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stringlengths 3
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|---|---|---|---|---|---|---|
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : P →A[R] Q
⊢ contLinear f = 0 ↔ ∃ q, f = const R P q
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
|
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
|
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.107_0.bJ3guivW1IqbUMZ
|
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : P →A[R] Q
⊢ contLinear f = 0 ↔ f.linear = 0
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
|
refine' ⟨fun h => _, fun h => _⟩
|
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.107_0.bJ3guivW1IqbUMZ
|
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
case refine'_1
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : P →A[R] Q
h : contLinear f = 0
⊢ f.linear = 0
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;>
|
ext
|
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;>
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.107_0.bJ3guivW1IqbUMZ
|
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
case refine'_2
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : P →A[R] Q
h : f.linear = 0
⊢ contLinear f = 0
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;>
|
ext
|
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;>
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.107_0.bJ3guivW1IqbUMZ
|
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
case refine'_1.h
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : P →A[R] Q
h : contLinear f = 0
x✝ : V
⊢ f.linear x✝ = 0 x✝
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
·
|
rw [← coe_contLinear_eq_linear, h]
|
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
·
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.107_0.bJ3guivW1IqbUMZ
|
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
case refine'_1.h
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : P →A[R] Q
h : contLinear f = 0
x✝ : V
⊢ ↑0 x✝ = 0 x✝
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h];
|
rfl
|
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h];
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.107_0.bJ3guivW1IqbUMZ
|
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
case refine'_2.h
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : P →A[R] Q
h : f.linear = 0
x✝ : V
⊢ (contLinear f) x✝ = 0 x✝
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
·
|
rw [← coe_linear_eq_coe_contLinear, h]
|
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
·
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.107_0.bJ3guivW1IqbUMZ
|
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
case refine'_2.h
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : P →A[R] Q
h : f.linear = 0
x✝ : V
⊢ 0 x✝ = 0 x✝
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h];
|
rfl
|
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h];
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.107_0.bJ3guivW1IqbUMZ
|
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : P →A[R] Q
h₁ : contLinear f = 0 ↔ f.linear = 0
⊢ contLinear f = 0 ↔ ∃ q, f = const R P q
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
|
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h]; rfl
|
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.107_0.bJ3guivW1IqbUMZ
|
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : P →A[R] Q
h₁ : contLinear f = 0 ↔ f.linear = 0
⊢ ∀ (q : Q), f = const R P q ↔ f.toAffineMap = AffineMap.const R P q
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
|
intro q
|
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.107_0.bJ3guivW1IqbUMZ
|
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : P →A[R] Q
h₁ : contLinear f = 0 ↔ f.linear = 0
q : Q
⊢ f = const R P q ↔ f.toAffineMap = AffineMap.const R P q
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
|
refine' ⟨fun h => _, fun h => _⟩
|
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.107_0.bJ3guivW1IqbUMZ
|
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
case refine'_1
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : P →A[R] Q
h₁ : contLinear f = 0 ↔ f.linear = 0
q : Q
h : f = const R P q
⊢ f.toAffineMap = AffineMap.const R P q
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;>
|
ext
|
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;>
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.107_0.bJ3guivW1IqbUMZ
|
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
case refine'_2
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : P →A[R] Q
h₁ : contLinear f = 0 ↔ f.linear = 0
q : Q
h : f.toAffineMap = AffineMap.const R P q
⊢ f = const R P q
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;>
|
ext
|
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;>
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.107_0.bJ3guivW1IqbUMZ
|
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
case refine'_1.h
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : P →A[R] Q
h₁ : contLinear f = 0 ↔ f.linear = 0
q : Q
h : f = const R P q
p✝ : P
⊢ f.toAffineMap p✝ = (AffineMap.const R P q) p✝
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
·
|
rw [h]
|
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
·
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.107_0.bJ3guivW1IqbUMZ
|
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
case refine'_1.h
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : P →A[R] Q
h₁ : contLinear f = 0 ↔ f.linear = 0
q : Q
h : f = const R P q
p✝ : P
⊢ (const R P q).toAffineMap p✝ = (AffineMap.const R P q) p✝
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h];
|
rfl
|
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h];
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.107_0.bJ3guivW1IqbUMZ
|
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
case refine'_2.h
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : P →A[R] Q
h₁ : contLinear f = 0 ↔ f.linear = 0
q : Q
h : f.toAffineMap = AffineMap.const R P q
x✝ : P
⊢ f x✝ = (const R P q) x✝
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
·
|
rw [← coe_to_affineMap, h]
|
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
·
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.107_0.bJ3guivW1IqbUMZ
|
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
case refine'_2.h
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : P →A[R] Q
h₁ : contLinear f = 0 ↔ f.linear = 0
q : Q
h : f.toAffineMap = AffineMap.const R P q
x✝ : P
⊢ (AffineMap.const R P q) x✝ = (const R P q) x✝
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h];
|
rfl
|
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h];
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.107_0.bJ3guivW1IqbUMZ
|
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : P →A[R] Q
h₁ : contLinear f = 0 ↔ f.linear = 0
h₂ : ∀ (q : Q), f = const R P q ↔ f.toAffineMap = AffineMap.const R P q
⊢ contLinear f = 0 ↔ ∃ q, f = const R P q
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h]; rfl
|
simp_rw [h₁, h₂]
|
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h]; rfl
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.107_0.bJ3guivW1IqbUMZ
|
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : P →A[R] Q
h₁ : contLinear f = 0 ↔ f.linear = 0
h₂ : ∀ (q : Q), f = const R P q ↔ f.toAffineMap = AffineMap.const R P q
⊢ f.linear = 0 ↔ ∃ q, f.toAffineMap = AffineMap.const R P q
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h]; rfl
simp_rw [h₁, h₂]
|
exact (f : P →ᵃ[R] Q).linear_eq_zero_iff_exists_const
|
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h]; rfl
simp_rw [h₁, h₂]
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.107_0.bJ3guivW1IqbUMZ
|
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : V →L[R] W
⊢ contLinear (ContinuousLinearMap.toContinuousAffineMap f) = f
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h]; rfl
simp_rw [h₁, h₂]
exact (f : P →ᵃ[R] Q).linear_eq_zero_iff_exists_const
#align continuous_affine_map.cont_linear_eq_zero_iff_exists_const ContinuousAffineMap.contLinear_eq_zero_iff_exists_const
@[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f := by
|
ext
|
@[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f := by
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.122_0.bJ3guivW1IqbUMZ
|
@[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
case h
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : V →L[R] W
x✝ : V
⊢ (contLinear (ContinuousLinearMap.toContinuousAffineMap f)) x✝ = f x✝
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h]; rfl
simp_rw [h₁, h₂]
exact (f : P →ᵃ[R] Q).linear_eq_zero_iff_exists_const
#align continuous_affine_map.cont_linear_eq_zero_iff_exists_const ContinuousAffineMap.contLinear_eq_zero_iff_exists_const
@[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f := by
ext
|
rfl
|
@[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f := by
ext
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.122_0.bJ3guivW1IqbUMZ
|
@[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : V →A[R] W
⊢ ⇑f = ⇑(contLinear f) + Function.const V (f 0)
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h]; rfl
simp_rw [h₁, h₂]
exact (f : P →ᵃ[R] Q).linear_eq_zero_iff_exists_const
#align continuous_affine_map.cont_linear_eq_zero_iff_exists_const ContinuousAffineMap.contLinear_eq_zero_iff_exists_const
@[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f := by
ext
rfl
#align continuous_affine_map.to_affine_map_cont_linear ContinuousAffineMap.to_affine_map_contLinear
@[simp]
theorem zero_contLinear : (0 : P →A[R] W).contLinear = 0 :=
rfl
#align continuous_affine_map.zero_cont_linear ContinuousAffineMap.zero_contLinear
@[simp]
theorem add_contLinear (f g : P →A[R] W) : (f + g).contLinear = f.contLinear + g.contLinear :=
rfl
#align continuous_affine_map.add_cont_linear ContinuousAffineMap.add_contLinear
@[simp]
theorem sub_contLinear (f g : P →A[R] W) : (f - g).contLinear = f.contLinear - g.contLinear :=
rfl
#align continuous_affine_map.sub_cont_linear ContinuousAffineMap.sub_contLinear
@[simp]
theorem neg_contLinear (f : P →A[R] W) : (-f).contLinear = -f.contLinear :=
rfl
#align continuous_affine_map.neg_cont_linear ContinuousAffineMap.neg_contLinear
@[simp]
theorem smul_contLinear (t : R) (f : P →A[R] W) : (t • f).contLinear = t • f.contLinear :=
rfl
#align continuous_affine_map.smul_cont_linear ContinuousAffineMap.smul_contLinear
theorem decomp (f : V →A[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) := by
|
rcases f with ⟨f, h⟩
|
theorem decomp (f : V →A[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) := by
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.153_0.bJ3guivW1IqbUMZ
|
theorem decomp (f : V →A[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0)
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
case mk
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : V →ᵃ[R] W
h : Continuous f.toFun
⊢ ⇑{ toAffineMap := f, cont := h } =
⇑(contLinear { toAffineMap := f, cont := h }) + Function.const V ({ toAffineMap := f, cont := h } 0)
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h]; rfl
simp_rw [h₁, h₂]
exact (f : P →ᵃ[R] Q).linear_eq_zero_iff_exists_const
#align continuous_affine_map.cont_linear_eq_zero_iff_exists_const ContinuousAffineMap.contLinear_eq_zero_iff_exists_const
@[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f := by
ext
rfl
#align continuous_affine_map.to_affine_map_cont_linear ContinuousAffineMap.to_affine_map_contLinear
@[simp]
theorem zero_contLinear : (0 : P →A[R] W).contLinear = 0 :=
rfl
#align continuous_affine_map.zero_cont_linear ContinuousAffineMap.zero_contLinear
@[simp]
theorem add_contLinear (f g : P →A[R] W) : (f + g).contLinear = f.contLinear + g.contLinear :=
rfl
#align continuous_affine_map.add_cont_linear ContinuousAffineMap.add_contLinear
@[simp]
theorem sub_contLinear (f g : P →A[R] W) : (f - g).contLinear = f.contLinear - g.contLinear :=
rfl
#align continuous_affine_map.sub_cont_linear ContinuousAffineMap.sub_contLinear
@[simp]
theorem neg_contLinear (f : P →A[R] W) : (-f).contLinear = -f.contLinear :=
rfl
#align continuous_affine_map.neg_cont_linear ContinuousAffineMap.neg_contLinear
@[simp]
theorem smul_contLinear (t : R) (f : P →A[R] W) : (t • f).contLinear = t • f.contLinear :=
rfl
#align continuous_affine_map.smul_cont_linear ContinuousAffineMap.smul_contLinear
theorem decomp (f : V →A[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) := by
rcases f with ⟨f, h⟩
|
rw [coe_mk_const_linear_eq_linear, coe_mk, f.decomp, Pi.add_apply, LinearMap.map_zero, zero_add,
← Function.const_def]
|
theorem decomp (f : V →A[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) := by
rcases f with ⟨f, h⟩
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.153_0.bJ3guivW1IqbUMZ
|
theorem decomp (f : V →A[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0)
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : V →A[𝕜] W
h : f 0 = 0
⊢ ‖f‖ = max ‖f 0‖ ‖contLinear f‖
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h]; rfl
simp_rw [h₁, h₂]
exact (f : P →ᵃ[R] Q).linear_eq_zero_iff_exists_const
#align continuous_affine_map.cont_linear_eq_zero_iff_exists_const ContinuousAffineMap.contLinear_eq_zero_iff_exists_const
@[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f := by
ext
rfl
#align continuous_affine_map.to_affine_map_cont_linear ContinuousAffineMap.to_affine_map_contLinear
@[simp]
theorem zero_contLinear : (0 : P →A[R] W).contLinear = 0 :=
rfl
#align continuous_affine_map.zero_cont_linear ContinuousAffineMap.zero_contLinear
@[simp]
theorem add_contLinear (f g : P →A[R] W) : (f + g).contLinear = f.contLinear + g.contLinear :=
rfl
#align continuous_affine_map.add_cont_linear ContinuousAffineMap.add_contLinear
@[simp]
theorem sub_contLinear (f g : P →A[R] W) : (f - g).contLinear = f.contLinear - g.contLinear :=
rfl
#align continuous_affine_map.sub_cont_linear ContinuousAffineMap.sub_contLinear
@[simp]
theorem neg_contLinear (f : P →A[R] W) : (-f).contLinear = -f.contLinear :=
rfl
#align continuous_affine_map.neg_cont_linear ContinuousAffineMap.neg_contLinear
@[simp]
theorem smul_contLinear (t : R) (f : P →A[R] W) : (t • f).contLinear = t • f.contLinear :=
rfl
#align continuous_affine_map.smul_cont_linear ContinuousAffineMap.smul_contLinear
theorem decomp (f : V →A[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) := by
rcases f with ⟨f, h⟩
rw [coe_mk_const_linear_eq_linear, coe_mk, f.decomp, Pi.add_apply, LinearMap.map_zero, zero_add,
← Function.const_def]
#align continuous_affine_map.decomp ContinuousAffineMap.decomp
section NormedSpaceStructure
variable (f : V →A[𝕜] W)
/-- Note that unlike the operator norm for linear maps, this norm is _not_ submultiplicative:
we do _not_ necessarily have `‖f.comp g‖ ≤ ‖f‖ * ‖g‖`. See `norm_comp_le` for what we can say. -/
noncomputable instance hasNorm : Norm (V →A[𝕜] W) :=
⟨fun f => max ‖f 0‖ ‖f.contLinear‖⟩
#align continuous_affine_map.has_norm ContinuousAffineMap.hasNorm
theorem norm_def : ‖f‖ = max ‖f 0‖ ‖f.contLinear‖ :=
rfl
#align continuous_affine_map.norm_def ContinuousAffineMap.norm_def
theorem norm_contLinear_le : ‖f.contLinear‖ ≤ ‖f‖ :=
le_max_right _ _
#align continuous_affine_map.norm_cont_linear_le ContinuousAffineMap.norm_contLinear_le
theorem norm_image_zero_le : ‖f 0‖ ≤ ‖f‖ :=
le_max_left _ _
#align continuous_affine_map.norm_image_zero_le ContinuousAffineMap.norm_image_zero_le
@[simp]
theorem norm_eq (h : f 0 = 0) : ‖f‖ = ‖f.contLinear‖ :=
calc
‖f‖ = max ‖f 0‖ ‖f.contLinear‖ := by
|
rw [norm_def]
|
@[simp]
theorem norm_eq (h : f 0 = 0) : ‖f‖ = ‖f.contLinear‖ :=
calc
‖f‖ = max ‖f 0‖ ‖f.contLinear‖ := by
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.181_0.bJ3guivW1IqbUMZ
|
@[simp]
theorem norm_eq (h : f 0 = 0) : ‖f‖ = ‖f.contLinear‖
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : V →A[𝕜] W
h : f 0 = 0
⊢ max ‖f 0‖ ‖contLinear f‖ = max 0 ‖contLinear f‖
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h]; rfl
simp_rw [h₁, h₂]
exact (f : P →ᵃ[R] Q).linear_eq_zero_iff_exists_const
#align continuous_affine_map.cont_linear_eq_zero_iff_exists_const ContinuousAffineMap.contLinear_eq_zero_iff_exists_const
@[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f := by
ext
rfl
#align continuous_affine_map.to_affine_map_cont_linear ContinuousAffineMap.to_affine_map_contLinear
@[simp]
theorem zero_contLinear : (0 : P →A[R] W).contLinear = 0 :=
rfl
#align continuous_affine_map.zero_cont_linear ContinuousAffineMap.zero_contLinear
@[simp]
theorem add_contLinear (f g : P →A[R] W) : (f + g).contLinear = f.contLinear + g.contLinear :=
rfl
#align continuous_affine_map.add_cont_linear ContinuousAffineMap.add_contLinear
@[simp]
theorem sub_contLinear (f g : P →A[R] W) : (f - g).contLinear = f.contLinear - g.contLinear :=
rfl
#align continuous_affine_map.sub_cont_linear ContinuousAffineMap.sub_contLinear
@[simp]
theorem neg_contLinear (f : P →A[R] W) : (-f).contLinear = -f.contLinear :=
rfl
#align continuous_affine_map.neg_cont_linear ContinuousAffineMap.neg_contLinear
@[simp]
theorem smul_contLinear (t : R) (f : P →A[R] W) : (t • f).contLinear = t • f.contLinear :=
rfl
#align continuous_affine_map.smul_cont_linear ContinuousAffineMap.smul_contLinear
theorem decomp (f : V →A[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) := by
rcases f with ⟨f, h⟩
rw [coe_mk_const_linear_eq_linear, coe_mk, f.decomp, Pi.add_apply, LinearMap.map_zero, zero_add,
← Function.const_def]
#align continuous_affine_map.decomp ContinuousAffineMap.decomp
section NormedSpaceStructure
variable (f : V →A[𝕜] W)
/-- Note that unlike the operator norm for linear maps, this norm is _not_ submultiplicative:
we do _not_ necessarily have `‖f.comp g‖ ≤ ‖f‖ * ‖g‖`. See `norm_comp_le` for what we can say. -/
noncomputable instance hasNorm : Norm (V →A[𝕜] W) :=
⟨fun f => max ‖f 0‖ ‖f.contLinear‖⟩
#align continuous_affine_map.has_norm ContinuousAffineMap.hasNorm
theorem norm_def : ‖f‖ = max ‖f 0‖ ‖f.contLinear‖ :=
rfl
#align continuous_affine_map.norm_def ContinuousAffineMap.norm_def
theorem norm_contLinear_le : ‖f.contLinear‖ ≤ ‖f‖ :=
le_max_right _ _
#align continuous_affine_map.norm_cont_linear_le ContinuousAffineMap.norm_contLinear_le
theorem norm_image_zero_le : ‖f 0‖ ≤ ‖f‖ :=
le_max_left _ _
#align continuous_affine_map.norm_image_zero_le ContinuousAffineMap.norm_image_zero_le
@[simp]
theorem norm_eq (h : f 0 = 0) : ‖f‖ = ‖f.contLinear‖ :=
calc
‖f‖ = max ‖f 0‖ ‖f.contLinear‖ := by rw [norm_def]
_ = max 0 ‖f.contLinear‖ := by
|
rw [h, norm_zero]
|
@[simp]
theorem norm_eq (h : f 0 = 0) : ‖f‖ = ‖f.contLinear‖ :=
calc
‖f‖ = max ‖f 0‖ ‖f.contLinear‖ := by rw [norm_def]
_ = max 0 ‖f.contLinear‖ := by
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.181_0.bJ3guivW1IqbUMZ
|
@[simp]
theorem norm_eq (h : f 0 = 0) : ‖f‖ = ‖f.contLinear‖
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : V →A[𝕜] W
⊢ (fun f => max ‖f 0‖ ‖contLinear f‖) 0 = 0
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h]; rfl
simp_rw [h₁, h₂]
exact (f : P →ᵃ[R] Q).linear_eq_zero_iff_exists_const
#align continuous_affine_map.cont_linear_eq_zero_iff_exists_const ContinuousAffineMap.contLinear_eq_zero_iff_exists_const
@[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f := by
ext
rfl
#align continuous_affine_map.to_affine_map_cont_linear ContinuousAffineMap.to_affine_map_contLinear
@[simp]
theorem zero_contLinear : (0 : P →A[R] W).contLinear = 0 :=
rfl
#align continuous_affine_map.zero_cont_linear ContinuousAffineMap.zero_contLinear
@[simp]
theorem add_contLinear (f g : P →A[R] W) : (f + g).contLinear = f.contLinear + g.contLinear :=
rfl
#align continuous_affine_map.add_cont_linear ContinuousAffineMap.add_contLinear
@[simp]
theorem sub_contLinear (f g : P →A[R] W) : (f - g).contLinear = f.contLinear - g.contLinear :=
rfl
#align continuous_affine_map.sub_cont_linear ContinuousAffineMap.sub_contLinear
@[simp]
theorem neg_contLinear (f : P →A[R] W) : (-f).contLinear = -f.contLinear :=
rfl
#align continuous_affine_map.neg_cont_linear ContinuousAffineMap.neg_contLinear
@[simp]
theorem smul_contLinear (t : R) (f : P →A[R] W) : (t • f).contLinear = t • f.contLinear :=
rfl
#align continuous_affine_map.smul_cont_linear ContinuousAffineMap.smul_contLinear
theorem decomp (f : V →A[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) := by
rcases f with ⟨f, h⟩
rw [coe_mk_const_linear_eq_linear, coe_mk, f.decomp, Pi.add_apply, LinearMap.map_zero, zero_add,
← Function.const_def]
#align continuous_affine_map.decomp ContinuousAffineMap.decomp
section NormedSpaceStructure
variable (f : V →A[𝕜] W)
/-- Note that unlike the operator norm for linear maps, this norm is _not_ submultiplicative:
we do _not_ necessarily have `‖f.comp g‖ ≤ ‖f‖ * ‖g‖`. See `norm_comp_le` for what we can say. -/
noncomputable instance hasNorm : Norm (V →A[𝕜] W) :=
⟨fun f => max ‖f 0‖ ‖f.contLinear‖⟩
#align continuous_affine_map.has_norm ContinuousAffineMap.hasNorm
theorem norm_def : ‖f‖ = max ‖f 0‖ ‖f.contLinear‖ :=
rfl
#align continuous_affine_map.norm_def ContinuousAffineMap.norm_def
theorem norm_contLinear_le : ‖f.contLinear‖ ≤ ‖f‖ :=
le_max_right _ _
#align continuous_affine_map.norm_cont_linear_le ContinuousAffineMap.norm_contLinear_le
theorem norm_image_zero_le : ‖f 0‖ ≤ ‖f‖ :=
le_max_left _ _
#align continuous_affine_map.norm_image_zero_le ContinuousAffineMap.norm_image_zero_le
@[simp]
theorem norm_eq (h : f 0 = 0) : ‖f‖ = ‖f.contLinear‖ :=
calc
‖f‖ = max ‖f 0‖ ‖f.contLinear‖ := by rw [norm_def]
_ = max 0 ‖f.contLinear‖ := by rw [h, norm_zero]
_ = ‖f.contLinear‖ := max_eq_right (norm_nonneg _)
#align continuous_affine_map.norm_eq ContinuousAffineMap.norm_eq
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by
|
simp [(ContinuousAffineMap.zero_apply)]
|
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.190_0.bJ3guivW1IqbUMZ
|
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W)
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f✝ f g : V →A[𝕜] W
⊢ (fun f => max ‖f 0‖ ‖contLinear f‖) (f + g) ≤
(fun f => max ‖f 0‖ ‖contLinear f‖) f + (fun f => max ‖f 0‖ ‖contLinear f‖) g
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h]; rfl
simp_rw [h₁, h₂]
exact (f : P →ᵃ[R] Q).linear_eq_zero_iff_exists_const
#align continuous_affine_map.cont_linear_eq_zero_iff_exists_const ContinuousAffineMap.contLinear_eq_zero_iff_exists_const
@[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f := by
ext
rfl
#align continuous_affine_map.to_affine_map_cont_linear ContinuousAffineMap.to_affine_map_contLinear
@[simp]
theorem zero_contLinear : (0 : P →A[R] W).contLinear = 0 :=
rfl
#align continuous_affine_map.zero_cont_linear ContinuousAffineMap.zero_contLinear
@[simp]
theorem add_contLinear (f g : P →A[R] W) : (f + g).contLinear = f.contLinear + g.contLinear :=
rfl
#align continuous_affine_map.add_cont_linear ContinuousAffineMap.add_contLinear
@[simp]
theorem sub_contLinear (f g : P →A[R] W) : (f - g).contLinear = f.contLinear - g.contLinear :=
rfl
#align continuous_affine_map.sub_cont_linear ContinuousAffineMap.sub_contLinear
@[simp]
theorem neg_contLinear (f : P →A[R] W) : (-f).contLinear = -f.contLinear :=
rfl
#align continuous_affine_map.neg_cont_linear ContinuousAffineMap.neg_contLinear
@[simp]
theorem smul_contLinear (t : R) (f : P →A[R] W) : (t • f).contLinear = t • f.contLinear :=
rfl
#align continuous_affine_map.smul_cont_linear ContinuousAffineMap.smul_contLinear
theorem decomp (f : V →A[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) := by
rcases f with ⟨f, h⟩
rw [coe_mk_const_linear_eq_linear, coe_mk, f.decomp, Pi.add_apply, LinearMap.map_zero, zero_add,
← Function.const_def]
#align continuous_affine_map.decomp ContinuousAffineMap.decomp
section NormedSpaceStructure
variable (f : V →A[𝕜] W)
/-- Note that unlike the operator norm for linear maps, this norm is _not_ submultiplicative:
we do _not_ necessarily have `‖f.comp g‖ ≤ ‖f‖ * ‖g‖`. See `norm_comp_le` for what we can say. -/
noncomputable instance hasNorm : Norm (V →A[𝕜] W) :=
⟨fun f => max ‖f 0‖ ‖f.contLinear‖⟩
#align continuous_affine_map.has_norm ContinuousAffineMap.hasNorm
theorem norm_def : ‖f‖ = max ‖f 0‖ ‖f.contLinear‖ :=
rfl
#align continuous_affine_map.norm_def ContinuousAffineMap.norm_def
theorem norm_contLinear_le : ‖f.contLinear‖ ≤ ‖f‖ :=
le_max_right _ _
#align continuous_affine_map.norm_cont_linear_le ContinuousAffineMap.norm_contLinear_le
theorem norm_image_zero_le : ‖f 0‖ ≤ ‖f‖ :=
le_max_left _ _
#align continuous_affine_map.norm_image_zero_le ContinuousAffineMap.norm_image_zero_le
@[simp]
theorem norm_eq (h : f 0 = 0) : ‖f‖ = ‖f.contLinear‖ :=
calc
‖f‖ = max ‖f 0‖ ‖f.contLinear‖ := by rw [norm_def]
_ = max 0 ‖f.contLinear‖ := by rw [h, norm_zero]
_ = ‖f.contLinear‖ := max_eq_right (norm_nonneg _)
#align continuous_affine_map.norm_eq ContinuousAffineMap.norm_eq
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
|
simp only [coe_add, max_le_iff]
|
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.190_0.bJ3guivW1IqbUMZ
|
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W)
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f✝ f g : V →A[𝕜] W
⊢ ‖(⇑f + ⇑g) 0‖ ≤ max ‖f 0‖ ‖contLinear f‖ + max ‖g 0‖ ‖contLinear g‖ ∧
‖contLinear (f + g)‖ ≤ max ‖f 0‖ ‖contLinear f‖ + max ‖g 0‖ ‖contLinear g‖
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h]; rfl
simp_rw [h₁, h₂]
exact (f : P →ᵃ[R] Q).linear_eq_zero_iff_exists_const
#align continuous_affine_map.cont_linear_eq_zero_iff_exists_const ContinuousAffineMap.contLinear_eq_zero_iff_exists_const
@[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f := by
ext
rfl
#align continuous_affine_map.to_affine_map_cont_linear ContinuousAffineMap.to_affine_map_contLinear
@[simp]
theorem zero_contLinear : (0 : P →A[R] W).contLinear = 0 :=
rfl
#align continuous_affine_map.zero_cont_linear ContinuousAffineMap.zero_contLinear
@[simp]
theorem add_contLinear (f g : P →A[R] W) : (f + g).contLinear = f.contLinear + g.contLinear :=
rfl
#align continuous_affine_map.add_cont_linear ContinuousAffineMap.add_contLinear
@[simp]
theorem sub_contLinear (f g : P →A[R] W) : (f - g).contLinear = f.contLinear - g.contLinear :=
rfl
#align continuous_affine_map.sub_cont_linear ContinuousAffineMap.sub_contLinear
@[simp]
theorem neg_contLinear (f : P →A[R] W) : (-f).contLinear = -f.contLinear :=
rfl
#align continuous_affine_map.neg_cont_linear ContinuousAffineMap.neg_contLinear
@[simp]
theorem smul_contLinear (t : R) (f : P →A[R] W) : (t • f).contLinear = t • f.contLinear :=
rfl
#align continuous_affine_map.smul_cont_linear ContinuousAffineMap.smul_contLinear
theorem decomp (f : V →A[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) := by
rcases f with ⟨f, h⟩
rw [coe_mk_const_linear_eq_linear, coe_mk, f.decomp, Pi.add_apply, LinearMap.map_zero, zero_add,
← Function.const_def]
#align continuous_affine_map.decomp ContinuousAffineMap.decomp
section NormedSpaceStructure
variable (f : V →A[𝕜] W)
/-- Note that unlike the operator norm for linear maps, this norm is _not_ submultiplicative:
we do _not_ necessarily have `‖f.comp g‖ ≤ ‖f‖ * ‖g‖`. See `norm_comp_le` for what we can say. -/
noncomputable instance hasNorm : Norm (V →A[𝕜] W) :=
⟨fun f => max ‖f 0‖ ‖f.contLinear‖⟩
#align continuous_affine_map.has_norm ContinuousAffineMap.hasNorm
theorem norm_def : ‖f‖ = max ‖f 0‖ ‖f.contLinear‖ :=
rfl
#align continuous_affine_map.norm_def ContinuousAffineMap.norm_def
theorem norm_contLinear_le : ‖f.contLinear‖ ≤ ‖f‖ :=
le_max_right _ _
#align continuous_affine_map.norm_cont_linear_le ContinuousAffineMap.norm_contLinear_le
theorem norm_image_zero_le : ‖f 0‖ ≤ ‖f‖ :=
le_max_left _ _
#align continuous_affine_map.norm_image_zero_le ContinuousAffineMap.norm_image_zero_le
@[simp]
theorem norm_eq (h : f 0 = 0) : ‖f‖ = ‖f.contLinear‖ :=
calc
‖f‖ = max ‖f 0‖ ‖f.contLinear‖ := by rw [norm_def]
_ = max 0 ‖f.contLinear‖ := by rw [h, norm_zero]
_ = ‖f.contLinear‖ := max_eq_right (norm_nonneg _)
#align continuous_affine_map.norm_eq ContinuousAffineMap.norm_eq
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
|
rw [add_contLinear]
|
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.190_0.bJ3guivW1IqbUMZ
|
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W)
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f✝ f g : V →A[𝕜] W
⊢ ‖(⇑f + ⇑g) 0‖ ≤ max ‖f 0‖ ‖contLinear f‖ + max ‖g 0‖ ‖contLinear g‖ ∧
‖contLinear f + contLinear g‖ ≤ max ‖f 0‖ ‖contLinear f‖ + max ‖g 0‖ ‖contLinear g‖
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h]; rfl
simp_rw [h₁, h₂]
exact (f : P →ᵃ[R] Q).linear_eq_zero_iff_exists_const
#align continuous_affine_map.cont_linear_eq_zero_iff_exists_const ContinuousAffineMap.contLinear_eq_zero_iff_exists_const
@[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f := by
ext
rfl
#align continuous_affine_map.to_affine_map_cont_linear ContinuousAffineMap.to_affine_map_contLinear
@[simp]
theorem zero_contLinear : (0 : P →A[R] W).contLinear = 0 :=
rfl
#align continuous_affine_map.zero_cont_linear ContinuousAffineMap.zero_contLinear
@[simp]
theorem add_contLinear (f g : P →A[R] W) : (f + g).contLinear = f.contLinear + g.contLinear :=
rfl
#align continuous_affine_map.add_cont_linear ContinuousAffineMap.add_contLinear
@[simp]
theorem sub_contLinear (f g : P →A[R] W) : (f - g).contLinear = f.contLinear - g.contLinear :=
rfl
#align continuous_affine_map.sub_cont_linear ContinuousAffineMap.sub_contLinear
@[simp]
theorem neg_contLinear (f : P →A[R] W) : (-f).contLinear = -f.contLinear :=
rfl
#align continuous_affine_map.neg_cont_linear ContinuousAffineMap.neg_contLinear
@[simp]
theorem smul_contLinear (t : R) (f : P →A[R] W) : (t • f).contLinear = t • f.contLinear :=
rfl
#align continuous_affine_map.smul_cont_linear ContinuousAffineMap.smul_contLinear
theorem decomp (f : V →A[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) := by
rcases f with ⟨f, h⟩
rw [coe_mk_const_linear_eq_linear, coe_mk, f.decomp, Pi.add_apply, LinearMap.map_zero, zero_add,
← Function.const_def]
#align continuous_affine_map.decomp ContinuousAffineMap.decomp
section NormedSpaceStructure
variable (f : V →A[𝕜] W)
/-- Note that unlike the operator norm for linear maps, this norm is _not_ submultiplicative:
we do _not_ necessarily have `‖f.comp g‖ ≤ ‖f‖ * ‖g‖`. See `norm_comp_le` for what we can say. -/
noncomputable instance hasNorm : Norm (V →A[𝕜] W) :=
⟨fun f => max ‖f 0‖ ‖f.contLinear‖⟩
#align continuous_affine_map.has_norm ContinuousAffineMap.hasNorm
theorem norm_def : ‖f‖ = max ‖f 0‖ ‖f.contLinear‖ :=
rfl
#align continuous_affine_map.norm_def ContinuousAffineMap.norm_def
theorem norm_contLinear_le : ‖f.contLinear‖ ≤ ‖f‖ :=
le_max_right _ _
#align continuous_affine_map.norm_cont_linear_le ContinuousAffineMap.norm_contLinear_le
theorem norm_image_zero_le : ‖f 0‖ ≤ ‖f‖ :=
le_max_left _ _
#align continuous_affine_map.norm_image_zero_le ContinuousAffineMap.norm_image_zero_le
@[simp]
theorem norm_eq (h : f 0 = 0) : ‖f‖ = ‖f.contLinear‖ :=
calc
‖f‖ = max ‖f 0‖ ‖f.contLinear‖ := by rw [norm_def]
_ = max 0 ‖f.contLinear‖ := by rw [h, norm_zero]
_ = ‖f.contLinear‖ := max_eq_right (norm_nonneg _)
#align continuous_affine_map.norm_eq ContinuousAffineMap.norm_eq
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
rw [add_contLinear]
|
exact
⟨(norm_add_le _ _).trans (add_le_add (le_max_left _ _) (le_max_left _ _)),
(norm_add_le _ _).trans (add_le_add (le_max_right _ _) (le_max_right _ _))⟩
|
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
rw [add_contLinear]
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.190_0.bJ3guivW1IqbUMZ
|
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W)
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f✝ f : V →A[𝕜] W
⊢ (fun f => max ‖f 0‖ ‖contLinear f‖) (-f) = (fun f => max ‖f 0‖ ‖contLinear f‖) f
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h]; rfl
simp_rw [h₁, h₂]
exact (f : P →ᵃ[R] Q).linear_eq_zero_iff_exists_const
#align continuous_affine_map.cont_linear_eq_zero_iff_exists_const ContinuousAffineMap.contLinear_eq_zero_iff_exists_const
@[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f := by
ext
rfl
#align continuous_affine_map.to_affine_map_cont_linear ContinuousAffineMap.to_affine_map_contLinear
@[simp]
theorem zero_contLinear : (0 : P →A[R] W).contLinear = 0 :=
rfl
#align continuous_affine_map.zero_cont_linear ContinuousAffineMap.zero_contLinear
@[simp]
theorem add_contLinear (f g : P →A[R] W) : (f + g).contLinear = f.contLinear + g.contLinear :=
rfl
#align continuous_affine_map.add_cont_linear ContinuousAffineMap.add_contLinear
@[simp]
theorem sub_contLinear (f g : P →A[R] W) : (f - g).contLinear = f.contLinear - g.contLinear :=
rfl
#align continuous_affine_map.sub_cont_linear ContinuousAffineMap.sub_contLinear
@[simp]
theorem neg_contLinear (f : P →A[R] W) : (-f).contLinear = -f.contLinear :=
rfl
#align continuous_affine_map.neg_cont_linear ContinuousAffineMap.neg_contLinear
@[simp]
theorem smul_contLinear (t : R) (f : P →A[R] W) : (t • f).contLinear = t • f.contLinear :=
rfl
#align continuous_affine_map.smul_cont_linear ContinuousAffineMap.smul_contLinear
theorem decomp (f : V →A[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) := by
rcases f with ⟨f, h⟩
rw [coe_mk_const_linear_eq_linear, coe_mk, f.decomp, Pi.add_apply, LinearMap.map_zero, zero_add,
← Function.const_def]
#align continuous_affine_map.decomp ContinuousAffineMap.decomp
section NormedSpaceStructure
variable (f : V →A[𝕜] W)
/-- Note that unlike the operator norm for linear maps, this norm is _not_ submultiplicative:
we do _not_ necessarily have `‖f.comp g‖ ≤ ‖f‖ * ‖g‖`. See `norm_comp_le` for what we can say. -/
noncomputable instance hasNorm : Norm (V →A[𝕜] W) :=
⟨fun f => max ‖f 0‖ ‖f.contLinear‖⟩
#align continuous_affine_map.has_norm ContinuousAffineMap.hasNorm
theorem norm_def : ‖f‖ = max ‖f 0‖ ‖f.contLinear‖ :=
rfl
#align continuous_affine_map.norm_def ContinuousAffineMap.norm_def
theorem norm_contLinear_le : ‖f.contLinear‖ ≤ ‖f‖ :=
le_max_right _ _
#align continuous_affine_map.norm_cont_linear_le ContinuousAffineMap.norm_contLinear_le
theorem norm_image_zero_le : ‖f 0‖ ≤ ‖f‖ :=
le_max_left _ _
#align continuous_affine_map.norm_image_zero_le ContinuousAffineMap.norm_image_zero_le
@[simp]
theorem norm_eq (h : f 0 = 0) : ‖f‖ = ‖f.contLinear‖ :=
calc
‖f‖ = max ‖f 0‖ ‖f.contLinear‖ := by rw [norm_def]
_ = max 0 ‖f.contLinear‖ := by rw [h, norm_zero]
_ = ‖f.contLinear‖ := max_eq_right (norm_nonneg _)
#align continuous_affine_map.norm_eq ContinuousAffineMap.norm_eq
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
|
simp [(ContinuousAffineMap.neg_apply)]
|
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.190_0.bJ3guivW1IqbUMZ
|
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W)
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f✝ f : V →A[𝕜] W
h₀ :
AddGroupSeminorm.toFun
{ toFun := fun f => max ‖f 0‖ ‖contLinear f‖, map_zero' := (_ : max ‖0 0‖ ‖0‖ = 0),
add_le' :=
(_ :
∀ (f g : V →A[𝕜] W),
(fun f => max ‖f 0‖ ‖contLinear f‖) (f + g) ≤
(fun f => max ‖f 0‖ ‖contLinear f‖) f + (fun f => max ‖f 0‖ ‖contLinear f‖) g),
neg' := (_ : ∀ (f : V →A[𝕜] W), max ‖(-f) 0‖ ‖-contLinear f‖ = max ‖f 0‖ ‖contLinear f‖) }
f =
0
⊢ f = 0
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h]; rfl
simp_rw [h₁, h₂]
exact (f : P →ᵃ[R] Q).linear_eq_zero_iff_exists_const
#align continuous_affine_map.cont_linear_eq_zero_iff_exists_const ContinuousAffineMap.contLinear_eq_zero_iff_exists_const
@[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f := by
ext
rfl
#align continuous_affine_map.to_affine_map_cont_linear ContinuousAffineMap.to_affine_map_contLinear
@[simp]
theorem zero_contLinear : (0 : P →A[R] W).contLinear = 0 :=
rfl
#align continuous_affine_map.zero_cont_linear ContinuousAffineMap.zero_contLinear
@[simp]
theorem add_contLinear (f g : P →A[R] W) : (f + g).contLinear = f.contLinear + g.contLinear :=
rfl
#align continuous_affine_map.add_cont_linear ContinuousAffineMap.add_contLinear
@[simp]
theorem sub_contLinear (f g : P →A[R] W) : (f - g).contLinear = f.contLinear - g.contLinear :=
rfl
#align continuous_affine_map.sub_cont_linear ContinuousAffineMap.sub_contLinear
@[simp]
theorem neg_contLinear (f : P →A[R] W) : (-f).contLinear = -f.contLinear :=
rfl
#align continuous_affine_map.neg_cont_linear ContinuousAffineMap.neg_contLinear
@[simp]
theorem smul_contLinear (t : R) (f : P →A[R] W) : (t • f).contLinear = t • f.contLinear :=
rfl
#align continuous_affine_map.smul_cont_linear ContinuousAffineMap.smul_contLinear
theorem decomp (f : V →A[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) := by
rcases f with ⟨f, h⟩
rw [coe_mk_const_linear_eq_linear, coe_mk, f.decomp, Pi.add_apply, LinearMap.map_zero, zero_add,
← Function.const_def]
#align continuous_affine_map.decomp ContinuousAffineMap.decomp
section NormedSpaceStructure
variable (f : V →A[𝕜] W)
/-- Note that unlike the operator norm for linear maps, this norm is _not_ submultiplicative:
we do _not_ necessarily have `‖f.comp g‖ ≤ ‖f‖ * ‖g‖`. See `norm_comp_le` for what we can say. -/
noncomputable instance hasNorm : Norm (V →A[𝕜] W) :=
⟨fun f => max ‖f 0‖ ‖f.contLinear‖⟩
#align continuous_affine_map.has_norm ContinuousAffineMap.hasNorm
theorem norm_def : ‖f‖ = max ‖f 0‖ ‖f.contLinear‖ :=
rfl
#align continuous_affine_map.norm_def ContinuousAffineMap.norm_def
theorem norm_contLinear_le : ‖f.contLinear‖ ≤ ‖f‖ :=
le_max_right _ _
#align continuous_affine_map.norm_cont_linear_le ContinuousAffineMap.norm_contLinear_le
theorem norm_image_zero_le : ‖f 0‖ ≤ ‖f‖ :=
le_max_left _ _
#align continuous_affine_map.norm_image_zero_le ContinuousAffineMap.norm_image_zero_le
@[simp]
theorem norm_eq (h : f 0 = 0) : ‖f‖ = ‖f.contLinear‖ :=
calc
‖f‖ = max ‖f 0‖ ‖f.contLinear‖ := by rw [norm_def]
_ = max 0 ‖f.contLinear‖ := by rw [h, norm_zero]
_ = ‖f.contLinear‖ := max_eq_right (norm_nonneg _)
#align continuous_affine_map.norm_eq ContinuousAffineMap.norm_eq
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
rw [add_contLinear]
exact
⟨(norm_add_le _ _).trans (add_le_add (le_max_left _ _) (le_max_left _ _)),
(norm_add_le _ _).trans (add_le_add (le_max_right _ _) (le_max_right _ _))⟩
eq_zero_of_map_eq_zero' := fun f h₀ => by
|
rcases max_eq_iff.mp h₀ with (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩)
|
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
rw [add_contLinear]
exact
⟨(norm_add_le _ _).trans (add_le_add (le_max_left _ _) (le_max_left _ _)),
(norm_add_le _ _).trans (add_le_add (le_max_right _ _) (le_max_right _ _))⟩
eq_zero_of_map_eq_zero' := fun f h₀ => by
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.190_0.bJ3guivW1IqbUMZ
|
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W)
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
case inl.intro
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f✝ f : V →A[𝕜] W
h₀ :
AddGroupSeminorm.toFun
{ toFun := fun f => max ‖f 0‖ ‖contLinear f‖, map_zero' := (_ : max ‖0 0‖ ‖0‖ = 0),
add_le' :=
(_ :
∀ (f g : V →A[𝕜] W),
(fun f => max ‖f 0‖ ‖contLinear f‖) (f + g) ≤
(fun f => max ‖f 0‖ ‖contLinear f‖) f + (fun f => max ‖f 0‖ ‖contLinear f‖) g),
neg' := (_ : ∀ (f : V →A[𝕜] W), max ‖(-f) 0‖ ‖-contLinear f‖ = max ‖f 0‖ ‖contLinear f‖) }
f =
0
h₁ : ‖f 0‖ = 0
h₂ : ‖contLinear f‖ ≤ ‖f 0‖
⊢ f = 0
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h]; rfl
simp_rw [h₁, h₂]
exact (f : P →ᵃ[R] Q).linear_eq_zero_iff_exists_const
#align continuous_affine_map.cont_linear_eq_zero_iff_exists_const ContinuousAffineMap.contLinear_eq_zero_iff_exists_const
@[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f := by
ext
rfl
