
@[simp]
theorem compRightContinuousMap_apply {X Y : Type*} (T : Type*) [TopologicalSpace X]
[CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y))
(g : C(Y, T)) : (compRightContinuousMap T f) g = g.comp f :=
rfl
#align continuous_map.comp_right_continuous_map_apply ContinuousMap.compRightContinuousMap_apply
/-- Precomposition by a homeomorphism is itself a homeomorphism between spaces of continuous maps.
-/
def compRightHomeomorph {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : X ≃ₜ Y) : C(Y, T) ≃ₜ C(X, T) where
toFun := compRightContinuousMap T f.toContinuousMap
invFun := compRightContinuousMap T f.symm.toContinuousMap
left_inv g := ext fun _ => congr_arg g (f.apply_symm_apply _)
right_inv g := ext fun _ => congr_arg g (f.symm_apply_apply _)
#align continuous_map.comp_right_homeomorph ContinuousMap.compRightHomeomorph
theorem compRightAlgHom_continuous {X Y : Type*} (R A : Type*) [TopologicalSpace X]
[CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [CommSemiring R] [Semiring A]
[MetricSpace A] [TopologicalSemiring A] [Algebra R A] (f : C(X, Y)) :
Continuous (compRightAlgHom R A f) :=
map_continuous (compRightContinuousMap A f)
#align continuous_map.comp_right_alg_hom_continuous ContinuousMap.compRightAlgHom_continuous
end CompRight
section LocalNormalConvergence
/-! ### Local normal convergence
A sum of continuous functions (on a locally compact space) is "locally normally convergent" if the
sum of its sup-norms on any compact subset is summable. This implies convergence in the topology
of `C(X, E)` (i.e. locally uniform convergence). -/
open TopologicalSpace
variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
variable {E : Type*} [NormedAddCommGroup E] [CompleteSpace E]
theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F := by
refine' (ContinuousMap.exists_tendsto_compactOpen_iff_forall _).2 fun K hK => _
lift K to Compacts X using hK
have A : ∀ s : Finset ι, restrict (↑K) (∑ i in s, F i) = ∑ i in s, restrict K (F i) := by
intro s
ext1 x
simp
-- This used to be the end of the proof before leanprover/lean4#2644
erw [restrict_apply, restrict_apply, restrict_apply, restrict_apply]
simp? says simp only [coe_sum, Finset.sum_apply]
congr!
simpa only [HasSum, A] using (hF K).of_norm
#align continuous_map.summable_of_locally_summable_norm ContinuousMap.summable_of_locally_summable_norm
end LocalNormalConvergence
/-!
### Star structures
In this section, if `β` is a normed ⋆-group, then so is the space of
continuous functions from `α` to `β`, by using the star operation pointwise.
Furthermore, if `α` is compact and `β` is a C⋆-ring, then `C(α, β)` is a C⋆-ring. -/
section NormedSpace
variable {α : Type*} {β : Type*}
variable [TopologicalSpace α] [NormedAddCommGroup β] [StarAddMonoid β] [NormedStarGroup β]
theorem _root_.BoundedContinuousFunction.mkOfCompact_star [CompactSpace α] (f : C(α, β)) :
mkOfCompact (star f) = star (mkOfCompact f) :=
rfl
#align bounded_continuous_function.mk_of_compact_star BoundedContinuousFunction.mkOfCompact_star
instance [CompactSpace α] : NormedStarGroup C(α, β) where
norm_star f := by
rw [← BoundedContinuousFunction.norm_mkOfCompact, BoundedContinuousFunction.mkOfCompact_star,
norm_star, BoundedContinuousFunction.norm_mkOfCompact]
end NormedSpace
section CstarRing
variable {α : Type*} {β : Type*}
variable [TopologicalSpace α] [NormedRing β] [StarRing β]
instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f} := by
|
refine' le_antisymm _ _
|
instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f} := by
|
Mathlib.Topology.ContinuousFunction.Compact.541_0.Mig2jTVnn2FLKEB
|
instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f}
|
Mathlib_Topology_ContinuousFunction_Compact
|
case refine'_1
α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : NormedRing β
inst✝² : StarRing β
inst✝¹ : CompactSpace α
inst✝ : CstarRing β
f : C(α, β)
⊢ ‖star f * f‖ ≤ ‖f‖ * ‖f‖
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compacts
#align_import topology.continuous_function.compact from "leanprover-community/mathlib"@"d3af0609f6db8691dffdc3e1fb7feb7da72698f2"
/-!
# Continuous functions on a compact space
Continuous functions `C(α, β)` from a compact space `α` to a metric space `β`
are automatically bounded, and so acquire various structures inherited from `α →ᵇ β`.
This file transfers these structures, and restates some lemmas
characterising these structures.
If you need a lemma which is proved about `α →ᵇ β` but not for `C(α, β)` when `α` is compact,
you should restate it here. You can also use
`ContinuousMap.equivBoundedOfCompact` to move functions back and forth.
-/
noncomputable section
open Topology Classical NNReal BoundedContinuousFunction BigOperators
open Set Filter Metric
open BoundedContinuousFunction
namespace ContinuousMap
variable {α β E : Type*} [TopologicalSpace α] [CompactSpace α] [MetricSpace β]
[NormedAddCommGroup E]
section
variable (α β)
/-- When `α` is compact, the bounded continuous maps `α →ᵇ β` are
equivalent to `C(α, β)`.
-/
@[simps (config := .asFn)]
def equivBoundedOfCompact : C(α, β) ≃ (α →ᵇ β) :=
⟨mkOfCompact, BoundedContinuousFunction.toContinuousMap, fun f => by
ext
rfl, fun f => by
ext
rfl⟩
#align continuous_map.equiv_bounded_of_compact ContinuousMap.equivBoundedOfCompact
theorem uniformInducing_equivBoundedOfCompact : UniformInducing (equivBoundedOfCompact α β) :=
UniformInducing.mk'
(by
simp only [hasBasis_compactConvergenceUniformity.mem_iff, uniformity_basis_dist_le.mem_iff]
exact fun s =>
⟨fun ⟨⟨a, b⟩, ⟨_, ⟨ε, hε, hb⟩⟩, hs⟩ =>
⟨{ p | ∀ x, (p.1 x, p.2 x) ∈ b }, ⟨ε, hε, fun _ h x => hb ((dist_le hε.le).mp h x)⟩,
fun f g h => hs fun x _ => h x⟩,
fun ⟨_, ⟨ε, hε, ht⟩, hs⟩ =>
⟨⟨Set.univ, { p | dist p.1 p.2 ≤ ε }⟩, ⟨isCompact_univ, ⟨ε, hε, fun _ h => h⟩⟩,
fun ⟨f, g⟩ h => hs _ _ (ht ((dist_le hε.le).mpr fun x => h x (mem_univ x)))⟩⟩)
#align continuous_map.uniform_inducing_equiv_bounded_of_compact ContinuousMap.uniformInducing_equivBoundedOfCompact
theorem uniformEmbedding_equivBoundedOfCompact : UniformEmbedding (equivBoundedOfCompact α β) :=
{ uniformInducing_equivBoundedOfCompact α β with inj := (equivBoundedOfCompact α β).injective }
#align continuous_map.uniform_embedding_equiv_bounded_of_compact ContinuousMap.uniformEmbedding_equivBoundedOfCompact
/-- When `α` is compact, the bounded continuous maps `α →ᵇ 𝕜` are
additively equivalent to `C(α, 𝕜)`.
-/
-- porting note: the following `simps` received a "maximum recursion depth" error
-- @[simps! (config := .asFn) apply symm_apply]
def addEquivBoundedOfCompact [AddMonoid β] [LipschitzAdd β] : C(α, β) ≃+ (α →ᵇ β) :=
({ toContinuousMapAddHom α β, (equivBoundedOfCompact α β).symm with } : (α →ᵇ β) ≃+ C(α, β)).symm
#align continuous_map.add_equiv_bounded_of_compact ContinuousMap.addEquivBoundedOfCompact
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_symm_apply [AddMonoid β] [LipschitzAdd β] :
⇑((addEquivBoundedOfCompact α β).symm) = toContinuousMapAddHom α β :=
rfl
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_apply [AddMonoid β] [LipschitzAdd β] :
⇑(addEquivBoundedOfCompact α β) = mkOfCompact :=
rfl
instance metricSpace : MetricSpace C(α, β) :=
(uniformEmbedding_equivBoundedOfCompact α β).comapMetricSpace _
#align continuous_map.metric_space ContinuousMap.metricSpace
/-- When `α` is compact, and `β` is a metric space, the bounded continuous maps `α →ᵇ β` are
isometric to `C(α, β)`.
-/
@[simps! (config := .asFn) toEquiv apply symm_apply]
def isometryEquivBoundedOfCompact : C(α, β) ≃ᵢ (α →ᵇ β) where
isometry_toFun _ _ := rfl
toEquiv := equivBoundedOfCompact α β
#align continuous_map.isometry_equiv_bounded_of_compact ContinuousMap.isometryEquivBoundedOfCompact
end
@[simp]
theorem _root_.BoundedContinuousFunction.dist_mkOfCompact (f g : C(α, β)) :
dist (mkOfCompact f) (mkOfCompact g) = dist f g :=
rfl
#align bounded_continuous_function.dist_mk_of_compact BoundedContinuousFunction.dist_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.dist_toContinuousMap (f g : α →ᵇ β) :
dist f.toContinuousMap g.toContinuousMap = dist f g :=
rfl
#align bounded_continuous_function.dist_to_continuous_map BoundedContinuousFunction.dist_toContinuousMap
open BoundedContinuousFunction
section
variable {f g : C(α, β)} {C : ℝ}
/-- The pointwise distance is controlled by the distance between functions, by definition. -/
theorem dist_apply_le_dist (x : α) : dist (f x) (g x) ≤ dist f g := by
simp only [← dist_mkOfCompact, dist_coe_le_dist, ← mkOfCompact_apply]
#align continuous_map.dist_apply_le_dist ContinuousMap.dist_apply_le_dist
/-- The distance between two functions is controlled by the supremum of the pointwise distances. -/
theorem dist_le (C0 : (0 : ℝ) ≤ C) : dist f g ≤ C ↔ ∀ x : α, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le C0, mkOfCompact_apply]
#align continuous_map.dist_le ContinuousMap.dist_le
theorem dist_le_iff_of_nonempty [Nonempty α] : dist f g ≤ C ↔ ∀ x, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le_iff_of_nonempty,
mkOfCompact_apply]
#align continuous_map.dist_le_iff_of_nonempty ContinuousMap.dist_le_iff_of_nonempty
theorem dist_lt_iff_of_nonempty [Nonempty α] : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
simp only [← dist_mkOfCompact, dist_lt_iff_of_nonempty_compact, mkOfCompact_apply]
#align continuous_map.dist_lt_iff_of_nonempty ContinuousMap.dist_lt_iff_of_nonempty
theorem dist_lt_of_nonempty [Nonempty α] (w : ∀ x : α, dist (f x) (g x) < C) : dist f g < C :=
dist_lt_iff_of_nonempty.2 w
#align continuous_map.dist_lt_of_nonempty ContinuousMap.dist_lt_of_nonempty
theorem dist_lt_iff (C0 : (0 : ℝ) < C) : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
rw [← dist_mkOfCompact, dist_lt_iff_of_compact C0]
simp only [mkOfCompact_apply]
#align continuous_map.dist_lt_iff ContinuousMap.dist_lt_iff
end
instance [CompleteSpace β] : CompleteSpace C(α, β) :=
(isometryEquivBoundedOfCompact α β).completeSpace
/-- See also `ContinuousMap.continuous_eval'`. -/
@[continuity]
theorem continuous_eval : Continuous fun p : C(α, β) × α => p.1 p.2 :=
continuous_eval.comp ((isometryEquivBoundedOfCompact α β).continuous.prod_map continuous_id)
#align continuous_map.continuous_eval ContinuousMap.continuous_eval
-- TODO at some point we will need lemmas characterising this norm!
-- At the moment the only way to reason about it is to transfer `f : C(α,E)` back to `α →ᵇ E`.
instance : Norm C(α, E) where norm x := dist x 0
@[simp]
theorem _root_.BoundedContinuousFunction.norm_mkOfCompact (f : C(α, E)) : ‖mkOfCompact f‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_mk_of_compact BoundedContinuousFunction.norm_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.norm_toContinuousMap_eq (f : α →ᵇ E) :
‖f.toContinuousMap‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_to_continuous_map_eq BoundedContinuousFunction.norm_toContinuousMap_eq
open BoundedContinuousFunction
instance : NormedAddCommGroup C(α, E) :=
{ ContinuousMap.metricSpace _ _,
ContinuousMap.instAddCommGroupContinuousMap with
dist_eq := fun x y => by
rw [← norm_mkOfCompact, ← dist_mkOfCompact, dist_eq_norm, mkOfCompact_sub]
dist := dist
norm := norm }
instance [Nonempty α] [One E] [NormOneClass E] : NormOneClass C(α, E) where
norm_one := by simp only [← norm_mkOfCompact, mkOfCompact_one, norm_one]
section
variable (f : C(α, E))
-- The corresponding lemmas for `BoundedContinuousFunction` are stated with `{f}`,
-- and so can not be used in dot notation.
theorem norm_coe_le_norm (x : α) : ‖f x‖ ≤ ‖f‖ :=
(mkOfCompact f).norm_coe_le_norm x
#align continuous_map.norm_coe_le_norm ContinuousMap.norm_coe_le_norm
/-- Distance between the images of any two points is at most twice the norm of the function. -/
theorem dist_le_two_norm (x y : α) : dist (f x) (f y) ≤ 2 * ‖f‖ :=
(mkOfCompact f).dist_le_two_norm x y
#align continuous_map.dist_le_two_norm ContinuousMap.dist_le_two_norm
/-- The norm of a function is controlled by the supremum of the pointwise norms. -/
theorem norm_le {C : ℝ} (C0 : (0 : ℝ) ≤ C) : ‖f‖ ≤ C ↔ ∀ x : α, ‖f x‖ ≤ C :=
@BoundedContinuousFunction.norm_le _ _ _ _ (mkOfCompact f) _ C0
#align continuous_map.norm_le ContinuousMap.norm_le
theorem norm_le_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ ≤ M ↔ ∀ x, ‖f x‖ ≤ M :=
@BoundedContinuousFunction.norm_le_of_nonempty _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_le_of_nonempty ContinuousMap.norm_le_of_nonempty
theorem norm_lt_iff {M : ℝ} (M0 : 0 < M) : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_compact _ _ _ _ _ (mkOfCompact f) _ M0
#align continuous_map.norm_lt_iff ContinuousMap.norm_lt_iff
theorem nnnorm_lt_iff {M : ℝ≥0} (M0 : 0 < M) : ‖f‖₊ < M ↔ ∀ x : α, ‖f x‖₊ < M :=
f.norm_lt_iff M0
#align continuous_map.nnnorm_lt_iff ContinuousMap.nnnorm_lt_iff
theorem norm_lt_iff_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_nonempty_compact _ _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_lt_iff_of_nonempty ContinuousMap.norm_lt_iff_of_nonempty
theorem nnnorm_lt_iff_of_nonempty [Nonempty α] {M : ℝ≥0} : ‖f‖₊ < M ↔ ∀ x, ‖f x‖₊ < M :=
f.norm_lt_iff_of_nonempty
#align continuous_map.nnnorm_lt_iff_of_nonempty ContinuousMap.nnnorm_lt_iff_of_nonempty
theorem apply_le_norm (f : C(α, ℝ)) (x : α) : f x ≤ ‖f‖ :=
le_trans (le_abs.mpr (Or.inl (le_refl (f x)))) (f.norm_coe_le_norm x)
#align continuous_map.apply_le_norm ContinuousMap.apply_le_norm
theorem neg_norm_le_apply (f : C(α, ℝ)) (x : α) : -‖f‖ ≤ f x :=
le_trans (neg_le_neg (f.norm_coe_le_norm x)) (neg_le.mp (neg_le_abs_self (f x)))
#align continuous_map.neg_norm_le_apply ContinuousMap.neg_norm_le_apply
theorem norm_eq_iSup_norm : ‖f‖ = ⨆ x : α, ‖f x‖ :=
(mkOfCompact f).norm_eq_iSup_norm
#align continuous_map.norm_eq_supr_norm ContinuousMap.norm_eq_iSup_norm
theorem norm_restrict_mono_set {X : Type*} [TopologicalSpace X] (f : C(X, E))
{K L : TopologicalSpace.Compacts X} (hKL : K ≤ L) : ‖f.restrict K‖ ≤ ‖f.restrict L‖ :=
(norm_le _ (norm_nonneg _)).mpr fun x => norm_coe_le_norm (f.restrict L) <| Set.inclusion hKL x
#align continuous_map.norm_restrict_mono_set ContinuousMap.norm_restrict_mono_set
end
section
variable {R : Type*} [NormedRing R]
instance : NormedRing C(α, R) :=
{ (inferInstance : NormedAddCommGroup C(α, R)), ContinuousMap.instRingContinuousMap with
norm_mul := fun f g => norm_mul_le (mkOfCompact f) (mkOfCompact g) }
end
section
variable {𝕜 : Type*} [NormedField 𝕜] [NormedSpace 𝕜 E]
instance normedSpace : NormedSpace 𝕜 C(α, E) where
norm_smul_le c f := (norm_smul_le c (mkOfCompact f) : _)
#align continuous_map.normed_space ContinuousMap.normedSpace
section
variable (α 𝕜 E)
/-- When `α` is compact and `𝕜` is a normed field,
the `𝕜`-algebra of bounded continuous maps `α →ᵇ β` is
`𝕜`-linearly isometric to `C(α, β)`.
-/
def linearIsometryBoundedOfCompact : C(α, E) ≃ₗᵢ[𝕜] α →ᵇ E :=
{ addEquivBoundedOfCompact α E with
map_smul' := fun c f => by
ext
norm_cast
norm_map' := fun f => rfl }
#align continuous_map.linear_isometry_bounded_of_compact ContinuousMap.linearIsometryBoundedOfCompact
variable {α E}
-- to match `BoundedContinuousFunction.evalClm`
/-- The evaluation at a point, as a continuous linear map from `C(α, 𝕜)` to `𝕜`. -/
def evalClm (x : α) : C(α, E) →L[𝕜] E :=
(BoundedContinuousFunction.evalClm 𝕜 x).comp
(linearIsometryBoundedOfCompact α E 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_map.eval_clm ContinuousMap.evalClm
end
-- this lemma and the next are the analogues of those autogenerated by `@[simps]` for
-- `equivBoundedOfCompact`, `addEquivBoundedOfCompact`
@[simp]
theorem linearIsometryBoundedOfCompact_symm_apply (f : α →ᵇ E) :
(linearIsometryBoundedOfCompact α E 𝕜).symm f = f.toContinuousMap :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_symm_apply ContinuousMap.linearIsometryBoundedOfCompact_symm_apply
@[simp]
theorem linearIsometryBoundedOfCompact_apply_apply (f : C(α, E)) (a : α) :
(linearIsometryBoundedOfCompact α E 𝕜 f) a = f a :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_apply_apply ContinuousMap.linearIsometryBoundedOfCompact_apply_apply
@[simp]
theorem linearIsometryBoundedOfCompact_toIsometryEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toIsometryEquiv = isometryEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_isometry_equiv ContinuousMap.linearIsometryBoundedOfCompact_toIsometryEquiv
@[simp] -- porting note: adjusted LHS because `simpNF` complained it simplified.
theorem linearIsometryBoundedOfCompact_toAddEquiv :
((linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv : C(α, E) ≃+ (α →ᵇ E)) =
addEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_add_equiv ContinuousMap.linearIsometryBoundedOfCompact_toAddEquiv
@[simp]
theorem linearIsometryBoundedOfCompact_of_compact_toEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv.toEquiv = equivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_of_compact_to_equiv ContinuousMap.linearIsometryBoundedOfCompact_of_compact_toEquiv
end
section
variable {𝕜 : Type*} {γ : Type*} [NormedField 𝕜] [NormedRing γ] [NormedAlgebra 𝕜 γ]
instance : NormedAlgebra 𝕜 C(α, γ) :=
{ ContinuousMap.normedSpace, ContinuousMap.algebra with }
end
end ContinuousMap
namespace ContinuousMap
section UniformContinuity
variable {α β : Type*}
variable [MetricSpace α] [CompactSpace α] [MetricSpace β]
/-!
We now set up some declarations making it convenient to use uniform continuity.
-/
theorem uniform_continuity (f : C(α, β)) (ε : ℝ) (h : 0 < ε) :
∃ δ > 0, ∀ {x y}, dist x y < δ → dist (f x) (f y) < ε :=
Metric.uniformContinuous_iff.mp (CompactSpace.uniformContinuous_of_continuous f.continuous) ε h
#align continuous_map.uniform_continuity ContinuousMap.uniform_continuity
-- This definition allows us to separate the choice of some `δ`,
-- and the corresponding use of `dist a b < δ → dist (f a) (f b) < ε`,
-- even across different declarations.
/-- An arbitrarily chosen modulus of uniform continuity for a given function `f` and `ε > 0`. -/
def modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) : ℝ :=
Classical.choose (uniform_continuity f ε h)
#align continuous_map.modulus ContinuousMap.modulus
theorem modulus_pos (f : C(α, β)) {ε : ℝ} {h : 0 < ε} : 0 < f.modulus ε h :=
(Classical.choose_spec (uniform_continuity f ε h)).1
#align continuous_map.modulus_pos ContinuousMap.modulus_pos
theorem dist_lt_of_dist_lt_modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) {a b : α}
(w : dist a b < f.modulus ε h) : dist (f a) (f b) < ε :=
(Classical.choose_spec (uniform_continuity f ε h)).2 w
#align continuous_map.dist_lt_of_dist_lt_modulus ContinuousMap.dist_lt_of_dist_lt_modulus
end UniformContinuity
end ContinuousMap
section CompLeft
variable (X : Type*) {𝕜 β γ : Type*} [TopologicalSpace X] [CompactSpace X]
[NontriviallyNormedField 𝕜]
variable [NormedAddCommGroup β] [NormedSpace 𝕜 β] [NormedAddCommGroup γ] [NormedSpace 𝕜 γ]
open ContinuousMap
/-- Postcomposition of continuous functions into a normed module by a continuous linear map is a
continuous linear map.
Transferred version of `ContinuousLinearMap.compLeftContinuousBounded`,
upgraded version of `ContinuousLinearMap.compLeftContinuous`,
similar to `LinearMap.compLeft`. -/
protected def ContinuousLinearMap.compLeftContinuousCompact (g : β →L[𝕜] γ) :
C(X, β) →L[𝕜] C(X, γ) :=
(linearIsometryBoundedOfCompact X γ 𝕜).symm.toLinearIsometry.toContinuousLinearMap.comp <|
(g.compLeftContinuousBounded X).comp <|
(linearIsometryBoundedOfCompact X β 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_linear_map.comp_left_continuous_compact ContinuousLinearMap.compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.toLinear_compLeftContinuousCompact (g : β →L[𝕜] γ) :
(g.compLeftContinuousCompact X : C(X, β) →ₗ[𝕜] C(X, γ)) = g.compLeftContinuous 𝕜 X := by
ext f
rfl
#align continuous_linear_map.to_linear_comp_left_continuous_compact ContinuousLinearMap.toLinear_compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.compLeftContinuousCompact_apply (g : β →L[𝕜] γ) (f : C(X, β)) (x : X) :
g.compLeftContinuousCompact X f x = g (f x) :=
rfl
#align continuous_linear_map.comp_left_continuous_compact_apply ContinuousLinearMap.compLeftContinuousCompact_apply
end CompLeft
namespace ContinuousMap
/-!
We now setup variations on `compRight* f`, where `f : C(X, Y)`
(that is, precomposition by a continuous map),
as a morphism `C(Y, T) → C(X, T)`, respecting various types of structure.
In particular:
* `compRightContinuousMap`, the bundled continuous map (for this we need `X Y` compact).
* `compRightHomeomorph`, when we precompose by a homeomorphism.
* `compRightAlgHom`, when `T = R` is a topological ring.
-/
section CompRight
/-- Precomposition by a continuous map is itself a continuous map between spaces of continuous maps.
-/
def compRightContinuousMap {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y)) : C(C(Y, T), C(X, T)) where
toFun g := g.comp f
continuous_toFun := by
refine' Metric.continuous_iff.mpr _
intro g ε ε_pos
refine' ⟨ε, ε_pos, fun g' h => _⟩
rw [ContinuousMap.dist_lt_iff ε_pos] at h ⊢
exact fun x => h (f x)
#align continuous_map.comp_right_continuous_map ContinuousMap.compRightContinuousMap
@[simp]
theorem compRightContinuousMap_apply {X Y : Type*} (T : Type*) [TopologicalSpace X]
[CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y))
(g : C(Y, T)) : (compRightContinuousMap T f) g = g.comp f :=
rfl
#align continuous_map.comp_right_continuous_map_apply ContinuousMap.compRightContinuousMap_apply
/-- Precomposition by a homeomorphism is itself a homeomorphism between spaces of continuous maps.
-/
def compRightHomeomorph {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : X ≃ₜ Y) : C(Y, T) ≃ₜ C(X, T) where
toFun := compRightContinuousMap T f.toContinuousMap
invFun := compRightContinuousMap T f.symm.toContinuousMap
left_inv g := ext fun _ => congr_arg g (f.apply_symm_apply _)
right_inv g := ext fun _ => congr_arg g (f.symm_apply_apply _)
#align continuous_map.comp_right_homeomorph ContinuousMap.compRightHomeomorph
theorem compRightAlgHom_continuous {X Y : Type*} (R A : Type*) [TopologicalSpace X]
[CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [CommSemiring R] [Semiring A]
[MetricSpace A] [TopologicalSemiring A] [Algebra R A] (f : C(X, Y)) :
Continuous (compRightAlgHom R A f) :=
map_continuous (compRightContinuousMap A f)
#align continuous_map.comp_right_alg_hom_continuous ContinuousMap.compRightAlgHom_continuous
end CompRight
section LocalNormalConvergence
/-! ### Local normal convergence
A sum of continuous functions (on a locally compact space) is "locally normally convergent" if the
sum of its sup-norms on any compact subset is summable. This implies convergence in the topology
of `C(X, E)` (i.e. locally uniform convergence). -/
open TopologicalSpace
variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
variable {E : Type*} [NormedAddCommGroup E] [CompleteSpace E]
theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F := by
refine' (ContinuousMap.exists_tendsto_compactOpen_iff_forall _).2 fun K hK => _
lift K to Compacts X using hK
have A : ∀ s : Finset ι, restrict (↑K) (∑ i in s, F i) = ∑ i in s, restrict K (F i) := by
intro s
ext1 x
simp
-- This used to be the end of the proof before leanprover/lean4#2644
erw [restrict_apply, restrict_apply, restrict_apply, restrict_apply]
simp? says simp only [coe_sum, Finset.sum_apply]
congr!
simpa only [HasSum, A] using (hF K).of_norm
#align continuous_map.summable_of_locally_summable_norm ContinuousMap.summable_of_locally_summable_norm
end LocalNormalConvergence
/-!
### Star structures
In this section, if `β` is a normed ⋆-group, then so is the space of
continuous functions from `α` to `β`, by using the star operation pointwise.
Furthermore, if `α` is compact and `β` is a C⋆-ring, then `C(α, β)` is a C⋆-ring. -/
section NormedSpace
variable {α : Type*} {β : Type*}
variable [TopologicalSpace α] [NormedAddCommGroup β] [StarAddMonoid β] [NormedStarGroup β]
theorem _root_.BoundedContinuousFunction.mkOfCompact_star [CompactSpace α] (f : C(α, β)) :
mkOfCompact (star f) = star (mkOfCompact f) :=
rfl
#align bounded_continuous_function.mk_of_compact_star BoundedContinuousFunction.mkOfCompact_star
instance [CompactSpace α] : NormedStarGroup C(α, β) where
norm_star f := by
rw [← BoundedContinuousFunction.norm_mkOfCompact, BoundedContinuousFunction.mkOfCompact_star,
norm_star, BoundedContinuousFunction.norm_mkOfCompact]
end NormedSpace
section CstarRing
variable {α : Type*} {β : Type*}
variable [TopologicalSpace α] [NormedRing β] [StarRing β]
instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f} := by
refine' le_antisymm _ _
·
|
rw [← sq, ContinuousMap.norm_le _ (sq_nonneg _)]
|
instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f} := by
refine' le_antisymm _ _
·
|
Mathlib.Topology.ContinuousFunction.Compact.541_0.Mig2jTVnn2FLKEB
|
instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f}
|
Mathlib_Topology_ContinuousFunction_Compact
|
case refine'_1
α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : NormedRing β
inst✝² : StarRing β
inst✝¹ : CompactSpace α
inst✝ : CstarRing β
f : C(α, β)
⊢ ∀ (x : α), ‖(star f * f) x‖ ≤ ‖f‖ ^ 2
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compacts
#align_import topology.continuous_function.compact from "leanprover-community/mathlib"@"d3af0609f6db8691dffdc3e1fb7feb7da72698f2"
/-!
# Continuous functions on a compact space
Continuous functions `C(α, β)` from a compact space `α` to a metric space `β`
are automatically bounded, and so acquire various structures inherited from `α →ᵇ β`.
This file transfers these structures, and restates some lemmas
characterising these structures.
If you need a lemma which is proved about `α →ᵇ β` but not for `C(α, β)` when `α` is compact,
you should restate it here. You can also use
`ContinuousMap.equivBoundedOfCompact` to move functions back and forth.
-/
noncomputable section
open Topology Classical NNReal BoundedContinuousFunction BigOperators
open Set Filter Metric
open BoundedContinuousFunction
namespace ContinuousMap
variable {α β E : Type*} [TopologicalSpace α] [CompactSpace α] [MetricSpace β]
[NormedAddCommGroup E]
section
variable (α β)
/-- When `α` is compact, the bounded continuous maps `α →ᵇ β` are
equivalent to `C(α, β)`.
-/
@[simps (config := .asFn)]
def equivBoundedOfCompact : C(α, β) ≃ (α →ᵇ β) :=
⟨mkOfCompact, BoundedContinuousFunction.toContinuousMap, fun f => by
ext
rfl, fun f => by
ext
rfl⟩
#align continuous_map.equiv_bounded_of_compact ContinuousMap.equivBoundedOfCompact
theorem uniformInducing_equivBoundedOfCompact : UniformInducing (equivBoundedOfCompact α β) :=
UniformInducing.mk'
(by
simp only [hasBasis_compactConvergenceUniformity.mem_iff, uniformity_basis_dist_le.mem_iff]
exact fun s =>
⟨fun ⟨⟨a, b⟩, ⟨_, ⟨ε, hε, hb⟩⟩, hs⟩ =>
⟨{ p | ∀ x, (p.1 x, p.2 x) ∈ b }, ⟨ε, hε, fun _ h x => hb ((dist_le hε.le).mp h x)⟩,
fun f g h => hs fun x _ => h x⟩,
fun ⟨_, ⟨ε, hε, ht⟩, hs⟩ =>
⟨⟨Set.univ, { p | dist p.1 p.2 ≤ ε }⟩, ⟨isCompact_univ, ⟨ε, hε, fun _ h => h⟩⟩,
fun ⟨f, g⟩ h => hs _ _ (ht ((dist_le hε.le).mpr fun x => h x (mem_univ x)))⟩⟩)
#align continuous_map.uniform_inducing_equiv_bounded_of_compact ContinuousMap.uniformInducing_equivBoundedOfCompact
theorem uniformEmbedding_equivBoundedOfCompact : UniformEmbedding (equivBoundedOfCompact α β) :=
{ uniformInducing_equivBoundedOfCompact α β with inj := (equivBoundedOfCompact α β).injective }
#align continuous_map.uniform_embedding_equiv_bounded_of_compact ContinuousMap.uniformEmbedding_equivBoundedOfCompact
/-- When `α` is compact, the bounded continuous maps `α →ᵇ 𝕜` are
additively equivalent to `C(α, 𝕜)`.
-/
-- porting note: the following `simps` received a "maximum recursion depth" error
-- @[simps! (config := .asFn) apply symm_apply]
def addEquivBoundedOfCompact [AddMonoid β] [LipschitzAdd β] : C(α, β) ≃+ (α →ᵇ β) :=
({ toContinuousMapAddHom α β, (equivBoundedOfCompact α β).symm with } : (α →ᵇ β) ≃+ C(α, β)).symm
#align continuous_map.add_equiv_bounded_of_compact ContinuousMap.addEquivBoundedOfCompact
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_symm_apply [AddMonoid β] [LipschitzAdd β] :
⇑((addEquivBoundedOfCompact α β).symm) = toContinuousMapAddHom α β :=
rfl
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_apply [AddMonoid β] [LipschitzAdd β] :
⇑(addEquivBoundedOfCompact α β) = mkOfCompact :=
rfl
instance metricSpace : MetricSpace C(α, β) :=
(uniformEmbedding_equivBoundedOfCompact α β).comapMetricSpace _
#align continuous_map.metric_space ContinuousMap.metricSpace
/-- When `α` is compact, and `β` is a metric space, the bounded continuous maps `α →ᵇ β` are
isometric to `C(α, β)`.
-/
@[simps! (config := .asFn) toEquiv apply symm_apply]
def isometryEquivBoundedOfCompact : C(α, β) ≃ᵢ (α →ᵇ β) where
isometry_toFun _ _ := rfl
toEquiv := equivBoundedOfCompact α β
#align continuous_map.isometry_equiv_bounded_of_compact ContinuousMap.isometryEquivBoundedOfCompact
end
@[simp]
theorem _root_.BoundedContinuousFunction.dist_mkOfCompact (f g : C(α, β)) :
dist (mkOfCompact f) (mkOfCompact g) = dist f g :=
rfl
#align bounded_continuous_function.dist_mk_of_compact BoundedContinuousFunction.dist_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.dist_toContinuousMap (f g : α →ᵇ β) :
dist f.toContinuousMap g.toContinuousMap = dist f g :=
rfl
#align bounded_continuous_function.dist_to_continuous_map BoundedContinuousFunction.dist_toContinuousMap
open BoundedContinuousFunction
section
variable {f g : C(α, β)} {C : ℝ}
/-- The pointwise distance is controlled by the distance between functions, by definition. -/
theorem dist_apply_le_dist (x : α) : dist (f x) (g x) ≤ dist f g := by
simp only [← dist_mkOfCompact, dist_coe_le_dist, ← mkOfCompact_apply]
#align continuous_map.dist_apply_le_dist ContinuousMap.dist_apply_le_dist
/-- The distance between two functions is controlled by the supremum of the pointwise distances. -/
theorem dist_le (C0 : (0 : ℝ) ≤ C) : dist f g ≤ C ↔ ∀ x : α, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le C0, mkOfCompact_apply]
#align continuous_map.dist_le ContinuousMap.dist_le
theorem dist_le_iff_of_nonempty [Nonempty α] : dist f g ≤ C ↔ ∀ x, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le_iff_of_nonempty,
mkOfCompact_apply]
#align continuous_map.dist_le_iff_of_nonempty ContinuousMap.dist_le_iff_of_nonempty
theorem dist_lt_iff_of_nonempty [Nonempty α] : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
simp only [← dist_mkOfCompact, dist_lt_iff_of_nonempty_compact, mkOfCompact_apply]
#align continuous_map.dist_lt_iff_of_nonempty ContinuousMap.dist_lt_iff_of_nonempty
theorem dist_lt_of_nonempty [Nonempty α] (w : ∀ x : α, dist (f x) (g x) < C) : dist f g < C :=
dist_lt_iff_of_nonempty.2 w
#align continuous_map.dist_lt_of_nonempty ContinuousMap.dist_lt_of_nonempty
theorem dist_lt_iff (C0 : (0 : ℝ) < C) : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
rw [← dist_mkOfCompact, dist_lt_iff_of_compact C0]
simp only [mkOfCompact_apply]
#align continuous_map.dist_lt_iff ContinuousMap.dist_lt_iff
end
instance [CompleteSpace β] : CompleteSpace C(α, β) :=
(isometryEquivBoundedOfCompact α β).completeSpace
/-- See also `ContinuousMap.continuous_eval'`. -/
@[continuity]
theorem continuous_eval : Continuous fun p : C(α, β) × α => p.1 p.2 :=
continuous_eval.comp ((isometryEquivBoundedOfCompact α β).continuous.prod_map continuous_id)
#align continuous_map.continuous_eval ContinuousMap.continuous_eval
-- TODO at some point we will need lemmas characterising this norm!
-- At the moment the only way to reason about it is to transfer `f : C(α,E)` back to `α →ᵇ E`.
instance : Norm C(α, E) where norm x := dist x 0
@[simp]
theorem _root_.BoundedContinuousFunction.norm_mkOfCompact (f : C(α, E)) : ‖mkOfCompact f‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_mk_of_compact BoundedContinuousFunction.norm_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.norm_toContinuousMap_eq (f : α →ᵇ E) :
‖f.toContinuousMap‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_to_continuous_map_eq BoundedContinuousFunction.norm_toContinuousMap_eq
open BoundedContinuousFunction
instance : NormedAddCommGroup C(α, E) :=
{ ContinuousMap.metricSpace _ _,
ContinuousMap.instAddCommGroupContinuousMap with
dist_eq := fun x y => by
rw [← norm_mkOfCompact, ← dist_mkOfCompact, dist_eq_norm, mkOfCompact_sub]
dist := dist
norm := norm }
instance [Nonempty α] [One E] [NormOneClass E] : NormOneClass C(α, E) where
norm_one := by simp only [← norm_mkOfCompact, mkOfCompact_one, norm_one]
section
variable (f : C(α, E))
-- The corresponding lemmas for `BoundedContinuousFunction` are stated with `{f}`,
-- and so can not be used in dot notation.
theorem norm_coe_le_norm (x : α) : ‖f x‖ ≤ ‖f‖ :=
(mkOfCompact f).norm_coe_le_norm x
#align continuous_map.norm_coe_le_norm ContinuousMap.norm_coe_le_norm
/-- Distance between the images of any two points is at most twice the norm of the function. -/
theorem dist_le_two_norm (x y : α) : dist (f x) (f y) ≤ 2 * ‖f‖ :=
(mkOfCompact f).dist_le_two_norm x y
#align continuous_map.dist_le_two_norm ContinuousMap.dist_le_two_norm
/-- The norm of a function is controlled by the supremum of the pointwise norms. -/
theorem norm_le {C : ℝ} (C0 : (0 : ℝ) ≤ C) : ‖f‖ ≤ C ↔ ∀ x : α, ‖f x‖ ≤ C :=
@BoundedContinuousFunction.norm_le _ _ _ _ (mkOfCompact f) _ C0
#align continuous_map.norm_le ContinuousMap.norm_le
theorem norm_le_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ ≤ M ↔ ∀ x, ‖f x‖ ≤ M :=
@BoundedContinuousFunction.norm_le_of_nonempty _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_le_of_nonempty ContinuousMap.norm_le_of_nonempty
theorem norm_lt_iff {M : ℝ} (M0 : 0 < M) : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_compact _ _ _ _ _ (mkOfCompact f) _ M0
#align continuous_map.norm_lt_iff ContinuousMap.norm_lt_iff
theorem nnnorm_lt_iff {M : ℝ≥0} (M0 : 0 < M) : ‖f‖₊ < M ↔ ∀ x : α, ‖f x‖₊ < M :=
f.norm_lt_iff M0
#align continuous_map.nnnorm_lt_iff ContinuousMap.nnnorm_lt_iff
theorem norm_lt_iff_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_nonempty_compact _ _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_lt_iff_of_nonempty ContinuousMap.norm_lt_iff_of_nonempty
theorem nnnorm_lt_iff_of_nonempty [Nonempty α] {M : ℝ≥0} : ‖f‖₊ < M ↔ ∀ x, ‖f x‖₊ < M :=
f.norm_lt_iff_of_nonempty
#align continuous_map.nnnorm_lt_iff_of_nonempty ContinuousMap.nnnorm_lt_iff_of_nonempty
theorem apply_le_norm (f : C(α, ℝ)) (x : α) : f x ≤ ‖f‖ :=
le_trans (le_abs.mpr (Or.inl (le_refl (f x)))) (f.norm_coe_le_norm x)
#align continuous_map.apply_le_norm ContinuousMap.apply_le_norm
theorem neg_norm_le_apply (f : C(α, ℝ)) (x : α) : -‖f‖ ≤ f x :=
le_trans (neg_le_neg (f.norm_coe_le_norm x)) (neg_le.mp (neg_le_abs_self (f x)))
#align continuous_map.neg_norm_le_apply ContinuousMap.neg_norm_le_apply
theorem norm_eq_iSup_norm : ‖f‖ = ⨆ x : α, ‖f x‖ :=
(mkOfCompact f).norm_eq_iSup_norm
#align continuous_map.norm_eq_supr_norm ContinuousMap.norm_eq_iSup_norm
theorem norm_restrict_mono_set {X : Type*} [TopologicalSpace X] (f : C(X, E))
{K L : TopologicalSpace.Compacts X} (hKL : K ≤ L) : ‖f.restrict K‖ ≤ ‖f.restrict L‖ :=
(norm_le _ (norm_nonneg _)).mpr fun x => norm_coe_le_norm (f.restrict L) <| Set.inclusion hKL x
#align continuous_map.norm_restrict_mono_set ContinuousMap.norm_restrict_mono_set
end
section
variable {R : Type*} [NormedRing R]
instance : NormedRing C(α, R) :=
{ (inferInstance : NormedAddCommGroup C(α, R)), ContinuousMap.instRingContinuousMap with
norm_mul := fun f g => norm_mul_le (mkOfCompact f) (mkOfCompact g) }
end
section
variable {𝕜 : Type*} [NormedField 𝕜] [NormedSpace 𝕜 E]
instance normedSpace : NormedSpace 𝕜 C(α, E) where
norm_smul_le c f := (norm_smul_le c (mkOfCompact f) : _)
#align continuous_map.normed_space ContinuousMap.normedSpace
section
variable (α 𝕜 E)
/-- When `α` is compact and `𝕜` is a normed field,
the `𝕜`-algebra of bounded continuous maps `α →ᵇ β` is
`𝕜`-linearly isometric to `C(α, β)`.
-/
def linearIsometryBoundedOfCompact : C(α, E) ≃ₗᵢ[𝕜] α →ᵇ E :=
{ addEquivBoundedOfCompact α E with
map_smul' := fun c f => by
ext
norm_cast
norm_map' := fun f => rfl }
#align continuous_map.linear_isometry_bounded_of_compact ContinuousMap.linearIsometryBoundedOfCompact
variable {α E}
-- to match `BoundedContinuousFunction.evalClm`
/-- The evaluation at a point, as a continuous linear map from `C(α, 𝕜)` to `𝕜`. -/
def evalClm (x : α) : C(α, E) →L[𝕜] E :=
(BoundedContinuousFunction.evalClm 𝕜 x).comp
(linearIsometryBoundedOfCompact α E 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_map.eval_clm ContinuousMap.evalClm
end
-- this lemma and the next are the analogues of those autogenerated by `@[simps]` for
-- `equivBoundedOfCompact`, `addEquivBoundedOfCompact`
@[simp]
theorem linearIsometryBoundedOfCompact_symm_apply (f : α →ᵇ E) :
(linearIsometryBoundedOfCompact α E 𝕜).symm f = f.toContinuousMap :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_symm_apply ContinuousMap.linearIsometryBoundedOfCompact_symm_apply
@[simp]
theorem linearIsometryBoundedOfCompact_apply_apply (f : C(α, E)) (a : α) :
(linearIsometryBoundedOfCompact α E 𝕜 f) a = f a :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_apply_apply ContinuousMap.linearIsometryBoundedOfCompact_apply_apply
@[simp]
theorem linearIsometryBoundedOfCompact_toIsometryEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toIsometryEquiv = isometryEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_isometry_equiv ContinuousMap.linearIsometryBoundedOfCompact_toIsometryEquiv
@[simp] -- porting note: adjusted LHS because `simpNF` complained it simplified.
theorem linearIsometryBoundedOfCompact_toAddEquiv :
((linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv : C(α, E) ≃+ (α →ᵇ E)) =
addEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_add_equiv ContinuousMap.linearIsometryBoundedOfCompact_toAddEquiv
@[simp]
theorem linearIsometryBoundedOfCompact_of_compact_toEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv.toEquiv = equivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_of_compact_to_equiv ContinuousMap.linearIsometryBoundedOfCompact_of_compact_toEquiv
end
section
variable {𝕜 : Type*} {γ : Type*} [NormedField 𝕜] [NormedRing γ] [NormedAlgebra 𝕜 γ]
instance : NormedAlgebra 𝕜 C(α, γ) :=
{ ContinuousMap.normedSpace, ContinuousMap.algebra with }
end
end ContinuousMap
namespace ContinuousMap
section UniformContinuity
variable {α β : Type*}
variable [MetricSpace α] [CompactSpace α] [MetricSpace β]
/-!
We now set up some declarations making it convenient to use uniform continuity.
-/
theorem uniform_continuity (f : C(α, β)) (ε : ℝ) (h : 0 < ε) :
∃ δ > 0, ∀ {x y}, dist x y < δ → dist (f x) (f y) < ε :=
Metric.uniformContinuous_iff.mp (CompactSpace.uniformContinuous_of_continuous f.continuous) ε h
#align continuous_map.uniform_continuity ContinuousMap.uniform_continuity
-- This definition allows us to separate the choice of some `δ`,
-- and the corresponding use of `dist a b < δ → dist (f a) (f b) < ε`,
-- even across different declarations.
/-- An arbitrarily chosen modulus of uniform continuity for a given function `f` and `ε > 0`. -/
def modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) : ℝ :=
Classical.choose (uniform_continuity f ε h)
#align continuous_map.modulus ContinuousMap.modulus
theorem modulus_pos (f : C(α, β)) {ε : ℝ} {h : 0 < ε} : 0 < f.modulus ε h :=
(Classical.choose_spec (uniform_continuity f ε h)).1
#align continuous_map.modulus_pos ContinuousMap.modulus_pos
theorem dist_lt_of_dist_lt_modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) {a b : α}
(w : dist a b < f.modulus ε h) : dist (f a) (f b) < ε :=
(Classical.choose_spec (uniform_continuity f ε h)).2 w
#align continuous_map.dist_lt_of_dist_lt_modulus ContinuousMap.dist_lt_of_dist_lt_modulus
end UniformContinuity
end ContinuousMap
section CompLeft
variable (X : Type*) {𝕜 β γ : Type*} [TopologicalSpace X] [CompactSpace X]
[NontriviallyNormedField 𝕜]
variable [NormedAddCommGroup β] [NormedSpace 𝕜 β] [NormedAddCommGroup γ] [NormedSpace 𝕜 γ]
open ContinuousMap
/-- Postcomposition of continuous functions into a normed module by a continuous linear map is a
continuous linear map.
Transferred version of `ContinuousLinearMap.compLeftContinuousBounded`,
upgraded version of `ContinuousLinearMap.compLeftContinuous`,
similar to `LinearMap.compLeft`. -/
protected def ContinuousLinearMap.compLeftContinuousCompact (g : β →L[𝕜] γ) :
C(X, β) →L[𝕜] C(X, γ) :=
(linearIsometryBoundedOfCompact X γ 𝕜).symm.toLinearIsometry.toContinuousLinearMap.comp <|
(g.compLeftContinuousBounded X).comp <|
(linearIsometryBoundedOfCompact X β 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_linear_map.comp_left_continuous_compact ContinuousLinearMap.compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.toLinear_compLeftContinuousCompact (g : β →L[𝕜] γ) :
(g.compLeftContinuousCompact X : C(X, β) →ₗ[𝕜] C(X, γ)) = g.compLeftContinuous 𝕜 X := by
ext f
rfl
#align continuous_linear_map.to_linear_comp_left_continuous_compact ContinuousLinearMap.toLinear_compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.compLeftContinuousCompact_apply (g : β →L[𝕜] γ) (f : C(X, β)) (x : X) :
g.compLeftContinuousCompact X f x = g (f x) :=
rfl
#align continuous_linear_map.comp_left_continuous_compact_apply ContinuousLinearMap.compLeftContinuousCompact_apply
end CompLeft
namespace ContinuousMap
/-!
We now setup variations on `compRight* f`, where `f : C(X, Y)`
(that is, precomposition by a continuous map),
as a morphism `C(Y, T) → C(X, T)`, respecting various types of structure.
In particular:
* `compRightContinuousMap`, the bundled continuous map (for this we need `X Y` compact).
* `compRightHomeomorph`, when we precompose by a homeomorphism.
* `compRightAlgHom`, when `T = R` is a topological ring.
-/
section CompRight
/-- Precomposition by a continuous map is itself a continuous map between spaces of continuous maps.
-/
def compRightContinuousMap {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y)) : C(C(Y, T), C(X, T)) where
toFun g := g.comp f
continuous_toFun := by
refine' Metric.continuous_iff.mpr _
intro g ε ε_pos
refine' ⟨ε, ε_pos, fun g' h => _⟩
rw [ContinuousMap.dist_lt_iff ε_pos] at h ⊢
exact fun x => h (f x)
#align continuous_map.comp_right_continuous_map ContinuousMap.compRightContinuousMap
@[simp]
theorem compRightContinuousMap_apply {X Y : Type*} (T : Type*) [TopologicalSpace X]
[CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y))
(g : C(Y, T)) : (compRightContinuousMap T f) g = g.comp f :=
rfl
#align continuous_map.comp_right_continuous_map_apply ContinuousMap.compRightContinuousMap_apply
/-- Precomposition by a homeomorphism is itself a homeomorphism between spaces of continuous maps.
-/
def compRightHomeomorph {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : X ≃ₜ Y) : C(Y, T) ≃ₜ C(X, T) where
toFun := compRightContinuousMap T f.toContinuousMap
invFun := compRightContinuousMap T f.symm.toContinuousMap
left_inv g := ext fun _ => congr_arg g (f.apply_symm_apply _)
right_inv g := ext fun _ => congr_arg g (f.symm_apply_apply _)
#align continuous_map.comp_right_homeomorph ContinuousMap.compRightHomeomorph
theorem compRightAlgHom_continuous {X Y : Type*} (R A : Type*) [TopologicalSpace X]
[CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [CommSemiring R] [Semiring A]
[MetricSpace A] [TopologicalSemiring A] [Algebra R A] (f : C(X, Y)) :
Continuous (compRightAlgHom R A f) :=
map_continuous (compRightContinuousMap A f)
#align continuous_map.comp_right_alg_hom_continuous ContinuousMap.compRightAlgHom_continuous
end CompRight
section LocalNormalConvergence
/-! ### Local normal convergence
A sum of continuous functions (on a locally compact space) is "locally normally convergent" if the
sum of its sup-norms on any compact subset is summable. This implies convergence in the topology
of `C(X, E)` (i.e. locally uniform convergence). -/
open TopologicalSpace
variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
variable {E : Type*} [NormedAddCommGroup E] [CompleteSpace E]
theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F := by
refine' (ContinuousMap.exists_tendsto_compactOpen_iff_forall _).2 fun K hK => _
lift K to Compacts X using hK
have A : ∀ s : Finset ι, restrict (↑K) (∑ i in s, F i) = ∑ i in s, restrict K (F i) := by
intro s
ext1 x
simp
-- This used to be the end of the proof before leanprover/lean4#2644
erw [restrict_apply, restrict_apply, restrict_apply, restrict_apply]
simp? says simp only [coe_sum, Finset.sum_apply]
congr!
simpa only [HasSum, A] using (hF K).of_norm
#align continuous_map.summable_of_locally_summable_norm ContinuousMap.summable_of_locally_summable_norm
end LocalNormalConvergence
/-!
### Star structures
In this section, if `β` is a normed ⋆-group, then so is the space of
continuous functions from `α` to `β`, by using the star operation pointwise.
Furthermore, if `α` is compact and `β` is a C⋆-ring, then `C(α, β)` is a C⋆-ring. -/
section NormedSpace
variable {α : Type*} {β : Type*}
variable [TopologicalSpace α] [NormedAddCommGroup β] [StarAddMonoid β] [NormedStarGroup β]
theorem _root_.BoundedContinuousFunction.mkOfCompact_star [CompactSpace α] (f : C(α, β)) :
mkOfCompact (star f) = star (mkOfCompact f) :=
rfl
#align bounded_continuous_function.mk_of_compact_star BoundedContinuousFunction.mkOfCompact_star
instance [CompactSpace α] : NormedStarGroup C(α, β) where
norm_star f := by
rw [← BoundedContinuousFunction.norm_mkOfCompact, BoundedContinuousFunction.mkOfCompact_star,
norm_star, BoundedContinuousFunction.norm_mkOfCompact]
end NormedSpace
section CstarRing
variable {α : Type*} {β : Type*}
variable [TopologicalSpace α] [NormedRing β] [StarRing β]
instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f} := by
refine' le_antisymm _ _
· rw [← sq, ContinuousMap.norm_le _ (sq_nonneg _)]
|
intro x
|
instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f} := by
refine' le_antisymm _ _
· rw [← sq, ContinuousMap.norm_le _ (sq_nonneg _)]
|
Mathlib.Topology.ContinuousFunction.Compact.541_0.Mig2jTVnn2FLKEB
|
instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f}
|
Mathlib_Topology_ContinuousFunction_Compact
|
case refine'_1
α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : NormedRing β
inst✝² : StarRing β
inst✝¹ : CompactSpace α
inst✝ : CstarRing β
f : C(α, β)
x : α
⊢ ‖(star f * f) x‖ ≤ ‖f‖ ^ 2
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compacts
#align_import topology.continuous_function.compact from "leanprover-community/mathlib"@"d3af0609f6db8691dffdc3e1fb7feb7da72698f2"
/-!
# Continuous functions on a compact space
Continuous functions `C(α, β)` from a compact space `α` to a metric space `β`
are automatically bounded, and so acquire various structures inherited from `α →ᵇ β`.
This file transfers these structures, and restates some lemmas
characterising these structures.
If you need a lemma which is proved about `α →ᵇ β` but not for `C(α, β)` when `α` is compact,
you should restate it here. You can also use
`ContinuousMap.equivBoundedOfCompact` to move functions back and forth.
-/
noncomputable section
open Topology Classical NNReal BoundedContinuousFunction BigOperators
open Set Filter Metric
open BoundedContinuousFunction
namespace ContinuousMap
variable {α β E : Type*} [TopologicalSpace α] [CompactSpace α] [MetricSpace β]
[NormedAddCommGroup E]
section
variable (α β)
/-- When `α` is compact, the bounded continuous maps `α →ᵇ β` are
equivalent to `C(α, β)`.
-/
@[simps (config := .asFn)]
def equivBoundedOfCompact : C(α, β) ≃ (α →ᵇ β) :=
⟨mkOfCompact, BoundedContinuousFunction.toContinuousMap, fun f => by
ext
rfl, fun f => by
ext
rfl⟩
#align continuous_map.equiv_bounded_of_compact ContinuousMap.equivBoundedOfCompact
theorem uniformInducing_equivBoundedOfCompact : UniformInducing (equivBoundedOfCompact α β) :=
UniformInducing.mk'
(by
simp only [hasBasis_compactConvergenceUniformity.mem_iff, uniformity_basis_dist_le.mem_iff]
exact fun s =>
⟨fun ⟨⟨a, b⟩, ⟨_, ⟨ε, hε, hb⟩⟩, hs⟩ =>
⟨{ p | ∀ x, (p.1 x, p.2 x) ∈ b }, ⟨ε, hε, fun _ h x => hb ((dist_le hε.le).mp h x)⟩,
fun f g h => hs fun x _ => h x⟩,
fun ⟨_, ⟨ε, hε, ht⟩, hs⟩ =>
⟨⟨Set.univ, { p | dist p.1 p.2 ≤ ε }⟩, ⟨isCompact_univ, ⟨ε, hε, fun _ h => h⟩⟩,
fun ⟨f, g⟩ h => hs _ _ (ht ((dist_le hε.le).mpr fun x => h x (mem_univ x)))⟩⟩)
#align continuous_map.uniform_inducing_equiv_bounded_of_compact ContinuousMap.uniformInducing_equivBoundedOfCompact
theorem uniformEmbedding_equivBoundedOfCompact : UniformEmbedding (equivBoundedOfCompact α β) :=
{ uniformInducing_equivBoundedOfCompact α β with inj := (equivBoundedOfCompact α β).injective }
#align continuous_map.uniform_embedding_equiv_bounded_of_compact ContinuousMap.uniformEmbedding_equivBoundedOfCompact
/-- When `α` is compact, the bounded continuous maps `α →ᵇ 𝕜` are
additively equivalent to `C(α, 𝕜)`.
-/
-- porting note: the following `simps` received a "maximum recursion depth" error
-- @[simps! (config := .asFn) apply symm_apply]
def addEquivBoundedOfCompact [AddMonoid β] [LipschitzAdd β] : C(α, β) ≃+ (α →ᵇ β) :=
({ toContinuousMapAddHom α β, (equivBoundedOfCompact α β).symm with } : (α →ᵇ β) ≃+ C(α, β)).symm
#align continuous_map.add_equiv_bounded_of_compact ContinuousMap.addEquivBoundedOfCompact
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_symm_apply [AddMonoid β] [LipschitzAdd β] :
⇑((addEquivBoundedOfCompact α β).symm) = toContinuousMapAddHom α β :=
rfl
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_apply [AddMonoid β] [LipschitzAdd β] :
⇑(addEquivBoundedOfCompact α β) = mkOfCompact :=
rfl
instance metricSpace : MetricSpace C(α, β) :=
(uniformEmbedding_equivBoundedOfCompact α β).comapMetricSpace _
#align continuous_map.metric_space ContinuousMap.metricSpace
/-- When `α` is compact, and `β` is a metric space, the bounded continuous maps `α →ᵇ β` are
isometric to `C(α, β)`.
-/
@[simps! (config := .asFn) toEquiv apply symm_apply]
def isometryEquivBoundedOfCompact : C(α, β) ≃ᵢ (α →ᵇ β) where
isometry_toFun _ _ := rfl
toEquiv := equivBoundedOfCompact α β
#align continuous_map.isometry_equiv_bounded_of_compact ContinuousMap.isometryEquivBoundedOfCompact
end
@[simp]
theorem _root_.BoundedContinuousFunction.dist_mkOfCompact (f g : C(α, β)) :
dist (mkOfCompact f) (mkOfCompact g) = dist f g :=
rfl
#align bounded_continuous_function.dist_mk_of_compact BoundedContinuousFunction.dist_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.dist_toContinuousMap (f g : α →ᵇ β) :
dist f.toContinuousMap g.toContinuousMap = dist f g :=
rfl
#align bounded_continuous_function.dist_to_continuous_map BoundedContinuousFunction.dist_toContinuousMap
open BoundedContinuousFunction
section
variable {f g : C(α, β)} {C : ℝ}
/-- The pointwise distance is controlled by the distance between functions, by definition. -/
theorem dist_apply_le_dist (x : α) : dist (f x) (g x) ≤ dist f g := by
simp only [← dist_mkOfCompact, dist_coe_le_dist, ← mkOfCompact_apply]
#align continuous_map.dist_apply_le_dist ContinuousMap.dist_apply_le_dist
/-- The distance between two functions is controlled by the supremum of the pointwise distances. -/
theorem dist_le (C0 : (0 : ℝ) ≤ C) : dist f g ≤ C ↔ ∀ x : α, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le C0, mkOfCompact_apply]
#align continuous_map.dist_le ContinuousMap.dist_le
theorem dist_le_iff_of_nonempty [Nonempty α] : dist f g ≤ C ↔ ∀ x, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le_iff_of_nonempty,
mkOfCompact_apply]
#align continuous_map.dist_le_iff_of_nonempty ContinuousMap.dist_le_iff_of_nonempty
theorem dist_lt_iff_of_nonempty [Nonempty α] : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
simp only [← dist_mkOfCompact, dist_lt_iff_of_nonempty_compact, mkOfCompact_apply]
#align continuous_map.dist_lt_iff_of_nonempty ContinuousMap.dist_lt_iff_of_nonempty
theorem dist_lt_of_nonempty [Nonempty α] (w : ∀ x : α, dist (f x) (g x) < C) : dist f g < C :=
dist_lt_iff_of_nonempty.2 w
#align continuous_map.dist_lt_of_nonempty ContinuousMap.dist_lt_of_nonempty
theorem dist_lt_iff (C0 : (0 : ℝ) < C) : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
rw [← dist_mkOfCompact, dist_lt_iff_of_compact C0]
simp only [mkOfCompact_apply]
#align continuous_map.dist_lt_iff ContinuousMap.dist_lt_iff
end
instance [CompleteSpace β] : CompleteSpace C(α, β) :=
(isometryEquivBoundedOfCompact α β).completeSpace
/-- See also `ContinuousMap.continuous_eval'`. -/
@[continuity]
theorem continuous_eval : Continuous fun p : C(α, β) × α => p.1 p.2 :=
continuous_eval.comp ((isometryEquivBoundedOfCompact α β).continuous.prod_map continuous_id)
#align continuous_map.continuous_eval ContinuousMap.continuous_eval
-- TODO at some point we will need lemmas characterising this norm!
-- At the moment the only way to reason about it is to transfer `f : C(α,E)` back to `α →ᵇ E`.
instance : Norm C(α, E) where norm x := dist x 0
@[simp]
theorem _root_.BoundedContinuousFunction.norm_mkOfCompact (f : C(α, E)) : ‖mkOfCompact f‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_mk_of_compact BoundedContinuousFunction.norm_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.norm_toContinuousMap_eq (f : α →ᵇ E) :
‖f.toContinuousMap‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_to_continuous_map_eq BoundedContinuousFunction.norm_toContinuousMap_eq
open BoundedContinuousFunction
instance : NormedAddCommGroup C(α, E) :=
{ ContinuousMap.metricSpace _ _,
ContinuousMap.instAddCommGroupContinuousMap with
dist_eq := fun x y => by
rw [← norm_mkOfCompact, ← dist_mkOfCompact, dist_eq_norm, mkOfCompact_sub]
dist := dist
norm := norm }
instance [Nonempty α] [One E] [NormOneClass E] : NormOneClass C(α, E) where
norm_one := by simp only [← norm_mkOfCompact, mkOfCompact_one, norm_one]
section
variable (f : C(α, E))
-- The corresponding lemmas for `BoundedContinuousFunction` are stated with `{f}`,
-- and so can not be used in dot notation.
theorem norm_coe_le_norm (x : α) : ‖f x‖ ≤ ‖f‖ :=
(mkOfCompact f).norm_coe_le_norm x
#align continuous_map.norm_coe_le_norm ContinuousMap.norm_coe_le_norm
/-- Distance between the images of any two points is at most twice the norm of the function. -/
theorem dist_le_two_norm (x y : α) : dist (f x) (f y) ≤ 2 * ‖f‖ :=
(mkOfCompact f).dist_le_two_norm x y
#align continuous_map.dist_le_two_norm ContinuousMap.dist_le_two_norm
/-- The norm of a function is controlled by the supremum of the pointwise norms. -/
theorem norm_le {C : ℝ} (C0 : (0 : ℝ) ≤ C) : ‖f‖ ≤ C ↔ ∀ x : α, ‖f x‖ ≤ C :=
@BoundedContinuousFunction.norm_le _ _ _ _ (mkOfCompact f) _ C0
#align continuous_map.norm_le ContinuousMap.norm_le
theorem norm_le_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ ≤ M ↔ ∀ x, ‖f x‖ ≤ M :=
@BoundedContinuousFunction.norm_le_of_nonempty _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_le_of_nonempty ContinuousMap.norm_le_of_nonempty
theorem norm_lt_iff {M : ℝ} (M0 : 0 < M) : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_compact _ _ _ _ _ (mkOfCompact f) _ M0
#align continuous_map.norm_lt_iff ContinuousMap.norm_lt_iff
theorem nnnorm_lt_iff {M : ℝ≥0} (M0 : 0 < M) : ‖f‖₊ < M ↔ ∀ x : α, ‖f x‖₊ < M :=
f.norm_lt_iff M0
#align continuous_map.nnnorm_lt_iff ContinuousMap.nnnorm_lt_iff
theorem norm_lt_iff_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_nonempty_compact _ _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_lt_iff_of_nonempty ContinuousMap.norm_lt_iff_of_nonempty
theorem nnnorm_lt_iff_of_nonempty [Nonempty α] {M : ℝ≥0} : ‖f‖₊ < M ↔ ∀ x, ‖f x‖₊ < M :=
f.norm_lt_iff_of_nonempty
#align continuous_map.nnnorm_lt_iff_of_nonempty ContinuousMap.nnnorm_lt_iff_of_nonempty
theorem apply_le_norm (f : C(α, ℝ)) (x : α) : f x ≤ ‖f‖ :=
le_trans (le_abs.mpr (Or.inl (le_refl (f x)))) (f.norm_coe_le_norm x)
#align continuous_map.apply_le_norm ContinuousMap.apply_le_norm
theorem neg_norm_le_apply (f : C(α, ℝ)) (x : α) : -‖f‖ ≤ f x :=
le_trans (neg_le_neg (f.norm_coe_le_norm x)) (neg_le.mp (neg_le_abs_self (f x)))
#align continuous_map.neg_norm_le_apply ContinuousMap.neg_norm_le_apply
theorem norm_eq_iSup_norm : ‖f‖ = ⨆ x : α, ‖f x‖ :=
(mkOfCompact f).norm_eq_iSup_norm
#align continuous_map.norm_eq_supr_norm ContinuousMap.norm_eq_iSup_norm
theorem norm_restrict_mono_set {X : Type*} [TopologicalSpace X] (f : C(X, E))
{K L : TopologicalSpace.Compacts X} (hKL : K ≤ L) : ‖f.restrict K‖ ≤ ‖f.restrict L‖ :=
(norm_le _ (norm_nonneg _)).mpr fun x => norm_coe_le_norm (f.restrict L) <| Set.inclusion hKL x
#align continuous_map.norm_restrict_mono_set ContinuousMap.norm_restrict_mono_set
end
section
variable {R : Type*} [NormedRing R]
instance : NormedRing C(α, R) :=
{ (inferInstance : NormedAddCommGroup C(α, R)), ContinuousMap.instRingContinuousMap with
norm_mul := fun f g => norm_mul_le (mkOfCompact f) (mkOfCompact g) }
end
section
variable {𝕜 : Type*} [NormedField 𝕜] [NormedSpace 𝕜 E]
instance normedSpace : NormedSpace 𝕜 C(α, E) where
norm_smul_le c f := (norm_smul_le c (mkOfCompact f) : _)
#align continuous_map.normed_space ContinuousMap.normedSpace
section
variable (α 𝕜 E)
/-- When `α` is compact and `𝕜` is a normed field,
the `𝕜`-algebra of bounded continuous maps `α →ᵇ β` is
`𝕜`-linearly isometric to `C(α, β)`.
-/
def linearIsometryBoundedOfCompact : C(α, E) ≃ₗᵢ[𝕜] α →ᵇ E :=
{ addEquivBoundedOfCompact α E with
map_smul' := fun c f => by
ext
norm_cast
norm_map' := fun f => rfl }
#align continuous_map.linear_isometry_bounded_of_compact ContinuousMap.linearIsometryBoundedOfCompact
variable {α E}
-- to match `BoundedContinuousFunction.evalClm`
/-- The evaluation at a point, as a continuous linear map from `C(α, 𝕜)` to `𝕜`. -/
def evalClm (x : α) : C(α, E) →L[𝕜] E :=
(BoundedContinuousFunction.evalClm 𝕜 x).comp
(linearIsometryBoundedOfCompact α E 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_map.eval_clm ContinuousMap.evalClm
end
-- this lemma and the next are the analogues of those autogenerated by `@[simps]` for
-- `equivBoundedOfCompact`, `addEquivBoundedOfCompact`
@[simp]
theorem linearIsometryBoundedOfCompact_symm_apply (f : α →ᵇ E) :
(linearIsometryBoundedOfCompact α E 𝕜).symm f = f.toContinuousMap :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_symm_apply ContinuousMap.linearIsometryBoundedOfCompact_symm_apply
@[simp]
theorem linearIsometryBoundedOfCompact_apply_apply (f : C(α, E)) (a : α) :
(linearIsometryBoundedOfCompact α E 𝕜 f) a = f a :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_apply_apply ContinuousMap.linearIsometryBoundedOfCompact_apply_apply
@[simp]
theorem linearIsometryBoundedOfCompact_toIsometryEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toIsometryEquiv = isometryEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_isometry_equiv ContinuousMap.linearIsometryBoundedOfCompact_toIsometryEquiv
@[simp] -- porting note: adjusted LHS because `simpNF` complained it simplified.
theorem linearIsometryBoundedOfCompact_toAddEquiv :
((linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv : C(α, E) ≃+ (α →ᵇ E)) =
addEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_add_equiv ContinuousMap.linearIsometryBoundedOfCompact_toAddEquiv
@[simp]
theorem linearIsometryBoundedOfCompact_of_compact_toEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv.toEquiv = equivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_of_compact_to_equiv ContinuousMap.linearIsometryBoundedOfCompact_of_compact_toEquiv
end
section
variable {𝕜 : Type*} {γ : Type*} [NormedField 𝕜] [NormedRing γ] [NormedAlgebra 𝕜 γ]
instance : NormedAlgebra 𝕜 C(α, γ) :=
{ ContinuousMap.normedSpace, ContinuousMap.algebra with }
end
end ContinuousMap
namespace ContinuousMap
section UniformContinuity
variable {α β : Type*}
variable [MetricSpace α] [CompactSpace α] [MetricSpace β]
/-!
We now set up some declarations making it convenient to use uniform continuity.
-/
theorem uniform_continuity (f : C(α, β)) (ε : ℝ) (h : 0 < ε) :
∃ δ > 0, ∀ {x y}, dist x y < δ → dist (f x) (f y) < ε :=
Metric.uniformContinuous_iff.mp (CompactSpace.uniformContinuous_of_continuous f.continuous) ε h
#align continuous_map.uniform_continuity ContinuousMap.uniform_continuity
-- This definition allows us to separate the choice of some `δ`,
-- and the corresponding use of `dist a b < δ → dist (f a) (f b) < ε`,
-- even across different declarations.
/-- An arbitrarily chosen modulus of uniform continuity for a given function `f` and `ε > 0`. -/
def modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) : ℝ :=
Classical.choose (uniform_continuity f ε h)
#align continuous_map.modulus ContinuousMap.modulus
theorem modulus_pos (f : C(α, β)) {ε : ℝ} {h : 0 < ε} : 0 < f.modulus ε h :=
(Classical.choose_spec (uniform_continuity f ε h)).1
#align continuous_map.modulus_pos ContinuousMap.modulus_pos
theorem dist_lt_of_dist_lt_modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) {a b : α}
(w : dist a b < f.modulus ε h) : dist (f a) (f b) < ε :=
(Classical.choose_spec (uniform_continuity f ε h)).2 w
#align continuous_map.dist_lt_of_dist_lt_modulus ContinuousMap.dist_lt_of_dist_lt_modulus
end UniformContinuity
end ContinuousMap
section CompLeft
variable (X : Type*) {𝕜 β γ : Type*} [TopologicalSpace X] [CompactSpace X]
[NontriviallyNormedField 𝕜]
variable [NormedAddCommGroup β] [NormedSpace 𝕜 β] [NormedAddCommGroup γ] [NormedSpace 𝕜 γ]
open ContinuousMap
/-- Postcomposition of continuous functions into a normed module by a continuous linear map is a
continuous linear map.
Transferred version of `ContinuousLinearMap.compLeftContinuousBounded`,
upgraded version of `ContinuousLinearMap.compLeftContinuous`,
similar to `LinearMap.compLeft`. -/
protected def ContinuousLinearMap.compLeftContinuousCompact (g : β →L[𝕜] γ) :
C(X, β) →L[𝕜] C(X, γ) :=
(linearIsometryBoundedOfCompact X γ 𝕜).symm.toLinearIsometry.toContinuousLinearMap.comp <|
(g.compLeftContinuousBounded X).comp <|
(linearIsometryBoundedOfCompact X β 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_linear_map.comp_left_continuous_compact ContinuousLinearMap.compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.toLinear_compLeftContinuousCompact (g : β →L[𝕜] γ) :
(g.compLeftContinuousCompact X : C(X, β) →ₗ[𝕜] C(X, γ)) = g.compLeftContinuous 𝕜 X := by
ext f
rfl
#align continuous_linear_map.to_linear_comp_left_continuous_compact ContinuousLinearMap.toLinear_compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.compLeftContinuousCompact_apply (g : β →L[𝕜] γ) (f : C(X, β)) (x : X) :
g.compLeftContinuousCompact X f x = g (f x) :=
rfl
#align continuous_linear_map.comp_left_continuous_compact_apply ContinuousLinearMap.compLeftContinuousCompact_apply
end CompLeft
namespace ContinuousMap
/-!
We now setup variations on `compRight* f`, where `f : C(X, Y)`
(that is, precomposition by a continuous map),
as a morphism `C(Y, T) → C(X, T)`, respecting various types of structure.
In particular:
* `compRightContinuousMap`, the bundled continuous map (for this we need `X Y` compact).
* `compRightHomeomorph`, when we precompose by a homeomorphism.
* `compRightAlgHom`, when `T = R` is a topological ring.
-/
section CompRight
/-- Precomposition by a continuous map is itself a continuous map between spaces of continuous maps.
-/
def compRightContinuousMap {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y)) : C(C(Y, T), C(X, T)) where
toFun g := g.comp f
continuous_toFun := by
refine' Metric.continuous_iff.mpr _
intro g ε ε_pos
refine' ⟨ε, ε_pos, fun g' h => _⟩
rw [ContinuousMap.dist_lt_iff ε_pos] at h ⊢
exact fun x => h (f x)
#align continuous_map.comp_right_continuous_map ContinuousMap.compRightContinuousMap
@[simp]
theorem compRightContinuousMap_apply {X Y : Type*} (T : Type*) [TopologicalSpace X]
[CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y))
(g : C(Y, T)) : (compRightContinuousMap T f) g = g.comp f :=
rfl
#align continuous_map.comp_right_continuous_map_apply ContinuousMap.compRightContinuousMap_apply
/-- Precomposition by a homeomorphism is itself a homeomorphism between spaces of continuous maps.
-/
def compRightHomeomorph {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : X ≃ₜ Y) : C(Y, T) ≃ₜ C(X, T) where
toFun := compRightContinuousMap T f.toContinuousMap
invFun := compRightContinuousMap T f.symm.toContinuousMap
left_inv g := ext fun _ => congr_arg g (f.apply_symm_apply _)
right_inv g := ext fun _ => congr_arg g (f.symm_apply_apply _)
#align continuous_map.comp_right_homeomorph ContinuousMap.compRightHomeomorph
theorem compRightAlgHom_continuous {X Y : Type*} (R A : Type*) [TopologicalSpace X]
[CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [CommSemiring R] [Semiring A]
[MetricSpace A] [TopologicalSemiring A] [Algebra R A] (f : C(X, Y)) :
Continuous (compRightAlgHom R A f) :=
map_continuous (compRightContinuousMap A f)
#align continuous_map.comp_right_alg_hom_continuous ContinuousMap.compRightAlgHom_continuous
end CompRight
section LocalNormalConvergence
/-! ### Local normal convergence
A sum of continuous functions (on a locally compact space) is "locally normally convergent" if the
sum of its sup-norms on any compact subset is summable. This implies convergence in the topology
of `C(X, E)` (i.e. locally uniform convergence). -/
open TopologicalSpace
variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
variable {E : Type*} [NormedAddCommGroup E] [CompleteSpace E]
theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F := by
refine' (ContinuousMap.exists_tendsto_compactOpen_iff_forall _).2 fun K hK => _
lift K to Compacts X using hK
have A : ∀ s : Finset ι, restrict (↑K) (∑ i in s, F i) = ∑ i in s, restrict K (F i) := by
intro s
ext1 x
simp
-- This used to be the end of the proof before leanprover/lean4#2644
erw [restrict_apply, restrict_apply, restrict_apply, restrict_apply]
simp? says simp only [coe_sum, Finset.sum_apply]
congr!
simpa only [HasSum, A] using (hF K).of_norm
#align continuous_map.summable_of_locally_summable_norm ContinuousMap.summable_of_locally_summable_norm
end LocalNormalConvergence
/-!
### Star structures
In this section, if `β` is a normed ⋆-group, then so is the space of
continuous functions from `α` to `β`, by using the star operation pointwise.
Furthermore, if `α` is compact and `β` is a C⋆-ring, then `C(α, β)` is a C⋆-ring. -/
section NormedSpace
variable {α : Type*} {β : Type*}
variable [TopologicalSpace α] [NormedAddCommGroup β] [StarAddMonoid β] [NormedStarGroup β]
theorem _root_.BoundedContinuousFunction.mkOfCompact_star [CompactSpace α] (f : C(α, β)) :
mkOfCompact (star f) = star (mkOfCompact f) :=
rfl
#align bounded_continuous_function.mk_of_compact_star BoundedContinuousFunction.mkOfCompact_star
instance [CompactSpace α] : NormedStarGroup C(α, β) where
norm_star f := by
rw [← BoundedContinuousFunction.norm_mkOfCompact, BoundedContinuousFunction.mkOfCompact_star,
norm_star, BoundedContinuousFunction.norm_mkOfCompact]
end NormedSpace
section CstarRing
variable {α : Type*} {β : Type*}
variable [TopologicalSpace α] [NormedRing β] [StarRing β]
instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f} := by
refine' le_antisymm _ _
· rw [← sq, ContinuousMap.norm_le _ (sq_nonneg _)]
intro x
|
simp only [ContinuousMap.coe_mul, coe_star, Pi.mul_apply, Pi.star_apply,
CstarRing.norm_star_mul_self, ← sq]
|
instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f} := by
refine' le_antisymm _ _
· rw [← sq, ContinuousMap.norm_le _ (sq_nonneg _)]
intro x
|
Mathlib.Topology.ContinuousFunction.Compact.541_0.Mig2jTVnn2FLKEB
|
instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f}
|
Mathlib_Topology_ContinuousFunction_Compact
|
case refine'_1
α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : NormedRing β
inst✝² : StarRing β
inst✝¹ : CompactSpace α
inst✝ : CstarRing β
f : C(α, β)
x : α
⊢ ‖f x‖ ^ 2 ≤ ‖f‖ ^ 2
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compacts
#align_import topology.continuous_function.compact from "leanprover-community/mathlib"@"d3af0609f6db8691dffdc3e1fb7feb7da72698f2"
/-!
# Continuous functions on a compact space
Continuous functions `C(α, β)` from a compact space `α` to a metric space `β`
are automatically bounded, and so acquire various structures inherited from `α →ᵇ β`.
This file transfers these structures, and restates some lemmas
characterising these structures.
If you need a lemma which is proved about `α →ᵇ β` but not for `C(α, β)` when `α` is compact,
you should restate it here. You can also use
`ContinuousMap.equivBoundedOfCompact` to move functions back and forth.
-/
noncomputable section
open Topology Classical NNReal BoundedContinuousFunction BigOperators
open Set Filter Metric
open BoundedContinuousFunction
namespace ContinuousMap
variable {α β E : Type*} [TopologicalSpace α] [CompactSpace α] [MetricSpace β]
[NormedAddCommGroup E]
section
variable (α β)
/-- When `α` is compact, the bounded continuous maps `α →ᵇ β` are
equivalent to `C(α, β)`.
-/
@[simps (config := .asFn)]
def equivBoundedOfCompact : C(α, β) ≃ (α →ᵇ β) :=
⟨mkOfCompact, BoundedContinuousFunction.toContinuousMap, fun f => by
ext
rfl, fun f => by
ext
rfl⟩
#align continuous_map.equiv_bounded_of_compact ContinuousMap.equivBoundedOfCompact
theorem uniformInducing_equivBoundedOfCompact : UniformInducing (equivBoundedOfCompact α β) :=
UniformInducing.mk'
(by
simp only [hasBasis_compactConvergenceUniformity.mem_iff, uniformity_basis_dist_le.mem_iff]
exact fun s =>
⟨fun ⟨⟨a, b⟩, ⟨_, ⟨ε, hε, hb⟩⟩, hs⟩ =>
⟨{ p | ∀ x, (p.1 x, p.2 x) ∈ b }, ⟨ε, hε, fun _ h x => hb ((dist_le hε.le).mp h x)⟩,
fun f g h => hs fun x _ => h x⟩,
fun ⟨_, ⟨ε, hε, ht⟩, hs⟩ =>
⟨⟨Set.univ, { p | dist p.1 p.2 ≤ ε }⟩, ⟨isCompact_univ, ⟨ε, hε, fun _ h => h⟩⟩,
fun ⟨f, g⟩ h => hs _ _ (ht ((dist_le hε.le).mpr fun x => h x (mem_univ x)))⟩⟩)
#align continuous_map.uniform_inducing_equiv_bounded_of_compact ContinuousMap.uniformInducing_equivBoundedOfCompact
theorem uniformEmbedding_equivBoundedOfCompact : UniformEmbedding (equivBoundedOfCompact α β) :=
{ uniformInducing_equivBoundedOfCompact α β with inj := (equivBoundedOfCompact α β).injective }
#align continuous_map.uniform_embedding_equiv_bounded_of_compact ContinuousMap.uniformEmbedding_equivBoundedOfCompact
/-- When `α` is compact, the bounded continuous maps `α →ᵇ 𝕜` are
additively equivalent to `C(α, 𝕜)`.
-/
-- porting note: the following `simps` received a "maximum recursion depth" error
-- @[simps! (config := .asFn) apply symm_apply]
def addEquivBoundedOfCompact [AddMonoid β] [LipschitzAdd β] : C(α, β) ≃+ (α →ᵇ β) :=
({ toContinuousMapAddHom α β, (equivBoundedOfCompact α β).symm with } : (α →ᵇ β) ≃+ C(α, β)).symm
#align continuous_map.add_equiv_bounded_of_compact ContinuousMap.addEquivBoundedOfCompact
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_symm_apply [AddMonoid β] [LipschitzAdd β] :
⇑((addEquivBoundedOfCompact α β).symm) = toContinuousMapAddHom α β :=
rfl
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_apply [AddMonoid β] [LipschitzAdd β] :
⇑(addEquivBoundedOfCompact α β) = mkOfCompact :=
rfl
instance metricSpace : MetricSpace C(α, β) :=
(uniformEmbedding_equivBoundedOfCompact α β).comapMetricSpace _
#align continuous_map.metric_space ContinuousMap.metricSpace
/-- When `α` is compact, and `β` is a metric space, the bounded continuous maps `α →ᵇ β` are
isometric to `C(α, β)`.
-/
@[simps! (config := .asFn) toEquiv apply symm_apply]
def isometryEquivBoundedOfCompact : C(α, β) ≃ᵢ (α →ᵇ β) where
isometry_toFun _ _ := rfl
toEquiv := equivBoundedOfCompact α β
#align continuous_map.isometry_equiv_bounded_of_compact ContinuousMap.isometryEquivBoundedOfCompact
end
@[simp]
theorem _root_.BoundedContinuousFunction.dist_mkOfCompact (f g : C(α, β)) :
dist (mkOfCompact f) (mkOfCompact g) = dist f g :=
rfl
#align bounded_continuous_function.dist_mk_of_compact BoundedContinuousFunction.dist_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.dist_toContinuousMap (f g : α →ᵇ β) :
dist f.toContinuousMap g.toContinuousMap = dist f g :=
rfl
#align bounded_continuous_function.dist_to_continuous_map BoundedContinuousFunction.dist_toContinuousMap
open BoundedContinuousFunction
section
variable {f g : C(α, β)} {C : ℝ}
/-- The pointwise distance is controlled by the distance between functions, by definition. -/
theorem dist_apply_le_dist (x : α) : dist (f x) (g x) ≤ dist f g := by
simp only [← dist_mkOfCompact, dist_coe_le_dist, ← mkOfCompact_apply]
#align continuous_map.dist_apply_le_dist ContinuousMap.dist_apply_le_dist
/-- The distance between two functions is controlled by the supremum of the pointwise distances. -/
theorem dist_le (C0 : (0 : ℝ) ≤ C) : dist f g ≤ C ↔ ∀ x : α, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le C0, mkOfCompact_apply]
#align continuous_map.dist_le ContinuousMap.dist_le
theorem dist_le_iff_of_nonempty [Nonempty α] : dist f g ≤ C ↔ ∀ x, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le_iff_of_nonempty,
mkOfCompact_apply]
#align continuous_map.dist_le_iff_of_nonempty ContinuousMap.dist_le_iff_of_nonempty
theorem dist_lt_iff_of_nonempty [Nonempty α] : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
simp only [← dist_mkOfCompact, dist_lt_iff_of_nonempty_compact, mkOfCompact_apply]
#align continuous_map.dist_lt_iff_of_nonempty ContinuousMap.dist_lt_iff_of_nonempty
theorem dist_lt_of_nonempty [Nonempty α] (w : ∀ x : α, dist (f x) (g x) < C) : dist f g < C :=
dist_lt_iff_of_nonempty.2 w
#align continuous_map.dist_lt_of_nonempty ContinuousMap.dist_lt_of_nonempty
theorem dist_lt_iff (C0 : (0 : ℝ) < C) : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
rw [← dist_mkOfCompact, dist_lt_iff_of_compact C0]
simp only [mkOfCompact_apply]
#align continuous_map.dist_lt_iff ContinuousMap.dist_lt_iff
end
instance [CompleteSpace β] : CompleteSpace C(α, β) :=
(isometryEquivBoundedOfCompact α β).completeSpace
/-- See also `ContinuousMap.continuous_eval'`. -/
@[continuity]
theorem continuous_eval : Continuous fun p : C(α, β) × α => p.1 p.2 :=
continuous_eval.comp ((isometryEquivBoundedOfCompact α β).continuous.prod_map continuous_id)
#align continuous_map.continuous_eval ContinuousMap.continuous_eval
-- TODO at some point we will need lemmas characterising this norm!
-- At the moment the only way to reason about it is to transfer `f : C(α,E)` back to `α →ᵇ E`.
instance : Norm C(α, E) where norm x := dist x 0
@[simp]
theorem _root_.BoundedContinuousFunction.norm_mkOfCompact (f : C(α, E)) : ‖mkOfCompact f‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_mk_of_compact BoundedContinuousFunction.norm_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.norm_toContinuousMap_eq (f : α →ᵇ E) :
‖f.toContinuousMap‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_to_continuous_map_eq BoundedContinuousFunction.norm_toContinuousMap_eq
open BoundedContinuousFunction
instance : NormedAddCommGroup C(α, E) :=
{ ContinuousMap.metricSpace _ _,
ContinuousMap.instAddCommGroupContinuousMap with
dist_eq := fun x y => by
rw [← norm_mkOfCompact, ← dist_mkOfCompact, dist_eq_norm, mkOfCompact_sub]
dist := dist
norm := norm }
instance [Nonempty α] [One E] [NormOneClass E] : NormOneClass C(α, E) where
norm_one := by simp only [← norm_mkOfCompact, mkOfCompact_one, norm_one]
section
variable (f : C(α, E))
-- The corresponding lemmas for `BoundedContinuousFunction` are stated with `{f}`,
-- and so can not be used in dot notation.
theorem norm_coe_le_norm (x : α) : ‖f x‖ ≤ ‖f‖ :=
(mkOfCompact f).norm_coe_le_norm x
#align continuous_map.norm_coe_le_norm ContinuousMap.norm_coe_le_norm
/-- Distance between the images of any two points is at most twice the norm of the function. -/
theorem dist_le_two_norm (x y : α) : dist (f x) (f y) ≤ 2 * ‖f‖ :=
(mkOfCompact f).dist_le_two_norm x y
#align continuous_map.dist_le_two_norm ContinuousMap.dist_le_two_norm
/-- The norm of a function is controlled by the supremum of the pointwise norms. -/
theorem norm_le {C : ℝ} (C0 : (0 : ℝ) ≤ C) : ‖f‖ ≤ C ↔ ∀ x : α, ‖f x‖ ≤ C :=
@BoundedContinuousFunction.norm_le _ _ _ _ (mkOfCompact f) _ C0
#align continuous_map.norm_le ContinuousMap.norm_le
theorem norm_le_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ ≤ M ↔ ∀ x, ‖f x‖ ≤ M :=
@BoundedContinuousFunction.norm_le_of_nonempty _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_le_of_nonempty ContinuousMap.norm_le_of_nonempty
theorem norm_lt_iff {M : ℝ} (M0 : 0 < M) : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_compact _ _ _ _ _ (mkOfCompact f) _ M0
#align continuous_map.norm_lt_iff ContinuousMap.norm_lt_iff
theorem nnnorm_lt_iff {M : ℝ≥0} (M0 : 0 < M) : ‖f‖₊ < M ↔ ∀ x : α, ‖f x‖₊ < M :=
f.norm_lt_iff M0
#align continuous_map.nnnorm_lt_iff ContinuousMap.nnnorm_lt_iff
theorem norm_lt_iff_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_nonempty_compact _ _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_lt_iff_of_nonempty ContinuousMap.norm_lt_iff_of_nonempty
theorem nnnorm_lt_iff_of_nonempty [Nonempty α] {M : ℝ≥0} : ‖f‖₊ < M ↔ ∀ x, ‖f x‖₊ < M :=
f.norm_lt_iff_of_nonempty
#align continuous_map.nnnorm_lt_iff_of_nonempty ContinuousMap.nnnorm_lt_iff_of_nonempty
theorem apply_le_norm (f : C(α, ℝ)) (x : α) : f x ≤ ‖f‖ :=
le_trans (le_abs.mpr (Or.inl (le_refl (f x)))) (f.norm_coe_le_norm x)
#align continuous_map.apply_le_norm ContinuousMap.apply_le_norm
theorem neg_norm_le_apply (f : C(α, ℝ)) (x : α) : -‖f‖ ≤ f x :=
le_trans (neg_le_neg (f.norm_coe_le_norm x)) (neg_le.mp (neg_le_abs_self (f x)))
#align continuous_map.neg_norm_le_apply ContinuousMap.neg_norm_le_apply
theorem norm_eq_iSup_norm : ‖f‖ = ⨆ x : α, ‖f x‖ :=
(mkOfCompact f).norm_eq_iSup_norm
#align continuous_map.norm_eq_supr_norm ContinuousMap.norm_eq_iSup_norm
theorem norm_restrict_mono_set {X : Type*} [TopologicalSpace X] (f : C(X, E))
{K L : TopologicalSpace.Compacts X} (hKL : K ≤ L) : ‖f.restrict K‖ ≤ ‖f.restrict L‖ :=
(norm_le _ (norm_nonneg _)).mpr fun x => norm_coe_le_norm (f.restrict L) <| Set.inclusion hKL x
#align continuous_map.norm_restrict_mono_set ContinuousMap.norm_restrict_mono_set
end
section
variable {R : Type*} [NormedRing R]
instance : NormedRing C(α, R) :=
{ (inferInstance : NormedAddCommGroup C(α, R)), ContinuousMap.instRingContinuousMap with
norm_mul := fun f g => norm_mul_le (mkOfCompact f) (mkOfCompact g) }
end
section
variable {𝕜 : Type*} [NormedField 𝕜] [NormedSpace 𝕜 E]
instance normedSpace : NormedSpace 𝕜 C(α, E) where
norm_smul_le c f := (norm_smul_le c (mkOfCompact f) : _)
#align continuous_map.normed_space ContinuousMap.normedSpace
section
variable (α 𝕜 E)
/-- When `α` is compact and `𝕜` is a normed field,
the `𝕜`-algebra of bounded continuous maps `α →ᵇ β` is
`𝕜`-linearly isometric to `C(α, β)`.
-/
def linearIsometryBoundedOfCompact : C(α, E) ≃ₗᵢ[𝕜] α →ᵇ E :=
{ addEquivBoundedOfCompact α E with
map_smul' := fun c f => by
ext
norm_cast
norm_map' := fun f => rfl }
#align continuous_map.linear_isometry_bounded_of_compact ContinuousMap.linearIsometryBoundedOfCompact
variable {α E}
-- to match `BoundedContinuousFunction.evalClm`
/-- The evaluation at a point, as a continuous linear map from `C(α, 𝕜)` to `𝕜`. -/
def evalClm (x : α) : C(α, E) →L[𝕜] E :=
(BoundedContinuousFunction.evalClm 𝕜 x).comp
(linearIsometryBoundedOfCompact α E 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_map.eval_clm ContinuousMap.evalClm
end
-- this lemma and the next are the analogues of those autogenerated by `@[simps]` for
-- `equivBoundedOfCompact`, `addEquivBoundedOfCompact`
@[simp]
theorem linearIsometryBoundedOfCompact_symm_apply (f : α →ᵇ E) :
(linearIsometryBoundedOfCompact α E 𝕜).symm f = f.toContinuousMap :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_symm_apply ContinuousMap.linearIsometryBoundedOfCompact_symm_apply
@[simp]
theorem linearIsometryBoundedOfCompact_apply_apply (f : C(α, E)) (a : α) :
(linearIsometryBoundedOfCompact α E 𝕜 f) a = f a :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_apply_apply ContinuousMap.linearIsometryBoundedOfCompact_apply_apply
@[simp]
theorem linearIsometryBoundedOfCompact_toIsometryEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toIsometryEquiv = isometryEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_isometry_equiv ContinuousMap.linearIsometryBoundedOfCompact_toIsometryEquiv
@[simp] -- porting note: adjusted LHS because `simpNF` complained it simplified.
theorem linearIsometryBoundedOfCompact_toAddEquiv :
((linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv : C(α, E) ≃+ (α →ᵇ E)) =
addEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_add_equiv ContinuousMap.linearIsometryBoundedOfCompact_toAddEquiv
@[simp]
theorem linearIsometryBoundedOfCompact_of_compact_toEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv.toEquiv = equivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_of_compact_to_equiv ContinuousMap.linearIsometryBoundedOfCompact_of_compact_toEquiv
end
section
variable {𝕜 : Type*} {γ : Type*} [NormedField 𝕜] [NormedRing γ] [NormedAlgebra 𝕜 γ]
instance : NormedAlgebra 𝕜 C(α, γ) :=
{ ContinuousMap.normedSpace, ContinuousMap.algebra with }
end
end ContinuousMap
namespace ContinuousMap
section UniformContinuity
variable {α β : Type*}
variable [MetricSpace α] [CompactSpace α] [MetricSpace β]
/-!
We now set up some declarations making it convenient to use uniform continuity.
-/
theorem uniform_continuity (f : C(α, β)) (ε : ℝ) (h : 0 < ε) :
∃ δ > 0, ∀ {x y}, dist x y < δ → dist (f x) (f y) < ε :=
Metric.uniformContinuous_iff.mp (CompactSpace.uniformContinuous_of_continuous f.continuous) ε h
#align continuous_map.uniform_continuity ContinuousMap.uniform_continuity
-- This definition allows us to separate the choice of some `δ`,
-- and the corresponding use of `dist a b < δ → dist (f a) (f b) < ε`,
-- even across different declarations.
/-- An arbitrarily chosen modulus of uniform continuity for a given function `f` and `ε > 0`. -/
def modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) : ℝ :=
Classical.choose (uniform_continuity f ε h)
#align continuous_map.modulus ContinuousMap.modulus
theorem modulus_pos (f : C(α, β)) {ε : ℝ} {h : 0 < ε} : 0 < f.modulus ε h :=
(Classical.choose_spec (uniform_continuity f ε h)).1
#align continuous_map.modulus_pos ContinuousMap.modulus_pos
theorem dist_lt_of_dist_lt_modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) {a b : α}
(w : dist a b < f.modulus ε h) : dist (f a) (f b) < ε :=
(Classical.choose_spec (uniform_continuity f ε h)).2 w
#align continuous_map.dist_lt_of_dist_lt_modulus ContinuousMap.dist_lt_of_dist_lt_modulus
end UniformContinuity
end ContinuousMap
section CompLeft
variable (X : Type*) {𝕜 β γ : Type*} [TopologicalSpace X] [CompactSpace X]
[NontriviallyNormedField 𝕜]
variable [NormedAddCommGroup β] [NormedSpace 𝕜 β] [NormedAddCommGroup γ] [NormedSpace 𝕜 γ]
open ContinuousMap
/-- Postcomposition of continuous functions into a normed module by a continuous linear map is a
continuous linear map.
Transferred version of `ContinuousLinearMap.compLeftContinuousBounded`,
upgraded version of `ContinuousLinearMap.compLeftContinuous`,
similar to `LinearMap.compLeft`. -/
protected def ContinuousLinearMap.compLeftContinuousCompact (g : β →L[𝕜] γ) :
C(X, β) →L[𝕜] C(X, γ) :=
(linearIsometryBoundedOfCompact X γ 𝕜).symm.toLinearIsometry.toContinuousLinearMap.comp <|
(g.compLeftContinuousBounded X).comp <|
(linearIsometryBoundedOfCompact X β 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_linear_map.comp_left_continuous_compact ContinuousLinearMap.compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.toLinear_compLeftContinuousCompact (g : β →L[𝕜] γ) :
(g.compLeftContinuousCompact X : C(X, β) →ₗ[𝕜] C(X, γ)) = g.compLeftContinuous 𝕜 X := by
ext f
rfl
#align continuous_linear_map.to_linear_comp_left_continuous_compact ContinuousLinearMap.toLinear_compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.compLeftContinuousCompact_apply (g : β →L[𝕜] γ) (f : C(X, β)) (x : X) :
g.compLeftContinuousCompact X f x = g (f x) :=
rfl
#align continuous_linear_map.comp_left_continuous_compact_apply ContinuousLinearMap.compLeftContinuousCompact_apply
end CompLeft
namespace ContinuousMap
/-!
We now setup variations on `compRight* f`, where `f : C(X, Y)`
(that is, precomposition by a continuous map),
as a morphism `C(Y, T) → C(X, T)`, respecting various types of structure.
In particular:
* `compRightContinuousMap`, the bundled continuous map (for this we need `X Y` compact).
* `compRightHomeomorph`, when we precompose by a homeomorphism.
* `compRightAlgHom`, when `T = R` is a topological ring.
-/
section CompRight
/-- Precomposition by a continuous map is itself a continuous map between spaces of continuous maps.
-/
def compRightContinuousMap {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y)) : C(C(Y, T), C(X, T)) where
toFun g := g.comp f
continuous_toFun := by
refine' Metric.continuous_iff.mpr _
intro g ε ε_pos
refine' ⟨ε, ε_pos, fun g' h => _⟩
rw [ContinuousMap.dist_lt_iff ε_pos] at h ⊢
exact fun x => h (f x)
#align continuous_map.comp_right_continuous_map ContinuousMap.compRightContinuousMap
@[simp]
theorem compRightContinuousMap_apply {X Y : Type*} (T : Type*) [TopologicalSpace X]
[CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y))
(g : C(Y, T)) : (compRightContinuousMap T f) g = g.comp f :=
rfl
#align continuous_map.comp_right_continuous_map_apply ContinuousMap.compRightContinuousMap_apply
/-- Precomposition by a homeomorphism is itself a homeomorphism between spaces of continuous maps.
-/
def compRightHomeomorph {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : X ≃ₜ Y) : C(Y, T) ≃ₜ C(X, T) where
toFun := compRightContinuousMap T f.toContinuousMap
invFun := compRightContinuousMap T f.symm.toContinuousMap
left_inv g := ext fun _ => congr_arg g (f.apply_symm_apply _)
right_inv g := ext fun _ => congr_arg g (f.symm_apply_apply _)
#align continuous_map.comp_right_homeomorph ContinuousMap.compRightHomeomorph
theorem compRightAlgHom_continuous {X Y : Type*} (R A : Type*) [TopologicalSpace X]
[CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [CommSemiring R] [Semiring A]
[MetricSpace A] [TopologicalSemiring A] [Algebra R A] (f : C(X, Y)) :
Continuous (compRightAlgHom R A f) :=
map_continuous (compRightContinuousMap A f)
#align continuous_map.comp_right_alg_hom_continuous ContinuousMap.compRightAlgHom_continuous
end CompRight
section LocalNormalConvergence
/-! ### Local normal convergence
A sum of continuous functions (on a locally compact space) is "locally normally convergent" if the
sum of its sup-norms on any compact subset is summable. This implies convergence in the topology
of `C(X, E)` (i.e. locally uniform convergence). -/
open TopologicalSpace
variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
variable {E : Type*} [NormedAddCommGroup E] [CompleteSpace E]
theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F := by
refine' (ContinuousMap.exists_tendsto_compactOpen_iff_forall _).2 fun K hK => _
lift K to Compacts X using hK
have A : ∀ s : Finset ι, restrict (↑K) (∑ i in s, F i) = ∑ i in s, restrict K (F i) := by
intro s
ext1 x
simp
-- This used to be the end of the proof before leanprover/lean4#2644
erw [restrict_apply, restrict_apply, restrict_apply, restrict_apply]
simp? says simp only [coe_sum, Finset.sum_apply]
congr!
simpa only [HasSum, A] using (hF K).of_norm
#align continuous_map.summable_of_locally_summable_norm ContinuousMap.summable_of_locally_summable_norm
end LocalNormalConvergence
/-!
### Star structures
In this section, if `β` is a normed ⋆-group, then so is the space of
continuous functions from `α` to `β`, by using the star operation pointwise.
Furthermore, if `α` is compact and `β` is a C⋆-ring, then `C(α, β)` is a C⋆-ring. -/
section NormedSpace
variable {α : Type*} {β : Type*}
variable [TopologicalSpace α] [NormedAddCommGroup β] [StarAddMonoid β] [NormedStarGroup β]
theorem _root_.BoundedContinuousFunction.mkOfCompact_star [CompactSpace α] (f : C(α, β)) :
mkOfCompact (star f) = star (mkOfCompact f) :=
rfl
#align bounded_continuous_function.mk_of_compact_star BoundedContinuousFunction.mkOfCompact_star
instance [CompactSpace α] : NormedStarGroup C(α, β) where
norm_star f := by
rw [← BoundedContinuousFunction.norm_mkOfCompact, BoundedContinuousFunction.mkOfCompact_star,
norm_star, BoundedContinuousFunction.norm_mkOfCompact]
end NormedSpace
section CstarRing
variable {α : Type*} {β : Type*}
variable [TopologicalSpace α] [NormedRing β] [StarRing β]
instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f} := by
refine' le_antisymm _ _
· rw [← sq, ContinuousMap.norm_le _ (sq_nonneg _)]
intro x
simp only [ContinuousMap.coe_mul, coe_star, Pi.mul_apply, Pi.star_apply,
CstarRing.norm_star_mul_self, ← sq]
|
refine' sq_le_sq' _ _
|
instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f} := by
refine' le_antisymm _ _
· rw [← sq, ContinuousMap.norm_le _ (sq_nonneg _)]
intro x
simp only [ContinuousMap.coe_mul, coe_star, Pi.mul_apply, Pi.star_apply,
CstarRing.norm_star_mul_self, ← sq]
|
Mathlib.Topology.ContinuousFunction.Compact.541_0.Mig2jTVnn2FLKEB
|
instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f}
|
Mathlib_Topology_ContinuousFunction_Compact
|
case refine'_1.refine'_1
α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : NormedRing β
inst✝² : StarRing β
inst✝¹ : CompactSpace α
inst✝ : CstarRing β
f : C(α, β)
x : α
⊢ -‖f‖ ≤ ‖f x‖
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compacts
#align_import topology.continuous_function.compact from "leanprover-community/mathlib"@"d3af0609f6db8691dffdc3e1fb7feb7da72698f2"
/-!
# Continuous functions on a compact space
Continuous functions `C(α, β)` from a compact space `α` to a metric space `β`
are automatically bounded, and so acquire various structures inherited from `α →ᵇ β`.
This file transfers these structures, and restates some lemmas
characterising these structures.
If you need a lemma which is proved about `α →ᵇ β` but not for `C(α, β)` when `α` is compact,
you should restate it here. You can also use
`ContinuousMap.equivBoundedOfCompact` to move functions back and forth.
-/
noncomputable section
open Topology Classical NNReal BoundedContinuousFunction BigOperators
open Set Filter Metric
open BoundedContinuousFunction
namespace ContinuousMap
variable {α β E : Type*} [TopologicalSpace α] [CompactSpace α] [MetricSpace β]
[NormedAddCommGroup E]
section
variable (α β)
/-- When `α` is compact, the bounded continuous maps `α →ᵇ β` are
equivalent to `C(α, β)`.
-/
@[simps (config := .asFn)]
def equivBoundedOfCompact : C(α, β) ≃ (α →ᵇ β) :=
⟨mkOfCompact, BoundedContinuousFunction.toContinuousMap, fun f => by
ext
rfl, fun f => by
ext
rfl⟩
#align continuous_map.equiv_bounded_of_compact ContinuousMap.equivBoundedOfCompact
theorem uniformInducing_equivBoundedOfCompact : UniformInducing (equivBoundedOfCompact α β) :=
UniformInducing.mk'
(by
simp only [hasBasis_compactConvergenceUniformity.mem_iff, uniformity_basis_dist_le.mem_iff]
exact fun s =>
⟨fun ⟨⟨a, b⟩, ⟨_, ⟨ε, hε, hb⟩⟩, hs⟩ =>
⟨{ p | ∀ x, (p.1 x, p.2 x) ∈ b }, ⟨ε, hε, fun _ h x => hb ((dist_le hε.le).mp h x)⟩,
fun f g h => hs fun x _ => h x⟩,
fun ⟨_, ⟨ε, hε, ht⟩, hs⟩ =>
⟨⟨Set.univ, { p | dist p.1 p.2 ≤ ε }⟩, ⟨isCompact_univ, ⟨ε, hε, fun _ h => h⟩⟩,
fun ⟨f, g⟩ h => hs _ _ (ht ((dist_le hε.le).mpr fun x => h x (mem_univ x)))⟩⟩)
#align continuous_map.uniform_inducing_equiv_bounded_of_compact ContinuousMap.uniformInducing_equivBoundedOfCompact
theorem uniformEmbedding_equivBoundedOfCompact : UniformEmbedding (equivBoundedOfCompact α β) :=
{ uniformInducing_equivBoundedOfCompact α β with inj := (equivBoundedOfCompact α β).injective }
#align continuous_map.uniform_embedding_equiv_bounded_of_compact ContinuousMap.uniformEmbedding_equivBoundedOfCompact
/-- When `α` is compact, the bounded continuous maps `α →ᵇ 𝕜` are
additively equivalent to `C(α, 𝕜)`.
-/
-- porting note: the following `simps` received a "maximum recursion depth" error
-- @[simps! (config := .asFn) apply symm_apply]
def addEquivBoundedOfCompact [AddMonoid β] [LipschitzAdd β] : C(α, β) ≃+ (α →ᵇ β) :=
({ toContinuousMapAddHom α β, (equivBoundedOfCompact α β).symm with } : (α →ᵇ β) ≃+ C(α, β)).symm
#align continuous_map.add_equiv_bounded_of_compact ContinuousMap.addEquivBoundedOfCompact
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_symm_apply [AddMonoid β] [LipschitzAdd β] :
⇑((addEquivBoundedOfCompact α β).symm) = toContinuousMapAddHom α β :=
rfl
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_apply [AddMonoid β] [LipschitzAdd β] :
⇑(addEquivBoundedOfCompact α β) = mkOfCompact :=
rfl
instance metricSpace : MetricSpace C(α, β) :=
(uniformEmbedding_equivBoundedOfCompact α β).comapMetricSpace _
#align continuous_map.metric_space ContinuousMap.metricSpace
/-- When `α` is compact, and `β` is a metric space, the bounded continuous maps `α →ᵇ β` are
isometric to `C(α, β)`.
-/
@[simps! (config := .asFn) toEquiv apply symm_apply]
def isometryEquivBoundedOfCompact : C(α, β) ≃ᵢ (α →ᵇ β) where
isometry_toFun _ _ := rfl
toEquiv := equivBoundedOfCompact α β
#align continuous_map.isometry_equiv_bounded_of_compact ContinuousMap.isometryEquivBoundedOfCompact
end
@[simp]
theorem _root_.BoundedContinuousFunction.dist_mkOfCompact (f g : C(α, β)) :
dist (mkOfCompact f) (mkOfCompact g) = dist f g :=
rfl
#align bounded_continuous_function.dist_mk_of_compact BoundedContinuousFunction.dist_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.dist_toContinuousMap (f g : α →ᵇ β) :
dist f.toContinuousMap g.toContinuousMap = dist f g :=
rfl
#align bounded_continuous_function.dist_to_continuous_map BoundedContinuousFunction.dist_toContinuousMap
open BoundedContinuousFunction
section
variable {f g : C(α, β)} {C : ℝ}
/-- The pointwise distance is controlled by the distance between functions, by definition. -/
theorem dist_apply_le_dist (x : α) : dist (f x) (g x) ≤ dist f g := by
simp only [← dist_mkOfCompact, dist_coe_le_dist, ← mkOfCompact_apply]
#align continuous_map.dist_apply_le_dist ContinuousMap.dist_apply_le_dist
/-- The distance between two functions is controlled by the supremum of the pointwise distances. -/
theorem dist_le (C0 : (0 : ℝ) ≤ C) : dist f g ≤ C ↔ ∀ x : α, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le C0, mkOfCompact_apply]
#align continuous_map.dist_le ContinuousMap.dist_le
theorem dist_le_iff_of_nonempty [Nonempty α] : dist f g ≤ C ↔ ∀ x, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le_iff_of_nonempty,
mkOfCompact_apply]
#align continuous_map.dist_le_iff_of_nonempty ContinuousMap.dist_le_iff_of_nonempty
theorem dist_lt_iff_of_nonempty [Nonempty α] : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
simp only [← dist_mkOfCompact, dist_lt_iff_of_nonempty_compact, mkOfCompact_apply]
#align continuous_map.dist_lt_iff_of_nonempty ContinuousMap.dist_lt_iff_of_nonempty
theorem dist_lt_of_nonempty [Nonempty α] (w : ∀ x : α, dist (f x) (g x) < C) : dist f g < C :=
dist_lt_iff_of_nonempty.2 w
#align continuous_map.dist_lt_of_nonempty ContinuousMap.dist_lt_of_nonempty
theorem dist_lt_iff (C0 : (0 : ℝ) < C) : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
rw [← dist_mkOfCompact, dist_lt_iff_of_compact C0]
simp only [mkOfCompact_apply]
#align continuous_map.dist_lt_iff ContinuousMap.dist_lt_iff
end
instance [CompleteSpace β] : CompleteSpace C(α, β) :=
(isometryEquivBoundedOfCompact α β).completeSpace
/-- See also `ContinuousMap.continuous_eval'`. -/
@[continuity]
theorem continuous_eval : Continuous fun p : C(α, β) × α => p.1 p.2 :=
continuous_eval.comp ((isometryEquivBoundedOfCompact α β).continuous.prod_map continuous_id)
#align continuous_map.continuous_eval ContinuousMap.continuous_eval
-- TODO at some point we will need lemmas characterising this norm!