#align continuous_affine_map.to_affine_map_cont_linear ContinuousAffineMap.to_affine_map_contLinear
@[simp]
theorem zero_contLinear : (0 : P →A[R] W).contLinear = 0 :=
rfl
#align continuous_affine_map.zero_cont_linear ContinuousAffineMap.zero_contLinear
@[simp]
theorem add_contLinear (f g : P →A[R] W) : (f + g).contLinear = f.contLinear + g.contLinear :=
rfl
#align continuous_affine_map.add_cont_linear ContinuousAffineMap.add_contLinear
@[simp]
theorem sub_contLinear (f g : P →A[R] W) : (f - g).contLinear = f.contLinear - g.contLinear :=
rfl
#align continuous_affine_map.sub_cont_linear ContinuousAffineMap.sub_contLinear
@[simp]
theorem neg_contLinear (f : P →A[R] W) : (-f).contLinear = -f.contLinear :=
rfl
#align continuous_affine_map.neg_cont_linear ContinuousAffineMap.neg_contLinear
@[simp]
theorem smul_contLinear (t : R) (f : P →A[R] W) : (t • f).contLinear = t • f.contLinear :=
rfl
#align continuous_affine_map.smul_cont_linear ContinuousAffineMap.smul_contLinear
theorem decomp (f : V →A[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) := by
rcases f with ⟨f, h⟩
rw [coe_mk_const_linear_eq_linear, coe_mk, f.decomp, Pi.add_apply, LinearMap.map_zero, zero_add,
← Function.const_def]
#align continuous_affine_map.decomp ContinuousAffineMap.decomp
section NormedSpaceStructure
variable (f : V →A[𝕜] W)
/-- Note that unlike the operator norm for linear maps, this norm is _not_ submultiplicative:
we do _not_ necessarily have `‖f.comp g‖ ≤ ‖f‖ * ‖g‖`. See `norm_comp_le` for what we can say. -/
noncomputable instance hasNorm : Norm (V →A[𝕜] W) :=
⟨fun f => max ‖f 0‖ ‖f.contLinear‖⟩
#align continuous_affine_map.has_norm ContinuousAffineMap.hasNorm
theorem norm_def : ‖f‖ = max ‖f 0‖ ‖f.contLinear‖ :=
rfl
#align continuous_affine_map.norm_def ContinuousAffineMap.norm_def
theorem norm_contLinear_le : ‖f.contLinear‖ ≤ ‖f‖ :=
le_max_right _ _
#align continuous_affine_map.norm_cont_linear_le ContinuousAffineMap.norm_contLinear_le
theorem norm_image_zero_le : ‖f 0‖ ≤ ‖f‖ :=
le_max_left _ _
#align continuous_affine_map.norm_image_zero_le ContinuousAffineMap.norm_image_zero_le
@[simp]
theorem norm_eq (h : f 0 = 0) : ‖f‖ = ‖f.contLinear‖ :=
calc
‖f‖ = max ‖f 0‖ ‖f.contLinear‖ := by rw [norm_def]
_ = max 0 ‖f.contLinear‖ := by rw [h, norm_zero]
_ = ‖f.contLinear‖ := max_eq_right (norm_nonneg _)
#align continuous_affine_map.norm_eq ContinuousAffineMap.norm_eq
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
rw [add_contLinear]
exact
⟨(norm_add_le _ _).trans (add_le_add (le_max_left _ _) (le_max_left _ _)),
(norm_add_le _ _).trans (add_le_add (le_max_right _ _) (le_max_right _ _))⟩
eq_zero_of_map_eq_zero' := fun f h₀ => by
rcases max_eq_iff.mp h₀ with (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩) <;>
|
rw [h₁] at h₂
|
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
rw [add_contLinear]
exact
⟨(norm_add_le _ _).trans (add_le_add (le_max_left _ _) (le_max_left _ _)),
(norm_add_le _ _).trans (add_le_add (le_max_right _ _) (le_max_right _ _))⟩
eq_zero_of_map_eq_zero' := fun f h₀ => by
rcases max_eq_iff.mp h₀ with (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩) <;>
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.190_0.bJ3guivW1IqbUMZ
|
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W)
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
case inr.intro
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f✝ f : V →A[𝕜] W
h₀ :
AddGroupSeminorm.toFun
{ toFun := fun f => max ‖f 0‖ ‖contLinear f‖, map_zero' := (_ : max ‖0 0‖ ‖0‖ = 0),
add_le' :=
(_ :
∀ (f g : V →A[𝕜] W),
(fun f => max ‖f 0‖ ‖contLinear f‖) (f + g) ≤
(fun f => max ‖f 0‖ ‖contLinear f‖) f + (fun f => max ‖f 0‖ ‖contLinear f‖) g),
neg' := (_ : ∀ (f : V →A[𝕜] W), max ‖(-f) 0‖ ‖-contLinear f‖ = max ‖f 0‖ ‖contLinear f‖) }
f =
0
h₁ : ‖contLinear f‖ = 0
h₂ : ‖f 0‖ ≤ ‖contLinear f‖
⊢ f = 0
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h]; rfl
simp_rw [h₁, h₂]
exact (f : P →ᵃ[R] Q).linear_eq_zero_iff_exists_const
#align continuous_affine_map.cont_linear_eq_zero_iff_exists_const ContinuousAffineMap.contLinear_eq_zero_iff_exists_const
@[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f := by
ext
rfl
#align continuous_affine_map.to_affine_map_cont_linear ContinuousAffineMap.to_affine_map_contLinear
@[simp]
theorem zero_contLinear : (0 : P →A[R] W).contLinear = 0 :=
rfl
#align continuous_affine_map.zero_cont_linear ContinuousAffineMap.zero_contLinear
@[simp]
theorem add_contLinear (f g : P →A[R] W) : (f + g).contLinear = f.contLinear + g.contLinear :=
rfl
#align continuous_affine_map.add_cont_linear ContinuousAffineMap.add_contLinear
@[simp]
theorem sub_contLinear (f g : P →A[R] W) : (f - g).contLinear = f.contLinear - g.contLinear :=
rfl
#align continuous_affine_map.sub_cont_linear ContinuousAffineMap.sub_contLinear
@[simp]
theorem neg_contLinear (f : P →A[R] W) : (-f).contLinear = -f.contLinear :=
rfl
#align continuous_affine_map.neg_cont_linear ContinuousAffineMap.neg_contLinear
@[simp]
theorem smul_contLinear (t : R) (f : P →A[R] W) : (t • f).contLinear = t • f.contLinear :=
rfl
#align continuous_affine_map.smul_cont_linear ContinuousAffineMap.smul_contLinear
theorem decomp (f : V →A[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) := by
rcases f with ⟨f, h⟩
rw [coe_mk_const_linear_eq_linear, coe_mk, f.decomp, Pi.add_apply, LinearMap.map_zero, zero_add,
← Function.const_def]
#align continuous_affine_map.decomp ContinuousAffineMap.decomp
section NormedSpaceStructure
variable (f : V →A[𝕜] W)
/-- Note that unlike the operator norm for linear maps, this norm is _not_ submultiplicative:
we do _not_ necessarily have `‖f.comp g‖ ≤ ‖f‖ * ‖g‖`. See `norm_comp_le` for what we can say. -/
noncomputable instance hasNorm : Norm (V →A[𝕜] W) :=
⟨fun f => max ‖f 0‖ ‖f.contLinear‖⟩
#align continuous_affine_map.has_norm ContinuousAffineMap.hasNorm
theorem norm_def : ‖f‖ = max ‖f 0‖ ‖f.contLinear‖ :=
rfl
#align continuous_affine_map.norm_def ContinuousAffineMap.norm_def
theorem norm_contLinear_le : ‖f.contLinear‖ ≤ ‖f‖ :=
le_max_right _ _
#align continuous_affine_map.norm_cont_linear_le ContinuousAffineMap.norm_contLinear_le
theorem norm_image_zero_le : ‖f 0‖ ≤ ‖f‖ :=
le_max_left _ _
#align continuous_affine_map.norm_image_zero_le ContinuousAffineMap.norm_image_zero_le
@[simp]
theorem norm_eq (h : f 0 = 0) : ‖f‖ = ‖f.contLinear‖ :=
calc
‖f‖ = max ‖f 0‖ ‖f.contLinear‖ := by rw [norm_def]
_ = max 0 ‖f.contLinear‖ := by rw [h, norm_zero]
_ = ‖f.contLinear‖ := max_eq_right (norm_nonneg _)
#align continuous_affine_map.norm_eq ContinuousAffineMap.norm_eq
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
rw [add_contLinear]
exact
⟨(norm_add_le _ _).trans (add_le_add (le_max_left _ _) (le_max_left _ _)),
(norm_add_le _ _).trans (add_le_add (le_max_right _ _) (le_max_right _ _))⟩
eq_zero_of_map_eq_zero' := fun f h₀ => by
rcases max_eq_iff.mp h₀ with (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩) <;>
|
rw [h₁] at h₂
|
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
rw [add_contLinear]
exact
⟨(norm_add_le _ _).trans (add_le_add (le_max_left _ _) (le_max_left _ _)),
(norm_add_le _ _).trans (add_le_add (le_max_right _ _) (le_max_right _ _))⟩
eq_zero_of_map_eq_zero' := fun f h₀ => by
rcases max_eq_iff.mp h₀ with (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩) <;>
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.190_0.bJ3guivW1IqbUMZ
|
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W)
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
case inl.intro
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f✝ f : V →A[𝕜] W
h₀ :
AddGroupSeminorm.toFun
{ toFun := fun f => max ‖f 0‖ ‖contLinear f‖, map_zero' := (_ : max ‖0 0‖ ‖0‖ = 0),
add_le' :=
(_ :
∀ (f g : V →A[𝕜] W),
(fun f => max ‖f 0‖ ‖contLinear f‖) (f + g) ≤
(fun f => max ‖f 0‖ ‖contLinear f‖) f + (fun f => max ‖f 0‖ ‖contLinear f‖) g),
neg' := (_ : ∀ (f : V →A[𝕜] W), max ‖(-f) 0‖ ‖-contLinear f‖ = max ‖f 0‖ ‖contLinear f‖) }
f =
0
h₁ : ‖f 0‖ = 0
h₂ : ‖contLinear f‖ ≤ 0
⊢ f = 0
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h]; rfl
simp_rw [h₁, h₂]
exact (f : P →ᵃ[R] Q).linear_eq_zero_iff_exists_const
#align continuous_affine_map.cont_linear_eq_zero_iff_exists_const ContinuousAffineMap.contLinear_eq_zero_iff_exists_const
@[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f := by
ext
rfl
#align continuous_affine_map.to_affine_map_cont_linear ContinuousAffineMap.to_affine_map_contLinear
@[simp]
theorem zero_contLinear : (0 : P →A[R] W).contLinear = 0 :=
rfl
#align continuous_affine_map.zero_cont_linear ContinuousAffineMap.zero_contLinear
@[simp]
theorem add_contLinear (f g : P →A[R] W) : (f + g).contLinear = f.contLinear + g.contLinear :=
rfl
#align continuous_affine_map.add_cont_linear ContinuousAffineMap.add_contLinear
@[simp]
theorem sub_contLinear (f g : P →A[R] W) : (f - g).contLinear = f.contLinear - g.contLinear :=
rfl
#align continuous_affine_map.sub_cont_linear ContinuousAffineMap.sub_contLinear
@[simp]
theorem neg_contLinear (f : P →A[R] W) : (-f).contLinear = -f.contLinear :=
rfl
#align continuous_affine_map.neg_cont_linear ContinuousAffineMap.neg_contLinear
@[simp]
theorem smul_contLinear (t : R) (f : P →A[R] W) : (t • f).contLinear = t • f.contLinear :=
rfl
#align continuous_affine_map.smul_cont_linear ContinuousAffineMap.smul_contLinear
theorem decomp (f : V →A[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) := by
rcases f with ⟨f, h⟩
rw [coe_mk_const_linear_eq_linear, coe_mk, f.decomp, Pi.add_apply, LinearMap.map_zero, zero_add,
← Function.const_def]
#align continuous_affine_map.decomp ContinuousAffineMap.decomp
section NormedSpaceStructure
variable (f : V →A[𝕜] W)
/-- Note that unlike the operator norm for linear maps, this norm is _not_ submultiplicative:
we do _not_ necessarily have `‖f.comp g‖ ≤ ‖f‖ * ‖g‖`. See `norm_comp_le` for what we can say. -/
noncomputable instance hasNorm : Norm (V →A[𝕜] W) :=
⟨fun f => max ‖f 0‖ ‖f.contLinear‖⟩
#align continuous_affine_map.has_norm ContinuousAffineMap.hasNorm
theorem norm_def : ‖f‖ = max ‖f 0‖ ‖f.contLinear‖ :=
rfl
#align continuous_affine_map.norm_def ContinuousAffineMap.norm_def
theorem norm_contLinear_le : ‖f.contLinear‖ ≤ ‖f‖ :=
le_max_right _ _
#align continuous_affine_map.norm_cont_linear_le ContinuousAffineMap.norm_contLinear_le
theorem norm_image_zero_le : ‖f 0‖ ≤ ‖f‖ :=
le_max_left _ _
#align continuous_affine_map.norm_image_zero_le ContinuousAffineMap.norm_image_zero_le
@[simp]
theorem norm_eq (h : f 0 = 0) : ‖f‖ = ‖f.contLinear‖ :=
calc
‖f‖ = max ‖f 0‖ ‖f.contLinear‖ := by rw [norm_def]
_ = max 0 ‖f.contLinear‖ := by rw [h, norm_zero]
_ = ‖f.contLinear‖ := max_eq_right (norm_nonneg _)
#align continuous_affine_map.norm_eq ContinuousAffineMap.norm_eq
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
rw [add_contLinear]
exact
⟨(norm_add_le _ _).trans (add_le_add (le_max_left _ _) (le_max_left _ _)),
(norm_add_le _ _).trans (add_le_add (le_max_right _ _) (le_max_right _ _))⟩
eq_zero_of_map_eq_zero' := fun f h₀ => by
rcases max_eq_iff.mp h₀ with (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩) <;> rw [h₁] at h₂
·
|
rw [norm_le_zero_iff, contLinear_eq_zero_iff_exists_const] at h₂
|
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
rw [add_contLinear]
exact
⟨(norm_add_le _ _).trans (add_le_add (le_max_left _ _) (le_max_left _ _)),
(norm_add_le _ _).trans (add_le_add (le_max_right _ _) (le_max_right _ _))⟩
eq_zero_of_map_eq_zero' := fun f h₀ => by
rcases max_eq_iff.mp h₀ with (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩) <;> rw [h₁] at h₂
·
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.190_0.bJ3guivW1IqbUMZ
|
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W)
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
case inl.intro
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f✝ f : V →A[𝕜] W
h₀ :
AddGroupSeminorm.toFun
{ toFun := fun f => max ‖f 0‖ ‖contLinear f‖, map_zero' := (_ : max ‖0 0‖ ‖0‖ = 0),
add_le' :=
(_ :
∀ (f g : V →A[𝕜] W),
(fun f => max ‖f 0‖ ‖contLinear f‖) (f + g) ≤
(fun f => max ‖f 0‖ ‖contLinear f‖) f + (fun f => max ‖f 0‖ ‖contLinear f‖) g),
neg' := (_ : ∀ (f : V →A[𝕜] W), max ‖(-f) 0‖ ‖-contLinear f‖ = max ‖f 0‖ ‖contLinear f‖) }
f =
0
h₁ : ‖f 0‖ = 0
h₂ : ∃ q, f = const 𝕜 V q
⊢ f = 0
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h]; rfl
simp_rw [h₁, h₂]
exact (f : P →ᵃ[R] Q).linear_eq_zero_iff_exists_const
#align continuous_affine_map.cont_linear_eq_zero_iff_exists_const ContinuousAffineMap.contLinear_eq_zero_iff_exists_const
@[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f := by
ext
rfl
#align continuous_affine_map.to_affine_map_cont_linear ContinuousAffineMap.to_affine_map_contLinear
@[simp]
theorem zero_contLinear : (0 : P →A[R] W).contLinear = 0 :=
rfl
#align continuous_affine_map.zero_cont_linear ContinuousAffineMap.zero_contLinear
@[simp]
theorem add_contLinear (f g : P →A[R] W) : (f + g).contLinear = f.contLinear + g.contLinear :=
rfl
#align continuous_affine_map.add_cont_linear ContinuousAffineMap.add_contLinear
@[simp]
theorem sub_contLinear (f g : P →A[R] W) : (f - g).contLinear = f.contLinear - g.contLinear :=
rfl
#align continuous_affine_map.sub_cont_linear ContinuousAffineMap.sub_contLinear
@[simp]
theorem neg_contLinear (f : P →A[R] W) : (-f).contLinear = -f.contLinear :=
rfl
#align continuous_affine_map.neg_cont_linear ContinuousAffineMap.neg_contLinear
@[simp]
theorem smul_contLinear (t : R) (f : P →A[R] W) : (t • f).contLinear = t • f.contLinear :=
rfl
#align continuous_affine_map.smul_cont_linear ContinuousAffineMap.smul_contLinear
theorem decomp (f : V →A[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) := by
rcases f with ⟨f, h⟩
rw [coe_mk_const_linear_eq_linear, coe_mk, f.decomp, Pi.add_apply, LinearMap.map_zero, zero_add,
← Function.const_def]
#align continuous_affine_map.decomp ContinuousAffineMap.decomp
section NormedSpaceStructure
variable (f : V →A[𝕜] W)
/-- Note that unlike the operator norm for linear maps, this norm is _not_ submultiplicative:
we do _not_ necessarily have `‖f.comp g‖ ≤ ‖f‖ * ‖g‖`. See `norm_comp_le` for what we can say. -/
noncomputable instance hasNorm : Norm (V →A[𝕜] W) :=
⟨fun f => max ‖f 0‖ ‖f.contLinear‖⟩
#align continuous_affine_map.has_norm ContinuousAffineMap.hasNorm
theorem norm_def : ‖f‖ = max ‖f 0‖ ‖f.contLinear‖ :=
rfl
#align continuous_affine_map.norm_def ContinuousAffineMap.norm_def
theorem norm_contLinear_le : ‖f.contLinear‖ ≤ ‖f‖ :=
le_max_right _ _
#align continuous_affine_map.norm_cont_linear_le ContinuousAffineMap.norm_contLinear_le
theorem norm_image_zero_le : ‖f 0‖ ≤ ‖f‖ :=
le_max_left _ _
#align continuous_affine_map.norm_image_zero_le ContinuousAffineMap.norm_image_zero_le
@[simp]
theorem norm_eq (h : f 0 = 0) : ‖f‖ = ‖f.contLinear‖ :=
calc
‖f‖ = max ‖f 0‖ ‖f.contLinear‖ := by rw [norm_def]
_ = max 0 ‖f.contLinear‖ := by rw [h, norm_zero]
_ = ‖f.contLinear‖ := max_eq_right (norm_nonneg _)
#align continuous_affine_map.norm_eq ContinuousAffineMap.norm_eq
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
rw [add_contLinear]
exact
⟨(norm_add_le _ _).trans (add_le_add (le_max_left _ _) (le_max_left _ _)),
(norm_add_le _ _).trans (add_le_add (le_max_right _ _) (le_max_right _ _))⟩
eq_zero_of_map_eq_zero' := fun f h₀ => by
rcases max_eq_iff.mp h₀ with (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩) <;> rw [h₁] at h₂
· rw [norm_le_zero_iff, contLinear_eq_zero_iff_exists_const] at h₂
|
obtain ⟨q, rfl⟩ := h₂
|
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
rw [add_contLinear]
exact
⟨(norm_add_le _ _).trans (add_le_add (le_max_left _ _) (le_max_left _ _)),
(norm_add_le _ _).trans (add_le_add (le_max_right _ _) (le_max_right _ _))⟩
eq_zero_of_map_eq_zero' := fun f h₀ => by
rcases max_eq_iff.mp h₀ with (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩) <;> rw [h₁] at h₂
· rw [norm_le_zero_iff, contLinear_eq_zero_iff_exists_const] at h₂
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.190_0.bJ3guivW1IqbUMZ
|
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W)
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
case inl.intro.intro
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : V →A[𝕜] W
q : W
h₀ :
AddGroupSeminorm.toFun
{ toFun := fun f => max ‖f 0‖ ‖contLinear f‖, map_zero' := (_ : max ‖0 0‖ ‖0‖ = 0),
add_le' :=
(_ :
∀ (f g : V →A[𝕜] W),
(fun f => max ‖f 0‖ ‖contLinear f‖) (f + g) ≤
(fun f => max ‖f 0‖ ‖contLinear f‖) f + (fun f => max ‖f 0‖ ‖contLinear f‖) g),
neg' := (_ : ∀ (f : V →A[𝕜] W), max ‖(-f) 0‖ ‖-contLinear f‖ = max ‖f 0‖ ‖contLinear f‖) }
(const 𝕜 V q) =
0
h₁ : ‖(const 𝕜 V q) 0‖ = 0
⊢ const 𝕜 V q = 0
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h]; rfl
simp_rw [h₁, h₂]
exact (f : P →ᵃ[R] Q).linear_eq_zero_iff_exists_const
#align continuous_affine_map.cont_linear_eq_zero_iff_exists_const ContinuousAffineMap.contLinear_eq_zero_iff_exists_const
@[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f := by
ext
rfl
#align continuous_affine_map.to_affine_map_cont_linear ContinuousAffineMap.to_affine_map_contLinear
@[simp]
theorem zero_contLinear : (0 : P →A[R] W).contLinear = 0 :=
rfl
#align continuous_affine_map.zero_cont_linear ContinuousAffineMap.zero_contLinear
@[simp]
theorem add_contLinear (f g : P →A[R] W) : (f + g).contLinear = f.contLinear + g.contLinear :=
rfl
#align continuous_affine_map.add_cont_linear ContinuousAffineMap.add_contLinear
@[simp]
theorem sub_contLinear (f g : P →A[R] W) : (f - g).contLinear = f.contLinear - g.contLinear :=
rfl
#align continuous_affine_map.sub_cont_linear ContinuousAffineMap.sub_contLinear
@[simp]
theorem neg_contLinear (f : P →A[R] W) : (-f).contLinear = -f.contLinear :=
rfl
#align continuous_affine_map.neg_cont_linear ContinuousAffineMap.neg_contLinear
@[simp]
theorem smul_contLinear (t : R) (f : P →A[R] W) : (t • f).contLinear = t • f.contLinear :=
rfl
#align continuous_affine_map.smul_cont_linear ContinuousAffineMap.smul_contLinear
theorem decomp (f : V →A[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) := by
rcases f with ⟨f, h⟩
rw [coe_mk_const_linear_eq_linear, coe_mk, f.decomp, Pi.add_apply, LinearMap.map_zero, zero_add,
← Function.const_def]
#align continuous_affine_map.decomp ContinuousAffineMap.decomp
section NormedSpaceStructure
variable (f : V →A[𝕜] W)
/-- Note that unlike the operator norm for linear maps, this norm is _not_ submultiplicative:
we do _not_ necessarily have `‖f.comp g‖ ≤ ‖f‖ * ‖g‖`. See `norm_comp_le` for what we can say. -/
noncomputable instance hasNorm : Norm (V →A[𝕜] W) :=
⟨fun f => max ‖f 0‖ ‖f.contLinear‖⟩
#align continuous_affine_map.has_norm ContinuousAffineMap.hasNorm
theorem norm_def : ‖f‖ = max ‖f 0‖ ‖f.contLinear‖ :=
rfl
#align continuous_affine_map.norm_def ContinuousAffineMap.norm_def
theorem norm_contLinear_le : ‖f.contLinear‖ ≤ ‖f‖ :=
le_max_right _ _
#align continuous_affine_map.norm_cont_linear_le ContinuousAffineMap.norm_contLinear_le
theorem norm_image_zero_le : ‖f 0‖ ≤ ‖f‖ :=
le_max_left _ _
#align continuous_affine_map.norm_image_zero_le ContinuousAffineMap.norm_image_zero_le
@[simp]
theorem norm_eq (h : f 0 = 0) : ‖f‖ = ‖f.contLinear‖ :=
calc
‖f‖ = max ‖f 0‖ ‖f.contLinear‖ := by rw [norm_def]
_ = max 0 ‖f.contLinear‖ := by rw [h, norm_zero]
_ = ‖f.contLinear‖ := max_eq_right (norm_nonneg _)
#align continuous_affine_map.norm_eq ContinuousAffineMap.norm_eq
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
rw [add_contLinear]
exact
⟨(norm_add_le _ _).trans (add_le_add (le_max_left _ _) (le_max_left _ _)),
(norm_add_le _ _).trans (add_le_add (le_max_right _ _) (le_max_right _ _))⟩
eq_zero_of_map_eq_zero' := fun f h₀ => by
rcases max_eq_iff.mp h₀ with (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩) <;> rw [h₁] at h₂
· rw [norm_le_zero_iff, contLinear_eq_zero_iff_exists_const] at h₂
obtain ⟨q, rfl⟩ := h₂
|
simp only [norm_eq_zero] at h₁
|
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
rw [add_contLinear]
exact
⟨(norm_add_le _ _).trans (add_le_add (le_max_left _ _) (le_max_left _ _)),
(norm_add_le _ _).trans (add_le_add (le_max_right _ _) (le_max_right _ _))⟩
eq_zero_of_map_eq_zero' := fun f h₀ => by
rcases max_eq_iff.mp h₀ with (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩) <;> rw [h₁] at h₂
· rw [norm_le_zero_iff, contLinear_eq_zero_iff_exists_const] at h₂
obtain ⟨q, rfl⟩ := h₂
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.190_0.bJ3guivW1IqbUMZ
|
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W)
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
case inl.intro.intro
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : V →A[𝕜] W
q : W
h₀ :
AddGroupSeminorm.toFun
{ toFun := fun f => max ‖f 0‖ ‖contLinear f‖, map_zero' := (_ : max ‖0 0‖ ‖0‖ = 0),
add_le' :=
(_ :
∀ (f g : V →A[𝕜] W),
(fun f => max ‖f 0‖ ‖contLinear f‖) (f + g) ≤
(fun f => max ‖f 0‖ ‖contLinear f‖) f + (fun f => max ‖f 0‖ ‖contLinear f‖) g),
neg' := (_ : ∀ (f : V →A[𝕜] W), max ‖(-f) 0‖ ‖-contLinear f‖ = max ‖f 0‖ ‖contLinear f‖) }
(const 𝕜 V q) =
0
h₁ : (const 𝕜 V q) 0 = 0
⊢ const 𝕜 V q = 0
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h]; rfl
simp_rw [h₁, h₂]
exact (f : P →ᵃ[R] Q).linear_eq_zero_iff_exists_const
#align continuous_affine_map.cont_linear_eq_zero_iff_exists_const ContinuousAffineMap.contLinear_eq_zero_iff_exists_const
@[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f := by
ext
rfl
#align continuous_affine_map.to_affine_map_cont_linear ContinuousAffineMap.to_affine_map_contLinear
@[simp]
theorem zero_contLinear : (0 : P →A[R] W).contLinear = 0 :=
rfl
#align continuous_affine_map.zero_cont_linear ContinuousAffineMap.zero_contLinear
@[simp]
theorem add_contLinear (f g : P →A[R] W) : (f + g).contLinear = f.contLinear + g.contLinear :=
rfl
#align continuous_affine_map.add_cont_linear ContinuousAffineMap.add_contLinear
@[simp]
theorem sub_contLinear (f g : P →A[R] W) : (f - g).contLinear = f.contLinear - g.contLinear :=
rfl
#align continuous_affine_map.sub_cont_linear ContinuousAffineMap.sub_contLinear
@[simp]
theorem neg_contLinear (f : P →A[R] W) : (-f).contLinear = -f.contLinear :=
rfl
#align continuous_affine_map.neg_cont_linear ContinuousAffineMap.neg_contLinear
@[simp]
theorem smul_contLinear (t : R) (f : P →A[R] W) : (t • f).contLinear = t • f.contLinear :=
rfl
#align continuous_affine_map.smul_cont_linear ContinuousAffineMap.smul_contLinear
theorem decomp (f : V →A[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) := by
rcases f with ⟨f, h⟩
rw [coe_mk_const_linear_eq_linear, coe_mk, f.decomp, Pi.add_apply, LinearMap.map_zero, zero_add,
← Function.const_def]
#align continuous_affine_map.decomp ContinuousAffineMap.decomp
section NormedSpaceStructure
variable (f : V →A[𝕜] W)
/-- Note that unlike the operator norm for linear maps, this norm is _not_ submultiplicative:
we do _not_ necessarily have `‖f.comp g‖ ≤ ‖f‖ * ‖g‖`. See `norm_comp_le` for what we can say. -/
noncomputable instance hasNorm : Norm (V →A[𝕜] W) :=
⟨fun f => max ‖f 0‖ ‖f.contLinear‖⟩
#align continuous_affine_map.has_norm ContinuousAffineMap.hasNorm
theorem norm_def : ‖f‖ = max ‖f 0‖ ‖f.contLinear‖ :=
rfl
#align continuous_affine_map.norm_def ContinuousAffineMap.norm_def
theorem norm_contLinear_le : ‖f.contLinear‖ ≤ ‖f‖ :=
le_max_right _ _
#align continuous_affine_map.norm_cont_linear_le ContinuousAffineMap.norm_contLinear_le
theorem norm_image_zero_le : ‖f 0‖ ≤ ‖f‖ :=
le_max_left _ _
#align continuous_affine_map.norm_image_zero_le ContinuousAffineMap.norm_image_zero_le
@[simp]
theorem norm_eq (h : f 0 = 0) : ‖f‖ = ‖f.contLinear‖ :=
calc
‖f‖ = max ‖f 0‖ ‖f.contLinear‖ := by rw [norm_def]
_ = max 0 ‖f.contLinear‖ := by rw [h, norm_zero]
_ = ‖f.contLinear‖ := max_eq_right (norm_nonneg _)
#align continuous_affine_map.norm_eq ContinuousAffineMap.norm_eq
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
rw [add_contLinear]
exact
⟨(norm_add_le _ _).trans (add_le_add (le_max_left _ _) (le_max_left _ _)),
(norm_add_le _ _).trans (add_le_add (le_max_right _ _) (le_max_right _ _))⟩
eq_zero_of_map_eq_zero' := fun f h₀ => by
rcases max_eq_iff.mp h₀ with (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩) <;> rw [h₁] at h₂
· rw [norm_le_zero_iff, contLinear_eq_zero_iff_exists_const] at h₂
obtain ⟨q, rfl⟩ := h₂
simp only [norm_eq_zero] at h₁
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
|
rw [coe_const, Function.const_apply] at h₁
|
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
rw [add_contLinear]
exact
⟨(norm_add_le _ _).trans (add_le_add (le_max_left _ _) (le_max_left _ _)),
(norm_add_le _ _).trans (add_le_add (le_max_right _ _) (le_max_right _ _))⟩
eq_zero_of_map_eq_zero' := fun f h₀ => by
rcases max_eq_iff.mp h₀ with (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩) <;> rw [h₁] at h₂
· rw [norm_le_zero_iff, contLinear_eq_zero_iff_exists_const] at h₂
obtain ⟨q, rfl⟩ := h₂
simp only [norm_eq_zero] at h₁
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.190_0.bJ3guivW1IqbUMZ
|
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W)
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
case inl.intro.intro
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : V →A[𝕜] W
q : W
h₀ :
AddGroupSeminorm.toFun
{ toFun := fun f => max ‖f 0‖ ‖contLinear f‖, map_zero' := (_ : max ‖0 0‖ ‖0‖ = 0),
add_le' :=
(_ :
∀ (f g : V →A[𝕜] W),
(fun f => max ‖f 0‖ ‖contLinear f‖) (f + g) ≤
(fun f => max ‖f 0‖ ‖contLinear f‖) f + (fun f => max ‖f 0‖ ‖contLinear f‖) g),
neg' := (_ : ∀ (f : V →A[𝕜] W), max ‖(-f) 0‖ ‖-contLinear f‖ = max ‖f 0‖ ‖contLinear f‖) }
(const 𝕜 V q) =
0
h₁ : q = 0
⊢ const 𝕜 V q = 0
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h]; rfl
simp_rw [h₁, h₂]
exact (f : P →ᵃ[R] Q).linear_eq_zero_iff_exists_const
#align continuous_affine_map.cont_linear_eq_zero_iff_exists_const ContinuousAffineMap.contLinear_eq_zero_iff_exists_const
@[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f := by
ext
rfl
#align continuous_affine_map.to_affine_map_cont_linear ContinuousAffineMap.to_affine_map_contLinear
@[simp]
theorem zero_contLinear : (0 : P →A[R] W).contLinear = 0 :=
rfl
#align continuous_affine_map.zero_cont_linear ContinuousAffineMap.zero_contLinear
@[simp]
theorem add_contLinear (f g : P →A[R] W) : (f + g).contLinear = f.contLinear + g.contLinear :=
rfl
#align continuous_affine_map.add_cont_linear ContinuousAffineMap.add_contLinear
@[simp]
theorem sub_contLinear (f g : P →A[R] W) : (f - g).contLinear = f.contLinear - g.contLinear :=
rfl
#align continuous_affine_map.sub_cont_linear ContinuousAffineMap.sub_contLinear
@[simp]
theorem neg_contLinear (f : P →A[R] W) : (-f).contLinear = -f.contLinear :=
rfl
#align continuous_affine_map.neg_cont_linear ContinuousAffineMap.neg_contLinear
@[simp]
theorem smul_contLinear (t : R) (f : P →A[R] W) : (t • f).contLinear = t • f.contLinear :=
rfl
#align continuous_affine_map.smul_cont_linear ContinuousAffineMap.smul_contLinear
theorem decomp (f : V →A[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) := by
rcases f with ⟨f, h⟩
rw [coe_mk_const_linear_eq_linear, coe_mk, f.decomp, Pi.add_apply, LinearMap.map_zero, zero_add,
← Function.const_def]
#align continuous_affine_map.decomp ContinuousAffineMap.decomp
section NormedSpaceStructure
variable (f : V →A[𝕜] W)
/-- Note that unlike the operator norm for linear maps, this norm is _not_ submultiplicative:
we do _not_ necessarily have `‖f.comp g‖ ≤ ‖f‖ * ‖g‖`. See `norm_comp_le` for what we can say. -/
noncomputable instance hasNorm : Norm (V →A[𝕜] W) :=
⟨fun f => max ‖f 0‖ ‖f.contLinear‖⟩
#align continuous_affine_map.has_norm ContinuousAffineMap.hasNorm
theorem norm_def : ‖f‖ = max ‖f 0‖ ‖f.contLinear‖ :=
rfl
#align continuous_affine_map.norm_def ContinuousAffineMap.norm_def
theorem norm_contLinear_le : ‖f.contLinear‖ ≤ ‖f‖ :=
le_max_right _ _
#align continuous_affine_map.norm_cont_linear_le ContinuousAffineMap.norm_contLinear_le
theorem norm_image_zero_le : ‖f 0‖ ≤ ‖f‖ :=
le_max_left _ _
#align continuous_affine_map.norm_image_zero_le ContinuousAffineMap.norm_image_zero_le
@[simp]
theorem norm_eq (h : f 0 = 0) : ‖f‖ = ‖f.contLinear‖ :=
calc
‖f‖ = max ‖f 0‖ ‖f.contLinear‖ := by rw [norm_def]
_ = max 0 ‖f.contLinear‖ := by rw [h, norm_zero]
_ = ‖f.contLinear‖ := max_eq_right (norm_nonneg _)
#align continuous_affine_map.norm_eq ContinuousAffineMap.norm_eq
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
rw [add_contLinear]
exact
⟨(norm_add_le _ _).trans (add_le_add (le_max_left _ _) (le_max_left _ _)),
(norm_add_le _ _).trans (add_le_add (le_max_right _ _) (le_max_right _ _))⟩
eq_zero_of_map_eq_zero' := fun f h₀ => by
rcases max_eq_iff.mp h₀ with (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩) <;> rw [h₁] at h₂
· rw [norm_le_zero_iff, contLinear_eq_zero_iff_exists_const] at h₂
obtain ⟨q, rfl⟩ := h₂
simp only [norm_eq_zero] at h₁
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₁
|
rw [h₁]
|
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
rw [add_contLinear]
exact
⟨(norm_add_le _ _).trans (add_le_add (le_max_left _ _) (le_max_left _ _)),
(norm_add_le _ _).trans (add_le_add (le_max_right _ _) (le_max_right _ _))⟩
eq_zero_of_map_eq_zero' := fun f h₀ => by
rcases max_eq_iff.mp h₀ with (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩) <;> rw [h₁] at h₂
· rw [norm_le_zero_iff, contLinear_eq_zero_iff_exists_const] at h₂
obtain ⟨q, rfl⟩ := h₂
simp only [norm_eq_zero] at h₁
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₁
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.190_0.bJ3guivW1IqbUMZ
|
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W)
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
case inl.intro.intro
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : V →A[𝕜] W
q : W
h₀ :
AddGroupSeminorm.toFun
{ toFun := fun f => max ‖f 0‖ ‖contLinear f‖, map_zero' := (_ : max ‖0 0‖ ‖0‖ = 0),
add_le' :=
(_ :
∀ (f g : V →A[𝕜] W),
(fun f => max ‖f 0‖ ‖contLinear f‖) (f + g) ≤
(fun f => max ‖f 0‖ ‖contLinear f‖) f + (fun f => max ‖f 0‖ ‖contLinear f‖) g),
neg' := (_ : ∀ (f : V →A[𝕜] W), max ‖(-f) 0‖ ‖-contLinear f‖ = max ‖f 0‖ ‖contLinear f‖) }
(const 𝕜 V q) =
0
h₁ : q = 0
⊢ const 𝕜 V 0 = 0
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h]; rfl
simp_rw [h₁, h₂]
exact (f : P →ᵃ[R] Q).linear_eq_zero_iff_exists_const
#align continuous_affine_map.cont_linear_eq_zero_iff_exists_const ContinuousAffineMap.contLinear_eq_zero_iff_exists_const
@[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f := by
ext
rfl
#align continuous_affine_map.to_affine_map_cont_linear ContinuousAffineMap.to_affine_map_contLinear
@[simp]
theorem zero_contLinear : (0 : P →A[R] W).contLinear = 0 :=
rfl
#align continuous_affine_map.zero_cont_linear ContinuousAffineMap.zero_contLinear
@[simp]
theorem add_contLinear (f g : P →A[R] W) : (f + g).contLinear = f.contLinear + g.contLinear :=
rfl
#align continuous_affine_map.add_cont_linear ContinuousAffineMap.add_contLinear
@[simp]
theorem sub_contLinear (f g : P →A[R] W) : (f - g).contLinear = f.contLinear - g.contLinear :=
rfl
#align continuous_affine_map.sub_cont_linear ContinuousAffineMap.sub_contLinear
@[simp]
theorem neg_contLinear (f : P →A[R] W) : (-f).contLinear = -f.contLinear :=
rfl
#align continuous_affine_map.neg_cont_linear ContinuousAffineMap.neg_contLinear
@[simp]
theorem smul_contLinear (t : R) (f : P →A[R] W) : (t • f).contLinear = t • f.contLinear :=
rfl
#align continuous_affine_map.smul_cont_linear ContinuousAffineMap.smul_contLinear
theorem decomp (f : V →A[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) := by
rcases f with ⟨f, h⟩
rw [coe_mk_const_linear_eq_linear, coe_mk, f.decomp, Pi.add_apply, LinearMap.map_zero, zero_add,
← Function.const_def]
#align continuous_affine_map.decomp ContinuousAffineMap.decomp
section NormedSpaceStructure
variable (f : V →A[𝕜] W)
/-- Note that unlike the operator norm for linear maps, this norm is _not_ submultiplicative:
we do _not_ necessarily have `‖f.comp g‖ ≤ ‖f‖ * ‖g‖`. See `norm_comp_le` for what we can say. -/
noncomputable instance hasNorm : Norm (V →A[𝕜] W) :=
⟨fun f => max ‖f 0‖ ‖f.contLinear‖⟩
#align continuous_affine_map.has_norm ContinuousAffineMap.hasNorm
theorem norm_def : ‖f‖ = max ‖f 0‖ ‖f.contLinear‖ :=
rfl
#align continuous_affine_map.norm_def ContinuousAffineMap.norm_def
theorem norm_contLinear_le : ‖f.contLinear‖ ≤ ‖f‖ :=
le_max_right _ _
#align continuous_affine_map.norm_cont_linear_le ContinuousAffineMap.norm_contLinear_le
theorem norm_image_zero_le : ‖f 0‖ ≤ ‖f‖ :=
le_max_left _ _
#align continuous_affine_map.norm_image_zero_le ContinuousAffineMap.norm_image_zero_le
@[simp]
theorem norm_eq (h : f 0 = 0) : ‖f‖ = ‖f.contLinear‖ :=
calc
‖f‖ = max ‖f 0‖ ‖f.contLinear‖ := by rw [norm_def]
_ = max 0 ‖f.contLinear‖ := by rw [h, norm_zero]
_ = ‖f.contLinear‖ := max_eq_right (norm_nonneg _)
#align continuous_affine_map.norm_eq ContinuousAffineMap.norm_eq
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
rw [add_contLinear]
exact
⟨(norm_add_le _ _).trans (add_le_add (le_max_left _ _) (le_max_left _ _)),
(norm_add_le _ _).trans (add_le_add (le_max_right _ _) (le_max_right _ _))⟩
eq_zero_of_map_eq_zero' := fun f h₀ => by
rcases max_eq_iff.mp h₀ with (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩) <;> rw [h₁] at h₂
· rw [norm_le_zero_iff, contLinear_eq_zero_iff_exists_const] at h₂
obtain ⟨q, rfl⟩ := h₂
simp only [norm_eq_zero] at h₁
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₁
rw [h₁]
|
rfl
|
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
rw [add_contLinear]
exact
⟨(norm_add_le _ _).trans (add_le_add (le_max_left _ _) (le_max_left _ _)),
(norm_add_le _ _).trans (add_le_add (le_max_right _ _) (le_max_right _ _))⟩
eq_zero_of_map_eq_zero' := fun f h₀ => by
rcases max_eq_iff.mp h₀ with (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩) <;> rw [h₁] at h₂
· rw [norm_le_zero_iff, contLinear_eq_zero_iff_exists_const] at h₂
obtain ⟨q, rfl⟩ := h₂
simp only [norm_eq_zero] at h₁
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₁
rw [h₁]
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.190_0.bJ3guivW1IqbUMZ
|
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W)
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
case inr.intro
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f✝ f : V →A[𝕜] W
h₀ :
AddGroupSeminorm.toFun
{ toFun := fun f => max ‖f 0‖ ‖contLinear f‖, map_zero' := (_ : max ‖0 0‖ ‖0‖ = 0),
add_le' :=
(_ :
∀ (f g : V →A[𝕜] W),
(fun f => max ‖f 0‖ ‖contLinear f‖) (f + g) ≤
(fun f => max ‖f 0‖ ‖contLinear f‖) f + (fun f => max ‖f 0‖ ‖contLinear f‖) g),
neg' := (_ : ∀ (f : V →A[𝕜] W), max ‖(-f) 0‖ ‖-contLinear f‖ = max ‖f 0‖ ‖contLinear f‖) }
f =
0
h₁ : ‖contLinear f‖ = 0
h₂ : ‖f 0‖ ≤ 0
⊢ f = 0
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h]; rfl
simp_rw [h₁, h₂]
exact (f : P →ᵃ[R] Q).linear_eq_zero_iff_exists_const
#align continuous_affine_map.cont_linear_eq_zero_iff_exists_const ContinuousAffineMap.contLinear_eq_zero_iff_exists_const
@[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f := by
ext
rfl
#align continuous_affine_map.to_affine_map_cont_linear ContinuousAffineMap.to_affine_map_contLinear
@[simp]
theorem zero_contLinear : (0 : P →A[R] W).contLinear = 0 :=
rfl
#align continuous_affine_map.zero_cont_linear ContinuousAffineMap.zero_contLinear
@[simp]
theorem add_contLinear (f g : P →A[R] W) : (f + g).contLinear = f.contLinear + g.contLinear :=
rfl
#align continuous_affine_map.add_cont_linear ContinuousAffineMap.add_contLinear
@[simp]
theorem sub_contLinear (f g : P →A[R] W) : (f - g).contLinear = f.contLinear - g.contLinear :=
rfl
#align continuous_affine_map.sub_cont_linear ContinuousAffineMap.sub_contLinear
@[simp]
theorem neg_contLinear (f : P →A[R] W) : (-f).contLinear = -f.contLinear :=
rfl
#align continuous_affine_map.neg_cont_linear ContinuousAffineMap.neg_contLinear
@[simp]
theorem smul_contLinear (t : R) (f : P →A[R] W) : (t • f).contLinear = t • f.contLinear :=
rfl
#align continuous_affine_map.smul_cont_linear ContinuousAffineMap.smul_contLinear
theorem decomp (f : V →A[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) := by
rcases f with ⟨f, h⟩
rw [coe_mk_const_linear_eq_linear, coe_mk, f.decomp, Pi.add_apply, LinearMap.map_zero, zero_add,
← Function.const_def]
#align continuous_affine_map.decomp ContinuousAffineMap.decomp
section NormedSpaceStructure
variable (f : V →A[𝕜] W)
/-- Note that unlike the operator norm for linear maps, this norm is _not_ submultiplicative:
we do _not_ necessarily have `‖f.comp g‖ ≤ ‖f‖ * ‖g‖`. See `norm_comp_le` for what we can say. -/
noncomputable instance hasNorm : Norm (V →A[𝕜] W) :=
⟨fun f => max ‖f 0‖ ‖f.contLinear‖⟩
#align continuous_affine_map.has_norm ContinuousAffineMap.hasNorm
theorem norm_def : ‖f‖ = max ‖f 0‖ ‖f.contLinear‖ :=
rfl
#align continuous_affine_map.norm_def ContinuousAffineMap.norm_def
theorem norm_contLinear_le : ‖f.contLinear‖ ≤ ‖f‖ :=
le_max_right _ _
#align continuous_affine_map.norm_cont_linear_le ContinuousAffineMap.norm_contLinear_le
theorem norm_image_zero_le : ‖f 0‖ ≤ ‖f‖ :=
le_max_left _ _
#align continuous_affine_map.norm_image_zero_le ContinuousAffineMap.norm_image_zero_le
@[simp]
theorem norm_eq (h : f 0 = 0) : ‖f‖ = ‖f.contLinear‖ :=
calc
‖f‖ = max ‖f 0‖ ‖f.contLinear‖ := by rw [norm_def]
_ = max 0 ‖f.contLinear‖ := by rw [h, norm_zero]
_ = ‖f.contLinear‖ := max_eq_right (norm_nonneg _)
#align continuous_affine_map.norm_eq ContinuousAffineMap.norm_eq
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
rw [add_contLinear]
exact
⟨(norm_add_le _ _).trans (add_le_add (le_max_left _ _) (le_max_left _ _)),
(norm_add_le _ _).trans (add_le_add (le_max_right _ _) (le_max_right _ _))⟩
eq_zero_of_map_eq_zero' := fun f h₀ => by
rcases max_eq_iff.mp h₀ with (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩) <;> rw [h₁] at h₂
· rw [norm_le_zero_iff, contLinear_eq_zero_iff_exists_const] at h₂
obtain ⟨q, rfl⟩ := h₂
simp only [norm_eq_zero] at h₁
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₁
rw [h₁]
rfl
·
|
rw [norm_eq_zero', contLinear_eq_zero_iff_exists_const] at h₁
|
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
rw [add_contLinear]
exact
⟨(norm_add_le _ _).trans (add_le_add (le_max_left _ _) (le_max_left _ _)),
(norm_add_le _ _).trans (add_le_add (le_max_right _ _) (le_max_right _ _))⟩
eq_zero_of_map_eq_zero' := fun f h₀ => by
rcases max_eq_iff.mp h₀ with (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩) <;> rw [h₁] at h₂
· rw [norm_le_zero_iff, contLinear_eq_zero_iff_exists_const] at h₂
obtain ⟨q, rfl⟩ := h₂
simp only [norm_eq_zero] at h₁
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₁
rw [h₁]
rfl
·
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.190_0.bJ3guivW1IqbUMZ
|
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W)
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
case inr.intro
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f✝ f : V →A[𝕜] W
h₀ :
AddGroupSeminorm.toFun
{ toFun := fun f => max ‖f 0‖ ‖contLinear f‖, map_zero' := (_ : max ‖0 0‖ ‖0‖ = 0),
add_le' :=
(_ :
∀ (f g : V →A[𝕜] W),
(fun f => max ‖f 0‖ ‖contLinear f‖) (f + g) ≤
(fun f => max ‖f 0‖ ‖contLinear f‖) f + (fun f => max ‖f 0‖ ‖contLinear f‖) g),
neg' := (_ : ∀ (f : V →A[𝕜] W), max ‖(-f) 0‖ ‖-contLinear f‖ = max ‖f 0‖ ‖contLinear f‖) }
f =
0
h₁ : ∃ q, f = const 𝕜 V q
h₂ : ‖f 0‖ ≤ 0
⊢ f = 0
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h]; rfl
simp_rw [h₁, h₂]
exact (f : P →ᵃ[R] Q).linear_eq_zero_iff_exists_const
#align continuous_affine_map.cont_linear_eq_zero_iff_exists_const ContinuousAffineMap.contLinear_eq_zero_iff_exists_const
@[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f := by
ext
rfl
#align continuous_affine_map.to_affine_map_cont_linear ContinuousAffineMap.to_affine_map_contLinear
@[simp]