-- At the moment the only way to reason about it is to transfer `f : C(α,E)` back to `α →ᵇ E`.
instance : Norm C(α, E) where norm x := dist x 0
@[simp]
theorem _root_.BoundedContinuousFunction.norm_mkOfCompact (f : C(α, E)) : ‖mkOfCompact f‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_mk_of_compact BoundedContinuousFunction.norm_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.norm_toContinuousMap_eq (f : α →ᵇ E) :
‖f.toContinuousMap‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_to_continuous_map_eq BoundedContinuousFunction.norm_toContinuousMap_eq
open BoundedContinuousFunction
instance : NormedAddCommGroup C(α, E) :=
{ ContinuousMap.metricSpace _ _,
ContinuousMap.instAddCommGroupContinuousMap with
dist_eq := fun x y => by
rw [← norm_mkOfCompact, ← dist_mkOfCompact, dist_eq_norm, mkOfCompact_sub]
dist := dist
norm := norm }
instance [Nonempty α] [One E] [NormOneClass E] : NormOneClass C(α, E) where
norm_one := by simp only [← norm_mkOfCompact, mkOfCompact_one, norm_one]
section
variable (f : C(α, E))
-- The corresponding lemmas for `BoundedContinuousFunction` are stated with `{f}`,
-- and so can not be used in dot notation.
theorem norm_coe_le_norm (x : α) : ‖f x‖ ≤ ‖f‖ :=
(mkOfCompact f).norm_coe_le_norm x
#align continuous_map.norm_coe_le_norm ContinuousMap.norm_coe_le_norm
/-- Distance between the images of any two points is at most twice the norm of the function. -/
theorem dist_le_two_norm (x y : α) : dist (f x) (f y) ≤ 2 * ‖f‖ :=
(mkOfCompact f).dist_le_two_norm x y
#align continuous_map.dist_le_two_norm ContinuousMap.dist_le_two_norm
/-- The norm of a function is controlled by the supremum of the pointwise norms. -/
theorem norm_le {C : ℝ} (C0 : (0 : ℝ) ≤ C) : ‖f‖ ≤ C ↔ ∀ x : α, ‖f x‖ ≤ C :=
@BoundedContinuousFunction.norm_le _ _ _ _ (mkOfCompact f) _ C0
#align continuous_map.norm_le ContinuousMap.norm_le
theorem norm_le_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ ≤ M ↔ ∀ x, ‖f x‖ ≤ M :=
@BoundedContinuousFunction.norm_le_of_nonempty _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_le_of_nonempty ContinuousMap.norm_le_of_nonempty
theorem norm_lt_iff {M : ℝ} (M0 : 0 < M) : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_compact _ _ _ _ _ (mkOfCompact f) _ M0
#align continuous_map.norm_lt_iff ContinuousMap.norm_lt_iff
theorem nnnorm_lt_iff {M : ℝ≥0} (M0 : 0 < M) : ‖f‖₊ < M ↔ ∀ x : α, ‖f x‖₊ < M :=
f.norm_lt_iff M0
#align continuous_map.nnnorm_lt_iff ContinuousMap.nnnorm_lt_iff
theorem norm_lt_iff_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_nonempty_compact _ _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_lt_iff_of_nonempty ContinuousMap.norm_lt_iff_of_nonempty
theorem nnnorm_lt_iff_of_nonempty [Nonempty α] {M : ℝ≥0} : ‖f‖₊ < M ↔ ∀ x, ‖f x‖₊ < M :=
f.norm_lt_iff_of_nonempty
#align continuous_map.nnnorm_lt_iff_of_nonempty ContinuousMap.nnnorm_lt_iff_of_nonempty
theorem apply_le_norm (f : C(α, ℝ)) (x : α) : f x ≤ ‖f‖ :=
le_trans (le_abs.mpr (Or.inl (le_refl (f x)))) (f.norm_coe_le_norm x)
#align continuous_map.apply_le_norm ContinuousMap.apply_le_norm
theorem neg_norm_le_apply (f : C(α, ℝ)) (x : α) : -‖f‖ ≤ f x :=
le_trans (neg_le_neg (f.norm_coe_le_norm x)) (neg_le.mp (neg_le_abs_self (f x)))
#align continuous_map.neg_norm_le_apply ContinuousMap.neg_norm_le_apply
theorem norm_eq_iSup_norm : ‖f‖ = ⨆ x : α, ‖f x‖ :=
(mkOfCompact f).norm_eq_iSup_norm
#align continuous_map.norm_eq_supr_norm ContinuousMap.norm_eq_iSup_norm
theorem norm_restrict_mono_set {X : Type*} [TopologicalSpace X] (f : C(X, E))
{K L : TopologicalSpace.Compacts X} (hKL : K ≤ L) : ‖f.restrict K‖ ≤ ‖f.restrict L‖ :=
(norm_le _ (norm_nonneg _)).mpr fun x => norm_coe_le_norm (f.restrict L) <| Set.inclusion hKL x
#align continuous_map.norm_restrict_mono_set ContinuousMap.norm_restrict_mono_set
end
section
variable {R : Type*} [NormedRing R]
instance : NormedRing C(α, R) :=
{ (inferInstance : NormedAddCommGroup C(α, R)), ContinuousMap.instRingContinuousMap with
norm_mul := fun f g => norm_mul_le (mkOfCompact f) (mkOfCompact g) }
end
section
variable {𝕜 : Type*} [NormedField 𝕜] [NormedSpace 𝕜 E]
instance normedSpace : NormedSpace 𝕜 C(α, E) where
norm_smul_le c f := (norm_smul_le c (mkOfCompact f) : _)
#align continuous_map.normed_space ContinuousMap.normedSpace
section
variable (α 𝕜 E)
/-- When `α` is compact and `𝕜` is a normed field,
the `𝕜`-algebra of bounded continuous maps `α →ᵇ β` is
`𝕜`-linearly isometric to `C(α, β)`.
-/
def linearIsometryBoundedOfCompact : C(α, E) ≃ₗᵢ[𝕜] α →ᵇ E :=
{ addEquivBoundedOfCompact α E with
map_smul' := fun c f => by
ext
norm_cast
norm_map' := fun f => rfl }
#align continuous_map.linear_isometry_bounded_of_compact ContinuousMap.linearIsometryBoundedOfCompact
variable {α E}
-- to match `BoundedContinuousFunction.evalClm`
/-- The evaluation at a point, as a continuous linear map from `C(α, 𝕜)` to `𝕜`. -/
def evalClm (x : α) : C(α, E) →L[𝕜] E :=
(BoundedContinuousFunction.evalClm 𝕜 x).comp
(linearIsometryBoundedOfCompact α E 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_map.eval_clm ContinuousMap.evalClm
end
-- this lemma and the next are the analogues of those autogenerated by `@[simps]` for
-- `equivBoundedOfCompact`, `addEquivBoundedOfCompact`
@[simp]
theorem linearIsometryBoundedOfCompact_symm_apply (f : α →ᵇ E) :
(linearIsometryBoundedOfCompact α E 𝕜).symm f = f.toContinuousMap :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_symm_apply ContinuousMap.linearIsometryBoundedOfCompact_symm_apply
@[simp]
theorem linearIsometryBoundedOfCompact_apply_apply (f : C(α, E)) (a : α) :
(linearIsometryBoundedOfCompact α E 𝕜 f) a = f a :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_apply_apply ContinuousMap.linearIsometryBoundedOfCompact_apply_apply
@[simp]
theorem linearIsometryBoundedOfCompact_toIsometryEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toIsometryEquiv = isometryEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_isometry_equiv ContinuousMap.linearIsometryBoundedOfCompact_toIsometryEquiv
@[simp] -- porting note: adjusted LHS because `simpNF` complained it simplified.
theorem linearIsometryBoundedOfCompact_toAddEquiv :
((linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv : C(α, E) ≃+ (α →ᵇ E)) =
addEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_add_equiv ContinuousMap.linearIsometryBoundedOfCompact_toAddEquiv
@[simp]
theorem linearIsometryBoundedOfCompact_of_compact_toEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv.toEquiv = equivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_of_compact_to_equiv ContinuousMap.linearIsometryBoundedOfCompact_of_compact_toEquiv
end
section
variable {𝕜 : Type*} {γ : Type*} [NormedField 𝕜] [NormedRing γ] [NormedAlgebra 𝕜 γ]
instance : NormedAlgebra 𝕜 C(α, γ) :=
{ ContinuousMap.normedSpace, ContinuousMap.algebra with }
end
end ContinuousMap
namespace ContinuousMap
section UniformContinuity
variable {α β : Type*}
variable [MetricSpace α] [CompactSpace α] [MetricSpace β]
/-!
We now set up some declarations making it convenient to use uniform continuity.
-/
theorem uniform_continuity (f : C(α, β)) (ε : ℝ) (h : 0 < ε) :
∃ δ > 0, ∀ {x y}, dist x y < δ → dist (f x) (f y) < ε :=
Metric.uniformContinuous_iff.mp (CompactSpace.uniformContinuous_of_continuous f.continuous) ε h
#align continuous_map.uniform_continuity ContinuousMap.uniform_continuity
-- This definition allows us to separate the choice of some `δ`,
-- and the corresponding use of `dist a b < δ → dist (f a) (f b) < ε`,
-- even across different declarations.
/-- An arbitrarily chosen modulus of uniform continuity for a given function `f` and `ε > 0`. -/
def modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) : ℝ :=
Classical.choose (uniform_continuity f ε h)
#align continuous_map.modulus ContinuousMap.modulus
theorem modulus_pos (f : C(α, β)) {ε : ℝ} {h : 0 < ε} : 0 < f.modulus ε h :=
(Classical.choose_spec (uniform_continuity f ε h)).1
#align continuous_map.modulus_pos ContinuousMap.modulus_pos
theorem dist_lt_of_dist_lt_modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) {a b : α}
(w : dist a b < f.modulus ε h) : dist (f a) (f b) < ε :=
(Classical.choose_spec (uniform_continuity f ε h)).2 w
#align continuous_map.dist_lt_of_dist_lt_modulus ContinuousMap.dist_lt_of_dist_lt_modulus
end UniformContinuity
end ContinuousMap
section CompLeft
variable (X : Type*) {𝕜 β γ : Type*} [TopologicalSpace X] [CompactSpace X]
[NontriviallyNormedField 𝕜]
variable [NormedAddCommGroup β] [NormedSpace 𝕜 β] [NormedAddCommGroup γ] [NormedSpace 𝕜 γ]
open ContinuousMap
/-- Postcomposition of continuous functions into a normed module by a continuous linear map is a
continuous linear map.
Transferred version of `ContinuousLinearMap.compLeftContinuousBounded`,
upgraded version of `ContinuousLinearMap.compLeftContinuous`,
similar to `LinearMap.compLeft`. -/
protected def ContinuousLinearMap.compLeftContinuousCompact (g : β →L[𝕜] γ) :
C(X, β) →L[𝕜] C(X, γ) :=
(linearIsometryBoundedOfCompact X γ 𝕜).symm.toLinearIsometry.toContinuousLinearMap.comp <|
(g.compLeftContinuousBounded X).comp <|
(linearIsometryBoundedOfCompact X β 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_linear_map.comp_left_continuous_compact ContinuousLinearMap.compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.toLinear_compLeftContinuousCompact (g : β →L[𝕜] γ) :
(g.compLeftContinuousCompact X : C(X, β) →ₗ[𝕜] C(X, γ)) = g.compLeftContinuous 𝕜 X := by
ext f
rfl
#align continuous_linear_map.to_linear_comp_left_continuous_compact ContinuousLinearMap.toLinear_compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.compLeftContinuousCompact_apply (g : β →L[𝕜] γ) (f : C(X, β)) (x : X) :
g.compLeftContinuousCompact X f x = g (f x) :=
rfl
#align continuous_linear_map.comp_left_continuous_compact_apply ContinuousLinearMap.compLeftContinuousCompact_apply
end CompLeft
namespace ContinuousMap
/-!
We now setup variations on `compRight* f`, where `f : C(X, Y)`
(that is, precomposition by a continuous map),
as a morphism `C(Y, T) → C(X, T)`, respecting various types of structure.
In particular:
* `compRightContinuousMap`, the bundled continuous map (for this we need `X Y` compact).
* `compRightHomeomorph`, when we precompose by a homeomorphism.
* `compRightAlgHom`, when `T = R` is a topological ring.
-/
section CompRight
/-- Precomposition by a continuous map is itself a continuous map between spaces of continuous maps.
-/
def compRightContinuousMap {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y)) : C(C(Y, T), C(X, T)) where
toFun g := g.comp f
continuous_toFun := by
refine' Metric.continuous_iff.mpr _
intro g ε ε_pos
refine' ⟨ε, ε_pos, fun g' h => _⟩
rw [ContinuousMap.dist_lt_iff ε_pos] at h ⊢
exact fun x => h (f x)
#align continuous_map.comp_right_continuous_map ContinuousMap.compRightContinuousMap
@[simp]
theorem compRightContinuousMap_apply {X Y : Type*} (T : Type*) [TopologicalSpace X]
[CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y))
(g : C(Y, T)) : (compRightContinuousMap T f) g = g.comp f :=
rfl
#align continuous_map.comp_right_continuous_map_apply ContinuousMap.compRightContinuousMap_apply
/-- Precomposition by a homeomorphism is itself a homeomorphism between spaces of continuous maps.
-/
def compRightHomeomorph {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : X ≃ₜ Y) : C(Y, T) ≃ₜ C(X, T) where
toFun := compRightContinuousMap T f.toContinuousMap
invFun := compRightContinuousMap T f.symm.toContinuousMap
left_inv g := ext fun _ => congr_arg g (f.apply_symm_apply _)
right_inv g := ext fun _ => congr_arg g (f.symm_apply_apply _)
#align continuous_map.comp_right_homeomorph ContinuousMap.compRightHomeomorph
theorem compRightAlgHom_continuous {X Y : Type*} (R A : Type*) [TopologicalSpace X]
[CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [CommSemiring R] [Semiring A]
[MetricSpace A] [TopologicalSemiring A] [Algebra R A] (f : C(X, Y)) :
Continuous (compRightAlgHom R A f) :=
map_continuous (compRightContinuousMap A f)
#align continuous_map.comp_right_alg_hom_continuous ContinuousMap.compRightAlgHom_continuous
end CompRight
section LocalNormalConvergence
/-! ### Local normal convergence
A sum of continuous functions (on a locally compact space) is "locally normally convergent" if the
sum of its sup-norms on any compact subset is summable. This implies convergence in the topology
of `C(X, E)` (i.e. locally uniform convergence). -/
open TopologicalSpace
variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
variable {E : Type*} [NormedAddCommGroup E] [CompleteSpace E]
theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F := by
refine' (ContinuousMap.exists_tendsto_compactOpen_iff_forall _).2 fun K hK => _
lift K to Compacts X using hK
have A : ∀ s : Finset ι, restrict (↑K) (∑ i in s, F i) = ∑ i in s, restrict K (F i) := by
intro s
ext1 x
simp
-- This used to be the end of the proof before leanprover/lean4#2644
erw [restrict_apply, restrict_apply, restrict_apply, restrict_apply]
simp? says simp only [coe_sum, Finset.sum_apply]
congr!
simpa only [HasSum, A] using (hF K).of_norm
#align continuous_map.summable_of_locally_summable_norm ContinuousMap.summable_of_locally_summable_norm
end LocalNormalConvergence
/-!
### Star structures
In this section, if `β` is a normed ⋆-group, then so is the space of
continuous functions from `α` to `β`, by using the star operation pointwise.
Furthermore, if `α` is compact and `β` is a C⋆-ring, then `C(α, β)` is a C⋆-ring. -/
section NormedSpace
variable {α : Type*} {β : Type*}
variable [TopologicalSpace α] [NormedAddCommGroup β] [StarAddMonoid β] [NormedStarGroup β]
theorem _root_.BoundedContinuousFunction.mkOfCompact_star [CompactSpace α] (f : C(α, β)) :
mkOfCompact (star f) = star (mkOfCompact f) :=
rfl
#align bounded_continuous_function.mk_of_compact_star BoundedContinuousFunction.mkOfCompact_star
instance [CompactSpace α] : NormedStarGroup C(α, β) where
norm_star f := by
rw [← BoundedContinuousFunction.norm_mkOfCompact, BoundedContinuousFunction.mkOfCompact_star,
norm_star, BoundedContinuousFunction.norm_mkOfCompact]
end NormedSpace
section CstarRing
variable {α : Type*} {β : Type*}
variable [TopologicalSpace α] [NormedRing β] [StarRing β]
instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f} := by
refine' le_antisymm _ _
· rw [← sq, ContinuousMap.norm_le _ (sq_nonneg _)]
intro x
simp only [ContinuousMap.coe_mul, coe_star, Pi.mul_apply, Pi.star_apply,
CstarRing.norm_star_mul_self, ← sq]
refine' sq_le_sq' _ _
·
|
linarith [norm_nonneg (f x), norm_nonneg f]
|
instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f} := by
refine' le_antisymm _ _
· rw [← sq, ContinuousMap.norm_le _ (sq_nonneg _)]
intro x
simp only [ContinuousMap.coe_mul, coe_star, Pi.mul_apply, Pi.star_apply,
CstarRing.norm_star_mul_self, ← sq]
refine' sq_le_sq' _ _
·
|
Mathlib.Topology.ContinuousFunction.Compact.541_0.Mig2jTVnn2FLKEB
|
instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f}
|
Mathlib_Topology_ContinuousFunction_Compact
|
case refine'_1.refine'_2
α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : NormedRing β
inst✝² : StarRing β
inst✝¹ : CompactSpace α
inst✝ : CstarRing β
f : C(α, β)
x : α
⊢ ‖f x‖ ≤ ‖f‖
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compacts
#align_import topology.continuous_function.compact from "leanprover-community/mathlib"@"d3af0609f6db8691dffdc3e1fb7feb7da72698f2"
/-!
# Continuous functions on a compact space
Continuous functions `C(α, β)` from a compact space `α` to a metric space `β`
are automatically bounded, and so acquire various structures inherited from `α →ᵇ β`.
This file transfers these structures, and restates some lemmas
characterising these structures.
If you need a lemma which is proved about `α →ᵇ β` but not for `C(α, β)` when `α` is compact,
you should restate it here. You can also use
`ContinuousMap.equivBoundedOfCompact` to move functions back and forth.
-/
noncomputable section
open Topology Classical NNReal BoundedContinuousFunction BigOperators
open Set Filter Metric
open BoundedContinuousFunction
namespace ContinuousMap
variable {α β E : Type*} [TopologicalSpace α] [CompactSpace α] [MetricSpace β]
[NormedAddCommGroup E]
section
variable (α β)
/-- When `α` is compact, the bounded continuous maps `α →ᵇ β` are
equivalent to `C(α, β)`.
-/
@[simps (config := .asFn)]
def equivBoundedOfCompact : C(α, β) ≃ (α →ᵇ β) :=
⟨mkOfCompact, BoundedContinuousFunction.toContinuousMap, fun f => by
ext
rfl, fun f => by
ext
rfl⟩
#align continuous_map.equiv_bounded_of_compact ContinuousMap.equivBoundedOfCompact
theorem uniformInducing_equivBoundedOfCompact : UniformInducing (equivBoundedOfCompact α β) :=
UniformInducing.mk'
(by
simp only [hasBasis_compactConvergenceUniformity.mem_iff, uniformity_basis_dist_le.mem_iff]
exact fun s =>
⟨fun ⟨⟨a, b⟩, ⟨_, ⟨ε, hε, hb⟩⟩, hs⟩ =>
⟨{ p | ∀ x, (p.1 x, p.2 x) ∈ b }, ⟨ε, hε, fun _ h x => hb ((dist_le hε.le).mp h x)⟩,
fun f g h => hs fun x _ => h x⟩,
fun ⟨_, ⟨ε, hε, ht⟩, hs⟩ =>
⟨⟨Set.univ, { p | dist p.1 p.2 ≤ ε }⟩, ⟨isCompact_univ, ⟨ε, hε, fun _ h => h⟩⟩,
fun ⟨f, g⟩ h => hs _ _ (ht ((dist_le hε.le).mpr fun x => h x (mem_univ x)))⟩⟩)
#align continuous_map.uniform_inducing_equiv_bounded_of_compact ContinuousMap.uniformInducing_equivBoundedOfCompact
theorem uniformEmbedding_equivBoundedOfCompact : UniformEmbedding (equivBoundedOfCompact α β) :=
{ uniformInducing_equivBoundedOfCompact α β with inj := (equivBoundedOfCompact α β).injective }
#align continuous_map.uniform_embedding_equiv_bounded_of_compact ContinuousMap.uniformEmbedding_equivBoundedOfCompact
/-- When `α` is compact, the bounded continuous maps `α →ᵇ 𝕜` are
additively equivalent to `C(α, 𝕜)`.
-/
-- porting note: the following `simps` received a "maximum recursion depth" error
-- @[simps! (config := .asFn) apply symm_apply]
def addEquivBoundedOfCompact [AddMonoid β] [LipschitzAdd β] : C(α, β) ≃+ (α →ᵇ β) :=
({ toContinuousMapAddHom α β, (equivBoundedOfCompact α β).symm with } : (α →ᵇ β) ≃+ C(α, β)).symm
#align continuous_map.add_equiv_bounded_of_compact ContinuousMap.addEquivBoundedOfCompact
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_symm_apply [AddMonoid β] [LipschitzAdd β] :
⇑((addEquivBoundedOfCompact α β).symm) = toContinuousMapAddHom α β :=
rfl
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_apply [AddMonoid β] [LipschitzAdd β] :
⇑(addEquivBoundedOfCompact α β) = mkOfCompact :=
rfl
instance metricSpace : MetricSpace C(α, β) :=
(uniformEmbedding_equivBoundedOfCompact α β).comapMetricSpace _
#align continuous_map.metric_space ContinuousMap.metricSpace
/-- When `α` is compact, and `β` is a metric space, the bounded continuous maps `α →ᵇ β` are
isometric to `C(α, β)`.
-/
@[simps! (config := .asFn) toEquiv apply symm_apply]
def isometryEquivBoundedOfCompact : C(α, β) ≃ᵢ (α →ᵇ β) where
isometry_toFun _ _ := rfl
toEquiv := equivBoundedOfCompact α β
#align continuous_map.isometry_equiv_bounded_of_compact ContinuousMap.isometryEquivBoundedOfCompact
end
@[simp]
theorem _root_.BoundedContinuousFunction.dist_mkOfCompact (f g : C(α, β)) :
dist (mkOfCompact f) (mkOfCompact g) = dist f g :=
rfl
#align bounded_continuous_function.dist_mk_of_compact BoundedContinuousFunction.dist_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.dist_toContinuousMap (f g : α →ᵇ β) :
dist f.toContinuousMap g.toContinuousMap = dist f g :=
rfl
#align bounded_continuous_function.dist_to_continuous_map BoundedContinuousFunction.dist_toContinuousMap
open BoundedContinuousFunction
section
variable {f g : C(α, β)} {C : ℝ}
/-- The pointwise distance is controlled by the distance between functions, by definition. -/
theorem dist_apply_le_dist (x : α) : dist (f x) (g x) ≤ dist f g := by
simp only [← dist_mkOfCompact, dist_coe_le_dist, ← mkOfCompact_apply]
#align continuous_map.dist_apply_le_dist ContinuousMap.dist_apply_le_dist
/-- The distance between two functions is controlled by the supremum of the pointwise distances. -/
theorem dist_le (C0 : (0 : ℝ) ≤ C) : dist f g ≤ C ↔ ∀ x : α, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le C0, mkOfCompact_apply]
#align continuous_map.dist_le ContinuousMap.dist_le
theorem dist_le_iff_of_nonempty [Nonempty α] : dist f g ≤ C ↔ ∀ x, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le_iff_of_nonempty,
mkOfCompact_apply]
#align continuous_map.dist_le_iff_of_nonempty ContinuousMap.dist_le_iff_of_nonempty
theorem dist_lt_iff_of_nonempty [Nonempty α] : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
simp only [← dist_mkOfCompact, dist_lt_iff_of_nonempty_compact, mkOfCompact_apply]
#align continuous_map.dist_lt_iff_of_nonempty ContinuousMap.dist_lt_iff_of_nonempty
theorem dist_lt_of_nonempty [Nonempty α] (w : ∀ x : α, dist (f x) (g x) < C) : dist f g < C :=
dist_lt_iff_of_nonempty.2 w
#align continuous_map.dist_lt_of_nonempty ContinuousMap.dist_lt_of_nonempty
theorem dist_lt_iff (C0 : (0 : ℝ) < C) : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
rw [← dist_mkOfCompact, dist_lt_iff_of_compact C0]
simp only [mkOfCompact_apply]
#align continuous_map.dist_lt_iff ContinuousMap.dist_lt_iff
end
instance [CompleteSpace β] : CompleteSpace C(α, β) :=
(isometryEquivBoundedOfCompact α β).completeSpace
/-- See also `ContinuousMap.continuous_eval'`. -/
@[continuity]
theorem continuous_eval : Continuous fun p : C(α, β) × α => p.1 p.2 :=
continuous_eval.comp ((isometryEquivBoundedOfCompact α β).continuous.prod_map continuous_id)
#align continuous_map.continuous_eval ContinuousMap.continuous_eval
-- TODO at some point we will need lemmas characterising this norm!
-- At the moment the only way to reason about it is to transfer `f : C(α,E)` back to `α →ᵇ E`.
instance : Norm C(α, E) where norm x := dist x 0
@[simp]
theorem _root_.BoundedContinuousFunction.norm_mkOfCompact (f : C(α, E)) : ‖mkOfCompact f‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_mk_of_compact BoundedContinuousFunction.norm_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.norm_toContinuousMap_eq (f : α →ᵇ E) :
‖f.toContinuousMap‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_to_continuous_map_eq BoundedContinuousFunction.norm_toContinuousMap_eq
open BoundedContinuousFunction
instance : NormedAddCommGroup C(α, E) :=
{ ContinuousMap.metricSpace _ _,
ContinuousMap.instAddCommGroupContinuousMap with
dist_eq := fun x y => by
rw [← norm_mkOfCompact, ← dist_mkOfCompact, dist_eq_norm, mkOfCompact_sub]
dist := dist
norm := norm }
instance [Nonempty α] [One E] [NormOneClass E] : NormOneClass C(α, E) where
norm_one := by simp only [← norm_mkOfCompact, mkOfCompact_one, norm_one]
section
variable (f : C(α, E))
-- The corresponding lemmas for `BoundedContinuousFunction` are stated with `{f}`,
-- and so can not be used in dot notation.
theorem norm_coe_le_norm (x : α) : ‖f x‖ ≤ ‖f‖ :=
(mkOfCompact f).norm_coe_le_norm x
#align continuous_map.norm_coe_le_norm ContinuousMap.norm_coe_le_norm
/-- Distance between the images of any two points is at most twice the norm of the function. -/
theorem dist_le_two_norm (x y : α) : dist (f x) (f y) ≤ 2 * ‖f‖ :=
(mkOfCompact f).dist_le_two_norm x y
#align continuous_map.dist_le_two_norm ContinuousMap.dist_le_two_norm
/-- The norm of a function is controlled by the supremum of the pointwise norms. -/
theorem norm_le {C : ℝ} (C0 : (0 : ℝ) ≤ C) : ‖f‖ ≤ C ↔ ∀ x : α, ‖f x‖ ≤ C :=
@BoundedContinuousFunction.norm_le _ _ _ _ (mkOfCompact f) _ C0
#align continuous_map.norm_le ContinuousMap.norm_le
theorem norm_le_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ ≤ M ↔ ∀ x, ‖f x‖ ≤ M :=
@BoundedContinuousFunction.norm_le_of_nonempty _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_le_of_nonempty ContinuousMap.norm_le_of_nonempty
theorem norm_lt_iff {M : ℝ} (M0 : 0 < M) : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_compact _ _ _ _ _ (mkOfCompact f) _ M0
#align continuous_map.norm_lt_iff ContinuousMap.norm_lt_iff
theorem nnnorm_lt_iff {M : ℝ≥0} (M0 : 0 < M) : ‖f‖₊ < M ↔ ∀ x : α, ‖f x‖₊ < M :=
f.norm_lt_iff M0
#align continuous_map.nnnorm_lt_iff ContinuousMap.nnnorm_lt_iff
theorem norm_lt_iff_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_nonempty_compact _ _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_lt_iff_of_nonempty ContinuousMap.norm_lt_iff_of_nonempty
theorem nnnorm_lt_iff_of_nonempty [Nonempty α] {M : ℝ≥0} : ‖f‖₊ < M ↔ ∀ x, ‖f x‖₊ < M :=
f.norm_lt_iff_of_nonempty
#align continuous_map.nnnorm_lt_iff_of_nonempty ContinuousMap.nnnorm_lt_iff_of_nonempty
theorem apply_le_norm (f : C(α, ℝ)) (x : α) : f x ≤ ‖f‖ :=
le_trans (le_abs.mpr (Or.inl (le_refl (f x)))) (f.norm_coe_le_norm x)
#align continuous_map.apply_le_norm ContinuousMap.apply_le_norm
theorem neg_norm_le_apply (f : C(α, ℝ)) (x : α) : -‖f‖ ≤ f x :=
le_trans (neg_le_neg (f.norm_coe_le_norm x)) (neg_le.mp (neg_le_abs_self (f x)))
#align continuous_map.neg_norm_le_apply ContinuousMap.neg_norm_le_apply
theorem norm_eq_iSup_norm : ‖f‖ = ⨆ x : α, ‖f x‖ :=
(mkOfCompact f).norm_eq_iSup_norm
#align continuous_map.norm_eq_supr_norm ContinuousMap.norm_eq_iSup_norm
theorem norm_restrict_mono_set {X : Type*} [TopologicalSpace X] (f : C(X, E))
{K L : TopologicalSpace.Compacts X} (hKL : K ≤ L) : ‖f.restrict K‖ ≤ ‖f.restrict L‖ :=
(norm_le _ (norm_nonneg _)).mpr fun x => norm_coe_le_norm (f.restrict L) <| Set.inclusion hKL x
#align continuous_map.norm_restrict_mono_set ContinuousMap.norm_restrict_mono_set
end
section
variable {R : Type*} [NormedRing R]
instance : NormedRing C(α, R) :=
{ (inferInstance : NormedAddCommGroup C(α, R)), ContinuousMap.instRingContinuousMap with
norm_mul := fun f g => norm_mul_le (mkOfCompact f) (mkOfCompact g) }
end
section
variable {𝕜 : Type*} [NormedField 𝕜] [NormedSpace 𝕜 E]
instance normedSpace : NormedSpace 𝕜 C(α, E) where
norm_smul_le c f := (norm_smul_le c (mkOfCompact f) : _)
#align continuous_map.normed_space ContinuousMap.normedSpace
section
variable (α 𝕜 E)
/-- When `α` is compact and `𝕜` is a normed field,
the `𝕜`-algebra of bounded continuous maps `α →ᵇ β` is
`𝕜`-linearly isometric to `C(α, β)`.
-/
def linearIsometryBoundedOfCompact : C(α, E) ≃ₗᵢ[𝕜] α →ᵇ E :=
{ addEquivBoundedOfCompact α E with
map_smul' := fun c f => by
ext
norm_cast
norm_map' := fun f => rfl }
#align continuous_map.linear_isometry_bounded_of_compact ContinuousMap.linearIsometryBoundedOfCompact
variable {α E}
-- to match `BoundedContinuousFunction.evalClm`
/-- The evaluation at a point, as a continuous linear map from `C(α, 𝕜)` to `𝕜`. -/
def evalClm (x : α) : C(α, E) →L[𝕜] E :=
(BoundedContinuousFunction.evalClm 𝕜 x).comp
(linearIsometryBoundedOfCompact α E 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_map.eval_clm ContinuousMap.evalClm
end
-- this lemma and the next are the analogues of those autogenerated by `@[simps]` for
-- `equivBoundedOfCompact`, `addEquivBoundedOfCompact`
@[simp]
theorem linearIsometryBoundedOfCompact_symm_apply (f : α →ᵇ E) :
(linearIsometryBoundedOfCompact α E 𝕜).symm f = f.toContinuousMap :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_symm_apply ContinuousMap.linearIsometryBoundedOfCompact_symm_apply
@[simp]
theorem linearIsometryBoundedOfCompact_apply_apply (f : C(α, E)) (a : α) :
(linearIsometryBoundedOfCompact α E 𝕜 f) a = f a :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_apply_apply ContinuousMap.linearIsometryBoundedOfCompact_apply_apply
@[simp]
theorem linearIsometryBoundedOfCompact_toIsometryEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toIsometryEquiv = isometryEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_isometry_equiv ContinuousMap.linearIsometryBoundedOfCompact_toIsometryEquiv
@[simp] -- porting note: adjusted LHS because `simpNF` complained it simplified.
theorem linearIsometryBoundedOfCompact_toAddEquiv :
((linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv : C(α, E) ≃+ (α →ᵇ E)) =
addEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_add_equiv ContinuousMap.linearIsometryBoundedOfCompact_toAddEquiv
@[simp]
theorem linearIsometryBoundedOfCompact_of_compact_toEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv.toEquiv = equivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_of_compact_to_equiv ContinuousMap.linearIsometryBoundedOfCompact_of_compact_toEquiv
end
section
variable {𝕜 : Type*} {γ : Type*} [NormedField 𝕜] [NormedRing γ] [NormedAlgebra 𝕜 γ]
instance : NormedAlgebra 𝕜 C(α, γ) :=
{ ContinuousMap.normedSpace, ContinuousMap.algebra with }
end
end ContinuousMap
namespace ContinuousMap
section UniformContinuity
variable {α β : Type*}
variable [MetricSpace α] [CompactSpace α] [MetricSpace β]
/-!
We now set up some declarations making it convenient to use uniform continuity.
-/
theorem uniform_continuity (f : C(α, β)) (ε : ℝ) (h : 0 < ε) :
∃ δ > 0, ∀ {x y}, dist x y < δ → dist (f x) (f y) < ε :=
Metric.uniformContinuous_iff.mp (CompactSpace.uniformContinuous_of_continuous f.continuous) ε h
#align continuous_map.uniform_continuity ContinuousMap.uniform_continuity
-- This definition allows us to separate the choice of some `δ`,
-- and the corresponding use of `dist a b < δ → dist (f a) (f b) < ε`,
-- even across different declarations.
/-- An arbitrarily chosen modulus of uniform continuity for a given function `f` and `ε > 0`. -/
def modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) : ℝ :=
Classical.choose (uniform_continuity f ε h)
#align continuous_map.modulus ContinuousMap.modulus
theorem modulus_pos (f : C(α, β)) {ε : ℝ} {h : 0 < ε} : 0 < f.modulus ε h :=
(Classical.choose_spec (uniform_continuity f ε h)).1
#align continuous_map.modulus_pos ContinuousMap.modulus_pos
theorem dist_lt_of_dist_lt_modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) {a b : α}
(w : dist a b < f.modulus ε h) : dist (f a) (f b) < ε :=
(Classical.choose_spec (uniform_continuity f ε h)).2 w
#align continuous_map.dist_lt_of_dist_lt_modulus ContinuousMap.dist_lt_of_dist_lt_modulus
end UniformContinuity
end ContinuousMap
section CompLeft
variable (X : Type*) {𝕜 β γ : Type*} [TopologicalSpace X] [CompactSpace X]
[NontriviallyNormedField 𝕜]
variable [NormedAddCommGroup β] [NormedSpace 𝕜 β] [NormedAddCommGroup γ] [NormedSpace 𝕜 γ]
open ContinuousMap
/-- Postcomposition of continuous functions into a normed module by a continuous linear map is a
continuous linear map.
Transferred version of `ContinuousLinearMap.compLeftContinuousBounded`,
upgraded version of `ContinuousLinearMap.compLeftContinuous`,
similar to `LinearMap.compLeft`. -/
protected def ContinuousLinearMap.compLeftContinuousCompact (g : β →L[𝕜] γ) :
C(X, β) →L[𝕜] C(X, γ) :=
(linearIsometryBoundedOfCompact X γ 𝕜).symm.toLinearIsometry.toContinuousLinearMap.comp <|
(g.compLeftContinuousBounded X).comp <|
(linearIsometryBoundedOfCompact X β 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_linear_map.comp_left_continuous_compact ContinuousLinearMap.compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.toLinear_compLeftContinuousCompact (g : β →L[𝕜] γ) :
(g.compLeftContinuousCompact X : C(X, β) →ₗ[𝕜] C(X, γ)) = g.compLeftContinuous 𝕜 X := by
ext f
rfl
#align continuous_linear_map.to_linear_comp_left_continuous_compact ContinuousLinearMap.toLinear_compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.compLeftContinuousCompact_apply (g : β →L[𝕜] γ) (f : C(X, β)) (x : X) :
g.compLeftContinuousCompact X f x = g (f x) :=
rfl
#align continuous_linear_map.comp_left_continuous_compact_apply ContinuousLinearMap.compLeftContinuousCompact_apply
end CompLeft
namespace ContinuousMap
/-!
We now setup variations on `compRight* f`, where `f : C(X, Y)`
(that is, precomposition by a continuous map),
as a morphism `C(Y, T) → C(X, T)`, respecting various types of structure.
In particular:
* `compRightContinuousMap`, the bundled continuous map (for this we need `X Y` compact).
* `compRightHomeomorph`, when we precompose by a homeomorphism.
* `compRightAlgHom`, when `T = R` is a topological ring.
-/
section CompRight
/-- Precomposition by a continuous map is itself a continuous map between spaces of continuous maps.
-/
def compRightContinuousMap {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y)) : C(C(Y, T), C(X, T)) where
toFun g := g.comp f
continuous_toFun := by
refine' Metric.continuous_iff.mpr _
intro g ε ε_pos
refine' ⟨ε, ε_pos, fun g' h => _⟩
rw [ContinuousMap.dist_lt_iff ε_pos] at h ⊢
exact fun x => h (f x)
#align continuous_map.comp_right_continuous_map ContinuousMap.compRightContinuousMap
@[simp]
theorem compRightContinuousMap_apply {X Y : Type*} (T : Type*) [TopologicalSpace X]
[CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y))
(g : C(Y, T)) : (compRightContinuousMap T f) g = g.comp f :=
rfl
#align continuous_map.comp_right_continuous_map_apply ContinuousMap.compRightContinuousMap_apply
/-- Precomposition by a homeomorphism is itself a homeomorphism between spaces of continuous maps.
-/
def compRightHomeomorph {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : X ≃ₜ Y) : C(Y, T) ≃ₜ C(X, T) where
toFun := compRightContinuousMap T f.toContinuousMap
invFun := compRightContinuousMap T f.symm.toContinuousMap
left_inv g := ext fun _ => congr_arg g (f.apply_symm_apply _)
right_inv g := ext fun _ => congr_arg g (f.symm_apply_apply _)
#align continuous_map.comp_right_homeomorph ContinuousMap.compRightHomeomorph
theorem compRightAlgHom_continuous {X Y : Type*} (R A : Type*) [TopologicalSpace X]
[CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [CommSemiring R] [Semiring A]
[MetricSpace A] [TopologicalSemiring A] [Algebra R A] (f : C(X, Y)) :
Continuous (compRightAlgHom R A f) :=
map_continuous (compRightContinuousMap A f)
#align continuous_map.comp_right_alg_hom_continuous ContinuousMap.compRightAlgHom_continuous
end CompRight
section LocalNormalConvergence
/-! ### Local normal convergence
A sum of continuous functions (on a locally compact space) is "locally normally convergent" if the
sum of its sup-norms on any compact subset is summable. This implies convergence in the topology
of `C(X, E)` (i.e. locally uniform convergence). -/
open TopologicalSpace
variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
variable {E : Type*} [NormedAddCommGroup E] [CompleteSpace E]
theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F := by
refine' (ContinuousMap.exists_tendsto_compactOpen_iff_forall _).2 fun K hK => _
lift K to Compacts X using hK
have A : ∀ s : Finset ι, restrict (↑K) (∑ i in s, F i) = ∑ i in s, restrict K (F i) := by
intro s
ext1 x
simp
-- This used to be the end of the proof before leanprover/lean4#2644
erw [restrict_apply, restrict_apply, restrict_apply, restrict_apply]
simp? says simp only [coe_sum, Finset.sum_apply]
congr!
simpa only [HasSum, A] using (hF K).of_norm
#align continuous_map.summable_of_locally_summable_norm ContinuousMap.summable_of_locally_summable_norm
end LocalNormalConvergence
/-!
### Star structures
In this section, if `β` is a normed ⋆-group, then so is the space of
continuous functions from `α` to `β`, by using the star operation pointwise.
Furthermore, if `α` is compact and `β` is a C⋆-ring, then `C(α, β)` is a C⋆-ring. -/
section NormedSpace
variable {α : Type*} {β : Type*}
variable [TopologicalSpace α] [NormedAddCommGroup β] [StarAddMonoid β] [NormedStarGroup β]
theorem _root_.BoundedContinuousFunction.mkOfCompact_star [CompactSpace α] (f : C(α, β)) :
mkOfCompact (star f) = star (mkOfCompact f) :=
rfl
#align bounded_continuous_function.mk_of_compact_star BoundedContinuousFunction.mkOfCompact_star
instance [CompactSpace α] : NormedStarGroup C(α, β) where
norm_star f := by
rw [← BoundedContinuousFunction.norm_mkOfCompact, BoundedContinuousFunction.mkOfCompact_star,
norm_star, BoundedContinuousFunction.norm_mkOfCompact]
end NormedSpace
section CstarRing
variable {α : Type*} {β : Type*}
variable [TopologicalSpace α] [NormedRing β] [StarRing β]
instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f} := by
refine' le_antisymm _ _
· rw [← sq, ContinuousMap.norm_le _ (sq_nonneg _)]
intro x
simp only [ContinuousMap.coe_mul, coe_star, Pi.mul_apply, Pi.star_apply,
CstarRing.norm_star_mul_self, ← sq]
refine' sq_le_sq' _ _
· linarith [norm_nonneg (f x), norm_nonneg f]
·
|
exact ContinuousMap.norm_coe_le_norm f x
|
instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f} := by
refine' le_antisymm _ _
· rw [← sq, ContinuousMap.norm_le _ (sq_nonneg _)]
intro x
simp only [ContinuousMap.coe_mul, coe_star, Pi.mul_apply, Pi.star_apply,
CstarRing.norm_star_mul_self, ← sq]
refine' sq_le_sq' _ _
· linarith [norm_nonneg (f x), norm_nonneg f]
·
|
Mathlib.Topology.ContinuousFunction.Compact.541_0.Mig2jTVnn2FLKEB
|
instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f}
|
Mathlib_Topology_ContinuousFunction_Compact
|
case refine'_2
α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : NormedRing β
inst✝² : StarRing β
inst✝¹ : CompactSpace α
inst✝ : CstarRing β
f : C(α, β)
⊢ ‖f‖ * ‖f‖ ≤ ‖star f * f‖
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compacts
#align_import topology.continuous_function.compact from "leanprover-community/mathlib"@"d3af0609f6db8691dffdc3e1fb7feb7da72698f2"
/-!
# Continuous functions on a compact space
Continuous functions `C(α, β)` from a compact space `α` to a metric space `β`
are automatically bounded, and so acquire various structures inherited from `α →ᵇ β`.
This file transfers these structures, and restates some lemmas
characterising these structures.
If you need a lemma which is proved about `α →ᵇ β` but not for `C(α, β)` when `α` is compact,
you should restate it here. You can also use
`ContinuousMap.equivBoundedOfCompact` to move functions back and forth.
-/
noncomputable section
open Topology Classical NNReal BoundedContinuousFunction BigOperators
open Set Filter Metric
open BoundedContinuousFunction
namespace ContinuousMap
variable {α β E : Type*} [TopologicalSpace α] [CompactSpace α] [MetricSpace β]
[NormedAddCommGroup E]
section
variable (α β)
/-- When `α` is compact, the bounded continuous maps `α →ᵇ β` are
equivalent to `C(α, β)`.
-/
@[simps (config := .asFn)]
def equivBoundedOfCompact : C(α, β) ≃ (α →ᵇ β) :=
⟨mkOfCompact, BoundedContinuousFunction.toContinuousMap, fun f => by
ext
rfl, fun f => by
ext
rfl⟩
#align continuous_map.equiv_bounded_of_compact ContinuousMap.equivBoundedOfCompact
theorem uniformInducing_equivBoundedOfCompact : UniformInducing (equivBoundedOfCompact α β) :=
UniformInducing.mk'
(by
simp only [hasBasis_compactConvergenceUniformity.mem_iff, uniformity_basis_dist_le.mem_iff]
exact fun s =>
⟨fun ⟨⟨a, b⟩, ⟨_, ⟨ε, hε, hb⟩⟩, hs⟩ =>
⟨{ p | ∀ x, (p.1 x, p.2 x) ∈ b }, ⟨ε, hε, fun _ h x => hb ((dist_le hε.le).mp h x)⟩,
fun f g h => hs fun x _ => h x⟩,
fun ⟨_, ⟨ε, hε, ht⟩, hs⟩ =>
⟨⟨Set.univ, { p | dist p.1 p.2 ≤ ε }⟩, ⟨isCompact_univ, ⟨ε, hε, fun _ h => h⟩⟩,
fun ⟨f, g⟩ h => hs _ _ (ht ((dist_le hε.le).mpr fun x => h x (mem_univ x)))⟩⟩)
#align continuous_map.uniform_inducing_equiv_bounded_of_compact ContinuousMap.uniformInducing_equivBoundedOfCompact
theorem uniformEmbedding_equivBoundedOfCompact : UniformEmbedding (equivBoundedOfCompact α β) :=
{ uniformInducing_equivBoundedOfCompact α β with inj := (equivBoundedOfCompact α β).injective }
#align continuous_map.uniform_embedding_equiv_bounded_of_compact ContinuousMap.uniformEmbedding_equivBoundedOfCompact
/-- When `α` is compact, the bounded continuous maps `α →ᵇ 𝕜` are
additively equivalent to `C(α, 𝕜)`.
-/
-- porting note: the following `simps` received a "maximum recursion depth" error
-- @[simps! (config := .asFn) apply symm_apply]
def addEquivBoundedOfCompact [AddMonoid β] [LipschitzAdd β] : C(α, β) ≃+ (α →ᵇ β) :=
({ toContinuousMapAddHom α β, (equivBoundedOfCompact α β).symm with } : (α →ᵇ β) ≃+ C(α, β)).symm
#align continuous_map.add_equiv_bounded_of_compact ContinuousMap.addEquivBoundedOfCompact
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_symm_apply [AddMonoid β] [LipschitzAdd β] :
⇑((addEquivBoundedOfCompact α β).symm) = toContinuousMapAddHom α β :=
rfl
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_apply [AddMonoid β] [LipschitzAdd β] :
⇑(addEquivBoundedOfCompact α β) = mkOfCompact :=
rfl
instance metricSpace : MetricSpace C(α, β) :=
(uniformEmbedding_equivBoundedOfCompact α β).comapMetricSpace _
#align continuous_map.metric_space ContinuousMap.metricSpace
/-- When `α` is compact, and `β` is a metric space, the bounded continuous maps `α →ᵇ β` are
isometric to `C(α, β)`.
-/
@[simps! (config := .asFn) toEquiv apply symm_apply]
def isometryEquivBoundedOfCompact : C(α, β) ≃ᵢ (α →ᵇ β) where
isometry_toFun _ _ := rfl
toEquiv := equivBoundedOfCompact α β
#align continuous_map.isometry_equiv_bounded_of_compact ContinuousMap.isometryEquivBoundedOfCompact
end
@[simp]
theorem _root_.BoundedContinuousFunction.dist_mkOfCompact (f g : C(α, β)) :
dist (mkOfCompact f) (mkOfCompact g) = dist f g :=
rfl
#align bounded_continuous_function.dist_mk_of_compact BoundedContinuousFunction.dist_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.dist_toContinuousMap (f g : α →ᵇ β) :
dist f.toContinuousMap g.toContinuousMap = dist f g :=
rfl
#align bounded_continuous_function.dist_to_continuous_map BoundedContinuousFunction.dist_toContinuousMap
open BoundedContinuousFunction
section
variable {f g : C(α, β)} {C : ℝ}
/-- The pointwise distance is controlled by the distance between functions, by definition. -/
theorem dist_apply_le_dist (x : α) : dist (f x) (g x) ≤ dist f g := by
simp only [← dist_mkOfCompact, dist_coe_le_dist, ← mkOfCompact_apply]
#align continuous_map.dist_apply_le_dist ContinuousMap.dist_apply_le_dist
/-- The distance between two functions is controlled by the supremum of the pointwise distances. -/
theorem dist_le (C0 : (0 : ℝ) ≤ C) : dist f g ≤ C ↔ ∀ x : α, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le C0, mkOfCompact_apply]
#align continuous_map.dist_le ContinuousMap.dist_le
theorem dist_le_iff_of_nonempty [Nonempty α] : dist f g ≤ C ↔ ∀ x, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le_iff_of_nonempty,
mkOfCompact_apply]
#align continuous_map.dist_le_iff_of_nonempty ContinuousMap.dist_le_iff_of_nonempty
theorem dist_lt_iff_of_nonempty [Nonempty α] : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
simp only [← dist_mkOfCompact, dist_lt_iff_of_nonempty_compact, mkOfCompact_apply]
#align continuous_map.dist_lt_iff_of_nonempty ContinuousMap.dist_lt_iff_of_nonempty
theorem dist_lt_of_nonempty [Nonempty α] (w : ∀ x : α, dist (f x) (g x) < C) : dist f g < C :=
dist_lt_iff_of_nonempty.2 w
#align continuous_map.dist_lt_of_nonempty ContinuousMap.dist_lt_of_nonempty
theorem dist_lt_iff (C0 : (0 : ℝ) < C) : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
rw [← dist_mkOfCompact, dist_lt_iff_of_compact C0]
simp only [mkOfCompact_apply]
#align continuous_map.dist_lt_iff ContinuousMap.dist_lt_iff
end
instance [CompleteSpace β] : CompleteSpace C(α, β) :=
(isometryEquivBoundedOfCompact α β).completeSpace
/-- See also `ContinuousMap.continuous_eval'`. -/
@[continuity]
theorem continuous_eval : Continuous fun p : C(α, β) × α => p.1 p.2 :=
continuous_eval.comp ((isometryEquivBoundedOfCompact α β).continuous.prod_map continuous_id)
#align continuous_map.continuous_eval ContinuousMap.continuous_eval
-- TODO at some point we will need lemmas characterising this norm!
-- At the moment the only way to reason about it is to transfer `f : C(α,E)` back to `α →ᵇ E`.
instance : Norm C(α, E) where norm x := dist x 0
@[simp]
theorem _root_.BoundedContinuousFunction.norm_mkOfCompact (f : C(α, E)) : ‖mkOfCompact f‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_mk_of_compact BoundedContinuousFunction.norm_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.norm_toContinuousMap_eq (f : α →ᵇ E) :
‖f.toContinuousMap‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_to_continuous_map_eq BoundedContinuousFunction.norm_toContinuousMap_eq
open BoundedContinuousFunction
instance : NormedAddCommGroup C(α, E) :=
{ ContinuousMap.metricSpace _ _,
ContinuousMap.instAddCommGroupContinuousMap with
dist_eq := fun x y => by
rw [← norm_mkOfCompact, ← dist_mkOfCompact, dist_eq_norm, mkOfCompact_sub]
dist := dist
norm := norm }
instance [Nonempty α] [One E] [NormOneClass E] : NormOneClass C(α, E) where
norm_one := by simp only [← norm_mkOfCompact, mkOfCompact_one, norm_one]
section
variable (f : C(α, E))
-- The corresponding lemmas for `BoundedContinuousFunction` are stated with `{f}`,
-- and so can not be used in dot notation.
theorem norm_coe_le_norm (x : α) : ‖f x‖ ≤ ‖f‖ :=
(mkOfCompact f).norm_coe_le_norm x
#align continuous_map.norm_coe_le_norm ContinuousMap.norm_coe_le_norm
/-- Distance between the images of any two points is at most twice the norm of the function. -/
theorem dist_le_two_norm (x y : α) : dist (f x) (f y) ≤ 2 * ‖f‖ :=
(mkOfCompact f).dist_le_two_norm x y
#align continuous_map.dist_le_two_norm ContinuousMap.dist_le_two_norm
/-- The norm of a function is controlled by the supremum of the pointwise norms. -/
theorem norm_le {C : ℝ} (C0 : (0 : ℝ) ≤ C) : ‖f‖ ≤ C ↔ ∀ x : α, ‖f x‖ ≤ C :=
@BoundedContinuousFunction.norm_le _ _ _ _ (mkOfCompact f) _ C0
#align continuous_map.norm_le ContinuousMap.norm_le
theorem norm_le_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ ≤ M ↔ ∀ x, ‖f x‖ ≤ M :=
@BoundedContinuousFunction.norm_le_of_nonempty _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_le_of_nonempty ContinuousMap.norm_le_of_nonempty
theorem norm_lt_iff {M : ℝ} (M0 : 0 < M) : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_compact _ _ _ _ _ (mkOfCompact f) _ M0
#align continuous_map.norm_lt_iff ContinuousMap.norm_lt_iff
theorem nnnorm_lt_iff {M : ℝ≥0} (M0 : 0 < M) : ‖f‖₊ < M ↔ ∀ x : α, ‖f x‖₊ < M :=
f.norm_lt_iff M0
#align continuous_map.nnnorm_lt_iff ContinuousMap.nnnorm_lt_iff
theorem norm_lt_iff_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_nonempty_compact _ _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_lt_iff_of_nonempty ContinuousMap.norm_lt_iff_of_nonempty
theorem nnnorm_lt_iff_of_nonempty [Nonempty α] {M : ℝ≥0} : ‖f‖₊ < M ↔ ∀ x, ‖f x‖₊ < M :=
f.norm_lt_iff_of_nonempty
#align continuous_map.nnnorm_lt_iff_of_nonempty ContinuousMap.nnnorm_lt_iff_of_nonempty
theorem apply_le_norm (f : C(α, ℝ)) (x : α) : f x ≤ ‖f‖ :=
le_trans (le_abs.mpr (Or.inl (le_refl (f x)))) (f.norm_coe_le_norm x)
#align continuous_map.apply_le_norm ContinuousMap.apply_le_norm
theorem neg_norm_le_apply (f : C(α, ℝ)) (x : α) : -‖f‖ ≤ f x :=
le_trans (neg_le_neg (f.norm_coe_le_norm x)) (neg_le.mp (neg_le_abs_self (f x)))
#align continuous_map.neg_norm_le_apply ContinuousMap.neg_norm_le_apply
theorem norm_eq_iSup_norm : ‖f‖ = ⨆ x : α, ‖f x‖ :=
(mkOfCompact f).norm_eq_iSup_norm
#align continuous_map.norm_eq_supr_norm ContinuousMap.norm_eq_iSup_norm
theorem norm_restrict_mono_set {X : Type*} [TopologicalSpace X] (f : C(X, E))
{K L : TopologicalSpace.Compacts X} (hKL : K ≤ L) : ‖f.restrict K‖ ≤ ‖f.restrict L‖ :=
(norm_le _ (norm_nonneg _)).mpr fun x => norm_coe_le_norm (f.restrict L) <| Set.inclusion hKL x
#align continuous_map.norm_restrict_mono_set ContinuousMap.norm_restrict_mono_set
end
section
variable {R : Type*} [NormedRing R]
instance : NormedRing C(α, R) :=
{ (inferInstance : NormedAddCommGroup C(α, R)), ContinuousMap.instRingContinuousMap with
norm_mul := fun f g => norm_mul_le (mkOfCompact f) (mkOfCompact g) }
end
section
variable {𝕜 : Type*} [NormedField 𝕜] [NormedSpace 𝕜 E]
instance normedSpace : NormedSpace 𝕜 C(α, E) where
norm_smul_le c f := (norm_smul_le c (mkOfCompact f) : _)
#align continuous_map.normed_space ContinuousMap.normedSpace
section
variable (α 𝕜 E)
/-- When `α` is compact and `𝕜` is a normed field,
the `𝕜`-algebra of bounded continuous maps `α →ᵇ β` is
`𝕜`-linearly isometric to `C(α, β)`.
-/
def linearIsometryBoundedOfCompact : C(α, E) ≃ₗᵢ[𝕜] α →ᵇ E :=
{ addEquivBoundedOfCompact α E with
map_smul' := fun c f => by
ext
norm_cast
norm_map' := fun f => rfl }
#align continuous_map.linear_isometry_bounded_of_compact ContinuousMap.linearIsometryBoundedOfCompact
variable {α E}
-- to match `BoundedContinuousFunction.evalClm`
/-- The evaluation at a point, as a continuous linear map from `C(α, 𝕜)` to `𝕜`. -/
def evalClm (x : α) : C(α, E) →L[𝕜] E :=
(BoundedContinuousFunction.evalClm 𝕜 x).comp
(linearIsometryBoundedOfCompact α E 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_map.eval_clm ContinuousMap.evalClm
end
-- this lemma and the next are the analogues of those autogenerated by `@[simps]` for
-- `equivBoundedOfCompact`, `addEquivBoundedOfCompact`
@[simp]
theorem linearIsometryBoundedOfCompact_symm_apply (f : α →ᵇ E) :
(linearIsometryBoundedOfCompact α E 𝕜).symm f = f.toContinuousMap :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_symm_apply ContinuousMap.linearIsometryBoundedOfCompact_symm_apply
@[simp]
theorem linearIsometryBoundedOfCompact_apply_apply (f : C(α, E)) (a : α) :
(linearIsometryBoundedOfCompact α E 𝕜 f) a = f a :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_apply_apply ContinuousMap.linearIsometryBoundedOfCompact_apply_apply
@[simp]
theorem linearIsometryBoundedOfCompact_toIsometryEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toIsometryEquiv = isometryEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_isometry_equiv ContinuousMap.linearIsometryBoundedOfCompact_toIsometryEquiv
@[simp] -- porting note: adjusted LHS because `simpNF` complained it simplified.
theorem linearIsometryBoundedOfCompact_toAddEquiv :
((linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv : C(α, E) ≃+ (α →ᵇ E)) =
addEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_add_equiv ContinuousMap.linearIsometryBoundedOfCompact_toAddEquiv
@[simp]
theorem linearIsometryBoundedOfCompact_of_compact_toEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv.toEquiv = equivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_of_compact_to_equiv ContinuousMap.linearIsometryBoundedOfCompact_of_compact_toEquiv
end
section
variable {𝕜 : Type*} {γ : Type*} [NormedField 𝕜] [NormedRing γ] [NormedAlgebra 𝕜 γ]
instance : NormedAlgebra 𝕜 C(α, γ) :=
{ ContinuousMap.normedSpace, ContinuousMap.algebra with }
end
end ContinuousMap
namespace ContinuousMap
section UniformContinuity
variable {α β : Type*}
variable [MetricSpace α] [CompactSpace α] [MetricSpace β]
/-!
We now set up some declarations making it convenient to use uniform continuity.
-/
theorem uniform_continuity (f : C(α, β)) (ε : ℝ) (h : 0 < ε) :
∃ δ > 0, ∀ {x y}, dist x y < δ → dist (f x) (f y) < ε :=
Metric.uniformContinuous_iff.mp (CompactSpace.uniformContinuous_of_continuous f.continuous) ε h
#align continuous_map.uniform_continuity ContinuousMap.uniform_continuity
-- This definition allows us to separate the choice of some `δ`,
-- and the corresponding use of `dist a b < δ → dist (f a) (f b) < ε`,
-- even across different declarations.
/-- An arbitrarily chosen modulus of uniform continuity for a given function `f` and `ε > 0`. -/
def modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) : ℝ :=
Classical.choose (uniform_continuity f ε h)
#align continuous_map.modulus ContinuousMap.modulus
theorem modulus_pos (f : C(α, β)) {ε : ℝ} {h : 0 < ε} : 0 < f.modulus ε h :=
(Classical.choose_spec (uniform_continuity f ε h)).1
#align continuous_map.modulus_pos ContinuousMap.modulus_pos
theorem dist_lt_of_dist_lt_modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) {a b : α}
(w : dist a b < f.modulus ε h) : dist (f a) (f b) < ε :=
(Classical.choose_spec (uniform_continuity f ε h)).2 w
#align continuous_map.dist_lt_of_dist_lt_modulus ContinuousMap.dist_lt_of_dist_lt_modulus
end UniformContinuity
end ContinuousMap
section CompLeft
variable (X : Type*) {𝕜 β γ : Type*} [TopologicalSpace X] [CompactSpace X]
[NontriviallyNormedField 𝕜]
variable [NormedAddCommGroup β] [NormedSpace 𝕜 β] [NormedAddCommGroup γ] [NormedSpace 𝕜 γ]
open ContinuousMap
/-- Postcomposition of continuous functions into a normed module by a continuous linear map is a
continuous linear map.
Transferred version of `ContinuousLinearMap.compLeftContinuousBounded`,
upgraded version of `ContinuousLinearMap.compLeftContinuous`,
similar to `LinearMap.compLeft`. -/
protected def ContinuousLinearMap.compLeftContinuousCompact (g : β →L[𝕜] γ) :
C(X, β) →L[𝕜] C(X, γ) :=
(linearIsometryBoundedOfCompact X γ 𝕜).symm.toLinearIsometry.toContinuousLinearMap.comp <|
(g.compLeftContinuousBounded X).comp <|
(linearIsometryBoundedOfCompact X β 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_linear_map.comp_left_continuous_compact ContinuousLinearMap.compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.toLinear_compLeftContinuousCompact (g : β →L[𝕜] γ) :
(g.compLeftContinuousCompact X : C(X, β) →ₗ[𝕜] C(X, γ)) = g.compLeftContinuous 𝕜 X := by
ext f
rfl
#align continuous_linear_map.to_linear_comp_left_continuous_compact ContinuousLinearMap.toLinear_compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.compLeftContinuousCompact_apply (g : β →L[𝕜] γ) (f : C(X, β)) (x : X) :
g.compLeftContinuousCompact X f x = g (f x) :=
rfl
#align continuous_linear_map.comp_left_continuous_compact_apply ContinuousLinearMap.compLeftContinuousCompact_apply
end CompLeft
namespace ContinuousMap
/-!
We now setup variations on `compRight* f`, where `f : C(X, Y)`
(that is, precomposition by a continuous map),
as a morphism `C(Y, T) → C(X, T)`, respecting various types of structure.
In particular:
* `compRightContinuousMap`, the bundled continuous map (for this we need `X Y` compact).
* `compRightHomeomorph`, when we precompose by a homeomorphism.
* `compRightAlgHom`, when `T = R` is a topological ring.
-/
section CompRight
/-- Precomposition by a continuous map is itself a continuous map between spaces of continuous maps.
-/
def compRightContinuousMap {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y)) : C(C(Y, T), C(X, T)) where
toFun g := g.comp f
continuous_toFun := by
refine' Metric.continuous_iff.mpr _
intro g ε ε_pos
refine' ⟨ε, ε_pos, fun g' h => _⟩
rw [ContinuousMap.dist_lt_iff ε_pos] at h ⊢
exact fun x => h (f x)
#align continuous_map.comp_right_continuous_map ContinuousMap.compRightContinuousMap
@[simp]
theorem compRightContinuousMap_apply {X Y : Type*} (T : Type*) [TopologicalSpace X]
[CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y))
(g : C(Y, T)) : (compRightContinuousMap T f) g = g.comp f :=
rfl
#align continuous_map.comp_right_continuous_map_apply ContinuousMap.compRightContinuousMap_apply
/-- Precomposition by a homeomorphism is itself a homeomorphism between spaces of continuous maps.
-/
def compRightHomeomorph {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : X ≃ₜ Y) : C(Y, T) ≃ₜ C(X, T) where
toFun := compRightContinuousMap T f.toContinuousMap
invFun := compRightContinuousMap T f.symm.toContinuousMap
left_inv g := ext fun _ => congr_arg g (f.apply_symm_apply _)
right_inv g := ext fun _ => congr_arg g (f.symm_apply_apply _)
#align continuous_map.comp_right_homeomorph ContinuousMap.compRightHomeomorph
theorem compRightAlgHom_continuous {X Y : Type*} (R A : Type*) [TopologicalSpace X]
[CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [CommSemiring R] [Semiring A]
[MetricSpace A] [TopologicalSemiring A] [Algebra R A] (f : C(X, Y)) :
Continuous (compRightAlgHom R A f) :=
map_continuous (compRightContinuousMap A f)
#align continuous_map.comp_right_alg_hom_continuous ContinuousMap.compRightAlgHom_continuous
end CompRight
section LocalNormalConvergence
/-! ### Local normal convergence
A sum of continuous functions (on a locally compact space) is "locally normally convergent" if the
sum of its sup-norms on any compact subset is summable. This implies convergence in the topology
of `C(X, E)` (i.e. locally uniform convergence). -/
open TopologicalSpace
variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
variable {E : Type*} [NormedAddCommGroup E] [CompleteSpace E]
theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F := by
refine' (ContinuousMap.exists_tendsto_compactOpen_iff_forall _).2 fun K hK => _
lift K to Compacts X using hK
have A : ∀ s : Finset ι, restrict (↑K) (∑ i in s, F i) = ∑ i in s, restrict K (F i) := by
intro s
ext1 x
simp
-- This used to be the end of the proof before leanprover/lean4#2644
erw [restrict_apply, restrict_apply, restrict_apply, restrict_apply]
simp? says simp only [coe_sum, Finset.sum_apply]
congr!
simpa only [HasSum, A] using (hF K).of_norm
#align continuous_map.summable_of_locally_summable_norm ContinuousMap.summable_of_locally_summable_norm
end LocalNormalConvergence
/-!
### Star structures
In this section, if `β` is a normed ⋆-group, then so is the space of
continuous functions from `α` to `β`, by using the star operation pointwise.
Furthermore, if `α` is compact and `β` is a C⋆-ring, then `C(α, β)` is a C⋆-ring. -/
section NormedSpace
variable {α : Type*} {β : Type*}
variable [TopologicalSpace α] [NormedAddCommGroup β] [StarAddMonoid β] [NormedStarGroup β]
theorem _root_.BoundedContinuousFunction.mkOfCompact_star [CompactSpace α] (f : C(α, β)) :
mkOfCompact (star f) = star (mkOfCompact f) :=
rfl
#align bounded_continuous_function.mk_of_compact_star BoundedContinuousFunction.mkOfCompact_star
instance [CompactSpace α] : NormedStarGroup C(α, β) where
norm_star f := by
rw [← BoundedContinuousFunction.norm_mkOfCompact, BoundedContinuousFunction.mkOfCompact_star,
norm_star, BoundedContinuousFunction.norm_mkOfCompact]
end NormedSpace
section CstarRing
variable {α : Type*} {β : Type*}
variable [TopologicalSpace α] [NormedRing β] [StarRing β]
instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f} := by
refine' le_antisymm _ _
· rw [← sq, ContinuousMap.norm_le _ (sq_nonneg _)]
intro x
simp only [ContinuousMap.coe_mul, coe_star, Pi.mul_apply, Pi.star_apply,
CstarRing.norm_star_mul_self, ← sq]
refine' sq_le_sq' _ _
· linarith [norm_nonneg (f x), norm_nonneg f]
· exact ContinuousMap.norm_coe_le_norm f x
·
|
rw [← sq, ← Real.le_sqrt (norm_nonneg _) (norm_nonneg _),
ContinuousMap.norm_le _ (Real.sqrt_nonneg _)]
|
instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f} := by
refine' le_antisymm _ _
· rw [← sq, ContinuousMap.norm_le _ (sq_nonneg _)]
intro x
simp only [ContinuousMap.coe_mul, coe_star, Pi.mul_apply, Pi.star_apply,
CstarRing.norm_star_mul_self, ← sq]
refine' sq_le_sq' _ _
· linarith [norm_nonneg (f x), norm_nonneg f]
· exact ContinuousMap.norm_coe_le_norm f x
·
|
Mathlib.Topology.ContinuousFunction.Compact.541_0.Mig2jTVnn2FLKEB
|
instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f}
|
Mathlib_Topology_ContinuousFunction_Compact
|
case refine'_2
α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : NormedRing β
inst✝² : StarRing β
inst✝¹ : CompactSpace α
inst✝ : CstarRing β
f : C(α, β)
⊢ ∀ (x : α), ‖f x‖ ≤ Real.sqrt ‖star f * f‖
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compacts
#align_import topology.continuous_function.compact from "leanprover-community/mathlib"@"d3af0609f6db8691dffdc3e1fb7feb7da72698f2"
/-!
# Continuous functions on a compact space
Continuous functions `C(α, β)` from a compact space `α` to a metric space `β`
are automatically bounded, and so acquire various structures inherited from `α →ᵇ β`.
This file transfers these structures, and restates some lemmas
characterising these structures.
If you need a lemma which is proved about `α →ᵇ β` but not for `C(α, β)` when `α` is compact,
you should restate it here. You can also use
`ContinuousMap.equivBoundedOfCompact` to move functions back and forth.
-/
noncomputable section
open Topology Classical NNReal BoundedContinuousFunction BigOperators
open Set Filter Metric
open BoundedContinuousFunction
namespace ContinuousMap
variable {α β E : Type*} [TopologicalSpace α] [CompactSpace α] [MetricSpace β]
[NormedAddCommGroup E]
section
variable (α β)
/-- When `α` is compact, the bounded continuous maps `α →ᵇ β` are
equivalent to `C(α, β)`.
-/
@[simps (config := .asFn)]
def equivBoundedOfCompact : C(α, β) ≃ (α →ᵇ β) :=
⟨mkOfCompact, BoundedContinuousFunction.toContinuousMap, fun f => by
ext
rfl, fun f => by
ext
rfl⟩
#align continuous_map.equiv_bounded_of_compact ContinuousMap.equivBoundedOfCompact
theorem uniformInducing_equivBoundedOfCompact : UniformInducing (equivBoundedOfCompact α β) :=
UniformInducing.mk'
(by
simp only [hasBasis_compactConvergenceUniformity.mem_iff, uniformity_basis_dist_le.mem_iff]
exact fun s =>
⟨fun ⟨⟨a, b⟩, ⟨_, ⟨ε, hε, hb⟩⟩, hs⟩ =>
⟨{ p | ∀ x, (p.1 x, p.2 x) ∈ b }, ⟨ε, hε, fun _ h x => hb ((dist_le hε.le).mp h x)⟩,
fun f g h => hs fun x _ => h x⟩,
fun ⟨_, ⟨ε, hε, ht⟩, hs⟩ =>
⟨⟨Set.univ, { p | dist p.1 p.2 ≤ ε }⟩, ⟨isCompact_univ, ⟨ε, hε, fun _ h => h⟩⟩,
fun ⟨f, g⟩ h => hs _ _ (ht ((dist_le hε.le).mpr fun x => h x (mem_univ x)))⟩⟩)
#align continuous_map.uniform_inducing_equiv_bounded_of_compact ContinuousMap.uniformInducing_equivBoundedOfCompact
theorem uniformEmbedding_equivBoundedOfCompact : UniformEmbedding (equivBoundedOfCompact α β) :=
{ uniformInducing_equivBoundedOfCompact α β with inj := (equivBoundedOfCompact α β).injective }
#align continuous_map.uniform_embedding_equiv_bounded_of_compact ContinuousMap.uniformEmbedding_equivBoundedOfCompact
/-- When `α` is compact, the bounded continuous maps `α →ᵇ 𝕜` are
additively equivalent to `C(α, 𝕜)`.
-/
-- porting note: the following `simps` received a "maximum recursion depth" error
-- @[simps! (config := .asFn) apply symm_apply]
def addEquivBoundedOfCompact [AddMonoid β] [LipschitzAdd β] : C(α, β) ≃+ (α →ᵇ β) :=
({ toContinuousMapAddHom α β, (equivBoundedOfCompact α β).symm with } : (α →ᵇ β) ≃+ C(α, β)).symm
#align continuous_map.add_equiv_bounded_of_compact ContinuousMap.addEquivBoundedOfCompact
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_symm_apply [AddMonoid β] [LipschitzAdd β] :
⇑((addEquivBoundedOfCompact α β).symm) = toContinuousMapAddHom α β :=
rfl
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_apply [AddMonoid β] [LipschitzAdd β] :
⇑(addEquivBoundedOfCompact α β) = mkOfCompact :=
rfl
instance metricSpace : MetricSpace C(α, β) :=
(uniformEmbedding_equivBoundedOfCompact α β).comapMetricSpace _
#align continuous_map.metric_space ContinuousMap.metricSpace
/-- When `α` is compact, and `β` is a metric space, the bounded continuous maps `α →ᵇ β` are
isometric to `C(α, β)`.
-/
@[simps! (config := .asFn) toEquiv apply symm_apply]
def isometryEquivBoundedOfCompact : C(α, β) ≃ᵢ (α →ᵇ β) where
isometry_toFun _ _ := rfl
toEquiv := equivBoundedOfCompact α β
#align continuous_map.isometry_equiv_bounded_of_compact ContinuousMap.isometryEquivBoundedOfCompact
end
@[simp]
theorem _root_.BoundedContinuousFunction.dist_mkOfCompact (f g : C(α, β)) :
dist (mkOfCompact f) (mkOfCompact g) = dist f g :=
rfl
#align bounded_continuous_function.dist_mk_of_compact BoundedContinuousFunction.dist_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.dist_toContinuousMap (f g : α →ᵇ β) :
dist f.toContinuousMap g.toContinuousMap = dist f g :=
rfl
#align bounded_continuous_function.dist_to_continuous_map BoundedContinuousFunction.dist_toContinuousMap
open BoundedContinuousFunction
section
variable {f g : C(α, β)} {C : ℝ}
/-- The pointwise distance is controlled by the distance between functions, by definition. -/
theorem dist_apply_le_dist (x : α) : dist (f x) (g x) ≤ dist f g := by
simp only [← dist_mkOfCompact, dist_coe_le_dist, ← mkOfCompact_apply]
#align continuous_map.dist_apply_le_dist ContinuousMap.dist_apply_le_dist
/-- The distance between two functions is controlled by the supremum of the pointwise distances. -/
theorem dist_le (C0 : (0 : ℝ) ≤ C) : dist f g ≤ C ↔ ∀ x : α, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le C0, mkOfCompact_apply]
#align continuous_map.dist_le ContinuousMap.dist_le
theorem dist_le_iff_of_nonempty [Nonempty α] : dist f g ≤ C ↔ ∀ x, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le_iff_of_nonempty,
mkOfCompact_apply]
#align continuous_map.dist_le_iff_of_nonempty ContinuousMap.dist_le_iff_of_nonempty
theorem dist_lt_iff_of_nonempty [Nonempty α] : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
simp only [← dist_mkOfCompact, dist_lt_iff_of_nonempty_compact, mkOfCompact_apply]
#align continuous_map.dist_lt_iff_of_nonempty ContinuousMap.dist_lt_iff_of_nonempty
theorem dist_lt_of_nonempty [Nonempty α] (w : ∀ x : α, dist (f x) (g x) < C) : dist f g < C :=
dist_lt_iff_of_nonempty.2 w
#align continuous_map.dist_lt_of_nonempty ContinuousMap.dist_lt_of_nonempty
theorem dist_lt_iff (C0 : (0 : ℝ) < C) : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
rw [← dist_mkOfCompact, dist_lt_iff_of_compact C0]
simp only [mkOfCompact_apply]
#align continuous_map.dist_lt_iff ContinuousMap.dist_lt_iff
end
instance [CompleteSpace β] : CompleteSpace C(α, β) :=
(isometryEquivBoundedOfCompact α β).completeSpace
/-- See also `ContinuousMap.continuous_eval'`. -/
@[continuity]
theorem continuous_eval : Continuous fun p : C(α, β) × α => p.1 p.2 :=
continuous_eval.comp ((isometryEquivBoundedOfCompact α β).continuous.prod_map continuous_id)
#align continuous_map.continuous_eval ContinuousMap.continuous_eval
-- TODO at some point we will need lemmas characterising this norm!
-- At the moment the only way to reason about it is to transfer `f : C(α,E)` back to `α →ᵇ E`.
instance : Norm C(α, E) where norm x := dist x 0
@[simp]
theorem _root_.BoundedContinuousFunction.norm_mkOfCompact (f : C(α, E)) : ‖mkOfCompact f‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_mk_of_compact BoundedContinuousFunction.norm_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.norm_toContinuousMap_eq (f : α →ᵇ E) :
‖f.toContinuousMap‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_to_continuous_map_eq BoundedContinuousFunction.norm_toContinuousMap_eq
open BoundedContinuousFunction
instance : NormedAddCommGroup C(α, E) :=
{ ContinuousMap.metricSpace _ _,
ContinuousMap.instAddCommGroupContinuousMap with
dist_eq := fun x y => by
rw [← norm_mkOfCompact, ← dist_mkOfCompact, dist_eq_norm, mkOfCompact_sub]
dist := dist
norm := norm }
instance [Nonempty α] [One E] [NormOneClass E] : NormOneClass C(α, E) where
norm_one := by simp only [← norm_mkOfCompact, mkOfCompact_one, norm_one]
section
variable (f : C(α, E))
-- The corresponding lemmas for `BoundedContinuousFunction` are stated with `{f}`,
-- and so can not be used in dot notation.
theorem norm_coe_le_norm (x : α) : ‖f x‖ ≤ ‖f‖ :=
(mkOfCompact f).norm_coe_le_norm x
#align continuous_map.norm_coe_le_norm ContinuousMap.norm_coe_le_norm
/-- Distance between the images of any two points is at most twice the norm of the function. -/
theorem dist_le_two_norm (x y : α) : dist (f x) (f y) ≤ 2 * ‖f‖ :=
(mkOfCompact f).dist_le_two_norm x y
#align continuous_map.dist_le_two_norm ContinuousMap.dist_le_two_norm
/-- The norm of a function is controlled by the supremum of the pointwise norms. -/
theorem norm_le {C : ℝ} (C0 : (0 : ℝ) ≤ C) : ‖f‖ ≤ C ↔ ∀ x : α, ‖f x‖ ≤ C :=
@BoundedContinuousFunction.norm_le _ _ _ _ (mkOfCompact f) _ C0
#align continuous_map.norm_le ContinuousMap.norm_le
theorem norm_le_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ ≤ M ↔ ∀ x, ‖f x‖ ≤ M :=
@BoundedContinuousFunction.norm_le_of_nonempty _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_le_of_nonempty ContinuousMap.norm_le_of_nonempty
theorem norm_lt_iff {M : ℝ} (M0 : 0 < M) : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_compact _ _ _ _ _ (mkOfCompact f) _ M0
#align continuous_map.norm_lt_iff ContinuousMap.norm_lt_iff
theorem nnnorm_lt_iff {M : ℝ≥0} (M0 : 0 < M) : ‖f‖₊ < M ↔ ∀ x : α, ‖f x‖₊ < M :=
f.norm_lt_iff M0
#align continuous_map.nnnorm_lt_iff ContinuousMap.nnnorm_lt_iff
theorem norm_lt_iff_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_nonempty_compact _ _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_lt_iff_of_nonempty ContinuousMap.norm_lt_iff_of_nonempty
theorem nnnorm_lt_iff_of_nonempty [Nonempty α] {M : ℝ≥0} : ‖f‖₊ < M ↔ ∀ x, ‖f x‖₊ < M :=
f.norm_lt_iff_of_nonempty
#align continuous_map.nnnorm_lt_iff_of_nonempty ContinuousMap.nnnorm_lt_iff_of_nonempty
theorem apply_le_norm (f : C(α, ℝ)) (x : α) : f x ≤ ‖f‖ :=
le_trans (le_abs.mpr (Or.inl (le_refl (f x)))) (f.norm_coe_le_norm x)
#align continuous_map.apply_le_norm ContinuousMap.apply_le_norm
theorem neg_norm_le_apply (f : C(α, ℝ)) (x : α) : -‖f‖ ≤ f x :=
le_trans (neg_le_neg (f.norm_coe_le_norm x)) (neg_le.mp (neg_le_abs_self (f x)))
#align continuous_map.neg_norm_le_apply ContinuousMap.neg_norm_le_apply
theorem norm_eq_iSup_norm : ‖f‖ = ⨆ x : α, ‖f x‖ :=
(mkOfCompact f).norm_eq_iSup_norm
#align continuous_map.norm_eq_supr_norm ContinuousMap.norm_eq_iSup_norm
theorem norm_restrict_mono_set {X : Type*} [TopologicalSpace X] (f : C(X, E))
{K L : TopologicalSpace.Compacts X} (hKL : K ≤ L) : ‖f.restrict K‖ ≤ ‖f.restrict L‖ :=
(norm_le _ (norm_nonneg _)).mpr fun x => norm_coe_le_norm (f.restrict L) <| Set.inclusion hKL x
#align continuous_map.norm_restrict_mono_set ContinuousMap.norm_restrict_mono_set
end
section
variable {R : Type*} [NormedRing R]
instance : NormedRing C(α, R) :=
{ (inferInstance : NormedAddCommGroup C(α, R)), ContinuousMap.instRingContinuousMap with
norm_mul := fun f g => norm_mul_le (mkOfCompact f) (mkOfCompact g) }
end
section
variable {𝕜 : Type*} [NormedField 𝕜] [NormedSpace 𝕜 E]
instance normedSpace : NormedSpace 𝕜 C(α, E) where
norm_smul_le c f := (norm_smul_le c (mkOfCompact f) : _)
#align continuous_map.normed_space ContinuousMap.normedSpace
section
variable (α 𝕜 E)
/-- When `α` is compact and `𝕜` is a normed field,
the `𝕜`-algebra of bounded continuous maps `α →ᵇ β` is
`𝕜`-linearly isometric to `C(α, β)`.
-/
def linearIsometryBoundedOfCompact : C(α, E) ≃ₗᵢ[𝕜] α →ᵇ E :=
{ addEquivBoundedOfCompact α E with
map_smul' := fun c f => by
ext
norm_cast
norm_map' := fun f => rfl }
#align continuous_map.linear_isometry_bounded_of_compact ContinuousMap.linearIsometryBoundedOfCompact
variable {α E}
-- to match `BoundedContinuousFunction.evalClm`
/-- The evaluation at a point, as a continuous linear map from `C(α, 𝕜)` to `𝕜`. -/
def evalClm (x : α) : C(α, E) →L[𝕜] E :=
(BoundedContinuousFunction.evalClm 𝕜 x).comp
(linearIsometryBoundedOfCompact α E 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_map.eval_clm ContinuousMap.evalClm
end
-- this lemma and the next are the analogues of those autogenerated by `@[simps]` for
-- `equivBoundedOfCompact`, `addEquivBoundedOfCompact`
@[simp]
theorem linearIsometryBoundedOfCompact_symm_apply (f : α →ᵇ E) :
(linearIsometryBoundedOfCompact α E 𝕜).symm f = f.toContinuousMap :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_symm_apply ContinuousMap.linearIsometryBoundedOfCompact_symm_apply
@[simp]
theorem linearIsometryBoundedOfCompact_apply_apply (f : C(α, E)) (a : α) :
(linearIsometryBoundedOfCompact α E 𝕜 f) a = f a :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_apply_apply ContinuousMap.linearIsometryBoundedOfCompact_apply_apply
@[simp]
theorem linearIsometryBoundedOfCompact_toIsometryEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toIsometryEquiv = isometryEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_isometry_equiv ContinuousMap.linearIsometryBoundedOfCompact_toIsometryEquiv
@[simp] -- porting note: adjusted LHS because `simpNF` complained it simplified.
theorem linearIsometryBoundedOfCompact_toAddEquiv :
((linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv : C(α, E) ≃+ (α →ᵇ E)) =
addEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_add_equiv ContinuousMap.linearIsometryBoundedOfCompact_toAddEquiv
@[simp]
theorem linearIsometryBoundedOfCompact_of_compact_toEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv.toEquiv = equivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_of_compact_to_equiv ContinuousMap.linearIsometryBoundedOfCompact_of_compact_toEquiv
end
section
variable {𝕜 : Type*} {γ : Type*} [NormedField 𝕜] [NormedRing γ] [NormedAlgebra 𝕜 γ]
instance : NormedAlgebra 𝕜 C(α, γ) :=
{ ContinuousMap.normedSpace, ContinuousMap.algebra with }
end
end ContinuousMap
namespace ContinuousMap
section UniformContinuity
variable {α β : Type*}
variable [MetricSpace α] [CompactSpace α] [MetricSpace β]
/-!
We now set up some declarations making it convenient to use uniform continuity.
-/
theorem uniform_continuity (f : C(α, β)) (ε : ℝ) (h : 0 < ε) :
∃ δ > 0, ∀ {x y}, dist x y < δ → dist (f x) (f y) < ε :=
Metric.uniformContinuous_iff.mp (CompactSpace.uniformContinuous_of_continuous f.continuous) ε h
#align continuous_map.uniform_continuity ContinuousMap.uniform_continuity
-- This definition allows us to separate the choice of some `δ`,
-- and the corresponding use of `dist a b < δ → dist (f a) (f b) < ε`,
-- even across different declarations.
/-- An arbitrarily chosen modulus of uniform continuity for a given function `f` and `ε > 0`. -/
def modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) : ℝ :=
Classical.choose (uniform_continuity f ε h)
#align continuous_map.modulus ContinuousMap.modulus
theorem modulus_pos (f : C(α, β)) {ε : ℝ} {h : 0 < ε} : 0 < f.modulus ε h :=
(Classical.choose_spec (uniform_continuity f ε h)).1
#align continuous_map.modulus_pos ContinuousMap.modulus_pos
theorem dist_lt_of_dist_lt_modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) {a b : α}
(w : dist a b < f.modulus ε h) : dist (f a) (f b) < ε :=
(Classical.choose_spec (uniform_continuity f ε h)).2 w
#align continuous_map.dist_lt_of_dist_lt_modulus ContinuousMap.dist_lt_of_dist_lt_modulus
end UniformContinuity
end ContinuousMap
section CompLeft
variable (X : Type*) {𝕜 β γ : Type*} [TopologicalSpace X] [CompactSpace X]
[NontriviallyNormedField 𝕜]
variable [NormedAddCommGroup β] [NormedSpace 𝕜 β] [NormedAddCommGroup γ] [NormedSpace 𝕜 γ]
open ContinuousMap
/-- Postcomposition of continuous functions into a normed module by a continuous linear map is a
continuous linear map.
Transferred version of `ContinuousLinearMap.compLeftContinuousBounded`,
upgraded version of `ContinuousLinearMap.compLeftContinuous`,
similar to `LinearMap.compLeft`. -/
protected def ContinuousLinearMap.compLeftContinuousCompact (g : β →L[𝕜] γ) :
C(X, β) →L[𝕜] C(X, γ) :=
(linearIsometryBoundedOfCompact X γ 𝕜).symm.toLinearIsometry.toContinuousLinearMap.comp <|
(g.compLeftContinuousBounded X).comp <|
(linearIsometryBoundedOfCompact X β 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_linear_map.comp_left_continuous_compact ContinuousLinearMap.compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.toLinear_compLeftContinuousCompact (g : β →L[𝕜] γ) :
(g.compLeftContinuousCompact X : C(X, β) →ₗ[𝕜] C(X, γ)) = g.compLeftContinuous 𝕜 X := by
ext f
rfl
#align continuous_linear_map.to_linear_comp_left_continuous_compact ContinuousLinearMap.toLinear_compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.compLeftContinuousCompact_apply (g : β →L[𝕜] γ) (f : C(X, β)) (x : X) :
g.compLeftContinuousCompact X f x = g (f x) :=
rfl
#align continuous_linear_map.comp_left_continuous_compact_apply ContinuousLinearMap.compLeftContinuousCompact_apply
end CompLeft
namespace ContinuousMap
/-!
We now setup variations on `compRight* f`, where `f : C(X, Y)`
(that is, precomposition by a continuous map),
as a morphism `C(Y, T) → C(X, T)`, respecting various types of structure.
In particular:
* `compRightContinuousMap`, the bundled continuous map (for this we need `X Y` compact).
* `compRightHomeomorph`, when we precompose by a homeomorphism.
* `compRightAlgHom`, when `T = R` is a topological ring.
-/
section CompRight
/-- Precomposition by a continuous map is itself a continuous map between spaces of continuous maps.
-/
def compRightContinuousMap {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y)) : C(C(Y, T), C(X, T)) where
toFun g := g.comp f
continuous_toFun := by
refine' Metric.continuous_iff.mpr _
intro g ε ε_pos
refine' ⟨ε, ε_pos, fun g' h => _⟩
rw [ContinuousMap.dist_lt_iff ε_pos] at h ⊢
exact fun x => h (f x)
#align continuous_map.comp_right_continuous_map ContinuousMap.compRightContinuousMap
@[simp]
theorem compRightContinuousMap_apply {X Y : Type*} (T : Type*) [TopologicalSpace X]
[CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y))
(g : C(Y, T)) : (compRightContinuousMap T f) g = g.comp f :=
rfl
#align continuous_map.comp_right_continuous_map_apply ContinuousMap.compRightContinuousMap_apply
/-- Precomposition by a homeomorphism is itself a homeomorphism between spaces of continuous maps.
-/
def compRightHomeomorph {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : X ≃ₜ Y) : C(Y, T) ≃ₜ C(X, T) where
toFun := compRightContinuousMap T f.toContinuousMap
invFun := compRightContinuousMap T f.symm.toContinuousMap
left_inv g := ext fun _ => congr_arg g (f.apply_symm_apply _)
right_inv g := ext fun _ => congr_arg g (f.symm_apply_apply _)
#align continuous_map.comp_right_homeomorph ContinuousMap.compRightHomeomorph
theorem compRightAlgHom_continuous {X Y : Type*} (R A : Type*) [TopologicalSpace X]
[CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [CommSemiring R] [Semiring A]
[MetricSpace A] [TopologicalSemiring A] [Algebra R A] (f : C(X, Y)) :
Continuous (compRightAlgHom R A f) :=
map_continuous (compRightContinuousMap A f)
#align continuous_map.comp_right_alg_hom_continuous ContinuousMap.compRightAlgHom_continuous
end CompRight
section LocalNormalConvergence
/-! ### Local normal convergence
A sum of continuous functions (on a locally compact space) is "locally normally convergent" if the
sum of its sup-norms on any compact subset is summable. This implies convergence in the topology
of `C(X, E)` (i.e. locally uniform convergence). -/
open TopologicalSpace
variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
variable {E : Type*} [NormedAddCommGroup E] [CompleteSpace E]
theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F := by
refine' (ContinuousMap.exists_tendsto_compactOpen_iff_forall _).2 fun K hK => _
lift K to Compacts X using hK
have A : ∀ s : Finset ι, restrict (↑K) (∑ i in s, F i) = ∑ i in s, restrict K (F i) := by
intro s
ext1 x
simp
-- This used to be the end of the proof before leanprover/lean4#2644
erw [restrict_apply, restrict_apply, restrict_apply, restrict_apply]
simp? says simp only [coe_sum, Finset.sum_apply]
congr!
simpa only [HasSum, A] using (hF K).of_norm
#align continuous_map.summable_of_locally_summable_norm ContinuousMap.summable_of_locally_summable_norm
end LocalNormalConvergence
/-!
### Star structures
In this section, if `β` is a normed ⋆-group, then so is the space of
continuous functions from `α` to `β`, by using the star operation pointwise.
Furthermore, if `α` is compact and `β` is a C⋆-ring, then `C(α, β)` is a C⋆-ring. -/
section NormedSpace
variable {α : Type*} {β : Type*}
variable [TopologicalSpace α] [NormedAddCommGroup β] [StarAddMonoid β] [NormedStarGroup β]
theorem _root_.BoundedContinuousFunction.mkOfCompact_star [CompactSpace α] (f : C(α, β)) :
mkOfCompact (star f) = star (mkOfCompact f) :=
rfl
#align bounded_continuous_function.mk_of_compact_star BoundedContinuousFunction.mkOfCompact_star
instance [CompactSpace α] : NormedStarGroup C(α, β) where
norm_star f := by
rw [← BoundedContinuousFunction.norm_mkOfCompact, BoundedContinuousFunction.mkOfCompact_star,
norm_star, BoundedContinuousFunction.norm_mkOfCompact]
end NormedSpace
section CstarRing
variable {α : Type*} {β : Type*}
variable [TopologicalSpace α] [NormedRing β] [StarRing β]
instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f} := by
refine' le_antisymm _ _
· rw [← sq, ContinuousMap.norm_le _ (sq_nonneg _)]
intro x
simp only [ContinuousMap.coe_mul, coe_star, Pi.mul_apply, Pi.star_apply,
CstarRing.norm_star_mul_self, ← sq]
refine' sq_le_sq' _ _
· linarith [norm_nonneg (f x), norm_nonneg f]
· exact ContinuousMap.norm_coe_le_norm f x
· rw [← sq, ← Real.le_sqrt (norm_nonneg _) (norm_nonneg _),
ContinuousMap.norm_le _ (Real.sqrt_nonneg _)]
|
intro x
|
instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f} := by
refine' le_antisymm _ _
· rw [← sq, ContinuousMap.norm_le _ (sq_nonneg _)]
intro x
simp only [ContinuousMap.coe_mul, coe_star, Pi.mul_apply, Pi.star_apply,
CstarRing.norm_star_mul_self, ← sq]
refine' sq_le_sq' _ _
· linarith [norm_nonneg (f x), norm_nonneg f]
· exact ContinuousMap.norm_coe_le_norm f x
· rw [← sq, ← Real.le_sqrt (norm_nonneg _) (norm_nonneg _),
ContinuousMap.norm_le _ (Real.sqrt_nonneg _)]
|
Mathlib.Topology.ContinuousFunction.Compact.541_0.Mig2jTVnn2FLKEB
|
instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f}
|
Mathlib_Topology_ContinuousFunction_Compact
|
case refine'_2
α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : NormedRing β
inst✝² : StarRing β
inst✝¹ : CompactSpace α
inst✝ : CstarRing β
f : C(α, β)
x : α
⊢ ‖f x‖ ≤ Real.sqrt ‖star f * f‖
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compacts
#align_import topology.continuous_function.compact from "leanprover-community/mathlib"@"d3af0609f6db8691dffdc3e1fb7feb7da72698f2"
/-!
# Continuous functions on a compact space
Continuous functions `C(α, β)` from a compact space `α` to a metric space `β`
are automatically bounded, and so acquire various structures inherited from `α →ᵇ β`.
This file transfers these structures, and restates some lemmas
characterising these structures.
If you need a lemma which is proved about `α →ᵇ β` but not for `C(α, β)` when `α` is compact,
you should restate it here. You can also use
`ContinuousMap.equivBoundedOfCompact` to move functions back and forth.
-/
noncomputable section
open Topology Classical NNReal BoundedContinuousFunction BigOperators
open Set Filter Metric
open BoundedContinuousFunction
namespace ContinuousMap
variable {α β E : Type*} [TopologicalSpace α] [CompactSpace α] [MetricSpace β]
[NormedAddCommGroup E]
section
variable (α β)
/-- When `α` is compact, the bounded continuous maps `α →ᵇ β` are
equivalent to `C(α, β)`.
-/
@[simps (config := .asFn)]
def equivBoundedOfCompact : C(α, β) ≃ (α →ᵇ β) :=
⟨mkOfCompact, BoundedContinuousFunction.toContinuousMap, fun f => by
ext
rfl, fun f => by
ext
rfl⟩
#align continuous_map.equiv_bounded_of_compact ContinuousMap.equivBoundedOfCompact
theorem uniformInducing_equivBoundedOfCompact : UniformInducing (equivBoundedOfCompact α β) :=
UniformInducing.mk'
(by
simp only [hasBasis_compactConvergenceUniformity.mem_iff, uniformity_basis_dist_le.mem_iff]
exact fun s =>
⟨fun ⟨⟨a, b⟩, ⟨_, ⟨ε, hε, hb⟩⟩, hs⟩ =>
⟨{ p | ∀ x, (p.1 x, p.2 x) ∈ b }, ⟨ε, hε, fun _ h x => hb ((dist_le hε.le).mp h x)⟩,
fun f g h => hs fun x _ => h x⟩,
fun ⟨_, ⟨ε, hε, ht⟩, hs⟩ =>
⟨⟨Set.univ, { p | dist p.1 p.2 ≤ ε }⟩, ⟨isCompact_univ, ⟨ε, hε, fun _ h => h⟩⟩,
fun ⟨f, g⟩ h => hs _ _ (ht ((dist_le hε.le).mpr fun x => h x (mem_univ x)))⟩⟩)
#align continuous_map.uniform_inducing_equiv_bounded_of_compact ContinuousMap.uniformInducing_equivBoundedOfCompact
theorem uniformEmbedding_equivBoundedOfCompact : UniformEmbedding (equivBoundedOfCompact α β) :=
{ uniformInducing_equivBoundedOfCompact α β with inj := (equivBoundedOfCompact α β).injective }
#align continuous_map.uniform_embedding_equiv_bounded_of_compact ContinuousMap.uniformEmbedding_equivBoundedOfCompact
/-- When `α` is compact, the bounded continuous maps `α →ᵇ 𝕜` are
additively equivalent to `C(α, 𝕜)`.
-/
-- porting note: the following `simps` received a "maximum recursion depth" error
-- @[simps! (config := .asFn) apply symm_apply]
def addEquivBoundedOfCompact [AddMonoid β] [LipschitzAdd β] : C(α, β) ≃+ (α →ᵇ β) :=
({ toContinuousMapAddHom α β, (equivBoundedOfCompact α β).symm with } : (α →ᵇ β) ≃+ C(α, β)).symm
#align continuous_map.add_equiv_bounded_of_compact ContinuousMap.addEquivBoundedOfCompact
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_symm_apply [AddMonoid β] [LipschitzAdd β] :
⇑((addEquivBoundedOfCompact α β).symm) = toContinuousMapAddHom α β :=
rfl
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_apply [AddMonoid β] [LipschitzAdd β] :
⇑(addEquivBoundedOfCompact α β) = mkOfCompact :=
rfl
instance metricSpace : MetricSpace C(α, β) :=
(uniformEmbedding_equivBoundedOfCompact α β).comapMetricSpace _
#align continuous_map.metric_space ContinuousMap.metricSpace
/-- When `α` is compact, and `β` is a metric space, the bounded continuous maps `α →ᵇ β` are
isometric to `C(α, β)`.
-/
@[simps! (config := .asFn) toEquiv apply symm_apply]
def isometryEquivBoundedOfCompact : C(α, β) ≃ᵢ (α →ᵇ β) where
isometry_toFun _ _ := rfl
toEquiv := equivBoundedOfCompact α β
#align continuous_map.isometry_equiv_bounded_of_compact ContinuousMap.isometryEquivBoundedOfCompact
end
@[simp]
theorem _root_.BoundedContinuousFunction.dist_mkOfCompact (f g : C(α, β)) :
dist (mkOfCompact f) (mkOfCompact g) = dist f g :=
rfl
#align bounded_continuous_function.dist_mk_of_compact BoundedContinuousFunction.dist_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.dist_toContinuousMap (f g : α →ᵇ β) :
dist f.toContinuousMap g.toContinuousMap = dist f g :=
rfl
#align bounded_continuous_function.dist_to_continuous_map BoundedContinuousFunction.dist_toContinuousMap
open BoundedContinuousFunction
section
variable {f g : C(α, β)} {C : ℝ}
/-- The pointwise distance is controlled by the distance between functions, by definition. -/
theorem dist_apply_le_dist (x : α) : dist (f x) (g x) ≤ dist f g := by
simp only [← dist_mkOfCompact, dist_coe_le_dist, ← mkOfCompact_apply]
#align continuous_map.dist_apply_le_dist ContinuousMap.dist_apply_le_dist
/-- The distance between two functions is controlled by the supremum of the pointwise distances. -/
theorem dist_le (C0 : (0 : ℝ) ≤ C) : dist f g ≤ C ↔ ∀ x : α, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le C0, mkOfCompact_apply]
#align continuous_map.dist_le ContinuousMap.dist_le
theorem dist_le_iff_of_nonempty [Nonempty α] : dist f g ≤ C ↔ ∀ x, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le_iff_of_nonempty,
mkOfCompact_apply]
#align continuous_map.dist_le_iff_of_nonempty ContinuousMap.dist_le_iff_of_nonempty
theorem dist_lt_iff_of_nonempty [Nonempty α] : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
simp only [← dist_mkOfCompact, dist_lt_iff_of_nonempty_compact, mkOfCompact_apply]
#align continuous_map.dist_lt_iff_of_nonempty ContinuousMap.dist_lt_iff_of_nonempty
theorem dist_lt_of_nonempty [Nonempty α] (w : ∀ x : α, dist (f x) (g x) < C) : dist f g < C :=
dist_lt_iff_of_nonempty.2 w
#align continuous_map.dist_lt_of_nonempty ContinuousMap.dist_lt_of_nonempty
theorem dist_lt_iff (C0 : (0 : ℝ) < C) : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
rw [← dist_mkOfCompact, dist_lt_iff_of_compact C0]
simp only [mkOfCompact_apply]
#align continuous_map.dist_lt_iff ContinuousMap.dist_lt_iff
end
instance [CompleteSpace β] : CompleteSpace C(α, β) :=
(isometryEquivBoundedOfCompact α β).completeSpace
/-- See also `ContinuousMap.continuous_eval'`. -/
@[continuity]
theorem continuous_eval : Continuous fun p : C(α, β) × α => p.1 p.2 :=
continuous_eval.comp ((isometryEquivBoundedOfCompact α β).continuous.prod_map continuous_id)
#align continuous_map.continuous_eval ContinuousMap.continuous_eval
-- TODO at some point we will need lemmas characterising this norm!
-- At the moment the only way to reason about it is to transfer `f : C(α,E)` back to `α →ᵇ E`.
instance : Norm C(α, E) where norm x := dist x 0
@[simp]
theorem _root_.BoundedContinuousFunction.norm_mkOfCompact (f : C(α, E)) : ‖mkOfCompact f‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_mk_of_compact BoundedContinuousFunction.norm_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.norm_toContinuousMap_eq (f : α →ᵇ E) :
‖f.toContinuousMap‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_to_continuous_map_eq BoundedContinuousFunction.norm_toContinuousMap_eq
open BoundedContinuousFunction
instance : NormedAddCommGroup C(α, E) :=
{ ContinuousMap.metricSpace _ _,
ContinuousMap.instAddCommGroupContinuousMap with
dist_eq := fun x y => by
rw [← norm_mkOfCompact, ← dist_mkOfCompact, dist_eq_norm, mkOfCompact_sub]
dist := dist
norm := norm }
instance [Nonempty α] [One E] [NormOneClass E] : NormOneClass C(α, E) where
norm_one := by simp only [← norm_mkOfCompact, mkOfCompact_one, norm_one]
section
variable (f : C(α, E))
-- The corresponding lemmas for `BoundedContinuousFunction` are stated with `{f}`,
-- and so can not be used in dot notation.
theorem norm_coe_le_norm (x : α) : ‖f x‖ ≤ ‖f‖ :=
(mkOfCompact f).norm_coe_le_norm x
#align continuous_map.norm_coe_le_norm ContinuousMap.norm_coe_le_norm
/-- Distance between the images of any two points is at most twice the norm of the function. -/
theorem dist_le_two_norm (x y : α) : dist (f x) (f y) ≤ 2 * ‖f‖ :=
(mkOfCompact f).dist_le_two_norm x y
#align continuous_map.dist_le_two_norm ContinuousMap.dist_le_two_norm
/-- The norm of a function is controlled by the supremum of the pointwise norms. -/
theorem norm_le {C : ℝ} (C0 : (0 : ℝ) ≤ C) : ‖f‖ ≤ C ↔ ∀ x : α, ‖f x‖ ≤ C :=
@BoundedContinuousFunction.norm_le _ _ _ _ (mkOfCompact f) _ C0
#align continuous_map.norm_le ContinuousMap.norm_le
theorem norm_le_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ ≤ M ↔ ∀ x, ‖f x‖ ≤ M :=
@BoundedContinuousFunction.norm_le_of_nonempty _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_le_of_nonempty ContinuousMap.norm_le_of_nonempty
theorem norm_lt_iff {M : ℝ} (M0 : 0 < M) : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_compact _ _ _ _ _ (mkOfCompact f) _ M0
#align continuous_map.norm_lt_iff ContinuousMap.norm_lt_iff
theorem nnnorm_lt_iff {M : ℝ≥0} (M0 : 0 < M) : ‖f‖₊ < M ↔ ∀ x : α, ‖f x‖₊ < M :=
f.norm_lt_iff M0
#align continuous_map.nnnorm_lt_iff ContinuousMap.nnnorm_lt_iff
theorem norm_lt_iff_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_nonempty_compact _ _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_lt_iff_of_nonempty ContinuousMap.norm_lt_iff_of_nonempty
theorem nnnorm_lt_iff_of_nonempty [Nonempty α] {M : ℝ≥0} : ‖f‖₊ < M ↔ ∀ x, ‖f x‖₊ < M :=
f.norm_lt_iff_of_nonempty
#align continuous_map.nnnorm_lt_iff_of_nonempty ContinuousMap.nnnorm_lt_iff_of_nonempty
theorem apply_le_norm (f : C(α, ℝ)) (x : α) : f x ≤ ‖f‖ :=
le_trans (le_abs.mpr (Or.inl (le_refl (f x)))) (f.norm_coe_le_norm x)
#align continuous_map.apply_le_norm ContinuousMap.apply_le_norm
theorem neg_norm_le_apply (f : C(α, ℝ)) (x : α) : -‖f‖ ≤ f x :=
le_trans (neg_le_neg (f.norm_coe_le_norm x)) (neg_le.mp (neg_le_abs_self (f x)))
#align continuous_map.neg_norm_le_apply ContinuousMap.neg_norm_le_apply
theorem norm_eq_iSup_norm : ‖f‖ = ⨆ x : α, ‖f x‖ :=
(mkOfCompact f).norm_eq_iSup_norm
#align continuous_map.norm_eq_supr_norm ContinuousMap.norm_eq_iSup_norm
theorem norm_restrict_mono_set {X : Type*} [TopologicalSpace X] (f : C(X, E))
{K L : TopologicalSpace.Compacts X} (hKL : K ≤ L) : ‖f.restrict K‖ ≤ ‖f.restrict L‖ :=
(norm_le _ (norm_nonneg _)).mpr fun x => norm_coe_le_norm (f.restrict L) <| Set.inclusion hKL x
#align continuous_map.norm_restrict_mono_set ContinuousMap.norm_restrict_mono_set
end
section
variable {R : Type*} [NormedRing R]
instance : NormedRing C(α, R) :=
{ (inferInstance : NormedAddCommGroup C(α, R)), ContinuousMap.instRingContinuousMap with
norm_mul := fun f g => norm_mul_le (mkOfCompact f) (mkOfCompact g) }
end
section
variable {𝕜 : Type*} [NormedField 𝕜] [NormedSpace 𝕜 E]
instance normedSpace : NormedSpace 𝕜 C(α, E) where
norm_smul_le c f := (norm_smul_le c (mkOfCompact f) : _)
#align continuous_map.normed_space ContinuousMap.normedSpace
section
variable (α 𝕜 E)
/-- When `α` is compact and `𝕜` is a normed field,
the `𝕜`-algebra of bounded continuous maps `α →ᵇ β` is
`𝕜`-linearly isometric to `C(α, β)`.
-/
def linearIsometryBoundedOfCompact : C(α, E) ≃ₗᵢ[𝕜] α →ᵇ E :=
{ addEquivBoundedOfCompact α E with
map_smul' := fun c f => by
ext
norm_cast
norm_map' := fun f => rfl }
#align continuous_map.linear_isometry_bounded_of_compact ContinuousMap.linearIsometryBoundedOfCompact
variable {α E}
-- to match `BoundedContinuousFunction.evalClm`
/-- The evaluation at a point, as a continuous linear map from `C(α, 𝕜)` to `𝕜`. -/
def evalClm (x : α) : C(α, E) →L[𝕜] E :=
(BoundedContinuousFunction.evalClm 𝕜 x).comp
(linearIsometryBoundedOfCompact α E 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_map.eval_clm ContinuousMap.evalClm
end
-- this lemma and the next are the analogues of those autogenerated by `@[simps]` for
-- `equivBoundedOfCompact`, `addEquivBoundedOfCompact`
@[simp]
theorem linearIsometryBoundedOfCompact_symm_apply (f : α →ᵇ E) :
(linearIsometryBoundedOfCompact α E 𝕜).symm f = f.toContinuousMap :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_symm_apply ContinuousMap.linearIsometryBoundedOfCompact_symm_apply
@[simp]
theorem linearIsometryBoundedOfCompact_apply_apply (f : C(α, E)) (a : α) :
(linearIsometryBoundedOfCompact α E 𝕜 f) a = f a :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_apply_apply ContinuousMap.linearIsometryBoundedOfCompact_apply_apply
@[simp]
theorem linearIsometryBoundedOfCompact_toIsometryEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toIsometryEquiv = isometryEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_isometry_equiv ContinuousMap.linearIsometryBoundedOfCompact_toIsometryEquiv
@[simp] -- porting note: adjusted LHS because `simpNF` complained it simplified.
theorem linearIsometryBoundedOfCompact_toAddEquiv :
((linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv : C(α, E) ≃+ (α →ᵇ E)) =
addEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_add_equiv ContinuousMap.linearIsometryBoundedOfCompact_toAddEquiv
@[simp]
theorem linearIsometryBoundedOfCompact_of_compact_toEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv.toEquiv = equivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_of_compact_to_equiv ContinuousMap.linearIsometryBoundedOfCompact_of_compact_toEquiv
end
section
variable {𝕜 : Type*} {γ : Type*} [NormedField 𝕜] [NormedRing γ] [NormedAlgebra 𝕜 γ]
instance : NormedAlgebra 𝕜 C(α, γ) :=
{ ContinuousMap.normedSpace, ContinuousMap.algebra with }
end
end ContinuousMap
namespace ContinuousMap
section UniformContinuity
variable {α β : Type*}
variable [MetricSpace α] [CompactSpace α] [MetricSpace β]
/-!
We now set up some declarations making it convenient to use uniform continuity.
-/
theorem uniform_continuity (f : C(α, β)) (ε : ℝ) (h : 0 < ε) :
∃ δ > 0, ∀ {x y}, dist x y < δ → dist (f x) (f y) < ε :=
Metric.uniformContinuous_iff.mp (CompactSpace.uniformContinuous_of_continuous f.continuous) ε h
#align continuous_map.uniform_continuity ContinuousMap.uniform_continuity
-- This definition allows us to separate the choice of some `δ`,
-- and the corresponding use of `dist a b < δ → dist (f a) (f b) < ε`,
-- even across different declarations.
/-- An arbitrarily chosen modulus of uniform continuity for a given function `f` and `ε > 0`. -/
def modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) : ℝ :=
Classical.choose (uniform_continuity f ε h)
#align continuous_map.modulus ContinuousMap.modulus
theorem modulus_pos (f : C(α, β)) {ε : ℝ} {h : 0 < ε} : 0 < f.modulus ε h :=
(Classical.choose_spec (uniform_continuity f ε h)).1
#align continuous_map.modulus_pos ContinuousMap.modulus_pos
theorem dist_lt_of_dist_lt_modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) {a b : α}
(w : dist a b < f.modulus ε h) : dist (f a) (f b) < ε :=
(Classical.choose_spec (uniform_continuity f ε h)).2 w
#align continuous_map.dist_lt_of_dist_lt_modulus ContinuousMap.dist_lt_of_dist_lt_modulus
end UniformContinuity
end ContinuousMap
section CompLeft
variable (X : Type*) {𝕜 β γ : Type*} [TopologicalSpace X] [CompactSpace X]
[NontriviallyNormedField 𝕜]
variable [NormedAddCommGroup β] [NormedSpace 𝕜 β] [NormedAddCommGroup γ] [NormedSpace 𝕜 γ]
open ContinuousMap
/-- Postcomposition of continuous functions into a normed module by a continuous linear map is a
continuous linear map.
Transferred version of `ContinuousLinearMap.compLeftContinuousBounded`,
upgraded version of `ContinuousLinearMap.compLeftContinuous`,
similar to `LinearMap.compLeft`. -/
protected def ContinuousLinearMap.compLeftContinuousCompact (g : β →L[𝕜] γ) :
C(X, β) →L[𝕜] C(X, γ) :=
(linearIsometryBoundedOfCompact X γ 𝕜).symm.toLinearIsometry.toContinuousLinearMap.comp <|
(g.compLeftContinuousBounded X).comp <|
(linearIsometryBoundedOfCompact X β 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_linear_map.comp_left_continuous_compact ContinuousLinearMap.compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.toLinear_compLeftContinuousCompact (g : β →L[𝕜] γ) :
(g.compLeftContinuousCompact X : C(X, β) →ₗ[𝕜] C(X, γ)) = g.compLeftContinuous 𝕜 X := by
ext f
rfl
#align continuous_linear_map.to_linear_comp_left_continuous_compact ContinuousLinearMap.toLinear_compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.compLeftContinuousCompact_apply (g : β →L[𝕜] γ) (f : C(X, β)) (x : X) :
g.compLeftContinuousCompact X f x = g (f x) :=
rfl
#align continuous_linear_map.comp_left_continuous_compact_apply ContinuousLinearMap.compLeftContinuousCompact_apply
end CompLeft
namespace ContinuousMap
/-!
We now setup variations on `compRight* f`, where `f : C(X, Y)`
(that is, precomposition by a continuous map),
as a morphism `C(Y, T) → C(X, T)`, respecting various types of structure.
In particular:
* `compRightContinuousMap`, the bundled continuous map (for this we need `X Y` compact).
* `compRightHomeomorph`, when we precompose by a homeomorphism.
* `compRightAlgHom`, when `T = R` is a topological ring.
-/
section CompRight
/-- Precomposition by a continuous map is itself a continuous map between spaces of continuous maps.
-/
def compRightContinuousMap {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y)) : C(C(Y, T), C(X, T)) where
toFun g := g.comp f
continuous_toFun := by
refine' Metric.continuous_iff.mpr _
intro g ε ε_pos
refine' ⟨ε, ε_pos, fun g' h => _⟩
rw [ContinuousMap.dist_lt_iff ε_pos] at h ⊢
exact fun x => h (f x)
#align continuous_map.comp_right_continuous_map ContinuousMap.compRightContinuousMap
@[simp]
theorem compRightContinuousMap_apply {X Y : Type*} (T : Type*) [TopologicalSpace X]
[CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y))
(g : C(Y, T)) : (compRightContinuousMap T f) g = g.comp f :=
rfl
#align continuous_map.comp_right_continuous_map_apply ContinuousMap.compRightContinuousMap_apply
/-- Precomposition by a homeomorphism is itself a homeomorphism between spaces of continuous maps.