theorem zero_contLinear : (0 : P →A[R] W).contLinear = 0 :=
rfl
#align continuous_affine_map.zero_cont_linear ContinuousAffineMap.zero_contLinear
@[simp]
theorem add_contLinear (f g : P →A[R] W) : (f + g).contLinear = f.contLinear + g.contLinear :=
rfl
#align continuous_affine_map.add_cont_linear ContinuousAffineMap.add_contLinear
@[simp]
theorem sub_contLinear (f g : P →A[R] W) : (f - g).contLinear = f.contLinear - g.contLinear :=
rfl
#align continuous_affine_map.sub_cont_linear ContinuousAffineMap.sub_contLinear
@[simp]
theorem neg_contLinear (f : P →A[R] W) : (-f).contLinear = -f.contLinear :=
rfl
#align continuous_affine_map.neg_cont_linear ContinuousAffineMap.neg_contLinear
@[simp]
theorem smul_contLinear (t : R) (f : P →A[R] W) : (t • f).contLinear = t • f.contLinear :=
rfl
#align continuous_affine_map.smul_cont_linear ContinuousAffineMap.smul_contLinear
theorem decomp (f : V →A[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) := by
rcases f with ⟨f, h⟩
rw [coe_mk_const_linear_eq_linear, coe_mk, f.decomp, Pi.add_apply, LinearMap.map_zero, zero_add,
← Function.const_def]
#align continuous_affine_map.decomp ContinuousAffineMap.decomp
section NormedSpaceStructure
variable (f : V →A[𝕜] W)
/-- Note that unlike the operator norm for linear maps, this norm is _not_ submultiplicative:
we do _not_ necessarily have `‖f.comp g‖ ≤ ‖f‖ * ‖g‖`. See `norm_comp_le` for what we can say. -/
noncomputable instance hasNorm : Norm (V →A[𝕜] W) :=
⟨fun f => max ‖f 0‖ ‖f.contLinear‖⟩
#align continuous_affine_map.has_norm ContinuousAffineMap.hasNorm
theorem norm_def : ‖f‖ = max ‖f 0‖ ‖f.contLinear‖ :=
rfl
#align continuous_affine_map.norm_def ContinuousAffineMap.norm_def
theorem norm_contLinear_le : ‖f.contLinear‖ ≤ ‖f‖ :=
le_max_right _ _
#align continuous_affine_map.norm_cont_linear_le ContinuousAffineMap.norm_contLinear_le
theorem norm_image_zero_le : ‖f 0‖ ≤ ‖f‖ :=
le_max_left _ _
#align continuous_affine_map.norm_image_zero_le ContinuousAffineMap.norm_image_zero_le
@[simp]
theorem norm_eq (h : f 0 = 0) : ‖f‖ = ‖f.contLinear‖ :=
calc
‖f‖ = max ‖f 0‖ ‖f.contLinear‖ := by rw [norm_def]
_ = max 0 ‖f.contLinear‖ := by rw [h, norm_zero]
_ = ‖f.contLinear‖ := max_eq_right (norm_nonneg _)
#align continuous_affine_map.norm_eq ContinuousAffineMap.norm_eq
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
rw [add_contLinear]
exact
⟨(norm_add_le _ _).trans (add_le_add (le_max_left _ _) (le_max_left _ _)),
(norm_add_le _ _).trans (add_le_add (le_max_right _ _) (le_max_right _ _))⟩
eq_zero_of_map_eq_zero' := fun f h₀ => by
rcases max_eq_iff.mp h₀ with (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩) <;> rw [h₁] at h₂
· rw [norm_le_zero_iff, contLinear_eq_zero_iff_exists_const] at h₂
obtain ⟨q, rfl⟩ := h₂
simp only [norm_eq_zero] at h₁
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₁
rw [h₁]
rfl
· rw [norm_eq_zero', contLinear_eq_zero_iff_exists_const] at h₁
|
obtain ⟨q, rfl⟩ := h₁
|
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
rw [add_contLinear]
exact
⟨(norm_add_le _ _).trans (add_le_add (le_max_left _ _) (le_max_left _ _)),
(norm_add_le _ _).trans (add_le_add (le_max_right _ _) (le_max_right _ _))⟩
eq_zero_of_map_eq_zero' := fun f h₀ => by
rcases max_eq_iff.mp h₀ with (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩) <;> rw [h₁] at h₂
· rw [norm_le_zero_iff, contLinear_eq_zero_iff_exists_const] at h₂
obtain ⟨q, rfl⟩ := h₂
simp only [norm_eq_zero] at h₁
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₁
rw [h₁]
rfl
· rw [norm_eq_zero', contLinear_eq_zero_iff_exists_const] at h₁
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.190_0.bJ3guivW1IqbUMZ
|
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W)
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
case inr.intro.intro
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : V →A[𝕜] W
q : W
h₀ :
AddGroupSeminorm.toFun
{ toFun := fun f => max ‖f 0‖ ‖contLinear f‖, map_zero' := (_ : max ‖0 0‖ ‖0‖ = 0),
add_le' :=
(_ :
∀ (f g : V →A[𝕜] W),
(fun f => max ‖f 0‖ ‖contLinear f‖) (f + g) ≤
(fun f => max ‖f 0‖ ‖contLinear f‖) f + (fun f => max ‖f 0‖ ‖contLinear f‖) g),
neg' := (_ : ∀ (f : V →A[𝕜] W), max ‖(-f) 0‖ ‖-contLinear f‖ = max ‖f 0‖ ‖contLinear f‖) }
(const 𝕜 V q) =
0
h₂ : ‖(const 𝕜 V q) 0‖ ≤ 0
⊢ const 𝕜 V q = 0
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h]; rfl
simp_rw [h₁, h₂]
exact (f : P →ᵃ[R] Q).linear_eq_zero_iff_exists_const
#align continuous_affine_map.cont_linear_eq_zero_iff_exists_const ContinuousAffineMap.contLinear_eq_zero_iff_exists_const
@[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f := by
ext
rfl
#align continuous_affine_map.to_affine_map_cont_linear ContinuousAffineMap.to_affine_map_contLinear
@[simp]
theorem zero_contLinear : (0 : P →A[R] W).contLinear = 0 :=
rfl
#align continuous_affine_map.zero_cont_linear ContinuousAffineMap.zero_contLinear
@[simp]
theorem add_contLinear (f g : P →A[R] W) : (f + g).contLinear = f.contLinear + g.contLinear :=
rfl
#align continuous_affine_map.add_cont_linear ContinuousAffineMap.add_contLinear
@[simp]
theorem sub_contLinear (f g : P →A[R] W) : (f - g).contLinear = f.contLinear - g.contLinear :=
rfl
#align continuous_affine_map.sub_cont_linear ContinuousAffineMap.sub_contLinear
@[simp]
theorem neg_contLinear (f : P →A[R] W) : (-f).contLinear = -f.contLinear :=
rfl
#align continuous_affine_map.neg_cont_linear ContinuousAffineMap.neg_contLinear
@[simp]
theorem smul_contLinear (t : R) (f : P →A[R] W) : (t • f).contLinear = t • f.contLinear :=
rfl
#align continuous_affine_map.smul_cont_linear ContinuousAffineMap.smul_contLinear
theorem decomp (f : V →A[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) := by
rcases f with ⟨f, h⟩
rw [coe_mk_const_linear_eq_linear, coe_mk, f.decomp, Pi.add_apply, LinearMap.map_zero, zero_add,
← Function.const_def]
#align continuous_affine_map.decomp ContinuousAffineMap.decomp
section NormedSpaceStructure
variable (f : V →A[𝕜] W)
/-- Note that unlike the operator norm for linear maps, this norm is _not_ submultiplicative:
we do _not_ necessarily have `‖f.comp g‖ ≤ ‖f‖ * ‖g‖`. See `norm_comp_le` for what we can say. -/
noncomputable instance hasNorm : Norm (V →A[𝕜] W) :=
⟨fun f => max ‖f 0‖ ‖f.contLinear‖⟩
#align continuous_affine_map.has_norm ContinuousAffineMap.hasNorm
theorem norm_def : ‖f‖ = max ‖f 0‖ ‖f.contLinear‖ :=
rfl
#align continuous_affine_map.norm_def ContinuousAffineMap.norm_def
theorem norm_contLinear_le : ‖f.contLinear‖ ≤ ‖f‖ :=
le_max_right _ _
#align continuous_affine_map.norm_cont_linear_le ContinuousAffineMap.norm_contLinear_le
theorem norm_image_zero_le : ‖f 0‖ ≤ ‖f‖ :=
le_max_left _ _
#align continuous_affine_map.norm_image_zero_le ContinuousAffineMap.norm_image_zero_le
@[simp]
theorem norm_eq (h : f 0 = 0) : ‖f‖ = ‖f.contLinear‖ :=
calc
‖f‖ = max ‖f 0‖ ‖f.contLinear‖ := by rw [norm_def]
_ = max 0 ‖f.contLinear‖ := by rw [h, norm_zero]
_ = ‖f.contLinear‖ := max_eq_right (norm_nonneg _)
#align continuous_affine_map.norm_eq ContinuousAffineMap.norm_eq
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
rw [add_contLinear]
exact
⟨(norm_add_le _ _).trans (add_le_add (le_max_left _ _) (le_max_left _ _)),
(norm_add_le _ _).trans (add_le_add (le_max_right _ _) (le_max_right _ _))⟩
eq_zero_of_map_eq_zero' := fun f h₀ => by
rcases max_eq_iff.mp h₀ with (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩) <;> rw [h₁] at h₂
· rw [norm_le_zero_iff, contLinear_eq_zero_iff_exists_const] at h₂
obtain ⟨q, rfl⟩ := h₂
simp only [norm_eq_zero] at h₁
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₁
rw [h₁]
rfl
· rw [norm_eq_zero', contLinear_eq_zero_iff_exists_const] at h₁
obtain ⟨q, rfl⟩ := h₁
|
simp only [norm_le_zero_iff] at h₂
|
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
rw [add_contLinear]
exact
⟨(norm_add_le _ _).trans (add_le_add (le_max_left _ _) (le_max_left _ _)),
(norm_add_le _ _).trans (add_le_add (le_max_right _ _) (le_max_right _ _))⟩
eq_zero_of_map_eq_zero' := fun f h₀ => by
rcases max_eq_iff.mp h₀ with (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩) <;> rw [h₁] at h₂
· rw [norm_le_zero_iff, contLinear_eq_zero_iff_exists_const] at h₂
obtain ⟨q, rfl⟩ := h₂
simp only [norm_eq_zero] at h₁
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₁
rw [h₁]
rfl
· rw [norm_eq_zero', contLinear_eq_zero_iff_exists_const] at h₁
obtain ⟨q, rfl⟩ := h₁
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.190_0.bJ3guivW1IqbUMZ
|
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W)
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
case inr.intro.intro
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : V →A[𝕜] W
q : W
h₀ :
AddGroupSeminorm.toFun
{ toFun := fun f => max ‖f 0‖ ‖contLinear f‖, map_zero' := (_ : max ‖0 0‖ ‖0‖ = 0),
add_le' :=
(_ :
∀ (f g : V →A[𝕜] W),
(fun f => max ‖f 0‖ ‖contLinear f‖) (f + g) ≤
(fun f => max ‖f 0‖ ‖contLinear f‖) f + (fun f => max ‖f 0‖ ‖contLinear f‖) g),
neg' := (_ : ∀ (f : V →A[𝕜] W), max ‖(-f) 0‖ ‖-contLinear f‖ = max ‖f 0‖ ‖contLinear f‖) }
(const 𝕜 V q) =
0
h₂ : (const 𝕜 V q) 0 = 0
⊢ const 𝕜 V q = 0
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h]; rfl
simp_rw [h₁, h₂]
exact (f : P →ᵃ[R] Q).linear_eq_zero_iff_exists_const
#align continuous_affine_map.cont_linear_eq_zero_iff_exists_const ContinuousAffineMap.contLinear_eq_zero_iff_exists_const
@[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f := by
ext
rfl
#align continuous_affine_map.to_affine_map_cont_linear ContinuousAffineMap.to_affine_map_contLinear
@[simp]
theorem zero_contLinear : (0 : P →A[R] W).contLinear = 0 :=
rfl
#align continuous_affine_map.zero_cont_linear ContinuousAffineMap.zero_contLinear
@[simp]
theorem add_contLinear (f g : P →A[R] W) : (f + g).contLinear = f.contLinear + g.contLinear :=
rfl
#align continuous_affine_map.add_cont_linear ContinuousAffineMap.add_contLinear
@[simp]
theorem sub_contLinear (f g : P →A[R] W) : (f - g).contLinear = f.contLinear - g.contLinear :=
rfl
#align continuous_affine_map.sub_cont_linear ContinuousAffineMap.sub_contLinear
@[simp]
theorem neg_contLinear (f : P →A[R] W) : (-f).contLinear = -f.contLinear :=
rfl
#align continuous_affine_map.neg_cont_linear ContinuousAffineMap.neg_contLinear
@[simp]
theorem smul_contLinear (t : R) (f : P →A[R] W) : (t • f).contLinear = t • f.contLinear :=
rfl
#align continuous_affine_map.smul_cont_linear ContinuousAffineMap.smul_contLinear
theorem decomp (f : V →A[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) := by
rcases f with ⟨f, h⟩
rw [coe_mk_const_linear_eq_linear, coe_mk, f.decomp, Pi.add_apply, LinearMap.map_zero, zero_add,
← Function.const_def]
#align continuous_affine_map.decomp ContinuousAffineMap.decomp
section NormedSpaceStructure
variable (f : V →A[𝕜] W)
/-- Note that unlike the operator norm for linear maps, this norm is _not_ submultiplicative:
we do _not_ necessarily have `‖f.comp g‖ ≤ ‖f‖ * ‖g‖`. See `norm_comp_le` for what we can say. -/
noncomputable instance hasNorm : Norm (V →A[𝕜] W) :=
⟨fun f => max ‖f 0‖ ‖f.contLinear‖⟩
#align continuous_affine_map.has_norm ContinuousAffineMap.hasNorm
theorem norm_def : ‖f‖ = max ‖f 0‖ ‖f.contLinear‖ :=
rfl
#align continuous_affine_map.norm_def ContinuousAffineMap.norm_def
theorem norm_contLinear_le : ‖f.contLinear‖ ≤ ‖f‖ :=
le_max_right _ _
#align continuous_affine_map.norm_cont_linear_le ContinuousAffineMap.norm_contLinear_le
theorem norm_image_zero_le : ‖f 0‖ ≤ ‖f‖ :=
le_max_left _ _
#align continuous_affine_map.norm_image_zero_le ContinuousAffineMap.norm_image_zero_le
@[simp]
theorem norm_eq (h : f 0 = 0) : ‖f‖ = ‖f.contLinear‖ :=
calc
‖f‖ = max ‖f 0‖ ‖f.contLinear‖ := by rw [norm_def]
_ = max 0 ‖f.contLinear‖ := by rw [h, norm_zero]
_ = ‖f.contLinear‖ := max_eq_right (norm_nonneg _)
#align continuous_affine_map.norm_eq ContinuousAffineMap.norm_eq
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
rw [add_contLinear]
exact
⟨(norm_add_le _ _).trans (add_le_add (le_max_left _ _) (le_max_left _ _)),
(norm_add_le _ _).trans (add_le_add (le_max_right _ _) (le_max_right _ _))⟩
eq_zero_of_map_eq_zero' := fun f h₀ => by
rcases max_eq_iff.mp h₀ with (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩) <;> rw [h₁] at h₂
· rw [norm_le_zero_iff, contLinear_eq_zero_iff_exists_const] at h₂
obtain ⟨q, rfl⟩ := h₂
simp only [norm_eq_zero] at h₁
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₁
rw [h₁]
rfl
· rw [norm_eq_zero', contLinear_eq_zero_iff_exists_const] at h₁
obtain ⟨q, rfl⟩ := h₁
simp only [norm_le_zero_iff] at h₂
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
|
rw [coe_const, Function.const_apply] at h₂
|
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
rw [add_contLinear]
exact
⟨(norm_add_le _ _).trans (add_le_add (le_max_left _ _) (le_max_left _ _)),
(norm_add_le _ _).trans (add_le_add (le_max_right _ _) (le_max_right _ _))⟩
eq_zero_of_map_eq_zero' := fun f h₀ => by
rcases max_eq_iff.mp h₀ with (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩) <;> rw [h₁] at h₂
· rw [norm_le_zero_iff, contLinear_eq_zero_iff_exists_const] at h₂
obtain ⟨q, rfl⟩ := h₂
simp only [norm_eq_zero] at h₁
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₁
rw [h₁]
rfl
· rw [norm_eq_zero', contLinear_eq_zero_iff_exists_const] at h₁
obtain ⟨q, rfl⟩ := h₁
simp only [norm_le_zero_iff] at h₂
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.190_0.bJ3guivW1IqbUMZ
|
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W)
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
case inr.intro.intro
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : V →A[𝕜] W
q : W
h₀ :
AddGroupSeminorm.toFun
{ toFun := fun f => max ‖f 0‖ ‖contLinear f‖, map_zero' := (_ : max ‖0 0‖ ‖0‖ = 0),
add_le' :=
(_ :
∀ (f g : V →A[𝕜] W),
(fun f => max ‖f 0‖ ‖contLinear f‖) (f + g) ≤
(fun f => max ‖f 0‖ ‖contLinear f‖) f + (fun f => max ‖f 0‖ ‖contLinear f‖) g),
neg' := (_ : ∀ (f : V →A[𝕜] W), max ‖(-f) 0‖ ‖-contLinear f‖ = max ‖f 0‖ ‖contLinear f‖) }
(const 𝕜 V q) =
0
h₂ : q = 0
⊢ const 𝕜 V q = 0
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h]; rfl
simp_rw [h₁, h₂]
exact (f : P →ᵃ[R] Q).linear_eq_zero_iff_exists_const
#align continuous_affine_map.cont_linear_eq_zero_iff_exists_const ContinuousAffineMap.contLinear_eq_zero_iff_exists_const
@[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f := by
ext
rfl
#align continuous_affine_map.to_affine_map_cont_linear ContinuousAffineMap.to_affine_map_contLinear
@[simp]
theorem zero_contLinear : (0 : P →A[R] W).contLinear = 0 :=
rfl
#align continuous_affine_map.zero_cont_linear ContinuousAffineMap.zero_contLinear
@[simp]
theorem add_contLinear (f g : P →A[R] W) : (f + g).contLinear = f.contLinear + g.contLinear :=
rfl
#align continuous_affine_map.add_cont_linear ContinuousAffineMap.add_contLinear
@[simp]
theorem sub_contLinear (f g : P →A[R] W) : (f - g).contLinear = f.contLinear - g.contLinear :=
rfl
#align continuous_affine_map.sub_cont_linear ContinuousAffineMap.sub_contLinear
@[simp]
theorem neg_contLinear (f : P →A[R] W) : (-f).contLinear = -f.contLinear :=
rfl
#align continuous_affine_map.neg_cont_linear ContinuousAffineMap.neg_contLinear
@[simp]
theorem smul_contLinear (t : R) (f : P →A[R] W) : (t • f).contLinear = t • f.contLinear :=
rfl
#align continuous_affine_map.smul_cont_linear ContinuousAffineMap.smul_contLinear
theorem decomp (f : V →A[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) := by
rcases f with ⟨f, h⟩
rw [coe_mk_const_linear_eq_linear, coe_mk, f.decomp, Pi.add_apply, LinearMap.map_zero, zero_add,
← Function.const_def]
#align continuous_affine_map.decomp ContinuousAffineMap.decomp
section NormedSpaceStructure
variable (f : V →A[𝕜] W)
/-- Note that unlike the operator norm for linear maps, this norm is _not_ submultiplicative:
we do _not_ necessarily have `‖f.comp g‖ ≤ ‖f‖ * ‖g‖`. See `norm_comp_le` for what we can say. -/
noncomputable instance hasNorm : Norm (V →A[𝕜] W) :=
⟨fun f => max ‖f 0‖ ‖f.contLinear‖⟩
#align continuous_affine_map.has_norm ContinuousAffineMap.hasNorm
theorem norm_def : ‖f‖ = max ‖f 0‖ ‖f.contLinear‖ :=
rfl
#align continuous_affine_map.norm_def ContinuousAffineMap.norm_def
theorem norm_contLinear_le : ‖f.contLinear‖ ≤ ‖f‖ :=
le_max_right _ _
#align continuous_affine_map.norm_cont_linear_le ContinuousAffineMap.norm_contLinear_le
theorem norm_image_zero_le : ‖f 0‖ ≤ ‖f‖ :=
le_max_left _ _
#align continuous_affine_map.norm_image_zero_le ContinuousAffineMap.norm_image_zero_le
@[simp]
theorem norm_eq (h : f 0 = 0) : ‖f‖ = ‖f.contLinear‖ :=
calc
‖f‖ = max ‖f 0‖ ‖f.contLinear‖ := by rw [norm_def]
_ = max 0 ‖f.contLinear‖ := by rw [h, norm_zero]
_ = ‖f.contLinear‖ := max_eq_right (norm_nonneg _)
#align continuous_affine_map.norm_eq ContinuousAffineMap.norm_eq
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
rw [add_contLinear]
exact
⟨(norm_add_le _ _).trans (add_le_add (le_max_left _ _) (le_max_left _ _)),
(norm_add_le _ _).trans (add_le_add (le_max_right _ _) (le_max_right _ _))⟩
eq_zero_of_map_eq_zero' := fun f h₀ => by
rcases max_eq_iff.mp h₀ with (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩) <;> rw [h₁] at h₂
· rw [norm_le_zero_iff, contLinear_eq_zero_iff_exists_const] at h₂
obtain ⟨q, rfl⟩ := h₂
simp only [norm_eq_zero] at h₁
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₁
rw [h₁]
rfl
· rw [norm_eq_zero', contLinear_eq_zero_iff_exists_const] at h₁
obtain ⟨q, rfl⟩ := h₁
simp only [norm_le_zero_iff] at h₂
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₂
|
rw [h₂]
|
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
rw [add_contLinear]
exact
⟨(norm_add_le _ _).trans (add_le_add (le_max_left _ _) (le_max_left _ _)),
(norm_add_le _ _).trans (add_le_add (le_max_right _ _) (le_max_right _ _))⟩
eq_zero_of_map_eq_zero' := fun f h₀ => by
rcases max_eq_iff.mp h₀ with (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩) <;> rw [h₁] at h₂
· rw [norm_le_zero_iff, contLinear_eq_zero_iff_exists_const] at h₂
obtain ⟨q, rfl⟩ := h₂
simp only [norm_eq_zero] at h₁
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₁
rw [h₁]
rfl
· rw [norm_eq_zero', contLinear_eq_zero_iff_exists_const] at h₁
obtain ⟨q, rfl⟩ := h₁
simp only [norm_le_zero_iff] at h₂
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₂
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.190_0.bJ3guivW1IqbUMZ
|
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W)
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
case inr.intro.intro
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : V →A[𝕜] W
q : W
h₀ :
AddGroupSeminorm.toFun
{ toFun := fun f => max ‖f 0‖ ‖contLinear f‖, map_zero' := (_ : max ‖0 0‖ ‖0‖ = 0),
add_le' :=
(_ :
∀ (f g : V →A[𝕜] W),
(fun f => max ‖f 0‖ ‖contLinear f‖) (f + g) ≤
(fun f => max ‖f 0‖ ‖contLinear f‖) f + (fun f => max ‖f 0‖ ‖contLinear f‖) g),
neg' := (_ : ∀ (f : V →A[𝕜] W), max ‖(-f) 0‖ ‖-contLinear f‖ = max ‖f 0‖ ‖contLinear f‖) }
(const 𝕜 V q) =
0
h₂ : q = 0
⊢ const 𝕜 V 0 = 0
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h]; rfl
simp_rw [h₁, h₂]
exact (f : P →ᵃ[R] Q).linear_eq_zero_iff_exists_const
#align continuous_affine_map.cont_linear_eq_zero_iff_exists_const ContinuousAffineMap.contLinear_eq_zero_iff_exists_const
@[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f := by
ext
rfl
#align continuous_affine_map.to_affine_map_cont_linear ContinuousAffineMap.to_affine_map_contLinear
@[simp]
theorem zero_contLinear : (0 : P →A[R] W).contLinear = 0 :=
rfl
#align continuous_affine_map.zero_cont_linear ContinuousAffineMap.zero_contLinear
@[simp]
theorem add_contLinear (f g : P →A[R] W) : (f + g).contLinear = f.contLinear + g.contLinear :=
rfl
#align continuous_affine_map.add_cont_linear ContinuousAffineMap.add_contLinear
@[simp]
theorem sub_contLinear (f g : P →A[R] W) : (f - g).contLinear = f.contLinear - g.contLinear :=
rfl
#align continuous_affine_map.sub_cont_linear ContinuousAffineMap.sub_contLinear
@[simp]
theorem neg_contLinear (f : P →A[R] W) : (-f).contLinear = -f.contLinear :=
rfl
#align continuous_affine_map.neg_cont_linear ContinuousAffineMap.neg_contLinear
@[simp]
theorem smul_contLinear (t : R) (f : P →A[R] W) : (t • f).contLinear = t • f.contLinear :=
rfl
#align continuous_affine_map.smul_cont_linear ContinuousAffineMap.smul_contLinear
theorem decomp (f : V →A[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) := by
rcases f with ⟨f, h⟩
rw [coe_mk_const_linear_eq_linear, coe_mk, f.decomp, Pi.add_apply, LinearMap.map_zero, zero_add,
← Function.const_def]
#align continuous_affine_map.decomp ContinuousAffineMap.decomp
section NormedSpaceStructure
variable (f : V →A[𝕜] W)
/-- Note that unlike the operator norm for linear maps, this norm is _not_ submultiplicative:
we do _not_ necessarily have `‖f.comp g‖ ≤ ‖f‖ * ‖g‖`. See `norm_comp_le` for what we can say. -/
noncomputable instance hasNorm : Norm (V →A[𝕜] W) :=
⟨fun f => max ‖f 0‖ ‖f.contLinear‖⟩
#align continuous_affine_map.has_norm ContinuousAffineMap.hasNorm
theorem norm_def : ‖f‖ = max ‖f 0‖ ‖f.contLinear‖ :=
rfl
#align continuous_affine_map.norm_def ContinuousAffineMap.norm_def
theorem norm_contLinear_le : ‖f.contLinear‖ ≤ ‖f‖ :=
le_max_right _ _
#align continuous_affine_map.norm_cont_linear_le ContinuousAffineMap.norm_contLinear_le
theorem norm_image_zero_le : ‖f 0‖ ≤ ‖f‖ :=
le_max_left _ _
#align continuous_affine_map.norm_image_zero_le ContinuousAffineMap.norm_image_zero_le
@[simp]
theorem norm_eq (h : f 0 = 0) : ‖f‖ = ‖f.contLinear‖ :=
calc
‖f‖ = max ‖f 0‖ ‖f.contLinear‖ := by rw [norm_def]
_ = max 0 ‖f.contLinear‖ := by rw [h, norm_zero]
_ = ‖f.contLinear‖ := max_eq_right (norm_nonneg _)
#align continuous_affine_map.norm_eq ContinuousAffineMap.norm_eq
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
rw [add_contLinear]
exact
⟨(norm_add_le _ _).trans (add_le_add (le_max_left _ _) (le_max_left _ _)),
(norm_add_le _ _).trans (add_le_add (le_max_right _ _) (le_max_right _ _))⟩
eq_zero_of_map_eq_zero' := fun f h₀ => by
rcases max_eq_iff.mp h₀ with (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩) <;> rw [h₁] at h₂
· rw [norm_le_zero_iff, contLinear_eq_zero_iff_exists_const] at h₂
obtain ⟨q, rfl⟩ := h₂
simp only [norm_eq_zero] at h₁
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₁
rw [h₁]
rfl
· rw [norm_eq_zero', contLinear_eq_zero_iff_exists_const] at h₁
obtain ⟨q, rfl⟩ := h₁
simp only [norm_le_zero_iff] at h₂
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₂
rw [h₂]
|
rfl
|
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
rw [add_contLinear]
exact
⟨(norm_add_le _ _).trans (add_le_add (le_max_left _ _) (le_max_left _ _)),
(norm_add_le _ _).trans (add_le_add (le_max_right _ _) (le_max_right _ _))⟩
eq_zero_of_map_eq_zero' := fun f h₀ => by
rcases max_eq_iff.mp h₀ with (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩) <;> rw [h₁] at h₂
· rw [norm_le_zero_iff, contLinear_eq_zero_iff_exists_const] at h₂
obtain ⟨q, rfl⟩ := h₂
simp only [norm_eq_zero] at h₁
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₁
rw [h₁]
rfl
· rw [norm_eq_zero', contLinear_eq_zero_iff_exists_const] at h₁
obtain ⟨q, rfl⟩ := h₁
simp only [norm_le_zero_iff] at h₂
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₂
rw [h₂]
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.190_0.bJ3guivW1IqbUMZ
|
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W)
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f✝ : V →A[𝕜] W
t : 𝕜
f : V →A[𝕜] W
⊢ ‖t • f‖ ≤ ‖t‖ * ‖f‖
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h]; rfl
simp_rw [h₁, h₂]
exact (f : P →ᵃ[R] Q).linear_eq_zero_iff_exists_const
#align continuous_affine_map.cont_linear_eq_zero_iff_exists_const ContinuousAffineMap.contLinear_eq_zero_iff_exists_const
@[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f := by
ext
rfl
#align continuous_affine_map.to_affine_map_cont_linear ContinuousAffineMap.to_affine_map_contLinear
@[simp]
theorem zero_contLinear : (0 : P →A[R] W).contLinear = 0 :=
rfl
#align continuous_affine_map.zero_cont_linear ContinuousAffineMap.zero_contLinear
@[simp]
theorem add_contLinear (f g : P →A[R] W) : (f + g).contLinear = f.contLinear + g.contLinear :=
rfl
#align continuous_affine_map.add_cont_linear ContinuousAffineMap.add_contLinear
@[simp]
theorem sub_contLinear (f g : P →A[R] W) : (f - g).contLinear = f.contLinear - g.contLinear :=
rfl
#align continuous_affine_map.sub_cont_linear ContinuousAffineMap.sub_contLinear
@[simp]
theorem neg_contLinear (f : P →A[R] W) : (-f).contLinear = -f.contLinear :=
rfl
#align continuous_affine_map.neg_cont_linear ContinuousAffineMap.neg_contLinear
@[simp]
theorem smul_contLinear (t : R) (f : P →A[R] W) : (t • f).contLinear = t • f.contLinear :=
rfl
#align continuous_affine_map.smul_cont_linear ContinuousAffineMap.smul_contLinear
theorem decomp (f : V →A[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) := by
rcases f with ⟨f, h⟩
rw [coe_mk_const_linear_eq_linear, coe_mk, f.decomp, Pi.add_apply, LinearMap.map_zero, zero_add,
← Function.const_def]
#align continuous_affine_map.decomp ContinuousAffineMap.decomp
section NormedSpaceStructure
variable (f : V →A[𝕜] W)
/-- Note that unlike the operator norm for linear maps, this norm is _not_ submultiplicative:
we do _not_ necessarily have `‖f.comp g‖ ≤ ‖f‖ * ‖g‖`. See `norm_comp_le` for what we can say. -/
noncomputable instance hasNorm : Norm (V →A[𝕜] W) :=
⟨fun f => max ‖f 0‖ ‖f.contLinear‖⟩
#align continuous_affine_map.has_norm ContinuousAffineMap.hasNorm
theorem norm_def : ‖f‖ = max ‖f 0‖ ‖f.contLinear‖ :=
rfl
#align continuous_affine_map.norm_def ContinuousAffineMap.norm_def
theorem norm_contLinear_le : ‖f.contLinear‖ ≤ ‖f‖ :=
le_max_right _ _
#align continuous_affine_map.norm_cont_linear_le ContinuousAffineMap.norm_contLinear_le
theorem norm_image_zero_le : ‖f 0‖ ≤ ‖f‖ :=
le_max_left _ _
#align continuous_affine_map.norm_image_zero_le ContinuousAffineMap.norm_image_zero_le
@[simp]
theorem norm_eq (h : f 0 = 0) : ‖f‖ = ‖f.contLinear‖ :=
calc
‖f‖ = max ‖f 0‖ ‖f.contLinear‖ := by rw [norm_def]
_ = max 0 ‖f.contLinear‖ := by rw [h, norm_zero]
_ = ‖f.contLinear‖ := max_eq_right (norm_nonneg _)
#align continuous_affine_map.norm_eq ContinuousAffineMap.norm_eq
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
rw [add_contLinear]
exact
⟨(norm_add_le _ _).trans (add_le_add (le_max_left _ _) (le_max_left _ _)),
(norm_add_le _ _).trans (add_le_add (le_max_right _ _) (le_max_right _ _))⟩
eq_zero_of_map_eq_zero' := fun f h₀ => by
rcases max_eq_iff.mp h₀ with (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩) <;> rw [h₁] at h₂
· rw [norm_le_zero_iff, contLinear_eq_zero_iff_exists_const] at h₂
obtain ⟨q, rfl⟩ := h₂
simp only [norm_eq_zero] at h₁
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₁
rw [h₁]
rfl
· rw [norm_eq_zero', contLinear_eq_zero_iff_exists_const] at h₁
obtain ⟨q, rfl⟩ := h₁
simp only [norm_le_zero_iff] at h₂
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₂
rw [h₂]
rfl }
instance : NormedSpace 𝕜 (V →A[𝕜] W) where
norm_smul_le t f := by
|
simp only [norm_def, (smul_contLinear), norm_smul]
|
instance : NormedSpace 𝕜 (V →A[𝕜] W) where
norm_smul_le t f := by
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.223_0.bJ3guivW1IqbUMZ
|
instance : NormedSpace 𝕜 (V →A[𝕜] W) where
norm_smul_le t f
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f✝ : V →A[𝕜] W
t : 𝕜
f : V →A[𝕜] W
⊢ max ‖(t • f) 0‖ (‖t‖ * ‖contLinear f‖) ≤ ‖t‖ * max ‖f 0‖ ‖contLinear f‖
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h]; rfl
simp_rw [h₁, h₂]
exact (f : P →ᵃ[R] Q).linear_eq_zero_iff_exists_const
#align continuous_affine_map.cont_linear_eq_zero_iff_exists_const ContinuousAffineMap.contLinear_eq_zero_iff_exists_const
@[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f := by
ext
rfl
#align continuous_affine_map.to_affine_map_cont_linear ContinuousAffineMap.to_affine_map_contLinear
@[simp]
theorem zero_contLinear : (0 : P →A[R] W).contLinear = 0 :=
rfl
#align continuous_affine_map.zero_cont_linear ContinuousAffineMap.zero_contLinear
@[simp]
theorem add_contLinear (f g : P →A[R] W) : (f + g).contLinear = f.contLinear + g.contLinear :=
rfl
#align continuous_affine_map.add_cont_linear ContinuousAffineMap.add_contLinear
@[simp]
theorem sub_contLinear (f g : P →A[R] W) : (f - g).contLinear = f.contLinear - g.contLinear :=
rfl
#align continuous_affine_map.sub_cont_linear ContinuousAffineMap.sub_contLinear
@[simp]
theorem neg_contLinear (f : P →A[R] W) : (-f).contLinear = -f.contLinear :=
rfl
#align continuous_affine_map.neg_cont_linear ContinuousAffineMap.neg_contLinear
@[simp]
theorem smul_contLinear (t : R) (f : P →A[R] W) : (t • f).contLinear = t • f.contLinear :=
rfl
#align continuous_affine_map.smul_cont_linear ContinuousAffineMap.smul_contLinear
theorem decomp (f : V →A[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) := by
rcases f with ⟨f, h⟩
rw [coe_mk_const_linear_eq_linear, coe_mk, f.decomp, Pi.add_apply, LinearMap.map_zero, zero_add,
← Function.const_def]
#align continuous_affine_map.decomp ContinuousAffineMap.decomp
section NormedSpaceStructure
variable (f : V →A[𝕜] W)
/-- Note that unlike the operator norm for linear maps, this norm is _not_ submultiplicative:
we do _not_ necessarily have `‖f.comp g‖ ≤ ‖f‖ * ‖g‖`. See `norm_comp_le` for what we can say. -/
noncomputable instance hasNorm : Norm (V →A[𝕜] W) :=
⟨fun f => max ‖f 0‖ ‖f.contLinear‖⟩
#align continuous_affine_map.has_norm ContinuousAffineMap.hasNorm
theorem norm_def : ‖f‖ = max ‖f 0‖ ‖f.contLinear‖ :=
rfl
#align continuous_affine_map.norm_def ContinuousAffineMap.norm_def
theorem norm_contLinear_le : ‖f.contLinear‖ ≤ ‖f‖ :=
le_max_right _ _
#align continuous_affine_map.norm_cont_linear_le ContinuousAffineMap.norm_contLinear_le
theorem norm_image_zero_le : ‖f 0‖ ≤ ‖f‖ :=
le_max_left _ _
#align continuous_affine_map.norm_image_zero_le ContinuousAffineMap.norm_image_zero_le
@[simp]
theorem norm_eq (h : f 0 = 0) : ‖f‖ = ‖f.contLinear‖ :=
calc
‖f‖ = max ‖f 0‖ ‖f.contLinear‖ := by rw [norm_def]
_ = max 0 ‖f.contLinear‖ := by rw [h, norm_zero]
_ = ‖f.contLinear‖ := max_eq_right (norm_nonneg _)
#align continuous_affine_map.norm_eq ContinuousAffineMap.norm_eq
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
rw [add_contLinear]
exact
⟨(norm_add_le _ _).trans (add_le_add (le_max_left _ _) (le_max_left _ _)),
(norm_add_le _ _).trans (add_le_add (le_max_right _ _) (le_max_right _ _))⟩
eq_zero_of_map_eq_zero' := fun f h₀ => by
rcases max_eq_iff.mp h₀ with (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩) <;> rw [h₁] at h₂
· rw [norm_le_zero_iff, contLinear_eq_zero_iff_exists_const] at h₂
obtain ⟨q, rfl⟩ := h₂
simp only [norm_eq_zero] at h₁
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₁
rw [h₁]
rfl
· rw [norm_eq_zero', contLinear_eq_zero_iff_exists_const] at h₁
obtain ⟨q, rfl⟩ := h₁
simp only [norm_le_zero_iff] at h₂
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₂
rw [h₂]
rfl }
instance : NormedSpace 𝕜 (V →A[𝕜] W) where
norm_smul_le t f := by
simp only [norm_def, (smul_contLinear), norm_smul]
-- Porting note: previously all these rewrites were in the `simp only`,
-- but now they don't fire.
-- (in fact, `norm_smul` fires, but only once rather than twice!)
|
rw [coe_smul, Pi.smul_apply, norm_smul, ← mul_max_of_nonneg _ _ (norm_nonneg t)]
|
instance : NormedSpace 𝕜 (V →A[𝕜] W) where
norm_smul_le t f := by
simp only [norm_def, (smul_contLinear), norm_smul]
-- Porting note: previously all these rewrites were in the `simp only`,
-- but now they don't fire.
-- (in fact, `norm_smul` fires, but only once rather than twice!)
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.223_0.bJ3guivW1IqbUMZ
|
instance : NormedSpace 𝕜 (V →A[𝕜] W) where
norm_smul_le t f
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : V →A[𝕜] W
g : W₂ →A[𝕜] V
⊢ ‖comp f g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h]; rfl
simp_rw [h₁, h₂]
exact (f : P →ᵃ[R] Q).linear_eq_zero_iff_exists_const
#align continuous_affine_map.cont_linear_eq_zero_iff_exists_const ContinuousAffineMap.contLinear_eq_zero_iff_exists_const
@[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f := by
ext
rfl
#align continuous_affine_map.to_affine_map_cont_linear ContinuousAffineMap.to_affine_map_contLinear
@[simp]
theorem zero_contLinear : (0 : P →A[R] W).contLinear = 0 :=
rfl
#align continuous_affine_map.zero_cont_linear ContinuousAffineMap.zero_contLinear
@[simp]
theorem add_contLinear (f g : P →A[R] W) : (f + g).contLinear = f.contLinear + g.contLinear :=
rfl
#align continuous_affine_map.add_cont_linear ContinuousAffineMap.add_contLinear
@[simp]
theorem sub_contLinear (f g : P →A[R] W) : (f - g).contLinear = f.contLinear - g.contLinear :=
rfl
#align continuous_affine_map.sub_cont_linear ContinuousAffineMap.sub_contLinear
@[simp]
theorem neg_contLinear (f : P →A[R] W) : (-f).contLinear = -f.contLinear :=
rfl
#align continuous_affine_map.neg_cont_linear ContinuousAffineMap.neg_contLinear
@[simp]
theorem smul_contLinear (t : R) (f : P →A[R] W) : (t • f).contLinear = t • f.contLinear :=
rfl
#align continuous_affine_map.smul_cont_linear ContinuousAffineMap.smul_contLinear
theorem decomp (f : V →A[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) := by
rcases f with ⟨f, h⟩
rw [coe_mk_const_linear_eq_linear, coe_mk, f.decomp, Pi.add_apply, LinearMap.map_zero, zero_add,
← Function.const_def]
#align continuous_affine_map.decomp ContinuousAffineMap.decomp
section NormedSpaceStructure
variable (f : V →A[𝕜] W)
/-- Note that unlike the operator norm for linear maps, this norm is _not_ submultiplicative:
we do _not_ necessarily have `‖f.comp g‖ ≤ ‖f‖ * ‖g‖`. See `norm_comp_le` for what we can say. -/
noncomputable instance hasNorm : Norm (V →A[𝕜] W) :=
⟨fun f => max ‖f 0‖ ‖f.contLinear‖⟩
#align continuous_affine_map.has_norm ContinuousAffineMap.hasNorm
theorem norm_def : ‖f‖ = max ‖f 0‖ ‖f.contLinear‖ :=
rfl
#align continuous_affine_map.norm_def ContinuousAffineMap.norm_def
theorem norm_contLinear_le : ‖f.contLinear‖ ≤ ‖f‖ :=
le_max_right _ _
#align continuous_affine_map.norm_cont_linear_le ContinuousAffineMap.norm_contLinear_le
theorem norm_image_zero_le : ‖f 0‖ ≤ ‖f‖ :=
le_max_left _ _
#align continuous_affine_map.norm_image_zero_le ContinuousAffineMap.norm_image_zero_le
@[simp]
theorem norm_eq (h : f 0 = 0) : ‖f‖ = ‖f.contLinear‖ :=
calc
‖f‖ = max ‖f 0‖ ‖f.contLinear‖ := by rw [norm_def]
_ = max 0 ‖f.contLinear‖ := by rw [h, norm_zero]
_ = ‖f.contLinear‖ := max_eq_right (norm_nonneg _)
#align continuous_affine_map.norm_eq ContinuousAffineMap.norm_eq
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
rw [add_contLinear]
exact
⟨(norm_add_le _ _).trans (add_le_add (le_max_left _ _) (le_max_left _ _)),
(norm_add_le _ _).trans (add_le_add (le_max_right _ _) (le_max_right _ _))⟩
eq_zero_of_map_eq_zero' := fun f h₀ => by
rcases max_eq_iff.mp h₀ with (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩) <;> rw [h₁] at h₂
· rw [norm_le_zero_iff, contLinear_eq_zero_iff_exists_const] at h₂
obtain ⟨q, rfl⟩ := h₂
simp only [norm_eq_zero] at h₁
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₁
rw [h₁]
rfl
· rw [norm_eq_zero', contLinear_eq_zero_iff_exists_const] at h₁
obtain ⟨q, rfl⟩ := h₁
simp only [norm_le_zero_iff] at h₂
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₂
rw [h₂]
rfl }
instance : NormedSpace 𝕜 (V →A[𝕜] W) where
norm_smul_le t f := by
simp only [norm_def, (smul_contLinear), norm_smul]
-- Porting note: previously all these rewrites were in the `simp only`,
-- but now they don't fire.
-- (in fact, `norm_smul` fires, but only once rather than twice!)
rw [coe_smul, Pi.smul_apply, norm_smul, ← mul_max_of_nonneg _ _ (norm_nonneg t)]
theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ := by
|
rw [norm_def, max_le_iff]
|
theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ := by
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.231_0.bJ3guivW1IqbUMZ
|
theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : V →A[𝕜] W
g : W₂ →A[𝕜] V
⊢ ‖(comp f g) 0‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ ∧ ‖contLinear (comp f g)‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h]; rfl
simp_rw [h₁, h₂]
exact (f : P →ᵃ[R] Q).linear_eq_zero_iff_exists_const
#align continuous_affine_map.cont_linear_eq_zero_iff_exists_const ContinuousAffineMap.contLinear_eq_zero_iff_exists_const
@[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f := by
ext
rfl
#align continuous_affine_map.to_affine_map_cont_linear ContinuousAffineMap.to_affine_map_contLinear
@[simp]
theorem zero_contLinear : (0 : P →A[R] W).contLinear = 0 :=
rfl
#align continuous_affine_map.zero_cont_linear ContinuousAffineMap.zero_contLinear
@[simp]
theorem add_contLinear (f g : P →A[R] W) : (f + g).contLinear = f.contLinear + g.contLinear :=
rfl
#align continuous_affine_map.add_cont_linear ContinuousAffineMap.add_contLinear
@[simp]
theorem sub_contLinear (f g : P →A[R] W) : (f - g).contLinear = f.contLinear - g.contLinear :=
rfl
#align continuous_affine_map.sub_cont_linear ContinuousAffineMap.sub_contLinear
@[simp]
theorem neg_contLinear (f : P →A[R] W) : (-f).contLinear = -f.contLinear :=
rfl
#align continuous_affine_map.neg_cont_linear ContinuousAffineMap.neg_contLinear
@[simp]
theorem smul_contLinear (t : R) (f : P →A[R] W) : (t • f).contLinear = t • f.contLinear :=
rfl
#align continuous_affine_map.smul_cont_linear ContinuousAffineMap.smul_contLinear
theorem decomp (f : V →A[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) := by
rcases f with ⟨f, h⟩
rw [coe_mk_const_linear_eq_linear, coe_mk, f.decomp, Pi.add_apply, LinearMap.map_zero, zero_add,
← Function.const_def]
#align continuous_affine_map.decomp ContinuousAffineMap.decomp
section NormedSpaceStructure
variable (f : V →A[𝕜] W)
/-- Note that unlike the operator norm for linear maps, this norm is _not_ submultiplicative:
we do _not_ necessarily have `‖f.comp g‖ ≤ ‖f‖ * ‖g‖`. See `norm_comp_le` for what we can say. -/
noncomputable instance hasNorm : Norm (V →A[𝕜] W) :=
⟨fun f => max ‖f 0‖ ‖f.contLinear‖⟩
#align continuous_affine_map.has_norm ContinuousAffineMap.hasNorm
theorem norm_def : ‖f‖ = max ‖f 0‖ ‖f.contLinear‖ :=
rfl
#align continuous_affine_map.norm_def ContinuousAffineMap.norm_def
theorem norm_contLinear_le : ‖f.contLinear‖ ≤ ‖f‖ :=
le_max_right _ _
#align continuous_affine_map.norm_cont_linear_le ContinuousAffineMap.norm_contLinear_le
theorem norm_image_zero_le : ‖f 0‖ ≤ ‖f‖ :=
le_max_left _ _
#align continuous_affine_map.norm_image_zero_le ContinuousAffineMap.norm_image_zero_le
@[simp]
theorem norm_eq (h : f 0 = 0) : ‖f‖ = ‖f.contLinear‖ :=
calc
‖f‖ = max ‖f 0‖ ‖f.contLinear‖ := by rw [norm_def]
_ = max 0 ‖f.contLinear‖ := by rw [h, norm_zero]
_ = ‖f.contLinear‖ := max_eq_right (norm_nonneg _)
#align continuous_affine_map.norm_eq ContinuousAffineMap.norm_eq
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
rw [add_contLinear]
exact
⟨(norm_add_le _ _).trans (add_le_add (le_max_left _ _) (le_max_left _ _)),
(norm_add_le _ _).trans (add_le_add (le_max_right _ _) (le_max_right _ _))⟩
eq_zero_of_map_eq_zero' := fun f h₀ => by
rcases max_eq_iff.mp h₀ with (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩) <;> rw [h₁] at h₂
· rw [norm_le_zero_iff, contLinear_eq_zero_iff_exists_const] at h₂
obtain ⟨q, rfl⟩ := h₂
simp only [norm_eq_zero] at h₁
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₁
rw [h₁]
rfl
· rw [norm_eq_zero', contLinear_eq_zero_iff_exists_const] at h₁
obtain ⟨q, rfl⟩ := h₁
simp only [norm_le_zero_iff] at h₂
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₂
rw [h₂]
rfl }
instance : NormedSpace 𝕜 (V →A[𝕜] W) where
norm_smul_le t f := by
simp only [norm_def, (smul_contLinear), norm_smul]
-- Porting note: previously all these rewrites were in the `simp only`,
-- but now they don't fire.