-/
def compRightHomeomorph {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : X ≃ₜ Y) : C(Y, T) ≃ₜ C(X, T) where
toFun := compRightContinuousMap T f.toContinuousMap
invFun := compRightContinuousMap T f.symm.toContinuousMap
left_inv g := ext fun _ => congr_arg g (f.apply_symm_apply _)
right_inv g := ext fun _ => congr_arg g (f.symm_apply_apply _)
#align continuous_map.comp_right_homeomorph ContinuousMap.compRightHomeomorph
theorem compRightAlgHom_continuous {X Y : Type*} (R A : Type*) [TopologicalSpace X]
[CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [CommSemiring R] [Semiring A]
[MetricSpace A] [TopologicalSemiring A] [Algebra R A] (f : C(X, Y)) :
Continuous (compRightAlgHom R A f) :=
map_continuous (compRightContinuousMap A f)
#align continuous_map.comp_right_alg_hom_continuous ContinuousMap.compRightAlgHom_continuous
end CompRight
section LocalNormalConvergence
/-! ### Local normal convergence
A sum of continuous functions (on a locally compact space) is "locally normally convergent" if the
sum of its sup-norms on any compact subset is summable. This implies convergence in the topology
of `C(X, E)` (i.e. locally uniform convergence). -/
open TopologicalSpace
variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
variable {E : Type*} [NormedAddCommGroup E] [CompleteSpace E]
theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F := by
refine' (ContinuousMap.exists_tendsto_compactOpen_iff_forall _).2 fun K hK => _
lift K to Compacts X using hK
have A : ∀ s : Finset ι, restrict (↑K) (∑ i in s, F i) = ∑ i in s, restrict K (F i) := by
intro s
ext1 x
simp
-- This used to be the end of the proof before leanprover/lean4#2644
erw [restrict_apply, restrict_apply, restrict_apply, restrict_apply]
simp? says simp only [coe_sum, Finset.sum_apply]
congr!
simpa only [HasSum, A] using (hF K).of_norm
#align continuous_map.summable_of_locally_summable_norm ContinuousMap.summable_of_locally_summable_norm
end LocalNormalConvergence
/-!
### Star structures
In this section, if `β` is a normed ⋆-group, then so is the space of
continuous functions from `α` to `β`, by using the star operation pointwise.
Furthermore, if `α` is compact and `β` is a C⋆-ring, then `C(α, β)` is a C⋆-ring. -/
section NormedSpace
variable {α : Type*} {β : Type*}
variable [TopologicalSpace α] [NormedAddCommGroup β] [StarAddMonoid β] [NormedStarGroup β]
theorem _root_.BoundedContinuousFunction.mkOfCompact_star [CompactSpace α] (f : C(α, β)) :
mkOfCompact (star f) = star (mkOfCompact f) :=
rfl
#align bounded_continuous_function.mk_of_compact_star BoundedContinuousFunction.mkOfCompact_star
instance [CompactSpace α] : NormedStarGroup C(α, β) where
norm_star f := by
rw [← BoundedContinuousFunction.norm_mkOfCompact, BoundedContinuousFunction.mkOfCompact_star,
norm_star, BoundedContinuousFunction.norm_mkOfCompact]
end NormedSpace
section CstarRing
variable {α : Type*} {β : Type*}
variable [TopologicalSpace α] [NormedRing β] [StarRing β]
instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f} := by
refine' le_antisymm _ _
· rw [← sq, ContinuousMap.norm_le _ (sq_nonneg _)]
intro x
simp only [ContinuousMap.coe_mul, coe_star, Pi.mul_apply, Pi.star_apply,
CstarRing.norm_star_mul_self, ← sq]
refine' sq_le_sq' _ _
· linarith [norm_nonneg (f x), norm_nonneg f]
· exact ContinuousMap.norm_coe_le_norm f x
· rw [← sq, ← Real.le_sqrt (norm_nonneg _) (norm_nonneg _),
ContinuousMap.norm_le _ (Real.sqrt_nonneg _)]
intro x
|
rw [Real.le_sqrt (norm_nonneg _) (norm_nonneg _), sq, ← CstarRing.norm_star_mul_self]
|
instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f} := by
refine' le_antisymm _ _
· rw [← sq, ContinuousMap.norm_le _ (sq_nonneg _)]
intro x
simp only [ContinuousMap.coe_mul, coe_star, Pi.mul_apply, Pi.star_apply,
CstarRing.norm_star_mul_self, ← sq]
refine' sq_le_sq' _ _
· linarith [norm_nonneg (f x), norm_nonneg f]
· exact ContinuousMap.norm_coe_le_norm f x
· rw [← sq, ← Real.le_sqrt (norm_nonneg _) (norm_nonneg _),
ContinuousMap.norm_le _ (Real.sqrt_nonneg _)]
intro x
|
Mathlib.Topology.ContinuousFunction.Compact.541_0.Mig2jTVnn2FLKEB
|
instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f}
|
Mathlib_Topology_ContinuousFunction_Compact
|
case refine'_2
α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : NormedRing β
inst✝² : StarRing β
inst✝¹ : CompactSpace α
inst✝ : CstarRing β
f : C(α, β)
x : α
⊢ ‖star (f x) * f x‖ ≤ ‖star f * f‖
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compacts
#align_import topology.continuous_function.compact from "leanprover-community/mathlib"@"d3af0609f6db8691dffdc3e1fb7feb7da72698f2"
/-!
# Continuous functions on a compact space
Continuous functions `C(α, β)` from a compact space `α` to a metric space `β`
are automatically bounded, and so acquire various structures inherited from `α →ᵇ β`.
This file transfers these structures, and restates some lemmas
characterising these structures.
If you need a lemma which is proved about `α →ᵇ β` but not for `C(α, β)` when `α` is compact,
you should restate it here. You can also use
`ContinuousMap.equivBoundedOfCompact` to move functions back and forth.
-/
noncomputable section
open Topology Classical NNReal BoundedContinuousFunction BigOperators
open Set Filter Metric
open BoundedContinuousFunction
namespace ContinuousMap
variable {α β E : Type*} [TopologicalSpace α] [CompactSpace α] [MetricSpace β]
[NormedAddCommGroup E]
section
variable (α β)
/-- When `α` is compact, the bounded continuous maps `α →ᵇ β` are
equivalent to `C(α, β)`.
-/
@[simps (config := .asFn)]
def equivBoundedOfCompact : C(α, β) ≃ (α →ᵇ β) :=
⟨mkOfCompact, BoundedContinuousFunction.toContinuousMap, fun f => by
ext
rfl, fun f => by
ext
rfl⟩
#align continuous_map.equiv_bounded_of_compact ContinuousMap.equivBoundedOfCompact
theorem uniformInducing_equivBoundedOfCompact : UniformInducing (equivBoundedOfCompact α β) :=
UniformInducing.mk'
(by
simp only [hasBasis_compactConvergenceUniformity.mem_iff, uniformity_basis_dist_le.mem_iff]
exact fun s =>
⟨fun ⟨⟨a, b⟩, ⟨_, ⟨ε, hε, hb⟩⟩, hs⟩ =>
⟨{ p | ∀ x, (p.1 x, p.2 x) ∈ b }, ⟨ε, hε, fun _ h x => hb ((dist_le hε.le).mp h x)⟩,
fun f g h => hs fun x _ => h x⟩,
fun ⟨_, ⟨ε, hε, ht⟩, hs⟩ =>
⟨⟨Set.univ, { p | dist p.1 p.2 ≤ ε }⟩, ⟨isCompact_univ, ⟨ε, hε, fun _ h => h⟩⟩,
fun ⟨f, g⟩ h => hs _ _ (ht ((dist_le hε.le).mpr fun x => h x (mem_univ x)))⟩⟩)
#align continuous_map.uniform_inducing_equiv_bounded_of_compact ContinuousMap.uniformInducing_equivBoundedOfCompact
theorem uniformEmbedding_equivBoundedOfCompact : UniformEmbedding (equivBoundedOfCompact α β) :=
{ uniformInducing_equivBoundedOfCompact α β with inj := (equivBoundedOfCompact α β).injective }
#align continuous_map.uniform_embedding_equiv_bounded_of_compact ContinuousMap.uniformEmbedding_equivBoundedOfCompact
/-- When `α` is compact, the bounded continuous maps `α →ᵇ 𝕜` are
additively equivalent to `C(α, 𝕜)`.
-/
-- porting note: the following `simps` received a "maximum recursion depth" error
-- @[simps! (config := .asFn) apply symm_apply]
def addEquivBoundedOfCompact [AddMonoid β] [LipschitzAdd β] : C(α, β) ≃+ (α →ᵇ β) :=
({ toContinuousMapAddHom α β, (equivBoundedOfCompact α β).symm with } : (α →ᵇ β) ≃+ C(α, β)).symm
#align continuous_map.add_equiv_bounded_of_compact ContinuousMap.addEquivBoundedOfCompact
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_symm_apply [AddMonoid β] [LipschitzAdd β] :
⇑((addEquivBoundedOfCompact α β).symm) = toContinuousMapAddHom α β :=
rfl
-- porting note: added this `simp` lemma manually because of the `simps` error above
@[simp]
theorem addEquivBoundedOfCompact_apply [AddMonoid β] [LipschitzAdd β] :
⇑(addEquivBoundedOfCompact α β) = mkOfCompact :=
rfl
instance metricSpace : MetricSpace C(α, β) :=
(uniformEmbedding_equivBoundedOfCompact α β).comapMetricSpace _
#align continuous_map.metric_space ContinuousMap.metricSpace
/-- When `α` is compact, and `β` is a metric space, the bounded continuous maps `α →ᵇ β` are
isometric to `C(α, β)`.
-/
@[simps! (config := .asFn) toEquiv apply symm_apply]
def isometryEquivBoundedOfCompact : C(α, β) ≃ᵢ (α →ᵇ β) where
isometry_toFun _ _ := rfl
toEquiv := equivBoundedOfCompact α β
#align continuous_map.isometry_equiv_bounded_of_compact ContinuousMap.isometryEquivBoundedOfCompact
end
@[simp]
theorem _root_.BoundedContinuousFunction.dist_mkOfCompact (f g : C(α, β)) :
dist (mkOfCompact f) (mkOfCompact g) = dist f g :=
rfl
#align bounded_continuous_function.dist_mk_of_compact BoundedContinuousFunction.dist_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.dist_toContinuousMap (f g : α →ᵇ β) :
dist f.toContinuousMap g.toContinuousMap = dist f g :=
rfl
#align bounded_continuous_function.dist_to_continuous_map BoundedContinuousFunction.dist_toContinuousMap
open BoundedContinuousFunction
section
variable {f g : C(α, β)} {C : ℝ}
/-- The pointwise distance is controlled by the distance between functions, by definition. -/
theorem dist_apply_le_dist (x : α) : dist (f x) (g x) ≤ dist f g := by
simp only [← dist_mkOfCompact, dist_coe_le_dist, ← mkOfCompact_apply]
#align continuous_map.dist_apply_le_dist ContinuousMap.dist_apply_le_dist
/-- The distance between two functions is controlled by the supremum of the pointwise distances. -/
theorem dist_le (C0 : (0 : ℝ) ≤ C) : dist f g ≤ C ↔ ∀ x : α, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le C0, mkOfCompact_apply]
#align continuous_map.dist_le ContinuousMap.dist_le
theorem dist_le_iff_of_nonempty [Nonempty α] : dist f g ≤ C ↔ ∀ x, dist (f x) (g x) ≤ C := by
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le_iff_of_nonempty,
mkOfCompact_apply]
#align continuous_map.dist_le_iff_of_nonempty ContinuousMap.dist_le_iff_of_nonempty
theorem dist_lt_iff_of_nonempty [Nonempty α] : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
simp only [← dist_mkOfCompact, dist_lt_iff_of_nonempty_compact, mkOfCompact_apply]
#align continuous_map.dist_lt_iff_of_nonempty ContinuousMap.dist_lt_iff_of_nonempty
theorem dist_lt_of_nonempty [Nonempty α] (w : ∀ x : α, dist (f x) (g x) < C) : dist f g < C :=
dist_lt_iff_of_nonempty.2 w
#align continuous_map.dist_lt_of_nonempty ContinuousMap.dist_lt_of_nonempty
theorem dist_lt_iff (C0 : (0 : ℝ) < C) : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
rw [← dist_mkOfCompact, dist_lt_iff_of_compact C0]
simp only [mkOfCompact_apply]
#align continuous_map.dist_lt_iff ContinuousMap.dist_lt_iff
end
instance [CompleteSpace β] : CompleteSpace C(α, β) :=
(isometryEquivBoundedOfCompact α β).completeSpace
/-- See also `ContinuousMap.continuous_eval'`. -/
@[continuity]
theorem continuous_eval : Continuous fun p : C(α, β) × α => p.1 p.2 :=
continuous_eval.comp ((isometryEquivBoundedOfCompact α β).continuous.prod_map continuous_id)
#align continuous_map.continuous_eval ContinuousMap.continuous_eval
-- TODO at some point we will need lemmas characterising this norm!
-- At the moment the only way to reason about it is to transfer `f : C(α,E)` back to `α →ᵇ E`.
instance : Norm C(α, E) where norm x := dist x 0
@[simp]
theorem _root_.BoundedContinuousFunction.norm_mkOfCompact (f : C(α, E)) : ‖mkOfCompact f‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_mk_of_compact BoundedContinuousFunction.norm_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.norm_toContinuousMap_eq (f : α →ᵇ E) :
‖f.toContinuousMap‖ = ‖f‖ :=
rfl
#align bounded_continuous_function.norm_to_continuous_map_eq BoundedContinuousFunction.norm_toContinuousMap_eq
open BoundedContinuousFunction
instance : NormedAddCommGroup C(α, E) :=
{ ContinuousMap.metricSpace _ _,
ContinuousMap.instAddCommGroupContinuousMap with
dist_eq := fun x y => by
rw [← norm_mkOfCompact, ← dist_mkOfCompact, dist_eq_norm, mkOfCompact_sub]
dist := dist
norm := norm }
instance [Nonempty α] [One E] [NormOneClass E] : NormOneClass C(α, E) where
norm_one := by simp only [← norm_mkOfCompact, mkOfCompact_one, norm_one]
section
variable (f : C(α, E))
-- The corresponding lemmas for `BoundedContinuousFunction` are stated with `{f}`,
-- and so can not be used in dot notation.
theorem norm_coe_le_norm (x : α) : ‖f x‖ ≤ ‖f‖ :=
(mkOfCompact f).norm_coe_le_norm x
#align continuous_map.norm_coe_le_norm ContinuousMap.norm_coe_le_norm
/-- Distance between the images of any two points is at most twice the norm of the function. -/
theorem dist_le_two_norm (x y : α) : dist (f x) (f y) ≤ 2 * ‖f‖ :=
(mkOfCompact f).dist_le_two_norm x y
#align continuous_map.dist_le_two_norm ContinuousMap.dist_le_two_norm
/-- The norm of a function is controlled by the supremum of the pointwise norms. -/
theorem norm_le {C : ℝ} (C0 : (0 : ℝ) ≤ C) : ‖f‖ ≤ C ↔ ∀ x : α, ‖f x‖ ≤ C :=
@BoundedContinuousFunction.norm_le _ _ _ _ (mkOfCompact f) _ C0
#align continuous_map.norm_le ContinuousMap.norm_le
theorem norm_le_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ ≤ M ↔ ∀ x, ‖f x‖ ≤ M :=
@BoundedContinuousFunction.norm_le_of_nonempty _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_le_of_nonempty ContinuousMap.norm_le_of_nonempty
theorem norm_lt_iff {M : ℝ} (M0 : 0 < M) : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_compact _ _ _ _ _ (mkOfCompact f) _ M0
#align continuous_map.norm_lt_iff ContinuousMap.norm_lt_iff
theorem nnnorm_lt_iff {M : ℝ≥0} (M0 : 0 < M) : ‖f‖₊ < M ↔ ∀ x : α, ‖f x‖₊ < M :=
f.norm_lt_iff M0
#align continuous_map.nnnorm_lt_iff ContinuousMap.nnnorm_lt_iff
theorem norm_lt_iff_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M :=
@BoundedContinuousFunction.norm_lt_iff_of_nonempty_compact _ _ _ _ _ _ (mkOfCompact f) _
#align continuous_map.norm_lt_iff_of_nonempty ContinuousMap.norm_lt_iff_of_nonempty
theorem nnnorm_lt_iff_of_nonempty [Nonempty α] {M : ℝ≥0} : ‖f‖₊ < M ↔ ∀ x, ‖f x‖₊ < M :=
f.norm_lt_iff_of_nonempty
#align continuous_map.nnnorm_lt_iff_of_nonempty ContinuousMap.nnnorm_lt_iff_of_nonempty
theorem apply_le_norm (f : C(α, ℝ)) (x : α) : f x ≤ ‖f‖ :=
le_trans (le_abs.mpr (Or.inl (le_refl (f x)))) (f.norm_coe_le_norm x)
#align continuous_map.apply_le_norm ContinuousMap.apply_le_norm
theorem neg_norm_le_apply (f : C(α, ℝ)) (x : α) : -‖f‖ ≤ f x :=
le_trans (neg_le_neg (f.norm_coe_le_norm x)) (neg_le.mp (neg_le_abs_self (f x)))
#align continuous_map.neg_norm_le_apply ContinuousMap.neg_norm_le_apply
theorem norm_eq_iSup_norm : ‖f‖ = ⨆ x : α, ‖f x‖ :=
(mkOfCompact f).norm_eq_iSup_norm
#align continuous_map.norm_eq_supr_norm ContinuousMap.norm_eq_iSup_norm
theorem norm_restrict_mono_set {X : Type*} [TopologicalSpace X] (f : C(X, E))
{K L : TopologicalSpace.Compacts X} (hKL : K ≤ L) : ‖f.restrict K‖ ≤ ‖f.restrict L‖ :=
(norm_le _ (norm_nonneg _)).mpr fun x => norm_coe_le_norm (f.restrict L) <| Set.inclusion hKL x
#align continuous_map.norm_restrict_mono_set ContinuousMap.norm_restrict_mono_set
end
section
variable {R : Type*} [NormedRing R]
instance : NormedRing C(α, R) :=
{ (inferInstance : NormedAddCommGroup C(α, R)), ContinuousMap.instRingContinuousMap with
norm_mul := fun f g => norm_mul_le (mkOfCompact f) (mkOfCompact g) }
end
section
variable {𝕜 : Type*} [NormedField 𝕜] [NormedSpace 𝕜 E]
instance normedSpace : NormedSpace 𝕜 C(α, E) where
norm_smul_le c f := (norm_smul_le c (mkOfCompact f) : _)
#align continuous_map.normed_space ContinuousMap.normedSpace
section
variable (α 𝕜 E)
/-- When `α` is compact and `𝕜` is a normed field,
the `𝕜`-algebra of bounded continuous maps `α →ᵇ β` is
`𝕜`-linearly isometric to `C(α, β)`.
-/
def linearIsometryBoundedOfCompact : C(α, E) ≃ₗᵢ[𝕜] α →ᵇ E :=
{ addEquivBoundedOfCompact α E with
map_smul' := fun c f => by
ext
norm_cast
norm_map' := fun f => rfl }
#align continuous_map.linear_isometry_bounded_of_compact ContinuousMap.linearIsometryBoundedOfCompact
variable {α E}
-- to match `BoundedContinuousFunction.evalClm`
/-- The evaluation at a point, as a continuous linear map from `C(α, 𝕜)` to `𝕜`. -/
def evalClm (x : α) : C(α, E) →L[𝕜] E :=
(BoundedContinuousFunction.evalClm 𝕜 x).comp
(linearIsometryBoundedOfCompact α E 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_map.eval_clm ContinuousMap.evalClm
end
-- this lemma and the next are the analogues of those autogenerated by `@[simps]` for
-- `equivBoundedOfCompact`, `addEquivBoundedOfCompact`
@[simp]
theorem linearIsometryBoundedOfCompact_symm_apply (f : α →ᵇ E) :
(linearIsometryBoundedOfCompact α E 𝕜).symm f = f.toContinuousMap :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_symm_apply ContinuousMap.linearIsometryBoundedOfCompact_symm_apply
@[simp]
theorem linearIsometryBoundedOfCompact_apply_apply (f : C(α, E)) (a : α) :
(linearIsometryBoundedOfCompact α E 𝕜 f) a = f a :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_apply_apply ContinuousMap.linearIsometryBoundedOfCompact_apply_apply
@[simp]
theorem linearIsometryBoundedOfCompact_toIsometryEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toIsometryEquiv = isometryEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_isometry_equiv ContinuousMap.linearIsometryBoundedOfCompact_toIsometryEquiv
@[simp] -- porting note: adjusted LHS because `simpNF` complained it simplified.
theorem linearIsometryBoundedOfCompact_toAddEquiv :
((linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv : C(α, E) ≃+ (α →ᵇ E)) =
addEquivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_to_add_equiv ContinuousMap.linearIsometryBoundedOfCompact_toAddEquiv
@[simp]
theorem linearIsometryBoundedOfCompact_of_compact_toEquiv :
(linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv.toEquiv = equivBoundedOfCompact α E :=
rfl
#align continuous_map.linear_isometry_bounded_of_compact_of_compact_to_equiv ContinuousMap.linearIsometryBoundedOfCompact_of_compact_toEquiv
end
section
variable {𝕜 : Type*} {γ : Type*} [NormedField 𝕜] [NormedRing γ] [NormedAlgebra 𝕜 γ]
instance : NormedAlgebra 𝕜 C(α, γ) :=
{ ContinuousMap.normedSpace, ContinuousMap.algebra with }
end
end ContinuousMap
namespace ContinuousMap
section UniformContinuity
variable {α β : Type*}
variable [MetricSpace α] [CompactSpace α] [MetricSpace β]
/-!
We now set up some declarations making it convenient to use uniform continuity.
-/
theorem uniform_continuity (f : C(α, β)) (ε : ℝ) (h : 0 < ε) :
∃ δ > 0, ∀ {x y}, dist x y < δ → dist (f x) (f y) < ε :=
Metric.uniformContinuous_iff.mp (CompactSpace.uniformContinuous_of_continuous f.continuous) ε h
#align continuous_map.uniform_continuity ContinuousMap.uniform_continuity
-- This definition allows us to separate the choice of some `δ`,
-- and the corresponding use of `dist a b < δ → dist (f a) (f b) < ε`,
-- even across different declarations.
/-- An arbitrarily chosen modulus of uniform continuity for a given function `f` and `ε > 0`. -/
def modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) : ℝ :=
Classical.choose (uniform_continuity f ε h)
#align continuous_map.modulus ContinuousMap.modulus
theorem modulus_pos (f : C(α, β)) {ε : ℝ} {h : 0 < ε} : 0 < f.modulus ε h :=
(Classical.choose_spec (uniform_continuity f ε h)).1
#align continuous_map.modulus_pos ContinuousMap.modulus_pos
theorem dist_lt_of_dist_lt_modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) {a b : α}
(w : dist a b < f.modulus ε h) : dist (f a) (f b) < ε :=
(Classical.choose_spec (uniform_continuity f ε h)).2 w
#align continuous_map.dist_lt_of_dist_lt_modulus ContinuousMap.dist_lt_of_dist_lt_modulus
end UniformContinuity
end ContinuousMap
section CompLeft
variable (X : Type*) {𝕜 β γ : Type*} [TopologicalSpace X] [CompactSpace X]
[NontriviallyNormedField 𝕜]
variable [NormedAddCommGroup β] [NormedSpace 𝕜 β] [NormedAddCommGroup γ] [NormedSpace 𝕜 γ]
open ContinuousMap
/-- Postcomposition of continuous functions into a normed module by a continuous linear map is a
continuous linear map.
Transferred version of `ContinuousLinearMap.compLeftContinuousBounded`,
upgraded version of `ContinuousLinearMap.compLeftContinuous`,
similar to `LinearMap.compLeft`. -/
protected def ContinuousLinearMap.compLeftContinuousCompact (g : β →L[𝕜] γ) :
C(X, β) →L[𝕜] C(X, γ) :=
(linearIsometryBoundedOfCompact X γ 𝕜).symm.toLinearIsometry.toContinuousLinearMap.comp <|
(g.compLeftContinuousBounded X).comp <|
(linearIsometryBoundedOfCompact X β 𝕜).toLinearIsometry.toContinuousLinearMap
#align continuous_linear_map.comp_left_continuous_compact ContinuousLinearMap.compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.toLinear_compLeftContinuousCompact (g : β →L[𝕜] γ) :
(g.compLeftContinuousCompact X : C(X, β) →ₗ[𝕜] C(X, γ)) = g.compLeftContinuous 𝕜 X := by
ext f
rfl
#align continuous_linear_map.to_linear_comp_left_continuous_compact ContinuousLinearMap.toLinear_compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.compLeftContinuousCompact_apply (g : β →L[𝕜] γ) (f : C(X, β)) (x : X) :
g.compLeftContinuousCompact X f x = g (f x) :=
rfl
#align continuous_linear_map.comp_left_continuous_compact_apply ContinuousLinearMap.compLeftContinuousCompact_apply
end CompLeft
namespace ContinuousMap
/-!
We now setup variations on `compRight* f`, where `f : C(X, Y)`
(that is, precomposition by a continuous map),
as a morphism `C(Y, T) → C(X, T)`, respecting various types of structure.
In particular:
* `compRightContinuousMap`, the bundled continuous map (for this we need `X Y` compact).
* `compRightHomeomorph`, when we precompose by a homeomorphism.
* `compRightAlgHom`, when `T = R` is a topological ring.
-/
section CompRight
/-- Precomposition by a continuous map is itself a continuous map between spaces of continuous maps.
-/
def compRightContinuousMap {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y)) : C(C(Y, T), C(X, T)) where
toFun g := g.comp f
continuous_toFun := by
refine' Metric.continuous_iff.mpr _
intro g ε ε_pos
refine' ⟨ε, ε_pos, fun g' h => _⟩
rw [ContinuousMap.dist_lt_iff ε_pos] at h ⊢
exact fun x => h (f x)
#align continuous_map.comp_right_continuous_map ContinuousMap.compRightContinuousMap
@[simp]
theorem compRightContinuousMap_apply {X Y : Type*} (T : Type*) [TopologicalSpace X]
[CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y))
(g : C(Y, T)) : (compRightContinuousMap T f) g = g.comp f :=
rfl
#align continuous_map.comp_right_continuous_map_apply ContinuousMap.compRightContinuousMap_apply
/-- Precomposition by a homeomorphism is itself a homeomorphism between spaces of continuous maps.
-/
def compRightHomeomorph {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
[TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : X ≃ₜ Y) : C(Y, T) ≃ₜ C(X, T) where
toFun := compRightContinuousMap T f.toContinuousMap
invFun := compRightContinuousMap T f.symm.toContinuousMap
left_inv g := ext fun _ => congr_arg g (f.apply_symm_apply _)
right_inv g := ext fun _ => congr_arg g (f.symm_apply_apply _)
#align continuous_map.comp_right_homeomorph ContinuousMap.compRightHomeomorph
theorem compRightAlgHom_continuous {X Y : Type*} (R A : Type*) [TopologicalSpace X]
[CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [CommSemiring R] [Semiring A]
[MetricSpace A] [TopologicalSemiring A] [Algebra R A] (f : C(X, Y)) :
Continuous (compRightAlgHom R A f) :=
map_continuous (compRightContinuousMap A f)
#align continuous_map.comp_right_alg_hom_continuous ContinuousMap.compRightAlgHom_continuous
end CompRight
section LocalNormalConvergence
/-! ### Local normal convergence
A sum of continuous functions (on a locally compact space) is "locally normally convergent" if the
sum of its sup-norms on any compact subset is summable. This implies convergence in the topology
of `C(X, E)` (i.e. locally uniform convergence). -/
open TopologicalSpace
variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X]
variable {E : Type*} [NormedAddCommGroup E] [CompleteSpace E]
theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
(hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F := by
refine' (ContinuousMap.exists_tendsto_compactOpen_iff_forall _).2 fun K hK => _
lift K to Compacts X using hK
have A : ∀ s : Finset ι, restrict (↑K) (∑ i in s, F i) = ∑ i in s, restrict K (F i) := by
intro s
ext1 x
simp
-- This used to be the end of the proof before leanprover/lean4#2644
erw [restrict_apply, restrict_apply, restrict_apply, restrict_apply]
simp? says simp only [coe_sum, Finset.sum_apply]
congr!
simpa only [HasSum, A] using (hF K).of_norm
#align continuous_map.summable_of_locally_summable_norm ContinuousMap.summable_of_locally_summable_norm
end LocalNormalConvergence
/-!
### Star structures
In this section, if `β` is a normed ⋆-group, then so is the space of
continuous functions from `α` to `β`, by using the star operation pointwise.
Furthermore, if `α` is compact and `β` is a C⋆-ring, then `C(α, β)` is a C⋆-ring. -/
section NormedSpace
variable {α : Type*} {β : Type*}
variable [TopologicalSpace α] [NormedAddCommGroup β] [StarAddMonoid β] [NormedStarGroup β]
theorem _root_.BoundedContinuousFunction.mkOfCompact_star [CompactSpace α] (f : C(α, β)) :
mkOfCompact (star f) = star (mkOfCompact f) :=
rfl
#align bounded_continuous_function.mk_of_compact_star BoundedContinuousFunction.mkOfCompact_star
instance [CompactSpace α] : NormedStarGroup C(α, β) where
norm_star f := by
rw [← BoundedContinuousFunction.norm_mkOfCompact, BoundedContinuousFunction.mkOfCompact_star,
norm_star, BoundedContinuousFunction.norm_mkOfCompact]
end NormedSpace
section CstarRing
variable {α : Type*} {β : Type*}
variable [TopologicalSpace α] [NormedRing β] [StarRing β]
instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f} := by
refine' le_antisymm _ _
· rw [← sq, ContinuousMap.norm_le _ (sq_nonneg _)]
intro x
simp only [ContinuousMap.coe_mul, coe_star, Pi.mul_apply, Pi.star_apply,
CstarRing.norm_star_mul_self, ← sq]
refine' sq_le_sq' _ _
· linarith [norm_nonneg (f x), norm_nonneg f]
· exact ContinuousMap.norm_coe_le_norm f x
· rw [← sq, ← Real.le_sqrt (norm_nonneg _) (norm_nonneg _),
ContinuousMap.norm_le _ (Real.sqrt_nonneg _)]
intro x
rw [Real.le_sqrt (norm_nonneg _) (norm_nonneg _), sq, ← CstarRing.norm_star_mul_self]
|
exact ContinuousMap.norm_coe_le_norm (star f * f) x
|
instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f} := by
refine' le_antisymm _ _
· rw [← sq, ContinuousMap.norm_le _ (sq_nonneg _)]
intro x
simp only [ContinuousMap.coe_mul, coe_star, Pi.mul_apply, Pi.star_apply,
CstarRing.norm_star_mul_self, ← sq]
refine' sq_le_sq' _ _
· linarith [norm_nonneg (f x), norm_nonneg f]
· exact ContinuousMap.norm_coe_le_norm f x
· rw [← sq, ← Real.le_sqrt (norm_nonneg _) (norm_nonneg _),
ContinuousMap.norm_le _ (Real.sqrt_nonneg _)]
intro x
rw [Real.le_sqrt (norm_nonneg _) (norm_nonneg _), sq, ← CstarRing.norm_star_mul_self]
|
Mathlib.Topology.ContinuousFunction.Compact.541_0.Mig2jTVnn2FLKEB
|
instance [CompactSpace α] [CstarRing β] : CstarRing C(α, β) where
norm_star_mul_self {f}
|
Mathlib_Topology_ContinuousFunction_Compact
|
⊢ Tendsto (fun n => ↑(numDerangements n) / ↑(Nat.factorial n)) atTop (𝓝 (Real.exp (-1)))
|
/-
Copyright (c) 2021 Henry Swanson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Henry Swanson, Patrick Massot
-/
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Combinatorics.Derangements.Finite
import Mathlib.Order.Filter.Basic
#align_import combinatorics.derangements.exponential from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# Derangement exponential series
This file proves that the probability of a permutation on n elements being a derangement is 1/e.
The specific lemma is `numDerangements_tendsto_inv_e`.
-/
open Filter NormedSpace
open scoped BigOperators
open scoped Topology
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the right way, e.g., that k ≤ n always
|
let s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial
|
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the right way, e.g., that k ≤ n always
|
Mathlib.Combinatorics.Derangements.Exponential.26_0.d2ZjWMu7TGZefpO
|
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1)))
|
Mathlib_Combinatorics_Derangements_Exponential
|
s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1) ^ k / ↑(Nat.factorial k)
⊢ Tendsto (fun n => ↑(numDerangements n) / ↑(Nat.factorial n)) atTop (𝓝 (Real.exp (-1)))
|
/-
Copyright (c) 2021 Henry Swanson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Henry Swanson, Patrick Massot
-/
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Combinatorics.Derangements.Finite
import Mathlib.Order.Filter.Basic
#align_import combinatorics.derangements.exponential from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# Derangement exponential series
This file proves that the probability of a permutation on n elements being a derangement is 1/e.
The specific lemma is `numDerangements_tendsto_inv_e`.
-/
open Filter NormedSpace
open scoped BigOperators
open scoped Topology
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the right way, e.g., that k ≤ n always
let s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial
|
suffices ∀ n : ℕ, (numDerangements n : ℝ) / n.factorial = s (n + 1) by
simp_rw [this]
-- shift the function by 1, and then use the fact that the partial sums
-- converge to the infinite sum
rw [tendsto_add_atTop_iff_nat
(f := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial) 1]
apply HasSum.tendsto_sum_nat
-- there's no specific lemma for ℝ that ∑ x^k/k! sums to exp(x), but it's
-- true in more general fields, so use that lemma
rw [Real.exp_eq_exp_ℝ]
exact expSeries_div_hasSum_exp ℝ (-1 : ℝ)
|
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the right way, e.g., that k ≤ n always
let s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial
|
Mathlib.Combinatorics.Derangements.Exponential.26_0.d2ZjWMu7TGZefpO
|
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1)))
|
Mathlib_Combinatorics_Derangements_Exponential
|
s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1) ^ k / ↑(Nat.factorial k)
this : ∀ (n : ℕ), ↑(numDerangements n) / ↑(Nat.factorial n) = s (n + 1)
⊢ Tendsto (fun n => ↑(numDerangements n) / ↑(Nat.factorial n)) atTop (𝓝 (Real.exp (-1)))
|
/-
Copyright (c) 2021 Henry Swanson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Henry Swanson, Patrick Massot
-/
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Combinatorics.Derangements.Finite
import Mathlib.Order.Filter.Basic
#align_import combinatorics.derangements.exponential from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# Derangement exponential series
This file proves that the probability of a permutation on n elements being a derangement is 1/e.
The specific lemma is `numDerangements_tendsto_inv_e`.
-/
open Filter NormedSpace
open scoped BigOperators
open scoped Topology
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the right way, e.g., that k ≤ n always
let s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial
suffices ∀ n : ℕ, (numDerangements n : ℝ) / n.factorial = s (n + 1) by
|
simp_rw [this]
|
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the right way, e.g., that k ≤ n always
let s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial
suffices ∀ n : ℕ, (numDerangements n : ℝ) / n.factorial = s (n + 1) by
|
Mathlib.Combinatorics.Derangements.Exponential.26_0.d2ZjWMu7TGZefpO
|
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1)))
|
Mathlib_Combinatorics_Derangements_Exponential
|
s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1) ^ k / ↑(Nat.factorial k)
this : ∀ (n : ℕ), ↑(numDerangements n) / ↑(Nat.factorial n) = s (n + 1)
⊢ Tendsto (fun n => ∑ k in Finset.range (n + 1), (-1) ^ k / ↑(Nat.factorial k)) atTop (𝓝 (Real.exp (-1)))
|
/-
Copyright (c) 2021 Henry Swanson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Henry Swanson, Patrick Massot
-/
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Combinatorics.Derangements.Finite
import Mathlib.Order.Filter.Basic
#align_import combinatorics.derangements.exponential from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# Derangement exponential series
This file proves that the probability of a permutation on n elements being a derangement is 1/e.
The specific lemma is `numDerangements_tendsto_inv_e`.
-/
open Filter NormedSpace
open scoped BigOperators
open scoped Topology
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the right way, e.g., that k ≤ n always
let s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial
suffices ∀ n : ℕ, (numDerangements n : ℝ) / n.factorial = s (n + 1) by
simp_rw [this]
-- shift the function by 1, and then use the fact that the partial sums
-- converge to the infinite sum
|
rw [tendsto_add_atTop_iff_nat
(f := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial) 1]
|
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the right way, e.g., that k ≤ n always
let s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial
suffices ∀ n : ℕ, (numDerangements n : ℝ) / n.factorial = s (n + 1) by
simp_rw [this]
-- shift the function by 1, and then use the fact that the partial sums
-- converge to the infinite sum
|
Mathlib.Combinatorics.Derangements.Exponential.26_0.d2ZjWMu7TGZefpO
|
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1)))
|
Mathlib_Combinatorics_Derangements_Exponential
|
s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1) ^ k / ↑(Nat.factorial k)
this : ∀ (n : ℕ), ↑(numDerangements n) / ↑(Nat.factorial n) = s (n + 1)
⊢ Tendsto (fun n => ∑ k in Finset.range n, (-1) ^ k / ↑(Nat.factorial k)) atTop (𝓝 (Real.exp (-1)))
|
/-
Copyright (c) 2021 Henry Swanson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Henry Swanson, Patrick Massot
-/
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Combinatorics.Derangements.Finite
import Mathlib.Order.Filter.Basic
#align_import combinatorics.derangements.exponential from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# Derangement exponential series
This file proves that the probability of a permutation on n elements being a derangement is 1/e.
The specific lemma is `numDerangements_tendsto_inv_e`.
-/
open Filter NormedSpace
open scoped BigOperators
open scoped Topology
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the right way, e.g., that k ≤ n always
let s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial
suffices ∀ n : ℕ, (numDerangements n : ℝ) / n.factorial = s (n + 1) by
simp_rw [this]
-- shift the function by 1, and then use the fact that the partial sums
-- converge to the infinite sum
rw [tendsto_add_atTop_iff_nat
(f := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial) 1]
|
apply HasSum.tendsto_sum_nat
|
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the right way, e.g., that k ≤ n always
let s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial
suffices ∀ n : ℕ, (numDerangements n : ℝ) / n.factorial = s (n + 1) by
simp_rw [this]
-- shift the function by 1, and then use the fact that the partial sums
-- converge to the infinite sum
rw [tendsto_add_atTop_iff_nat
(f := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial) 1]
|
Mathlib.Combinatorics.Derangements.Exponential.26_0.d2ZjWMu7TGZefpO
|
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1)))
|
Mathlib_Combinatorics_Derangements_Exponential
|
case h
s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1) ^ k / ↑(Nat.factorial k)
this : ∀ (n : ℕ), ↑(numDerangements n) / ↑(Nat.factorial n) = s (n + 1)
⊢ HasSum (fun i => (-1) ^ i / ↑(Nat.factorial i)) (Real.exp (-1))
|
/-
Copyright (c) 2021 Henry Swanson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Henry Swanson, Patrick Massot
-/
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Combinatorics.Derangements.Finite
import Mathlib.Order.Filter.Basic
#align_import combinatorics.derangements.exponential from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# Derangement exponential series
This file proves that the probability of a permutation on n elements being a derangement is 1/e.
The specific lemma is `numDerangements_tendsto_inv_e`.
-/
open Filter NormedSpace
open scoped BigOperators
open scoped Topology
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the right way, e.g., that k ≤ n always
let s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial
suffices ∀ n : ℕ, (numDerangements n : ℝ) / n.factorial = s (n + 1) by
simp_rw [this]
-- shift the function by 1, and then use the fact that the partial sums
-- converge to the infinite sum
rw [tendsto_add_atTop_iff_nat
(f := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial) 1]
apply HasSum.tendsto_sum_nat
-- there's no specific lemma for ℝ that ∑ x^k/k! sums to exp(x), but it's
-- true in more general fields, so use that lemma
|
rw [Real.exp_eq_exp_ℝ]
|
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the right way, e.g., that k ≤ n always
let s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial
suffices ∀ n : ℕ, (numDerangements n : ℝ) / n.factorial = s (n + 1) by
simp_rw [this]
-- shift the function by 1, and then use the fact that the partial sums
-- converge to the infinite sum
rw [tendsto_add_atTop_iff_nat
(f := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial) 1]
apply HasSum.tendsto_sum_nat
-- there's no specific lemma for ℝ that ∑ x^k/k! sums to exp(x), but it's
-- true in more general fields, so use that lemma
|
Mathlib.Combinatorics.Derangements.Exponential.26_0.d2ZjWMu7TGZefpO
|
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1)))
|
Mathlib_Combinatorics_Derangements_Exponential
|
case h
s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1) ^ k / ↑(Nat.factorial k)
this : ∀ (n : ℕ), ↑(numDerangements n) / ↑(Nat.factorial n) = s (n + 1)
⊢ HasSum (fun i => (-1) ^ i / ↑(Nat.factorial i)) (exp ℝ (-1))
|
/-
Copyright (c) 2021 Henry Swanson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Henry Swanson, Patrick Massot
-/
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Combinatorics.Derangements.Finite
import Mathlib.Order.Filter.Basic
#align_import combinatorics.derangements.exponential from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# Derangement exponential series
This file proves that the probability of a permutation on n elements being a derangement is 1/e.
The specific lemma is `numDerangements_tendsto_inv_e`.
-/
open Filter NormedSpace
open scoped BigOperators
open scoped Topology
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the right way, e.g., that k ≤ n always
let s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial
suffices ∀ n : ℕ, (numDerangements n : ℝ) / n.factorial = s (n + 1) by
simp_rw [this]
-- shift the function by 1, and then use the fact that the partial sums
-- converge to the infinite sum
rw [tendsto_add_atTop_iff_nat
(f := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial) 1]
apply HasSum.tendsto_sum_nat
-- there's no specific lemma for ℝ that ∑ x^k/k! sums to exp(x), but it's
-- true in more general fields, so use that lemma
rw [Real.exp_eq_exp_ℝ]
|
exact expSeries_div_hasSum_exp ℝ (-1 : ℝ)
|
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the right way, e.g., that k ≤ n always
let s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial
suffices ∀ n : ℕ, (numDerangements n : ℝ) / n.factorial = s (n + 1) by
simp_rw [this]
-- shift the function by 1, and then use the fact that the partial sums
-- converge to the infinite sum
rw [tendsto_add_atTop_iff_nat
(f := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial) 1]
apply HasSum.tendsto_sum_nat
-- there's no specific lemma for ℝ that ∑ x^k/k! sums to exp(x), but it's
-- true in more general fields, so use that lemma
rw [Real.exp_eq_exp_ℝ]
|
Mathlib.Combinatorics.Derangements.Exponential.26_0.d2ZjWMu7TGZefpO
|
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1)))
|
Mathlib_Combinatorics_Derangements_Exponential
|
s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1) ^ k / ↑(Nat.factorial k)
⊢ ∀ (n : ℕ), ↑(numDerangements n) / ↑(Nat.factorial n) = s (n + 1)
|
/-
Copyright (c) 2021 Henry Swanson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Henry Swanson, Patrick Massot
-/
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Combinatorics.Derangements.Finite
import Mathlib.Order.Filter.Basic
#align_import combinatorics.derangements.exponential from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# Derangement exponential series
This file proves that the probability of a permutation on n elements being a derangement is 1/e.
The specific lemma is `numDerangements_tendsto_inv_e`.
-/
open Filter NormedSpace
open scoped BigOperators
open scoped Topology
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the right way, e.g., that k ≤ n always
let s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial
suffices ∀ n : ℕ, (numDerangements n : ℝ) / n.factorial = s (n + 1) by
simp_rw [this]
-- shift the function by 1, and then use the fact that the partial sums
-- converge to the infinite sum
rw [tendsto_add_atTop_iff_nat
(f := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial) 1]
apply HasSum.tendsto_sum_nat
-- there's no specific lemma for ℝ that ∑ x^k/k! sums to exp(x), but it's
-- true in more general fields, so use that lemma
rw [Real.exp_eq_exp_ℝ]
exact expSeries_div_hasSum_exp ℝ (-1 : ℝ)
|
intro n
|
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the right way, e.g., that k ≤ n always
let s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial
suffices ∀ n : ℕ, (numDerangements n : ℝ) / n.factorial = s (n + 1) by
simp_rw [this]
-- shift the function by 1, and then use the fact that the partial sums
-- converge to the infinite sum
rw [tendsto_add_atTop_iff_nat
(f := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial) 1]
apply HasSum.tendsto_sum_nat
-- there's no specific lemma for ℝ that ∑ x^k/k! sums to exp(x), but it's
-- true in more general fields, so use that lemma
rw [Real.exp_eq_exp_ℝ]
exact expSeries_div_hasSum_exp ℝ (-1 : ℝ)
|
Mathlib.Combinatorics.Derangements.Exponential.26_0.d2ZjWMu7TGZefpO
|
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1)))
|
Mathlib_Combinatorics_Derangements_Exponential
|
s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1) ^ k / ↑(Nat.factorial k)
n : ℕ
⊢ ↑(numDerangements n) / ↑(Nat.factorial n) = s (n + 1)
|
/-
Copyright (c) 2021 Henry Swanson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Henry Swanson, Patrick Massot
-/
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Combinatorics.Derangements.Finite
import Mathlib.Order.Filter.Basic
#align_import combinatorics.derangements.exponential from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# Derangement exponential series
This file proves that the probability of a permutation on n elements being a derangement is 1/e.
The specific lemma is `numDerangements_tendsto_inv_e`.
-/
open Filter NormedSpace
open scoped BigOperators
open scoped Topology
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the right way, e.g., that k ≤ n always
let s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial
suffices ∀ n : ℕ, (numDerangements n : ℝ) / n.factorial = s (n + 1) by
simp_rw [this]
-- shift the function by 1, and then use the fact that the partial sums
-- converge to the infinite sum
rw [tendsto_add_atTop_iff_nat
(f := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial) 1]
apply HasSum.tendsto_sum_nat
-- there's no specific lemma for ℝ that ∑ x^k/k! sums to exp(x), but it's
-- true in more general fields, so use that lemma
rw [Real.exp_eq_exp_ℝ]
exact expSeries_div_hasSum_exp ℝ (-1 : ℝ)
intro n
|
rw [← Int.cast_ofNat, numDerangements_sum]
|
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the right way, e.g., that k ≤ n always
let s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial
suffices ∀ n : ℕ, (numDerangements n : ℝ) / n.factorial = s (n + 1) by
simp_rw [this]
-- shift the function by 1, and then use the fact that the partial sums
-- converge to the infinite sum
rw [tendsto_add_atTop_iff_nat
(f := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial) 1]
apply HasSum.tendsto_sum_nat
-- there's no specific lemma for ℝ that ∑ x^k/k! sums to exp(x), but it's
-- true in more general fields, so use that lemma
rw [Real.exp_eq_exp_ℝ]
exact expSeries_div_hasSum_exp ℝ (-1 : ℝ)
intro n
|
Mathlib.Combinatorics.Derangements.Exponential.26_0.d2ZjWMu7TGZefpO
|
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1)))
|
Mathlib_Combinatorics_Derangements_Exponential
|
s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1) ^ k / ↑(Nat.factorial k)
n : ℕ
⊢ ↑(∑ k in Finset.range (n + 1), (-1) ^ k * ↑(Nat.ascFactorial k (n - k))) / ↑(Nat.factorial n) = s (n + 1)
|
/-
Copyright (c) 2021 Henry Swanson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Henry Swanson, Patrick Massot
-/
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Combinatorics.Derangements.Finite
import Mathlib.Order.Filter.Basic
#align_import combinatorics.derangements.exponential from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# Derangement exponential series
This file proves that the probability of a permutation on n elements being a derangement is 1/e.
The specific lemma is `numDerangements_tendsto_inv_e`.
-/
open Filter NormedSpace
open scoped BigOperators
open scoped Topology
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the right way, e.g., that k ≤ n always
let s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial
suffices ∀ n : ℕ, (numDerangements n : ℝ) / n.factorial = s (n + 1) by
simp_rw [this]
-- shift the function by 1, and then use the fact that the partial sums
-- converge to the infinite sum
rw [tendsto_add_atTop_iff_nat
(f := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial) 1]
apply HasSum.tendsto_sum_nat
-- there's no specific lemma for ℝ that ∑ x^k/k! sums to exp(x), but it's
-- true in more general fields, so use that lemma
rw [Real.exp_eq_exp_ℝ]
exact expSeries_div_hasSum_exp ℝ (-1 : ℝ)
intro n
rw [← Int.cast_ofNat, numDerangements_sum]
|
push_cast
|
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the right way, e.g., that k ≤ n always
let s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial
suffices ∀ n : ℕ, (numDerangements n : ℝ) / n.factorial = s (n + 1) by
simp_rw [this]
-- shift the function by 1, and then use the fact that the partial sums
-- converge to the infinite sum
rw [tendsto_add_atTop_iff_nat
(f := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial) 1]
apply HasSum.tendsto_sum_nat
-- there's no specific lemma for ℝ that ∑ x^k/k! sums to exp(x), but it's
-- true in more general fields, so use that lemma
rw [Real.exp_eq_exp_ℝ]
exact expSeries_div_hasSum_exp ℝ (-1 : ℝ)
intro n
rw [← Int.cast_ofNat, numDerangements_sum]
|
Mathlib.Combinatorics.Derangements.Exponential.26_0.d2ZjWMu7TGZefpO
|
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1)))
|
Mathlib_Combinatorics_Derangements_Exponential
|
s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1) ^ k / ↑(Nat.factorial k)
n : ℕ
⊢ (∑ x in Finset.range (n + 1), (-1) ^ x * ↑(Nat.ascFactorial x (n - x))) / ↑(Nat.factorial n) =
∑ k in Finset.range (n + 1), (-1) ^ k / ↑(Nat.factorial k)
|
/-
Copyright (c) 2021 Henry Swanson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Henry Swanson, Patrick Massot
-/
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Combinatorics.Derangements.Finite
import Mathlib.Order.Filter.Basic
#align_import combinatorics.derangements.exponential from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# Derangement exponential series
This file proves that the probability of a permutation on n elements being a derangement is 1/e.
The specific lemma is `numDerangements_tendsto_inv_e`.
-/
open Filter NormedSpace
open scoped BigOperators
open scoped Topology
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the right way, e.g., that k ≤ n always
let s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial
suffices ∀ n : ℕ, (numDerangements n : ℝ) / n.factorial = s (n + 1) by
simp_rw [this]
-- shift the function by 1, and then use the fact that the partial sums
-- converge to the infinite sum
rw [tendsto_add_atTop_iff_nat
(f := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial) 1]
apply HasSum.tendsto_sum_nat
-- there's no specific lemma for ℝ that ∑ x^k/k! sums to exp(x), but it's
-- true in more general fields, so use that lemma
rw [Real.exp_eq_exp_ℝ]
exact expSeries_div_hasSum_exp ℝ (-1 : ℝ)
intro n
rw [← Int.cast_ofNat, numDerangements_sum]
push_cast
|
rw [Finset.sum_div]
|
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the right way, e.g., that k ≤ n always
let s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial
suffices ∀ n : ℕ, (numDerangements n : ℝ) / n.factorial = s (n + 1) by
simp_rw [this]
-- shift the function by 1, and then use the fact that the partial sums
-- converge to the infinite sum
rw [tendsto_add_atTop_iff_nat
(f := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial) 1]
apply HasSum.tendsto_sum_nat
-- there's no specific lemma for ℝ that ∑ x^k/k! sums to exp(x), but it's
-- true in more general fields, so use that lemma
rw [Real.exp_eq_exp_ℝ]
exact expSeries_div_hasSum_exp ℝ (-1 : ℝ)
intro n
rw [← Int.cast_ofNat, numDerangements_sum]
push_cast
|
Mathlib.Combinatorics.Derangements.Exponential.26_0.d2ZjWMu7TGZefpO
|
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1)))
|
Mathlib_Combinatorics_Derangements_Exponential
|
s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1) ^ k / ↑(Nat.factorial k)
n : ℕ
⊢ ∑ x in Finset.range (n + 1), (-1) ^ x * ↑(Nat.ascFactorial x (n - x)) / ↑(Nat.factorial n) =
∑ k in Finset.range (n + 1), (-1) ^ k / ↑(Nat.factorial k)
|
/-
Copyright (c) 2021 Henry Swanson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Henry Swanson, Patrick Massot
-/
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Combinatorics.Derangements.Finite
import Mathlib.Order.Filter.Basic
#align_import combinatorics.derangements.exponential from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# Derangement exponential series
This file proves that the probability of a permutation on n elements being a derangement is 1/e.
The specific lemma is `numDerangements_tendsto_inv_e`.
-/
open Filter NormedSpace
open scoped BigOperators
open scoped Topology
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the right way, e.g., that k ≤ n always
let s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial
suffices ∀ n : ℕ, (numDerangements n : ℝ) / n.factorial = s (n + 1) by
simp_rw [this]
-- shift the function by 1, and then use the fact that the partial sums
-- converge to the infinite sum
rw [tendsto_add_atTop_iff_nat
(f := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial) 1]
apply HasSum.tendsto_sum_nat
-- there's no specific lemma for ℝ that ∑ x^k/k! sums to exp(x), but it's
-- true in more general fields, so use that lemma
rw [Real.exp_eq_exp_ℝ]
exact expSeries_div_hasSum_exp ℝ (-1 : ℝ)
intro n
rw [← Int.cast_ofNat, numDerangements_sum]
push_cast
rw [Finset.sum_div]
-- get down to individual terms
|
refine' Finset.sum_congr (refl _) _
|
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the right way, e.g., that k ≤ n always
let s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial
suffices ∀ n : ℕ, (numDerangements n : ℝ) / n.factorial = s (n + 1) by
simp_rw [this]
-- shift the function by 1, and then use the fact that the partial sums
-- converge to the infinite sum
rw [tendsto_add_atTop_iff_nat
(f := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial) 1]
apply HasSum.tendsto_sum_nat
-- there's no specific lemma for ℝ that ∑ x^k/k! sums to exp(x), but it's
-- true in more general fields, so use that lemma
rw [Real.exp_eq_exp_ℝ]
exact expSeries_div_hasSum_exp ℝ (-1 : ℝ)
intro n
rw [← Int.cast_ofNat, numDerangements_sum]
push_cast
rw [Finset.sum_div]
-- get down to individual terms
|
Mathlib.Combinatorics.Derangements.Exponential.26_0.d2ZjWMu7TGZefpO
|
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1)))
|
Mathlib_Combinatorics_Derangements_Exponential
|
s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1) ^ k / ↑(Nat.factorial k)
n : ℕ
⊢ ∀ x ∈ Finset.range (n + 1),
(-1) ^ x * ↑(Nat.ascFactorial x (n - x)) / ↑(Nat.factorial n) = (-1) ^ x / ↑(Nat.factorial x)
|
/-
Copyright (c) 2021 Henry Swanson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Henry Swanson, Patrick Massot
-/
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Combinatorics.Derangements.Finite
import Mathlib.Order.Filter.Basic
#align_import combinatorics.derangements.exponential from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# Derangement exponential series
This file proves that the probability of a permutation on n elements being a derangement is 1/e.
The specific lemma is `numDerangements_tendsto_inv_e`.
-/
open Filter NormedSpace
open scoped BigOperators
open scoped Topology
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the right way, e.g., that k ≤ n always
let s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial
suffices ∀ n : ℕ, (numDerangements n : ℝ) / n.factorial = s (n + 1) by
simp_rw [this]
-- shift the function by 1, and then use the fact that the partial sums
-- converge to the infinite sum
rw [tendsto_add_atTop_iff_nat
(f := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial) 1]
apply HasSum.tendsto_sum_nat
-- there's no specific lemma for ℝ that ∑ x^k/k! sums to exp(x), but it's
-- true in more general fields, so use that lemma
rw [Real.exp_eq_exp_ℝ]
exact expSeries_div_hasSum_exp ℝ (-1 : ℝ)
intro n
rw [← Int.cast_ofNat, numDerangements_sum]
push_cast
rw [Finset.sum_div]
-- get down to individual terms
refine' Finset.sum_congr (refl _) _
|
intro k hk
|
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the right way, e.g., that k ≤ n always
let s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial
suffices ∀ n : ℕ, (numDerangements n : ℝ) / n.factorial = s (n + 1) by
simp_rw [this]
-- shift the function by 1, and then use the fact that the partial sums
-- converge to the infinite sum
rw [tendsto_add_atTop_iff_nat
(f := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial) 1]
apply HasSum.tendsto_sum_nat
-- there's no specific lemma for ℝ that ∑ x^k/k! sums to exp(x), but it's
-- true in more general fields, so use that lemma
rw [Real.exp_eq_exp_ℝ]
exact expSeries_div_hasSum_exp ℝ (-1 : ℝ)
intro n
rw [← Int.cast_ofNat, numDerangements_sum]
push_cast
rw [Finset.sum_div]
-- get down to individual terms
refine' Finset.sum_congr (refl _) _
|
Mathlib.Combinatorics.Derangements.Exponential.26_0.d2ZjWMu7TGZefpO
|
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1)))
|
Mathlib_Combinatorics_Derangements_Exponential
|
s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1) ^ k / ↑(Nat.factorial k)
n k : ℕ
hk : k ∈ Finset.range (n + 1)
⊢ (-1) ^ k * ↑(Nat.ascFactorial k (n - k)) / ↑(Nat.factorial n) = (-1) ^ k / ↑(Nat.factorial k)
|
/-
Copyright (c) 2021 Henry Swanson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Henry Swanson, Patrick Massot
-/
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Combinatorics.Derangements.Finite
import Mathlib.Order.Filter.Basic
#align_import combinatorics.derangements.exponential from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# Derangement exponential series
This file proves that the probability of a permutation on n elements being a derangement is 1/e.
The specific lemma is `numDerangements_tendsto_inv_e`.
-/
open Filter NormedSpace
open scoped BigOperators
open scoped Topology
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the right way, e.g., that k ≤ n always
let s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial
suffices ∀ n : ℕ, (numDerangements n : ℝ) / n.factorial = s (n + 1) by
simp_rw [this]
-- shift the function by 1, and then use the fact that the partial sums
-- converge to the infinite sum
rw [tendsto_add_atTop_iff_nat
(f := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial) 1]
apply HasSum.tendsto_sum_nat
-- there's no specific lemma for ℝ that ∑ x^k/k! sums to exp(x), but it's
-- true in more general fields, so use that lemma
rw [Real.exp_eq_exp_ℝ]
exact expSeries_div_hasSum_exp ℝ (-1 : ℝ)
intro n
rw [← Int.cast_ofNat, numDerangements_sum]
push_cast
rw [Finset.sum_div]
-- get down to individual terms
refine' Finset.sum_congr (refl _) _
intro k hk
|
have h_le : k ≤ n := Finset.mem_range_succ_iff.mp hk
|
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the right way, e.g., that k ≤ n always
let s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial
suffices ∀ n : ℕ, (numDerangements n : ℝ) / n.factorial = s (n + 1) by
simp_rw [this]
-- shift the function by 1, and then use the fact that the partial sums
-- converge to the infinite sum
rw [tendsto_add_atTop_iff_nat
(f := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial) 1]
apply HasSum.tendsto_sum_nat
-- there's no specific lemma for ℝ that ∑ x^k/k! sums to exp(x), but it's
-- true in more general fields, so use that lemma
rw [Real.exp_eq_exp_ℝ]
exact expSeries_div_hasSum_exp ℝ (-1 : ℝ)
intro n
rw [← Int.cast_ofNat, numDerangements_sum]
push_cast
rw [Finset.sum_div]
-- get down to individual terms
refine' Finset.sum_congr (refl _) _
intro k hk
|
Mathlib.Combinatorics.Derangements.Exponential.26_0.d2ZjWMu7TGZefpO
|
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1)))
|
Mathlib_Combinatorics_Derangements_Exponential
|
s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1) ^ k / ↑(Nat.factorial k)
n k : ℕ
hk : k ∈ Finset.range (n + 1)
h_le : k ≤ n
⊢ (-1) ^ k * ↑(Nat.ascFactorial k (n - k)) / ↑(Nat.factorial n) = (-1) ^ k / ↑(Nat.factorial k)
|
/-
Copyright (c) 2021 Henry Swanson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Henry Swanson, Patrick Massot
-/
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Combinatorics.Derangements.Finite
import Mathlib.Order.Filter.Basic
#align_import combinatorics.derangements.exponential from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# Derangement exponential series
This file proves that the probability of a permutation on n elements being a derangement is 1/e.
The specific lemma is `numDerangements_tendsto_inv_e`.
-/
open Filter NormedSpace
open scoped BigOperators
open scoped Topology
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the right way, e.g., that k ≤ n always
let s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial
suffices ∀ n : ℕ, (numDerangements n : ℝ) / n.factorial = s (n + 1) by
simp_rw [this]
-- shift the function by 1, and then use the fact that the partial sums
-- converge to the infinite sum
rw [tendsto_add_atTop_iff_nat
(f := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial) 1]
apply HasSum.tendsto_sum_nat
-- there's no specific lemma for ℝ that ∑ x^k/k! sums to exp(x), but it's
-- true in more general fields, so use that lemma
rw [Real.exp_eq_exp_ℝ]
exact expSeries_div_hasSum_exp ℝ (-1 : ℝ)
intro n
rw [← Int.cast_ofNat, numDerangements_sum]
push_cast
rw [Finset.sum_div]
-- get down to individual terms
refine' Finset.sum_congr (refl _) _
intro k hk
have h_le : k ≤ n := Finset.mem_range_succ_iff.mp hk
|
rw [Nat.ascFactorial_eq_div, add_tsub_cancel_of_le h_le]
|
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the right way, e.g., that k ≤ n always
let s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial
suffices ∀ n : ℕ, (numDerangements n : ℝ) / n.factorial = s (n + 1) by
simp_rw [this]
-- shift the function by 1, and then use the fact that the partial sums
-- converge to the infinite sum
rw [tendsto_add_atTop_iff_nat
(f := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial) 1]
apply HasSum.tendsto_sum_nat
-- there's no specific lemma for ℝ that ∑ x^k/k! sums to exp(x), but it's
-- true in more general fields, so use that lemma
rw [Real.exp_eq_exp_ℝ]
exact expSeries_div_hasSum_exp ℝ (-1 : ℝ)
intro n
rw [← Int.cast_ofNat, numDerangements_sum]
push_cast
rw [Finset.sum_div]
-- get down to individual terms
refine' Finset.sum_congr (refl _) _
intro k hk
have h_le : k ≤ n := Finset.mem_range_succ_iff.mp hk
|
Mathlib.Combinatorics.Derangements.Exponential.26_0.d2ZjWMu7TGZefpO
|
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1)))
|
Mathlib_Combinatorics_Derangements_Exponential
|
s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1) ^ k / ↑(Nat.factorial k)
n k : ℕ
hk : k ∈ Finset.range (n + 1)
h_le : k ≤ n
⊢ (-1) ^ k * ↑(Nat.factorial n / Nat.factorial k) / ↑(Nat.factorial n) = (-1) ^ k / ↑(Nat.factorial k)
|
/-
Copyright (c) 2021 Henry Swanson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Henry Swanson, Patrick Massot
-/
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Combinatorics.Derangements.Finite
import Mathlib.Order.Filter.Basic
#align_import combinatorics.derangements.exponential from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# Derangement exponential series
This file proves that the probability of a permutation on n elements being a derangement is 1/e.
The specific lemma is `numDerangements_tendsto_inv_e`.
-/
open Filter NormedSpace
open scoped BigOperators
open scoped Topology
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the right way, e.g., that k ≤ n always
let s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial
suffices ∀ n : ℕ, (numDerangements n : ℝ) / n.factorial = s (n + 1) by
simp_rw [this]
-- shift the function by 1, and then use the fact that the partial sums
-- converge to the infinite sum
rw [tendsto_add_atTop_iff_nat
(f := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial) 1]
apply HasSum.tendsto_sum_nat
-- there's no specific lemma for ℝ that ∑ x^k/k! sums to exp(x), but it's
-- true in more general fields, so use that lemma
rw [Real.exp_eq_exp_ℝ]
exact expSeries_div_hasSum_exp ℝ (-1 : ℝ)
intro n
rw [← Int.cast_ofNat, numDerangements_sum]
push_cast
rw [Finset.sum_div]
-- get down to individual terms
refine' Finset.sum_congr (refl _) _
intro k hk
have h_le : k ≤ n := Finset.mem_range_succ_iff.mp hk
rw [Nat.ascFactorial_eq_div, add_tsub_cancel_of_le h_le]
|
push_cast [Nat.factorial_dvd_factorial h_le]
|
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the right way, e.g., that k ≤ n always
let s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial
suffices ∀ n : ℕ, (numDerangements n : ℝ) / n.factorial = s (n + 1) by
simp_rw [this]
-- shift the function by 1, and then use the fact that the partial sums
-- converge to the infinite sum
rw [tendsto_add_atTop_iff_nat
(f := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial) 1]
apply HasSum.tendsto_sum_nat
-- there's no specific lemma for ℝ that ∑ x^k/k! sums to exp(x), but it's
-- true in more general fields, so use that lemma
rw [Real.exp_eq_exp_ℝ]
exact expSeries_div_hasSum_exp ℝ (-1 : ℝ)
intro n
rw [← Int.cast_ofNat, numDerangements_sum]
push_cast
rw [Finset.sum_div]
-- get down to individual terms
refine' Finset.sum_congr (refl _) _
intro k hk
have h_le : k ≤ n := Finset.mem_range_succ_iff.mp hk
rw [Nat.ascFactorial_eq_div, add_tsub_cancel_of_le h_le]
|
Mathlib.Combinatorics.Derangements.Exponential.26_0.d2ZjWMu7TGZefpO
|
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1)))
|
Mathlib_Combinatorics_Derangements_Exponential
|
s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1) ^ k / ↑(Nat.factorial k)
n k : ℕ
hk : k ∈ Finset.range (n + 1)
h_le : k ≤ n
⊢ (-1) ^ k * (↑(Nat.factorial n) / ↑(Nat.factorial k)) / ↑(Nat.factorial n) = (-1) ^ k / ↑(Nat.factorial k)
|
/-
Copyright (c) 2021 Henry Swanson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Henry Swanson, Patrick Massot
-/
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Combinatorics.Derangements.Finite
import Mathlib.Order.Filter.Basic
#align_import combinatorics.derangements.exponential from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# Derangement exponential series
This file proves that the probability of a permutation on n elements being a derangement is 1/e.
The specific lemma is `numDerangements_tendsto_inv_e`.
-/
open Filter NormedSpace
open scoped BigOperators
open scoped Topology
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the right way, e.g., that k ≤ n always
let s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial
suffices ∀ n : ℕ, (numDerangements n : ℝ) / n.factorial = s (n + 1) by
simp_rw [this]
-- shift the function by 1, and then use the fact that the partial sums
-- converge to the infinite sum
rw [tendsto_add_atTop_iff_nat
(f := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial) 1]
apply HasSum.tendsto_sum_nat
-- there's no specific lemma for ℝ that ∑ x^k/k! sums to exp(x), but it's
-- true in more general fields, so use that lemma
rw [Real.exp_eq_exp_ℝ]
exact expSeries_div_hasSum_exp ℝ (-1 : ℝ)
intro n
rw [← Int.cast_ofNat, numDerangements_sum]
push_cast
rw [Finset.sum_div]
-- get down to individual terms
refine' Finset.sum_congr (refl _) _
intro k hk
have h_le : k ≤ n := Finset.mem_range_succ_iff.mp hk
rw [Nat.ascFactorial_eq_div, add_tsub_cancel_of_le h_le]
push_cast [Nat.factorial_dvd_factorial h_le]
|
field_simp [Nat.factorial_ne_zero]
|
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the right way, e.g., that k ≤ n always
let s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial
suffices ∀ n : ℕ, (numDerangements n : ℝ) / n.factorial = s (n + 1) by
simp_rw [this]
-- shift the function by 1, and then use the fact that the partial sums
-- converge to the infinite sum
rw [tendsto_add_atTop_iff_nat
(f := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial) 1]
apply HasSum.tendsto_sum_nat
-- there's no specific lemma for ℝ that ∑ x^k/k! sums to exp(x), but it's
-- true in more general fields, so use that lemma
rw [Real.exp_eq_exp_ℝ]
exact expSeries_div_hasSum_exp ℝ (-1 : ℝ)
intro n
rw [← Int.cast_ofNat, numDerangements_sum]
push_cast
rw [Finset.sum_div]
-- get down to individual terms
refine' Finset.sum_congr (refl _) _
intro k hk
have h_le : k ≤ n := Finset.mem_range_succ_iff.mp hk
rw [Nat.ascFactorial_eq_div, add_tsub_cancel_of_le h_le]
push_cast [Nat.factorial_dvd_factorial h_le]
|
Mathlib.Combinatorics.Derangements.Exponential.26_0.d2ZjWMu7TGZefpO
|
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1)))
|
Mathlib_Combinatorics_Derangements_Exponential
|
s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1) ^ k / ↑(Nat.factorial k)
n k : ℕ
hk : k ∈ Finset.range (n + 1)
h_le : k ≤ n
⊢ (-1) ^ k * ↑(Nat.factorial n) * ↑(Nat.factorial k) = (-1) ^ k * (↑(Nat.factorial k) * ↑(Nat.factorial n))
|
/-
Copyright (c) 2021 Henry Swanson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Henry Swanson, Patrick Massot
-/
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Combinatorics.Derangements.Finite
import Mathlib.Order.Filter.Basic
#align_import combinatorics.derangements.exponential from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# Derangement exponential series
This file proves that the probability of a permutation on n elements being a derangement is 1/e.
The specific lemma is `numDerangements_tendsto_inv_e`.
-/
open Filter NormedSpace
open scoped BigOperators
open scoped Topology
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the right way, e.g., that k ≤ n always
let s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial
suffices ∀ n : ℕ, (numDerangements n : ℝ) / n.factorial = s (n + 1) by
simp_rw [this]
-- shift the function by 1, and then use the fact that the partial sums
-- converge to the infinite sum
rw [tendsto_add_atTop_iff_nat
(f := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial) 1]
apply HasSum.tendsto_sum_nat
-- there's no specific lemma for ℝ that ∑ x^k/k! sums to exp(x), but it's
-- true in more general fields, so use that lemma
rw [Real.exp_eq_exp_ℝ]
exact expSeries_div_hasSum_exp ℝ (-1 : ℝ)
intro n
rw [← Int.cast_ofNat, numDerangements_sum]
push_cast
rw [Finset.sum_div]
-- get down to individual terms
refine' Finset.sum_congr (refl _) _
intro k hk
have h_le : k ≤ n := Finset.mem_range_succ_iff.mp hk
rw [Nat.ascFactorial_eq_div, add_tsub_cancel_of_le h_le]
push_cast [Nat.factorial_dvd_factorial h_le]
field_simp [Nat.factorial_ne_zero]
|
ring
|
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the right way, e.g., that k ≤ n always
let s : ℕ → ℝ := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial
suffices ∀ n : ℕ, (numDerangements n : ℝ) / n.factorial = s (n + 1) by
simp_rw [this]
-- shift the function by 1, and then use the fact that the partial sums
-- converge to the infinite sum
rw [tendsto_add_atTop_iff_nat
(f := fun n => ∑ k in Finset.range n, (-1 : ℝ) ^ k / k.factorial) 1]
apply HasSum.tendsto_sum_nat
-- there's no specific lemma for ℝ that ∑ x^k/k! sums to exp(x), but it's
-- true in more general fields, so use that lemma
rw [Real.exp_eq_exp_ℝ]
exact expSeries_div_hasSum_exp ℝ (-1 : ℝ)
intro n
rw [← Int.cast_ofNat, numDerangements_sum]
push_cast
rw [Finset.sum_div]
-- get down to individual terms
refine' Finset.sum_congr (refl _) _
intro k hk
have h_le : k ≤ n := Finset.mem_range_succ_iff.mp hk
rw [Nat.ascFactorial_eq_div, add_tsub_cancel_of_le h_le]
push_cast [Nat.factorial_dvd_factorial h_le]
field_simp [Nat.factorial_ne_zero]
|
Mathlib.Combinatorics.Derangements.Exponential.26_0.d2ZjWMu7TGZefpO
|
theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1)))
|
Mathlib_Combinatorics_Derangements_Exponential
|
α : Sort u_1
p q r : Prop
inst✝¹ : Decidable p
inst✝ : Decidable q
a✝ b✝ c✝ : α
a : p → α
b : ¬p → q → α
c : ¬p → ¬q → α
⊢ (dite p a fun hp => dite q (b hp) (c hp)) = if hq : q then dite p a fun hp => b hp hq else dite p a fun hp => c hp hq
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a7c3ac4ac7873a7a43ead17c"
/-!
# More basic logic properties
A few more logic lemmas. These are in their own file, rather than `Logic.Basic`, because it is
convenient to be able to use the `split_ifs` tactic.
## Implementation notes
We spell those lemmas out with `dite` and `ite` rather than the `if then else` notation because this
would result in less delta-reduced statements.
-/
protected alias ⟨HEq.eq, Eq.heq⟩ := heq_iff_eq
#align heq.eq HEq.eq
#align eq.heq Eq.heq
variable {α : Sort*} {p q r : Prop} [Decidable p] [Decidable q] {a b c : α}
theorem dite_dite_distrib_left {a : p → α} {b : ¬p → q → α} {c : ¬p → ¬q → α} :
(dite p a fun hp ↦ dite q (b hp) (c hp)) =
dite q (fun hq ↦ (dite p a) fun hp ↦ b hp hq) fun hq ↦ (dite p a) fun hp ↦ c hp hq := by
|
split_ifs
|
theorem dite_dite_distrib_left {a : p → α} {b : ¬p → q → α} {c : ¬p → ¬q → α} :
(dite p a fun hp ↦ dite q (b hp) (c hp)) =
dite q (fun hq ↦ (dite p a) fun hp ↦ b hp hq) fun hq ↦ (dite p a) fun hp ↦ c hp hq := by
|
Mathlib.Logic.Lemmas.28_0.END1WfxnNs4h6Zj
|
theorem dite_dite_distrib_left {a : p → α} {b : ¬p → q → α} {c : ¬p → ¬q → α} :
(dite p a fun hp ↦ dite q (b hp) (c hp)) =
dite q (fun hq ↦ (dite p a) fun hp ↦ b hp hq) fun hq ↦ (dite p a) fun hp ↦ c hp hq
|
Mathlib_Logic_Lemmas
|
case pos
α : Sort u_1
p q r : Prop
inst✝¹ : Decidable p
inst✝ : Decidable q
a✝ b✝ c✝ : α
a : p → α
b : ¬p → q → α
c : ¬p → ¬q → α
h✝¹ : p
h✝ : q
⊢ a h✝¹ = a h✝¹
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a7c3ac4ac7873a7a43ead17c"
/-!
# More basic logic properties
A few more logic lemmas. These are in their own file, rather than `Logic.Basic`, because it is
convenient to be able to use the `split_ifs` tactic.
## Implementation notes
We spell those lemmas out with `dite` and `ite` rather than the `if then else` notation because this
would result in less delta-reduced statements.
-/
protected alias ⟨HEq.eq, Eq.heq⟩ := heq_iff_eq
#align heq.eq HEq.eq
#align eq.heq Eq.heq
variable {α : Sort*} {p q r : Prop} [Decidable p] [Decidable q] {a b c : α}
theorem dite_dite_distrib_left {a : p → α} {b : ¬p → q → α} {c : ¬p → ¬q → α} :
(dite p a fun hp ↦ dite q (b hp) (c hp)) =
dite q (fun hq ↦ (dite p a) fun hp ↦ b hp hq) fun hq ↦ (dite p a) fun hp ↦ c hp hq := by
split_ifs <;>
|
rfl
|
theorem dite_dite_distrib_left {a : p → α} {b : ¬p → q → α} {c : ¬p → ¬q → α} :
(dite p a fun hp ↦ dite q (b hp) (c hp)) =
dite q (fun hq ↦ (dite p a) fun hp ↦ b hp hq) fun hq ↦ (dite p a) fun hp ↦ c hp hq := by
split_ifs <;>
|
Mathlib.Logic.Lemmas.28_0.END1WfxnNs4h6Zj
|
theorem dite_dite_distrib_left {a : p → α} {b : ¬p → q → α} {c : ¬p → ¬q → α} :
(dite p a fun hp ↦ dite q (b hp) (c hp)) =
dite q (fun hq ↦ (dite p a) fun hp ↦ b hp hq) fun hq ↦ (dite p a) fun hp ↦ c hp hq
|
Mathlib_Logic_Lemmas
|
case neg
α : Sort u_1
p q r : Prop
inst✝¹ : Decidable p
inst✝ : Decidable q
a✝ b✝ c✝ : α
a : p → α
b : ¬p → q → α
c : ¬p → ¬q → α
h✝¹ : p
h✝ : ¬q
⊢ a h✝¹ = a h✝¹
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a7c3ac4ac7873a7a43ead17c"
/-!
# More basic logic properties
A few more logic lemmas. These are in their own file, rather than `Logic.Basic`, because it is
convenient to be able to use the `split_ifs` tactic.
## Implementation notes
We spell those lemmas out with `dite` and `ite` rather than the `if then else` notation because this
would result in less delta-reduced statements.
-/
protected alias ⟨HEq.eq, Eq.heq⟩ := heq_iff_eq
#align heq.eq HEq.eq
#align eq.heq Eq.heq
variable {α : Sort*} {p q r : Prop} [Decidable p] [Decidable q] {a b c : α}
theorem dite_dite_distrib_left {a : p → α} {b : ¬p → q → α} {c : ¬p → ¬q → α} :
(dite p a fun hp ↦ dite q (b hp) (c hp)) =
dite q (fun hq ↦ (dite p a) fun hp ↦ b hp hq) fun hq ↦ (dite p a) fun hp ↦ c hp hq := by
split_ifs <;>
|
rfl
|
theorem dite_dite_distrib_left {a : p → α} {b : ¬p → q → α} {c : ¬p → ¬q → α} :
(dite p a fun hp ↦ dite q (b hp) (c hp)) =
dite q (fun hq ↦ (dite p a) fun hp ↦ b hp hq) fun hq ↦ (dite p a) fun hp ↦ c hp hq := by
split_ifs <;>
|
Mathlib.Logic.Lemmas.28_0.END1WfxnNs4h6Zj
|
theorem dite_dite_distrib_left {a : p → α} {b : ¬p → q → α} {c : ¬p → ¬q → α} :
(dite p a fun hp ↦ dite q (b hp) (c hp)) =
dite q (fun hq ↦ (dite p a) fun hp ↦ b hp hq) fun hq ↦ (dite p a) fun hp ↦ c hp hq
|
Mathlib_Logic_Lemmas
|
case pos
α : Sort u_1
p q r : Prop
inst✝¹ : Decidable p
inst✝ : Decidable q
a✝ b✝ c✝ : α
a : p → α
b : ¬p → q → α
c : ¬p → ¬q → α
h✝¹ : ¬p
h✝ : q
⊢ b h✝¹ h✝ = b h✝¹ (_ : q)
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a7c3ac4ac7873a7a43ead17c"
/-!
# More basic logic properties
A few more logic lemmas. These are in their own file, rather than `Logic.Basic`, because it is
convenient to be able to use the `split_ifs` tactic.
## Implementation notes
We spell those lemmas out with `dite` and `ite` rather than the `if then else` notation because this
would result in less delta-reduced statements.
-/
protected alias ⟨HEq.eq, Eq.heq⟩ := heq_iff_eq
#align heq.eq HEq.eq
#align eq.heq Eq.heq
variable {α : Sort*} {p q r : Prop} [Decidable p] [Decidable q] {a b c : α}
theorem dite_dite_distrib_left {a : p → α} {b : ¬p → q → α} {c : ¬p → ¬q → α} :
(dite p a fun hp ↦ dite q (b hp) (c hp)) =
dite q (fun hq ↦ (dite p a) fun hp ↦ b hp hq) fun hq ↦ (dite p a) fun hp ↦ c hp hq := by
split_ifs <;>
|
rfl
|
theorem dite_dite_distrib_left {a : p → α} {b : ¬p → q → α} {c : ¬p → ¬q → α} :
(dite p a fun hp ↦ dite q (b hp) (c hp)) =
dite q (fun hq ↦ (dite p a) fun hp ↦ b hp hq) fun hq ↦ (dite p a) fun hp ↦ c hp hq := by
split_ifs <;>
|
Mathlib.Logic.Lemmas.28_0.END1WfxnNs4h6Zj
|
theorem dite_dite_distrib_left {a : p → α} {b : ¬p → q → α} {c : ¬p → ¬q → α} :
(dite p a fun hp ↦ dite q (b hp) (c hp)) =
dite q (fun hq ↦ (dite p a) fun hp ↦ b hp hq) fun hq ↦ (dite p a) fun hp ↦ c hp hq
|
Mathlib_Logic_Lemmas
|
case neg
α : Sort u_1
p q r : Prop
inst✝¹ : Decidable p
inst✝ : Decidable q
a✝ b✝ c✝ : α
a : p → α
b : ¬p → q → α
c : ¬p → ¬q → α
h✝¹ : ¬p
h✝ : ¬q
⊢ c h✝¹ h✝ = c h✝¹ (_ : ¬q)
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a7c3ac4ac7873a7a43ead17c"
/-!
# More basic logic properties
A few more logic lemmas. These are in their own file, rather than `Logic.Basic`, because it is
convenient to be able to use the `split_ifs` tactic.
## Implementation notes
We spell those lemmas out with `dite` and `ite` rather than the `if then else` notation because this
would result in less delta-reduced statements.
-/
protected alias ⟨HEq.eq, Eq.heq⟩ := heq_iff_eq
#align heq.eq HEq.eq
#align eq.heq Eq.heq
variable {α : Sort*} {p q r : Prop} [Decidable p] [Decidable q] {a b c : α}
theorem dite_dite_distrib_left {a : p → α} {b : ¬p → q → α} {c : ¬p → ¬q → α} :
(dite p a fun hp ↦ dite q (b hp) (c hp)) =
dite q (fun hq ↦ (dite p a) fun hp ↦ b hp hq) fun hq ↦ (dite p a) fun hp ↦ c hp hq := by
split_ifs <;>
|
rfl
|
theorem dite_dite_distrib_left {a : p → α} {b : ¬p → q → α} {c : ¬p → ¬q → α} :
(dite p a fun hp ↦ dite q (b hp) (c hp)) =
dite q (fun hq ↦ (dite p a) fun hp ↦ b hp hq) fun hq ↦ (dite p a) fun hp ↦ c hp hq := by
split_ifs <;>
|
Mathlib.Logic.Lemmas.28_0.END1WfxnNs4h6Zj
|
theorem dite_dite_distrib_left {a : p → α} {b : ¬p → q → α} {c : ¬p → ¬q → α} :
(dite p a fun hp ↦ dite q (b hp) (c hp)) =
dite q (fun hq ↦ (dite p a) fun hp ↦ b hp hq) fun hq ↦ (dite p a) fun hp ↦ c hp hq
|
Mathlib_Logic_Lemmas
|
α : Sort u_1
p q r : Prop
inst✝¹ : Decidable p
inst✝ : Decidable q
a✝ b✝ c✝ : α
a : p → q → α
b : p → ¬q → α
c : ¬p → α
⊢ dite p (fun hp => dite q (a hp) (b hp)) c =
if hq : q then dite p (fun hp => a hp hq) c else dite p (fun hp => b hp hq) c
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a7c3ac4ac7873a7a43ead17c"
/-!
# More basic logic properties
A few more logic lemmas. These are in their own file, rather than `Logic.Basic`, because it is
convenient to be able to use the `split_ifs` tactic.
## Implementation notes
We spell those lemmas out with `dite` and `ite` rather than the `if then else` notation because this
would result in less delta-reduced statements.
-/
protected alias ⟨HEq.eq, Eq.heq⟩ := heq_iff_eq
#align heq.eq HEq.eq
#align eq.heq Eq.heq
variable {α : Sort*} {p q r : Prop} [Decidable p] [Decidable q] {a b c : α}
theorem dite_dite_distrib_left {a : p → α} {b : ¬p → q → α} {c : ¬p → ¬q → α} :
(dite p a fun hp ↦ dite q (b hp) (c hp)) =
dite q (fun hq ↦ (dite p a) fun hp ↦ b hp hq) fun hq ↦ (dite p a) fun hp ↦ c hp hq := by
split_ifs <;> rfl
#align dite_dite_distrib_left dite_dite_distrib_left
theorem dite_dite_distrib_right {a : p → q → α} {b : p → ¬q → α} {c : ¬p → α} :
dite p (fun hp ↦ dite q (a hp) (b hp)) c =
dite q (fun hq ↦ dite p (fun hp ↦ a hp hq) c) fun hq ↦ dite p (fun hp ↦ b hp hq) c := by
|
split_ifs
|
theorem dite_dite_distrib_right {a : p → q → α} {b : p → ¬q → α} {c : ¬p → α} :
dite p (fun hp ↦ dite q (a hp) (b hp)) c =
dite q (fun hq ↦ dite p (fun hp ↦ a hp hq) c) fun hq ↦ dite p (fun hp ↦ b hp hq) c := by
|
Mathlib.Logic.Lemmas.34_0.END1WfxnNs4h6Zj
|
theorem dite_dite_distrib_right {a : p → q → α} {b : p → ¬q → α} {c : ¬p → α} :
dite p (fun hp ↦ dite q (a hp) (b hp)) c =
dite q (fun hq ↦ dite p (fun hp ↦ a hp hq) c) fun hq ↦ dite p (fun hp ↦ b hp hq) c
|
Mathlib_Logic_Lemmas
|
case pos
α : Sort u_1
p q r : Prop
inst✝¹ : Decidable p
inst✝ : Decidable q
a✝ b✝ c✝ : α
a : p → q → α
b : p → ¬q → α
c : ¬p → α
h✝¹ : p
h✝ : q
⊢ a h✝¹ h✝ = a h✝¹ (_ : q)
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a7c3ac4ac7873a7a43ead17c"
/-!
# More basic logic properties
A few more logic lemmas. These are in their own file, rather than `Logic.Basic`, because it is
convenient to be able to use the `split_ifs` tactic.
## Implementation notes
We spell those lemmas out with `dite` and `ite` rather than the `if then else` notation because this
would result in less delta-reduced statements.
-/
protected alias ⟨HEq.eq, Eq.heq⟩ := heq_iff_eq
#align heq.eq HEq.eq
#align eq.heq Eq.heq
variable {α : Sort*} {p q r : Prop} [Decidable p] [Decidable q] {a b c : α}
theorem dite_dite_distrib_left {a : p → α} {b : ¬p → q → α} {c : ¬p → ¬q → α} :
(dite p a fun hp ↦ dite q (b hp) (c hp)) =
dite q (fun hq ↦ (dite p a) fun hp ↦ b hp hq) fun hq ↦ (dite p a) fun hp ↦ c hp hq := by
split_ifs <;> rfl
#align dite_dite_distrib_left dite_dite_distrib_left
theorem dite_dite_distrib_right {a : p → q → α} {b : p → ¬q → α} {c : ¬p → α} :
dite p (fun hp ↦ dite q (a hp) (b hp)) c =
dite q (fun hq ↦ dite p (fun hp ↦ a hp hq) c) fun hq ↦ dite p (fun hp ↦ b hp hq) c := by
split_ifs <;>
|
rfl
|
theorem dite_dite_distrib_right {a : p → q → α} {b : p → ¬q → α} {c : ¬p → α} :
dite p (fun hp ↦ dite q (a hp) (b hp)) c =
dite q (fun hq ↦ dite p (fun hp ↦ a hp hq) c) fun hq ↦ dite p (fun hp ↦ b hp hq) c := by
split_ifs <;>
|
Mathlib.Logic.Lemmas.34_0.END1WfxnNs4h6Zj
|
theorem dite_dite_distrib_right {a : p → q → α} {b : p → ¬q → α} {c : ¬p → α} :
dite p (fun hp ↦ dite q (a hp) (b hp)) c =
dite q (fun hq ↦ dite p (fun hp ↦ a hp hq) c) fun hq ↦ dite p (fun hp ↦ b hp hq) c
|
Mathlib_Logic_Lemmas
|
case neg
α : Sort u_1
p q r : Prop
inst✝¹ : Decidable p
inst✝ : Decidable q
a✝ b✝ c✝ : α
a : p → q → α
b : p → ¬q → α
c : ¬p → α
h✝¹ : p
h✝ : ¬q
⊢ b h✝¹ h✝ = b h✝¹ (_ : ¬q)
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a7c3ac4ac7873a7a43ead17c"
/-!
# More basic logic properties
A few more logic lemmas. These are in their own file, rather than `Logic.Basic`, because it is
convenient to be able to use the `split_ifs` tactic.
## Implementation notes
We spell those lemmas out with `dite` and `ite` rather than the `if then else` notation because this
would result in less delta-reduced statements.
-/
protected alias ⟨HEq.eq, Eq.heq⟩ := heq_iff_eq
#align heq.eq HEq.eq
#align eq.heq Eq.heq
variable {α : Sort*} {p q r : Prop} [Decidable p] [Decidable q] {a b c : α}
theorem dite_dite_distrib_left {a : p → α} {b : ¬p → q → α} {c : ¬p → ¬q → α} :
(dite p a fun hp ↦ dite q (b hp) (c hp)) =
dite q (fun hq ↦ (dite p a) fun hp ↦ b hp hq) fun hq ↦ (dite p a) fun hp ↦ c hp hq := by
split_ifs <;> rfl
#align dite_dite_distrib_left dite_dite_distrib_left
theorem dite_dite_distrib_right {a : p → q → α} {b : p → ¬q → α} {c : ¬p → α} :
dite p (fun hp ↦ dite q (a hp) (b hp)) c =
dite q (fun hq ↦ dite p (fun hp ↦ a hp hq) c) fun hq ↦ dite p (fun hp ↦ b hp hq) c := by
split_ifs <;>
|
rfl
|
theorem dite_dite_distrib_right {a : p → q → α} {b : p → ¬q → α} {c : ¬p → α} :
dite p (fun hp ↦ dite q (a hp) (b hp)) c =
dite q (fun hq ↦ dite p (fun hp ↦ a hp hq) c) fun hq ↦ dite p (fun hp ↦ b hp hq) c := by
split_ifs <;>
|
Mathlib.Logic.Lemmas.34_0.END1WfxnNs4h6Zj
|
theorem dite_dite_distrib_right {a : p → q → α} {b : p → ¬q → α} {c : ¬p → α} :
dite p (fun hp ↦ dite q (a hp) (b hp)) c =
dite q (fun hq ↦ dite p (fun hp ↦ a hp hq) c) fun hq ↦ dite p (fun hp ↦ b hp hq) c
|
Mathlib_Logic_Lemmas
|
case pos
α : Sort u_1
p q r : Prop
inst✝¹ : Decidable p
inst✝ : Decidable q
a✝ b✝ c✝ : α
a : p → q → α
b : p → ¬q → α
c : ¬p → α
h✝¹ : ¬p
h✝ : q
⊢ c h✝¹ = c h✝¹
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a7c3ac4ac7873a7a43ead17c"
/-!
# More basic logic properties
A few more logic lemmas. These are in their own file, rather than `Logic.Basic`, because it is
convenient to be able to use the `split_ifs` tactic.
## Implementation notes
We spell those lemmas out with `dite` and `ite` rather than the `if then else` notation because this
would result in less delta-reduced statements.
-/
protected alias ⟨HEq.eq, Eq.heq⟩ := heq_iff_eq
#align heq.eq HEq.eq
#align eq.heq Eq.heq
variable {α : Sort*} {p q r : Prop} [Decidable p] [Decidable q] {a b c : α}
theorem dite_dite_distrib_left {a : p → α} {b : ¬p → q → α} {c : ¬p → ¬q → α} :
(dite p a fun hp ↦ dite q (b hp) (c hp)) =
dite q (fun hq ↦ (dite p a) fun hp ↦ b hp hq) fun hq ↦ (dite p a) fun hp ↦ c hp hq := by
split_ifs <;> rfl
#align dite_dite_distrib_left dite_dite_distrib_left
theorem dite_dite_distrib_right {a : p → q → α} {b : p → ¬q → α} {c : ¬p → α} :
dite p (fun hp ↦ dite q (a hp) (b hp)) c =
dite q (fun hq ↦ dite p (fun hp ↦ a hp hq) c) fun hq ↦ dite p (fun hp ↦ b hp hq) c := by
split_ifs <;>
|
rfl
|
theorem dite_dite_distrib_right {a : p → q → α} {b : p → ¬q → α} {c : ¬p → α} :
dite p (fun hp ↦ dite q (a hp) (b hp)) c =
dite q (fun hq ↦ dite p (fun hp ↦ a hp hq) c) fun hq ↦ dite p (fun hp ↦ b hp hq) c := by
split_ifs <;>
|
Mathlib.Logic.Lemmas.34_0.END1WfxnNs4h6Zj
|
theorem dite_dite_distrib_right {a : p → q → α} {b : p → ¬q → α} {c : ¬p → α} :
dite p (fun hp ↦ dite q (a hp) (b hp)) c =
dite q (fun hq ↦ dite p (fun hp ↦ a hp hq) c) fun hq ↦ dite p (fun hp ↦ b hp hq) c
|
Mathlib_Logic_Lemmas
|
case neg
α : Sort u_1
p q r : Prop
inst✝¹ : Decidable p
inst✝ : Decidable q
a✝ b✝ c✝ : α
a : p → q → α
b : p → ¬q → α
c : ¬p → α
h✝¹ : ¬p
h✝ : ¬q
⊢ c h✝¹ = c h✝¹
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a7c3ac4ac7873a7a43ead17c"
/-!
# More basic logic properties
A few more logic lemmas. These are in their own file, rather than `Logic.Basic`, because it is
convenient to be able to use the `split_ifs` tactic.
## Implementation notes
We spell those lemmas out with `dite` and `ite` rather than the `if then else` notation because this
would result in less delta-reduced statements.
-/
protected alias ⟨HEq.eq, Eq.heq⟩ := heq_iff_eq
#align heq.eq HEq.eq
#align eq.heq Eq.heq
variable {α : Sort*} {p q r : Prop} [Decidable p] [Decidable q] {a b c : α}
theorem dite_dite_distrib_left {a : p → α} {b : ¬p → q → α} {c : ¬p → ¬q → α} :
(dite p a fun hp ↦ dite q (b hp) (c hp)) =
dite q (fun hq ↦ (dite p a) fun hp ↦ b hp hq) fun hq ↦ (dite p a) fun hp ↦ c hp hq := by
split_ifs <;> rfl
#align dite_dite_distrib_left dite_dite_distrib_left
theorem dite_dite_distrib_right {a : p → q → α} {b : p → ¬q → α} {c : ¬p → α} :
dite p (fun hp ↦ dite q (a hp) (b hp)) c =
dite q (fun hq ↦ dite p (fun hp ↦ a hp hq) c) fun hq ↦ dite p (fun hp ↦ b hp hq) c := by
split_ifs <;>
|
rfl
|
theorem dite_dite_distrib_right {a : p → q → α} {b : p → ¬q → α} {c : ¬p → α} :
dite p (fun hp ↦ dite q (a hp) (b hp)) c =
dite q (fun hq ↦ dite p (fun hp ↦ a hp hq) c) fun hq ↦ dite p (fun hp ↦ b hp hq) c := by
split_ifs <;>
|
Mathlib.Logic.Lemmas.34_0.END1WfxnNs4h6Zj
|
theorem dite_dite_distrib_right {a : p → q → α} {b : p → ¬q → α} {c : ¬p → α} :
dite p (fun hp ↦ dite q (a hp) (b hp)) c =
dite q (fun hq ↦ dite p (fun hp ↦ a hp hq) c) fun hq ↦ dite p (fun hp ↦ b hp hq) c
|
Mathlib_Logic_Lemmas
|
α : Sort u_1
p q r : Prop
inst✝¹ : Decidable p
inst✝ : Decidable q
a b c : α
f : Prop → Prop
⊢ f True ∧ f False → ∀ (p : Prop), f p
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a7c3ac4ac7873a7a43ead17c"
/-!
# More basic logic properties
A few more logic lemmas. These are in their own file, rather than `Logic.Basic`, because it is
convenient to be able to use the `split_ifs` tactic.
## Implementation notes
We spell those lemmas out with `dite` and `ite` rather than the `if then else` notation because this
would result in less delta-reduced statements.
-/
protected alias ⟨HEq.eq, Eq.heq⟩ := heq_iff_eq
#align heq.eq HEq.eq
#align eq.heq Eq.heq
variable {α : Sort*} {p q r : Prop} [Decidable p] [Decidable q] {a b c : α}
theorem dite_dite_distrib_left {a : p → α} {b : ¬p → q → α} {c : ¬p → ¬q → α} :
(dite p a fun hp ↦ dite q (b hp) (c hp)) =
dite q (fun hq ↦ (dite p a) fun hp ↦ b hp hq) fun hq ↦ (dite p a) fun hp ↦ c hp hq := by
split_ifs <;> rfl
#align dite_dite_distrib_left dite_dite_distrib_left
theorem dite_dite_distrib_right {a : p → q → α} {b : p → ¬q → α} {c : ¬p → α} :
dite p (fun hp ↦ dite q (a hp) (b hp)) c =
dite q (fun hq ↦ dite p (fun hp ↦ a hp hq) c) fun hq ↦ dite p (fun hp ↦ b hp hq) c := by
split_ifs <;> rfl
#align dite_dite_distrib_right dite_dite_distrib_right
theorem ite_dite_distrib_left {a : α} {b : q → α} {c : ¬q → α} :
ite p a (dite q b c) = dite q (fun hq ↦ ite p a <| b hq) fun hq ↦ ite p a <| c hq :=
dite_dite_distrib_left
#align ite_dite_distrib_left ite_dite_distrib_left
theorem ite_dite_distrib_right {a : q → α} {b : ¬q → α} {c : α} :
ite p (dite q a b) c = dite q (fun hq ↦ ite p (a hq) c) fun hq ↦ ite p (b hq) c :=
dite_dite_distrib_right
#align ite_dite_distrib_right ite_dite_distrib_right
theorem dite_ite_distrib_left {a : p → α} {b : ¬p → α} {c : ¬p → α} :
(dite p a fun hp ↦ ite q (b hp) (c hp)) = ite q (dite p a b) (dite p a c) :=
dite_dite_distrib_left
#align dite_ite_distrib_left dite_ite_distrib_left
theorem dite_ite_distrib_right {a : p → α} {b : p → α} {c : ¬p → α} :
dite p (fun hp ↦ ite q (a hp) (b hp)) c = ite q (dite p a c) (dite p b c) :=
dite_dite_distrib_right
#align dite_ite_distrib_right dite_ite_distrib_right
theorem ite_ite_distrib_left : ite p a (ite q b c) = ite q (ite p a b) (ite p a c) :=
dite_dite_distrib_left
#align ite_ite_distrib_left ite_ite_distrib_left
theorem ite_ite_distrib_right : ite p (ite q a b) c = ite q (ite p a c) (ite p b c) :=
dite_dite_distrib_right
#align ite_ite_distrib_right ite_ite_distrib_right
lemma Prop.forall {f : Prop → Prop} : (∀ p, f p) ↔ f True ∧ f False :=
⟨fun h ↦ ⟨h _, h _⟩, by
|
rintro ⟨h₁, h₀⟩ p
|
lemma Prop.forall {f : Prop → Prop} : (∀ p, f p) ↔ f True ∧ f False :=
⟨fun h ↦ ⟨h _, h _⟩, by
|
Mathlib.Logic.Lemmas.68_0.END1WfxnNs4h6Zj
|
lemma Prop.forall {f : Prop → Prop} : (∀ p, f p) ↔ f True ∧ f False
|
Mathlib_Logic_Lemmas
|
case intro
α : Sort u_1
p✝ q r : Prop
inst✝¹ : Decidable p✝
inst✝ : Decidable q
a b c : α
f : Prop → Prop
h₁ : f True
h₀ : f False
p : Prop
⊢ f p
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a7c3ac4ac7873a7a43ead17c"
/-!
# More basic logic properties
A few more logic lemmas. These are in their own file, rather than `Logic.Basic`, because it is
convenient to be able to use the `split_ifs` tactic.
## Implementation notes
We spell those lemmas out with `dite` and `ite` rather than the `if then else` notation because this
would result in less delta-reduced statements.
-/
protected alias ⟨HEq.eq, Eq.heq⟩ := heq_iff_eq
#align heq.eq HEq.eq
#align eq.heq Eq.heq
variable {α : Sort*} {p q r : Prop} [Decidable p] [Decidable q] {a b c : α}
theorem dite_dite_distrib_left {a : p → α} {b : ¬p → q → α} {c : ¬p → ¬q → α} :
(dite p a fun hp ↦ dite q (b hp) (c hp)) =
dite q (fun hq ↦ (dite p a) fun hp ↦ b hp hq) fun hq ↦ (dite p a) fun hp ↦ c hp hq := by
split_ifs <;> rfl
#align dite_dite_distrib_left dite_dite_distrib_left
theorem dite_dite_distrib_right {a : p → q → α} {b : p → ¬q → α} {c : ¬p → α} :
dite p (fun hp ↦ dite q (a hp) (b hp)) c =
dite q (fun hq ↦ dite p (fun hp ↦ a hp hq) c) fun hq ↦ dite p (fun hp ↦ b hp hq) c := by
split_ifs <;> rfl
#align dite_dite_distrib_right dite_dite_distrib_right
theorem ite_dite_distrib_left {a : α} {b : q → α} {c : ¬q → α} :
ite p a (dite q b c) = dite q (fun hq ↦ ite p a <| b hq) fun hq ↦ ite p a <| c hq :=
dite_dite_distrib_left
#align ite_dite_distrib_left ite_dite_distrib_left
theorem ite_dite_distrib_right {a : q → α} {b : ¬q → α} {c : α} :
ite p (dite q a b) c = dite q (fun hq ↦ ite p (a hq) c) fun hq ↦ ite p (b hq) c :=
dite_dite_distrib_right
#align ite_dite_distrib_right ite_dite_distrib_right
theorem dite_ite_distrib_left {a : p → α} {b : ¬p → α} {c : ¬p → α} :
(dite p a fun hp ↦ ite q (b hp) (c hp)) = ite q (dite p a b) (dite p a c) :=
dite_dite_distrib_left
#align dite_ite_distrib_left dite_ite_distrib_left
theorem dite_ite_distrib_right {a : p → α} {b : p → α} {c : ¬p → α} :
dite p (fun hp ↦ ite q (a hp) (b hp)) c = ite q (dite p a c) (dite p b c) :=
dite_dite_distrib_right
#align dite_ite_distrib_right dite_ite_distrib_right
theorem ite_ite_distrib_left : ite p a (ite q b c) = ite q (ite p a b) (ite p a c) :=
dite_dite_distrib_left
#align ite_ite_distrib_left ite_ite_distrib_left
theorem ite_ite_distrib_right : ite p (ite q a b) c = ite q (ite p a c) (ite p b c) :=
dite_dite_distrib_right
#align ite_ite_distrib_right ite_ite_distrib_right
lemma Prop.forall {f : Prop → Prop} : (∀ p, f p) ↔ f True ∧ f False :=
⟨fun h ↦ ⟨h _, h _⟩, by rintro ⟨h₁, h₀⟩ p;
|
by_cases hp : p
|
lemma Prop.forall {f : Prop → Prop} : (∀ p, f p) ↔ f True ∧ f False :=
⟨fun h ↦ ⟨h _, h _⟩, by rintro ⟨h₁, h₀⟩ p;
|
Mathlib.Logic.Lemmas.68_0.END1WfxnNs4h6Zj
|
lemma Prop.forall {f : Prop → Prop} : (∀ p, f p) ↔ f True ∧ f False
|
Mathlib_Logic_Lemmas
|
case pos
α : Sort u_1
p✝ q r : Prop
inst✝¹ : Decidable p✝
inst✝ : Decidable q
a b c : α
f : Prop → Prop
h₁ : f True
h₀ : f False
p : Prop
hp : p
⊢ f p
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a7c3ac4ac7873a7a43ead17c"
/-!
# More basic logic properties
A few more logic lemmas. These are in their own file, rather than `Logic.Basic`, because it is
convenient to be able to use the `split_ifs` tactic.
## Implementation notes
We spell those lemmas out with `dite` and `ite` rather than the `if then else` notation because this
would result in less delta-reduced statements.
-/
protected alias ⟨HEq.eq, Eq.heq⟩ := heq_iff_eq
#align heq.eq HEq.eq
#align eq.heq Eq.heq
variable {α : Sort*} {p q r : Prop} [Decidable p] [Decidable q] {a b c : α}
theorem dite_dite_distrib_left {a : p → α} {b : ¬p → q → α} {c : ¬p → ¬q → α} :
(dite p a fun hp ↦ dite q (b hp) (c hp)) =
dite q (fun hq ↦ (dite p a) fun hp ↦ b hp hq) fun hq ↦ (dite p a) fun hp ↦ c hp hq := by
split_ifs <;> rfl
#align dite_dite_distrib_left dite_dite_distrib_left
theorem dite_dite_distrib_right {a : p → q → α} {b : p → ¬q → α} {c : ¬p → α} :
dite p (fun hp ↦ dite q (a hp) (b hp)) c =
dite q (fun hq ↦ dite p (fun hp ↦ a hp hq) c) fun hq ↦ dite p (fun hp ↦ b hp hq) c := by
split_ifs <;> rfl
#align dite_dite_distrib_right dite_dite_distrib_right
theorem ite_dite_distrib_left {a : α} {b : q → α} {c : ¬q → α} :
ite p a (dite q b c) = dite q (fun hq ↦ ite p a <| b hq) fun hq ↦ ite p a <| c hq :=
dite_dite_distrib_left
#align ite_dite_distrib_left ite_dite_distrib_left
theorem ite_dite_distrib_right {a : q → α} {b : ¬q → α} {c : α} :
ite p (dite q a b) c = dite q (fun hq ↦ ite p (a hq) c) fun hq ↦ ite p (b hq) c :=
dite_dite_distrib_right
#align ite_dite_distrib_right ite_dite_distrib_right
theorem dite_ite_distrib_left {a : p → α} {b : ¬p → α} {c : ¬p → α} :
(dite p a fun hp ↦ ite q (b hp) (c hp)) = ite q (dite p a b) (dite p a c) :=
dite_dite_distrib_left
#align dite_ite_distrib_left dite_ite_distrib_left
theorem dite_ite_distrib_right {a : p → α} {b : p → α} {c : ¬p → α} :
dite p (fun hp ↦ ite q (a hp) (b hp)) c = ite q (dite p a c) (dite p b c) :=
dite_dite_distrib_right
#align dite_ite_distrib_right dite_ite_distrib_right
theorem ite_ite_distrib_left : ite p a (ite q b c) = ite q (ite p a b) (ite p a c) :=
dite_dite_distrib_left
#align ite_ite_distrib_left ite_ite_distrib_left
theorem ite_ite_distrib_right : ite p (ite q a b) c = ite q (ite p a c) (ite p b c) :=
dite_dite_distrib_right
#align ite_ite_distrib_right ite_ite_distrib_right
lemma Prop.forall {f : Prop → Prop} : (∀ p, f p) ↔ f True ∧ f False :=
⟨fun h ↦ ⟨h _, h _⟩, by rintro ⟨h₁, h₀⟩ p; by_cases hp : p <;>
|
simp only [hp]
|
lemma Prop.forall {f : Prop → Prop} : (∀ p, f p) ↔ f True ∧ f False :=
⟨fun h ↦ ⟨h _, h _⟩, by rintro ⟨h₁, h₀⟩ p; by_cases hp : p <;>
|
Mathlib.Logic.Lemmas.68_0.END1WfxnNs4h6Zj
|
lemma Prop.forall {f : Prop → Prop} : (∀ p, f p) ↔ f True ∧ f False
|
Mathlib_Logic_Lemmas
|
case neg
α : Sort u_1
p✝ q r : Prop
inst✝¹ : Decidable p✝
inst✝ : Decidable q
a b c : α
f : Prop → Prop
h₁ : f True
h₀ : f False
p : Prop
hp : ¬p
⊢ f p
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a7c3ac4ac7873a7a43ead17c"
/-!
# More basic logic properties
A few more logic lemmas. These are in their own file, rather than `Logic.Basic`, because it is
convenient to be able to use the `split_ifs` tactic.
## Implementation notes
We spell those lemmas out with `dite` and `ite` rather than the `if then else` notation because this
would result in less delta-reduced statements.
-/
protected alias ⟨HEq.eq, Eq.heq⟩ := heq_iff_eq
#align heq.eq HEq.eq
#align eq.heq Eq.heq
variable {α : Sort*} {p q r : Prop} [Decidable p] [Decidable q] {a b c : α}
theorem dite_dite_distrib_left {a : p → α} {b : ¬p → q → α} {c : ¬p → ¬q → α} :
(dite p a fun hp ↦ dite q (b hp) (c hp)) =
dite q (fun hq ↦ (dite p a) fun hp ↦ b hp hq) fun hq ↦ (dite p a) fun hp ↦ c hp hq := by
split_ifs <;> rfl
#align dite_dite_distrib_left dite_dite_distrib_left
theorem dite_dite_distrib_right {a : p → q → α} {b : p → ¬q → α} {c : ¬p → α} :
dite p (fun hp ↦ dite q (a hp) (b hp)) c =
dite q (fun hq ↦ dite p (fun hp ↦ a hp hq) c) fun hq ↦ dite p (fun hp ↦ b hp hq) c := by
split_ifs <;> rfl
#align dite_dite_distrib_right dite_dite_distrib_right
theorem ite_dite_distrib_left {a : α} {b : q → α} {c : ¬q → α} :
ite p a (dite q b c) = dite q (fun hq ↦ ite p a <| b hq) fun hq ↦ ite p a <| c hq :=
dite_dite_distrib_left
#align ite_dite_distrib_left ite_dite_distrib_left
theorem ite_dite_distrib_right {a : q → α} {b : ¬q → α} {c : α} :
ite p (dite q a b) c = dite q (fun hq ↦ ite p (a hq) c) fun hq ↦ ite p (b hq) c :=
dite_dite_distrib_right
#align ite_dite_distrib_right ite_dite_distrib_right
theorem dite_ite_distrib_left {a : p → α} {b : ¬p → α} {c : ¬p → α} :
(dite p a fun hp ↦ ite q (b hp) (c hp)) = ite q (dite p a b) (dite p a c) :=
dite_dite_distrib_left
#align dite_ite_distrib_left dite_ite_distrib_left
theorem dite_ite_distrib_right {a : p → α} {b : p → α} {c : ¬p → α} :
dite p (fun hp ↦ ite q (a hp) (b hp)) c = ite q (dite p a c) (dite p b c) :=
dite_dite_distrib_right
#align dite_ite_distrib_right dite_ite_distrib_right
theorem ite_ite_distrib_left : ite p a (ite q b c) = ite q (ite p a b) (ite p a c) :=
dite_dite_distrib_left
#align ite_ite_distrib_left ite_ite_distrib_left
theorem ite_ite_distrib_right : ite p (ite q a b) c = ite q (ite p a c) (ite p b c) :=
dite_dite_distrib_right
#align ite_ite_distrib_right ite_ite_distrib_right
lemma Prop.forall {f : Prop → Prop} : (∀ p, f p) ↔ f True ∧ f False :=
⟨fun h ↦ ⟨h _, h _⟩, by rintro ⟨h₁, h₀⟩ p; by_cases hp : p <;>
|
simp only [hp]
|
lemma Prop.forall {f : Prop → Prop} : (∀ p, f p) ↔ f True ∧ f False :=
⟨fun h ↦ ⟨h _, h _⟩, by rintro ⟨h₁, h₀⟩ p; by_cases hp : p <;>
|
Mathlib.Logic.Lemmas.68_0.END1WfxnNs4h6Zj
|
lemma Prop.forall {f : Prop → Prop} : (∀ p, f p) ↔ f True ∧ f False
|
Mathlib_Logic_Lemmas
|
case pos
α : Sort u_1
p✝ q r : Prop
inst✝¹ : Decidable p✝
inst✝ : Decidable q
a b c : α
f : Prop → Prop
h₁ : f True
h₀ : f False
p : Prop
hp : p
⊢ f True
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a7c3ac4ac7873a7a43ead17c"
/-!