-- (in fact, `norm_smul` fires, but only once rather than twice!)
rw [coe_smul, Pi.smul_apply, norm_smul, ← mul_max_of_nonneg _ _ (norm_nonneg t)]
theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ := by
rw [norm_def, max_le_iff]
|
constructor
|
theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ := by
rw [norm_def, max_le_iff]
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.231_0.bJ3guivW1IqbUMZ
|
theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
case left
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : V →A[𝕜] W
g : W₂ →A[𝕜] V
⊢ ‖(comp f g) 0‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h]; rfl
simp_rw [h₁, h₂]
exact (f : P →ᵃ[R] Q).linear_eq_zero_iff_exists_const
#align continuous_affine_map.cont_linear_eq_zero_iff_exists_const ContinuousAffineMap.contLinear_eq_zero_iff_exists_const
@[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f := by
ext
rfl
#align continuous_affine_map.to_affine_map_cont_linear ContinuousAffineMap.to_affine_map_contLinear
@[simp]
theorem zero_contLinear : (0 : P →A[R] W).contLinear = 0 :=
rfl
#align continuous_affine_map.zero_cont_linear ContinuousAffineMap.zero_contLinear
@[simp]
theorem add_contLinear (f g : P →A[R] W) : (f + g).contLinear = f.contLinear + g.contLinear :=
rfl
#align continuous_affine_map.add_cont_linear ContinuousAffineMap.add_contLinear
@[simp]
theorem sub_contLinear (f g : P →A[R] W) : (f - g).contLinear = f.contLinear - g.contLinear :=
rfl
#align continuous_affine_map.sub_cont_linear ContinuousAffineMap.sub_contLinear
@[simp]
theorem neg_contLinear (f : P →A[R] W) : (-f).contLinear = -f.contLinear :=
rfl
#align continuous_affine_map.neg_cont_linear ContinuousAffineMap.neg_contLinear
@[simp]
theorem smul_contLinear (t : R) (f : P →A[R] W) : (t • f).contLinear = t • f.contLinear :=
rfl
#align continuous_affine_map.smul_cont_linear ContinuousAffineMap.smul_contLinear
theorem decomp (f : V →A[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) := by
rcases f with ⟨f, h⟩
rw [coe_mk_const_linear_eq_linear, coe_mk, f.decomp, Pi.add_apply, LinearMap.map_zero, zero_add,
← Function.const_def]
#align continuous_affine_map.decomp ContinuousAffineMap.decomp
section NormedSpaceStructure
variable (f : V →A[𝕜] W)
/-- Note that unlike the operator norm for linear maps, this norm is _not_ submultiplicative:
we do _not_ necessarily have `‖f.comp g‖ ≤ ‖f‖ * ‖g‖`. See `norm_comp_le` for what we can say. -/
noncomputable instance hasNorm : Norm (V →A[𝕜] W) :=
⟨fun f => max ‖f 0‖ ‖f.contLinear‖⟩
#align continuous_affine_map.has_norm ContinuousAffineMap.hasNorm
theorem norm_def : ‖f‖ = max ‖f 0‖ ‖f.contLinear‖ :=
rfl
#align continuous_affine_map.norm_def ContinuousAffineMap.norm_def
theorem norm_contLinear_le : ‖f.contLinear‖ ≤ ‖f‖ :=
le_max_right _ _
#align continuous_affine_map.norm_cont_linear_le ContinuousAffineMap.norm_contLinear_le
theorem norm_image_zero_le : ‖f 0‖ ≤ ‖f‖ :=
le_max_left _ _
#align continuous_affine_map.norm_image_zero_le ContinuousAffineMap.norm_image_zero_le
@[simp]
theorem norm_eq (h : f 0 = 0) : ‖f‖ = ‖f.contLinear‖ :=
calc
‖f‖ = max ‖f 0‖ ‖f.contLinear‖ := by rw [norm_def]
_ = max 0 ‖f.contLinear‖ := by rw [h, norm_zero]
_ = ‖f.contLinear‖ := max_eq_right (norm_nonneg _)
#align continuous_affine_map.norm_eq ContinuousAffineMap.norm_eq
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
rw [add_contLinear]
exact
⟨(norm_add_le _ _).trans (add_le_add (le_max_left _ _) (le_max_left _ _)),
(norm_add_le _ _).trans (add_le_add (le_max_right _ _) (le_max_right _ _))⟩
eq_zero_of_map_eq_zero' := fun f h₀ => by
rcases max_eq_iff.mp h₀ with (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩) <;> rw [h₁] at h₂
· rw [norm_le_zero_iff, contLinear_eq_zero_iff_exists_const] at h₂
obtain ⟨q, rfl⟩ := h₂
simp only [norm_eq_zero] at h₁
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₁
rw [h₁]
rfl
· rw [norm_eq_zero', contLinear_eq_zero_iff_exists_const] at h₁
obtain ⟨q, rfl⟩ := h₁
simp only [norm_le_zero_iff] at h₂
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₂
rw [h₂]
rfl }
instance : NormedSpace 𝕜 (V →A[𝕜] W) where
norm_smul_le t f := by
simp only [norm_def, (smul_contLinear), norm_smul]
-- Porting note: previously all these rewrites were in the `simp only`,
-- but now they don't fire.
-- (in fact, `norm_smul` fires, but only once rather than twice!)
rw [coe_smul, Pi.smul_apply, norm_smul, ← mul_max_of_nonneg _ _ (norm_nonneg t)]
theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ := by
rw [norm_def, max_le_iff]
constructor
·
|
calc
‖f.comp g 0‖ = ‖f (g 0)‖ := by simp
_ = ‖f.contLinear (g 0) + f 0‖ := by rw [f.decomp]; simp
_ ≤ ‖f.contLinear‖ * ‖g 0‖ + ‖f 0‖ :=
((norm_add_le _ _).trans (add_le_add_right (f.contLinear.le_op_norm _) _))
_ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ :=
add_le_add_right
(mul_le_mul f.norm_contLinear_le g.norm_image_zero_le (norm_nonneg _) (norm_nonneg _)) _
|
theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ := by
rw [norm_def, max_le_iff]
constructor
·
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.231_0.bJ3guivW1IqbUMZ
|
theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : V →A[𝕜] W
g : W₂ →A[𝕜] V
⊢ ‖(comp f g) 0‖ = ‖f (g 0)‖
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h]; rfl
simp_rw [h₁, h₂]
exact (f : P →ᵃ[R] Q).linear_eq_zero_iff_exists_const
#align continuous_affine_map.cont_linear_eq_zero_iff_exists_const ContinuousAffineMap.contLinear_eq_zero_iff_exists_const
@[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f := by
ext
rfl
#align continuous_affine_map.to_affine_map_cont_linear ContinuousAffineMap.to_affine_map_contLinear
@[simp]
theorem zero_contLinear : (0 : P →A[R] W).contLinear = 0 :=
rfl
#align continuous_affine_map.zero_cont_linear ContinuousAffineMap.zero_contLinear
@[simp]
theorem add_contLinear (f g : P →A[R] W) : (f + g).contLinear = f.contLinear + g.contLinear :=
rfl
#align continuous_affine_map.add_cont_linear ContinuousAffineMap.add_contLinear
@[simp]
theorem sub_contLinear (f g : P →A[R] W) : (f - g).contLinear = f.contLinear - g.contLinear :=
rfl
#align continuous_affine_map.sub_cont_linear ContinuousAffineMap.sub_contLinear
@[simp]
theorem neg_contLinear (f : P →A[R] W) : (-f).contLinear = -f.contLinear :=
rfl
#align continuous_affine_map.neg_cont_linear ContinuousAffineMap.neg_contLinear
@[simp]
theorem smul_contLinear (t : R) (f : P →A[R] W) : (t • f).contLinear = t • f.contLinear :=
rfl
#align continuous_affine_map.smul_cont_linear ContinuousAffineMap.smul_contLinear
theorem decomp (f : V →A[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) := by
rcases f with ⟨f, h⟩
rw [coe_mk_const_linear_eq_linear, coe_mk, f.decomp, Pi.add_apply, LinearMap.map_zero, zero_add,
← Function.const_def]
#align continuous_affine_map.decomp ContinuousAffineMap.decomp
section NormedSpaceStructure
variable (f : V →A[𝕜] W)
/-- Note that unlike the operator norm for linear maps, this norm is _not_ submultiplicative:
we do _not_ necessarily have `‖f.comp g‖ ≤ ‖f‖ * ‖g‖`. See `norm_comp_le` for what we can say. -/
noncomputable instance hasNorm : Norm (V →A[𝕜] W) :=
⟨fun f => max ‖f 0‖ ‖f.contLinear‖⟩
#align continuous_affine_map.has_norm ContinuousAffineMap.hasNorm
theorem norm_def : ‖f‖ = max ‖f 0‖ ‖f.contLinear‖ :=
rfl
#align continuous_affine_map.norm_def ContinuousAffineMap.norm_def
theorem norm_contLinear_le : ‖f.contLinear‖ ≤ ‖f‖ :=
le_max_right _ _
#align continuous_affine_map.norm_cont_linear_le ContinuousAffineMap.norm_contLinear_le
theorem norm_image_zero_le : ‖f 0‖ ≤ ‖f‖ :=
le_max_left _ _
#align continuous_affine_map.norm_image_zero_le ContinuousAffineMap.norm_image_zero_le
@[simp]
theorem norm_eq (h : f 0 = 0) : ‖f‖ = ‖f.contLinear‖ :=
calc
‖f‖ = max ‖f 0‖ ‖f.contLinear‖ := by rw [norm_def]
_ = max 0 ‖f.contLinear‖ := by rw [h, norm_zero]
_ = ‖f.contLinear‖ := max_eq_right (norm_nonneg _)
#align continuous_affine_map.norm_eq ContinuousAffineMap.norm_eq
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
rw [add_contLinear]
exact
⟨(norm_add_le _ _).trans (add_le_add (le_max_left _ _) (le_max_left _ _)),
(norm_add_le _ _).trans (add_le_add (le_max_right _ _) (le_max_right _ _))⟩
eq_zero_of_map_eq_zero' := fun f h₀ => by
rcases max_eq_iff.mp h₀ with (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩) <;> rw [h₁] at h₂
· rw [norm_le_zero_iff, contLinear_eq_zero_iff_exists_const] at h₂
obtain ⟨q, rfl⟩ := h₂
simp only [norm_eq_zero] at h₁
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₁
rw [h₁]
rfl
· rw [norm_eq_zero', contLinear_eq_zero_iff_exists_const] at h₁
obtain ⟨q, rfl⟩ := h₁
simp only [norm_le_zero_iff] at h₂
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₂
rw [h₂]
rfl }
instance : NormedSpace 𝕜 (V →A[𝕜] W) where
norm_smul_le t f := by
simp only [norm_def, (smul_contLinear), norm_smul]
-- Porting note: previously all these rewrites were in the `simp only`,
-- but now they don't fire.
-- (in fact, `norm_smul` fires, but only once rather than twice!)
rw [coe_smul, Pi.smul_apply, norm_smul, ← mul_max_of_nonneg _ _ (norm_nonneg t)]
theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ := by
rw [norm_def, max_le_iff]
constructor
· calc
‖f.comp g 0‖ = ‖f (g 0)‖ := by
|
simp
|
theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ := by
rw [norm_def, max_le_iff]
constructor
· calc
‖f.comp g 0‖ = ‖f (g 0)‖ := by
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.231_0.bJ3guivW1IqbUMZ
|
theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : V →A[𝕜] W
g : W₂ →A[𝕜] V
⊢ ‖f (g 0)‖ = ‖(contLinear f) (g 0) + f 0‖
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h]; rfl
simp_rw [h₁, h₂]
exact (f : P →ᵃ[R] Q).linear_eq_zero_iff_exists_const
#align continuous_affine_map.cont_linear_eq_zero_iff_exists_const ContinuousAffineMap.contLinear_eq_zero_iff_exists_const
@[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f := by
ext
rfl
#align continuous_affine_map.to_affine_map_cont_linear ContinuousAffineMap.to_affine_map_contLinear
@[simp]
theorem zero_contLinear : (0 : P →A[R] W).contLinear = 0 :=
rfl
#align continuous_affine_map.zero_cont_linear ContinuousAffineMap.zero_contLinear
@[simp]
theorem add_contLinear (f g : P →A[R] W) : (f + g).contLinear = f.contLinear + g.contLinear :=
rfl
#align continuous_affine_map.add_cont_linear ContinuousAffineMap.add_contLinear
@[simp]
theorem sub_contLinear (f g : P →A[R] W) : (f - g).contLinear = f.contLinear - g.contLinear :=
rfl
#align continuous_affine_map.sub_cont_linear ContinuousAffineMap.sub_contLinear
@[simp]
theorem neg_contLinear (f : P →A[R] W) : (-f).contLinear = -f.contLinear :=
rfl
#align continuous_affine_map.neg_cont_linear ContinuousAffineMap.neg_contLinear
@[simp]
theorem smul_contLinear (t : R) (f : P →A[R] W) : (t • f).contLinear = t • f.contLinear :=
rfl
#align continuous_affine_map.smul_cont_linear ContinuousAffineMap.smul_contLinear
theorem decomp (f : V →A[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) := by
rcases f with ⟨f, h⟩
rw [coe_mk_const_linear_eq_linear, coe_mk, f.decomp, Pi.add_apply, LinearMap.map_zero, zero_add,
← Function.const_def]
#align continuous_affine_map.decomp ContinuousAffineMap.decomp
section NormedSpaceStructure
variable (f : V →A[𝕜] W)
/-- Note that unlike the operator norm for linear maps, this norm is _not_ submultiplicative:
we do _not_ necessarily have `‖f.comp g‖ ≤ ‖f‖ * ‖g‖`. See `norm_comp_le` for what we can say. -/
noncomputable instance hasNorm : Norm (V →A[𝕜] W) :=
⟨fun f => max ‖f 0‖ ‖f.contLinear‖⟩
#align continuous_affine_map.has_norm ContinuousAffineMap.hasNorm
theorem norm_def : ‖f‖ = max ‖f 0‖ ‖f.contLinear‖ :=
rfl
#align continuous_affine_map.norm_def ContinuousAffineMap.norm_def
theorem norm_contLinear_le : ‖f.contLinear‖ ≤ ‖f‖ :=
le_max_right _ _
#align continuous_affine_map.norm_cont_linear_le ContinuousAffineMap.norm_contLinear_le
theorem norm_image_zero_le : ‖f 0‖ ≤ ‖f‖ :=
le_max_left _ _
#align continuous_affine_map.norm_image_zero_le ContinuousAffineMap.norm_image_zero_le
@[simp]
theorem norm_eq (h : f 0 = 0) : ‖f‖ = ‖f.contLinear‖ :=
calc
‖f‖ = max ‖f 0‖ ‖f.contLinear‖ := by rw [norm_def]
_ = max 0 ‖f.contLinear‖ := by rw [h, norm_zero]
_ = ‖f.contLinear‖ := max_eq_right (norm_nonneg _)
#align continuous_affine_map.norm_eq ContinuousAffineMap.norm_eq
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
rw [add_contLinear]
exact
⟨(norm_add_le _ _).trans (add_le_add (le_max_left _ _) (le_max_left _ _)),
(norm_add_le _ _).trans (add_le_add (le_max_right _ _) (le_max_right _ _))⟩
eq_zero_of_map_eq_zero' := fun f h₀ => by
rcases max_eq_iff.mp h₀ with (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩) <;> rw [h₁] at h₂
· rw [norm_le_zero_iff, contLinear_eq_zero_iff_exists_const] at h₂
obtain ⟨q, rfl⟩ := h₂
simp only [norm_eq_zero] at h₁
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₁
rw [h₁]
rfl
· rw [norm_eq_zero', contLinear_eq_zero_iff_exists_const] at h₁
obtain ⟨q, rfl⟩ := h₁
simp only [norm_le_zero_iff] at h₂
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₂
rw [h₂]
rfl }
instance : NormedSpace 𝕜 (V →A[𝕜] W) where
norm_smul_le t f := by
simp only [norm_def, (smul_contLinear), norm_smul]
-- Porting note: previously all these rewrites were in the `simp only`,
-- but now they don't fire.
-- (in fact, `norm_smul` fires, but only once rather than twice!)
rw [coe_smul, Pi.smul_apply, norm_smul, ← mul_max_of_nonneg _ _ (norm_nonneg t)]
theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ := by
rw [norm_def, max_le_iff]
constructor
· calc
‖f.comp g 0‖ = ‖f (g 0)‖ := by simp
_ = ‖f.contLinear (g 0) + f 0‖ := by
|
rw [f.decomp]
|
theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ := by
rw [norm_def, max_le_iff]
constructor
· calc
‖f.comp g 0‖ = ‖f (g 0)‖ := by simp
_ = ‖f.contLinear (g 0) + f 0‖ := by
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.231_0.bJ3guivW1IqbUMZ
|
theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : V →A[𝕜] W
g : W₂ →A[𝕜] V
⊢ ‖(⇑(contLinear f) + Function.const V (f 0)) (g 0)‖ =
‖(contLinear f) (g 0) + (⇑(contLinear f) + Function.const V (f 0)) 0‖
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h]; rfl
simp_rw [h₁, h₂]
exact (f : P →ᵃ[R] Q).linear_eq_zero_iff_exists_const
#align continuous_affine_map.cont_linear_eq_zero_iff_exists_const ContinuousAffineMap.contLinear_eq_zero_iff_exists_const
@[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f := by
ext
rfl
#align continuous_affine_map.to_affine_map_cont_linear ContinuousAffineMap.to_affine_map_contLinear
@[simp]
theorem zero_contLinear : (0 : P →A[R] W).contLinear = 0 :=
rfl
#align continuous_affine_map.zero_cont_linear ContinuousAffineMap.zero_contLinear
@[simp]
theorem add_contLinear (f g : P →A[R] W) : (f + g).contLinear = f.contLinear + g.contLinear :=
rfl
#align continuous_affine_map.add_cont_linear ContinuousAffineMap.add_contLinear
@[simp]
theorem sub_contLinear (f g : P →A[R] W) : (f - g).contLinear = f.contLinear - g.contLinear :=
rfl
#align continuous_affine_map.sub_cont_linear ContinuousAffineMap.sub_contLinear
@[simp]
theorem neg_contLinear (f : P →A[R] W) : (-f).contLinear = -f.contLinear :=
rfl
#align continuous_affine_map.neg_cont_linear ContinuousAffineMap.neg_contLinear
@[simp]
theorem smul_contLinear (t : R) (f : P →A[R] W) : (t • f).contLinear = t • f.contLinear :=
rfl
#align continuous_affine_map.smul_cont_linear ContinuousAffineMap.smul_contLinear
theorem decomp (f : V →A[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) := by
rcases f with ⟨f, h⟩
rw [coe_mk_const_linear_eq_linear, coe_mk, f.decomp, Pi.add_apply, LinearMap.map_zero, zero_add,
← Function.const_def]
#align continuous_affine_map.decomp ContinuousAffineMap.decomp
section NormedSpaceStructure
variable (f : V →A[𝕜] W)
/-- Note that unlike the operator norm for linear maps, this norm is _not_ submultiplicative:
we do _not_ necessarily have `‖f.comp g‖ ≤ ‖f‖ * ‖g‖`. See `norm_comp_le` for what we can say. -/
noncomputable instance hasNorm : Norm (V →A[𝕜] W) :=
⟨fun f => max ‖f 0‖ ‖f.contLinear‖⟩
#align continuous_affine_map.has_norm ContinuousAffineMap.hasNorm
theorem norm_def : ‖f‖ = max ‖f 0‖ ‖f.contLinear‖ :=
rfl
#align continuous_affine_map.norm_def ContinuousAffineMap.norm_def
theorem norm_contLinear_le : ‖f.contLinear‖ ≤ ‖f‖ :=
le_max_right _ _
#align continuous_affine_map.norm_cont_linear_le ContinuousAffineMap.norm_contLinear_le
theorem norm_image_zero_le : ‖f 0‖ ≤ ‖f‖ :=
le_max_left _ _
#align continuous_affine_map.norm_image_zero_le ContinuousAffineMap.norm_image_zero_le
@[simp]
theorem norm_eq (h : f 0 = 0) : ‖f‖ = ‖f.contLinear‖ :=
calc
‖f‖ = max ‖f 0‖ ‖f.contLinear‖ := by rw [norm_def]
_ = max 0 ‖f.contLinear‖ := by rw [h, norm_zero]
_ = ‖f.contLinear‖ := max_eq_right (norm_nonneg _)
#align continuous_affine_map.norm_eq ContinuousAffineMap.norm_eq
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
rw [add_contLinear]
exact
⟨(norm_add_le _ _).trans (add_le_add (le_max_left _ _) (le_max_left _ _)),
(norm_add_le _ _).trans (add_le_add (le_max_right _ _) (le_max_right _ _))⟩
eq_zero_of_map_eq_zero' := fun f h₀ => by
rcases max_eq_iff.mp h₀ with (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩) <;> rw [h₁] at h₂
· rw [norm_le_zero_iff, contLinear_eq_zero_iff_exists_const] at h₂
obtain ⟨q, rfl⟩ := h₂
simp only [norm_eq_zero] at h₁
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₁
rw [h₁]
rfl
· rw [norm_eq_zero', contLinear_eq_zero_iff_exists_const] at h₁
obtain ⟨q, rfl⟩ := h₁
simp only [norm_le_zero_iff] at h₂
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₂
rw [h₂]
rfl }
instance : NormedSpace 𝕜 (V →A[𝕜] W) where
norm_smul_le t f := by
simp only [norm_def, (smul_contLinear), norm_smul]
-- Porting note: previously all these rewrites were in the `simp only`,
-- but now they don't fire.
-- (in fact, `norm_smul` fires, but only once rather than twice!)
rw [coe_smul, Pi.smul_apply, norm_smul, ← mul_max_of_nonneg _ _ (norm_nonneg t)]
theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ := by
rw [norm_def, max_le_iff]
constructor
· calc
‖f.comp g 0‖ = ‖f (g 0)‖ := by simp
_ = ‖f.contLinear (g 0) + f 0‖ := by rw [f.decomp];
|
simp
|
theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ := by
rw [norm_def, max_le_iff]
constructor
· calc
‖f.comp g 0‖ = ‖f (g 0)‖ := by simp
_ = ‖f.contLinear (g 0) + f 0‖ := by rw [f.decomp];
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.231_0.bJ3guivW1IqbUMZ
|
theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
case right
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : V →A[𝕜] W
g : W₂ →A[𝕜] V
⊢ ‖contLinear (comp f g)‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h]; rfl
simp_rw [h₁, h₂]
exact (f : P →ᵃ[R] Q).linear_eq_zero_iff_exists_const
#align continuous_affine_map.cont_linear_eq_zero_iff_exists_const ContinuousAffineMap.contLinear_eq_zero_iff_exists_const
@[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f := by
ext
rfl
#align continuous_affine_map.to_affine_map_cont_linear ContinuousAffineMap.to_affine_map_contLinear
@[simp]
theorem zero_contLinear : (0 : P →A[R] W).contLinear = 0 :=
rfl
#align continuous_affine_map.zero_cont_linear ContinuousAffineMap.zero_contLinear
@[simp]
theorem add_contLinear (f g : P →A[R] W) : (f + g).contLinear = f.contLinear + g.contLinear :=
rfl
#align continuous_affine_map.add_cont_linear ContinuousAffineMap.add_contLinear
@[simp]
theorem sub_contLinear (f g : P →A[R] W) : (f - g).contLinear = f.contLinear - g.contLinear :=
rfl
#align continuous_affine_map.sub_cont_linear ContinuousAffineMap.sub_contLinear
@[simp]
theorem neg_contLinear (f : P →A[R] W) : (-f).contLinear = -f.contLinear :=
rfl
#align continuous_affine_map.neg_cont_linear ContinuousAffineMap.neg_contLinear
@[simp]
theorem smul_contLinear (t : R) (f : P →A[R] W) : (t • f).contLinear = t • f.contLinear :=
rfl
#align continuous_affine_map.smul_cont_linear ContinuousAffineMap.smul_contLinear
theorem decomp (f : V →A[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) := by
rcases f with ⟨f, h⟩
rw [coe_mk_const_linear_eq_linear, coe_mk, f.decomp, Pi.add_apply, LinearMap.map_zero, zero_add,
← Function.const_def]
#align continuous_affine_map.decomp ContinuousAffineMap.decomp
section NormedSpaceStructure
variable (f : V →A[𝕜] W)
/-- Note that unlike the operator norm for linear maps, this norm is _not_ submultiplicative:
we do _not_ necessarily have `‖f.comp g‖ ≤ ‖f‖ * ‖g‖`. See `norm_comp_le` for what we can say. -/
noncomputable instance hasNorm : Norm (V →A[𝕜] W) :=
⟨fun f => max ‖f 0‖ ‖f.contLinear‖⟩
#align continuous_affine_map.has_norm ContinuousAffineMap.hasNorm
theorem norm_def : ‖f‖ = max ‖f 0‖ ‖f.contLinear‖ :=
rfl
#align continuous_affine_map.norm_def ContinuousAffineMap.norm_def
theorem norm_contLinear_le : ‖f.contLinear‖ ≤ ‖f‖ :=
le_max_right _ _
#align continuous_affine_map.norm_cont_linear_le ContinuousAffineMap.norm_contLinear_le
theorem norm_image_zero_le : ‖f 0‖ ≤ ‖f‖ :=
le_max_left _ _
#align continuous_affine_map.norm_image_zero_le ContinuousAffineMap.norm_image_zero_le
@[simp]
theorem norm_eq (h : f 0 = 0) : ‖f‖ = ‖f.contLinear‖ :=
calc
‖f‖ = max ‖f 0‖ ‖f.contLinear‖ := by rw [norm_def]
_ = max 0 ‖f.contLinear‖ := by rw [h, norm_zero]
_ = ‖f.contLinear‖ := max_eq_right (norm_nonneg _)
#align continuous_affine_map.norm_eq ContinuousAffineMap.norm_eq
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
rw [add_contLinear]
exact
⟨(norm_add_le _ _).trans (add_le_add (le_max_left _ _) (le_max_left _ _)),
(norm_add_le _ _).trans (add_le_add (le_max_right _ _) (le_max_right _ _))⟩
eq_zero_of_map_eq_zero' := fun f h₀ => by
rcases max_eq_iff.mp h₀ with (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩) <;> rw [h₁] at h₂
· rw [norm_le_zero_iff, contLinear_eq_zero_iff_exists_const] at h₂
obtain ⟨q, rfl⟩ := h₂
simp only [norm_eq_zero] at h₁
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₁
rw [h₁]
rfl
· rw [norm_eq_zero', contLinear_eq_zero_iff_exists_const] at h₁
obtain ⟨q, rfl⟩ := h₁
simp only [norm_le_zero_iff] at h₂
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₂
rw [h₂]
rfl }
instance : NormedSpace 𝕜 (V →A[𝕜] W) where
norm_smul_le t f := by
simp only [norm_def, (smul_contLinear), norm_smul]
-- Porting note: previously all these rewrites were in the `simp only`,
-- but now they don't fire.
-- (in fact, `norm_smul` fires, but only once rather than twice!)
rw [coe_smul, Pi.smul_apply, norm_smul, ← mul_max_of_nonneg _ _ (norm_nonneg t)]
theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ := by
rw [norm_def, max_le_iff]
constructor
· calc
‖f.comp g 0‖ = ‖f (g 0)‖ := by simp
_ = ‖f.contLinear (g 0) + f 0‖ := by rw [f.decomp]; simp
_ ≤ ‖f.contLinear‖ * ‖g 0‖ + ‖f 0‖ :=
((norm_add_le _ _).trans (add_le_add_right (f.contLinear.le_op_norm _) _))
_ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ :=
add_le_add_right
(mul_le_mul f.norm_contLinear_le g.norm_image_zero_le (norm_nonneg _) (norm_nonneg _)) _
·
|
calc
‖(f.comp g).contLinear‖ ≤ ‖f.contLinear‖ * ‖g.contLinear‖ :=
(g.comp_contLinear f).symm ▸ f.contLinear.op_norm_comp_le _
_ ≤ ‖f‖ * ‖g‖ :=
(mul_le_mul f.norm_contLinear_le g.norm_contLinear_le (norm_nonneg _) (norm_nonneg _))
_ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ := by rw [le_add_iff_nonneg_right]; apply norm_nonneg
|
theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ := by
rw [norm_def, max_le_iff]
constructor
· calc
‖f.comp g 0‖ = ‖f (g 0)‖ := by simp
_ = ‖f.contLinear (g 0) + f 0‖ := by rw [f.decomp]; simp
_ ≤ ‖f.contLinear‖ * ‖g 0‖ + ‖f 0‖ :=
((norm_add_le _ _).trans (add_le_add_right (f.contLinear.le_op_norm _) _))
_ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ :=
add_le_add_right
(mul_le_mul f.norm_contLinear_le g.norm_image_zero_le (norm_nonneg _) (norm_nonneg _)) _
·
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.231_0.bJ3guivW1IqbUMZ
|
theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : V →A[𝕜] W
g : W₂ →A[𝕜] V
⊢ ‖f‖ * ‖g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h]; rfl
simp_rw [h₁, h₂]
exact (f : P →ᵃ[R] Q).linear_eq_zero_iff_exists_const
#align continuous_affine_map.cont_linear_eq_zero_iff_exists_const ContinuousAffineMap.contLinear_eq_zero_iff_exists_const
@[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f := by
ext
rfl
#align continuous_affine_map.to_affine_map_cont_linear ContinuousAffineMap.to_affine_map_contLinear
@[simp]
theorem zero_contLinear : (0 : P →A[R] W).contLinear = 0 :=
rfl
#align continuous_affine_map.zero_cont_linear ContinuousAffineMap.zero_contLinear
@[simp]
theorem add_contLinear (f g : P →A[R] W) : (f + g).contLinear = f.contLinear + g.contLinear :=
rfl
#align continuous_affine_map.add_cont_linear ContinuousAffineMap.add_contLinear
@[simp]
theorem sub_contLinear (f g : P →A[R] W) : (f - g).contLinear = f.contLinear - g.contLinear :=
rfl
#align continuous_affine_map.sub_cont_linear ContinuousAffineMap.sub_contLinear
@[simp]
theorem neg_contLinear (f : P →A[R] W) : (-f).contLinear = -f.contLinear :=
rfl
#align continuous_affine_map.neg_cont_linear ContinuousAffineMap.neg_contLinear
@[simp]
theorem smul_contLinear (t : R) (f : P →A[R] W) : (t • f).contLinear = t • f.contLinear :=
rfl
#align continuous_affine_map.smul_cont_linear ContinuousAffineMap.smul_contLinear
theorem decomp (f : V →A[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) := by
rcases f with ⟨f, h⟩
rw [coe_mk_const_linear_eq_linear, coe_mk, f.decomp, Pi.add_apply, LinearMap.map_zero, zero_add,
← Function.const_def]
#align continuous_affine_map.decomp ContinuousAffineMap.decomp
section NormedSpaceStructure
variable (f : V →A[𝕜] W)
/-- Note that unlike the operator norm for linear maps, this norm is _not_ submultiplicative:
we do _not_ necessarily have `‖f.comp g‖ ≤ ‖f‖ * ‖g‖`. See `norm_comp_le` for what we can say. -/
noncomputable instance hasNorm : Norm (V →A[𝕜] W) :=
⟨fun f => max ‖f 0‖ ‖f.contLinear‖⟩
#align continuous_affine_map.has_norm ContinuousAffineMap.hasNorm
theorem norm_def : ‖f‖ = max ‖f 0‖ ‖f.contLinear‖ :=
rfl
#align continuous_affine_map.norm_def ContinuousAffineMap.norm_def
theorem norm_contLinear_le : ‖f.contLinear‖ ≤ ‖f‖ :=
le_max_right _ _
#align continuous_affine_map.norm_cont_linear_le ContinuousAffineMap.norm_contLinear_le
theorem norm_image_zero_le : ‖f 0‖ ≤ ‖f‖ :=
le_max_left _ _
#align continuous_affine_map.norm_image_zero_le ContinuousAffineMap.norm_image_zero_le
@[simp]
theorem norm_eq (h : f 0 = 0) : ‖f‖ = ‖f.contLinear‖ :=
calc
‖f‖ = max ‖f 0‖ ‖f.contLinear‖ := by rw [norm_def]
_ = max 0 ‖f.contLinear‖ := by rw [h, norm_zero]
_ = ‖f.contLinear‖ := max_eq_right (norm_nonneg _)
#align continuous_affine_map.norm_eq ContinuousAffineMap.norm_eq
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
rw [add_contLinear]
exact
⟨(norm_add_le _ _).trans (add_le_add (le_max_left _ _) (le_max_left _ _)),
(norm_add_le _ _).trans (add_le_add (le_max_right _ _) (le_max_right _ _))⟩
eq_zero_of_map_eq_zero' := fun f h₀ => by
rcases max_eq_iff.mp h₀ with (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩) <;> rw [h₁] at h₂
· rw [norm_le_zero_iff, contLinear_eq_zero_iff_exists_const] at h₂
obtain ⟨q, rfl⟩ := h₂
simp only [norm_eq_zero] at h₁
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₁
rw [h₁]
rfl
· rw [norm_eq_zero', contLinear_eq_zero_iff_exists_const] at h₁
obtain ⟨q, rfl⟩ := h₁
simp only [norm_le_zero_iff] at h₂
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₂
rw [h₂]
rfl }
instance : NormedSpace 𝕜 (V →A[𝕜] W) where
norm_smul_le t f := by
simp only [norm_def, (smul_contLinear), norm_smul]
-- Porting note: previously all these rewrites were in the `simp only`,
-- but now they don't fire.
-- (in fact, `norm_smul` fires, but only once rather than twice!)
rw [coe_smul, Pi.smul_apply, norm_smul, ← mul_max_of_nonneg _ _ (norm_nonneg t)]
theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ := by
rw [norm_def, max_le_iff]
constructor
· calc
‖f.comp g 0‖ = ‖f (g 0)‖ := by simp
_ = ‖f.contLinear (g 0) + f 0‖ := by rw [f.decomp]; simp
_ ≤ ‖f.contLinear‖ * ‖g 0‖ + ‖f 0‖ :=
((norm_add_le _ _).trans (add_le_add_right (f.contLinear.le_op_norm _) _))
_ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ :=
add_le_add_right
(mul_le_mul f.norm_contLinear_le g.norm_image_zero_le (norm_nonneg _) (norm_nonneg _)) _
· calc
‖(f.comp g).contLinear‖ ≤ ‖f.contLinear‖ * ‖g.contLinear‖ :=
(g.comp_contLinear f).symm ▸ f.contLinear.op_norm_comp_le _
_ ≤ ‖f‖ * ‖g‖ :=
(mul_le_mul f.norm_contLinear_le g.norm_contLinear_le (norm_nonneg _) (norm_nonneg _))
_ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ := by
|
rw [le_add_iff_nonneg_right]
|
theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ := by
rw [norm_def, max_le_iff]
constructor
· calc
‖f.comp g 0‖ = ‖f (g 0)‖ := by simp
_ = ‖f.contLinear (g 0) + f 0‖ := by rw [f.decomp]; simp
_ ≤ ‖f.contLinear‖ * ‖g 0‖ + ‖f 0‖ :=
((norm_add_le _ _).trans (add_le_add_right (f.contLinear.le_op_norm _) _))
_ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ :=
add_le_add_right
(mul_le_mul f.norm_contLinear_le g.norm_image_zero_le (norm_nonneg _) (norm_nonneg _)) _
· calc
‖(f.comp g).contLinear‖ ≤ ‖f.contLinear‖ * ‖g.contLinear‖ :=
(g.comp_contLinear f).symm ▸ f.contLinear.op_norm_comp_le _
_ ≤ ‖f‖ * ‖g‖ :=
(mul_le_mul f.norm_contLinear_le g.norm_contLinear_le (norm_nonneg _) (norm_nonneg _))
_ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ := by
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.231_0.bJ3guivW1IqbUMZ
|
theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : V →A[𝕜] W
g : W₂ →A[𝕜] V
⊢ 0 ≤ ‖f 0‖
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h]; rfl
simp_rw [h₁, h₂]
exact (f : P →ᵃ[R] Q).linear_eq_zero_iff_exists_const
#align continuous_affine_map.cont_linear_eq_zero_iff_exists_const ContinuousAffineMap.contLinear_eq_zero_iff_exists_const
@[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f := by
ext
rfl
#align continuous_affine_map.to_affine_map_cont_linear ContinuousAffineMap.to_affine_map_contLinear
@[simp]
theorem zero_contLinear : (0 : P →A[R] W).contLinear = 0 :=
rfl
#align continuous_affine_map.zero_cont_linear ContinuousAffineMap.zero_contLinear
@[simp]
theorem add_contLinear (f g : P →A[R] W) : (f + g).contLinear = f.contLinear + g.contLinear :=
rfl
#align continuous_affine_map.add_cont_linear ContinuousAffineMap.add_contLinear
@[simp]
theorem sub_contLinear (f g : P →A[R] W) : (f - g).contLinear = f.contLinear - g.contLinear :=
rfl
#align continuous_affine_map.sub_cont_linear ContinuousAffineMap.sub_contLinear
@[simp]
theorem neg_contLinear (f : P →A[R] W) : (-f).contLinear = -f.contLinear :=
rfl
#align continuous_affine_map.neg_cont_linear ContinuousAffineMap.neg_contLinear
@[simp]
theorem smul_contLinear (t : R) (f : P →A[R] W) : (t • f).contLinear = t • f.contLinear :=
rfl
#align continuous_affine_map.smul_cont_linear ContinuousAffineMap.smul_contLinear
theorem decomp (f : V →A[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) := by
rcases f with ⟨f, h⟩
rw [coe_mk_const_linear_eq_linear, coe_mk, f.decomp, Pi.add_apply, LinearMap.map_zero, zero_add,
← Function.const_def]
#align continuous_affine_map.decomp ContinuousAffineMap.decomp
section NormedSpaceStructure
variable (f : V →A[𝕜] W)
/-- Note that unlike the operator norm for linear maps, this norm is _not_ submultiplicative:
we do _not_ necessarily have `‖f.comp g‖ ≤ ‖f‖ * ‖g‖`. See `norm_comp_le` for what we can say. -/
noncomputable instance hasNorm : Norm (V →A[𝕜] W) :=
⟨fun f => max ‖f 0‖ ‖f.contLinear‖⟩
#align continuous_affine_map.has_norm ContinuousAffineMap.hasNorm
theorem norm_def : ‖f‖ = max ‖f 0‖ ‖f.contLinear‖ :=
rfl
#align continuous_affine_map.norm_def ContinuousAffineMap.norm_def
theorem norm_contLinear_le : ‖f.contLinear‖ ≤ ‖f‖ :=
le_max_right _ _
#align continuous_affine_map.norm_cont_linear_le ContinuousAffineMap.norm_contLinear_le
theorem norm_image_zero_le : ‖f 0‖ ≤ ‖f‖ :=
le_max_left _ _
#align continuous_affine_map.norm_image_zero_le ContinuousAffineMap.norm_image_zero_le
@[simp]
theorem norm_eq (h : f 0 = 0) : ‖f‖ = ‖f.contLinear‖ :=
calc
‖f‖ = max ‖f 0‖ ‖f.contLinear‖ := by rw [norm_def]
_ = max 0 ‖f.contLinear‖ := by rw [h, norm_zero]
_ = ‖f.contLinear‖ := max_eq_right (norm_nonneg _)
#align continuous_affine_map.norm_eq ContinuousAffineMap.norm_eq
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
rw [add_contLinear]
exact
⟨(norm_add_le _ _).trans (add_le_add (le_max_left _ _) (le_max_left _ _)),
(norm_add_le _ _).trans (add_le_add (le_max_right _ _) (le_max_right _ _))⟩
eq_zero_of_map_eq_zero' := fun f h₀ => by
rcases max_eq_iff.mp h₀ with (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩) <;> rw [h₁] at h₂
· rw [norm_le_zero_iff, contLinear_eq_zero_iff_exists_const] at h₂
obtain ⟨q, rfl⟩ := h₂
simp only [norm_eq_zero] at h₁
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₁
rw [h₁]
rfl
· rw [norm_eq_zero', contLinear_eq_zero_iff_exists_const] at h₁
obtain ⟨q, rfl⟩ := h₁
simp only [norm_le_zero_iff] at h₂
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₂
rw [h₂]
rfl }
instance : NormedSpace 𝕜 (V →A[𝕜] W) where
norm_smul_le t f := by
simp only [norm_def, (smul_contLinear), norm_smul]
-- Porting note: previously all these rewrites were in the `simp only`,
-- but now they don't fire.
-- (in fact, `norm_smul` fires, but only once rather than twice!)
rw [coe_smul, Pi.smul_apply, norm_smul, ← mul_max_of_nonneg _ _ (norm_nonneg t)]
theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ := by
rw [norm_def, max_le_iff]
constructor
· calc
‖f.comp g 0‖ = ‖f (g 0)‖ := by simp
_ = ‖f.contLinear (g 0) + f 0‖ := by rw [f.decomp]; simp
_ ≤ ‖f.contLinear‖ * ‖g 0‖ + ‖f 0‖ :=
((norm_add_le _ _).trans (add_le_add_right (f.contLinear.le_op_norm _) _))
_ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ :=
add_le_add_right
(mul_le_mul f.norm_contLinear_le g.norm_image_zero_le (norm_nonneg _) (norm_nonneg _)) _
· calc
‖(f.comp g).contLinear‖ ≤ ‖f.contLinear‖ * ‖g.contLinear‖ :=
(g.comp_contLinear f).symm ▸ f.contLinear.op_norm_comp_le _
_ ≤ ‖f‖ * ‖g‖ :=
(mul_le_mul f.norm_contLinear_le g.norm_contLinear_le (norm_nonneg _) (norm_nonneg _))
_ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ := by rw [le_add_iff_nonneg_right];
|
apply norm_nonneg
|
theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ := by
rw [norm_def, max_le_iff]
constructor
· calc
‖f.comp g 0‖ = ‖f (g 0)‖ := by simp
_ = ‖f.contLinear (g 0) + f 0‖ := by rw [f.decomp]; simp
_ ≤ ‖f.contLinear‖ * ‖g 0‖ + ‖f 0‖ :=
((norm_add_le _ _).trans (add_le_add_right (f.contLinear.le_op_norm _) _))
_ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ :=
add_le_add_right
(mul_le_mul f.norm_contLinear_le g.norm_image_zero_le (norm_nonneg _) (norm_nonneg _)) _
· calc
‖(f.comp g).contLinear‖ ≤ ‖f.contLinear‖ * ‖g.contLinear‖ :=
(g.comp_contLinear f).symm ▸ f.contLinear.op_norm_comp_le _
_ ≤ ‖f‖ * ‖g‖ :=
(mul_le_mul f.norm_contLinear_le g.norm_contLinear_le (norm_nonneg _) (norm_nonneg _))
_ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ := by rw [le_add_iff_nonneg_right];
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.231_0.bJ3guivW1IqbUMZ
|
theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f✝ f : V →A[𝕜] W
⊢ (fun p => ContinuousLinearMap.toContinuousAffineMap p.2 + const 𝕜 V p.1)
(AddHom.toFun
{
toAddHom :=
{ toFun := fun f => (f 0, contLinear f),
map_add' :=
(_ :
∀ (x x_1 : V →A[𝕜] W),
(fun f => (f 0, contLinear f)) (x + x_1) = (fun f => (f 0, contLinear f)) (x + x_1)) },
map_smul' :=
(_ :
∀ (x : 𝕜) (x_1 : V →A[𝕜] W),
AddHom.toFun
{ toFun := fun f => (f 0, contLinear f),
map_add' :=
(_ :
∀ (x x_2 : V →A[𝕜] W),
(fun f => (f 0, contLinear f)) (x + x_2) = (fun f => (f 0, contLinear f)) (x + x_2)) }
(x • x_1) =
AddHom.toFun
{ toFun := fun f => (f 0, contLinear f),
map_add' :=
(_ :
∀ (x x_2 : V →A[𝕜] W),
(fun f => (f 0, contLinear f)) (x + x_2) = (fun f => (f 0, contLinear f)) (x + x_2)) }
(x • x_1)) }.toAddHom
f) =
f
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h]; rfl
simp_rw [h₁, h₂]
exact (f : P →ᵃ[R] Q).linear_eq_zero_iff_exists_const
#align continuous_affine_map.cont_linear_eq_zero_iff_exists_const ContinuousAffineMap.contLinear_eq_zero_iff_exists_const
@[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f := by
ext
rfl
#align continuous_affine_map.to_affine_map_cont_linear ContinuousAffineMap.to_affine_map_contLinear
@[simp]
theorem zero_contLinear : (0 : P →A[R] W).contLinear = 0 :=
rfl
#align continuous_affine_map.zero_cont_linear ContinuousAffineMap.zero_contLinear
@[simp]
theorem add_contLinear (f g : P →A[R] W) : (f + g).contLinear = f.contLinear + g.contLinear :=
rfl
#align continuous_affine_map.add_cont_linear ContinuousAffineMap.add_contLinear
@[simp]
theorem sub_contLinear (f g : P →A[R] W) : (f - g).contLinear = f.contLinear - g.contLinear :=
rfl
#align continuous_affine_map.sub_cont_linear ContinuousAffineMap.sub_contLinear
@[simp]
theorem neg_contLinear (f : P →A[R] W) : (-f).contLinear = -f.contLinear :=
rfl
#align continuous_affine_map.neg_cont_linear ContinuousAffineMap.neg_contLinear
@[simp]
theorem smul_contLinear (t : R) (f : P →A[R] W) : (t • f).contLinear = t • f.contLinear :=
rfl
#align continuous_affine_map.smul_cont_linear ContinuousAffineMap.smul_contLinear
theorem decomp (f : V →A[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) := by
rcases f with ⟨f, h⟩
rw [coe_mk_const_linear_eq_linear, coe_mk, f.decomp, Pi.add_apply, LinearMap.map_zero, zero_add,
← Function.const_def]
#align continuous_affine_map.decomp ContinuousAffineMap.decomp
section NormedSpaceStructure
variable (f : V →A[𝕜] W)
/-- Note that unlike the operator norm for linear maps, this norm is _not_ submultiplicative:
we do _not_ necessarily have `‖f.comp g‖ ≤ ‖f‖ * ‖g‖`. See `norm_comp_le` for what we can say. -/
noncomputable instance hasNorm : Norm (V →A[𝕜] W) :=
⟨fun f => max ‖f 0‖ ‖f.contLinear‖⟩
#align continuous_affine_map.has_norm ContinuousAffineMap.hasNorm
theorem norm_def : ‖f‖ = max ‖f 0‖ ‖f.contLinear‖ :=
rfl
#align continuous_affine_map.norm_def ContinuousAffineMap.norm_def
theorem norm_contLinear_le : ‖f.contLinear‖ ≤ ‖f‖ :=
le_max_right _ _
#align continuous_affine_map.norm_cont_linear_le ContinuousAffineMap.norm_contLinear_le
theorem norm_image_zero_le : ‖f 0‖ ≤ ‖f‖ :=
le_max_left _ _
#align continuous_affine_map.norm_image_zero_le ContinuousAffineMap.norm_image_zero_le
@[simp]
theorem norm_eq (h : f 0 = 0) : ‖f‖ = ‖f.contLinear‖ :=
calc
‖f‖ = max ‖f 0‖ ‖f.contLinear‖ := by rw [norm_def]
_ = max 0 ‖f.contLinear‖ := by rw [h, norm_zero]
_ = ‖f.contLinear‖ := max_eq_right (norm_nonneg _)
#align continuous_affine_map.norm_eq ContinuousAffineMap.norm_eq
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
rw [add_contLinear]
exact
⟨(norm_add_le _ _).trans (add_le_add (le_max_left _ _) (le_max_left _ _)),
(norm_add_le _ _).trans (add_le_add (le_max_right _ _) (le_max_right _ _))⟩
eq_zero_of_map_eq_zero' := fun f h₀ => by
rcases max_eq_iff.mp h₀ with (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩) <;> rw [h₁] at h₂
· rw [norm_le_zero_iff, contLinear_eq_zero_iff_exists_const] at h₂
obtain ⟨q, rfl⟩ := h₂
simp only [norm_eq_zero] at h₁
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₁
rw [h₁]
rfl
· rw [norm_eq_zero', contLinear_eq_zero_iff_exists_const] at h₁
obtain ⟨q, rfl⟩ := h₁
simp only [norm_le_zero_iff] at h₂
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₂
rw [h₂]
rfl }
instance : NormedSpace 𝕜 (V →A[𝕜] W) where
norm_smul_le t f := by
simp only [norm_def, (smul_contLinear), norm_smul]
-- Porting note: previously all these rewrites were in the `simp only`,
-- but now they don't fire.
-- (in fact, `norm_smul` fires, but only once rather than twice!)
rw [coe_smul, Pi.smul_apply, norm_smul, ← mul_max_of_nonneg _ _ (norm_nonneg t)]
theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ := by
rw [norm_def, max_le_iff]
constructor
· calc
‖f.comp g 0‖ = ‖f (g 0)‖ := by simp
_ = ‖f.contLinear (g 0) + f 0‖ := by rw [f.decomp]; simp
_ ≤ ‖f.contLinear‖ * ‖g 0‖ + ‖f 0‖ :=
((norm_add_le _ _).trans (add_le_add_right (f.contLinear.le_op_norm _) _))
_ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ :=
add_le_add_right
(mul_le_mul f.norm_contLinear_le g.norm_image_zero_le (norm_nonneg _) (norm_nonneg _)) _
· calc
‖(f.comp g).contLinear‖ ≤ ‖f.contLinear‖ * ‖g.contLinear‖ :=
(g.comp_contLinear f).symm ▸ f.contLinear.op_norm_comp_le _
_ ≤ ‖f‖ * ‖g‖ :=
(mul_le_mul f.norm_contLinear_le g.norm_contLinear_le (norm_nonneg _) (norm_nonneg _))
_ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ := by rw [le_add_iff_nonneg_right]; apply norm_nonneg
#align continuous_affine_map.norm_comp_le ContinuousAffineMap.norm_comp_le
variable (𝕜 V W)
/-- The space of affine maps between two normed spaces is linearly isometric to the product of the
codomain with the space of linear maps, by taking the value of the affine map at `(0 : V)` and the
linear part. -/
def toConstProdContinuousLinearMap : (V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W) where
toFun f := ⟨f 0, f.contLinear⟩
invFun p := p.2.toContinuousAffineMap + const 𝕜 V p.1
left_inv f := by
|
ext
|
/-- The space of affine maps between two normed spaces is linearly isometric to the product of the
codomain with the space of linear maps, by taking the value of the affine map at `(0 : V)` and the
linear part. -/
def toConstProdContinuousLinearMap : (V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W) where
toFun f := ⟨f 0, f.contLinear⟩
invFun p := p.2.toContinuousAffineMap + const 𝕜 V p.1
left_inv f := by
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.253_0.bJ3guivW1IqbUMZ
|
/-- The space of affine maps between two normed spaces is linearly isometric to the product of the
codomain with the space of linear maps, by taking the value of the affine map at `(0 : V)` and the
linear part. -/
def toConstProdContinuousLinearMap : (V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W) where
toFun f
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
case h
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f✝ f : V →A[𝕜] W
x✝ : V
⊢ ((fun p => ContinuousLinearMap.toContinuousAffineMap p.2 + const 𝕜 V p.1)
(AddHom.toFun
{
toAddHom :=
{ toFun := fun f => (f 0, contLinear f),
map_add' :=
(_ :
∀ (x x_1 : V →A[𝕜] W),
(fun f => (f 0, contLinear f)) (x + x_1) = (fun f => (f 0, contLinear f)) (x + x_1)) },
map_smul' :=
(_ :
∀ (x : 𝕜) (x_1 : V →A[𝕜] W),
AddHom.toFun
{ toFun := fun f => (f 0, contLinear f),
map_add' :=
(_ :
∀ (x x_2 : V →A[𝕜] W),
(fun f => (f 0, contLinear f)) (x + x_2) = (fun f => (f 0, contLinear f)) (x + x_2)) }
(x • x_1) =
AddHom.toFun
{ toFun := fun f => (f 0, contLinear f),
map_add' :=
(_ :
∀ (x x_2 : V →A[𝕜] W),
(fun f => (f 0, contLinear f)) (x + x_2) = (fun f => (f 0, contLinear f)) (x + x_2)) }
(x • x_1)) }.toAddHom
f))
x✝ =
f x✝
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h]; rfl
simp_rw [h₁, h₂]
exact (f : P →ᵃ[R] Q).linear_eq_zero_iff_exists_const
#align continuous_affine_map.cont_linear_eq_zero_iff_exists_const ContinuousAffineMap.contLinear_eq_zero_iff_exists_const
@[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f := by
ext
rfl
#align continuous_affine_map.to_affine_map_cont_linear ContinuousAffineMap.to_affine_map_contLinear
@[simp]
theorem zero_contLinear : (0 : P →A[R] W).contLinear = 0 :=
rfl
#align continuous_affine_map.zero_cont_linear ContinuousAffineMap.zero_contLinear
@[simp]
theorem add_contLinear (f g : P →A[R] W) : (f + g).contLinear = f.contLinear + g.contLinear :=
rfl
#align continuous_affine_map.add_cont_linear ContinuousAffineMap.add_contLinear
@[simp]
theorem sub_contLinear (f g : P →A[R] W) : (f - g).contLinear = f.contLinear - g.contLinear :=
rfl
#align continuous_affine_map.sub_cont_linear ContinuousAffineMap.sub_contLinear
@[simp]
theorem neg_contLinear (f : P →A[R] W) : (-f).contLinear = -f.contLinear :=
rfl
#align continuous_affine_map.neg_cont_linear ContinuousAffineMap.neg_contLinear
@[simp]
theorem smul_contLinear (t : R) (f : P →A[R] W) : (t • f).contLinear = t • f.contLinear :=
rfl
#align continuous_affine_map.smul_cont_linear ContinuousAffineMap.smul_contLinear
theorem decomp (f : V →A[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) := by
rcases f with ⟨f, h⟩
rw [coe_mk_const_linear_eq_linear, coe_mk, f.decomp, Pi.add_apply, LinearMap.map_zero, zero_add,
← Function.const_def]
#align continuous_affine_map.decomp ContinuousAffineMap.decomp
section NormedSpaceStructure
variable (f : V →A[𝕜] W)
/-- Note that unlike the operator norm for linear maps, this norm is _not_ submultiplicative:
we do _not_ necessarily have `‖f.comp g‖ ≤ ‖f‖ * ‖g‖`. See `norm_comp_le` for what we can say. -/
noncomputable instance hasNorm : Norm (V →A[𝕜] W) :=
⟨fun f => max ‖f 0‖ ‖f.contLinear‖⟩
#align continuous_affine_map.has_norm ContinuousAffineMap.hasNorm
theorem norm_def : ‖f‖ = max ‖f 0‖ ‖f.contLinear‖ :=
rfl
#align continuous_affine_map.norm_def ContinuousAffineMap.norm_def
theorem norm_contLinear_le : ‖f.contLinear‖ ≤ ‖f‖ :=
le_max_right _ _
#align continuous_affine_map.norm_cont_linear_le ContinuousAffineMap.norm_contLinear_le
theorem norm_image_zero_le : ‖f 0‖ ≤ ‖f‖ :=
le_max_left _ _
#align continuous_affine_map.norm_image_zero_le ContinuousAffineMap.norm_image_zero_le
@[simp]
theorem norm_eq (h : f 0 = 0) : ‖f‖ = ‖f.contLinear‖ :=
calc
‖f‖ = max ‖f 0‖ ‖f.contLinear‖ := by rw [norm_def]
_ = max 0 ‖f.contLinear‖ := by rw [h, norm_zero]
_ = ‖f.contLinear‖ := max_eq_right (norm_nonneg _)
#align continuous_affine_map.norm_eq ContinuousAffineMap.norm_eq
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
rw [add_contLinear]
exact
⟨(norm_add_le _ _).trans (add_le_add (le_max_left _ _) (le_max_left _ _)),
(norm_add_le _ _).trans (add_le_add (le_max_right _ _) (le_max_right _ _))⟩
eq_zero_of_map_eq_zero' := fun f h₀ => by
rcases max_eq_iff.mp h₀ with (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩) <;> rw [h₁] at h₂
· rw [norm_le_zero_iff, contLinear_eq_zero_iff_exists_const] at h₂
obtain ⟨q, rfl⟩ := h₂
simp only [norm_eq_zero] at h₁
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₁
rw [h₁]
rfl
· rw [norm_eq_zero', contLinear_eq_zero_iff_exists_const] at h₁
obtain ⟨q, rfl⟩ := h₁
simp only [norm_le_zero_iff] at h₂
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₂
rw [h₂]
rfl }
instance : NormedSpace 𝕜 (V →A[𝕜] W) where
norm_smul_le t f := by
simp only [norm_def, (smul_contLinear), norm_smul]
-- Porting note: previously all these rewrites were in the `simp only`,
-- but now they don't fire.
-- (in fact, `norm_smul` fires, but only once rather than twice!)
rw [coe_smul, Pi.smul_apply, norm_smul, ← mul_max_of_nonneg _ _ (norm_nonneg t)]
theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ := by
rw [norm_def, max_le_iff]
constructor
· calc
‖f.comp g 0‖ = ‖f (g 0)‖ := by simp
_ = ‖f.contLinear (g 0) + f 0‖ := by rw [f.decomp]; simp
_ ≤ ‖f.contLinear‖ * ‖g 0‖ + ‖f 0‖ :=
((norm_add_le _ _).trans (add_le_add_right (f.contLinear.le_op_norm _) _))
_ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ :=
add_le_add_right
(mul_le_mul f.norm_contLinear_le g.norm_image_zero_le (norm_nonneg _) (norm_nonneg _)) _
· calc
‖(f.comp g).contLinear‖ ≤ ‖f.contLinear‖ * ‖g.contLinear‖ :=
(g.comp_contLinear f).symm ▸ f.contLinear.op_norm_comp_le _
_ ≤ ‖f‖ * ‖g‖ :=
(mul_le_mul f.norm_contLinear_le g.norm_contLinear_le (norm_nonneg _) (norm_nonneg _))
_ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ := by rw [le_add_iff_nonneg_right]; apply norm_nonneg
#align continuous_affine_map.norm_comp_le ContinuousAffineMap.norm_comp_le
variable (𝕜 V W)
/-- The space of affine maps between two normed spaces is linearly isometric to the product of the
codomain with the space of linear maps, by taking the value of the affine map at `(0 : V)` and the
linear part. -/
def toConstProdContinuousLinearMap : (V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W) where
toFun f := ⟨f 0, f.contLinear⟩
invFun p := p.2.toContinuousAffineMap + const 𝕜 V p.1
left_inv f := by
ext
|
rw [f.decomp]
|
/-- The space of affine maps between two normed spaces is linearly isometric to the product of the
codomain with the space of linear maps, by taking the value of the affine map at `(0 : V)` and the
linear part. -/
def toConstProdContinuousLinearMap : (V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W) where
toFun f := ⟨f 0, f.contLinear⟩
invFun p := p.2.toContinuousAffineMap + const 𝕜 V p.1
left_inv f := by
ext
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.253_0.bJ3guivW1IqbUMZ
|
/-- The space of affine maps between two normed spaces is linearly isometric to the product of the
codomain with the space of linear maps, by taking the value of the affine map at `(0 : V)` and the
linear part. -/
def toConstProdContinuousLinearMap : (V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W) where
toFun f
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
case h
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f✝ f : V →A[𝕜] W
x✝ : V
⊢ ((fun p => ContinuousLinearMap.toContinuousAffineMap p.2 + const 𝕜 V p.1)
(AddHom.toFun
{
toAddHom :=
{ toFun := fun f => (f 0, contLinear f),
map_add' :=
(_ :
∀ (x x_1 : V →A[𝕜] W),
(fun f => (f 0, contLinear f)) (x + x_1) = (fun f => (f 0, contLinear f)) (x + x_1)) },
map_smul' :=
(_ :
∀ (x : 𝕜) (x_1 : V →A[𝕜] W),
AddHom.toFun
{ toFun := fun f => (f 0, contLinear f),
map_add' :=
(_ :
∀ (x x_2 : V →A[𝕜] W),
(fun f => (f 0, contLinear f)) (x + x_2) = (fun f => (f 0, contLinear f)) (x + x_2)) }
(x • x_1) =
AddHom.toFun
{ toFun := fun f => (f 0, contLinear f),
map_add' :=
(_ :
∀ (x x_2 : V →A[𝕜] W),
(fun f => (f 0, contLinear f)) (x + x_2) = (fun f => (f 0, contLinear f)) (x + x_2)) }
(x • x_1)) }.toAddHom
f))
x✝ =
(⇑(contLinear f) + Function.const V (f 0)) x✝
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h]; rfl
simp_rw [h₁, h₂]
exact (f : P →ᵃ[R] Q).linear_eq_zero_iff_exists_const
#align continuous_affine_map.cont_linear_eq_zero_iff_exists_const ContinuousAffineMap.contLinear_eq_zero_iff_exists_const
@[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f := by
ext
rfl
#align continuous_affine_map.to_affine_map_cont_linear ContinuousAffineMap.to_affine_map_contLinear
@[simp]
theorem zero_contLinear : (0 : P →A[R] W).contLinear = 0 :=
rfl
#align continuous_affine_map.zero_cont_linear ContinuousAffineMap.zero_contLinear
@[simp]
theorem add_contLinear (f g : P →A[R] W) : (f + g).contLinear = f.contLinear + g.contLinear :=
rfl
#align continuous_affine_map.add_cont_linear ContinuousAffineMap.add_contLinear
@[simp]
theorem sub_contLinear (f g : P →A[R] W) : (f - g).contLinear = f.contLinear - g.contLinear :=
rfl
#align continuous_affine_map.sub_cont_linear ContinuousAffineMap.sub_contLinear
@[simp]
theorem neg_contLinear (f : P →A[R] W) : (-f).contLinear = -f.contLinear :=
rfl
#align continuous_affine_map.neg_cont_linear ContinuousAffineMap.neg_contLinear
@[simp]
theorem smul_contLinear (t : R) (f : P →A[R] W) : (t • f).contLinear = t • f.contLinear :=
rfl
#align continuous_affine_map.smul_cont_linear ContinuousAffineMap.smul_contLinear
theorem decomp (f : V →A[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) := by
rcases f with ⟨f, h⟩
rw [coe_mk_const_linear_eq_linear, coe_mk, f.decomp, Pi.add_apply, LinearMap.map_zero, zero_add,
← Function.const_def]
#align continuous_affine_map.decomp ContinuousAffineMap.decomp
section NormedSpaceStructure
variable (f : V →A[𝕜] W)
/-- Note that unlike the operator norm for linear maps, this norm is _not_ submultiplicative:
we do _not_ necessarily have `‖f.comp g‖ ≤ ‖f‖ * ‖g‖`. See `norm_comp_le` for what we can say. -/
noncomputable instance hasNorm : Norm (V →A[𝕜] W) :=
⟨fun f => max ‖f 0‖ ‖f.contLinear‖⟩
#align continuous_affine_map.has_norm ContinuousAffineMap.hasNorm
theorem norm_def : ‖f‖ = max ‖f 0‖ ‖f.contLinear‖ :=
rfl
#align continuous_affine_map.norm_def ContinuousAffineMap.norm_def
theorem norm_contLinear_le : ‖f.contLinear‖ ≤ ‖f‖ :=
le_max_right _ _
#align continuous_affine_map.norm_cont_linear_le ContinuousAffineMap.norm_contLinear_le
theorem norm_image_zero_le : ‖f 0‖ ≤ ‖f‖ :=
le_max_left _ _
#align continuous_affine_map.norm_image_zero_le ContinuousAffineMap.norm_image_zero_le
@[simp]
theorem norm_eq (h : f 0 = 0) : ‖f‖ = ‖f.contLinear‖ :=
calc
‖f‖ = max ‖f 0‖ ‖f.contLinear‖ := by rw [norm_def]
_ = max 0 ‖f.contLinear‖ := by rw [h, norm_zero]
_ = ‖f.contLinear‖ := max_eq_right (norm_nonneg _)
#align continuous_affine_map.norm_eq ContinuousAffineMap.norm_eq
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
rw [add_contLinear]
exact
⟨(norm_add_le _ _).trans (add_le_add (le_max_left _ _) (le_max_left _ _)),
(norm_add_le _ _).trans (add_le_add (le_max_right _ _) (le_max_right _ _))⟩
eq_zero_of_map_eq_zero' := fun f h₀ => by
rcases max_eq_iff.mp h₀ with (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩) <;> rw [h₁] at h₂
· rw [norm_le_zero_iff, contLinear_eq_zero_iff_exists_const] at h₂
obtain ⟨q, rfl⟩ := h₂
simp only [norm_eq_zero] at h₁
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₁
rw [h₁]
rfl
· rw [norm_eq_zero', contLinear_eq_zero_iff_exists_const] at h₁
obtain ⟨q, rfl⟩ := h₁
simp only [norm_le_zero_iff] at h₂
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₂
rw [h₂]
rfl }
instance : NormedSpace 𝕜 (V →A[𝕜] W) where
norm_smul_le t f := by
simp only [norm_def, (smul_contLinear), norm_smul]
-- Porting note: previously all these rewrites were in the `simp only`,
-- but now they don't fire.
-- (in fact, `norm_smul` fires, but only once rather than twice!)
rw [coe_smul, Pi.smul_apply, norm_smul, ← mul_max_of_nonneg _ _ (norm_nonneg t)]
theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ := by
rw [norm_def, max_le_iff]
constructor
· calc
‖f.comp g 0‖ = ‖f (g 0)‖ := by simp
_ = ‖f.contLinear (g 0) + f 0‖ := by rw [f.decomp]; simp
_ ≤ ‖f.contLinear‖ * ‖g 0‖ + ‖f 0‖ :=
((norm_add_le _ _).trans (add_le_add_right (f.contLinear.le_op_norm _) _))
_ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ :=
add_le_add_right
(mul_le_mul f.norm_contLinear_le g.norm_image_zero_le (norm_nonneg _) (norm_nonneg _)) _
· calc
‖(f.comp g).contLinear‖ ≤ ‖f.contLinear‖ * ‖g.contLinear‖ :=
(g.comp_contLinear f).symm ▸ f.contLinear.op_norm_comp_le _
_ ≤ ‖f‖ * ‖g‖ :=
(mul_le_mul f.norm_contLinear_le g.norm_contLinear_le (norm_nonneg _) (norm_nonneg _))
_ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ := by rw [le_add_iff_nonneg_right]; apply norm_nonneg
#align continuous_affine_map.norm_comp_le ContinuousAffineMap.norm_comp_le
variable (𝕜 V W)
/-- The space of affine maps between two normed spaces is linearly isometric to the product of the
codomain with the space of linear maps, by taking the value of the affine map at `(0 : V)` and the
linear part. -/
def toConstProdContinuousLinearMap : (V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W) where
toFun f := ⟨f 0, f.contLinear⟩
invFun p := p.2.toContinuousAffineMap + const 𝕜 V p.1
left_inv f := by
ext
rw [f.decomp]
-- Porting note: previously `simp` closed the goal, but now we need to rewrite:
|
simp only [coe_add, ContinuousLinearMap.coe_toContinuousAffineMap, Pi.add_apply]
|
/-- The space of affine maps between two normed spaces is linearly isometric to the product of the
codomain with the space of linear maps, by taking the value of the affine map at `(0 : V)` and the
linear part. -/
def toConstProdContinuousLinearMap : (V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W) where
toFun f := ⟨f 0, f.contLinear⟩
invFun p := p.2.toContinuousAffineMap + const 𝕜 V p.1
left_inv f := by
ext
rw [f.decomp]
-- Porting note: previously `simp` closed the goal, but now we need to rewrite:
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.253_0.bJ3guivW1IqbUMZ
|
/-- The space of affine maps between two normed spaces is linearly isometric to the product of the
codomain with the space of linear maps, by taking the value of the affine map at `(0 : V)` and the
linear part. -/
def toConstProdContinuousLinearMap : (V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W) where
toFun f
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
case h
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f✝ f : V →A[𝕜] W
x✝ : V
⊢ (contLinear f) x✝ + (const 𝕜 V (f 0)) x✝ = (contLinear f) x✝ + Function.const V (f 0) x✝
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h]; rfl
simp_rw [h₁, h₂]
exact (f : P →ᵃ[R] Q).linear_eq_zero_iff_exists_const
#align continuous_affine_map.cont_linear_eq_zero_iff_exists_const ContinuousAffineMap.contLinear_eq_zero_iff_exists_const
@[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f := by
ext
rfl
#align continuous_affine_map.to_affine_map_cont_linear ContinuousAffineMap.to_affine_map_contLinear
@[simp]
theorem zero_contLinear : (0 : P →A[R] W).contLinear = 0 :=
rfl
#align continuous_affine_map.zero_cont_linear ContinuousAffineMap.zero_contLinear
@[simp]
theorem add_contLinear (f g : P →A[R] W) : (f + g).contLinear = f.contLinear + g.contLinear :=
rfl
#align continuous_affine_map.add_cont_linear ContinuousAffineMap.add_contLinear
@[simp]
theorem sub_contLinear (f g : P →A[R] W) : (f - g).contLinear = f.contLinear - g.contLinear :=
rfl
#align continuous_affine_map.sub_cont_linear ContinuousAffineMap.sub_contLinear
@[simp]
theorem neg_contLinear (f : P →A[R] W) : (-f).contLinear = -f.contLinear :=
rfl
#align continuous_affine_map.neg_cont_linear ContinuousAffineMap.neg_contLinear
@[simp]
theorem smul_contLinear (t : R) (f : P →A[R] W) : (t • f).contLinear = t • f.contLinear :=
rfl
#align continuous_affine_map.smul_cont_linear ContinuousAffineMap.smul_contLinear
theorem decomp (f : V →A[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) := by
rcases f with ⟨f, h⟩
rw [coe_mk_const_linear_eq_linear, coe_mk, f.decomp, Pi.add_apply, LinearMap.map_zero, zero_add,
← Function.const_def]
#align continuous_affine_map.decomp ContinuousAffineMap.decomp
section NormedSpaceStructure
variable (f : V →A[𝕜] W)
/-- Note that unlike the operator norm for linear maps, this norm is _not_ submultiplicative:
we do _not_ necessarily have `‖f.comp g‖ ≤ ‖f‖ * ‖g‖`. See `norm_comp_le` for what we can say. -/
noncomputable instance hasNorm : Norm (V →A[𝕜] W) :=
⟨fun f => max ‖f 0‖ ‖f.contLinear‖⟩
#align continuous_affine_map.has_norm ContinuousAffineMap.hasNorm
theorem norm_def : ‖f‖ = max ‖f 0‖ ‖f.contLinear‖ :=
rfl
#align continuous_affine_map.norm_def ContinuousAffineMap.norm_def
theorem norm_contLinear_le : ‖f.contLinear‖ ≤ ‖f‖ :=
le_max_right _ _
#align continuous_affine_map.norm_cont_linear_le ContinuousAffineMap.norm_contLinear_le
theorem norm_image_zero_le : ‖f 0‖ ≤ ‖f‖ :=
le_max_left _ _
#align continuous_affine_map.norm_image_zero_le ContinuousAffineMap.norm_image_zero_le
@[simp]
theorem norm_eq (h : f 0 = 0) : ‖f‖ = ‖f.contLinear‖ :=
calc
‖f‖ = max ‖f 0‖ ‖f.contLinear‖ := by rw [norm_def]
_ = max 0 ‖f.contLinear‖ := by rw [h, norm_zero]
_ = ‖f.contLinear‖ := max_eq_right (norm_nonneg _)
#align continuous_affine_map.norm_eq ContinuousAffineMap.norm_eq
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
rw [add_contLinear]
exact
⟨(norm_add_le _ _).trans (add_le_add (le_max_left _ _) (le_max_left _ _)),
(norm_add_le _ _).trans (add_le_add (le_max_right _ _) (le_max_right _ _))⟩
eq_zero_of_map_eq_zero' := fun f h₀ => by
rcases max_eq_iff.mp h₀ with (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩) <;> rw [h₁] at h₂
· rw [norm_le_zero_iff, contLinear_eq_zero_iff_exists_const] at h₂
obtain ⟨q, rfl⟩ := h₂
simp only [norm_eq_zero] at h₁
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₁
rw [h₁]
rfl
· rw [norm_eq_zero', contLinear_eq_zero_iff_exists_const] at h₁
obtain ⟨q, rfl⟩ := h₁
simp only [norm_le_zero_iff] at h₂
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₂
rw [h₂]
rfl }
instance : NormedSpace 𝕜 (V →A[𝕜] W) where
norm_smul_le t f := by
simp only [norm_def, (smul_contLinear), norm_smul]
-- Porting note: previously all these rewrites were in the `simp only`,
-- but now they don't fire.
-- (in fact, `norm_smul` fires, but only once rather than twice!)
rw [coe_smul, Pi.smul_apply, norm_smul, ← mul_max_of_nonneg _ _ (norm_nonneg t)]
theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ := by
rw [norm_def, max_le_iff]
constructor
· calc
‖f.comp g 0‖ = ‖f (g 0)‖ := by simp
_ = ‖f.contLinear (g 0) + f 0‖ := by rw [f.decomp]; simp
_ ≤ ‖f.contLinear‖ * ‖g 0‖ + ‖f 0‖ :=
((norm_add_le _ _).trans (add_le_add_right (f.contLinear.le_op_norm _) _))
_ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ :=
add_le_add_right
(mul_le_mul f.norm_contLinear_le g.norm_image_zero_le (norm_nonneg _) (norm_nonneg _)) _
· calc
‖(f.comp g).contLinear‖ ≤ ‖f.contLinear‖ * ‖g.contLinear‖ :=
(g.comp_contLinear f).symm ▸ f.contLinear.op_norm_comp_le _
_ ≤ ‖f‖ * ‖g‖ :=
(mul_le_mul f.norm_contLinear_le g.norm_contLinear_le (norm_nonneg _) (norm_nonneg _))
_ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ := by rw [le_add_iff_nonneg_right]; apply norm_nonneg
#align continuous_affine_map.norm_comp_le ContinuousAffineMap.norm_comp_le
variable (𝕜 V W)
/-- The space of affine maps between two normed spaces is linearly isometric to the product of the
codomain with the space of linear maps, by taking the value of the affine map at `(0 : V)` and the
linear part. -/
def toConstProdContinuousLinearMap : (V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W) where
toFun f := ⟨f 0, f.contLinear⟩
invFun p := p.2.toContinuousAffineMap + const 𝕜 V p.1
left_inv f := by
ext
rw [f.decomp]
-- Porting note: previously `simp` closed the goal, but now we need to rewrite:
simp only [coe_add, ContinuousLinearMap.coe_toContinuousAffineMap, Pi.add_apply]
|
rw [ContinuousAffineMap.coe_const, Function.const_apply]
|
/-- The space of affine maps between two normed spaces is linearly isometric to the product of the
codomain with the space of linear maps, by taking the value of the affine map at `(0 : V)` and the
linear part. -/
def toConstProdContinuousLinearMap : (V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W) where
toFun f := ⟨f 0, f.contLinear⟩
invFun p := p.2.toContinuousAffineMap + const 𝕜 V p.1
left_inv f := by
ext
rw [f.decomp]
-- Porting note: previously `simp` closed the goal, but now we need to rewrite:
simp only [coe_add, ContinuousLinearMap.coe_toContinuousAffineMap, Pi.add_apply]
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.253_0.bJ3guivW1IqbUMZ
|
/-- The space of affine maps between two normed spaces is linearly isometric to the product of the
codomain with the space of linear maps, by taking the value of the affine map at `(0 : V)` and the
linear part. -/
def toConstProdContinuousLinearMap : (V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W) where
toFun f
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : V →A[𝕜] W
⊢ Function.RightInverse (fun p => ContinuousLinearMap.toContinuousAffineMap p.2 + const 𝕜 V p.1)
{
toAddHom :=
{ toFun := fun f => (f 0, contLinear f),
map_add' :=
(_ :
∀ (x x_1 : V →A[𝕜] W),
(fun f => (f 0, contLinear f)) (x + x_1) = (fun f => (f 0, contLinear f)) (x + x_1)) },
map_smul' :=
(_ :
∀ (x : 𝕜) (x_1 : V →A[𝕜] W),
AddHom.toFun
{ toFun := fun f => (f 0, contLinear f),
map_add' :=
(_ :
∀ (x x_2 : V →A[𝕜] W),
(fun f => (f 0, contLinear f)) (x + x_2) = (fun f => (f 0, contLinear f)) (x + x_2)) }
(x • x_1) =
AddHom.toFun
{ toFun := fun f => (f 0, contLinear f),
map_add' :=
(_ :
∀ (x x_2 : V →A[𝕜] W),
(fun f => (f 0, contLinear f)) (x + x_2) = (fun f => (f 0, contLinear f)) (x + x_2)) }
(x • x_1)) }.toAddHom.toFun
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h]; rfl
simp_rw [h₁, h₂]
exact (f : P →ᵃ[R] Q).linear_eq_zero_iff_exists_const
#align continuous_affine_map.cont_linear_eq_zero_iff_exists_const ContinuousAffineMap.contLinear_eq_zero_iff_exists_const
@[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f := by
ext
rfl
#align continuous_affine_map.to_affine_map_cont_linear ContinuousAffineMap.to_affine_map_contLinear
@[simp]
theorem zero_contLinear : (0 : P →A[R] W).contLinear = 0 :=
rfl
#align continuous_affine_map.zero_cont_linear ContinuousAffineMap.zero_contLinear
@[simp]
theorem add_contLinear (f g : P →A[R] W) : (f + g).contLinear = f.contLinear + g.contLinear :=
rfl
#align continuous_affine_map.add_cont_linear ContinuousAffineMap.add_contLinear
@[simp]
theorem sub_contLinear (f g : P →A[R] W) : (f - g).contLinear = f.contLinear - g.contLinear :=
rfl
#align continuous_affine_map.sub_cont_linear ContinuousAffineMap.sub_contLinear
@[simp]
theorem neg_contLinear (f : P →A[R] W) : (-f).contLinear = -f.contLinear :=
rfl
#align continuous_affine_map.neg_cont_linear ContinuousAffineMap.neg_contLinear
@[simp]
theorem smul_contLinear (t : R) (f : P →A[R] W) : (t • f).contLinear = t • f.contLinear :=
rfl
#align continuous_affine_map.smul_cont_linear ContinuousAffineMap.smul_contLinear
theorem decomp (f : V →A[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) := by
rcases f with ⟨f, h⟩
rw [coe_mk_const_linear_eq_linear, coe_mk, f.decomp, Pi.add_apply, LinearMap.map_zero, zero_add,
← Function.const_def]
#align continuous_affine_map.decomp ContinuousAffineMap.decomp
section NormedSpaceStructure
variable (f : V →A[𝕜] W)
/-- Note that unlike the operator norm for linear maps, this norm is _not_ submultiplicative:
we do _not_ necessarily have `‖f.comp g‖ ≤ ‖f‖ * ‖g‖`. See `norm_comp_le` for what we can say. -/
noncomputable instance hasNorm : Norm (V →A[𝕜] W) :=
⟨fun f => max ‖f 0‖ ‖f.contLinear‖⟩
#align continuous_affine_map.has_norm ContinuousAffineMap.hasNorm
theorem norm_def : ‖f‖ = max ‖f 0‖ ‖f.contLinear‖ :=
rfl
#align continuous_affine_map.norm_def ContinuousAffineMap.norm_def
theorem norm_contLinear_le : ‖f.contLinear‖ ≤ ‖f‖ :=
le_max_right _ _
#align continuous_affine_map.norm_cont_linear_le ContinuousAffineMap.norm_contLinear_le
theorem norm_image_zero_le : ‖f 0‖ ≤ ‖f‖ :=
le_max_left _ _
#align continuous_affine_map.norm_image_zero_le ContinuousAffineMap.norm_image_zero_le
@[simp]
theorem norm_eq (h : f 0 = 0) : ‖f‖ = ‖f.contLinear‖ :=
calc
‖f‖ = max ‖f 0‖ ‖f.contLinear‖ := by rw [norm_def]
_ = max 0 ‖f.contLinear‖ := by rw [h, norm_zero]
_ = ‖f.contLinear‖ := max_eq_right (norm_nonneg _)
#align continuous_affine_map.norm_eq ContinuousAffineMap.norm_eq
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
rw [add_contLinear]
exact
⟨(norm_add_le _ _).trans (add_le_add (le_max_left _ _) (le_max_left _ _)),
(norm_add_le _ _).trans (add_le_add (le_max_right _ _) (le_max_right _ _))⟩
eq_zero_of_map_eq_zero' := fun f h₀ => by
rcases max_eq_iff.mp h₀ with (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩) <;> rw [h₁] at h₂
· rw [norm_le_zero_iff, contLinear_eq_zero_iff_exists_const] at h₂
obtain ⟨q, rfl⟩ := h₂
simp only [norm_eq_zero] at h₁
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₁
rw [h₁]
rfl
· rw [norm_eq_zero', contLinear_eq_zero_iff_exists_const] at h₁
obtain ⟨q, rfl⟩ := h₁
simp only [norm_le_zero_iff] at h₂
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₂
rw [h₂]
rfl }
instance : NormedSpace 𝕜 (V →A[𝕜] W) where
norm_smul_le t f := by
simp only [norm_def, (smul_contLinear), norm_smul]
-- Porting note: previously all these rewrites were in the `simp only`,
-- but now they don't fire.
-- (in fact, `norm_smul` fires, but only once rather than twice!)
rw [coe_smul, Pi.smul_apply, norm_smul, ← mul_max_of_nonneg _ _ (norm_nonneg t)]
theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ := by
rw [norm_def, max_le_iff]
constructor
· calc
‖f.comp g 0‖ = ‖f (g 0)‖ := by simp
_ = ‖f.contLinear (g 0) + f 0‖ := by rw [f.decomp]; simp
_ ≤ ‖f.contLinear‖ * ‖g 0‖ + ‖f 0‖ :=
((norm_add_le _ _).trans (add_le_add_right (f.contLinear.le_op_norm _) _))
_ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ :=
add_le_add_right
(mul_le_mul f.norm_contLinear_le g.norm_image_zero_le (norm_nonneg _) (norm_nonneg _)) _
· calc
‖(f.comp g).contLinear‖ ≤ ‖f.contLinear‖ * ‖g.contLinear‖ :=
(g.comp_contLinear f).symm ▸ f.contLinear.op_norm_comp_le _
_ ≤ ‖f‖ * ‖g‖ :=
(mul_le_mul f.norm_contLinear_le g.norm_contLinear_le (norm_nonneg _) (norm_nonneg _))
_ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ := by rw [le_add_iff_nonneg_right]; apply norm_nonneg
#align continuous_affine_map.norm_comp_le ContinuousAffineMap.norm_comp_le
variable (𝕜 V W)
/-- The space of affine maps between two normed spaces is linearly isometric to the product of the
codomain with the space of linear maps, by taking the value of the affine map at `(0 : V)` and the
linear part. -/
def toConstProdContinuousLinearMap : (V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W) where
toFun f := ⟨f 0, f.contLinear⟩
invFun p := p.2.toContinuousAffineMap + const 𝕜 V p.1
left_inv f := by
ext
rw [f.decomp]
-- Porting note: previously `simp` closed the goal, but now we need to rewrite:
simp only [coe_add, ContinuousLinearMap.coe_toContinuousAffineMap, Pi.add_apply]
rw [ContinuousAffineMap.coe_const, Function.const_apply]
right_inv := by
|
rintro ⟨v, f⟩
|
/-- The space of affine maps between two normed spaces is linearly isometric to the product of the
codomain with the space of linear maps, by taking the value of the affine map at `(0 : V)` and the
linear part. -/
def toConstProdContinuousLinearMap : (V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W) where
toFun f := ⟨f 0, f.contLinear⟩
invFun p := p.2.toContinuousAffineMap + const 𝕜 V p.1
left_inv f := by
ext
rw [f.decomp]
-- Porting note: previously `simp` closed the goal, but now we need to rewrite:
simp only [coe_add, ContinuousLinearMap.coe_toContinuousAffineMap, Pi.add_apply]
rw [ContinuousAffineMap.coe_const, Function.const_apply]
right_inv := by
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.253_0.bJ3guivW1IqbUMZ
|
/-- The space of affine maps between two normed spaces is linearly isometric to the product of the
codomain with the space of linear maps, by taking the value of the affine map at `(0 : V)` and the
linear part. -/
def toConstProdContinuousLinearMap : (V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W) where
toFun f
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
case mk
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f✝ : V →A[𝕜] W
v : W
f : V →L[𝕜] W
⊢ AddHom.toFun
{
toAddHom :=
{ toFun := fun f => (f 0, contLinear f),
map_add' :=
(_ :
∀ (x x_1 : V →A[𝕜] W),
(fun f => (f 0, contLinear f)) (x + x_1) = (fun f => (f 0, contLinear f)) (x + x_1)) },
map_smul' :=
(_ :
∀ (x : 𝕜) (x_1 : V →A[𝕜] W),
AddHom.toFun
{ toFun := fun f => (f 0, contLinear f),
map_add' :=
(_ :
∀ (x x_2 : V →A[𝕜] W),
(fun f => (f 0, contLinear f)) (x + x_2) = (fun f => (f 0, contLinear f)) (x + x_2)) }
(x • x_1) =
AddHom.toFun
{ toFun := fun f => (f 0, contLinear f),
map_add' :=
(_ :
∀ (x x_2 : V →A[𝕜] W),
(fun f => (f 0, contLinear f)) (x + x_2) = (fun f => (f 0, contLinear f)) (x + x_2)) }
(x • x_1)) }.toAddHom
((fun p => ContinuousLinearMap.toContinuousAffineMap p.2 + const 𝕜 V p.1) (v, f)) =
(v, f)
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h]; rfl
simp_rw [h₁, h₂]
exact (f : P →ᵃ[R] Q).linear_eq_zero_iff_exists_const
#align continuous_affine_map.cont_linear_eq_zero_iff_exists_const ContinuousAffineMap.contLinear_eq_zero_iff_exists_const
@[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f := by
ext
rfl
#align continuous_affine_map.to_affine_map_cont_linear ContinuousAffineMap.to_affine_map_contLinear
@[simp]
theorem zero_contLinear : (0 : P →A[R] W).contLinear = 0 :=
rfl
#align continuous_affine_map.zero_cont_linear ContinuousAffineMap.zero_contLinear
@[simp]
theorem add_contLinear (f g : P →A[R] W) : (f + g).contLinear = f.contLinear + g.contLinear :=
rfl
#align continuous_affine_map.add_cont_linear ContinuousAffineMap.add_contLinear
@[simp]
theorem sub_contLinear (f g : P →A[R] W) : (f - g).contLinear = f.contLinear - g.contLinear :=
rfl
#align continuous_affine_map.sub_cont_linear ContinuousAffineMap.sub_contLinear
@[simp]
theorem neg_contLinear (f : P →A[R] W) : (-f).contLinear = -f.contLinear :=
rfl
#align continuous_affine_map.neg_cont_linear ContinuousAffineMap.neg_contLinear
@[simp]
theorem smul_contLinear (t : R) (f : P →A[R] W) : (t • f).contLinear = t • f.contLinear :=
rfl
#align continuous_affine_map.smul_cont_linear ContinuousAffineMap.smul_contLinear
theorem decomp (f : V →A[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) := by
rcases f with ⟨f, h⟩
rw [coe_mk_const_linear_eq_linear, coe_mk, f.decomp, Pi.add_apply, LinearMap.map_zero, zero_add,
← Function.const_def]
#align continuous_affine_map.decomp ContinuousAffineMap.decomp
section NormedSpaceStructure
variable (f : V →A[𝕜] W)
/-- Note that unlike the operator norm for linear maps, this norm is _not_ submultiplicative:
we do _not_ necessarily have `‖f.comp g‖ ≤ ‖f‖ * ‖g‖`. See `norm_comp_le` for what we can say. -/
noncomputable instance hasNorm : Norm (V →A[𝕜] W) :=
⟨fun f => max ‖f 0‖ ‖f.contLinear‖⟩
#align continuous_affine_map.has_norm ContinuousAffineMap.hasNorm
theorem norm_def : ‖f‖ = max ‖f 0‖ ‖f.contLinear‖ :=
rfl
#align continuous_affine_map.norm_def ContinuousAffineMap.norm_def
theorem norm_contLinear_le : ‖f.contLinear‖ ≤ ‖f‖ :=
le_max_right _ _
#align continuous_affine_map.norm_cont_linear_le ContinuousAffineMap.norm_contLinear_le
theorem norm_image_zero_le : ‖f 0‖ ≤ ‖f‖ :=
le_max_left _ _
#align continuous_affine_map.norm_image_zero_le ContinuousAffineMap.norm_image_zero_le
@[simp]
theorem norm_eq (h : f 0 = 0) : ‖f‖ = ‖f.contLinear‖ :=
calc
‖f‖ = max ‖f 0‖ ‖f.contLinear‖ := by rw [norm_def]
_ = max 0 ‖f.contLinear‖ := by rw [h, norm_zero]
_ = ‖f.contLinear‖ := max_eq_right (norm_nonneg _)
#align continuous_affine_map.norm_eq ContinuousAffineMap.norm_eq
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
rw [add_contLinear]
exact
⟨(norm_add_le _ _).trans (add_le_add (le_max_left _ _) (le_max_left _ _)),
(norm_add_le _ _).trans (add_le_add (le_max_right _ _) (le_max_right _ _))⟩
eq_zero_of_map_eq_zero' := fun f h₀ => by
rcases max_eq_iff.mp h₀ with (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩) <;> rw [h₁] at h₂
· rw [norm_le_zero_iff, contLinear_eq_zero_iff_exists_const] at h₂
obtain ⟨q, rfl⟩ := h₂
simp only [norm_eq_zero] at h₁
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₁
rw [h₁]
rfl
· rw [norm_eq_zero', contLinear_eq_zero_iff_exists_const] at h₁
obtain ⟨q, rfl⟩ := h₁
simp only [norm_le_zero_iff] at h₂
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₂
rw [h₂]
rfl }
instance : NormedSpace 𝕜 (V →A[𝕜] W) where
norm_smul_le t f := by
simp only [norm_def, (smul_contLinear), norm_smul]
-- Porting note: previously all these rewrites were in the `simp only`,
-- but now they don't fire.
-- (in fact, `norm_smul` fires, but only once rather than twice!)
rw [coe_smul, Pi.smul_apply, norm_smul, ← mul_max_of_nonneg _ _ (norm_nonneg t)]
theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ := by
rw [norm_def, max_le_iff]
constructor
· calc
‖f.comp g 0‖ = ‖f (g 0)‖ := by simp
_ = ‖f.contLinear (g 0) + f 0‖ := by rw [f.decomp]; simp
_ ≤ ‖f.contLinear‖ * ‖g 0‖ + ‖f 0‖ :=
((norm_add_le _ _).trans (add_le_add_right (f.contLinear.le_op_norm _) _))
_ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ :=
add_le_add_right
(mul_le_mul f.norm_contLinear_le g.norm_image_zero_le (norm_nonneg _) (norm_nonneg _)) _
· calc
‖(f.comp g).contLinear‖ ≤ ‖f.contLinear‖ * ‖g.contLinear‖ :=
(g.comp_contLinear f).symm ▸ f.contLinear.op_norm_comp_le _
_ ≤ ‖f‖ * ‖g‖ :=
(mul_le_mul f.norm_contLinear_le g.norm_contLinear_le (norm_nonneg _) (norm_nonneg _))
_ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ := by rw [le_add_iff_nonneg_right]; apply norm_nonneg
#align continuous_affine_map.norm_comp_le ContinuousAffineMap.norm_comp_le
variable (𝕜 V W)
/-- The space of affine maps between two normed spaces is linearly isometric to the product of the
codomain with the space of linear maps, by taking the value of the affine map at `(0 : V)` and the
linear part. -/
def toConstProdContinuousLinearMap : (V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W) where
toFun f := ⟨f 0, f.contLinear⟩
invFun p := p.2.toContinuousAffineMap + const 𝕜 V p.1
left_inv f := by
ext
rw [f.decomp]
-- Porting note: previously `simp` closed the goal, but now we need to rewrite:
simp only [coe_add, ContinuousLinearMap.coe_toContinuousAffineMap, Pi.add_apply]
rw [ContinuousAffineMap.coe_const, Function.const_apply]
right_inv := by rintro ⟨v, f⟩;
|
ext
|
/-- The space of affine maps between two normed spaces is linearly isometric to the product of the
codomain with the space of linear maps, by taking the value of the affine map at `(0 : V)` and the
linear part. -/
def toConstProdContinuousLinearMap : (V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W) where
toFun f := ⟨f 0, f.contLinear⟩
invFun p := p.2.toContinuousAffineMap + const 𝕜 V p.1
left_inv f := by
ext
rw [f.decomp]
-- Porting note: previously `simp` closed the goal, but now we need to rewrite:
simp only [coe_add, ContinuousLinearMap.coe_toContinuousAffineMap, Pi.add_apply]
rw [ContinuousAffineMap.coe_const, Function.const_apply]
right_inv := by rintro ⟨v, f⟩;
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.253_0.bJ3guivW1IqbUMZ
|
/-- The space of affine maps between two normed spaces is linearly isometric to the product of the
codomain with the space of linear maps, by taking the value of the affine map at `(0 : V)` and the
linear part. -/
def toConstProdContinuousLinearMap : (V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W) where
toFun f
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
case mk.a
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f✝ : V →A[𝕜] W
v : W
f : V →L[𝕜] W
⊢ (AddHom.toFun
{
toAddHom :=
{ toFun := fun f => (f 0, contLinear f),
map_add' :=
(_ :
∀ (x x_1 : V →A[𝕜] W),
(fun f => (f 0, contLinear f)) (x + x_1) = (fun f => (f 0, contLinear f)) (x + x_1)) },
map_smul' :=
(_ :
∀ (x : 𝕜) (x_1 : V →A[𝕜] W),
AddHom.toFun
{ toFun := fun f => (f 0, contLinear f),
map_add' :=
(_ :
∀ (x x_2 : V →A[𝕜] W),
(fun f => (f 0, contLinear f)) (x + x_2) = (fun f => (f 0, contLinear f)) (x + x_2)) }
(x • x_1) =
AddHom.toFun
{ toFun := fun f => (f 0, contLinear f),
map_add' :=
(_ :
∀ (x x_2 : V →A[𝕜] W),
(fun f => (f 0, contLinear f)) (x + x_2) = (fun f => (f 0, contLinear f)) (x + x_2)) }
(x • x_1)) }.toAddHom
((fun p => ContinuousLinearMap.toContinuousAffineMap p.2 + const 𝕜 V p.1) (v, f))).1 =
(v, f).1
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h]; rfl
simp_rw [h₁, h₂]
exact (f : P →ᵃ[R] Q).linear_eq_zero_iff_exists_const
#align continuous_affine_map.cont_linear_eq_zero_iff_exists_const ContinuousAffineMap.contLinear_eq_zero_iff_exists_const
@[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f := by
ext
rfl
#align continuous_affine_map.to_affine_map_cont_linear ContinuousAffineMap.to_affine_map_contLinear
@[simp]
theorem zero_contLinear : (0 : P →A[R] W).contLinear = 0 :=
rfl
#align continuous_affine_map.zero_cont_linear ContinuousAffineMap.zero_contLinear
@[simp]
theorem add_contLinear (f g : P →A[R] W) : (f + g).contLinear = f.contLinear + g.contLinear :=
rfl
#align continuous_affine_map.add_cont_linear ContinuousAffineMap.add_contLinear
@[simp]
theorem sub_contLinear (f g : P →A[R] W) : (f - g).contLinear = f.contLinear - g.contLinear :=
rfl
#align continuous_affine_map.sub_cont_linear ContinuousAffineMap.sub_contLinear
@[simp]
theorem neg_contLinear (f : P →A[R] W) : (-f).contLinear = -f.contLinear :=
rfl
#align continuous_affine_map.neg_cont_linear ContinuousAffineMap.neg_contLinear
@[simp]
theorem smul_contLinear (t : R) (f : P →A[R] W) : (t • f).contLinear = t • f.contLinear :=
rfl
#align continuous_affine_map.smul_cont_linear ContinuousAffineMap.smul_contLinear
theorem decomp (f : V →A[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) := by
rcases f with ⟨f, h⟩
rw [coe_mk_const_linear_eq_linear, coe_mk, f.decomp, Pi.add_apply, LinearMap.map_zero, zero_add,
← Function.const_def]
#align continuous_affine_map.decomp ContinuousAffineMap.decomp
section NormedSpaceStructure
variable (f : V →A[𝕜] W)
/-- Note that unlike the operator norm for linear maps, this norm is _not_ submultiplicative:
we do _not_ necessarily have `‖f.comp g‖ ≤ ‖f‖ * ‖g‖`. See `norm_comp_le` for what we can say. -/
noncomputable instance hasNorm : Norm (V →A[𝕜] W) :=
⟨fun f => max ‖f 0‖ ‖f.contLinear‖⟩
#align continuous_affine_map.has_norm ContinuousAffineMap.hasNorm
theorem norm_def : ‖f‖ = max ‖f 0‖ ‖f.contLinear‖ :=
rfl
#align continuous_affine_map.norm_def ContinuousAffineMap.norm_def
theorem norm_contLinear_le : ‖f.contLinear‖ ≤ ‖f‖ :=
le_max_right _ _
#align continuous_affine_map.norm_cont_linear_le ContinuousAffineMap.norm_contLinear_le
theorem norm_image_zero_le : ‖f 0‖ ≤ ‖f‖ :=
le_max_left _ _
#align continuous_affine_map.norm_image_zero_le ContinuousAffineMap.norm_image_zero_le
@[simp]
theorem norm_eq (h : f 0 = 0) : ‖f‖ = ‖f.contLinear‖ :=
calc
‖f‖ = max ‖f 0‖ ‖f.contLinear‖ := by rw [norm_def]
_ = max 0 ‖f.contLinear‖ := by rw [h, norm_zero]
_ = ‖f.contLinear‖ := max_eq_right (norm_nonneg _)
#align continuous_affine_map.norm_eq ContinuousAffineMap.norm_eq
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
rw [add_contLinear]
exact
⟨(norm_add_le _ _).trans (add_le_add (le_max_left _ _) (le_max_left _ _)),
(norm_add_le _ _).trans (add_le_add (le_max_right _ _) (le_max_right _ _))⟩
eq_zero_of_map_eq_zero' := fun f h₀ => by
rcases max_eq_iff.mp h₀ with (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩) <;> rw [h₁] at h₂
· rw [norm_le_zero_iff, contLinear_eq_zero_iff_exists_const] at h₂
obtain ⟨q, rfl⟩ := h₂
simp only [norm_eq_zero] at h₁
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₁
rw [h₁]
rfl
· rw [norm_eq_zero', contLinear_eq_zero_iff_exists_const] at h₁
obtain ⟨q, rfl⟩ := h₁
simp only [norm_le_zero_iff] at h₂
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₂
rw [h₂]
rfl }
instance : NormedSpace 𝕜 (V →A[𝕜] W) where
norm_smul_le t f := by
simp only [norm_def, (smul_contLinear), norm_smul]
-- Porting note: previously all these rewrites were in the `simp only`,
-- but now they don't fire.