# More basic logic properties
A few more logic lemmas. These are in their own file, rather than `Logic.Basic`, because it is
convenient to be able to use the `split_ifs` tactic.
## Implementation notes
We spell those lemmas out with `dite` and `ite` rather than the `if then else` notation because this
would result in less delta-reduced statements.
-/
protected alias ⟨HEq.eq, Eq.heq⟩ := heq_iff_eq
#align heq.eq HEq.eq
#align eq.heq Eq.heq
variable {α : Sort*} {p q r : Prop} [Decidable p] [Decidable q] {a b c : α}
theorem dite_dite_distrib_left {a : p → α} {b : ¬p → q → α} {c : ¬p → ¬q → α} :
(dite p a fun hp ↦ dite q (b hp) (c hp)) =
dite q (fun hq ↦ (dite p a) fun hp ↦ b hp hq) fun hq ↦ (dite p a) fun hp ↦ c hp hq := by
split_ifs <;> rfl
#align dite_dite_distrib_left dite_dite_distrib_left
theorem dite_dite_distrib_right {a : p → q → α} {b : p → ¬q → α} {c : ¬p → α} :
dite p (fun hp ↦ dite q (a hp) (b hp)) c =
dite q (fun hq ↦ dite p (fun hp ↦ a hp hq) c) fun hq ↦ dite p (fun hp ↦ b hp hq) c := by
split_ifs <;> rfl
#align dite_dite_distrib_right dite_dite_distrib_right
theorem ite_dite_distrib_left {a : α} {b : q → α} {c : ¬q → α} :
ite p a (dite q b c) = dite q (fun hq ↦ ite p a <| b hq) fun hq ↦ ite p a <| c hq :=
dite_dite_distrib_left
#align ite_dite_distrib_left ite_dite_distrib_left
theorem ite_dite_distrib_right {a : q → α} {b : ¬q → α} {c : α} :
ite p (dite q a b) c = dite q (fun hq ↦ ite p (a hq) c) fun hq ↦ ite p (b hq) c :=
dite_dite_distrib_right
#align ite_dite_distrib_right ite_dite_distrib_right
theorem dite_ite_distrib_left {a : p → α} {b : ¬p → α} {c : ¬p → α} :
(dite p a fun hp ↦ ite q (b hp) (c hp)) = ite q (dite p a b) (dite p a c) :=
dite_dite_distrib_left
#align dite_ite_distrib_left dite_ite_distrib_left
theorem dite_ite_distrib_right {a : p → α} {b : p → α} {c : ¬p → α} :
dite p (fun hp ↦ ite q (a hp) (b hp)) c = ite q (dite p a c) (dite p b c) :=
dite_dite_distrib_right
#align dite_ite_distrib_right dite_ite_distrib_right
theorem ite_ite_distrib_left : ite p a (ite q b c) = ite q (ite p a b) (ite p a c) :=
dite_dite_distrib_left
#align ite_ite_distrib_left ite_ite_distrib_left
theorem ite_ite_distrib_right : ite p (ite q a b) c = ite q (ite p a c) (ite p b c) :=
dite_dite_distrib_right
#align ite_ite_distrib_right ite_ite_distrib_right
lemma Prop.forall {f : Prop → Prop} : (∀ p, f p) ↔ f True ∧ f False :=
⟨fun h ↦ ⟨h _, h _⟩, by rintro ⟨h₁, h₀⟩ p; by_cases hp : p <;> simp only [hp] <;>
|
assumption
|
lemma Prop.forall {f : Prop → Prop} : (∀ p, f p) ↔ f True ∧ f False :=
⟨fun h ↦ ⟨h _, h _⟩, by rintro ⟨h₁, h₀⟩ p; by_cases hp : p <;> simp only [hp] <;>
|
Mathlib.Logic.Lemmas.68_0.END1WfxnNs4h6Zj
|
lemma Prop.forall {f : Prop → Prop} : (∀ p, f p) ↔ f True ∧ f False
|
Mathlib_Logic_Lemmas
|
case neg
α : Sort u_1
p✝ q r : Prop
inst✝¹ : Decidable p✝
inst✝ : Decidable q
a b c : α
f : Prop → Prop
h₁ : f True
h₀ : f False
p : Prop
hp : ¬p
⊢ f False
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a7c3ac4ac7873a7a43ead17c"
/-!
# More basic logic properties
A few more logic lemmas. These are in their own file, rather than `Logic.Basic`, because it is
convenient to be able to use the `split_ifs` tactic.
## Implementation notes
We spell those lemmas out with `dite` and `ite` rather than the `if then else` notation because this
would result in less delta-reduced statements.
-/
protected alias ⟨HEq.eq, Eq.heq⟩ := heq_iff_eq
#align heq.eq HEq.eq
#align eq.heq Eq.heq
variable {α : Sort*} {p q r : Prop} [Decidable p] [Decidable q] {a b c : α}
theorem dite_dite_distrib_left {a : p → α} {b : ¬p → q → α} {c : ¬p → ¬q → α} :
(dite p a fun hp ↦ dite q (b hp) (c hp)) =
dite q (fun hq ↦ (dite p a) fun hp ↦ b hp hq) fun hq ↦ (dite p a) fun hp ↦ c hp hq := by
split_ifs <;> rfl
#align dite_dite_distrib_left dite_dite_distrib_left
theorem dite_dite_distrib_right {a : p → q → α} {b : p → ¬q → α} {c : ¬p → α} :
dite p (fun hp ↦ dite q (a hp) (b hp)) c =
dite q (fun hq ↦ dite p (fun hp ↦ a hp hq) c) fun hq ↦ dite p (fun hp ↦ b hp hq) c := by
split_ifs <;> rfl
#align dite_dite_distrib_right dite_dite_distrib_right
theorem ite_dite_distrib_left {a : α} {b : q → α} {c : ¬q → α} :
ite p a (dite q b c) = dite q (fun hq ↦ ite p a <| b hq) fun hq ↦ ite p a <| c hq :=
dite_dite_distrib_left
#align ite_dite_distrib_left ite_dite_distrib_left
theorem ite_dite_distrib_right {a : q → α} {b : ¬q → α} {c : α} :
ite p (dite q a b) c = dite q (fun hq ↦ ite p (a hq) c) fun hq ↦ ite p (b hq) c :=
dite_dite_distrib_right
#align ite_dite_distrib_right ite_dite_distrib_right
theorem dite_ite_distrib_left {a : p → α} {b : ¬p → α} {c : ¬p → α} :
(dite p a fun hp ↦ ite q (b hp) (c hp)) = ite q (dite p a b) (dite p a c) :=
dite_dite_distrib_left
#align dite_ite_distrib_left dite_ite_distrib_left
theorem dite_ite_distrib_right {a : p → α} {b : p → α} {c : ¬p → α} :
dite p (fun hp ↦ ite q (a hp) (b hp)) c = ite q (dite p a c) (dite p b c) :=
dite_dite_distrib_right
#align dite_ite_distrib_right dite_ite_distrib_right
theorem ite_ite_distrib_left : ite p a (ite q b c) = ite q (ite p a b) (ite p a c) :=
dite_dite_distrib_left
#align ite_ite_distrib_left ite_ite_distrib_left
theorem ite_ite_distrib_right : ite p (ite q a b) c = ite q (ite p a c) (ite p b c) :=
dite_dite_distrib_right
#align ite_ite_distrib_right ite_ite_distrib_right
lemma Prop.forall {f : Prop → Prop} : (∀ p, f p) ↔ f True ∧ f False :=
⟨fun h ↦ ⟨h _, h _⟩, by rintro ⟨h₁, h₀⟩ p; by_cases hp : p <;> simp only [hp] <;>
|
assumption
|
lemma Prop.forall {f : Prop → Prop} : (∀ p, f p) ↔ f True ∧ f False :=
⟨fun h ↦ ⟨h _, h _⟩, by rintro ⟨h₁, h₀⟩ p; by_cases hp : p <;> simp only [hp] <;>
|
Mathlib.Logic.Lemmas.68_0.END1WfxnNs4h6Zj
|
lemma Prop.forall {f : Prop → Prop} : (∀ p, f p) ↔ f True ∧ f False
|
Mathlib_Logic_Lemmas
|
α : Sort u_1
p✝ q r : Prop
inst✝¹ : Decidable p✝
inst✝ : Decidable q
a b c : α
f : Prop → Prop
x✝ : ∃ p, f p
p : Prop
h : f p
⊢ f True ∨ f False
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a7c3ac4ac7873a7a43ead17c"
/-!
# More basic logic properties
A few more logic lemmas. These are in their own file, rather than `Logic.Basic`, because it is
convenient to be able to use the `split_ifs` tactic.
## Implementation notes
We spell those lemmas out with `dite` and `ite` rather than the `if then else` notation because this
would result in less delta-reduced statements.
-/
protected alias ⟨HEq.eq, Eq.heq⟩ := heq_iff_eq
#align heq.eq HEq.eq
#align eq.heq Eq.heq
variable {α : Sort*} {p q r : Prop} [Decidable p] [Decidable q] {a b c : α}
theorem dite_dite_distrib_left {a : p → α} {b : ¬p → q → α} {c : ¬p → ¬q → α} :
(dite p a fun hp ↦ dite q (b hp) (c hp)) =
dite q (fun hq ↦ (dite p a) fun hp ↦ b hp hq) fun hq ↦ (dite p a) fun hp ↦ c hp hq := by
split_ifs <;> rfl
#align dite_dite_distrib_left dite_dite_distrib_left
theorem dite_dite_distrib_right {a : p → q → α} {b : p → ¬q → α} {c : ¬p → α} :
dite p (fun hp ↦ dite q (a hp) (b hp)) c =
dite q (fun hq ↦ dite p (fun hp ↦ a hp hq) c) fun hq ↦ dite p (fun hp ↦ b hp hq) c := by
split_ifs <;> rfl
#align dite_dite_distrib_right dite_dite_distrib_right
theorem ite_dite_distrib_left {a : α} {b : q → α} {c : ¬q → α} :
ite p a (dite q b c) = dite q (fun hq ↦ ite p a <| b hq) fun hq ↦ ite p a <| c hq :=
dite_dite_distrib_left
#align ite_dite_distrib_left ite_dite_distrib_left
theorem ite_dite_distrib_right {a : q → α} {b : ¬q → α} {c : α} :
ite p (dite q a b) c = dite q (fun hq ↦ ite p (a hq) c) fun hq ↦ ite p (b hq) c :=
dite_dite_distrib_right
#align ite_dite_distrib_right ite_dite_distrib_right
theorem dite_ite_distrib_left {a : p → α} {b : ¬p → α} {c : ¬p → α} :
(dite p a fun hp ↦ ite q (b hp) (c hp)) = ite q (dite p a b) (dite p a c) :=
dite_dite_distrib_left
#align dite_ite_distrib_left dite_ite_distrib_left
theorem dite_ite_distrib_right {a : p → α} {b : p → α} {c : ¬p → α} :
dite p (fun hp ↦ ite q (a hp) (b hp)) c = ite q (dite p a c) (dite p b c) :=
dite_dite_distrib_right
#align dite_ite_distrib_right dite_ite_distrib_right
theorem ite_ite_distrib_left : ite p a (ite q b c) = ite q (ite p a b) (ite p a c) :=
dite_dite_distrib_left
#align ite_ite_distrib_left ite_ite_distrib_left
theorem ite_ite_distrib_right : ite p (ite q a b) c = ite q (ite p a c) (ite p b c) :=
dite_dite_distrib_right
#align ite_ite_distrib_right ite_ite_distrib_right
lemma Prop.forall {f : Prop → Prop} : (∀ p, f p) ↔ f True ∧ f False :=
⟨fun h ↦ ⟨h _, h _⟩, by rintro ⟨h₁, h₀⟩ p; by_cases hp : p <;> simp only [hp] <;> assumption⟩
#align Prop.forall Prop.forall
lemma Prop.exists {f : Prop → Prop} : (∃ p, f p) ↔ f True ∨ f False :=
⟨fun ⟨p, h⟩ ↦ by
|
refine' (em p).imp _ _
|
lemma Prop.exists {f : Prop → Prop} : (∃ p, f p) ↔ f True ∨ f False :=
⟨fun ⟨p, h⟩ ↦ by
|
Mathlib.Logic.Lemmas.72_0.END1WfxnNs4h6Zj
|
lemma Prop.exists {f : Prop → Prop} : (∃ p, f p) ↔ f True ∨ f False
|
Mathlib_Logic_Lemmas
|
case refine'_1
α : Sort u_1
p✝ q r : Prop
inst✝¹ : Decidable p✝
inst✝ : Decidable q
a b c : α
f : Prop → Prop
x✝ : ∃ p, f p
p : Prop
h : f p
⊢ p → f True
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a7c3ac4ac7873a7a43ead17c"
/-!
# More basic logic properties
A few more logic lemmas. These are in their own file, rather than `Logic.Basic`, because it is
convenient to be able to use the `split_ifs` tactic.
## Implementation notes
We spell those lemmas out with `dite` and `ite` rather than the `if then else` notation because this
would result in less delta-reduced statements.
-/
protected alias ⟨HEq.eq, Eq.heq⟩ := heq_iff_eq
#align heq.eq HEq.eq
#align eq.heq Eq.heq
variable {α : Sort*} {p q r : Prop} [Decidable p] [Decidable q] {a b c : α}
theorem dite_dite_distrib_left {a : p → α} {b : ¬p → q → α} {c : ¬p → ¬q → α} :
(dite p a fun hp ↦ dite q (b hp) (c hp)) =
dite q (fun hq ↦ (dite p a) fun hp ↦ b hp hq) fun hq ↦ (dite p a) fun hp ↦ c hp hq := by
split_ifs <;> rfl
#align dite_dite_distrib_left dite_dite_distrib_left
theorem dite_dite_distrib_right {a : p → q → α} {b : p → ¬q → α} {c : ¬p → α} :
dite p (fun hp ↦ dite q (a hp) (b hp)) c =
dite q (fun hq ↦ dite p (fun hp ↦ a hp hq) c) fun hq ↦ dite p (fun hp ↦ b hp hq) c := by
split_ifs <;> rfl
#align dite_dite_distrib_right dite_dite_distrib_right
theorem ite_dite_distrib_left {a : α} {b : q → α} {c : ¬q → α} :
ite p a (dite q b c) = dite q (fun hq ↦ ite p a <| b hq) fun hq ↦ ite p a <| c hq :=
dite_dite_distrib_left
#align ite_dite_distrib_left ite_dite_distrib_left
theorem ite_dite_distrib_right {a : q → α} {b : ¬q → α} {c : α} :
ite p (dite q a b) c = dite q (fun hq ↦ ite p (a hq) c) fun hq ↦ ite p (b hq) c :=
dite_dite_distrib_right
#align ite_dite_distrib_right ite_dite_distrib_right
theorem dite_ite_distrib_left {a : p → α} {b : ¬p → α} {c : ¬p → α} :
(dite p a fun hp ↦ ite q (b hp) (c hp)) = ite q (dite p a b) (dite p a c) :=
dite_dite_distrib_left
#align dite_ite_distrib_left dite_ite_distrib_left
theorem dite_ite_distrib_right {a : p → α} {b : p → α} {c : ¬p → α} :
dite p (fun hp ↦ ite q (a hp) (b hp)) c = ite q (dite p a c) (dite p b c) :=
dite_dite_distrib_right
#align dite_ite_distrib_right dite_ite_distrib_right
theorem ite_ite_distrib_left : ite p a (ite q b c) = ite q (ite p a b) (ite p a c) :=
dite_dite_distrib_left
#align ite_ite_distrib_left ite_ite_distrib_left
theorem ite_ite_distrib_right : ite p (ite q a b) c = ite q (ite p a c) (ite p b c) :=
dite_dite_distrib_right
#align ite_ite_distrib_right ite_ite_distrib_right
lemma Prop.forall {f : Prop → Prop} : (∀ p, f p) ↔ f True ∧ f False :=
⟨fun h ↦ ⟨h _, h _⟩, by rintro ⟨h₁, h₀⟩ p; by_cases hp : p <;> simp only [hp] <;> assumption⟩
#align Prop.forall Prop.forall
lemma Prop.exists {f : Prop → Prop} : (∃ p, f p) ↔ f True ∨ f False :=
⟨fun ⟨p, h⟩ ↦ by refine' (em p).imp _ _ <;>
|
intro H
|
lemma Prop.exists {f : Prop → Prop} : (∃ p, f p) ↔ f True ∨ f False :=
⟨fun ⟨p, h⟩ ↦ by refine' (em p).imp _ _ <;>
|
Mathlib.Logic.Lemmas.72_0.END1WfxnNs4h6Zj
|
lemma Prop.exists {f : Prop → Prop} : (∃ p, f p) ↔ f True ∨ f False
|
Mathlib_Logic_Lemmas
|
case refine'_2
α : Sort u_1
p✝ q r : Prop
inst✝¹ : Decidable p✝
inst✝ : Decidable q
a b c : α
f : Prop → Prop
x✝ : ∃ p, f p
p : Prop
h : f p
⊢ ¬p → f False
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a7c3ac4ac7873a7a43ead17c"
/-!
# More basic logic properties
A few more logic lemmas. These are in their own file, rather than `Logic.Basic`, because it is
convenient to be able to use the `split_ifs` tactic.
## Implementation notes
We spell those lemmas out with `dite` and `ite` rather than the `if then else` notation because this
would result in less delta-reduced statements.
-/
protected alias ⟨HEq.eq, Eq.heq⟩ := heq_iff_eq
#align heq.eq HEq.eq
#align eq.heq Eq.heq
variable {α : Sort*} {p q r : Prop} [Decidable p] [Decidable q] {a b c : α}
theorem dite_dite_distrib_left {a : p → α} {b : ¬p → q → α} {c : ¬p → ¬q → α} :
(dite p a fun hp ↦ dite q (b hp) (c hp)) =
dite q (fun hq ↦ (dite p a) fun hp ↦ b hp hq) fun hq ↦ (dite p a) fun hp ↦ c hp hq := by
split_ifs <;> rfl
#align dite_dite_distrib_left dite_dite_distrib_left
theorem dite_dite_distrib_right {a : p → q → α} {b : p → ¬q → α} {c : ¬p → α} :
dite p (fun hp ↦ dite q (a hp) (b hp)) c =
dite q (fun hq ↦ dite p (fun hp ↦ a hp hq) c) fun hq ↦ dite p (fun hp ↦ b hp hq) c := by
split_ifs <;> rfl
#align dite_dite_distrib_right dite_dite_distrib_right
theorem ite_dite_distrib_left {a : α} {b : q → α} {c : ¬q → α} :
ite p a (dite q b c) = dite q (fun hq ↦ ite p a <| b hq) fun hq ↦ ite p a <| c hq :=
dite_dite_distrib_left
#align ite_dite_distrib_left ite_dite_distrib_left
theorem ite_dite_distrib_right {a : q → α} {b : ¬q → α} {c : α} :
ite p (dite q a b) c = dite q (fun hq ↦ ite p (a hq) c) fun hq ↦ ite p (b hq) c :=
dite_dite_distrib_right
#align ite_dite_distrib_right ite_dite_distrib_right
theorem dite_ite_distrib_left {a : p → α} {b : ¬p → α} {c : ¬p → α} :
(dite p a fun hp ↦ ite q (b hp) (c hp)) = ite q (dite p a b) (dite p a c) :=
dite_dite_distrib_left
#align dite_ite_distrib_left dite_ite_distrib_left
theorem dite_ite_distrib_right {a : p → α} {b : p → α} {c : ¬p → α} :
dite p (fun hp ↦ ite q (a hp) (b hp)) c = ite q (dite p a c) (dite p b c) :=
dite_dite_distrib_right
#align dite_ite_distrib_right dite_ite_distrib_right
theorem ite_ite_distrib_left : ite p a (ite q b c) = ite q (ite p a b) (ite p a c) :=
dite_dite_distrib_left
#align ite_ite_distrib_left ite_ite_distrib_left
theorem ite_ite_distrib_right : ite p (ite q a b) c = ite q (ite p a c) (ite p b c) :=
dite_dite_distrib_right
#align ite_ite_distrib_right ite_ite_distrib_right
lemma Prop.forall {f : Prop → Prop} : (∀ p, f p) ↔ f True ∧ f False :=
⟨fun h ↦ ⟨h _, h _⟩, by rintro ⟨h₁, h₀⟩ p; by_cases hp : p <;> simp only [hp] <;> assumption⟩
#align Prop.forall Prop.forall
lemma Prop.exists {f : Prop → Prop} : (∃ p, f p) ↔ f True ∨ f False :=
⟨fun ⟨p, h⟩ ↦ by refine' (em p).imp _ _ <;>
|
intro H
|
lemma Prop.exists {f : Prop → Prop} : (∃ p, f p) ↔ f True ∨ f False :=
⟨fun ⟨p, h⟩ ↦ by refine' (em p).imp _ _ <;>
|
Mathlib.Logic.Lemmas.72_0.END1WfxnNs4h6Zj
|
lemma Prop.exists {f : Prop → Prop} : (∃ p, f p) ↔ f True ∨ f False
|
Mathlib_Logic_Lemmas
|
case refine'_1
α : Sort u_1
p✝ q r : Prop
inst✝¹ : Decidable p✝
inst✝ : Decidable q
a b c : α
f : Prop → Prop
x✝ : ∃ p, f p
p : Prop
h : f p
H : p
⊢ f True
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a7c3ac4ac7873a7a43ead17c"
/-!
# More basic logic properties
A few more logic lemmas. These are in their own file, rather than `Logic.Basic`, because it is
convenient to be able to use the `split_ifs` tactic.
## Implementation notes
We spell those lemmas out with `dite` and `ite` rather than the `if then else` notation because this
would result in less delta-reduced statements.
-/
protected alias ⟨HEq.eq, Eq.heq⟩ := heq_iff_eq
#align heq.eq HEq.eq
#align eq.heq Eq.heq
variable {α : Sort*} {p q r : Prop} [Decidable p] [Decidable q] {a b c : α}
theorem dite_dite_distrib_left {a : p → α} {b : ¬p → q → α} {c : ¬p → ¬q → α} :
(dite p a fun hp ↦ dite q (b hp) (c hp)) =
dite q (fun hq ↦ (dite p a) fun hp ↦ b hp hq) fun hq ↦ (dite p a) fun hp ↦ c hp hq := by
split_ifs <;> rfl
#align dite_dite_distrib_left dite_dite_distrib_left
theorem dite_dite_distrib_right {a : p → q → α} {b : p → ¬q → α} {c : ¬p → α} :
dite p (fun hp ↦ dite q (a hp) (b hp)) c =
dite q (fun hq ↦ dite p (fun hp ↦ a hp hq) c) fun hq ↦ dite p (fun hp ↦ b hp hq) c := by
split_ifs <;> rfl
#align dite_dite_distrib_right dite_dite_distrib_right
theorem ite_dite_distrib_left {a : α} {b : q → α} {c : ¬q → α} :
ite p a (dite q b c) = dite q (fun hq ↦ ite p a <| b hq) fun hq ↦ ite p a <| c hq :=
dite_dite_distrib_left
#align ite_dite_distrib_left ite_dite_distrib_left
theorem ite_dite_distrib_right {a : q → α} {b : ¬q → α} {c : α} :
ite p (dite q a b) c = dite q (fun hq ↦ ite p (a hq) c) fun hq ↦ ite p (b hq) c :=
dite_dite_distrib_right
#align ite_dite_distrib_right ite_dite_distrib_right
theorem dite_ite_distrib_left {a : p → α} {b : ¬p → α} {c : ¬p → α} :
(dite p a fun hp ↦ ite q (b hp) (c hp)) = ite q (dite p a b) (dite p a c) :=
dite_dite_distrib_left
#align dite_ite_distrib_left dite_ite_distrib_left
theorem dite_ite_distrib_right {a : p → α} {b : p → α} {c : ¬p → α} :
dite p (fun hp ↦ ite q (a hp) (b hp)) c = ite q (dite p a c) (dite p b c) :=
dite_dite_distrib_right
#align dite_ite_distrib_right dite_ite_distrib_right
theorem ite_ite_distrib_left : ite p a (ite q b c) = ite q (ite p a b) (ite p a c) :=
dite_dite_distrib_left
#align ite_ite_distrib_left ite_ite_distrib_left
theorem ite_ite_distrib_right : ite p (ite q a b) c = ite q (ite p a c) (ite p b c) :=
dite_dite_distrib_right
#align ite_ite_distrib_right ite_ite_distrib_right
lemma Prop.forall {f : Prop → Prop} : (∀ p, f p) ↔ f True ∧ f False :=
⟨fun h ↦ ⟨h _, h _⟩, by rintro ⟨h₁, h₀⟩ p; by_cases hp : p <;> simp only [hp] <;> assumption⟩
#align Prop.forall Prop.forall
lemma Prop.exists {f : Prop → Prop} : (∃ p, f p) ↔ f True ∨ f False :=
⟨fun ⟨p, h⟩ ↦ by refine' (em p).imp _ _ <;> intro H <;>
|
convert h
|
lemma Prop.exists {f : Prop → Prop} : (∃ p, f p) ↔ f True ∨ f False :=
⟨fun ⟨p, h⟩ ↦ by refine' (em p).imp _ _ <;> intro H <;>
|
Mathlib.Logic.Lemmas.72_0.END1WfxnNs4h6Zj
|
lemma Prop.exists {f : Prop → Prop} : (∃ p, f p) ↔ f True ∨ f False
|
Mathlib_Logic_Lemmas
|
case refine'_2
α : Sort u_1
p✝ q r : Prop
inst✝¹ : Decidable p✝
inst✝ : Decidable q
a b c : α
f : Prop → Prop
x✝ : ∃ p, f p
p : Prop
h : f p
H : ¬p
⊢ f False
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a7c3ac4ac7873a7a43ead17c"
/-!
# More basic logic properties
A few more logic lemmas. These are in their own file, rather than `Logic.Basic`, because it is
convenient to be able to use the `split_ifs` tactic.
## Implementation notes
We spell those lemmas out with `dite` and `ite` rather than the `if then else` notation because this
would result in less delta-reduced statements.
-/
protected alias ⟨HEq.eq, Eq.heq⟩ := heq_iff_eq
#align heq.eq HEq.eq
#align eq.heq Eq.heq
variable {α : Sort*} {p q r : Prop} [Decidable p] [Decidable q] {a b c : α}
theorem dite_dite_distrib_left {a : p → α} {b : ¬p → q → α} {c : ¬p → ¬q → α} :
(dite p a fun hp ↦ dite q (b hp) (c hp)) =
dite q (fun hq ↦ (dite p a) fun hp ↦ b hp hq) fun hq ↦ (dite p a) fun hp ↦ c hp hq := by
split_ifs <;> rfl
#align dite_dite_distrib_left dite_dite_distrib_left
theorem dite_dite_distrib_right {a : p → q → α} {b : p → ¬q → α} {c : ¬p → α} :
dite p (fun hp ↦ dite q (a hp) (b hp)) c =
dite q (fun hq ↦ dite p (fun hp ↦ a hp hq) c) fun hq ↦ dite p (fun hp ↦ b hp hq) c := by
split_ifs <;> rfl
#align dite_dite_distrib_right dite_dite_distrib_right
theorem ite_dite_distrib_left {a : α} {b : q → α} {c : ¬q → α} :
ite p a (dite q b c) = dite q (fun hq ↦ ite p a <| b hq) fun hq ↦ ite p a <| c hq :=
dite_dite_distrib_left
#align ite_dite_distrib_left ite_dite_distrib_left
theorem ite_dite_distrib_right {a : q → α} {b : ¬q → α} {c : α} :
ite p (dite q a b) c = dite q (fun hq ↦ ite p (a hq) c) fun hq ↦ ite p (b hq) c :=
dite_dite_distrib_right
#align ite_dite_distrib_right ite_dite_distrib_right
theorem dite_ite_distrib_left {a : p → α} {b : ¬p → α} {c : ¬p → α} :
(dite p a fun hp ↦ ite q (b hp) (c hp)) = ite q (dite p a b) (dite p a c) :=
dite_dite_distrib_left
#align dite_ite_distrib_left dite_ite_distrib_left
theorem dite_ite_distrib_right {a : p → α} {b : p → α} {c : ¬p → α} :
dite p (fun hp ↦ ite q (a hp) (b hp)) c = ite q (dite p a c) (dite p b c) :=
dite_dite_distrib_right
#align dite_ite_distrib_right dite_ite_distrib_right
theorem ite_ite_distrib_left : ite p a (ite q b c) = ite q (ite p a b) (ite p a c) :=
dite_dite_distrib_left
#align ite_ite_distrib_left ite_ite_distrib_left
theorem ite_ite_distrib_right : ite p (ite q a b) c = ite q (ite p a c) (ite p b c) :=
dite_dite_distrib_right
#align ite_ite_distrib_right ite_ite_distrib_right
lemma Prop.forall {f : Prop → Prop} : (∀ p, f p) ↔ f True ∧ f False :=
⟨fun h ↦ ⟨h _, h _⟩, by rintro ⟨h₁, h₀⟩ p; by_cases hp : p <;> simp only [hp] <;> assumption⟩
#align Prop.forall Prop.forall
lemma Prop.exists {f : Prop → Prop} : (∃ p, f p) ↔ f True ∨ f False :=
⟨fun ⟨p, h⟩ ↦ by refine' (em p).imp _ _ <;> intro H <;>
|
convert h
|
lemma Prop.exists {f : Prop → Prop} : (∃ p, f p) ↔ f True ∨ f False :=
⟨fun ⟨p, h⟩ ↦ by refine' (em p).imp _ _ <;> intro H <;>
|
Mathlib.Logic.Lemmas.72_0.END1WfxnNs4h6Zj
|
lemma Prop.exists {f : Prop → Prop} : (∃ p, f p) ↔ f True ∨ f False
|
Mathlib_Logic_Lemmas
|
case h.e'_1.a
α : Sort u_1
p✝ q r : Prop
inst✝¹ : Decidable p✝
inst✝ : Decidable q
a b c : α
f : Prop → Prop
x✝ : ∃ p, f p
p : Prop
h : f p
H : p
⊢ True ↔ p
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a7c3ac4ac7873a7a43ead17c"
/-!
# More basic logic properties
A few more logic lemmas. These are in their own file, rather than `Logic.Basic`, because it is
convenient to be able to use the `split_ifs` tactic.
## Implementation notes
We spell those lemmas out with `dite` and `ite` rather than the `if then else` notation because this
would result in less delta-reduced statements.
-/
protected alias ⟨HEq.eq, Eq.heq⟩ := heq_iff_eq
#align heq.eq HEq.eq
#align eq.heq Eq.heq
variable {α : Sort*} {p q r : Prop} [Decidable p] [Decidable q] {a b c : α}
theorem dite_dite_distrib_left {a : p → α} {b : ¬p → q → α} {c : ¬p → ¬q → α} :
(dite p a fun hp ↦ dite q (b hp) (c hp)) =
dite q (fun hq ↦ (dite p a) fun hp ↦ b hp hq) fun hq ↦ (dite p a) fun hp ↦ c hp hq := by
split_ifs <;> rfl
#align dite_dite_distrib_left dite_dite_distrib_left
theorem dite_dite_distrib_right {a : p → q → α} {b : p → ¬q → α} {c : ¬p → α} :
dite p (fun hp ↦ dite q (a hp) (b hp)) c =
dite q (fun hq ↦ dite p (fun hp ↦ a hp hq) c) fun hq ↦ dite p (fun hp ↦ b hp hq) c := by
split_ifs <;> rfl
#align dite_dite_distrib_right dite_dite_distrib_right
theorem ite_dite_distrib_left {a : α} {b : q → α} {c : ¬q → α} :
ite p a (dite q b c) = dite q (fun hq ↦ ite p a <| b hq) fun hq ↦ ite p a <| c hq :=
dite_dite_distrib_left
#align ite_dite_distrib_left ite_dite_distrib_left
theorem ite_dite_distrib_right {a : q → α} {b : ¬q → α} {c : α} :
ite p (dite q a b) c = dite q (fun hq ↦ ite p (a hq) c) fun hq ↦ ite p (b hq) c :=
dite_dite_distrib_right
#align ite_dite_distrib_right ite_dite_distrib_right
theorem dite_ite_distrib_left {a : p → α} {b : ¬p → α} {c : ¬p → α} :
(dite p a fun hp ↦ ite q (b hp) (c hp)) = ite q (dite p a b) (dite p a c) :=
dite_dite_distrib_left
#align dite_ite_distrib_left dite_ite_distrib_left
theorem dite_ite_distrib_right {a : p → α} {b : p → α} {c : ¬p → α} :
dite p (fun hp ↦ ite q (a hp) (b hp)) c = ite q (dite p a c) (dite p b c) :=
dite_dite_distrib_right
#align dite_ite_distrib_right dite_ite_distrib_right
theorem ite_ite_distrib_left : ite p a (ite q b c) = ite q (ite p a b) (ite p a c) :=
dite_dite_distrib_left
#align ite_ite_distrib_left ite_ite_distrib_left
theorem ite_ite_distrib_right : ite p (ite q a b) c = ite q (ite p a c) (ite p b c) :=
dite_dite_distrib_right
#align ite_ite_distrib_right ite_ite_distrib_right
lemma Prop.forall {f : Prop → Prop} : (∀ p, f p) ↔ f True ∧ f False :=
⟨fun h ↦ ⟨h _, h _⟩, by rintro ⟨h₁, h₀⟩ p; by_cases hp : p <;> simp only [hp] <;> assumption⟩
#align Prop.forall Prop.forall
lemma Prop.exists {f : Prop → Prop} : (∃ p, f p) ↔ f True ∨ f False :=
⟨fun ⟨p, h⟩ ↦ by refine' (em p).imp _ _ <;> intro H <;> convert h <;>
|
simp [H]
|
lemma Prop.exists {f : Prop → Prop} : (∃ p, f p) ↔ f True ∨ f False :=
⟨fun ⟨p, h⟩ ↦ by refine' (em p).imp _ _ <;> intro H <;> convert h <;>
|
Mathlib.Logic.Lemmas.72_0.END1WfxnNs4h6Zj
|
lemma Prop.exists {f : Prop → Prop} : (∃ p, f p) ↔ f True ∨ f False
|
Mathlib_Logic_Lemmas
|
case h.e'_1.a
α : Sort u_1
p✝ q r : Prop
inst✝¹ : Decidable p✝
inst✝ : Decidable q
a b c : α
f : Prop → Prop
x✝ : ∃ p, f p
p : Prop
h : f p
H : ¬p
⊢ False ↔ p
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a7c3ac4ac7873a7a43ead17c"
/-!
# More basic logic properties
A few more logic lemmas. These are in their own file, rather than `Logic.Basic`, because it is
convenient to be able to use the `split_ifs` tactic.
## Implementation notes
We spell those lemmas out with `dite` and `ite` rather than the `if then else` notation because this
would result in less delta-reduced statements.
-/
protected alias ⟨HEq.eq, Eq.heq⟩ := heq_iff_eq
#align heq.eq HEq.eq
#align eq.heq Eq.heq
variable {α : Sort*} {p q r : Prop} [Decidable p] [Decidable q] {a b c : α}
theorem dite_dite_distrib_left {a : p → α} {b : ¬p → q → α} {c : ¬p → ¬q → α} :
(dite p a fun hp ↦ dite q (b hp) (c hp)) =
dite q (fun hq ↦ (dite p a) fun hp ↦ b hp hq) fun hq ↦ (dite p a) fun hp ↦ c hp hq := by
split_ifs <;> rfl
#align dite_dite_distrib_left dite_dite_distrib_left
theorem dite_dite_distrib_right {a : p → q → α} {b : p → ¬q → α} {c : ¬p → α} :
dite p (fun hp ↦ dite q (a hp) (b hp)) c =
dite q (fun hq ↦ dite p (fun hp ↦ a hp hq) c) fun hq ↦ dite p (fun hp ↦ b hp hq) c := by
split_ifs <;> rfl
#align dite_dite_distrib_right dite_dite_distrib_right
theorem ite_dite_distrib_left {a : α} {b : q → α} {c : ¬q → α} :
ite p a (dite q b c) = dite q (fun hq ↦ ite p a <| b hq) fun hq ↦ ite p a <| c hq :=
dite_dite_distrib_left
#align ite_dite_distrib_left ite_dite_distrib_left
theorem ite_dite_distrib_right {a : q → α} {b : ¬q → α} {c : α} :
ite p (dite q a b) c = dite q (fun hq ↦ ite p (a hq) c) fun hq ↦ ite p (b hq) c :=
dite_dite_distrib_right
#align ite_dite_distrib_right ite_dite_distrib_right
theorem dite_ite_distrib_left {a : p → α} {b : ¬p → α} {c : ¬p → α} :
(dite p a fun hp ↦ ite q (b hp) (c hp)) = ite q (dite p a b) (dite p a c) :=
dite_dite_distrib_left
#align dite_ite_distrib_left dite_ite_distrib_left
theorem dite_ite_distrib_right {a : p → α} {b : p → α} {c : ¬p → α} :
dite p (fun hp ↦ ite q (a hp) (b hp)) c = ite q (dite p a c) (dite p b c) :=
dite_dite_distrib_right
#align dite_ite_distrib_right dite_ite_distrib_right
theorem ite_ite_distrib_left : ite p a (ite q b c) = ite q (ite p a b) (ite p a c) :=
dite_dite_distrib_left
#align ite_ite_distrib_left ite_ite_distrib_left
theorem ite_ite_distrib_right : ite p (ite q a b) c = ite q (ite p a c) (ite p b c) :=
dite_dite_distrib_right
#align ite_ite_distrib_right ite_ite_distrib_right
lemma Prop.forall {f : Prop → Prop} : (∀ p, f p) ↔ f True ∧ f False :=
⟨fun h ↦ ⟨h _, h _⟩, by rintro ⟨h₁, h₀⟩ p; by_cases hp : p <;> simp only [hp] <;> assumption⟩
#align Prop.forall Prop.forall
lemma Prop.exists {f : Prop → Prop} : (∃ p, f p) ↔ f True ∨ f False :=
⟨fun ⟨p, h⟩ ↦ by refine' (em p).imp _ _ <;> intro H <;> convert h <;>
|
simp [H]
|
lemma Prop.exists {f : Prop → Prop} : (∃ p, f p) ↔ f True ∨ f False :=
⟨fun ⟨p, h⟩ ↦ by refine' (em p).imp _ _ <;> intro H <;> convert h <;>
|
Mathlib.Logic.Lemmas.72_0.END1WfxnNs4h6Zj
|
lemma Prop.exists {f : Prop → Prop} : (∃ p, f p) ↔ f True ∨ f False
|
Mathlib_Logic_Lemmas
|
α : Sort u_1
p q r : Prop
inst✝¹ : Decidable p
inst✝ : Decidable q
a b c : α
f : Prop → Prop
⊢ f True ∨ f False → ∃ p, f p
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a7c3ac4ac7873a7a43ead17c"
/-!
# More basic logic properties
A few more logic lemmas. These are in their own file, rather than `Logic.Basic`, because it is
convenient to be able to use the `split_ifs` tactic.
## Implementation notes
We spell those lemmas out with `dite` and `ite` rather than the `if then else` notation because this
would result in less delta-reduced statements.
-/
protected alias ⟨HEq.eq, Eq.heq⟩ := heq_iff_eq
#align heq.eq HEq.eq
#align eq.heq Eq.heq
variable {α : Sort*} {p q r : Prop} [Decidable p] [Decidable q] {a b c : α}
theorem dite_dite_distrib_left {a : p → α} {b : ¬p → q → α} {c : ¬p → ¬q → α} :
(dite p a fun hp ↦ dite q (b hp) (c hp)) =
dite q (fun hq ↦ (dite p a) fun hp ↦ b hp hq) fun hq ↦ (dite p a) fun hp ↦ c hp hq := by
split_ifs <;> rfl
#align dite_dite_distrib_left dite_dite_distrib_left
theorem dite_dite_distrib_right {a : p → q → α} {b : p → ¬q → α} {c : ¬p → α} :
dite p (fun hp ↦ dite q (a hp) (b hp)) c =
dite q (fun hq ↦ dite p (fun hp ↦ a hp hq) c) fun hq ↦ dite p (fun hp ↦ b hp hq) c := by
split_ifs <;> rfl
#align dite_dite_distrib_right dite_dite_distrib_right
theorem ite_dite_distrib_left {a : α} {b : q → α} {c : ¬q → α} :
ite p a (dite q b c) = dite q (fun hq ↦ ite p a <| b hq) fun hq ↦ ite p a <| c hq :=
dite_dite_distrib_left
#align ite_dite_distrib_left ite_dite_distrib_left
theorem ite_dite_distrib_right {a : q → α} {b : ¬q → α} {c : α} :
ite p (dite q a b) c = dite q (fun hq ↦ ite p (a hq) c) fun hq ↦ ite p (b hq) c :=
dite_dite_distrib_right
#align ite_dite_distrib_right ite_dite_distrib_right
theorem dite_ite_distrib_left {a : p → α} {b : ¬p → α} {c : ¬p → α} :
(dite p a fun hp ↦ ite q (b hp) (c hp)) = ite q (dite p a b) (dite p a c) :=
dite_dite_distrib_left
#align dite_ite_distrib_left dite_ite_distrib_left
theorem dite_ite_distrib_right {a : p → α} {b : p → α} {c : ¬p → α} :
dite p (fun hp ↦ ite q (a hp) (b hp)) c = ite q (dite p a c) (dite p b c) :=
dite_dite_distrib_right
#align dite_ite_distrib_right dite_ite_distrib_right
theorem ite_ite_distrib_left : ite p a (ite q b c) = ite q (ite p a b) (ite p a c) :=
dite_dite_distrib_left
#align ite_ite_distrib_left ite_ite_distrib_left
theorem ite_ite_distrib_right : ite p (ite q a b) c = ite q (ite p a c) (ite p b c) :=
dite_dite_distrib_right
#align ite_ite_distrib_right ite_ite_distrib_right
lemma Prop.forall {f : Prop → Prop} : (∀ p, f p) ↔ f True ∧ f False :=
⟨fun h ↦ ⟨h _, h _⟩, by rintro ⟨h₁, h₀⟩ p; by_cases hp : p <;> simp only [hp] <;> assumption⟩
#align Prop.forall Prop.forall
lemma Prop.exists {f : Prop → Prop} : (∃ p, f p) ↔ f True ∨ f False :=
⟨fun ⟨p, h⟩ ↦ by refine' (em p).imp _ _ <;> intro H <;> convert h <;> simp [H],
by
|
rintro (h | h)
|
lemma Prop.exists {f : Prop → Prop} : (∃ p, f p) ↔ f True ∨ f False :=
⟨fun ⟨p, h⟩ ↦ by refine' (em p).imp _ _ <;> intro H <;> convert h <;> simp [H],
by
|
Mathlib.Logic.Lemmas.72_0.END1WfxnNs4h6Zj
|
lemma Prop.exists {f : Prop → Prop} : (∃ p, f p) ↔ f True ∨ f False
|
Mathlib_Logic_Lemmas
|
case inl
α : Sort u_1
p q r : Prop
inst✝¹ : Decidable p
inst✝ : Decidable q
a b c : α
f : Prop → Prop
h : f True
⊢ ∃ p, f p
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a7c3ac4ac7873a7a43ead17c"
/-!
# More basic logic properties
A few more logic lemmas. These are in their own file, rather than `Logic.Basic`, because it is
convenient to be able to use the `split_ifs` tactic.
## Implementation notes
We spell those lemmas out with `dite` and `ite` rather than the `if then else` notation because this
would result in less delta-reduced statements.
-/
protected alias ⟨HEq.eq, Eq.heq⟩ := heq_iff_eq
#align heq.eq HEq.eq
#align eq.heq Eq.heq
variable {α : Sort*} {p q r : Prop} [Decidable p] [Decidable q] {a b c : α}
theorem dite_dite_distrib_left {a : p → α} {b : ¬p → q → α} {c : ¬p → ¬q → α} :
(dite p a fun hp ↦ dite q (b hp) (c hp)) =
dite q (fun hq ↦ (dite p a) fun hp ↦ b hp hq) fun hq ↦ (dite p a) fun hp ↦ c hp hq := by
split_ifs <;> rfl
#align dite_dite_distrib_left dite_dite_distrib_left
theorem dite_dite_distrib_right {a : p → q → α} {b : p → ¬q → α} {c : ¬p → α} :
dite p (fun hp ↦ dite q (a hp) (b hp)) c =
dite q (fun hq ↦ dite p (fun hp ↦ a hp hq) c) fun hq ↦ dite p (fun hp ↦ b hp hq) c := by
split_ifs <;> rfl
#align dite_dite_distrib_right dite_dite_distrib_right
theorem ite_dite_distrib_left {a : α} {b : q → α} {c : ¬q → α} :
ite p a (dite q b c) = dite q (fun hq ↦ ite p a <| b hq) fun hq ↦ ite p a <| c hq :=
dite_dite_distrib_left
#align ite_dite_distrib_left ite_dite_distrib_left
theorem ite_dite_distrib_right {a : q → α} {b : ¬q → α} {c : α} :
ite p (dite q a b) c = dite q (fun hq ↦ ite p (a hq) c) fun hq ↦ ite p (b hq) c :=
dite_dite_distrib_right
#align ite_dite_distrib_right ite_dite_distrib_right
theorem dite_ite_distrib_left {a : p → α} {b : ¬p → α} {c : ¬p → α} :
(dite p a fun hp ↦ ite q (b hp) (c hp)) = ite q (dite p a b) (dite p a c) :=
dite_dite_distrib_left
#align dite_ite_distrib_left dite_ite_distrib_left
theorem dite_ite_distrib_right {a : p → α} {b : p → α} {c : ¬p → α} :
dite p (fun hp ↦ ite q (a hp) (b hp)) c = ite q (dite p a c) (dite p b c) :=
dite_dite_distrib_right
#align dite_ite_distrib_right dite_ite_distrib_right
theorem ite_ite_distrib_left : ite p a (ite q b c) = ite q (ite p a b) (ite p a c) :=
dite_dite_distrib_left
#align ite_ite_distrib_left ite_ite_distrib_left
theorem ite_ite_distrib_right : ite p (ite q a b) c = ite q (ite p a c) (ite p b c) :=
dite_dite_distrib_right
#align ite_ite_distrib_right ite_ite_distrib_right
lemma Prop.forall {f : Prop → Prop} : (∀ p, f p) ↔ f True ∧ f False :=
⟨fun h ↦ ⟨h _, h _⟩, by rintro ⟨h₁, h₀⟩ p; by_cases hp : p <;> simp only [hp] <;> assumption⟩
#align Prop.forall Prop.forall
lemma Prop.exists {f : Prop → Prop} : (∃ p, f p) ↔ f True ∨ f False :=
⟨fun ⟨p, h⟩ ↦ by refine' (em p).imp _ _ <;> intro H <;> convert h <;> simp [H],
by rintro (h | h) <;>
|
exact ⟨_, h⟩
|
lemma Prop.exists {f : Prop → Prop} : (∃ p, f p) ↔ f True ∨ f False :=
⟨fun ⟨p, h⟩ ↦ by refine' (em p).imp _ _ <;> intro H <;> convert h <;> simp [H],
by rintro (h | h) <;>
|
Mathlib.Logic.Lemmas.72_0.END1WfxnNs4h6Zj
|
lemma Prop.exists {f : Prop → Prop} : (∃ p, f p) ↔ f True ∨ f False
|
Mathlib_Logic_Lemmas
|
case inr
α : Sort u_1
p q r : Prop
inst✝¹ : Decidable p
inst✝ : Decidable q
a b c : α
f : Prop → Prop
h : f False
⊢ ∃ p, f p
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
#align_import logic.lemmas from "leanprover-community/mathlib"@"2ed7e4aec72395b6a7c3ac4ac7873a7a43ead17c"
/-!
# More basic logic properties
A few more logic lemmas. These are in their own file, rather than `Logic.Basic`, because it is
convenient to be able to use the `split_ifs` tactic.
## Implementation notes
We spell those lemmas out with `dite` and `ite` rather than the `if then else` notation because this
would result in less delta-reduced statements.
-/
protected alias ⟨HEq.eq, Eq.heq⟩ := heq_iff_eq
#align heq.eq HEq.eq
#align eq.heq Eq.heq
variable {α : Sort*} {p q r : Prop} [Decidable p] [Decidable q] {a b c : α}
theorem dite_dite_distrib_left {a : p → α} {b : ¬p → q → α} {c : ¬p → ¬q → α} :
(dite p a fun hp ↦ dite q (b hp) (c hp)) =
dite q (fun hq ↦ (dite p a) fun hp ↦ b hp hq) fun hq ↦ (dite p a) fun hp ↦ c hp hq := by
split_ifs <;> rfl
#align dite_dite_distrib_left dite_dite_distrib_left
theorem dite_dite_distrib_right {a : p → q → α} {b : p → ¬q → α} {c : ¬p → α} :
dite p (fun hp ↦ dite q (a hp) (b hp)) c =
dite q (fun hq ↦ dite p (fun hp ↦ a hp hq) c) fun hq ↦ dite p (fun hp ↦ b hp hq) c := by
split_ifs <;> rfl
#align dite_dite_distrib_right dite_dite_distrib_right
theorem ite_dite_distrib_left {a : α} {b : q → α} {c : ¬q → α} :
ite p a (dite q b c) = dite q (fun hq ↦ ite p a <| b hq) fun hq ↦ ite p a <| c hq :=
dite_dite_distrib_left
#align ite_dite_distrib_left ite_dite_distrib_left
theorem ite_dite_distrib_right {a : q → α} {b : ¬q → α} {c : α} :
ite p (dite q a b) c = dite q (fun hq ↦ ite p (a hq) c) fun hq ↦ ite p (b hq) c :=
dite_dite_distrib_right
#align ite_dite_distrib_right ite_dite_distrib_right
theorem dite_ite_distrib_left {a : p → α} {b : ¬p → α} {c : ¬p → α} :
(dite p a fun hp ↦ ite q (b hp) (c hp)) = ite q (dite p a b) (dite p a c) :=
dite_dite_distrib_left
#align dite_ite_distrib_left dite_ite_distrib_left
theorem dite_ite_distrib_right {a : p → α} {b : p → α} {c : ¬p → α} :
dite p (fun hp ↦ ite q (a hp) (b hp)) c = ite q (dite p a c) (dite p b c) :=
dite_dite_distrib_right
#align dite_ite_distrib_right dite_ite_distrib_right
theorem ite_ite_distrib_left : ite p a (ite q b c) = ite q (ite p a b) (ite p a c) :=
dite_dite_distrib_left
#align ite_ite_distrib_left ite_ite_distrib_left
theorem ite_ite_distrib_right : ite p (ite q a b) c = ite q (ite p a c) (ite p b c) :=
dite_dite_distrib_right
#align ite_ite_distrib_right ite_ite_distrib_right
lemma Prop.forall {f : Prop → Prop} : (∀ p, f p) ↔ f True ∧ f False :=
⟨fun h ↦ ⟨h _, h _⟩, by rintro ⟨h₁, h₀⟩ p; by_cases hp : p <;> simp only [hp] <;> assumption⟩
#align Prop.forall Prop.forall
lemma Prop.exists {f : Prop → Prop} : (∃ p, f p) ↔ f True ∨ f False :=
⟨fun ⟨p, h⟩ ↦ by refine' (em p).imp _ _ <;> intro H <;> convert h <;> simp [H],
by rintro (h | h) <;>
|
exact ⟨_, h⟩
|
lemma Prop.exists {f : Prop → Prop} : (∃ p, f p) ↔ f True ∨ f False :=
⟨fun ⟨p, h⟩ ↦ by refine' (em p).imp _ _ <;> intro H <;> convert h <;> simp [H],
by rintro (h | h) <;>
|
Mathlib.Logic.Lemmas.72_0.END1WfxnNs4h6Zj
|
lemma Prop.exists {f : Prop → Prop} : (∃ p, f p) ↔ f True ∨ f False
|
Mathlib_Logic_Lemmas
|
C : Type u_1
inst✝¹ : Category.{?u.28, u_1} C
inst✝ : Abelian C
⊢ Abelian Cᵒᵖ
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_theory.abelian.opposite from "leanprover-community/mathlib"@"a5ff45a1c92c278b03b52459a620cfd9c49ebc80"
/-!
# The opposite of an abelian category is abelian.
-/
noncomputable section
namespace CategoryTheory
open CategoryTheory.Limits
variable (C : Type*) [Category C] [Abelian C]
-- porting note: these local instances do not seem to be necessary
--attribute [local instance]
-- hasFiniteLimits_of_hasEqualizers_and_finite_products
-- hasFiniteColimits_of_hasCoequalizers_and_finite_coproducts
-- Abelian.hasFiniteBiproducts
instance : Abelian Cᵒᵖ := by
-- porting note: priorities of `Abelian.has_kernels` and `Abelian.has_cokernels` have
-- been set to 90 in `Abelian.Basic` in order to prevent a timeout here
|
exact {
normalMonoOfMono := fun f => normalMonoOfNormalEpiUnop _ (normalEpiOfEpi f.unop)
normalEpiOfEpi := fun f => normalEpiOfNormalMonoUnop _ (normalMonoOfMono f.unop) }
|
instance : Abelian Cᵒᵖ := by
-- porting note: priorities of `Abelian.has_kernels` and `Abelian.has_cokernels` have
-- been set to 90 in `Abelian.Basic` in order to prevent a timeout here
|
Mathlib.CategoryTheory.Abelian.Opposite.31_0.3nBRs3fSYrCoEsT
|
instance : Abelian Cᵒᵖ
|
Mathlib_CategoryTheory_Abelian_Opposite
|
C : Type u_1
inst✝¹ : Category.{?u.2298, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ (cokernel.π f).op ≫ f.op = 0
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_theory.abelian.opposite from "leanprover-community/mathlib"@"a5ff45a1c92c278b03b52459a620cfd9c49ebc80"
/-!
# The opposite of an abelian category is abelian.
-/
noncomputable section
namespace CategoryTheory
open CategoryTheory.Limits
variable (C : Type*) [Category C] [Abelian C]
-- porting note: these local instances do not seem to be necessary
--attribute [local instance]
-- hasFiniteLimits_of_hasEqualizers_and_finite_products
-- hasFiniteColimits_of_hasCoequalizers_and_finite_coproducts
-- Abelian.hasFiniteBiproducts
instance : Abelian Cᵒᵖ := by
-- porting note: priorities of `Abelian.has_kernels` and `Abelian.has_cokernels` have
-- been set to 90 in `Abelian.Basic` in order to prevent a timeout here
exact {
normalMonoOfMono := fun f => normalMonoOfNormalEpiUnop _ (normalEpiOfEpi f.unop)
normalEpiOfEpi := fun f => normalEpiOfNormalMonoUnop _ (normalMonoOfMono f.unop) }
section
variable {C}
variable {X Y : C} (f : X ⟶ Y) {A B : Cᵒᵖ} (g : A ⟶ B)
-- TODO: Generalize (this will work whenever f has a cokernel)
-- (The abelian case is probably sufficient for most applications.)
/-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom := (kernel.lift f.op (cokernel.π f).op <| by
|
simp [← op_comp]
|
/-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom := (kernel.lift f.op (cokernel.π f).op <| by
|
Mathlib.CategoryTheory.Abelian.Opposite.45_0.3nBRs3fSYrCoEsT
|
/-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom
|
Mathlib_CategoryTheory_Abelian_Opposite
|
C : Type u_1
inst✝¹ : Category.{?u.2298, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ f ≫ (kernel.ι f.op).unop = 0
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_theory.abelian.opposite from "leanprover-community/mathlib"@"a5ff45a1c92c278b03b52459a620cfd9c49ebc80"
/-!
# The opposite of an abelian category is abelian.
-/
noncomputable section
namespace CategoryTheory
open CategoryTheory.Limits
variable (C : Type*) [Category C] [Abelian C]
-- porting note: these local instances do not seem to be necessary
--attribute [local instance]
-- hasFiniteLimits_of_hasEqualizers_and_finite_products
-- hasFiniteColimits_of_hasCoequalizers_and_finite_coproducts
-- Abelian.hasFiniteBiproducts
instance : Abelian Cᵒᵖ := by
-- porting note: priorities of `Abelian.has_kernels` and `Abelian.has_cokernels` have
-- been set to 90 in `Abelian.Basic` in order to prevent a timeout here
exact {
normalMonoOfMono := fun f => normalMonoOfNormalEpiUnop _ (normalEpiOfEpi f.unop)
normalEpiOfEpi := fun f => normalEpiOfNormalMonoUnop _ (normalMonoOfMono f.unop) }
section
variable {C}
variable {X Y : C} (f : X ⟶ Y) {A B : Cᵒᵖ} (g : A ⟶ B)
-- TODO: Generalize (this will work whenever f has a cokernel)
-- (The abelian case is probably sufficient for most applications.)