-- (in fact, `norm_smul` fires, but only once rather than twice!)
rw [coe_smul, Pi.smul_apply, norm_smul, ← mul_max_of_nonneg _ _ (norm_nonneg t)]
theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ := by
rw [norm_def, max_le_iff]
constructor
· calc
‖f.comp g 0‖ = ‖f (g 0)‖ := by simp
_ = ‖f.contLinear (g 0) + f 0‖ := by rw [f.decomp]; simp
_ ≤ ‖f.contLinear‖ * ‖g 0‖ + ‖f 0‖ :=
((norm_add_le _ _).trans (add_le_add_right (f.contLinear.le_op_norm _) _))
_ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ :=
add_le_add_right
(mul_le_mul f.norm_contLinear_le g.norm_image_zero_le (norm_nonneg _) (norm_nonneg _)) _
· calc
‖(f.comp g).contLinear‖ ≤ ‖f.contLinear‖ * ‖g.contLinear‖ :=
(g.comp_contLinear f).symm ▸ f.contLinear.op_norm_comp_le _
_ ≤ ‖f‖ * ‖g‖ :=
(mul_le_mul f.norm_contLinear_le g.norm_contLinear_le (norm_nonneg _) (norm_nonneg _))
_ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ := by rw [le_add_iff_nonneg_right]; apply norm_nonneg
#align continuous_affine_map.norm_comp_le ContinuousAffineMap.norm_comp_le
variable (𝕜 V W)
/-- The space of affine maps between two normed spaces is linearly isometric to the product of the
codomain with the space of linear maps, by taking the value of the affine map at `(0 : V)` and the
linear part. -/
def toConstProdContinuousLinearMap : (V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W) where
toFun f := ⟨f 0, f.contLinear⟩
invFun p := p.2.toContinuousAffineMap + const 𝕜 V p.1
left_inv f := by
ext
rw [f.decomp]
-- Porting note: previously `simp` closed the goal, but now we need to rewrite:
simp only [coe_add, ContinuousLinearMap.coe_toContinuousAffineMap, Pi.add_apply]
rw [ContinuousAffineMap.coe_const, Function.const_apply]
right_inv := by rintro ⟨v, f⟩; ext <;>
|
simp
|
/-- The space of affine maps between two normed spaces is linearly isometric to the product of the
codomain with the space of linear maps, by taking the value of the affine map at `(0 : V)` and the
linear part. -/
def toConstProdContinuousLinearMap : (V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W) where
toFun f := ⟨f 0, f.contLinear⟩
invFun p := p.2.toContinuousAffineMap + const 𝕜 V p.1
left_inv f := by
ext
rw [f.decomp]
-- Porting note: previously `simp` closed the goal, but now we need to rewrite:
simp only [coe_add, ContinuousLinearMap.coe_toContinuousAffineMap, Pi.add_apply]
rw [ContinuousAffineMap.coe_const, Function.const_apply]
right_inv := by rintro ⟨v, f⟩; ext <;>
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.253_0.bJ3guivW1IqbUMZ
|
/-- The space of affine maps between two normed spaces is linearly isometric to the product of the
codomain with the space of linear maps, by taking the value of the affine map at `(0 : V)` and the
linear part. -/
def toConstProdContinuousLinearMap : (V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W) where
toFun f
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
case mk.a.h
𝕜 : Type u_1
R : Type u_2
V : Type u_3
W : Type u_4
W₂ : Type u_5
P : Type u_6
Q : Type u_7
Q₂ : Type u_8
inst✝¹⁶ : NormedAddCommGroup V
inst✝¹⁵ : MetricSpace P
inst✝¹⁴ : NormedAddTorsor V P
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : MetricSpace Q
inst✝¹¹ : NormedAddTorsor W Q
inst✝¹⁰ : NormedAddCommGroup W₂
inst✝⁹ : MetricSpace Q₂
inst✝⁸ : NormedAddTorsor W₂ Q₂
inst✝⁷ : NormedField R
inst✝⁶ : NormedSpace R V
inst✝⁵ : NormedSpace R W
inst✝⁴ : NormedSpace R W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f✝ : V →A[𝕜] W
v : W
f : V →L[𝕜] W
x✝ : V
⊢ (AddHom.toFun
{
toAddHom :=
{ toFun := fun f => (f 0, contLinear f),
map_add' :=
(_ :
∀ (x x_1 : V →A[𝕜] W),
(fun f => (f 0, contLinear f)) (x + x_1) = (fun f => (f 0, contLinear f)) (x + x_1)) },
map_smul' :=
(_ :
∀ (x : 𝕜) (x_1 : V →A[𝕜] W),
AddHom.toFun
{ toFun := fun f => (f 0, contLinear f),
map_add' :=
(_ :
∀ (x x_2 : V →A[𝕜] W),
(fun f => (f 0, contLinear f)) (x + x_2) = (fun f => (f 0, contLinear f)) (x + x_2)) }
(x • x_1) =
AddHom.toFun
{ toFun := fun f => (f 0, contLinear f),
map_add' :=
(_ :
∀ (x x_2 : V →A[𝕜] W),
(fun f => (f 0, contLinear f)) (x + x_2) = (fun f => (f 0, contLinear f)) (x + x_2)) }
(x • x_1)) }.toAddHom
((fun p => ContinuousLinearMap.toContinuousAffineMap p.2 + const 𝕜 V p.1) (v, f))).2
x✝ =
(v, f).2 x✝
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!
# Continuous affine maps between normed spaces.
This file develops the theory of continuous affine maps between affine spaces modelled on normed
spaces.
In the particular case that the affine spaces are just normed vector spaces `V`, `W`, we define a
norm on the space of continuous affine maps by defining the norm of `f : V →A[𝕜] W` to be
`‖f‖ = max ‖f 0‖ ‖f.cont_linear‖`. This is chosen so that we have a linear isometry:
`(V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W)`.
The abstract picture is that for an affine space `P` modelled on a vector space `V`, together with
a vector space `W`, there is an exact sequence of `𝕜`-modules: `0 → C → A → L → 0` where `C`, `A`
are the spaces of constant and affine maps `P → W` and `L` is the space of linear maps `V → W`.
Any choice of a base point in `P` corresponds to a splitting of this sequence so in particular if we
take `P = V`, using `0 : V` as the base point provides a splitting, and we prove this is an
isometric decomposition.
On the other hand, choosing a base point breaks the affine invariance so the norm fails to be
submultiplicative: for a composition of maps, we have only `‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖`.
## Main definitions:
* `ContinuousAffineMap.contLinear`
* `ContinuousAffineMap.hasNorm`
* `ContinuousAffineMap.norm_comp_le`
* `ContinuousAffineMap.toConstProdContinuousLinearMap`
-/
namespace ContinuousAffineMap
variable {𝕜 R V W W₂ P Q Q₂ : Type*}
variable [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P]
variable [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q]
variable [NormedAddCommGroup W₂] [MetricSpace Q₂] [NormedAddTorsor W₂ Q₂]
variable [NormedField R] [NormedSpace R V] [NormedSpace R W] [NormedSpace R W₂]
variable [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂]
/-- The linear map underlying a continuous affine map is continuous. -/
def contLinear (f : P →A[R] Q) : V →L[R] W :=
{ f.linear with
toFun := f.linear
cont := by rw [AffineMap.continuous_linear_iff]; exact f.cont }
#align continuous_affine_map.cont_linear ContinuousAffineMap.contLinear
@[simp]
theorem coe_contLinear (f : P →A[R] Q) : (f.contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_cont_linear ContinuousAffineMap.coe_contLinear
@[simp]
theorem coe_contLinear_eq_linear (f : P →A[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by ext; rfl
#align continuous_affine_map.coe_cont_linear_eq_linear ContinuousAffineMap.coe_contLinear_eq_linear
@[simp]
theorem coe_mk_const_linear_eq_linear (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q).contLinear : V → W) = f.linear :=
rfl
#align continuous_affine_map.coe_mk_const_linear_eq_linear ContinuousAffineMap.coe_mk_const_linear_eq_linear
theorem coe_linear_eq_coe_contLinear (f : P →A[R] Q) :
((f : P →ᵃ[R] Q).linear : V → W) = (⇑f.contLinear : V → W) :=
rfl
#align continuous_affine_map.coe_linear_eq_coe_cont_linear ContinuousAffineMap.coe_linear_eq_coe_contLinear
@[simp]
theorem comp_contLinear (f : P →A[R] Q) (g : Q →A[R] Q₂) :
(g.comp f).contLinear = g.contLinear.comp f.contLinear :=
rfl
#align continuous_affine_map.comp_cont_linear ContinuousAffineMap.comp_contLinear
@[simp]
theorem map_vadd (f : P →A[R] Q) (p : P) (v : V) : f (v +ᵥ p) = f.contLinear v +ᵥ f p :=
f.map_vadd' p v
#align continuous_affine_map.map_vadd ContinuousAffineMap.map_vadd
@[simp]
theorem contLinear_map_vsub (f : P →A[R] Q) (p₁ p₂ : P) : f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂ :=
f.toAffineMap.linearMap_vsub p₁ p₂
#align continuous_affine_map.cont_linear_map_vsub ContinuousAffineMap.contLinear_map_vsub
@[simp]
theorem const_contLinear (q : Q) : (const R P q).contLinear = 0 :=
rfl
#align continuous_affine_map.const_cont_linear ContinuousAffineMap.const_contLinear
theorem contLinear_eq_zero_iff_exists_const (f : P →A[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine' ⟨fun h => _, fun h => _⟩ <;> ext
· rw [h]; rfl
· rw [← coe_to_affineMap, h]; rfl
simp_rw [h₁, h₂]
exact (f : P →ᵃ[R] Q).linear_eq_zero_iff_exists_const
#align continuous_affine_map.cont_linear_eq_zero_iff_exists_const ContinuousAffineMap.contLinear_eq_zero_iff_exists_const
@[simp]
theorem to_affine_map_contLinear (f : V →L[R] W) : f.toContinuousAffineMap.contLinear = f := by
ext
rfl
#align continuous_affine_map.to_affine_map_cont_linear ContinuousAffineMap.to_affine_map_contLinear
@[simp]
theorem zero_contLinear : (0 : P →A[R] W).contLinear = 0 :=
rfl
#align continuous_affine_map.zero_cont_linear ContinuousAffineMap.zero_contLinear
@[simp]
theorem add_contLinear (f g : P →A[R] W) : (f + g).contLinear = f.contLinear + g.contLinear :=
rfl
#align continuous_affine_map.add_cont_linear ContinuousAffineMap.add_contLinear
@[simp]
theorem sub_contLinear (f g : P →A[R] W) : (f - g).contLinear = f.contLinear - g.contLinear :=
rfl
#align continuous_affine_map.sub_cont_linear ContinuousAffineMap.sub_contLinear
@[simp]
theorem neg_contLinear (f : P →A[R] W) : (-f).contLinear = -f.contLinear :=
rfl
#align continuous_affine_map.neg_cont_linear ContinuousAffineMap.neg_contLinear
@[simp]
theorem smul_contLinear (t : R) (f : P →A[R] W) : (t • f).contLinear = t • f.contLinear :=
rfl
#align continuous_affine_map.smul_cont_linear ContinuousAffineMap.smul_contLinear
theorem decomp (f : V →A[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) := by
rcases f with ⟨f, h⟩
rw [coe_mk_const_linear_eq_linear, coe_mk, f.decomp, Pi.add_apply, LinearMap.map_zero, zero_add,
← Function.const_def]
#align continuous_affine_map.decomp ContinuousAffineMap.decomp
section NormedSpaceStructure
variable (f : V →A[𝕜] W)
/-- Note that unlike the operator norm for linear maps, this norm is _not_ submultiplicative:
we do _not_ necessarily have `‖f.comp g‖ ≤ ‖f‖ * ‖g‖`. See `norm_comp_le` for what we can say. -/
noncomputable instance hasNorm : Norm (V →A[𝕜] W) :=
⟨fun f => max ‖f 0‖ ‖f.contLinear‖⟩
#align continuous_affine_map.has_norm ContinuousAffineMap.hasNorm
theorem norm_def : ‖f‖ = max ‖f 0‖ ‖f.contLinear‖ :=
rfl
#align continuous_affine_map.norm_def ContinuousAffineMap.norm_def
theorem norm_contLinear_le : ‖f.contLinear‖ ≤ ‖f‖ :=
le_max_right _ _
#align continuous_affine_map.norm_cont_linear_le ContinuousAffineMap.norm_contLinear_le
theorem norm_image_zero_le : ‖f 0‖ ≤ ‖f‖ :=
le_max_left _ _
#align continuous_affine_map.norm_image_zero_le ContinuousAffineMap.norm_image_zero_le
@[simp]
theorem norm_eq (h : f 0 = 0) : ‖f‖ = ‖f.contLinear‖ :=
calc
‖f‖ = max ‖f 0‖ ‖f.contLinear‖ := by rw [norm_def]
_ = max 0 ‖f.contLinear‖ := by rw [h, norm_zero]
_ = ‖f.contLinear‖ := max_eq_right (norm_nonneg _)
#align continuous_affine_map.norm_eq ContinuousAffineMap.norm_eq
noncomputable instance : NormedAddCommGroup (V →A[𝕜] W) :=
AddGroupNorm.toNormedAddCommGroup
{ toFun := fun f => max ‖f 0‖ ‖f.contLinear‖
map_zero' := by simp [(ContinuousAffineMap.zero_apply)]
neg' := fun f => by
simp [(ContinuousAffineMap.neg_apply)]
add_le' := fun f g => by
simp only [coe_add, max_le_iff]
-- Porting note: previously `Pi.add_apply, add_contLinear, ` in the previous `simp only`
-- suffices, but now they don't fire.
rw [add_contLinear]
exact
⟨(norm_add_le _ _).trans (add_le_add (le_max_left _ _) (le_max_left _ _)),
(norm_add_le _ _).trans (add_le_add (le_max_right _ _) (le_max_right _ _))⟩
eq_zero_of_map_eq_zero' := fun f h₀ => by
rcases max_eq_iff.mp h₀ with (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩) <;> rw [h₁] at h₂
· rw [norm_le_zero_iff, contLinear_eq_zero_iff_exists_const] at h₂
obtain ⟨q, rfl⟩ := h₂
simp only [norm_eq_zero] at h₁
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₁
rw [h₁]
rfl
· rw [norm_eq_zero', contLinear_eq_zero_iff_exists_const] at h₁
obtain ⟨q, rfl⟩ := h₁
simp only [norm_le_zero_iff] at h₂
-- Porting note: prevously `coe_const, Function.const_apply` were in the previous
-- `simp only`, but now they don't fire.
rw [coe_const, Function.const_apply] at h₂
rw [h₂]
rfl }
instance : NormedSpace 𝕜 (V →A[𝕜] W) where
norm_smul_le t f := by
simp only [norm_def, (smul_contLinear), norm_smul]
-- Porting note: previously all these rewrites were in the `simp only`,
-- but now they don't fire.
-- (in fact, `norm_smul` fires, but only once rather than twice!)
rw [coe_smul, Pi.smul_apply, norm_smul, ← mul_max_of_nonneg _ _ (norm_nonneg t)]
theorem norm_comp_le (g : W₂ →A[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ := by
rw [norm_def, max_le_iff]
constructor
· calc
‖f.comp g 0‖ = ‖f (g 0)‖ := by simp
_ = ‖f.contLinear (g 0) + f 0‖ := by rw [f.decomp]; simp
_ ≤ ‖f.contLinear‖ * ‖g 0‖ + ‖f 0‖ :=
((norm_add_le _ _).trans (add_le_add_right (f.contLinear.le_op_norm _) _))
_ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ :=
add_le_add_right
(mul_le_mul f.norm_contLinear_le g.norm_image_zero_le (norm_nonneg _) (norm_nonneg _)) _
· calc
‖(f.comp g).contLinear‖ ≤ ‖f.contLinear‖ * ‖g.contLinear‖ :=
(g.comp_contLinear f).symm ▸ f.contLinear.op_norm_comp_le _
_ ≤ ‖f‖ * ‖g‖ :=
(mul_le_mul f.norm_contLinear_le g.norm_contLinear_le (norm_nonneg _) (norm_nonneg _))
_ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ := by rw [le_add_iff_nonneg_right]; apply norm_nonneg
#align continuous_affine_map.norm_comp_le ContinuousAffineMap.norm_comp_le
variable (𝕜 V W)
/-- The space of affine maps between two normed spaces is linearly isometric to the product of the
codomain with the space of linear maps, by taking the value of the affine map at `(0 : V)` and the
linear part. -/
def toConstProdContinuousLinearMap : (V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W) where
toFun f := ⟨f 0, f.contLinear⟩
invFun p := p.2.toContinuousAffineMap + const 𝕜 V p.1
left_inv f := by
ext
rw [f.decomp]
-- Porting note: previously `simp` closed the goal, but now we need to rewrite:
simp only [coe_add, ContinuousLinearMap.coe_toContinuousAffineMap, Pi.add_apply]
rw [ContinuousAffineMap.coe_const, Function.const_apply]
right_inv := by rintro ⟨v, f⟩; ext <;>
|
simp
|
/-- The space of affine maps between two normed spaces is linearly isometric to the product of the
codomain with the space of linear maps, by taking the value of the affine map at `(0 : V)` and the
linear part. -/
def toConstProdContinuousLinearMap : (V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W) where
toFun f := ⟨f 0, f.contLinear⟩
invFun p := p.2.toContinuousAffineMap + const 𝕜 V p.1
left_inv f := by
ext
rw [f.decomp]
-- Porting note: previously `simp` closed the goal, but now we need to rewrite:
simp only [coe_add, ContinuousLinearMap.coe_toContinuousAffineMap, Pi.add_apply]
rw [ContinuousAffineMap.coe_const, Function.const_apply]
right_inv := by rintro ⟨v, f⟩; ext <;>
|
Mathlib.Analysis.NormedSpace.ContinuousAffineMap.253_0.bJ3guivW1IqbUMZ
|
/-- The space of affine maps between two normed spaces is linearly isometric to the product of the
codomain with the space of linear maps, by taking the value of the affine map at `(0 : V)` and the
linear part. -/
def toConstProdContinuousLinearMap : (V →A[𝕜] W) ≃ₗᵢ[𝕜] W × (V →L[𝕜] W) where
toFun f
|
Mathlib_Analysis_NormedSpace_ContinuousAffineMap
|
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : Preorder ι
G : ι → Type u_3
inst✝³ : (i : ι) → AddCommGroup (G i)
inst✝² : (i : ι) → Module R (G i)
f : (i j : ι) → i ≤ j → G i →ₗ[R] G j
M : Type u_4
inst✝¹ : AddCommGroup M
inst✝ : Module R M
x✝² x✝¹ : ι
x✝ : x✝² ≤ x✝¹
x : G x✝² ⊗[R] M
⊢ ((fun x => LinearMap.rTensor M (of R ι G f x)) x✝¹) ((LinearMap.rTensor M (f x✝² x✝¹ x✝)) x) =
((fun x => LinearMap.rTensor M (of R ι G f x)) x✝²) x
|
/-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.Algebra.DirectLimit
/-!
# Tensor product and direct limits commute with each other.
Given a family of `R`-modules `Gᵢ` with a family of compatible `R`-linear maps `fᵢⱼ : Gᵢ → Gⱼ` for
every `i ≤ j` and another `R`-module `M`, we have `(limᵢ Gᵢ) ⊗ M` and `lim (Gᵢ ⊗ M)` are isomorphic
as `R`-modules.
## Main definitions:
* `TensorProduct.directLimitLeft : DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M)`
* `TensorProduct.directLimitRight : M ⊗[R] DirectLimit G f ≃ₗ[R] DirectLimit (M ⊗[R] G ·) (M ◁ f)`
-/
open TensorProduct Module Module.DirectLimit
variable {R : Type*} [CommRing R]
variable {ι : Type*}
variable [DecidableEq ι] [Preorder ι]
variable {G : ι → Type*}
variable [∀ i, AddCommGroup (G i)] [∀ i, Module R (G i)]
variable (f : ∀ i j, i ≤ j → G i →ₗ[R] G j)
variable (M : Type*) [AddCommGroup M] [Module R M]
-- alluding to the notation in `CategoryTheory.Monoidal`
local notation M " ◁ " f => fun i j h ↦ LinearMap.lTensor M (f _ _ h)
local notation f " ▷ " N => fun i j h ↦ LinearMap.rTensor N (f _ _ h)
namespace TensorProduct
/--
the map `limᵢ (Gᵢ ⊗ M) → (limᵢ Gᵢ) ⊗ M` induced by the family of maps `Gᵢ ⊗ M → (limᵢ Gᵢ) ⊗ M`
given by `gᵢ ⊗ m ↦ [gᵢ] ⊗ m`.
-/
noncomputable def fromDirectLimit :
DirectLimit (G · ⊗[R] M) (f ▷ M) →ₗ[R] DirectLimit G f ⊗[R] M :=
DirectLimit.lift _ _ _ _ (fun _ ↦ (of _ _ _ _ _).rTensor M)
fun _ _ _ x ↦ by
|
refine' x.induction_on _ _ _
|
/--
the map `limᵢ (Gᵢ ⊗ M) → (limᵢ Gᵢ) ⊗ M` induced by the family of maps `Gᵢ ⊗ M → (limᵢ Gᵢ) ⊗ M`
given by `gᵢ ⊗ m ↦ [gᵢ] ⊗ m`.
-/
noncomputable def fromDirectLimit :
DirectLimit (G · ⊗[R] M) (f ▷ M) →ₗ[R] DirectLimit G f ⊗[R] M :=
DirectLimit.lift _ _ _ _ (fun _ ↦ (of _ _ _ _ _).rTensor M)
fun _ _ _ x ↦ by
|
Mathlib.Algebra.Module.DirectLimitAndTensorProduct.41_0.NNGkUEQVeDhgQVm
|
/--
the map `limᵢ (Gᵢ ⊗ M) → (limᵢ Gᵢ) ⊗ M` induced by the family of maps `Gᵢ ⊗ M → (limᵢ Gᵢ) ⊗ M`
given by `gᵢ ⊗ m ↦ [gᵢ] ⊗ m`.
-/
noncomputable def fromDirectLimit :
DirectLimit (G · ⊗[R] M) (f ▷ M) →ₗ[R] DirectLimit G f ⊗[R] M
|
Mathlib_Algebra_Module_DirectLimitAndTensorProduct
|
case refine'_1
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : Preorder ι
G : ι → Type u_3
inst✝³ : (i : ι) → AddCommGroup (G i)
inst✝² : (i : ι) → Module R (G i)
f : (i j : ι) → i ≤ j → G i →ₗ[R] G j
M : Type u_4
inst✝¹ : AddCommGroup M
inst✝ : Module R M
x✝² x✝¹ : ι
x✝ : x✝² ≤ x✝¹
x : G x✝² ⊗[R] M
⊢ ((fun x => LinearMap.rTensor M (of R ι G f x)) x✝¹) ((LinearMap.rTensor M (f x✝² x✝¹ x✝)) 0) =
((fun x => LinearMap.rTensor M (of R ι G f x)) x✝²) 0
|
/-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.Algebra.DirectLimit
/-!
# Tensor product and direct limits commute with each other.
Given a family of `R`-modules `Gᵢ` with a family of compatible `R`-linear maps `fᵢⱼ : Gᵢ → Gⱼ` for
every `i ≤ j` and another `R`-module `M`, we have `(limᵢ Gᵢ) ⊗ M` and `lim (Gᵢ ⊗ M)` are isomorphic
as `R`-modules.
## Main definitions:
* `TensorProduct.directLimitLeft : DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M)`
* `TensorProduct.directLimitRight : M ⊗[R] DirectLimit G f ≃ₗ[R] DirectLimit (M ⊗[R] G ·) (M ◁ f)`
-/
open TensorProduct Module Module.DirectLimit
variable {R : Type*} [CommRing R]
variable {ι : Type*}
variable [DecidableEq ι] [Preorder ι]
variable {G : ι → Type*}
variable [∀ i, AddCommGroup (G i)] [∀ i, Module R (G i)]
variable (f : ∀ i j, i ≤ j → G i →ₗ[R] G j)
variable (M : Type*) [AddCommGroup M] [Module R M]
-- alluding to the notation in `CategoryTheory.Monoidal`
local notation M " ◁ " f => fun i j h ↦ LinearMap.lTensor M (f _ _ h)
local notation f " ▷ " N => fun i j h ↦ LinearMap.rTensor N (f _ _ h)
namespace TensorProduct
/--
the map `limᵢ (Gᵢ ⊗ M) → (limᵢ Gᵢ) ⊗ M` induced by the family of maps `Gᵢ ⊗ M → (limᵢ Gᵢ) ⊗ M`
given by `gᵢ ⊗ m ↦ [gᵢ] ⊗ m`.
-/
noncomputable def fromDirectLimit :
DirectLimit (G · ⊗[R] M) (f ▷ M) →ₗ[R] DirectLimit G f ⊗[R] M :=
DirectLimit.lift _ _ _ _ (fun _ ↦ (of _ _ _ _ _).rTensor M)
fun _ _ _ x ↦ by refine' x.induction_on _ _ _ <;>
|
aesop
|
/--
the map `limᵢ (Gᵢ ⊗ M) → (limᵢ Gᵢ) ⊗ M` induced by the family of maps `Gᵢ ⊗ M → (limᵢ Gᵢ) ⊗ M`
given by `gᵢ ⊗ m ↦ [gᵢ] ⊗ m`.
-/
noncomputable def fromDirectLimit :
DirectLimit (G · ⊗[R] M) (f ▷ M) →ₗ[R] DirectLimit G f ⊗[R] M :=
DirectLimit.lift _ _ _ _ (fun _ ↦ (of _ _ _ _ _).rTensor M)
fun _ _ _ x ↦ by refine' x.induction_on _ _ _ <;>
|
Mathlib.Algebra.Module.DirectLimitAndTensorProduct.41_0.NNGkUEQVeDhgQVm
|
/--
the map `limᵢ (Gᵢ ⊗ M) → (limᵢ Gᵢ) ⊗ M` induced by the family of maps `Gᵢ ⊗ M → (limᵢ Gᵢ) ⊗ M`
given by `gᵢ ⊗ m ↦ [gᵢ] ⊗ m`.
-/
noncomputable def fromDirectLimit :
DirectLimit (G · ⊗[R] M) (f ▷ M) →ₗ[R] DirectLimit G f ⊗[R] M
|
Mathlib_Algebra_Module_DirectLimitAndTensorProduct
|
case refine'_2
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : Preorder ι
G : ι → Type u_3
inst✝³ : (i : ι) → AddCommGroup (G i)
inst✝² : (i : ι) → Module R (G i)
f : (i j : ι) → i ≤ j → G i →ₗ[R] G j
M : Type u_4
inst✝¹ : AddCommGroup M
inst✝ : Module R M
x✝² x✝¹ : ι
x✝ : x✝² ≤ x✝¹
x : G x✝² ⊗[R] M
⊢ ∀ (x : G x✝²) (y : M),
((fun x => LinearMap.rTensor M (of R ι G f x)) x✝¹) ((LinearMap.rTensor M (f x✝² x✝¹ x✝)) (x ⊗ₜ[R] y)) =
((fun x => LinearMap.rTensor M (of R ι G f x)) x✝²) (x ⊗ₜ[R] y)
|
/-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.Algebra.DirectLimit
/-!
# Tensor product and direct limits commute with each other.
Given a family of `R`-modules `Gᵢ` with a family of compatible `R`-linear maps `fᵢⱼ : Gᵢ → Gⱼ` for
every `i ≤ j` and another `R`-module `M`, we have `(limᵢ Gᵢ) ⊗ M` and `lim (Gᵢ ⊗ M)` are isomorphic
as `R`-modules.
## Main definitions:
* `TensorProduct.directLimitLeft : DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M)`
* `TensorProduct.directLimitRight : M ⊗[R] DirectLimit G f ≃ₗ[R] DirectLimit (M ⊗[R] G ·) (M ◁ f)`
-/
open TensorProduct Module Module.DirectLimit
variable {R : Type*} [CommRing R]
variable {ι : Type*}
variable [DecidableEq ι] [Preorder ι]
variable {G : ι → Type*}
variable [∀ i, AddCommGroup (G i)] [∀ i, Module R (G i)]
variable (f : ∀ i j, i ≤ j → G i →ₗ[R] G j)
variable (M : Type*) [AddCommGroup M] [Module R M]
-- alluding to the notation in `CategoryTheory.Monoidal`
local notation M " ◁ " f => fun i j h ↦ LinearMap.lTensor M (f _ _ h)
local notation f " ▷ " N => fun i j h ↦ LinearMap.rTensor N (f _ _ h)
namespace TensorProduct
/--
the map `limᵢ (Gᵢ ⊗ M) → (limᵢ Gᵢ) ⊗ M` induced by the family of maps `Gᵢ ⊗ M → (limᵢ Gᵢ) ⊗ M`
given by `gᵢ ⊗ m ↦ [gᵢ] ⊗ m`.
-/
noncomputable def fromDirectLimit :
DirectLimit (G · ⊗[R] M) (f ▷ M) →ₗ[R] DirectLimit G f ⊗[R] M :=
DirectLimit.lift _ _ _ _ (fun _ ↦ (of _ _ _ _ _).rTensor M)
fun _ _ _ x ↦ by refine' x.induction_on _ _ _ <;>
|
aesop
|
/--
the map `limᵢ (Gᵢ ⊗ M) → (limᵢ Gᵢ) ⊗ M` induced by the family of maps `Gᵢ ⊗ M → (limᵢ Gᵢ) ⊗ M`
given by `gᵢ ⊗ m ↦ [gᵢ] ⊗ m`.
-/
noncomputable def fromDirectLimit :
DirectLimit (G · ⊗[R] M) (f ▷ M) →ₗ[R] DirectLimit G f ⊗[R] M :=
DirectLimit.lift _ _ _ _ (fun _ ↦ (of _ _ _ _ _).rTensor M)
fun _ _ _ x ↦ by refine' x.induction_on _ _ _ <;>
|
Mathlib.Algebra.Module.DirectLimitAndTensorProduct.41_0.NNGkUEQVeDhgQVm
|
/--
the map `limᵢ (Gᵢ ⊗ M) → (limᵢ Gᵢ) ⊗ M` induced by the family of maps `Gᵢ ⊗ M → (limᵢ Gᵢ) ⊗ M`
given by `gᵢ ⊗ m ↦ [gᵢ] ⊗ m`.
-/
noncomputable def fromDirectLimit :
DirectLimit (G · ⊗[R] M) (f ▷ M) →ₗ[R] DirectLimit G f ⊗[R] M
|
Mathlib_Algebra_Module_DirectLimitAndTensorProduct
|
case refine'_3
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : Preorder ι
G : ι → Type u_3
inst✝³ : (i : ι) → AddCommGroup (G i)
inst✝² : (i : ι) → Module R (G i)
f : (i j : ι) → i ≤ j → G i →ₗ[R] G j
M : Type u_4
inst✝¹ : AddCommGroup M
inst✝ : Module R M
x✝² x✝¹ : ι
x✝ : x✝² ≤ x✝¹
x : G x✝² ⊗[R] M
⊢ ∀ (x y : G x✝² ⊗[R] M),
((fun x => LinearMap.rTensor M (of R ι G f x)) x✝¹) ((LinearMap.rTensor M (f x✝² x✝¹ x✝)) x) =
((fun x => LinearMap.rTensor M (of R ι G f x)) x✝²) x →
((fun x => LinearMap.rTensor M (of R ι G f x)) x✝¹) ((LinearMap.rTensor M (f x✝² x✝¹ x✝)) y) =
((fun x => LinearMap.rTensor M (of R ι G f x)) x✝²) y →
((fun x => LinearMap.rTensor M (of R ι G f x)) x✝¹) ((LinearMap.rTensor M (f x✝² x✝¹ x✝)) (x + y)) =
((fun x => LinearMap.rTensor M (of R ι G f x)) x✝²) (x + y)
|
/-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.Algebra.DirectLimit
/-!
# Tensor product and direct limits commute with each other.
Given a family of `R`-modules `Gᵢ` with a family of compatible `R`-linear maps `fᵢⱼ : Gᵢ → Gⱼ` for
every `i ≤ j` and another `R`-module `M`, we have `(limᵢ Gᵢ) ⊗ M` and `lim (Gᵢ ⊗ M)` are isomorphic
as `R`-modules.
## Main definitions:
* `TensorProduct.directLimitLeft : DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M)`
* `TensorProduct.directLimitRight : M ⊗[R] DirectLimit G f ≃ₗ[R] DirectLimit (M ⊗[R] G ·) (M ◁ f)`
-/
open TensorProduct Module Module.DirectLimit
variable {R : Type*} [CommRing R]
variable {ι : Type*}
variable [DecidableEq ι] [Preorder ι]
variable {G : ι → Type*}
variable [∀ i, AddCommGroup (G i)] [∀ i, Module R (G i)]
variable (f : ∀ i j, i ≤ j → G i →ₗ[R] G j)
variable (M : Type*) [AddCommGroup M] [Module R M]
-- alluding to the notation in `CategoryTheory.Monoidal`
local notation M " ◁ " f => fun i j h ↦ LinearMap.lTensor M (f _ _ h)
local notation f " ▷ " N => fun i j h ↦ LinearMap.rTensor N (f _ _ h)
namespace TensorProduct
/--
the map `limᵢ (Gᵢ ⊗ M) → (limᵢ Gᵢ) ⊗ M` induced by the family of maps `Gᵢ ⊗ M → (limᵢ Gᵢ) ⊗ M`
given by `gᵢ ⊗ m ↦ [gᵢ] ⊗ m`.
-/
noncomputable def fromDirectLimit :
DirectLimit (G · ⊗[R] M) (f ▷ M) →ₗ[R] DirectLimit G f ⊗[R] M :=
DirectLimit.lift _ _ _ _ (fun _ ↦ (of _ _ _ _ _).rTensor M)
fun _ _ _ x ↦ by refine' x.induction_on _ _ _ <;>
|
aesop
|
/--
the map `limᵢ (Gᵢ ⊗ M) → (limᵢ Gᵢ) ⊗ M` induced by the family of maps `Gᵢ ⊗ M → (limᵢ Gᵢ) ⊗ M`
given by `gᵢ ⊗ m ↦ [gᵢ] ⊗ m`.
-/
noncomputable def fromDirectLimit :
DirectLimit (G · ⊗[R] M) (f ▷ M) →ₗ[R] DirectLimit G f ⊗[R] M :=
DirectLimit.lift _ _ _ _ (fun _ ↦ (of _ _ _ _ _).rTensor M)
fun _ _ _ x ↦ by refine' x.induction_on _ _ _ <;>
|
Mathlib.Algebra.Module.DirectLimitAndTensorProduct.41_0.NNGkUEQVeDhgQVm
|
/--
the map `limᵢ (Gᵢ ⊗ M) → (limᵢ Gᵢ) ⊗ M` induced by the family of maps `Gᵢ ⊗ M → (limᵢ Gᵢ) ⊗ M`
given by `gᵢ ⊗ m ↦ [gᵢ] ⊗ m`.
-/
noncomputable def fromDirectLimit :
DirectLimit (G · ⊗[R] M) (f ▷ M) →ₗ[R] DirectLimit G f ⊗[R] M
|
Mathlib_Algebra_Module_DirectLimitAndTensorProduct
|
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : Preorder ι
G : ι → Type u_3
inst✝³ : (i : ι) → AddCommGroup (G i)
inst✝² : (i : ι) → Module R (G i)
f : (i j : ι) → i ≤ j → G i →ₗ[R] G j
M : Type u_4
inst✝¹ : AddCommGroup M
inst✝ : Module R M
i : ι
g : G i
m : M
⊢ (toDirectLimit f M) ((of R ι G f i) g ⊗ₜ[R] m) =
(of R ι (fun x => G x ⊗[R] M) (fun i j h => LinearMap.rTensor M (f i j h)) i) (g ⊗ₜ[R] m)
|
/-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.Algebra.DirectLimit
/-!
# Tensor product and direct limits commute with each other.
Given a family of `R`-modules `Gᵢ` with a family of compatible `R`-linear maps `fᵢⱼ : Gᵢ → Gⱼ` for
every `i ≤ j` and another `R`-module `M`, we have `(limᵢ Gᵢ) ⊗ M` and `lim (Gᵢ ⊗ M)` are isomorphic
as `R`-modules.
## Main definitions:
* `TensorProduct.directLimitLeft : DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M)`
* `TensorProduct.directLimitRight : M ⊗[R] DirectLimit G f ≃ₗ[R] DirectLimit (M ⊗[R] G ·) (M ◁ f)`
-/
open TensorProduct Module Module.DirectLimit
variable {R : Type*} [CommRing R]
variable {ι : Type*}
variable [DecidableEq ι] [Preorder ι]
variable {G : ι → Type*}
variable [∀ i, AddCommGroup (G i)] [∀ i, Module R (G i)]
variable (f : ∀ i j, i ≤ j → G i →ₗ[R] G j)
variable (M : Type*) [AddCommGroup M] [Module R M]
-- alluding to the notation in `CategoryTheory.Monoidal`
local notation M " ◁ " f => fun i j h ↦ LinearMap.lTensor M (f _ _ h)
local notation f " ▷ " N => fun i j h ↦ LinearMap.rTensor N (f _ _ h)
namespace TensorProduct
/--
the map `limᵢ (Gᵢ ⊗ M) → (limᵢ Gᵢ) ⊗ M` induced by the family of maps `Gᵢ ⊗ M → (limᵢ Gᵢ) ⊗ M`
given by `gᵢ ⊗ m ↦ [gᵢ] ⊗ m`.
-/
noncomputable def fromDirectLimit :
DirectLimit (G · ⊗[R] M) (f ▷ M) →ₗ[R] DirectLimit G f ⊗[R] M :=
DirectLimit.lift _ _ _ _ (fun _ ↦ (of _ _ _ _ _).rTensor M)
fun _ _ _ x ↦ by refine' x.induction_on _ _ _ <;> aesop
variable {M} in
@[simp] lemma fromDirectLimit_of_tmul {i : ι} (g : G i) (m : M) :
fromDirectLimit f M (of _ _ _ _ i (g ⊗ₜ m)) = (of _ _ _ f i g) ⊗ₜ m :=
lift_of (G := (G · ⊗[R] M)) _ _ (g ⊗ₜ m)
/--
the map `(limᵢ Gᵢ) ⊗ M → limᵢ (Gᵢ ⊗ M)` from the bilinear map `limᵢ Gᵢ → M → limᵢ (Gᵢ ⊗ M)` given
by the family of maps `Gᵢ → M → limᵢ (Gᵢ ⊗ M)` where `gᵢ ↦ m ↦ [gᵢ ⊗ m]`
-/
noncomputable def toDirectLimit : DirectLimit G f ⊗[R] M →ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) :=
TensorProduct.lift <| DirectLimit.lift _ _ _ _
(fun i ↦
(TensorProduct.mk R _ _).compr₂ (of R ι _ (fun _i _j h ↦ (f _ _ h).rTensor M) i))
fun _ _ _ g ↦ FunLike.ext _ _ (of_f (G := (G · ⊗[R] M)) (x := g ⊗ₜ ·))
variable {M} in
@[simp] lemma toDirectLimit_tmul_of
{i : ι} (g : G i) (m : M) :
(toDirectLimit f M <| (of _ _ G f i g) ⊗ₜ m) = (of _ _ _ _ i (g ⊗ₜ m)) := by
|
rw [toDirectLimit, lift.tmul, lift_of]
|
variable {M} in
@[simp] lemma toDirectLimit_tmul_of
{i : ι} (g : G i) (m : M) :
(toDirectLimit f M <| (of _ _ G f i g) ⊗ₜ m) = (of _ _ _ _ i (g ⊗ₜ m)) := by
|
Mathlib.Algebra.Module.DirectLimitAndTensorProduct.66_0.NNGkUEQVeDhgQVm
|
variable {M} in
@[simp] lemma toDirectLimit_tmul_of
{i : ι} (g : G i) (m : M) :
(toDirectLimit f M <| (of _ _ G f i g) ⊗ₜ m) = (of _ _ _ _ i (g ⊗ₜ m))
|
Mathlib_Algebra_Module_DirectLimitAndTensorProduct
|
R : Type u_1
inst✝⁶ : CommRing R
ι : Type u_2
inst✝⁵ : DecidableEq ι
inst✝⁴ : Preorder ι
G : ι → Type u_3
inst✝³ : (i : ι) → AddCommGroup (G i)
inst✝² : (i : ι) → Module R (G i)
f : (i j : ι) → i ≤ j → G i →ₗ[R] G j
M : Type u_4
inst✝¹ : AddCommGroup M
inst✝ : Module R M
i : ι
g : G i
m : M
⊢ ((LinearMap.compr₂ (mk R (G i) M) (of R ι (fun _i => G _i ⊗[R] M) (fun _i _j h => LinearMap.rTensor M (f _i _j h)) i))
g)
m =
(of R ι (fun x => G x ⊗[R] M) (fun i j h => LinearMap.rTensor M (f i j h)) i) (g ⊗ₜ[R] m)
|
/-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.Algebra.DirectLimit
/-!
# Tensor product and direct limits commute with each other.
Given a family of `R`-modules `Gᵢ` with a family of compatible `R`-linear maps `fᵢⱼ : Gᵢ → Gⱼ` for
every `i ≤ j` and another `R`-module `M`, we have `(limᵢ Gᵢ) ⊗ M` and `lim (Gᵢ ⊗ M)` are isomorphic
as `R`-modules.
## Main definitions:
* `TensorProduct.directLimitLeft : DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M)`
* `TensorProduct.directLimitRight : M ⊗[R] DirectLimit G f ≃ₗ[R] DirectLimit (M ⊗[R] G ·) (M ◁ f)`
-/
open TensorProduct Module Module.DirectLimit
variable {R : Type*} [CommRing R]
variable {ι : Type*}
variable [DecidableEq ι] [Preorder ι]
variable {G : ι → Type*}
variable [∀ i, AddCommGroup (G i)] [∀ i, Module R (G i)]
variable (f : ∀ i j, i ≤ j → G i →ₗ[R] G j)
variable (M : Type*) [AddCommGroup M] [Module R M]
-- alluding to the notation in `CategoryTheory.Monoidal`
local notation M " ◁ " f => fun i j h ↦ LinearMap.lTensor M (f _ _ h)
local notation f " ▷ " N => fun i j h ↦ LinearMap.rTensor N (f _ _ h)
namespace TensorProduct
/--
the map `limᵢ (Gᵢ ⊗ M) → (limᵢ Gᵢ) ⊗ M` induced by the family of maps `Gᵢ ⊗ M → (limᵢ Gᵢ) ⊗ M`
given by `gᵢ ⊗ m ↦ [gᵢ] ⊗ m`.
-/
noncomputable def fromDirectLimit :
DirectLimit (G · ⊗[R] M) (f ▷ M) →ₗ[R] DirectLimit G f ⊗[R] M :=
DirectLimit.lift _ _ _ _ (fun _ ↦ (of _ _ _ _ _).rTensor M)
fun _ _ _ x ↦ by refine' x.induction_on _ _ _ <;> aesop
variable {M} in
@[simp] lemma fromDirectLimit_of_tmul {i : ι} (g : G i) (m : M) :
fromDirectLimit f M (of _ _ _ _ i (g ⊗ₜ m)) = (of _ _ _ f i g) ⊗ₜ m :=
lift_of (G := (G · ⊗[R] M)) _ _ (g ⊗ₜ m)
/--
the map `(limᵢ Gᵢ) ⊗ M → limᵢ (Gᵢ ⊗ M)` from the bilinear map `limᵢ Gᵢ → M → limᵢ (Gᵢ ⊗ M)` given
by the family of maps `Gᵢ → M → limᵢ (Gᵢ ⊗ M)` where `gᵢ ↦ m ↦ [gᵢ ⊗ m]`
-/
noncomputable def toDirectLimit : DirectLimit G f ⊗[R] M →ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) :=
TensorProduct.lift <| DirectLimit.lift _ _ _ _
(fun i ↦
(TensorProduct.mk R _ _).compr₂ (of R ι _ (fun _i _j h ↦ (f _ _ h).rTensor M) i))
fun _ _ _ g ↦ FunLike.ext _ _ (of_f (G := (G · ⊗[R] M)) (x := g ⊗ₜ ·))
variable {M} in
@[simp] lemma toDirectLimit_tmul_of
{i : ι} (g : G i) (m : M) :
(toDirectLimit f M <| (of _ _ G f i g) ⊗ₜ m) = (of _ _ _ _ i (g ⊗ₜ m)) := by
rw [toDirectLimit, lift.tmul, lift_of]
|
rfl
|
variable {M} in
@[simp] lemma toDirectLimit_tmul_of
{i : ι} (g : G i) (m : M) :
(toDirectLimit f M <| (of _ _ G f i g) ⊗ₜ m) = (of _ _ _ _ i (g ⊗ₜ m)) := by
rw [toDirectLimit, lift.tmul, lift_of]
|
Mathlib.Algebra.Module.DirectLimitAndTensorProduct.66_0.NNGkUEQVeDhgQVm
|
variable {M} in
@[simp] lemma toDirectLimit_tmul_of
{i : ι} (g : G i) (m : M) :
(toDirectLimit f M <| (of _ _ G f i g) ⊗ₜ m) = (of _ _ _ _ i (g ⊗ₜ m))
|
Mathlib_Algebra_Module_DirectLimitAndTensorProduct
|
R : Type u_1
inst✝⁷ : CommRing R
ι : Type u_2
inst✝⁶ : DecidableEq ι
inst✝⁵ : Preorder ι
G : ι → Type u_3
inst✝⁴ : (i : ι) → AddCommGroup (G i)
inst✝³ : (i : ι) → Module R (G i)
f : (i j : ι) → i ≤ j → G i →ₗ[R] G j
M : Type u_4
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsDirected ι fun x x_1 => x ≤ x_1
⊢ DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (fun x => G x ⊗[R] M) fun i j h => LinearMap.rTensor M (f i j h)
|
/-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.Algebra.DirectLimit
/-!
# Tensor product and direct limits commute with each other.
Given a family of `R`-modules `Gᵢ` with a family of compatible `R`-linear maps `fᵢⱼ : Gᵢ → Gⱼ` for
every `i ≤ j` and another `R`-module `M`, we have `(limᵢ Gᵢ) ⊗ M` and `lim (Gᵢ ⊗ M)` are isomorphic
as `R`-modules.
## Main definitions:
* `TensorProduct.directLimitLeft : DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M)`
* `TensorProduct.directLimitRight : M ⊗[R] DirectLimit G f ≃ₗ[R] DirectLimit (M ⊗[R] G ·) (M ◁ f)`
-/
open TensorProduct Module Module.DirectLimit
variable {R : Type*} [CommRing R]
variable {ι : Type*}
variable [DecidableEq ι] [Preorder ι]
variable {G : ι → Type*}
variable [∀ i, AddCommGroup (G i)] [∀ i, Module R (G i)]
variable (f : ∀ i j, i ≤ j → G i →ₗ[R] G j)
variable (M : Type*) [AddCommGroup M] [Module R M]
-- alluding to the notation in `CategoryTheory.Monoidal`
local notation M " ◁ " f => fun i j h ↦ LinearMap.lTensor M (f _ _ h)
local notation f " ▷ " N => fun i j h ↦ LinearMap.rTensor N (f _ _ h)
namespace TensorProduct
/--
the map `limᵢ (Gᵢ ⊗ M) → (limᵢ Gᵢ) ⊗ M` induced by the family of maps `Gᵢ ⊗ M → (limᵢ Gᵢ) ⊗ M`
given by `gᵢ ⊗ m ↦ [gᵢ] ⊗ m`.
-/
noncomputable def fromDirectLimit :
DirectLimit (G · ⊗[R] M) (f ▷ M) →ₗ[R] DirectLimit G f ⊗[R] M :=
DirectLimit.lift _ _ _ _ (fun _ ↦ (of _ _ _ _ _).rTensor M)
fun _ _ _ x ↦ by refine' x.induction_on _ _ _ <;> aesop
variable {M} in
@[simp] lemma fromDirectLimit_of_tmul {i : ι} (g : G i) (m : M) :
fromDirectLimit f M (of _ _ _ _ i (g ⊗ₜ m)) = (of _ _ _ f i g) ⊗ₜ m :=
lift_of (G := (G · ⊗[R] M)) _ _ (g ⊗ₜ m)
/--
the map `(limᵢ Gᵢ) ⊗ M → limᵢ (Gᵢ ⊗ M)` from the bilinear map `limᵢ Gᵢ → M → limᵢ (Gᵢ ⊗ M)` given
by the family of maps `Gᵢ → M → limᵢ (Gᵢ ⊗ M)` where `gᵢ ↦ m ↦ [gᵢ ⊗ m]`
-/
noncomputable def toDirectLimit : DirectLimit G f ⊗[R] M →ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) :=
TensorProduct.lift <| DirectLimit.lift _ _ _ _
(fun i ↦
(TensorProduct.mk R _ _).compr₂ (of R ι _ (fun _i _j h ↦ (f _ _ h).rTensor M) i))
fun _ _ _ g ↦ FunLike.ext _ _ (of_f (G := (G · ⊗[R] M)) (x := g ⊗ₜ ·))
variable {M} in
@[simp] lemma toDirectLimit_tmul_of
{i : ι} (g : G i) (m : M) :
(toDirectLimit f M <| (of _ _ G f i g) ⊗ₜ m) = (of _ _ _ _ i (g ⊗ₜ m)) := by
rw [toDirectLimit, lift.tmul, lift_of]
rfl
variable [IsDirected ι (· ≤ ·)]
/--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) := by
|
refine LinearEquiv.ofLinear (toDirectLimit f M) (fromDirectLimit f M) ?_ ?_
|
/--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) := by
|
Mathlib.Algebra.Module.DirectLimitAndTensorProduct.75_0.NNGkUEQVeDhgQVm
|
/--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M)
|
Mathlib_Algebra_Module_DirectLimitAndTensorProduct
|
case refine_1
R : Type u_1
inst✝⁷ : CommRing R
ι : Type u_2
inst✝⁶ : DecidableEq ι
inst✝⁵ : Preorder ι
G : ι → Type u_3
inst✝⁴ : (i : ι) → AddCommGroup (G i)
inst✝³ : (i : ι) → Module R (G i)
f : (i j : ι) → i ≤ j → G i →ₗ[R] G j
M : Type u_4
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsDirected ι fun x x_1 => x ≤ x_1
⊢ LinearMap.comp (toDirectLimit f M) (fromDirectLimit f M) = LinearMap.id
|
/-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.Algebra.DirectLimit
/-!
# Tensor product and direct limits commute with each other.
Given a family of `R`-modules `Gᵢ` with a family of compatible `R`-linear maps `fᵢⱼ : Gᵢ → Gⱼ` for
every `i ≤ j` and another `R`-module `M`, we have `(limᵢ Gᵢ) ⊗ M` and `lim (Gᵢ ⊗ M)` are isomorphic
as `R`-modules.
## Main definitions:
* `TensorProduct.directLimitLeft : DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M)`
* `TensorProduct.directLimitRight : M ⊗[R] DirectLimit G f ≃ₗ[R] DirectLimit (M ⊗[R] G ·) (M ◁ f)`
-/
open TensorProduct Module Module.DirectLimit
variable {R : Type*} [CommRing R]
variable {ι : Type*}
variable [DecidableEq ι] [Preorder ι]
variable {G : ι → Type*}
variable [∀ i, AddCommGroup (G i)] [∀ i, Module R (G i)]
variable (f : ∀ i j, i ≤ j → G i →ₗ[R] G j)
variable (M : Type*) [AddCommGroup M] [Module R M]
-- alluding to the notation in `CategoryTheory.Monoidal`
local notation M " ◁ " f => fun i j h ↦ LinearMap.lTensor M (f _ _ h)
local notation f " ▷ " N => fun i j h ↦ LinearMap.rTensor N (f _ _ h)
namespace TensorProduct
/--
the map `limᵢ (Gᵢ ⊗ M) → (limᵢ Gᵢ) ⊗ M` induced by the family of maps `Gᵢ ⊗ M → (limᵢ Gᵢ) ⊗ M`
given by `gᵢ ⊗ m ↦ [gᵢ] ⊗ m`.
-/
noncomputable def fromDirectLimit :
DirectLimit (G · ⊗[R] M) (f ▷ M) →ₗ[R] DirectLimit G f ⊗[R] M :=
DirectLimit.lift _ _ _ _ (fun _ ↦ (of _ _ _ _ _).rTensor M)
fun _ _ _ x ↦ by refine' x.induction_on _ _ _ <;> aesop
variable {M} in
@[simp] lemma fromDirectLimit_of_tmul {i : ι} (g : G i) (m : M) :
fromDirectLimit f M (of _ _ _ _ i (g ⊗ₜ m)) = (of _ _ _ f i g) ⊗ₜ m :=
lift_of (G := (G · ⊗[R] M)) _ _ (g ⊗ₜ m)
/--
the map `(limᵢ Gᵢ) ⊗ M → limᵢ (Gᵢ ⊗ M)` from the bilinear map `limᵢ Gᵢ → M → limᵢ (Gᵢ ⊗ M)` given
by the family of maps `Gᵢ → M → limᵢ (Gᵢ ⊗ M)` where `gᵢ ↦ m ↦ [gᵢ ⊗ m]`
-/
noncomputable def toDirectLimit : DirectLimit G f ⊗[R] M →ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) :=
TensorProduct.lift <| DirectLimit.lift _ _ _ _
(fun i ↦
(TensorProduct.mk R _ _).compr₂ (of R ι _ (fun _i _j h ↦ (f _ _ h).rTensor M) i))
fun _ _ _ g ↦ FunLike.ext _ _ (of_f (G := (G · ⊗[R] M)) (x := g ⊗ₜ ·))
variable {M} in
@[simp] lemma toDirectLimit_tmul_of
{i : ι} (g : G i) (m : M) :
(toDirectLimit f M <| (of _ _ G f i g) ⊗ₜ m) = (of _ _ _ _ i (g ⊗ₜ m)) := by
rw [toDirectLimit, lift.tmul, lift_of]
rfl
variable [IsDirected ι (· ≤ ·)]
/--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) := by
refine LinearEquiv.ofLinear (toDirectLimit f M) (fromDirectLimit f M) ?_ ?_
<;>
|
cases isEmpty_or_nonempty ι
|
/--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) := by
refine LinearEquiv.ofLinear (toDirectLimit f M) (fromDirectLimit f M) ?_ ?_
<;>
|
Mathlib.Algebra.Module.DirectLimitAndTensorProduct.75_0.NNGkUEQVeDhgQVm
|
/--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M)
|
Mathlib_Algebra_Module_DirectLimitAndTensorProduct
|
case refine_2
R : Type u_1
inst✝⁷ : CommRing R
ι : Type u_2
inst✝⁶ : DecidableEq ι
inst✝⁵ : Preorder ι
G : ι → Type u_3
inst✝⁴ : (i : ι) → AddCommGroup (G i)
inst✝³ : (i : ι) → Module R (G i)
f : (i j : ι) → i ≤ j → G i →ₗ[R] G j
M : Type u_4
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsDirected ι fun x x_1 => x ≤ x_1
⊢ LinearMap.comp (fromDirectLimit f M) (toDirectLimit f M) = LinearMap.id
|
/-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.Algebra.DirectLimit
/-!
# Tensor product and direct limits commute with each other.
Given a family of `R`-modules `Gᵢ` with a family of compatible `R`-linear maps `fᵢⱼ : Gᵢ → Gⱼ` for
every `i ≤ j` and another `R`-module `M`, we have `(limᵢ Gᵢ) ⊗ M` and `lim (Gᵢ ⊗ M)` are isomorphic
as `R`-modules.
## Main definitions:
* `TensorProduct.directLimitLeft : DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M)`
* `TensorProduct.directLimitRight : M ⊗[R] DirectLimit G f ≃ₗ[R] DirectLimit (M ⊗[R] G ·) (M ◁ f)`
-/
open TensorProduct Module Module.DirectLimit
variable {R : Type*} [CommRing R]
variable {ι : Type*}
variable [DecidableEq ι] [Preorder ι]
variable {G : ι → Type*}
variable [∀ i, AddCommGroup (G i)] [∀ i, Module R (G i)]
variable (f : ∀ i j, i ≤ j → G i →ₗ[R] G j)
variable (M : Type*) [AddCommGroup M] [Module R M]
-- alluding to the notation in `CategoryTheory.Monoidal`
local notation M " ◁ " f => fun i j h ↦ LinearMap.lTensor M (f _ _ h)
local notation f " ▷ " N => fun i j h ↦ LinearMap.rTensor N (f _ _ h)
namespace TensorProduct
/--
the map `limᵢ (Gᵢ ⊗ M) → (limᵢ Gᵢ) ⊗ M` induced by the family of maps `Gᵢ ⊗ M → (limᵢ Gᵢ) ⊗ M`
given by `gᵢ ⊗ m ↦ [gᵢ] ⊗ m`.
-/
noncomputable def fromDirectLimit :
DirectLimit (G · ⊗[R] M) (f ▷ M) →ₗ[R] DirectLimit G f ⊗[R] M :=
DirectLimit.lift _ _ _ _ (fun _ ↦ (of _ _ _ _ _).rTensor M)
fun _ _ _ x ↦ by refine' x.induction_on _ _ _ <;> aesop
variable {M} in
@[simp] lemma fromDirectLimit_of_tmul {i : ι} (g : G i) (m : M) :
fromDirectLimit f M (of _ _ _ _ i (g ⊗ₜ m)) = (of _ _ _ f i g) ⊗ₜ m :=
lift_of (G := (G · ⊗[R] M)) _ _ (g ⊗ₜ m)
/--
the map `(limᵢ Gᵢ) ⊗ M → limᵢ (Gᵢ ⊗ M)` from the bilinear map `limᵢ Gᵢ → M → limᵢ (Gᵢ ⊗ M)` given
by the family of maps `Gᵢ → M → limᵢ (Gᵢ ⊗ M)` where `gᵢ ↦ m ↦ [gᵢ ⊗ m]`
-/
noncomputable def toDirectLimit : DirectLimit G f ⊗[R] M →ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) :=
TensorProduct.lift <| DirectLimit.lift _ _ _ _
(fun i ↦
(TensorProduct.mk R _ _).compr₂ (of R ι _ (fun _i _j h ↦ (f _ _ h).rTensor M) i))
fun _ _ _ g ↦ FunLike.ext _ _ (of_f (G := (G · ⊗[R] M)) (x := g ⊗ₜ ·))
variable {M} in
@[simp] lemma toDirectLimit_tmul_of
{i : ι} (g : G i) (m : M) :
(toDirectLimit f M <| (of _ _ G f i g) ⊗ₜ m) = (of _ _ _ _ i (g ⊗ₜ m)) := by
rw [toDirectLimit, lift.tmul, lift_of]
rfl
variable [IsDirected ι (· ≤ ·)]
/--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) := by
refine LinearEquiv.ofLinear (toDirectLimit f M) (fromDirectLimit f M) ?_ ?_
<;>
|
cases isEmpty_or_nonempty ι
|
/--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) := by
refine LinearEquiv.ofLinear (toDirectLimit f M) (fromDirectLimit f M) ?_ ?_
<;>
|
Mathlib.Algebra.Module.DirectLimitAndTensorProduct.75_0.NNGkUEQVeDhgQVm
|
/--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M)
|
Mathlib_Algebra_Module_DirectLimitAndTensorProduct
|
case refine_1.inl
R : Type u_1
inst✝⁷ : CommRing R
ι : Type u_2
inst✝⁶ : DecidableEq ι
inst✝⁵ : Preorder ι
G : ι → Type u_3
inst✝⁴ : (i : ι) → AddCommGroup (G i)
inst✝³ : (i : ι) → Module R (G i)
f : (i j : ι) → i ≤ j → G i →ₗ[R] G j
M : Type u_4
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsDirected ι fun x x_1 => x ≤ x_1
h✝ : IsEmpty ι
⊢ LinearMap.comp (toDirectLimit f M) (fromDirectLimit f M) = LinearMap.id
|
/-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.Algebra.DirectLimit
/-!
# Tensor product and direct limits commute with each other.
Given a family of `R`-modules `Gᵢ` with a family of compatible `R`-linear maps `fᵢⱼ : Gᵢ → Gⱼ` for
every `i ≤ j` and another `R`-module `M`, we have `(limᵢ Gᵢ) ⊗ M` and `lim (Gᵢ ⊗ M)` are isomorphic
as `R`-modules.
## Main definitions:
* `TensorProduct.directLimitLeft : DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M)`
* `TensorProduct.directLimitRight : M ⊗[R] DirectLimit G f ≃ₗ[R] DirectLimit (M ⊗[R] G ·) (M ◁ f)`
-/
open TensorProduct Module Module.DirectLimit
variable {R : Type*} [CommRing R]
variable {ι : Type*}
variable [DecidableEq ι] [Preorder ι]
variable {G : ι → Type*}
variable [∀ i, AddCommGroup (G i)] [∀ i, Module R (G i)]
variable (f : ∀ i j, i ≤ j → G i →ₗ[R] G j)
variable (M : Type*) [AddCommGroup M] [Module R M]
-- alluding to the notation in `CategoryTheory.Monoidal`
local notation M " ◁ " f => fun i j h ↦ LinearMap.lTensor M (f _ _ h)
local notation f " ▷ " N => fun i j h ↦ LinearMap.rTensor N (f _ _ h)
namespace TensorProduct
/--
the map `limᵢ (Gᵢ ⊗ M) → (limᵢ Gᵢ) ⊗ M` induced by the family of maps `Gᵢ ⊗ M → (limᵢ Gᵢ) ⊗ M`
given by `gᵢ ⊗ m ↦ [gᵢ] ⊗ m`.
-/
noncomputable def fromDirectLimit :
DirectLimit (G · ⊗[R] M) (f ▷ M) →ₗ[R] DirectLimit G f ⊗[R] M :=
DirectLimit.lift _ _ _ _ (fun _ ↦ (of _ _ _ _ _).rTensor M)
fun _ _ _ x ↦ by refine' x.induction_on _ _ _ <;> aesop
variable {M} in
@[simp] lemma fromDirectLimit_of_tmul {i : ι} (g : G i) (m : M) :
fromDirectLimit f M (of _ _ _ _ i (g ⊗ₜ m)) = (of _ _ _ f i g) ⊗ₜ m :=
lift_of (G := (G · ⊗[R] M)) _ _ (g ⊗ₜ m)
/--
the map `(limᵢ Gᵢ) ⊗ M → limᵢ (Gᵢ ⊗ M)` from the bilinear map `limᵢ Gᵢ → M → limᵢ (Gᵢ ⊗ M)` given
by the family of maps `Gᵢ → M → limᵢ (Gᵢ ⊗ M)` where `gᵢ ↦ m ↦ [gᵢ ⊗ m]`
-/
noncomputable def toDirectLimit : DirectLimit G f ⊗[R] M →ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) :=
TensorProduct.lift <| DirectLimit.lift _ _ _ _
(fun i ↦
(TensorProduct.mk R _ _).compr₂ (of R ι _ (fun _i _j h ↦ (f _ _ h).rTensor M) i))
fun _ _ _ g ↦ FunLike.ext _ _ (of_f (G := (G · ⊗[R] M)) (x := g ⊗ₜ ·))
variable {M} in
@[simp] lemma toDirectLimit_tmul_of
{i : ι} (g : G i) (m : M) :
(toDirectLimit f M <| (of _ _ G f i g) ⊗ₜ m) = (of _ _ _ _ i (g ⊗ₜ m)) := by
rw [toDirectLimit, lift.tmul, lift_of]
rfl
variable [IsDirected ι (· ≤ ·)]
/--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) := by
refine LinearEquiv.ofLinear (toDirectLimit f M) (fromDirectLimit f M) ?_ ?_
<;> cases isEmpty_or_nonempty ι
·
|
ext
|
/--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) := by
refine LinearEquiv.ofLinear (toDirectLimit f M) (fromDirectLimit f M) ?_ ?_
<;> cases isEmpty_or_nonempty ι
·
|
Mathlib.Algebra.Module.DirectLimitAndTensorProduct.75_0.NNGkUEQVeDhgQVm
|
/--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M)
|
Mathlib_Algebra_Module_DirectLimitAndTensorProduct
|
case refine_1.inl.h
R : Type u_1
inst✝⁷ : CommRing R
ι : Type u_2
inst✝⁶ : DecidableEq ι
inst✝⁵ : Preorder ι
G : ι → Type u_3
inst✝⁴ : (i : ι) → AddCommGroup (G i)
inst✝³ : (i : ι) → Module R (G i)
f : (i j : ι) → i ≤ j → G i →ₗ[R] G j
M : Type u_4
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsDirected ι fun x x_1 => x ≤ x_1
h✝ : IsEmpty ι
x✝ : DirectLimit (fun x => G x ⊗[R] M) fun i j h => LinearMap.rTensor M (f i j h)
⊢ (LinearMap.comp (toDirectLimit f M) (fromDirectLimit f M)) x✝ = LinearMap.id x✝
|
/-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.Algebra.DirectLimit
/-!
# Tensor product and direct limits commute with each other.
Given a family of `R`-modules `Gᵢ` with a family of compatible `R`-linear maps `fᵢⱼ : Gᵢ → Gⱼ` for
every `i ≤ j` and another `R`-module `M`, we have `(limᵢ Gᵢ) ⊗ M` and `lim (Gᵢ ⊗ M)` are isomorphic
as `R`-modules.
## Main definitions:
* `TensorProduct.directLimitLeft : DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M)`
* `TensorProduct.directLimitRight : M ⊗[R] DirectLimit G f ≃ₗ[R] DirectLimit (M ⊗[R] G ·) (M ◁ f)`
-/
open TensorProduct Module Module.DirectLimit
variable {R : Type*} [CommRing R]
variable {ι : Type*}
variable [DecidableEq ι] [Preorder ι]
variable {G : ι → Type*}
variable [∀ i, AddCommGroup (G i)] [∀ i, Module R (G i)]
variable (f : ∀ i j, i ≤ j → G i →ₗ[R] G j)
variable (M : Type*) [AddCommGroup M] [Module R M]
-- alluding to the notation in `CategoryTheory.Monoidal`
local notation M " ◁ " f => fun i j h ↦ LinearMap.lTensor M (f _ _ h)
local notation f " ▷ " N => fun i j h ↦ LinearMap.rTensor N (f _ _ h)
namespace TensorProduct
/--
the map `limᵢ (Gᵢ ⊗ M) → (limᵢ Gᵢ) ⊗ M` induced by the family of maps `Gᵢ ⊗ M → (limᵢ Gᵢ) ⊗ M`
given by `gᵢ ⊗ m ↦ [gᵢ] ⊗ m`.
-/
noncomputable def fromDirectLimit :
DirectLimit (G · ⊗[R] M) (f ▷ M) →ₗ[R] DirectLimit G f ⊗[R] M :=
DirectLimit.lift _ _ _ _ (fun _ ↦ (of _ _ _ _ _).rTensor M)
fun _ _ _ x ↦ by refine' x.induction_on _ _ _ <;> aesop
variable {M} in
@[simp] lemma fromDirectLimit_of_tmul {i : ι} (g : G i) (m : M) :
fromDirectLimit f M (of _ _ _ _ i (g ⊗ₜ m)) = (of _ _ _ f i g) ⊗ₜ m :=
lift_of (G := (G · ⊗[R] M)) _ _ (g ⊗ₜ m)
/--
the map `(limᵢ Gᵢ) ⊗ M → limᵢ (Gᵢ ⊗ M)` from the bilinear map `limᵢ Gᵢ → M → limᵢ (Gᵢ ⊗ M)` given
by the family of maps `Gᵢ → M → limᵢ (Gᵢ ⊗ M)` where `gᵢ ↦ m ↦ [gᵢ ⊗ m]`
-/
noncomputable def toDirectLimit : DirectLimit G f ⊗[R] M →ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) :=
TensorProduct.lift <| DirectLimit.lift _ _ _ _
(fun i ↦
(TensorProduct.mk R _ _).compr₂ (of R ι _ (fun _i _j h ↦ (f _ _ h).rTensor M) i))
fun _ _ _ g ↦ FunLike.ext _ _ (of_f (G := (G · ⊗[R] M)) (x := g ⊗ₜ ·))
variable {M} in
@[simp] lemma toDirectLimit_tmul_of
{i : ι} (g : G i) (m : M) :
(toDirectLimit f M <| (of _ _ G f i g) ⊗ₜ m) = (of _ _ _ _ i (g ⊗ₜ m)) := by
rw [toDirectLimit, lift.tmul, lift_of]
rfl
variable [IsDirected ι (· ≤ ·)]
/--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) := by
refine LinearEquiv.ofLinear (toDirectLimit f M) (fromDirectLimit f M) ?_ ?_
<;> cases isEmpty_or_nonempty ι
· ext;
|
apply Subsingleton.elim
|
/--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) := by
refine LinearEquiv.ofLinear (toDirectLimit f M) (fromDirectLimit f M) ?_ ?_
<;> cases isEmpty_or_nonempty ι
· ext;
|
Mathlib.Algebra.Module.DirectLimitAndTensorProduct.75_0.NNGkUEQVeDhgQVm
|
/--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M)
|
Mathlib_Algebra_Module_DirectLimitAndTensorProduct
|
case refine_1.inr
R : Type u_1
inst✝⁷ : CommRing R
ι : Type u_2
inst✝⁶ : DecidableEq ι
inst✝⁵ : Preorder ι
G : ι → Type u_3
inst✝⁴ : (i : ι) → AddCommGroup (G i)
inst✝³ : (i : ι) → Module R (G i)
f : (i j : ι) → i ≤ j → G i →ₗ[R] G j
M : Type u_4
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsDirected ι fun x x_1 => x ≤ x_1
h✝ : Nonempty ι
⊢ LinearMap.comp (toDirectLimit f M) (fromDirectLimit f M) = LinearMap.id
|
/-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.Algebra.DirectLimit
/-!
# Tensor product and direct limits commute with each other.
Given a family of `R`-modules `Gᵢ` with a family of compatible `R`-linear maps `fᵢⱼ : Gᵢ → Gⱼ` for
every `i ≤ j` and another `R`-module `M`, we have `(limᵢ Gᵢ) ⊗ M` and `lim (Gᵢ ⊗ M)` are isomorphic
as `R`-modules.
## Main definitions:
* `TensorProduct.directLimitLeft : DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M)`
* `TensorProduct.directLimitRight : M ⊗[R] DirectLimit G f ≃ₗ[R] DirectLimit (M ⊗[R] G ·) (M ◁ f)`
-/
open TensorProduct Module Module.DirectLimit
variable {R : Type*} [CommRing R]
variable {ι : Type*}
variable [DecidableEq ι] [Preorder ι]
variable {G : ι → Type*}
variable [∀ i, AddCommGroup (G i)] [∀ i, Module R (G i)]
variable (f : ∀ i j, i ≤ j → G i →ₗ[R] G j)
variable (M : Type*) [AddCommGroup M] [Module R M]
-- alluding to the notation in `CategoryTheory.Monoidal`
local notation M " ◁ " f => fun i j h ↦ LinearMap.lTensor M (f _ _ h)
local notation f " ▷ " N => fun i j h ↦ LinearMap.rTensor N (f _ _ h)
namespace TensorProduct
/--
the map `limᵢ (Gᵢ ⊗ M) → (limᵢ Gᵢ) ⊗ M` induced by the family of maps `Gᵢ ⊗ M → (limᵢ Gᵢ) ⊗ M`
given by `gᵢ ⊗ m ↦ [gᵢ] ⊗ m`.
-/
noncomputable def fromDirectLimit :
DirectLimit (G · ⊗[R] M) (f ▷ M) →ₗ[R] DirectLimit G f ⊗[R] M :=
DirectLimit.lift _ _ _ _ (fun _ ↦ (of _ _ _ _ _).rTensor M)
fun _ _ _ x ↦ by refine' x.induction_on _ _ _ <;> aesop
variable {M} in
@[simp] lemma fromDirectLimit_of_tmul {i : ι} (g : G i) (m : M) :
fromDirectLimit f M (of _ _ _ _ i (g ⊗ₜ m)) = (of _ _ _ f i g) ⊗ₜ m :=
lift_of (G := (G · ⊗[R] M)) _ _ (g ⊗ₜ m)
/--
the map `(limᵢ Gᵢ) ⊗ M → limᵢ (Gᵢ ⊗ M)` from the bilinear map `limᵢ Gᵢ → M → limᵢ (Gᵢ ⊗ M)` given
by the family of maps `Gᵢ → M → limᵢ (Gᵢ ⊗ M)` where `gᵢ ↦ m ↦ [gᵢ ⊗ m]`
-/
noncomputable def toDirectLimit : DirectLimit G f ⊗[R] M →ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) :=
TensorProduct.lift <| DirectLimit.lift _ _ _ _
(fun i ↦
(TensorProduct.mk R _ _).compr₂ (of R ι _ (fun _i _j h ↦ (f _ _ h).rTensor M) i))
fun _ _ _ g ↦ FunLike.ext _ _ (of_f (G := (G · ⊗[R] M)) (x := g ⊗ₜ ·))
variable {M} in
@[simp] lemma toDirectLimit_tmul_of
{i : ι} (g : G i) (m : M) :
(toDirectLimit f M <| (of _ _ G f i g) ⊗ₜ m) = (of _ _ _ _ i (g ⊗ₜ m)) := by
rw [toDirectLimit, lift.tmul, lift_of]
rfl
variable [IsDirected ι (· ≤ ·)]
/--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) := by
refine LinearEquiv.ofLinear (toDirectLimit f M) (fromDirectLimit f M) ?_ ?_
<;> cases isEmpty_or_nonempty ι
· ext; apply Subsingleton.elim
·
|
refine FunLike.ext _ _ fun x ↦ x.induction_on fun i g ↦ g.induction_on ?_ ?_ ?_
|
/--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) := by
refine LinearEquiv.ofLinear (toDirectLimit f M) (fromDirectLimit f M) ?_ ?_
<;> cases isEmpty_or_nonempty ι
· ext; apply Subsingleton.elim
·
|
Mathlib.Algebra.Module.DirectLimitAndTensorProduct.75_0.NNGkUEQVeDhgQVm
|
/--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M)
|
Mathlib_Algebra_Module_DirectLimitAndTensorProduct
|
case refine_1.inr.refine_1
R : Type u_1
inst✝⁷ : CommRing R
ι : Type u_2
inst✝⁶ : DecidableEq ι
inst✝⁵ : Preorder ι
G : ι → Type u_3
inst✝⁴ : (i : ι) → AddCommGroup (G i)
inst✝³ : (i : ι) → Module R (G i)
f : (i j : ι) → i ≤ j → G i →ₗ[R] G j
M : Type u_4
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsDirected ι fun x x_1 => x ≤ x_1
h✝ : Nonempty ι
x : DirectLimit (fun x => G x ⊗[R] M) fun i j h => LinearMap.rTensor M (f i j h)
i : ι
g : G i ⊗[R] M
⊢ (LinearMap.comp (toDirectLimit f M) (fromDirectLimit f M))
((of R ι (fun x => G x ⊗[R] M) (fun i j h => LinearMap.rTensor M (f i j h)) i) 0) =
LinearMap.id ((of R ι (fun x => G x ⊗[R] M) (fun i j h => LinearMap.rTensor M (f i j h)) i) 0)
|
/-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.Algebra.DirectLimit
/-!
# Tensor product and direct limits commute with each other.
Given a family of `R`-modules `Gᵢ` with a family of compatible `R`-linear maps `fᵢⱼ : Gᵢ → Gⱼ` for
every `i ≤ j` and another `R`-module `M`, we have `(limᵢ Gᵢ) ⊗ M` and `lim (Gᵢ ⊗ M)` are isomorphic
as `R`-modules.
## Main definitions:
* `TensorProduct.directLimitLeft : DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M)`
* `TensorProduct.directLimitRight : M ⊗[R] DirectLimit G f ≃ₗ[R] DirectLimit (M ⊗[R] G ·) (M ◁ f)`
-/
open TensorProduct Module Module.DirectLimit
variable {R : Type*} [CommRing R]
variable {ι : Type*}
variable [DecidableEq ι] [Preorder ι]
variable {G : ι → Type*}
variable [∀ i, AddCommGroup (G i)] [∀ i, Module R (G i)]
variable (f : ∀ i j, i ≤ j → G i →ₗ[R] G j)
variable (M : Type*) [AddCommGroup M] [Module R M]
-- alluding to the notation in `CategoryTheory.Monoidal`
local notation M " ◁ " f => fun i j h ↦ LinearMap.lTensor M (f _ _ h)
local notation f " ▷ " N => fun i j h ↦ LinearMap.rTensor N (f _ _ h)
namespace TensorProduct
/--
the map `limᵢ (Gᵢ ⊗ M) → (limᵢ Gᵢ) ⊗ M` induced by the family of maps `Gᵢ ⊗ M → (limᵢ Gᵢ) ⊗ M`
given by `gᵢ ⊗ m ↦ [gᵢ] ⊗ m`.
-/
noncomputable def fromDirectLimit :
DirectLimit (G · ⊗[R] M) (f ▷ M) →ₗ[R] DirectLimit G f ⊗[R] M :=
DirectLimit.lift _ _ _ _ (fun _ ↦ (of _ _ _ _ _).rTensor M)
fun _ _ _ x ↦ by refine' x.induction_on _ _ _ <;> aesop
variable {M} in
@[simp] lemma fromDirectLimit_of_tmul {i : ι} (g : G i) (m : M) :
fromDirectLimit f M (of _ _ _ _ i (g ⊗ₜ m)) = (of _ _ _ f i g) ⊗ₜ m :=
lift_of (G := (G · ⊗[R] M)) _ _ (g ⊗ₜ m)
/--
the map `(limᵢ Gᵢ) ⊗ M → limᵢ (Gᵢ ⊗ M)` from the bilinear map `limᵢ Gᵢ → M → limᵢ (Gᵢ ⊗ M)` given
by the family of maps `Gᵢ → M → limᵢ (Gᵢ ⊗ M)` where `gᵢ ↦ m ↦ [gᵢ ⊗ m]`
-/
noncomputable def toDirectLimit : DirectLimit G f ⊗[R] M →ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) :=
TensorProduct.lift <| DirectLimit.lift _ _ _ _
(fun i ↦
(TensorProduct.mk R _ _).compr₂ (of R ι _ (fun _i _j h ↦ (f _ _ h).rTensor M) i))
fun _ _ _ g ↦ FunLike.ext _ _ (of_f (G := (G · ⊗[R] M)) (x := g ⊗ₜ ·))
variable {M} in
@[simp] lemma toDirectLimit_tmul_of
{i : ι} (g : G i) (m : M) :
(toDirectLimit f M <| (of _ _ G f i g) ⊗ₜ m) = (of _ _ _ _ i (g ⊗ₜ m)) := by
rw [toDirectLimit, lift.tmul, lift_of]
rfl
variable [IsDirected ι (· ≤ ·)]
/--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) := by
refine LinearEquiv.ofLinear (toDirectLimit f M) (fromDirectLimit f M) ?_ ?_
<;> cases isEmpty_or_nonempty ι
· ext; apply Subsingleton.elim
· refine FunLike.ext _ _ fun x ↦ x.induction_on fun i g ↦ g.induction_on ?_ ?_ ?_ <;>
|
aesop
|
/--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) := by
refine LinearEquiv.ofLinear (toDirectLimit f M) (fromDirectLimit f M) ?_ ?_
<;> cases isEmpty_or_nonempty ι
· ext; apply Subsingleton.elim
· refine FunLike.ext _ _ fun x ↦ x.induction_on fun i g ↦ g.induction_on ?_ ?_ ?_ <;>
|
Mathlib.Algebra.Module.DirectLimitAndTensorProduct.75_0.NNGkUEQVeDhgQVm
|
/--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M)
|
Mathlib_Algebra_Module_DirectLimitAndTensorProduct
|
case refine_1.inr.refine_2
R : Type u_1
inst✝⁷ : CommRing R
ι : Type u_2
inst✝⁶ : DecidableEq ι
inst✝⁵ : Preorder ι
G : ι → Type u_3
inst✝⁴ : (i : ι) → AddCommGroup (G i)
inst✝³ : (i : ι) → Module R (G i)
f : (i j : ι) → i ≤ j → G i →ₗ[R] G j
M : Type u_4
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsDirected ι fun x x_1 => x ≤ x_1
h✝ : Nonempty ι
x : DirectLimit (fun x => G x ⊗[R] M) fun i j h => LinearMap.rTensor M (f i j h)
i : ι
g : G i ⊗[R] M
⊢ ∀ (x : G i) (y : M),
(LinearMap.comp (toDirectLimit f M) (fromDirectLimit f M))
((of R ι (fun x => G x ⊗[R] M) (fun i j h => LinearMap.rTensor M (f i j h)) i) (x ⊗ₜ[R] y)) =
LinearMap.id ((of R ι (fun x => G x ⊗[R] M) (fun i j h => LinearMap.rTensor M (f i j h)) i) (x ⊗ₜ[R] y))
|
/-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.Algebra.DirectLimit
/-!
# Tensor product and direct limits commute with each other.
Given a family of `R`-modules `Gᵢ` with a family of compatible `R`-linear maps `fᵢⱼ : Gᵢ → Gⱼ` for
every `i ≤ j` and another `R`-module `M`, we have `(limᵢ Gᵢ) ⊗ M` and `lim (Gᵢ ⊗ M)` are isomorphic
as `R`-modules.
## Main definitions:
* `TensorProduct.directLimitLeft : DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M)`
* `TensorProduct.directLimitRight : M ⊗[R] DirectLimit G f ≃ₗ[R] DirectLimit (M ⊗[R] G ·) (M ◁ f)`
-/
open TensorProduct Module Module.DirectLimit
variable {R : Type*} [CommRing R]
variable {ι : Type*}
variable [DecidableEq ι] [Preorder ι]
variable {G : ι → Type*}
variable [∀ i, AddCommGroup (G i)] [∀ i, Module R (G i)]
variable (f : ∀ i j, i ≤ j → G i →ₗ[R] G j)
variable (M : Type*) [AddCommGroup M] [Module R M]
-- alluding to the notation in `CategoryTheory.Monoidal`
local notation M " ◁ " f => fun i j h ↦ LinearMap.lTensor M (f _ _ h)
local notation f " ▷ " N => fun i j h ↦ LinearMap.rTensor N (f _ _ h)
namespace TensorProduct
/--
the map `limᵢ (Gᵢ ⊗ M) → (limᵢ Gᵢ) ⊗ M` induced by the family of maps `Gᵢ ⊗ M → (limᵢ Gᵢ) ⊗ M`
given by `gᵢ ⊗ m ↦ [gᵢ] ⊗ m`.
-/
noncomputable def fromDirectLimit :
DirectLimit (G · ⊗[R] M) (f ▷ M) →ₗ[R] DirectLimit G f ⊗[R] M :=
DirectLimit.lift _ _ _ _ (fun _ ↦ (of _ _ _ _ _).rTensor M)
fun _ _ _ x ↦ by refine' x.induction_on _ _ _ <;> aesop
variable {M} in
@[simp] lemma fromDirectLimit_of_tmul {i : ι} (g : G i) (m : M) :
fromDirectLimit f M (of _ _ _ _ i (g ⊗ₜ m)) = (of _ _ _ f i g) ⊗ₜ m :=
lift_of (G := (G · ⊗[R] M)) _ _ (g ⊗ₜ m)
/--
the map `(limᵢ Gᵢ) ⊗ M → limᵢ (Gᵢ ⊗ M)` from the bilinear map `limᵢ Gᵢ → M → limᵢ (Gᵢ ⊗ M)` given
by the family of maps `Gᵢ → M → limᵢ (Gᵢ ⊗ M)` where `gᵢ ↦ m ↦ [gᵢ ⊗ m]`
-/
noncomputable def toDirectLimit : DirectLimit G f ⊗[R] M →ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) :=
TensorProduct.lift <| DirectLimit.lift _ _ _ _
(fun i ↦
(TensorProduct.mk R _ _).compr₂ (of R ι _ (fun _i _j h ↦ (f _ _ h).rTensor M) i))
fun _ _ _ g ↦ FunLike.ext _ _ (of_f (G := (G · ⊗[R] M)) (x := g ⊗ₜ ·))
variable {M} in
@[simp] lemma toDirectLimit_tmul_of
{i : ι} (g : G i) (m : M) :
(toDirectLimit f M <| (of _ _ G f i g) ⊗ₜ m) = (of _ _ _ _ i (g ⊗ₜ m)) := by
rw [toDirectLimit, lift.tmul, lift_of]
rfl
variable [IsDirected ι (· ≤ ·)]
/--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) := by
refine LinearEquiv.ofLinear (toDirectLimit f M) (fromDirectLimit f M) ?_ ?_
<;> cases isEmpty_or_nonempty ι
· ext; apply Subsingleton.elim
· refine FunLike.ext _ _ fun x ↦ x.induction_on fun i g ↦ g.induction_on ?_ ?_ ?_ <;>
|
aesop
|
/--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) := by
refine LinearEquiv.ofLinear (toDirectLimit f M) (fromDirectLimit f M) ?_ ?_
<;> cases isEmpty_or_nonempty ι
· ext; apply Subsingleton.elim
· refine FunLike.ext _ _ fun x ↦ x.induction_on fun i g ↦ g.induction_on ?_ ?_ ?_ <;>
|
Mathlib.Algebra.Module.DirectLimitAndTensorProduct.75_0.NNGkUEQVeDhgQVm
|
/--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M)
|
Mathlib_Algebra_Module_DirectLimitAndTensorProduct
|
case refine_1.inr.refine_3
R : Type u_1
inst✝⁷ : CommRing R
ι : Type u_2
inst✝⁶ : DecidableEq ι
inst✝⁵ : Preorder ι
G : ι → Type u_3
inst✝⁴ : (i : ι) → AddCommGroup (G i)
inst✝³ : (i : ι) → Module R (G i)
f : (i j : ι) → i ≤ j → G i →ₗ[R] G j
M : Type u_4
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsDirected ι fun x x_1 => x ≤ x_1
h✝ : Nonempty ι
x : DirectLimit (fun x => G x ⊗[R] M) fun i j h => LinearMap.rTensor M (f i j h)
i : ι
g : G i ⊗[R] M
⊢ ∀ (x y : G i ⊗[R] M),
(LinearMap.comp (toDirectLimit f M) (fromDirectLimit f M))
((of R ι (fun x => G x ⊗[R] M) (fun i j h => LinearMap.rTensor M (f i j h)) i) x) =
LinearMap.id ((of R ι (fun x => G x ⊗[R] M) (fun i j h => LinearMap.rTensor M (f i j h)) i) x) →
(LinearMap.comp (toDirectLimit f M) (fromDirectLimit f M))
((of R ι (fun x => G x ⊗[R] M) (fun i j h => LinearMap.rTensor M (f i j h)) i) y) =
LinearMap.id ((of R ι (fun x => G x ⊗[R] M) (fun i j h => LinearMap.rTensor M (f i j h)) i) y) →
(LinearMap.comp (toDirectLimit f M) (fromDirectLimit f M))
((of R ι (fun x => G x ⊗[R] M) (fun i j h => LinearMap.rTensor M (f i j h)) i) (x + y)) =
LinearMap.id ((of R ι (fun x => G x ⊗[R] M) (fun i j h => LinearMap.rTensor M (f i j h)) i) (x + y))
|
/-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.Algebra.DirectLimit
/-!
# Tensor product and direct limits commute with each other.
Given a family of `R`-modules `Gᵢ` with a family of compatible `R`-linear maps `fᵢⱼ : Gᵢ → Gⱼ` for
every `i ≤ j` and another `R`-module `M`, we have `(limᵢ Gᵢ) ⊗ M` and `lim (Gᵢ ⊗ M)` are isomorphic
as `R`-modules.
## Main definitions:
* `TensorProduct.directLimitLeft : DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M)`
* `TensorProduct.directLimitRight : M ⊗[R] DirectLimit G f ≃ₗ[R] DirectLimit (M ⊗[R] G ·) (M ◁ f)`
-/
open TensorProduct Module Module.DirectLimit
variable {R : Type*} [CommRing R]
variable {ι : Type*}
variable [DecidableEq ι] [Preorder ι]
variable {G : ι → Type*}
variable [∀ i, AddCommGroup (G i)] [∀ i, Module R (G i)]
variable (f : ∀ i j, i ≤ j → G i →ₗ[R] G j)
variable (M : Type*) [AddCommGroup M] [Module R M]
-- alluding to the notation in `CategoryTheory.Monoidal`
local notation M " ◁ " f => fun i j h ↦ LinearMap.lTensor M (f _ _ h)
local notation f " ▷ " N => fun i j h ↦ LinearMap.rTensor N (f _ _ h)
namespace TensorProduct
/--
the map `limᵢ (Gᵢ ⊗ M) → (limᵢ Gᵢ) ⊗ M` induced by the family of maps `Gᵢ ⊗ M → (limᵢ Gᵢ) ⊗ M`
given by `gᵢ ⊗ m ↦ [gᵢ] ⊗ m`.
-/
noncomputable def fromDirectLimit :
DirectLimit (G · ⊗[R] M) (f ▷ M) →ₗ[R] DirectLimit G f ⊗[R] M :=
DirectLimit.lift _ _ _ _ (fun _ ↦ (of _ _ _ _ _).rTensor M)
fun _ _ _ x ↦ by refine' x.induction_on _ _ _ <;> aesop
variable {M} in
@[simp] lemma fromDirectLimit_of_tmul {i : ι} (g : G i) (m : M) :
fromDirectLimit f M (of _ _ _ _ i (g ⊗ₜ m)) = (of _ _ _ f i g) ⊗ₜ m :=
lift_of (G := (G · ⊗[R] M)) _ _ (g ⊗ₜ m)
/--
the map `(limᵢ Gᵢ) ⊗ M → limᵢ (Gᵢ ⊗ M)` from the bilinear map `limᵢ Gᵢ → M → limᵢ (Gᵢ ⊗ M)` given
by the family of maps `Gᵢ → M → limᵢ (Gᵢ ⊗ M)` where `gᵢ ↦ m ↦ [gᵢ ⊗ m]`
-/
noncomputable def toDirectLimit : DirectLimit G f ⊗[R] M →ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) :=
TensorProduct.lift <| DirectLimit.lift _ _ _ _
(fun i ↦
(TensorProduct.mk R _ _).compr₂ (of R ι _ (fun _i _j h ↦ (f _ _ h).rTensor M) i))
fun _ _ _ g ↦ FunLike.ext _ _ (of_f (G := (G · ⊗[R] M)) (x := g ⊗ₜ ·))
variable {M} in
@[simp] lemma toDirectLimit_tmul_of
{i : ι} (g : G i) (m : M) :
(toDirectLimit f M <| (of _ _ G f i g) ⊗ₜ m) = (of _ _ _ _ i (g ⊗ₜ m)) := by
rw [toDirectLimit, lift.tmul, lift_of]
rfl
variable [IsDirected ι (· ≤ ·)]
/--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) := by
refine LinearEquiv.ofLinear (toDirectLimit f M) (fromDirectLimit f M) ?_ ?_
<;> cases isEmpty_or_nonempty ι
· ext; apply Subsingleton.elim
· refine FunLike.ext _ _ fun x ↦ x.induction_on fun i g ↦ g.induction_on ?_ ?_ ?_ <;>
|
aesop
|
/--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) := by
refine LinearEquiv.ofLinear (toDirectLimit f M) (fromDirectLimit f M) ?_ ?_
<;> cases isEmpty_or_nonempty ι
· ext; apply Subsingleton.elim
· refine FunLike.ext _ _ fun x ↦ x.induction_on fun i g ↦ g.induction_on ?_ ?_ ?_ <;>
|
Mathlib.Algebra.Module.DirectLimitAndTensorProduct.75_0.NNGkUEQVeDhgQVm
|
/--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M)
|
Mathlib_Algebra_Module_DirectLimitAndTensorProduct
|
case refine_2.inl
R : Type u_1
inst✝⁷ : CommRing R
ι : Type u_2
inst✝⁶ : DecidableEq ι
inst✝⁵ : Preorder ι
G : ι → Type u_3
inst✝⁴ : (i : ι) → AddCommGroup (G i)
inst✝³ : (i : ι) → Module R (G i)
f : (i j : ι) → i ≤ j → G i →ₗ[R] G j
M : Type u_4
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsDirected ι fun x x_1 => x ≤ x_1
h✝ : IsEmpty ι
⊢ LinearMap.comp (fromDirectLimit f M) (toDirectLimit f M) = LinearMap.id
|
/-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.Algebra.DirectLimit
/-!
# Tensor product and direct limits commute with each other.
Given a family of `R`-modules `Gᵢ` with a family of compatible `R`-linear maps `fᵢⱼ : Gᵢ → Gⱼ` for
every `i ≤ j` and another `R`-module `M`, we have `(limᵢ Gᵢ) ⊗ M` and `lim (Gᵢ ⊗ M)` are isomorphic
as `R`-modules.
## Main definitions:
* `TensorProduct.directLimitLeft : DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M)`
* `TensorProduct.directLimitRight : M ⊗[R] DirectLimit G f ≃ₗ[R] DirectLimit (M ⊗[R] G ·) (M ◁ f)`
-/
open TensorProduct Module Module.DirectLimit
variable {R : Type*} [CommRing R]
variable {ι : Type*}
variable [DecidableEq ι] [Preorder ι]
variable {G : ι → Type*}
variable [∀ i, AddCommGroup (G i)] [∀ i, Module R (G i)]
variable (f : ∀ i j, i ≤ j → G i →ₗ[R] G j)
variable (M : Type*) [AddCommGroup M] [Module R M]
-- alluding to the notation in `CategoryTheory.Monoidal`
local notation M " ◁ " f => fun i j h ↦ LinearMap.lTensor M (f _ _ h)
local notation f " ▷ " N => fun i j h ↦ LinearMap.rTensor N (f _ _ h)
namespace TensorProduct
/--
the map `limᵢ (Gᵢ ⊗ M) → (limᵢ Gᵢ) ⊗ M` induced by the family of maps `Gᵢ ⊗ M → (limᵢ Gᵢ) ⊗ M`
given by `gᵢ ⊗ m ↦ [gᵢ] ⊗ m`.