/-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom := (kernel.lift f.op (cokernel.π f).op <| by simp [← op_comp]).unop
inv :=
cokernel.desc f (kernel.ι f.op).unop <| by
|
rw [← f.unop_op, ← unop_comp, f.unop_op]
|
/-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom := (kernel.lift f.op (cokernel.π f).op <| by simp [← op_comp]).unop
inv :=
cokernel.desc f (kernel.ι f.op).unop <| by
|
Mathlib.CategoryTheory.Abelian.Opposite.45_0.3nBRs3fSYrCoEsT
|
/-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom
|
Mathlib_CategoryTheory_Abelian_Opposite
|
C : Type u_1
inst✝¹ : Category.{?u.2298, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ (kernel.ι f.op ≫ f.op).unop = 0
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_theory.abelian.opposite from "leanprover-community/mathlib"@"a5ff45a1c92c278b03b52459a620cfd9c49ebc80"
/-!
# The opposite of an abelian category is abelian.
-/
noncomputable section
namespace CategoryTheory
open CategoryTheory.Limits
variable (C : Type*) [Category C] [Abelian C]
-- porting note: these local instances do not seem to be necessary
--attribute [local instance]
-- hasFiniteLimits_of_hasEqualizers_and_finite_products
-- hasFiniteColimits_of_hasCoequalizers_and_finite_coproducts
-- Abelian.hasFiniteBiproducts
instance : Abelian Cᵒᵖ := by
-- porting note: priorities of `Abelian.has_kernels` and `Abelian.has_cokernels` have
-- been set to 90 in `Abelian.Basic` in order to prevent a timeout here
exact {
normalMonoOfMono := fun f => normalMonoOfNormalEpiUnop _ (normalEpiOfEpi f.unop)
normalEpiOfEpi := fun f => normalEpiOfNormalMonoUnop _ (normalMonoOfMono f.unop) }
section
variable {C}
variable {X Y : C} (f : X ⟶ Y) {A B : Cᵒᵖ} (g : A ⟶ B)
-- TODO: Generalize (this will work whenever f has a cokernel)
-- (The abelian case is probably sufficient for most applications.)
/-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom := (kernel.lift f.op (cokernel.π f).op <| by simp [← op_comp]).unop
inv :=
cokernel.desc f (kernel.ι f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
|
simp
|
/-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom := (kernel.lift f.op (cokernel.π f).op <| by simp [← op_comp]).unop
inv :=
cokernel.desc f (kernel.ι f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
|
Mathlib.CategoryTheory.Abelian.Opposite.45_0.3nBRs3fSYrCoEsT
|
/-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom
|
Mathlib_CategoryTheory_Abelian_Opposite
|
C : Type u_1
inst✝¹ : Category.{?u.2298, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ (kernel.lift f.op (cokernel.π f).op (_ : (f ≫ cokernel.π f).op = 0)).unop ≫
cokernel.desc f (kernel.ι f.op).unop (_ : f ≫ (kernel.ι f.op).unop = 0) =
𝟙 (kernel f.op).unop
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_theory.abelian.opposite from "leanprover-community/mathlib"@"a5ff45a1c92c278b03b52459a620cfd9c49ebc80"
/-!
# The opposite of an abelian category is abelian.
-/
noncomputable section
namespace CategoryTheory
open CategoryTheory.Limits
variable (C : Type*) [Category C] [Abelian C]
-- porting note: these local instances do not seem to be necessary
--attribute [local instance]
-- hasFiniteLimits_of_hasEqualizers_and_finite_products
-- hasFiniteColimits_of_hasCoequalizers_and_finite_coproducts
-- Abelian.hasFiniteBiproducts
instance : Abelian Cᵒᵖ := by
-- porting note: priorities of `Abelian.has_kernels` and `Abelian.has_cokernels` have
-- been set to 90 in `Abelian.Basic` in order to prevent a timeout here
exact {
normalMonoOfMono := fun f => normalMonoOfNormalEpiUnop _ (normalEpiOfEpi f.unop)
normalEpiOfEpi := fun f => normalEpiOfNormalMonoUnop _ (normalMonoOfMono f.unop) }
section
variable {C}
variable {X Y : C} (f : X ⟶ Y) {A B : Cᵒᵖ} (g : A ⟶ B)
-- TODO: Generalize (this will work whenever f has a cokernel)
-- (The abelian case is probably sufficient for most applications.)
/-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom := (kernel.lift f.op (cokernel.π f).op <| by simp [← op_comp]).unop
inv :=
cokernel.desc f (kernel.ι f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
simp
hom_inv_id := by
|
rw [← unop_id, ← (cokernel.desc f _ _).unop_op, ← unop_comp]
|
/-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom := (kernel.lift f.op (cokernel.π f).op <| by simp [← op_comp]).unop
inv :=
cokernel.desc f (kernel.ι f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
simp
hom_inv_id := by
|
Mathlib.CategoryTheory.Abelian.Opposite.45_0.3nBRs3fSYrCoEsT
|
/-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom
|
Mathlib_CategoryTheory_Abelian_Opposite
|
C : Type u_1
inst✝¹ : Category.{?u.2298, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ ((cokernel.desc f (kernel.ι f.op).unop (_ : f ≫ (kernel.ι f.op).unop = 0)).op ≫
kernel.lift f.op (cokernel.π f).op (_ : (f ≫ cokernel.π f).op = 0)).unop =
(𝟙 (kernel f.op)).unop
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_theory.abelian.opposite from "leanprover-community/mathlib"@"a5ff45a1c92c278b03b52459a620cfd9c49ebc80"
/-!
# The opposite of an abelian category is abelian.
-/
noncomputable section
namespace CategoryTheory
open CategoryTheory.Limits
variable (C : Type*) [Category C] [Abelian C]
-- porting note: these local instances do not seem to be necessary
--attribute [local instance]
-- hasFiniteLimits_of_hasEqualizers_and_finite_products
-- hasFiniteColimits_of_hasCoequalizers_and_finite_coproducts
-- Abelian.hasFiniteBiproducts
instance : Abelian Cᵒᵖ := by
-- porting note: priorities of `Abelian.has_kernels` and `Abelian.has_cokernels` have
-- been set to 90 in `Abelian.Basic` in order to prevent a timeout here
exact {
normalMonoOfMono := fun f => normalMonoOfNormalEpiUnop _ (normalEpiOfEpi f.unop)
normalEpiOfEpi := fun f => normalEpiOfNormalMonoUnop _ (normalMonoOfMono f.unop) }
section
variable {C}
variable {X Y : C} (f : X ⟶ Y) {A B : Cᵒᵖ} (g : A ⟶ B)
-- TODO: Generalize (this will work whenever f has a cokernel)
-- (The abelian case is probably sufficient for most applications.)
/-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom := (kernel.lift f.op (cokernel.π f).op <| by simp [← op_comp]).unop
inv :=
cokernel.desc f (kernel.ι f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
simp
hom_inv_id := by
rw [← unop_id, ← (cokernel.desc f _ _).unop_op, ← unop_comp]
|
congr 1
|
/-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom := (kernel.lift f.op (cokernel.π f).op <| by simp [← op_comp]).unop
inv :=
cokernel.desc f (kernel.ι f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
simp
hom_inv_id := by
rw [← unop_id, ← (cokernel.desc f _ _).unop_op, ← unop_comp]
|
Mathlib.CategoryTheory.Abelian.Opposite.45_0.3nBRs3fSYrCoEsT
|
/-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom
|
Mathlib_CategoryTheory_Abelian_Opposite
|
case e_f
C : Type u_1
inst✝¹ : Category.{?u.2298, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ (cokernel.desc f (kernel.ι f.op).unop (_ : f ≫ (kernel.ι f.op).unop = 0)).op ≫
kernel.lift f.op (cokernel.π f).op (_ : (f ≫ cokernel.π f).op = 0) =
𝟙 (kernel f.op)
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_theory.abelian.opposite from "leanprover-community/mathlib"@"a5ff45a1c92c278b03b52459a620cfd9c49ebc80"
/-!
# The opposite of an abelian category is abelian.
-/
noncomputable section
namespace CategoryTheory
open CategoryTheory.Limits
variable (C : Type*) [Category C] [Abelian C]
-- porting note: these local instances do not seem to be necessary
--attribute [local instance]
-- hasFiniteLimits_of_hasEqualizers_and_finite_products
-- hasFiniteColimits_of_hasCoequalizers_and_finite_coproducts
-- Abelian.hasFiniteBiproducts
instance : Abelian Cᵒᵖ := by
-- porting note: priorities of `Abelian.has_kernels` and `Abelian.has_cokernels` have
-- been set to 90 in `Abelian.Basic` in order to prevent a timeout here
exact {
normalMonoOfMono := fun f => normalMonoOfNormalEpiUnop _ (normalEpiOfEpi f.unop)
normalEpiOfEpi := fun f => normalEpiOfNormalMonoUnop _ (normalMonoOfMono f.unop) }
section
variable {C}
variable {X Y : C} (f : X ⟶ Y) {A B : Cᵒᵖ} (g : A ⟶ B)
-- TODO: Generalize (this will work whenever f has a cokernel)
-- (The abelian case is probably sufficient for most applications.)
/-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom := (kernel.lift f.op (cokernel.π f).op <| by simp [← op_comp]).unop
inv :=
cokernel.desc f (kernel.ι f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
simp
hom_inv_id := by
rw [← unop_id, ← (cokernel.desc f _ _).unop_op, ← unop_comp]
congr 1
|
ext
|
/-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom := (kernel.lift f.op (cokernel.π f).op <| by simp [← op_comp]).unop
inv :=
cokernel.desc f (kernel.ι f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
simp
hom_inv_id := by
rw [← unop_id, ← (cokernel.desc f _ _).unop_op, ← unop_comp]
congr 1
|
Mathlib.CategoryTheory.Abelian.Opposite.45_0.3nBRs3fSYrCoEsT
|
/-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom
|
Mathlib_CategoryTheory_Abelian_Opposite
|
case e_f.h
C : Type u_1
inst✝¹ : Category.{?u.2298, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ ((cokernel.desc f (kernel.ι f.op).unop (_ : f ≫ (kernel.ι f.op).unop = 0)).op ≫
kernel.lift f.op (cokernel.π f).op (_ : (f ≫ cokernel.π f).op = 0)) ≫
equalizer.ι f.op 0 =
𝟙 (kernel f.op) ≫ equalizer.ι f.op 0
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_theory.abelian.opposite from "leanprover-community/mathlib"@"a5ff45a1c92c278b03b52459a620cfd9c49ebc80"
/-!
# The opposite of an abelian category is abelian.
-/
noncomputable section
namespace CategoryTheory
open CategoryTheory.Limits
variable (C : Type*) [Category C] [Abelian C]
-- porting note: these local instances do not seem to be necessary
--attribute [local instance]
-- hasFiniteLimits_of_hasEqualizers_and_finite_products
-- hasFiniteColimits_of_hasCoequalizers_and_finite_coproducts
-- Abelian.hasFiniteBiproducts
instance : Abelian Cᵒᵖ := by
-- porting note: priorities of `Abelian.has_kernels` and `Abelian.has_cokernels` have
-- been set to 90 in `Abelian.Basic` in order to prevent a timeout here
exact {
normalMonoOfMono := fun f => normalMonoOfNormalEpiUnop _ (normalEpiOfEpi f.unop)
normalEpiOfEpi := fun f => normalEpiOfNormalMonoUnop _ (normalMonoOfMono f.unop) }
section
variable {C}
variable {X Y : C} (f : X ⟶ Y) {A B : Cᵒᵖ} (g : A ⟶ B)
-- TODO: Generalize (this will work whenever f has a cokernel)
-- (The abelian case is probably sufficient for most applications.)
/-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom := (kernel.lift f.op (cokernel.π f).op <| by simp [← op_comp]).unop
inv :=
cokernel.desc f (kernel.ι f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
simp
hom_inv_id := by
rw [← unop_id, ← (cokernel.desc f _ _).unop_op, ← unop_comp]
congr 1
ext
|
simp [← op_comp]
|
/-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom := (kernel.lift f.op (cokernel.π f).op <| by simp [← op_comp]).unop
inv :=
cokernel.desc f (kernel.ι f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
simp
hom_inv_id := by
rw [← unop_id, ← (cokernel.desc f _ _).unop_op, ← unop_comp]
congr 1
ext
|
Mathlib.CategoryTheory.Abelian.Opposite.45_0.3nBRs3fSYrCoEsT
|
/-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom
|
Mathlib_CategoryTheory_Abelian_Opposite
|
C : Type u_1
inst✝¹ : Category.{?u.2298, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ cokernel.desc f (kernel.ι f.op).unop (_ : f ≫ (kernel.ι f.op).unop = 0) ≫
(kernel.lift f.op (cokernel.π f).op (_ : (f ≫ cokernel.π f).op = 0)).unop =
𝟙 (cokernel f)
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_theory.abelian.opposite from "leanprover-community/mathlib"@"a5ff45a1c92c278b03b52459a620cfd9c49ebc80"
/-!
# The opposite of an abelian category is abelian.
-/
noncomputable section
namespace CategoryTheory
open CategoryTheory.Limits
variable (C : Type*) [Category C] [Abelian C]
-- porting note: these local instances do not seem to be necessary
--attribute [local instance]
-- hasFiniteLimits_of_hasEqualizers_and_finite_products
-- hasFiniteColimits_of_hasCoequalizers_and_finite_coproducts
-- Abelian.hasFiniteBiproducts
instance : Abelian Cᵒᵖ := by
-- porting note: priorities of `Abelian.has_kernels` and `Abelian.has_cokernels` have
-- been set to 90 in `Abelian.Basic` in order to prevent a timeout here
exact {
normalMonoOfMono := fun f => normalMonoOfNormalEpiUnop _ (normalEpiOfEpi f.unop)
normalEpiOfEpi := fun f => normalEpiOfNormalMonoUnop _ (normalMonoOfMono f.unop) }
section
variable {C}
variable {X Y : C} (f : X ⟶ Y) {A B : Cᵒᵖ} (g : A ⟶ B)
-- TODO: Generalize (this will work whenever f has a cokernel)
-- (The abelian case is probably sufficient for most applications.)
/-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom := (kernel.lift f.op (cokernel.π f).op <| by simp [← op_comp]).unop
inv :=
cokernel.desc f (kernel.ι f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
simp
hom_inv_id := by
rw [← unop_id, ← (cokernel.desc f _ _).unop_op, ← unop_comp]
congr 1
ext
simp [← op_comp]
inv_hom_id := by
|
ext
|
/-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom := (kernel.lift f.op (cokernel.π f).op <| by simp [← op_comp]).unop
inv :=
cokernel.desc f (kernel.ι f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
simp
hom_inv_id := by
rw [← unop_id, ← (cokernel.desc f _ _).unop_op, ← unop_comp]
congr 1
ext
simp [← op_comp]
inv_hom_id := by
|
Mathlib.CategoryTheory.Abelian.Opposite.45_0.3nBRs3fSYrCoEsT
|
/-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom
|
Mathlib_CategoryTheory_Abelian_Opposite
|
case h
C : Type u_1
inst✝¹ : Category.{?u.2298, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ coequalizer.π f 0 ≫
cokernel.desc f (kernel.ι f.op).unop (_ : f ≫ (kernel.ι f.op).unop = 0) ≫
(kernel.lift f.op (cokernel.π f).op (_ : (f ≫ cokernel.π f).op = 0)).unop =
coequalizer.π f 0 ≫ 𝟙 (cokernel f)
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_theory.abelian.opposite from "leanprover-community/mathlib"@"a5ff45a1c92c278b03b52459a620cfd9c49ebc80"
/-!
# The opposite of an abelian category is abelian.
-/
noncomputable section
namespace CategoryTheory
open CategoryTheory.Limits
variable (C : Type*) [Category C] [Abelian C]
-- porting note: these local instances do not seem to be necessary
--attribute [local instance]
-- hasFiniteLimits_of_hasEqualizers_and_finite_products
-- hasFiniteColimits_of_hasCoequalizers_and_finite_coproducts
-- Abelian.hasFiniteBiproducts
instance : Abelian Cᵒᵖ := by
-- porting note: priorities of `Abelian.has_kernels` and `Abelian.has_cokernels` have
-- been set to 90 in `Abelian.Basic` in order to prevent a timeout here
exact {
normalMonoOfMono := fun f => normalMonoOfNormalEpiUnop _ (normalEpiOfEpi f.unop)
normalEpiOfEpi := fun f => normalEpiOfNormalMonoUnop _ (normalMonoOfMono f.unop) }
section
variable {C}
variable {X Y : C} (f : X ⟶ Y) {A B : Cᵒᵖ} (g : A ⟶ B)
-- TODO: Generalize (this will work whenever f has a cokernel)
-- (The abelian case is probably sufficient for most applications.)
/-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom := (kernel.lift f.op (cokernel.π f).op <| by simp [← op_comp]).unop
inv :=
cokernel.desc f (kernel.ι f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
simp
hom_inv_id := by
rw [← unop_id, ← (cokernel.desc f _ _).unop_op, ← unop_comp]
congr 1
ext
simp [← op_comp]
inv_hom_id := by
ext
|
simp [← unop_comp]
|
/-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom := (kernel.lift f.op (cokernel.π f).op <| by simp [← op_comp]).unop
inv :=
cokernel.desc f (kernel.ι f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
simp
hom_inv_id := by
rw [← unop_id, ← (cokernel.desc f _ _).unop_op, ← unop_comp]
congr 1
ext
simp [← op_comp]
inv_hom_id := by
ext
|
Mathlib.CategoryTheory.Abelian.Opposite.45_0.3nBRs3fSYrCoEsT
|
/-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom
|
Mathlib_CategoryTheory_Abelian_Opposite
|
C : Type u_1
inst✝¹ : Category.{?u.12173, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ (cokernel.π f.op).unop ≫ f = 0
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_theory.abelian.opposite from "leanprover-community/mathlib"@"a5ff45a1c92c278b03b52459a620cfd9c49ebc80"
/-!
# The opposite of an abelian category is abelian.
-/
noncomputable section
namespace CategoryTheory
open CategoryTheory.Limits
variable (C : Type*) [Category C] [Abelian C]
-- porting note: these local instances do not seem to be necessary
--attribute [local instance]
-- hasFiniteLimits_of_hasEqualizers_and_finite_products
-- hasFiniteColimits_of_hasCoequalizers_and_finite_coproducts
-- Abelian.hasFiniteBiproducts
instance : Abelian Cᵒᵖ := by
-- porting note: priorities of `Abelian.has_kernels` and `Abelian.has_cokernels` have
-- been set to 90 in `Abelian.Basic` in order to prevent a timeout here
exact {
normalMonoOfMono := fun f => normalMonoOfNormalEpiUnop _ (normalEpiOfEpi f.unop)
normalEpiOfEpi := fun f => normalEpiOfNormalMonoUnop _ (normalMonoOfMono f.unop) }
section
variable {C}
variable {X Y : C} (f : X ⟶ Y) {A B : Cᵒᵖ} (g : A ⟶ B)
-- TODO: Generalize (this will work whenever f has a cokernel)
-- (The abelian case is probably sufficient for most applications.)
/-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom := (kernel.lift f.op (cokernel.π f).op <| by simp [← op_comp]).unop
inv :=
cokernel.desc f (kernel.ι f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
simp
hom_inv_id := by
rw [← unop_id, ← (cokernel.desc f _ _).unop_op, ← unop_comp]
congr 1
ext
simp [← op_comp]
inv_hom_id := by
ext
simp [← unop_comp]
#align category_theory.kernel_op_unop CategoryTheory.kernelOpUnop
-- TODO: Generalize (this will work whenever f has a kernel)
-- (The abelian case is probably sufficient for most applications.)
/-- The cokernel of `f.op` is the opposite of `kernel f`. -/
@[simps]
def cokernelOpUnop : (cokernel f.op).unop ≅ kernel f where
hom :=
kernel.lift f (cokernel.π f.op).unop <| by
|
rw [← f.unop_op, ← unop_comp, f.unop_op]
|
/-- The cokernel of `f.op` is the opposite of `kernel f`. -/
@[simps]
def cokernelOpUnop : (cokernel f.op).unop ≅ kernel f where
hom :=
kernel.lift f (cokernel.π f.op).unop <| by
|
Mathlib.CategoryTheory.Abelian.Opposite.65_0.3nBRs3fSYrCoEsT
|
/-- The cokernel of `f.op` is the opposite of `kernel f`. -/
@[simps]
def cokernelOpUnop : (cokernel f.op).unop ≅ kernel f where
hom
|
Mathlib_CategoryTheory_Abelian_Opposite
|
C : Type u_1
inst✝¹ : Category.{?u.12173, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ (f.op ≫ cokernel.π f.op).unop = 0
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_theory.abelian.opposite from "leanprover-community/mathlib"@"a5ff45a1c92c278b03b52459a620cfd9c49ebc80"
/-!
# The opposite of an abelian category is abelian.
-/
noncomputable section
namespace CategoryTheory
open CategoryTheory.Limits
variable (C : Type*) [Category C] [Abelian C]
-- porting note: these local instances do not seem to be necessary
--attribute [local instance]
-- hasFiniteLimits_of_hasEqualizers_and_finite_products
-- hasFiniteColimits_of_hasCoequalizers_and_finite_coproducts
-- Abelian.hasFiniteBiproducts
instance : Abelian Cᵒᵖ := by
-- porting note: priorities of `Abelian.has_kernels` and `Abelian.has_cokernels` have
-- been set to 90 in `Abelian.Basic` in order to prevent a timeout here
exact {
normalMonoOfMono := fun f => normalMonoOfNormalEpiUnop _ (normalEpiOfEpi f.unop)
normalEpiOfEpi := fun f => normalEpiOfNormalMonoUnop _ (normalMonoOfMono f.unop) }
section
variable {C}
variable {X Y : C} (f : X ⟶ Y) {A B : Cᵒᵖ} (g : A ⟶ B)
-- TODO: Generalize (this will work whenever f has a cokernel)
-- (The abelian case is probably sufficient for most applications.)
/-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom := (kernel.lift f.op (cokernel.π f).op <| by simp [← op_comp]).unop
inv :=
cokernel.desc f (kernel.ι f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
simp
hom_inv_id := by
rw [← unop_id, ← (cokernel.desc f _ _).unop_op, ← unop_comp]
congr 1
ext
simp [← op_comp]
inv_hom_id := by
ext
simp [← unop_comp]
#align category_theory.kernel_op_unop CategoryTheory.kernelOpUnop
-- TODO: Generalize (this will work whenever f has a kernel)
-- (The abelian case is probably sufficient for most applications.)
/-- The cokernel of `f.op` is the opposite of `kernel f`. -/
@[simps]
def cokernelOpUnop : (cokernel f.op).unop ≅ kernel f where
hom :=
kernel.lift f (cokernel.π f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
|
simp
|
/-- The cokernel of `f.op` is the opposite of `kernel f`. -/
@[simps]
def cokernelOpUnop : (cokernel f.op).unop ≅ kernel f where
hom :=
kernel.lift f (cokernel.π f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
|
Mathlib.CategoryTheory.Abelian.Opposite.65_0.3nBRs3fSYrCoEsT
|
/-- The cokernel of `f.op` is the opposite of `kernel f`. -/
@[simps]
def cokernelOpUnop : (cokernel f.op).unop ≅ kernel f where
hom
|
Mathlib_CategoryTheory_Abelian_Opposite
|
C : Type u_1
inst✝¹ : Category.{?u.12173, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ f.op ≫ (kernel.ι f).op = 0
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_theory.abelian.opposite from "leanprover-community/mathlib"@"a5ff45a1c92c278b03b52459a620cfd9c49ebc80"
/-!
# The opposite of an abelian category is abelian.
-/
noncomputable section
namespace CategoryTheory
open CategoryTheory.Limits
variable (C : Type*) [Category C] [Abelian C]
-- porting note: these local instances do not seem to be necessary
--attribute [local instance]
-- hasFiniteLimits_of_hasEqualizers_and_finite_products
-- hasFiniteColimits_of_hasCoequalizers_and_finite_coproducts
-- Abelian.hasFiniteBiproducts
instance : Abelian Cᵒᵖ := by
-- porting note: priorities of `Abelian.has_kernels` and `Abelian.has_cokernels` have
-- been set to 90 in `Abelian.Basic` in order to prevent a timeout here
exact {
normalMonoOfMono := fun f => normalMonoOfNormalEpiUnop _ (normalEpiOfEpi f.unop)
normalEpiOfEpi := fun f => normalEpiOfNormalMonoUnop _ (normalMonoOfMono f.unop) }
section
variable {C}
variable {X Y : C} (f : X ⟶ Y) {A B : Cᵒᵖ} (g : A ⟶ B)
-- TODO: Generalize (this will work whenever f has a cokernel)
-- (The abelian case is probably sufficient for most applications.)
/-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom := (kernel.lift f.op (cokernel.π f).op <| by simp [← op_comp]).unop
inv :=
cokernel.desc f (kernel.ι f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
simp
hom_inv_id := by
rw [← unop_id, ← (cokernel.desc f _ _).unop_op, ← unop_comp]
congr 1
ext
simp [← op_comp]
inv_hom_id := by
ext
simp [← unop_comp]
#align category_theory.kernel_op_unop CategoryTheory.kernelOpUnop
-- TODO: Generalize (this will work whenever f has a kernel)
-- (The abelian case is probably sufficient for most applications.)
/-- The cokernel of `f.op` is the opposite of `kernel f`. -/
@[simps]
def cokernelOpUnop : (cokernel f.op).unop ≅ kernel f where
hom :=
kernel.lift f (cokernel.π f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
simp
inv := (cokernel.desc f.op (kernel.ι f).op <| by
|
simp [← op_comp]
|
/-- The cokernel of `f.op` is the opposite of `kernel f`. -/
@[simps]
def cokernelOpUnop : (cokernel f.op).unop ≅ kernel f where
hom :=
kernel.lift f (cokernel.π f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
simp
inv := (cokernel.desc f.op (kernel.ι f).op <| by
|
Mathlib.CategoryTheory.Abelian.Opposite.65_0.3nBRs3fSYrCoEsT
|
/-- The cokernel of `f.op` is the opposite of `kernel f`. -/
@[simps]
def cokernelOpUnop : (cokernel f.op).unop ≅ kernel f where
hom
|
Mathlib_CategoryTheory_Abelian_Opposite
|
C : Type u_1
inst✝¹ : Category.{?u.12173, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ kernel.lift f (cokernel.π f.op).unop (_ : (cokernel.π f.op).unop ≫ f = 0) ≫
(cokernel.desc f.op (kernel.ι f).op (_ : (kernel.ι f ≫ f).op = 0)).unop =
𝟙 (cokernel f.op).unop
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_theory.abelian.opposite from "leanprover-community/mathlib"@"a5ff45a1c92c278b03b52459a620cfd9c49ebc80"
/-!
# The opposite of an abelian category is abelian.
-/
noncomputable section
namespace CategoryTheory
open CategoryTheory.Limits
variable (C : Type*) [Category C] [Abelian C]
-- porting note: these local instances do not seem to be necessary
--attribute [local instance]
-- hasFiniteLimits_of_hasEqualizers_and_finite_products
-- hasFiniteColimits_of_hasCoequalizers_and_finite_coproducts
-- Abelian.hasFiniteBiproducts
instance : Abelian Cᵒᵖ := by
-- porting note: priorities of `Abelian.has_kernels` and `Abelian.has_cokernels` have
-- been set to 90 in `Abelian.Basic` in order to prevent a timeout here
exact {
normalMonoOfMono := fun f => normalMonoOfNormalEpiUnop _ (normalEpiOfEpi f.unop)
normalEpiOfEpi := fun f => normalEpiOfNormalMonoUnop _ (normalMonoOfMono f.unop) }
section
variable {C}
variable {X Y : C} (f : X ⟶ Y) {A B : Cᵒᵖ} (g : A ⟶ B)
-- TODO: Generalize (this will work whenever f has a cokernel)
-- (The abelian case is probably sufficient for most applications.)
/-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom := (kernel.lift f.op (cokernel.π f).op <| by simp [← op_comp]).unop
inv :=
cokernel.desc f (kernel.ι f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
simp
hom_inv_id := by
rw [← unop_id, ← (cokernel.desc f _ _).unop_op, ← unop_comp]
congr 1
ext
simp [← op_comp]
inv_hom_id := by
ext
simp [← unop_comp]
#align category_theory.kernel_op_unop CategoryTheory.kernelOpUnop
-- TODO: Generalize (this will work whenever f has a kernel)
-- (The abelian case is probably sufficient for most applications.)
/-- The cokernel of `f.op` is the opposite of `kernel f`. -/
@[simps]
def cokernelOpUnop : (cokernel f.op).unop ≅ kernel f where
hom :=
kernel.lift f (cokernel.π f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
simp
inv := (cokernel.desc f.op (kernel.ι f).op <| by simp [← op_comp]).unop
hom_inv_id := by
|
rw [← unop_id, ← (kernel.lift f _ _).unop_op, ← unop_comp]
|
/-- The cokernel of `f.op` is the opposite of `kernel f`. -/
@[simps]
def cokernelOpUnop : (cokernel f.op).unop ≅ kernel f where
hom :=
kernel.lift f (cokernel.π f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
simp
inv := (cokernel.desc f.op (kernel.ι f).op <| by simp [← op_comp]).unop
hom_inv_id := by
|
Mathlib.CategoryTheory.Abelian.Opposite.65_0.3nBRs3fSYrCoEsT
|
/-- The cokernel of `f.op` is the opposite of `kernel f`. -/
@[simps]
def cokernelOpUnop : (cokernel f.op).unop ≅ kernel f where
hom
|
Mathlib_CategoryTheory_Abelian_Opposite
|
C : Type u_1
inst✝¹ : Category.{?u.12173, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ (cokernel.desc f.op (kernel.ι f).op (_ : (kernel.ι f ≫ f).op = 0) ≫
(kernel.lift f (cokernel.π f.op).unop (_ : (cokernel.π f.op).unop ≫ f = 0)).op).unop =
(𝟙 (cokernel f.op)).unop
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_theory.abelian.opposite from "leanprover-community/mathlib"@"a5ff45a1c92c278b03b52459a620cfd9c49ebc80"
/-!
# The opposite of an abelian category is abelian.
-/
noncomputable section
namespace CategoryTheory
open CategoryTheory.Limits
variable (C : Type*) [Category C] [Abelian C]
-- porting note: these local instances do not seem to be necessary
--attribute [local instance]
-- hasFiniteLimits_of_hasEqualizers_and_finite_products
-- hasFiniteColimits_of_hasCoequalizers_and_finite_coproducts
-- Abelian.hasFiniteBiproducts
instance : Abelian Cᵒᵖ := by
-- porting note: priorities of `Abelian.has_kernels` and `Abelian.has_cokernels` have
-- been set to 90 in `Abelian.Basic` in order to prevent a timeout here
exact {
normalMonoOfMono := fun f => normalMonoOfNormalEpiUnop _ (normalEpiOfEpi f.unop)
normalEpiOfEpi := fun f => normalEpiOfNormalMonoUnop _ (normalMonoOfMono f.unop) }
section
variable {C}
variable {X Y : C} (f : X ⟶ Y) {A B : Cᵒᵖ} (g : A ⟶ B)
-- TODO: Generalize (this will work whenever f has a cokernel)
-- (The abelian case is probably sufficient for most applications.)
/-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom := (kernel.lift f.op (cokernel.π f).op <| by simp [← op_comp]).unop
inv :=
cokernel.desc f (kernel.ι f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
simp
hom_inv_id := by
rw [← unop_id, ← (cokernel.desc f _ _).unop_op, ← unop_comp]
congr 1
ext
simp [← op_comp]
inv_hom_id := by
ext
simp [← unop_comp]
#align category_theory.kernel_op_unop CategoryTheory.kernelOpUnop
-- TODO: Generalize (this will work whenever f has a kernel)
-- (The abelian case is probably sufficient for most applications.)
/-- The cokernel of `f.op` is the opposite of `kernel f`. -/
@[simps]
def cokernelOpUnop : (cokernel f.op).unop ≅ kernel f where
hom :=
kernel.lift f (cokernel.π f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
simp
inv := (cokernel.desc f.op (kernel.ι f).op <| by simp [← op_comp]).unop
hom_inv_id := by
rw [← unop_id, ← (kernel.lift f _ _).unop_op, ← unop_comp]
|
congr 1
|
/-- The cokernel of `f.op` is the opposite of `kernel f`. -/
@[simps]
def cokernelOpUnop : (cokernel f.op).unop ≅ kernel f where
hom :=
kernel.lift f (cokernel.π f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
simp
inv := (cokernel.desc f.op (kernel.ι f).op <| by simp [← op_comp]).unop
hom_inv_id := by
rw [← unop_id, ← (kernel.lift f _ _).unop_op, ← unop_comp]
|
Mathlib.CategoryTheory.Abelian.Opposite.65_0.3nBRs3fSYrCoEsT
|
/-- The cokernel of `f.op` is the opposite of `kernel f`. -/
@[simps]
def cokernelOpUnop : (cokernel f.op).unop ≅ kernel f where
hom
|
Mathlib_CategoryTheory_Abelian_Opposite
|
case e_f
C : Type u_1
inst✝¹ : Category.{?u.12173, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ cokernel.desc f.op (kernel.ι f).op (_ : (kernel.ι f ≫ f).op = 0) ≫
(kernel.lift f (cokernel.π f.op).unop (_ : (cokernel.π f.op).unop ≫ f = 0)).op =
𝟙 (cokernel f.op)
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_theory.abelian.opposite from "leanprover-community/mathlib"@"a5ff45a1c92c278b03b52459a620cfd9c49ebc80"
/-!
# The opposite of an abelian category is abelian.
-/
noncomputable section
namespace CategoryTheory
open CategoryTheory.Limits
variable (C : Type*) [Category C] [Abelian C]
-- porting note: these local instances do not seem to be necessary
--attribute [local instance]
-- hasFiniteLimits_of_hasEqualizers_and_finite_products
-- hasFiniteColimits_of_hasCoequalizers_and_finite_coproducts
-- Abelian.hasFiniteBiproducts
instance : Abelian Cᵒᵖ := by
-- porting note: priorities of `Abelian.has_kernels` and `Abelian.has_cokernels` have
-- been set to 90 in `Abelian.Basic` in order to prevent a timeout here
exact {
normalMonoOfMono := fun f => normalMonoOfNormalEpiUnop _ (normalEpiOfEpi f.unop)
normalEpiOfEpi := fun f => normalEpiOfNormalMonoUnop _ (normalMonoOfMono f.unop) }
section
variable {C}
variable {X Y : C} (f : X ⟶ Y) {A B : Cᵒᵖ} (g : A ⟶ B)
-- TODO: Generalize (this will work whenever f has a cokernel)
-- (The abelian case is probably sufficient for most applications.)
/-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom := (kernel.lift f.op (cokernel.π f).op <| by simp [← op_comp]).unop
inv :=
cokernel.desc f (kernel.ι f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
simp
hom_inv_id := by
rw [← unop_id, ← (cokernel.desc f _ _).unop_op, ← unop_comp]
congr 1
ext
simp [← op_comp]
inv_hom_id := by
ext
simp [← unop_comp]
#align category_theory.kernel_op_unop CategoryTheory.kernelOpUnop
-- TODO: Generalize (this will work whenever f has a kernel)
-- (The abelian case is probably sufficient for most applications.)
/-- The cokernel of `f.op` is the opposite of `kernel f`. -/
@[simps]
def cokernelOpUnop : (cokernel f.op).unop ≅ kernel f where
hom :=
kernel.lift f (cokernel.π f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
simp
inv := (cokernel.desc f.op (kernel.ι f).op <| by simp [← op_comp]).unop
hom_inv_id := by
rw [← unop_id, ← (kernel.lift f _ _).unop_op, ← unop_comp]
congr 1
|
ext
|
/-- The cokernel of `f.op` is the opposite of `kernel f`. -/
@[simps]
def cokernelOpUnop : (cokernel f.op).unop ≅ kernel f where
hom :=
kernel.lift f (cokernel.π f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
simp
inv := (cokernel.desc f.op (kernel.ι f).op <| by simp [← op_comp]).unop
hom_inv_id := by
rw [← unop_id, ← (kernel.lift f _ _).unop_op, ← unop_comp]
congr 1
|
Mathlib.CategoryTheory.Abelian.Opposite.65_0.3nBRs3fSYrCoEsT
|
/-- The cokernel of `f.op` is the opposite of `kernel f`. -/
@[simps]
def cokernelOpUnop : (cokernel f.op).unop ≅ kernel f where
hom
|
Mathlib_CategoryTheory_Abelian_Opposite
|
case e_f.h
C : Type u_1
inst✝¹ : Category.{?u.12173, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ coequalizer.π f.op 0 ≫
cokernel.desc f.op (kernel.ι f).op (_ : (kernel.ι f ≫ f).op = 0) ≫
(kernel.lift f (cokernel.π f.op).unop (_ : (cokernel.π f.op).unop ≫ f = 0)).op =
coequalizer.π f.op 0 ≫ 𝟙 (cokernel f.op)
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_theory.abelian.opposite from "leanprover-community/mathlib"@"a5ff45a1c92c278b03b52459a620cfd9c49ebc80"
/-!
# The opposite of an abelian category is abelian.
-/
noncomputable section
namespace CategoryTheory
open CategoryTheory.Limits
variable (C : Type*) [Category C] [Abelian C]
-- porting note: these local instances do not seem to be necessary
--attribute [local instance]
-- hasFiniteLimits_of_hasEqualizers_and_finite_products
-- hasFiniteColimits_of_hasCoequalizers_and_finite_coproducts
-- Abelian.hasFiniteBiproducts
instance : Abelian Cᵒᵖ := by
-- porting note: priorities of `Abelian.has_kernels` and `Abelian.has_cokernels` have
-- been set to 90 in `Abelian.Basic` in order to prevent a timeout here
exact {
normalMonoOfMono := fun f => normalMonoOfNormalEpiUnop _ (normalEpiOfEpi f.unop)
normalEpiOfEpi := fun f => normalEpiOfNormalMonoUnop _ (normalMonoOfMono f.unop) }
section
variable {C}
variable {X Y : C} (f : X ⟶ Y) {A B : Cᵒᵖ} (g : A ⟶ B)
-- TODO: Generalize (this will work whenever f has a cokernel)
-- (The abelian case is probably sufficient for most applications.)
/-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom := (kernel.lift f.op (cokernel.π f).op <| by simp [← op_comp]).unop
inv :=
cokernel.desc f (kernel.ι f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
simp
hom_inv_id := by
rw [← unop_id, ← (cokernel.desc f _ _).unop_op, ← unop_comp]
congr 1
ext
simp [← op_comp]
inv_hom_id := by
ext
simp [← unop_comp]
#align category_theory.kernel_op_unop CategoryTheory.kernelOpUnop
-- TODO: Generalize (this will work whenever f has a kernel)
-- (The abelian case is probably sufficient for most applications.)
/-- The cokernel of `f.op` is the opposite of `kernel f`. -/
@[simps]
def cokernelOpUnop : (cokernel f.op).unop ≅ kernel f where
hom :=
kernel.lift f (cokernel.π f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
simp
inv := (cokernel.desc f.op (kernel.ι f).op <| by simp [← op_comp]).unop
hom_inv_id := by
rw [← unop_id, ← (kernel.lift f _ _).unop_op, ← unop_comp]
congr 1
ext
|
simp [← op_comp]
|
/-- The cokernel of `f.op` is the opposite of `kernel f`. -/
@[simps]
def cokernelOpUnop : (cokernel f.op).unop ≅ kernel f where
hom :=
kernel.lift f (cokernel.π f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
simp
inv := (cokernel.desc f.op (kernel.ι f).op <| by simp [← op_comp]).unop
hom_inv_id := by
rw [← unop_id, ← (kernel.lift f _ _).unop_op, ← unop_comp]
congr 1
ext
|
Mathlib.CategoryTheory.Abelian.Opposite.65_0.3nBRs3fSYrCoEsT
|
/-- The cokernel of `f.op` is the opposite of `kernel f`. -/
@[simps]
def cokernelOpUnop : (cokernel f.op).unop ≅ kernel f where
hom
|
Mathlib_CategoryTheory_Abelian_Opposite
|
C : Type u_1
inst✝¹ : Category.{?u.12173, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ (cokernel.desc f.op (kernel.ι f).op (_ : (kernel.ι f ≫ f).op = 0)).unop ≫
kernel.lift f (cokernel.π f.op).unop (_ : (cokernel.π f.op).unop ≫ f = 0) =
𝟙 (kernel f)
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_theory.abelian.opposite from "leanprover-community/mathlib"@"a5ff45a1c92c278b03b52459a620cfd9c49ebc80"
/-!
# The opposite of an abelian category is abelian.
-/
noncomputable section
namespace CategoryTheory
open CategoryTheory.Limits
variable (C : Type*) [Category C] [Abelian C]
-- porting note: these local instances do not seem to be necessary
--attribute [local instance]
-- hasFiniteLimits_of_hasEqualizers_and_finite_products
-- hasFiniteColimits_of_hasCoequalizers_and_finite_coproducts
-- Abelian.hasFiniteBiproducts
instance : Abelian Cᵒᵖ := by
-- porting note: priorities of `Abelian.has_kernels` and `Abelian.has_cokernels` have
-- been set to 90 in `Abelian.Basic` in order to prevent a timeout here
exact {
normalMonoOfMono := fun f => normalMonoOfNormalEpiUnop _ (normalEpiOfEpi f.unop)
normalEpiOfEpi := fun f => normalEpiOfNormalMonoUnop _ (normalMonoOfMono f.unop) }
section
variable {C}
variable {X Y : C} (f : X ⟶ Y) {A B : Cᵒᵖ} (g : A ⟶ B)
-- TODO: Generalize (this will work whenever f has a cokernel)
-- (The abelian case is probably sufficient for most applications.)
/-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom := (kernel.lift f.op (cokernel.π f).op <| by simp [← op_comp]).unop
inv :=
cokernel.desc f (kernel.ι f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
simp
hom_inv_id := by
rw [← unop_id, ← (cokernel.desc f _ _).unop_op, ← unop_comp]
congr 1
ext
simp [← op_comp]
inv_hom_id := by
ext
simp [← unop_comp]
#align category_theory.kernel_op_unop CategoryTheory.kernelOpUnop
-- TODO: Generalize (this will work whenever f has a kernel)
-- (The abelian case is probably sufficient for most applications.)
/-- The cokernel of `f.op` is the opposite of `kernel f`. -/
@[simps]
def cokernelOpUnop : (cokernel f.op).unop ≅ kernel f where
hom :=
kernel.lift f (cokernel.π f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
simp
inv := (cokernel.desc f.op (kernel.ι f).op <| by simp [← op_comp]).unop
hom_inv_id := by
rw [← unop_id, ← (kernel.lift f _ _).unop_op, ← unop_comp]
congr 1
ext
simp [← op_comp]
inv_hom_id := by
|
ext
|
/-- The cokernel of `f.op` is the opposite of `kernel f`. -/
@[simps]
def cokernelOpUnop : (cokernel f.op).unop ≅ kernel f where
hom :=
kernel.lift f (cokernel.π f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
simp
inv := (cokernel.desc f.op (kernel.ι f).op <| by simp [← op_comp]).unop
hom_inv_id := by
rw [← unop_id, ← (kernel.lift f _ _).unop_op, ← unop_comp]
congr 1
ext
simp [← op_comp]
inv_hom_id := by
|
Mathlib.CategoryTheory.Abelian.Opposite.65_0.3nBRs3fSYrCoEsT
|
/-- The cokernel of `f.op` is the opposite of `kernel f`. -/
@[simps]
def cokernelOpUnop : (cokernel f.op).unop ≅ kernel f where
hom
|
Mathlib_CategoryTheory_Abelian_Opposite
|
case h
C : Type u_1
inst✝¹ : Category.{?u.12173, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ ((cokernel.desc f.op (kernel.ι f).op (_ : (kernel.ι f ≫ f).op = 0)).unop ≫
kernel.lift f (cokernel.π f.op).unop (_ : (cokernel.π f.op).unop ≫ f = 0)) ≫
equalizer.ι f 0 =
𝟙 (kernel f) ≫ equalizer.ι f 0
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_theory.abelian.opposite from "leanprover-community/mathlib"@"a5ff45a1c92c278b03b52459a620cfd9c49ebc80"
/-!
# The opposite of an abelian category is abelian.
-/
noncomputable section
namespace CategoryTheory
open CategoryTheory.Limits
variable (C : Type*) [Category C] [Abelian C]
-- porting note: these local instances do not seem to be necessary
--attribute [local instance]
-- hasFiniteLimits_of_hasEqualizers_and_finite_products
-- hasFiniteColimits_of_hasCoequalizers_and_finite_coproducts
-- Abelian.hasFiniteBiproducts
instance : Abelian Cᵒᵖ := by
-- porting note: priorities of `Abelian.has_kernels` and `Abelian.has_cokernels` have
-- been set to 90 in `Abelian.Basic` in order to prevent a timeout here
exact {
normalMonoOfMono := fun f => normalMonoOfNormalEpiUnop _ (normalEpiOfEpi f.unop)
normalEpiOfEpi := fun f => normalEpiOfNormalMonoUnop _ (normalMonoOfMono f.unop) }
section
variable {C}
variable {X Y : C} (f : X ⟶ Y) {A B : Cᵒᵖ} (g : A ⟶ B)
-- TODO: Generalize (this will work whenever f has a cokernel)
-- (The abelian case is probably sufficient for most applications.)
/-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom := (kernel.lift f.op (cokernel.π f).op <| by simp [← op_comp]).unop
inv :=
cokernel.desc f (kernel.ι f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
simp
hom_inv_id := by
rw [← unop_id, ← (cokernel.desc f _ _).unop_op, ← unop_comp]
congr 1
ext
simp [← op_comp]
inv_hom_id := by
ext
simp [← unop_comp]
#align category_theory.kernel_op_unop CategoryTheory.kernelOpUnop
-- TODO: Generalize (this will work whenever f has a kernel)
-- (The abelian case is probably sufficient for most applications.)
/-- The cokernel of `f.op` is the opposite of `kernel f`. -/
@[simps]
def cokernelOpUnop : (cokernel f.op).unop ≅ kernel f where
hom :=
kernel.lift f (cokernel.π f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
simp
inv := (cokernel.desc f.op (kernel.ι f).op <| by simp [← op_comp]).unop
hom_inv_id := by
rw [← unop_id, ← (kernel.lift f _ _).unop_op, ← unop_comp]
congr 1
ext
simp [← op_comp]
inv_hom_id := by
ext
|
simp [← unop_comp]
|
/-- The cokernel of `f.op` is the opposite of `kernel f`. -/
@[simps]
def cokernelOpUnop : (cokernel f.op).unop ≅ kernel f where
hom :=
kernel.lift f (cokernel.π f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
simp
inv := (cokernel.desc f.op (kernel.ι f).op <| by simp [← op_comp]).unop
hom_inv_id := by
rw [← unop_id, ← (kernel.lift f _ _).unop_op, ← unop_comp]
congr 1
ext
simp [← op_comp]
inv_hom_id := by
ext
|
Mathlib.CategoryTheory.Abelian.Opposite.65_0.3nBRs3fSYrCoEsT
|
/-- The cokernel of `f.op` is the opposite of `kernel f`. -/
@[simps]
def cokernelOpUnop : (cokernel f.op).unop ≅ kernel f where
hom
|
Mathlib_CategoryTheory_Abelian_Opposite
|
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ (cokernel.π f.op).unop = (cokernelOpUnop f).hom ≫ kernel.ι f ≫ eqToHom (_ : X = (Opposite.op X).unop)
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_theory.abelian.opposite from "leanprover-community/mathlib"@"a5ff45a1c92c278b03b52459a620cfd9c49ebc80"
/-!
# The opposite of an abelian category is abelian.
-/
noncomputable section
namespace CategoryTheory
open CategoryTheory.Limits
variable (C : Type*) [Category C] [Abelian C]
-- porting note: these local instances do not seem to be necessary
--attribute [local instance]
-- hasFiniteLimits_of_hasEqualizers_and_finite_products
-- hasFiniteColimits_of_hasCoequalizers_and_finite_coproducts
-- Abelian.hasFiniteBiproducts
instance : Abelian Cᵒᵖ := by
-- porting note: priorities of `Abelian.has_kernels` and `Abelian.has_cokernels` have
-- been set to 90 in `Abelian.Basic` in order to prevent a timeout here
exact {
normalMonoOfMono := fun f => normalMonoOfNormalEpiUnop _ (normalEpiOfEpi f.unop)
normalEpiOfEpi := fun f => normalEpiOfNormalMonoUnop _ (normalMonoOfMono f.unop) }
section
variable {C}
variable {X Y : C} (f : X ⟶ Y) {A B : Cᵒᵖ} (g : A ⟶ B)
-- TODO: Generalize (this will work whenever f has a cokernel)
-- (The abelian case is probably sufficient for most applications.)
/-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom := (kernel.lift f.op (cokernel.π f).op <| by simp [← op_comp]).unop
inv :=
cokernel.desc f (kernel.ι f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
simp
hom_inv_id := by
rw [← unop_id, ← (cokernel.desc f _ _).unop_op, ← unop_comp]
congr 1
ext
simp [← op_comp]
inv_hom_id := by
ext
simp [← unop_comp]
#align category_theory.kernel_op_unop CategoryTheory.kernelOpUnop
-- TODO: Generalize (this will work whenever f has a kernel)
-- (The abelian case is probably sufficient for most applications.)
/-- The cokernel of `f.op` is the opposite of `kernel f`. -/
@[simps]
def cokernelOpUnop : (cokernel f.op).unop ≅ kernel f where
hom :=
kernel.lift f (cokernel.π f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
simp
inv := (cokernel.desc f.op (kernel.ι f).op <| by simp [← op_comp]).unop
hom_inv_id := by
rw [← unop_id, ← (kernel.lift f _ _).unop_op, ← unop_comp]
congr 1
ext
simp [← op_comp]
inv_hom_id := by
ext
simp [← unop_comp]
#align category_theory.cokernel_op_unop CategoryTheory.cokernelOpUnop
/-- The kernel of `g.unop` is the opposite of `cokernel g`. -/
@[simps!]
def kernelUnopOp : Opposite.op (kernel g.unop) ≅ cokernel g :=
(cokernelOpUnop g.unop).op
#align category_theory.kernel_unop_op CategoryTheory.kernelUnopOp
/-- The cokernel of `g.unop` is the opposite of `kernel g`. -/
@[simps!]
def cokernelUnopOp : Opposite.op (cokernel g.unop) ≅ kernel g :=
(kernelOpUnop g.unop).op
#align category_theory.cokernel_unop_op CategoryTheory.cokernelUnopOp
theorem cokernel.π_op :
(cokernel.π f.op).unop =
(cokernelOpUnop f).hom ≫ kernel.ι f ≫ eqToHom (Opposite.unop_op _).symm :=
by
|
simp [cokernelOpUnop]
|
theorem cokernel.π_op :
(cokernel.π f.op).unop =
(cokernelOpUnop f).hom ≫ kernel.ι f ≫ eqToHom (Opposite.unop_op _).symm :=
by
|
Mathlib.CategoryTheory.Abelian.Opposite.95_0.3nBRs3fSYrCoEsT
|
theorem cokernel.π_op :
(cokernel.π f.op).unop =
(cokernelOpUnop f).hom ≫ kernel.ι f ≫ eqToHom (Opposite.unop_op _).symm
|
Mathlib_CategoryTheory_Abelian_Opposite
|
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
⊢ (kernel.ι f.op).unop = eqToHom (_ : (Opposite.op Y).unop = Y) ≫ cokernel.π f ≫ (kernelOpUnop f).inv
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_theory.abelian.opposite from "leanprover-community/mathlib"@"a5ff45a1c92c278b03b52459a620cfd9c49ebc80"
/-!
# The opposite of an abelian category is abelian.
-/
noncomputable section
namespace CategoryTheory
open CategoryTheory.Limits
variable (C : Type*) [Category C] [Abelian C]
-- porting note: these local instances do not seem to be necessary
--attribute [local instance]
-- hasFiniteLimits_of_hasEqualizers_and_finite_products
-- hasFiniteColimits_of_hasCoequalizers_and_finite_coproducts
-- Abelian.hasFiniteBiproducts
instance : Abelian Cᵒᵖ := by
-- porting note: priorities of `Abelian.has_kernels` and `Abelian.has_cokernels` have
-- been set to 90 in `Abelian.Basic` in order to prevent a timeout here
exact {
normalMonoOfMono := fun f => normalMonoOfNormalEpiUnop _ (normalEpiOfEpi f.unop)
normalEpiOfEpi := fun f => normalEpiOfNormalMonoUnop _ (normalMonoOfMono f.unop) }
section
variable {C}
variable {X Y : C} (f : X ⟶ Y) {A B : Cᵒᵖ} (g : A ⟶ B)
-- TODO: Generalize (this will work whenever f has a cokernel)
-- (The abelian case is probably sufficient for most applications.)
/-- The kernel of `f.op` is the opposite of `cokernel f`. -/
@[simps]
def kernelOpUnop : (kernel f.op).unop ≅ cokernel f where
hom := (kernel.lift f.op (cokernel.π f).op <| by simp [← op_comp]).unop
inv :=
cokernel.desc f (kernel.ι f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
simp
hom_inv_id := by
rw [← unop_id, ← (cokernel.desc f _ _).unop_op, ← unop_comp]
congr 1
ext
simp [← op_comp]
inv_hom_id := by
ext
simp [← unop_comp]
#align category_theory.kernel_op_unop CategoryTheory.kernelOpUnop
-- TODO: Generalize (this will work whenever f has a kernel)
-- (The abelian case is probably sufficient for most applications.)
/-- The cokernel of `f.op` is the opposite of `kernel f`. -/
@[simps]
def cokernelOpUnop : (cokernel f.op).unop ≅ kernel f where
hom :=
kernel.lift f (cokernel.π f.op).unop <| by
rw [← f.unop_op, ← unop_comp, f.unop_op]
simp
inv := (cokernel.desc f.op (kernel.ι f).op <| by simp [← op_comp]).unop
hom_inv_id := by
rw [← unop_id, ← (kernel.lift f _ _).unop_op, ← unop_comp]
congr 1
ext
simp [← op_comp]
inv_hom_id := by
ext
simp [← unop_comp]
#align category_theory.cokernel_op_unop CategoryTheory.cokernelOpUnop
/-- The kernel of `g.unop` is the opposite of `cokernel g`. -/
@[simps!]
def kernelUnopOp : Opposite.op (kernel g.unop) ≅ cokernel g :=
(cokernelOpUnop g.unop).op
#align category_theory.kernel_unop_op CategoryTheory.kernelUnopOp
/-- The cokernel of `g.unop` is the opposite of `kernel g`. -/
@[simps!]
def cokernelUnopOp : Opposite.op (cokernel g.unop) ≅ kernel g :=
(kernelOpUnop g.unop).op
#align category_theory.cokernel_unop_op CategoryTheory.cokernelUnopOp
theorem cokernel.π_op :
(cokernel.π f.op).unop =
(cokernelOpUnop f).hom ≫ kernel.ι f ≫ eqToHom (Opposite.unop_op _).symm :=
by simp [cokernelOpUnop]
#align category_theory.cokernel.π_op CategoryTheory.cokernel.π_op
theorem kernel.ι_op :
(kernel.ι f.op).unop = eqToHom (Opposite.unop_op _) ≫ cokernel.π f ≫ (kernelOpUnop f).inv := by
|
simp [kernelOpUnop]
|
theorem kernel.ι_op :
(kernel.ι f.op).unop = eqToHom (Opposite.unop_op _) ≫ cokernel.π f ≫ (kernelOpUnop f).inv := by
|
Mathlib.CategoryTheory.Abelian.Opposite.101_0.3nBRs3fSYrCoEsT
|
theorem kernel.ι_op :
(kernel.ι f.op).unop = eqToHom (Opposite.unop_op _) ≫ cokernel.π f ≫ (kernelOpUnop f).inv
|
Mathlib_CategoryTheory_Abelian_Opposite
|
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