-/
noncomputable def fromDirectLimit :
DirectLimit (G · ⊗[R] M) (f ▷ M) →ₗ[R] DirectLimit G f ⊗[R] M :=
DirectLimit.lift _ _ _ _ (fun _ ↦ (of _ _ _ _ _).rTensor M)
fun _ _ _ x ↦ by refine' x.induction_on _ _ _ <;> aesop
variable {M} in
@[simp] lemma fromDirectLimit_of_tmul {i : ι} (g : G i) (m : M) :
fromDirectLimit f M (of _ _ _ _ i (g ⊗ₜ m)) = (of _ _ _ f i g) ⊗ₜ m :=
lift_of (G := (G · ⊗[R] M)) _ _ (g ⊗ₜ m)
/--
the map `(limᵢ Gᵢ) ⊗ M → limᵢ (Gᵢ ⊗ M)` from the bilinear map `limᵢ Gᵢ → M → limᵢ (Gᵢ ⊗ M)` given
by the family of maps `Gᵢ → M → limᵢ (Gᵢ ⊗ M)` where `gᵢ ↦ m ↦ [gᵢ ⊗ m]`
-/
noncomputable def toDirectLimit : DirectLimit G f ⊗[R] M →ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) :=
TensorProduct.lift <| DirectLimit.lift _ _ _ _
(fun i ↦
(TensorProduct.mk R _ _).compr₂ (of R ι _ (fun _i _j h ↦ (f _ _ h).rTensor M) i))
fun _ _ _ g ↦ FunLike.ext _ _ (of_f (G := (G · ⊗[R] M)) (x := g ⊗ₜ ·))
variable {M} in
@[simp] lemma toDirectLimit_tmul_of
{i : ι} (g : G i) (m : M) :
(toDirectLimit f M <| (of _ _ G f i g) ⊗ₜ m) = (of _ _ _ _ i (g ⊗ₜ m)) := by
rw [toDirectLimit, lift.tmul, lift_of]
rfl
variable [IsDirected ι (· ≤ ·)]
/--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) := by
refine LinearEquiv.ofLinear (toDirectLimit f M) (fromDirectLimit f M) ?_ ?_
<;> cases isEmpty_or_nonempty ι
· ext; apply Subsingleton.elim
· refine FunLike.ext _ _ fun x ↦ x.induction_on fun i g ↦ g.induction_on ?_ ?_ ?_ <;> aesop
·
|
ext
|
/--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) := by
refine LinearEquiv.ofLinear (toDirectLimit f M) (fromDirectLimit f M) ?_ ?_
<;> cases isEmpty_or_nonempty ι
· ext; apply Subsingleton.elim
· refine FunLike.ext _ _ fun x ↦ x.induction_on fun i g ↦ g.induction_on ?_ ?_ ?_ <;> aesop
·
|
Mathlib.Algebra.Module.DirectLimitAndTensorProduct.75_0.NNGkUEQVeDhgQVm
|
/--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M)
|
Mathlib_Algebra_Module_DirectLimitAndTensorProduct
|
case refine_2.inl.h
R : Type u_1
inst✝⁷ : CommRing R
ι : Type u_2
inst✝⁶ : DecidableEq ι
inst✝⁵ : Preorder ι
G : ι → Type u_3
inst✝⁴ : (i : ι) → AddCommGroup (G i)
inst✝³ : (i : ι) → Module R (G i)
f : (i j : ι) → i ≤ j → G i →ₗ[R] G j
M : Type u_4
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsDirected ι fun x x_1 => x ≤ x_1
h✝ : IsEmpty ι
x✝ : DirectLimit G f ⊗[R] M
⊢ (LinearMap.comp (fromDirectLimit f M) (toDirectLimit f M)) x✝ = LinearMap.id x✝
|
/-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.Algebra.DirectLimit
/-!
# Tensor product and direct limits commute with each other.
Given a family of `R`-modules `Gᵢ` with a family of compatible `R`-linear maps `fᵢⱼ : Gᵢ → Gⱼ` for
every `i ≤ j` and another `R`-module `M`, we have `(limᵢ Gᵢ) ⊗ M` and `lim (Gᵢ ⊗ M)` are isomorphic
as `R`-modules.
## Main definitions:
* `TensorProduct.directLimitLeft : DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M)`
* `TensorProduct.directLimitRight : M ⊗[R] DirectLimit G f ≃ₗ[R] DirectLimit (M ⊗[R] G ·) (M ◁ f)`
-/
open TensorProduct Module Module.DirectLimit
variable {R : Type*} [CommRing R]
variable {ι : Type*}
variable [DecidableEq ι] [Preorder ι]
variable {G : ι → Type*}
variable [∀ i, AddCommGroup (G i)] [∀ i, Module R (G i)]
variable (f : ∀ i j, i ≤ j → G i →ₗ[R] G j)
variable (M : Type*) [AddCommGroup M] [Module R M]
-- alluding to the notation in `CategoryTheory.Monoidal`
local notation M " ◁ " f => fun i j h ↦ LinearMap.lTensor M (f _ _ h)
local notation f " ▷ " N => fun i j h ↦ LinearMap.rTensor N (f _ _ h)
namespace TensorProduct
/--
the map `limᵢ (Gᵢ ⊗ M) → (limᵢ Gᵢ) ⊗ M` induced by the family of maps `Gᵢ ⊗ M → (limᵢ Gᵢ) ⊗ M`
given by `gᵢ ⊗ m ↦ [gᵢ] ⊗ m`.
-/
noncomputable def fromDirectLimit :
DirectLimit (G · ⊗[R] M) (f ▷ M) →ₗ[R] DirectLimit G f ⊗[R] M :=
DirectLimit.lift _ _ _ _ (fun _ ↦ (of _ _ _ _ _).rTensor M)
fun _ _ _ x ↦ by refine' x.induction_on _ _ _ <;> aesop
variable {M} in
@[simp] lemma fromDirectLimit_of_tmul {i : ι} (g : G i) (m : M) :
fromDirectLimit f M (of _ _ _ _ i (g ⊗ₜ m)) = (of _ _ _ f i g) ⊗ₜ m :=
lift_of (G := (G · ⊗[R] M)) _ _ (g ⊗ₜ m)
/--
the map `(limᵢ Gᵢ) ⊗ M → limᵢ (Gᵢ ⊗ M)` from the bilinear map `limᵢ Gᵢ → M → limᵢ (Gᵢ ⊗ M)` given
by the family of maps `Gᵢ → M → limᵢ (Gᵢ ⊗ M)` where `gᵢ ↦ m ↦ [gᵢ ⊗ m]`
-/
noncomputable def toDirectLimit : DirectLimit G f ⊗[R] M →ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) :=
TensorProduct.lift <| DirectLimit.lift _ _ _ _
(fun i ↦
(TensorProduct.mk R _ _).compr₂ (of R ι _ (fun _i _j h ↦ (f _ _ h).rTensor M) i))
fun _ _ _ g ↦ FunLike.ext _ _ (of_f (G := (G · ⊗[R] M)) (x := g ⊗ₜ ·))
variable {M} in
@[simp] lemma toDirectLimit_tmul_of
{i : ι} (g : G i) (m : M) :
(toDirectLimit f M <| (of _ _ G f i g) ⊗ₜ m) = (of _ _ _ _ i (g ⊗ₜ m)) := by
rw [toDirectLimit, lift.tmul, lift_of]
rfl
variable [IsDirected ι (· ≤ ·)]
/--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) := by
refine LinearEquiv.ofLinear (toDirectLimit f M) (fromDirectLimit f M) ?_ ?_
<;> cases isEmpty_or_nonempty ι
· ext; apply Subsingleton.elim
· refine FunLike.ext _ _ fun x ↦ x.induction_on fun i g ↦ g.induction_on ?_ ?_ ?_ <;> aesop
· ext;
|
apply Subsingleton.elim
|
/--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) := by
refine LinearEquiv.ofLinear (toDirectLimit f M) (fromDirectLimit f M) ?_ ?_
<;> cases isEmpty_or_nonempty ι
· ext; apply Subsingleton.elim
· refine FunLike.ext _ _ fun x ↦ x.induction_on fun i g ↦ g.induction_on ?_ ?_ ?_ <;> aesop
· ext;
|
Mathlib.Algebra.Module.DirectLimitAndTensorProduct.75_0.NNGkUEQVeDhgQVm
|
/--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M)
|
Mathlib_Algebra_Module_DirectLimitAndTensorProduct
|
case refine_2.inr
R : Type u_1
inst✝⁷ : CommRing R
ι : Type u_2
inst✝⁶ : DecidableEq ι
inst✝⁵ : Preorder ι
G : ι → Type u_3
inst✝⁴ : (i : ι) → AddCommGroup (G i)
inst✝³ : (i : ι) → Module R (G i)
f : (i j : ι) → i ≤ j → G i →ₗ[R] G j
M : Type u_4
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsDirected ι fun x x_1 => x ≤ x_1
h✝ : Nonempty ι
⊢ LinearMap.comp (fromDirectLimit f M) (toDirectLimit f M) = LinearMap.id
|
/-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.Algebra.DirectLimit
/-!
# Tensor product and direct limits commute with each other.
Given a family of `R`-modules `Gᵢ` with a family of compatible `R`-linear maps `fᵢⱼ : Gᵢ → Gⱼ` for
every `i ≤ j` and another `R`-module `M`, we have `(limᵢ Gᵢ) ⊗ M` and `lim (Gᵢ ⊗ M)` are isomorphic
as `R`-modules.
## Main definitions:
* `TensorProduct.directLimitLeft : DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M)`
* `TensorProduct.directLimitRight : M ⊗[R] DirectLimit G f ≃ₗ[R] DirectLimit (M ⊗[R] G ·) (M ◁ f)`
-/
open TensorProduct Module Module.DirectLimit
variable {R : Type*} [CommRing R]
variable {ι : Type*}
variable [DecidableEq ι] [Preorder ι]
variable {G : ι → Type*}
variable [∀ i, AddCommGroup (G i)] [∀ i, Module R (G i)]
variable (f : ∀ i j, i ≤ j → G i →ₗ[R] G j)
variable (M : Type*) [AddCommGroup M] [Module R M]
-- alluding to the notation in `CategoryTheory.Monoidal`
local notation M " ◁ " f => fun i j h ↦ LinearMap.lTensor M (f _ _ h)
local notation f " ▷ " N => fun i j h ↦ LinearMap.rTensor N (f _ _ h)
namespace TensorProduct
/--
the map `limᵢ (Gᵢ ⊗ M) → (limᵢ Gᵢ) ⊗ M` induced by the family of maps `Gᵢ ⊗ M → (limᵢ Gᵢ) ⊗ M`
given by `gᵢ ⊗ m ↦ [gᵢ] ⊗ m`.
-/
noncomputable def fromDirectLimit :
DirectLimit (G · ⊗[R] M) (f ▷ M) →ₗ[R] DirectLimit G f ⊗[R] M :=
DirectLimit.lift _ _ _ _ (fun _ ↦ (of _ _ _ _ _).rTensor M)
fun _ _ _ x ↦ by refine' x.induction_on _ _ _ <;> aesop
variable {M} in
@[simp] lemma fromDirectLimit_of_tmul {i : ι} (g : G i) (m : M) :
fromDirectLimit f M (of _ _ _ _ i (g ⊗ₜ m)) = (of _ _ _ f i g) ⊗ₜ m :=
lift_of (G := (G · ⊗[R] M)) _ _ (g ⊗ₜ m)
/--
the map `(limᵢ Gᵢ) ⊗ M → limᵢ (Gᵢ ⊗ M)` from the bilinear map `limᵢ Gᵢ → M → limᵢ (Gᵢ ⊗ M)` given
by the family of maps `Gᵢ → M → limᵢ (Gᵢ ⊗ M)` where `gᵢ ↦ m ↦ [gᵢ ⊗ m]`
-/
noncomputable def toDirectLimit : DirectLimit G f ⊗[R] M →ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) :=
TensorProduct.lift <| DirectLimit.lift _ _ _ _
(fun i ↦
(TensorProduct.mk R _ _).compr₂ (of R ι _ (fun _i _j h ↦ (f _ _ h).rTensor M) i))
fun _ _ _ g ↦ FunLike.ext _ _ (of_f (G := (G · ⊗[R] M)) (x := g ⊗ₜ ·))
variable {M} in
@[simp] lemma toDirectLimit_tmul_of
{i : ι} (g : G i) (m : M) :
(toDirectLimit f M <| (of _ _ G f i g) ⊗ₜ m) = (of _ _ _ _ i (g ⊗ₜ m)) := by
rw [toDirectLimit, lift.tmul, lift_of]
rfl
variable [IsDirected ι (· ≤ ·)]
/--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) := by
refine LinearEquiv.ofLinear (toDirectLimit f M) (fromDirectLimit f M) ?_ ?_
<;> cases isEmpty_or_nonempty ι
· ext; apply Subsingleton.elim
· refine FunLike.ext _ _ fun x ↦ x.induction_on fun i g ↦ g.induction_on ?_ ?_ ?_ <;> aesop
· ext; apply Subsingleton.elim
·
|
exact ext (FunLike.ext _ _ fun g ↦ FunLike.ext _ _ fun _ ↦ g.induction_on <| by aesop)
|
/--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) := by
refine LinearEquiv.ofLinear (toDirectLimit f M) (fromDirectLimit f M) ?_ ?_
<;> cases isEmpty_or_nonempty ι
· ext; apply Subsingleton.elim
· refine FunLike.ext _ _ fun x ↦ x.induction_on fun i g ↦ g.induction_on ?_ ?_ ?_ <;> aesop
· ext; apply Subsingleton.elim
·
|
Mathlib.Algebra.Module.DirectLimitAndTensorProduct.75_0.NNGkUEQVeDhgQVm
|
/--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M)
|
Mathlib_Algebra_Module_DirectLimitAndTensorProduct
|
R : Type u_1
inst✝⁷ : CommRing R
ι : Type u_2
inst✝⁶ : DecidableEq ι
inst✝⁵ : Preorder ι
G : ι → Type u_3
inst✝⁴ : (i : ι) → AddCommGroup (G i)
inst✝³ : (i : ι) → Module R (G i)
f : (i j : ι) → i ≤ j → G i →ₗ[R] G j
M : Type u_4
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsDirected ι fun x x_1 => x ≤ x_1
h✝ : Nonempty ι
g : DirectLimit G f
x✝ : M
⊢ ∀ (i : ι) (x : G i),
((LinearMap.compr₂ (mk R (DirectLimit G f) M) (LinearMap.comp (fromDirectLimit f M) (toDirectLimit f M)))
((of R ι G f i) x))
x✝ =
((LinearMap.compr₂ (mk R (DirectLimit G f) M) LinearMap.id) ((of R ι G f i) x)) x✝
|
/-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.Algebra.DirectLimit
/-!
# Tensor product and direct limits commute with each other.
Given a family of `R`-modules `Gᵢ` with a family of compatible `R`-linear maps `fᵢⱼ : Gᵢ → Gⱼ` for
every `i ≤ j` and another `R`-module `M`, we have `(limᵢ Gᵢ) ⊗ M` and `lim (Gᵢ ⊗ M)` are isomorphic
as `R`-modules.
## Main definitions:
* `TensorProduct.directLimitLeft : DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M)`
* `TensorProduct.directLimitRight : M ⊗[R] DirectLimit G f ≃ₗ[R] DirectLimit (M ⊗[R] G ·) (M ◁ f)`
-/
open TensorProduct Module Module.DirectLimit
variable {R : Type*} [CommRing R]
variable {ι : Type*}
variable [DecidableEq ι] [Preorder ι]
variable {G : ι → Type*}
variable [∀ i, AddCommGroup (G i)] [∀ i, Module R (G i)]
variable (f : ∀ i j, i ≤ j → G i →ₗ[R] G j)
variable (M : Type*) [AddCommGroup M] [Module R M]
-- alluding to the notation in `CategoryTheory.Monoidal`
local notation M " ◁ " f => fun i j h ↦ LinearMap.lTensor M (f _ _ h)
local notation f " ▷ " N => fun i j h ↦ LinearMap.rTensor N (f _ _ h)
namespace TensorProduct
/--
the map `limᵢ (Gᵢ ⊗ M) → (limᵢ Gᵢ) ⊗ M` induced by the family of maps `Gᵢ ⊗ M → (limᵢ Gᵢ) ⊗ M`
given by `gᵢ ⊗ m ↦ [gᵢ] ⊗ m`.
-/
noncomputable def fromDirectLimit :
DirectLimit (G · ⊗[R] M) (f ▷ M) →ₗ[R] DirectLimit G f ⊗[R] M :=
DirectLimit.lift _ _ _ _ (fun _ ↦ (of _ _ _ _ _).rTensor M)
fun _ _ _ x ↦ by refine' x.induction_on _ _ _ <;> aesop
variable {M} in
@[simp] lemma fromDirectLimit_of_tmul {i : ι} (g : G i) (m : M) :
fromDirectLimit f M (of _ _ _ _ i (g ⊗ₜ m)) = (of _ _ _ f i g) ⊗ₜ m :=
lift_of (G := (G · ⊗[R] M)) _ _ (g ⊗ₜ m)
/--
the map `(limᵢ Gᵢ) ⊗ M → limᵢ (Gᵢ ⊗ M)` from the bilinear map `limᵢ Gᵢ → M → limᵢ (Gᵢ ⊗ M)` given
by the family of maps `Gᵢ → M → limᵢ (Gᵢ ⊗ M)` where `gᵢ ↦ m ↦ [gᵢ ⊗ m]`
-/
noncomputable def toDirectLimit : DirectLimit G f ⊗[R] M →ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) :=
TensorProduct.lift <| DirectLimit.lift _ _ _ _
(fun i ↦
(TensorProduct.mk R _ _).compr₂ (of R ι _ (fun _i _j h ↦ (f _ _ h).rTensor M) i))
fun _ _ _ g ↦ FunLike.ext _ _ (of_f (G := (G · ⊗[R] M)) (x := g ⊗ₜ ·))
variable {M} in
@[simp] lemma toDirectLimit_tmul_of
{i : ι} (g : G i) (m : M) :
(toDirectLimit f M <| (of _ _ G f i g) ⊗ₜ m) = (of _ _ _ _ i (g ⊗ₜ m)) := by
rw [toDirectLimit, lift.tmul, lift_of]
rfl
variable [IsDirected ι (· ≤ ·)]
/--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) := by
refine LinearEquiv.ofLinear (toDirectLimit f M) (fromDirectLimit f M) ?_ ?_
<;> cases isEmpty_or_nonempty ι
· ext; apply Subsingleton.elim
· refine FunLike.ext _ _ fun x ↦ x.induction_on fun i g ↦ g.induction_on ?_ ?_ ?_ <;> aesop
· ext; apply Subsingleton.elim
· exact ext (FunLike.ext _ _ fun g ↦ FunLike.ext _ _ fun _ ↦ g.induction_on <| by
|
aesop
|
/--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) := by
refine LinearEquiv.ofLinear (toDirectLimit f M) (fromDirectLimit f M) ?_ ?_
<;> cases isEmpty_or_nonempty ι
· ext; apply Subsingleton.elim
· refine FunLike.ext _ _ fun x ↦ x.induction_on fun i g ↦ g.induction_on ?_ ?_ ?_ <;> aesop
· ext; apply Subsingleton.elim
· exact ext (FunLike.ext _ _ fun g ↦ FunLike.ext _ _ fun _ ↦ g.induction_on <| by
|
Mathlib.Algebra.Module.DirectLimitAndTensorProduct.75_0.NNGkUEQVeDhgQVm
|
/--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M)
|
Mathlib_Algebra_Module_DirectLimitAndTensorProduct
|
R : Type u_1
inst✝⁷ : CommRing R
ι : Type u_2
inst✝⁶ : DecidableEq ι
inst✝⁵ : Preorder ι
G : ι → Type u_3
inst✝⁴ : (i : ι) → AddCommGroup (G i)
inst✝³ : (i : ι) → Module R (G i)
f : (i j : ι) → i ≤ j → G i →ₗ[R] G j
M : Type u_4
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsDirected ι fun x x_1 => x ≤ x_1
i j : ι
h : i ≤ j
⊢ ∀ (x : G i),
(LinearMap.compr₂ (mk R (G i) M) (↑((fun i => TensorProduct.comm R (G i) M) j) ∘ₗ LinearMap.rTensor M (f i j h)))
x =
(LinearMap.compr₂ (mk R (G i) M) (LinearMap.lTensor M (f i j h) ∘ₗ ↑((fun i => TensorProduct.comm R (G i) M) i)))
x
|
/-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.Algebra.DirectLimit
/-!
# Tensor product and direct limits commute with each other.
Given a family of `R`-modules `Gᵢ` with a family of compatible `R`-linear maps `fᵢⱼ : Gᵢ → Gⱼ` for
every `i ≤ j` and another `R`-module `M`, we have `(limᵢ Gᵢ) ⊗ M` and `lim (Gᵢ ⊗ M)` are isomorphic
as `R`-modules.
## Main definitions:
* `TensorProduct.directLimitLeft : DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M)`
* `TensorProduct.directLimitRight : M ⊗[R] DirectLimit G f ≃ₗ[R] DirectLimit (M ⊗[R] G ·) (M ◁ f)`
-/
open TensorProduct Module Module.DirectLimit
variable {R : Type*} [CommRing R]
variable {ι : Type*}
variable [DecidableEq ι] [Preorder ι]
variable {G : ι → Type*}
variable [∀ i, AddCommGroup (G i)] [∀ i, Module R (G i)]
variable (f : ∀ i j, i ≤ j → G i →ₗ[R] G j)
variable (M : Type*) [AddCommGroup M] [Module R M]
-- alluding to the notation in `CategoryTheory.Monoidal`
local notation M " ◁ " f => fun i j h ↦ LinearMap.lTensor M (f _ _ h)
local notation f " ▷ " N => fun i j h ↦ LinearMap.rTensor N (f _ _ h)
namespace TensorProduct
/--
the map `limᵢ (Gᵢ ⊗ M) → (limᵢ Gᵢ) ⊗ M` induced by the family of maps `Gᵢ ⊗ M → (limᵢ Gᵢ) ⊗ M`
given by `gᵢ ⊗ m ↦ [gᵢ] ⊗ m`.
-/
noncomputable def fromDirectLimit :
DirectLimit (G · ⊗[R] M) (f ▷ M) →ₗ[R] DirectLimit G f ⊗[R] M :=
DirectLimit.lift _ _ _ _ (fun _ ↦ (of _ _ _ _ _).rTensor M)
fun _ _ _ x ↦ by refine' x.induction_on _ _ _ <;> aesop
variable {M} in
@[simp] lemma fromDirectLimit_of_tmul {i : ι} (g : G i) (m : M) :
fromDirectLimit f M (of _ _ _ _ i (g ⊗ₜ m)) = (of _ _ _ f i g) ⊗ₜ m :=
lift_of (G := (G · ⊗[R] M)) _ _ (g ⊗ₜ m)
/--
the map `(limᵢ Gᵢ) ⊗ M → limᵢ (Gᵢ ⊗ M)` from the bilinear map `limᵢ Gᵢ → M → limᵢ (Gᵢ ⊗ M)` given
by the family of maps `Gᵢ → M → limᵢ (Gᵢ ⊗ M)` where `gᵢ ↦ m ↦ [gᵢ ⊗ m]`
-/
noncomputable def toDirectLimit : DirectLimit G f ⊗[R] M →ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) :=
TensorProduct.lift <| DirectLimit.lift _ _ _ _
(fun i ↦
(TensorProduct.mk R _ _).compr₂ (of R ι _ (fun _i _j h ↦ (f _ _ h).rTensor M) i))
fun _ _ _ g ↦ FunLike.ext _ _ (of_f (G := (G · ⊗[R] M)) (x := g ⊗ₜ ·))
variable {M} in
@[simp] lemma toDirectLimit_tmul_of
{i : ι} (g : G i) (m : M) :
(toDirectLimit f M <| (of _ _ G f i g) ⊗ₜ m) = (of _ _ _ _ i (g ⊗ₜ m)) := by
rw [toDirectLimit, lift.tmul, lift_of]
rfl
variable [IsDirected ι (· ≤ ·)]
/--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) := by
refine LinearEquiv.ofLinear (toDirectLimit f M) (fromDirectLimit f M) ?_ ?_
<;> cases isEmpty_or_nonempty ι
· ext; apply Subsingleton.elim
· refine FunLike.ext _ _ fun x ↦ x.induction_on fun i g ↦ g.induction_on ?_ ?_ ?_ <;> aesop
· ext; apply Subsingleton.elim
· exact ext (FunLike.ext _ _ fun g ↦ FunLike.ext _ _ fun _ ↦ g.induction_on <| by aesop)
@[simp] lemma directLimitLeft_tmul_of {i : ι} (g : G i) (m : M) :
directLimitLeft f M (of _ _ _ _ _ g ⊗ₜ m) = of _ _ _ (f ▷ M) _ (g ⊗ₜ m) :=
toDirectLimit_tmul_of f g m
@[simp] lemma directLimitLeft_symm_of_tmul {i : ι} (g : G i) (m : M) :
(directLimitLeft f M).symm (of _ _ _ _ _ (g ⊗ₜ m)) = of _ _ _ f _ g ⊗ₜ m :=
fromDirectLimit_of_tmul f g m
/--
`M ⊗ (limᵢ Gᵢ)` and `limᵢ (M ⊗ Gᵢ)` are isomorphic as modules
-/
noncomputable def directLimitRight :
M ⊗[R] DirectLimit G f ≃ₗ[R] DirectLimit (M ⊗[R] G ·) (M ◁ f) :=
TensorProduct.comm _ _ _ ≪≫ₗ directLimitLeft f M ≪≫ₗ
Module.DirectLimit.congr (fun i ↦ TensorProduct.comm _ _ _)
(fun i j h ↦ TensorProduct.ext <| FunLike.ext _ _ <| by
|
aesop
|
/--
`M ⊗ (limᵢ Gᵢ)` and `limᵢ (M ⊗ Gᵢ)` are isomorphic as modules
-/
noncomputable def directLimitRight :
M ⊗[R] DirectLimit G f ≃ₗ[R] DirectLimit (M ⊗[R] G ·) (M ◁ f) :=
TensorProduct.comm _ _ _ ≪≫ₗ directLimitLeft f M ≪≫ₗ
Module.DirectLimit.congr (fun i ↦ TensorProduct.comm _ _ _)
(fun i j h ↦ TensorProduct.ext <| FunLike.ext _ _ <| by
|
Mathlib.Algebra.Module.DirectLimitAndTensorProduct.95_0.NNGkUEQVeDhgQVm
|
/--
`M ⊗ (limᵢ Gᵢ)` and `limᵢ (M ⊗ Gᵢ)` are isomorphic as modules
-/
noncomputable def directLimitRight :
M ⊗[R] DirectLimit G f ≃ₗ[R] DirectLimit (M ⊗[R] G ·) (M ◁ f)
|
Mathlib_Algebra_Module_DirectLimitAndTensorProduct
|
R : Type u_1
inst✝⁷ : CommRing R
ι : Type u_2
inst✝⁶ : DecidableEq ι
inst✝⁵ : Preorder ι
G : ι → Type u_3
inst✝⁴ : (i : ι) → AddCommGroup (G i)
inst✝³ : (i : ι) → Module R (G i)
f : (i j : ι) → i ≤ j → G i →ₗ[R] G j
M : Type u_4
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsDirected ι fun x x_1 => x ≤ x_1
i : ι
m : M
g : G i
⊢ (directLimitRight f M) (m ⊗ₜ[R] (of R ι (fun i => G i) f i) g) =
(of R ι (fun x => M ⊗[R] G x) (fun i j h => LinearMap.lTensor M (f i j h)) i) (m ⊗ₜ[R] g)
|
/-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.Algebra.DirectLimit
/-!
# Tensor product and direct limits commute with each other.
Given a family of `R`-modules `Gᵢ` with a family of compatible `R`-linear maps `fᵢⱼ : Gᵢ → Gⱼ` for
every `i ≤ j` and another `R`-module `M`, we have `(limᵢ Gᵢ) ⊗ M` and `lim (Gᵢ ⊗ M)` are isomorphic
as `R`-modules.
## Main definitions:
* `TensorProduct.directLimitLeft : DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M)`
* `TensorProduct.directLimitRight : M ⊗[R] DirectLimit G f ≃ₗ[R] DirectLimit (M ⊗[R] G ·) (M ◁ f)`
-/
open TensorProduct Module Module.DirectLimit
variable {R : Type*} [CommRing R]
variable {ι : Type*}
variable [DecidableEq ι] [Preorder ι]
variable {G : ι → Type*}
variable [∀ i, AddCommGroup (G i)] [∀ i, Module R (G i)]
variable (f : ∀ i j, i ≤ j → G i →ₗ[R] G j)
variable (M : Type*) [AddCommGroup M] [Module R M]
-- alluding to the notation in `CategoryTheory.Monoidal`
local notation M " ◁ " f => fun i j h ↦ LinearMap.lTensor M (f _ _ h)
local notation f " ▷ " N => fun i j h ↦ LinearMap.rTensor N (f _ _ h)
namespace TensorProduct
/--
the map `limᵢ (Gᵢ ⊗ M) → (limᵢ Gᵢ) ⊗ M` induced by the family of maps `Gᵢ ⊗ M → (limᵢ Gᵢ) ⊗ M`
given by `gᵢ ⊗ m ↦ [gᵢ] ⊗ m`.
-/
noncomputable def fromDirectLimit :
DirectLimit (G · ⊗[R] M) (f ▷ M) →ₗ[R] DirectLimit G f ⊗[R] M :=
DirectLimit.lift _ _ _ _ (fun _ ↦ (of _ _ _ _ _).rTensor M)
fun _ _ _ x ↦ by refine' x.induction_on _ _ _ <;> aesop
variable {M} in
@[simp] lemma fromDirectLimit_of_tmul {i : ι} (g : G i) (m : M) :
fromDirectLimit f M (of _ _ _ _ i (g ⊗ₜ m)) = (of _ _ _ f i g) ⊗ₜ m :=
lift_of (G := (G · ⊗[R] M)) _ _ (g ⊗ₜ m)
/--
the map `(limᵢ Gᵢ) ⊗ M → limᵢ (Gᵢ ⊗ M)` from the bilinear map `limᵢ Gᵢ → M → limᵢ (Gᵢ ⊗ M)` given
by the family of maps `Gᵢ → M → limᵢ (Gᵢ ⊗ M)` where `gᵢ ↦ m ↦ [gᵢ ⊗ m]`
-/
noncomputable def toDirectLimit : DirectLimit G f ⊗[R] M →ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) :=
TensorProduct.lift <| DirectLimit.lift _ _ _ _
(fun i ↦
(TensorProduct.mk R _ _).compr₂ (of R ι _ (fun _i _j h ↦ (f _ _ h).rTensor M) i))
fun _ _ _ g ↦ FunLike.ext _ _ (of_f (G := (G · ⊗[R] M)) (x := g ⊗ₜ ·))
variable {M} in
@[simp] lemma toDirectLimit_tmul_of
{i : ι} (g : G i) (m : M) :
(toDirectLimit f M <| (of _ _ G f i g) ⊗ₜ m) = (of _ _ _ _ i (g ⊗ₜ m)) := by
rw [toDirectLimit, lift.tmul, lift_of]
rfl
variable [IsDirected ι (· ≤ ·)]
/--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) := by
refine LinearEquiv.ofLinear (toDirectLimit f M) (fromDirectLimit f M) ?_ ?_
<;> cases isEmpty_or_nonempty ι
· ext; apply Subsingleton.elim
· refine FunLike.ext _ _ fun x ↦ x.induction_on fun i g ↦ g.induction_on ?_ ?_ ?_ <;> aesop
· ext; apply Subsingleton.elim
· exact ext (FunLike.ext _ _ fun g ↦ FunLike.ext _ _ fun _ ↦ g.induction_on <| by aesop)
@[simp] lemma directLimitLeft_tmul_of {i : ι} (g : G i) (m : M) :
directLimitLeft f M (of _ _ _ _ _ g ⊗ₜ m) = of _ _ _ (f ▷ M) _ (g ⊗ₜ m) :=
toDirectLimit_tmul_of f g m
@[simp] lemma directLimitLeft_symm_of_tmul {i : ι} (g : G i) (m : M) :
(directLimitLeft f M).symm (of _ _ _ _ _ (g ⊗ₜ m)) = of _ _ _ f _ g ⊗ₜ m :=
fromDirectLimit_of_tmul f g m
/--
`M ⊗ (limᵢ Gᵢ)` and `limᵢ (M ⊗ Gᵢ)` are isomorphic as modules
-/
noncomputable def directLimitRight :
M ⊗[R] DirectLimit G f ≃ₗ[R] DirectLimit (M ⊗[R] G ·) (M ◁ f) :=
TensorProduct.comm _ _ _ ≪≫ₗ directLimitLeft f M ≪≫ₗ
Module.DirectLimit.congr (fun i ↦ TensorProduct.comm _ _ _)
(fun i j h ↦ TensorProduct.ext <| FunLike.ext _ _ <| by aesop)
@[simp] lemma directLimitRight_tmul_of {i : ι} (m : M) (g : G i):
directLimitRight f M (m ⊗ₜ of _ _ _ _ _ g) = of _ _ _ _ i (m ⊗ₜ g) := by
|
simp [directLimitRight, congr_apply_of]
|
@[simp] lemma directLimitRight_tmul_of {i : ι} (m : M) (g : G i):
directLimitRight f M (m ⊗ₜ of _ _ _ _ _ g) = of _ _ _ _ i (m ⊗ₜ g) := by
|
Mathlib.Algebra.Module.DirectLimitAndTensorProduct.104_0.NNGkUEQVeDhgQVm
|
@[simp] lemma directLimitRight_tmul_of {i : ι} (m : M) (g : G i):
directLimitRight f M (m ⊗ₜ of _ _ _ _ _ g) = of _ _ _ _ i (m ⊗ₜ g)
|
Mathlib_Algebra_Module_DirectLimitAndTensorProduct
|
R : Type u_1
inst✝⁷ : CommRing R
ι : Type u_2
inst✝⁶ : DecidableEq ι
inst✝⁵ : Preorder ι
G : ι → Type u_3
inst✝⁴ : (i : ι) → AddCommGroup (G i)
inst✝³ : (i : ι) → Module R (G i)
f : (i j : ι) → i ≤ j → G i →ₗ[R] G j
M : Type u_4
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsDirected ι fun x x_1 => x ≤ x_1
i : ι
m : M
g : G i
⊢ (LinearEquiv.symm (directLimitRight f M))
((of R ι (fun x => M ⊗[R] G x) (fun i j h => LinearMap.lTensor M (f i j h)) i) (m ⊗ₜ[R] g)) =
m ⊗ₜ[R] (of R ι (fun i => G i) f i) g
|
/-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.Algebra.DirectLimit
/-!
# Tensor product and direct limits commute with each other.
Given a family of `R`-modules `Gᵢ` with a family of compatible `R`-linear maps `fᵢⱼ : Gᵢ → Gⱼ` for
every `i ≤ j` and another `R`-module `M`, we have `(limᵢ Gᵢ) ⊗ M` and `lim (Gᵢ ⊗ M)` are isomorphic
as `R`-modules.
## Main definitions:
* `TensorProduct.directLimitLeft : DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M)`
* `TensorProduct.directLimitRight : M ⊗[R] DirectLimit G f ≃ₗ[R] DirectLimit (M ⊗[R] G ·) (M ◁ f)`
-/
open TensorProduct Module Module.DirectLimit
variable {R : Type*} [CommRing R]
variable {ι : Type*}
variable [DecidableEq ι] [Preorder ι]
variable {G : ι → Type*}
variable [∀ i, AddCommGroup (G i)] [∀ i, Module R (G i)]
variable (f : ∀ i j, i ≤ j → G i →ₗ[R] G j)
variable (M : Type*) [AddCommGroup M] [Module R M]
-- alluding to the notation in `CategoryTheory.Monoidal`
local notation M " ◁ " f => fun i j h ↦ LinearMap.lTensor M (f _ _ h)
local notation f " ▷ " N => fun i j h ↦ LinearMap.rTensor N (f _ _ h)
namespace TensorProduct
/--
the map `limᵢ (Gᵢ ⊗ M) → (limᵢ Gᵢ) ⊗ M` induced by the family of maps `Gᵢ ⊗ M → (limᵢ Gᵢ) ⊗ M`
given by `gᵢ ⊗ m ↦ [gᵢ] ⊗ m`.
-/
noncomputable def fromDirectLimit :
DirectLimit (G · ⊗[R] M) (f ▷ M) →ₗ[R] DirectLimit G f ⊗[R] M :=
DirectLimit.lift _ _ _ _ (fun _ ↦ (of _ _ _ _ _).rTensor M)
fun _ _ _ x ↦ by refine' x.induction_on _ _ _ <;> aesop
variable {M} in
@[simp] lemma fromDirectLimit_of_tmul {i : ι} (g : G i) (m : M) :
fromDirectLimit f M (of _ _ _ _ i (g ⊗ₜ m)) = (of _ _ _ f i g) ⊗ₜ m :=
lift_of (G := (G · ⊗[R] M)) _ _ (g ⊗ₜ m)
/--
the map `(limᵢ Gᵢ) ⊗ M → limᵢ (Gᵢ ⊗ M)` from the bilinear map `limᵢ Gᵢ → M → limᵢ (Gᵢ ⊗ M)` given
by the family of maps `Gᵢ → M → limᵢ (Gᵢ ⊗ M)` where `gᵢ ↦ m ↦ [gᵢ ⊗ m]`
-/
noncomputable def toDirectLimit : DirectLimit G f ⊗[R] M →ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) :=
TensorProduct.lift <| DirectLimit.lift _ _ _ _
(fun i ↦
(TensorProduct.mk R _ _).compr₂ (of R ι _ (fun _i _j h ↦ (f _ _ h).rTensor M) i))
fun _ _ _ g ↦ FunLike.ext _ _ (of_f (G := (G · ⊗[R] M)) (x := g ⊗ₜ ·))
variable {M} in
@[simp] lemma toDirectLimit_tmul_of
{i : ι} (g : G i) (m : M) :
(toDirectLimit f M <| (of _ _ G f i g) ⊗ₜ m) = (of _ _ _ _ i (g ⊗ₜ m)) := by
rw [toDirectLimit, lift.tmul, lift_of]
rfl
variable [IsDirected ι (· ≤ ·)]
/--
`limᵢ (Gᵢ ⊗ M)` and `(limᵢ Gᵢ) ⊗ M` are isomorphic as modules
-/
noncomputable def directLimitLeft :
DirectLimit G f ⊗[R] M ≃ₗ[R] DirectLimit (G · ⊗[R] M) (f ▷ M) := by
refine LinearEquiv.ofLinear (toDirectLimit f M) (fromDirectLimit f M) ?_ ?_
<;> cases isEmpty_or_nonempty ι
· ext; apply Subsingleton.elim
· refine FunLike.ext _ _ fun x ↦ x.induction_on fun i g ↦ g.induction_on ?_ ?_ ?_ <;> aesop
· ext; apply Subsingleton.elim
· exact ext (FunLike.ext _ _ fun g ↦ FunLike.ext _ _ fun _ ↦ g.induction_on <| by aesop)
@[simp] lemma directLimitLeft_tmul_of {i : ι} (g : G i) (m : M) :
directLimitLeft f M (of _ _ _ _ _ g ⊗ₜ m) = of _ _ _ (f ▷ M) _ (g ⊗ₜ m) :=
toDirectLimit_tmul_of f g m
@[simp] lemma directLimitLeft_symm_of_tmul {i : ι} (g : G i) (m : M) :
(directLimitLeft f M).symm (of _ _ _ _ _ (g ⊗ₜ m)) = of _ _ _ f _ g ⊗ₜ m :=
fromDirectLimit_of_tmul f g m
/--
`M ⊗ (limᵢ Gᵢ)` and `limᵢ (M ⊗ Gᵢ)` are isomorphic as modules
-/
noncomputable def directLimitRight :
M ⊗[R] DirectLimit G f ≃ₗ[R] DirectLimit (M ⊗[R] G ·) (M ◁ f) :=
TensorProduct.comm _ _ _ ≪≫ₗ directLimitLeft f M ≪≫ₗ
Module.DirectLimit.congr (fun i ↦ TensorProduct.comm _ _ _)
(fun i j h ↦ TensorProduct.ext <| FunLike.ext _ _ <| by aesop)
@[simp] lemma directLimitRight_tmul_of {i : ι} (m : M) (g : G i):
directLimitRight f M (m ⊗ₜ of _ _ _ _ _ g) = of _ _ _ _ i (m ⊗ₜ g) := by
simp [directLimitRight, congr_apply_of]
@[simp] lemma directLimitRight_symm_of_tmul {i : ι} (m : M) (g : G i) :
(directLimitRight f M).symm (of _ _ _ _ _ (m ⊗ₜ g)) = m ⊗ₜ of _ _ _ f _ g := by
|
simp [directLimitRight, congr_symm_apply_of]
|
@[simp] lemma directLimitRight_symm_of_tmul {i : ι} (m : M) (g : G i) :
(directLimitRight f M).symm (of _ _ _ _ _ (m ⊗ₜ g)) = m ⊗ₜ of _ _ _ f _ g := by
|
Mathlib.Algebra.Module.DirectLimitAndTensorProduct.108_0.NNGkUEQVeDhgQVm
|
@[simp] lemma directLimitRight_symm_of_tmul {i : ι} (m : M) (g : G i) :
(directLimitRight f M).symm (of _ _ _ _ _ (m ⊗ₜ g)) = m ⊗ₜ of _ _ _ f _ g
|
Mathlib_Algebra_Module_DirectLimitAndTensorProduct
|
A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra ℂ A
inst✝³ : StarRing A
inst✝² : ContinuousStar A
inst✝¹ : CompleteSpace A
inst✝ : StarModule ℂ A
a b : ↥(selfAdjoint A)
h : Commute ↑a ↑b
⊢ expUnitary (a + b) = expUnitary a * expUnitary b
|
/-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Analysis.NormedSpace.Exponential
#align_import analysis.normed_space.star.exponential from "leanprover-community/mathlib"@"1e3201306d4d9eb1fd54c60d7c4510ad5126f6f9"
/-! # The exponential map from selfadjoint to unitary
In this file, we establish various properties related to the map `λ a, exp ℂ A (I • a)` between the
subtypes `selfAdjoint A` and `unitary A`.
## TODO
* Show that any exponential unitary is path-connected in `unitary A` to `1 : unitary A`.
* Prove any unitary whose distance to `1 : unitary A` is less than `1` can be expressed as an
exponential unitary.
* A unitary is in the path component of `1` if and only if it is a finite product of exponential
unitaries.
-/
open NormedSpace -- For `exp`.
section Star
variable {A : Type*} [NormedRing A] [NormedAlgebra ℂ A] [StarRing A] [ContinuousStar A]
[CompleteSpace A] [StarModule ℂ A]
open Complex
/-- The map from the selfadjoint real subspace to the unitary group. This map only makes sense
over ℂ. -/
@[simps]
noncomputable def selfAdjoint.expUnitary (a : selfAdjoint A) : unitary A :=
⟨exp ℂ ((I • a.val) : A),
exp_mem_unitary_of_mem_skewAdjoint _ (a.prop.smul_mem_skewAdjoint conj_I)⟩
#align self_adjoint.exp_unitary selfAdjoint.expUnitary
open selfAdjoint
theorem Commute.expUnitary_add {a b : selfAdjoint A} (h : Commute (a : A) (b : A)) :
expUnitary (a + b) = expUnitary a * expUnitary b := by
|
ext
|
theorem Commute.expUnitary_add {a b : selfAdjoint A} (h : Commute (a : A) (b : A)) :
expUnitary (a + b) = expUnitary a * expUnitary b := by
|
Mathlib.Analysis.NormedSpace.Star.Exponential.42_0.QAfFqeOSAt19Prd
|
theorem Commute.expUnitary_add {a b : selfAdjoint A} (h : Commute (a : A) (b : A)) :
expUnitary (a + b) = expUnitary a * expUnitary b
|
Mathlib_Analysis_NormedSpace_Star_Exponential
|
case a
A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra ℂ A
inst✝³ : StarRing A
inst✝² : ContinuousStar A
inst✝¹ : CompleteSpace A
inst✝ : StarModule ℂ A
a b : ↥(selfAdjoint A)
h : Commute ↑a ↑b
⊢ ↑(expUnitary (a + b)) = ↑(expUnitary a * expUnitary b)
|
/-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Analysis.NormedSpace.Exponential
#align_import analysis.normed_space.star.exponential from "leanprover-community/mathlib"@"1e3201306d4d9eb1fd54c60d7c4510ad5126f6f9"
/-! # The exponential map from selfadjoint to unitary
In this file, we establish various properties related to the map `λ a, exp ℂ A (I • a)` between the
subtypes `selfAdjoint A` and `unitary A`.
## TODO
* Show that any exponential unitary is path-connected in `unitary A` to `1 : unitary A`.
* Prove any unitary whose distance to `1 : unitary A` is less than `1` can be expressed as an
exponential unitary.
* A unitary is in the path component of `1` if and only if it is a finite product of exponential
unitaries.
-/
open NormedSpace -- For `exp`.
section Star
variable {A : Type*} [NormedRing A] [NormedAlgebra ℂ A] [StarRing A] [ContinuousStar A]
[CompleteSpace A] [StarModule ℂ A]
open Complex
/-- The map from the selfadjoint real subspace to the unitary group. This map only makes sense
over ℂ. -/
@[simps]
noncomputable def selfAdjoint.expUnitary (a : selfAdjoint A) : unitary A :=
⟨exp ℂ ((I • a.val) : A),
exp_mem_unitary_of_mem_skewAdjoint _ (a.prop.smul_mem_skewAdjoint conj_I)⟩
#align self_adjoint.exp_unitary selfAdjoint.expUnitary
open selfAdjoint
theorem Commute.expUnitary_add {a b : selfAdjoint A} (h : Commute (a : A) (b : A)) :
expUnitary (a + b) = expUnitary a * expUnitary b := by
ext
|
have hcomm : Commute (I • (a : A)) (I • (b : A)) := by
unfold Commute SemiconjBy
simp only [h.eq, Algebra.smul_mul_assoc, Algebra.mul_smul_comm]
|
theorem Commute.expUnitary_add {a b : selfAdjoint A} (h : Commute (a : A) (b : A)) :
expUnitary (a + b) = expUnitary a * expUnitary b := by
ext
|
Mathlib.Analysis.NormedSpace.Star.Exponential.42_0.QAfFqeOSAt19Prd
|
theorem Commute.expUnitary_add {a b : selfAdjoint A} (h : Commute (a : A) (b : A)) :
expUnitary (a + b) = expUnitary a * expUnitary b
|
Mathlib_Analysis_NormedSpace_Star_Exponential
|
A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra ℂ A
inst✝³ : StarRing A
inst✝² : ContinuousStar A
inst✝¹ : CompleteSpace A
inst✝ : StarModule ℂ A
a b : ↥(selfAdjoint A)
h : Commute ↑a ↑b
⊢ Commute (I • ↑a) (I • ↑b)
|
/-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Analysis.NormedSpace.Exponential
#align_import analysis.normed_space.star.exponential from "leanprover-community/mathlib"@"1e3201306d4d9eb1fd54c60d7c4510ad5126f6f9"
/-! # The exponential map from selfadjoint to unitary
In this file, we establish various properties related to the map `λ a, exp ℂ A (I • a)` between the
subtypes `selfAdjoint A` and `unitary A`.
## TODO
* Show that any exponential unitary is path-connected in `unitary A` to `1 : unitary A`.
* Prove any unitary whose distance to `1 : unitary A` is less than `1` can be expressed as an
exponential unitary.
* A unitary is in the path component of `1` if and only if it is a finite product of exponential
unitaries.
-/
open NormedSpace -- For `exp`.
section Star
variable {A : Type*} [NormedRing A] [NormedAlgebra ℂ A] [StarRing A] [ContinuousStar A]
[CompleteSpace A] [StarModule ℂ A]
open Complex
/-- The map from the selfadjoint real subspace to the unitary group. This map only makes sense
over ℂ. -/
@[simps]
noncomputable def selfAdjoint.expUnitary (a : selfAdjoint A) : unitary A :=
⟨exp ℂ ((I • a.val) : A),
exp_mem_unitary_of_mem_skewAdjoint _ (a.prop.smul_mem_skewAdjoint conj_I)⟩
#align self_adjoint.exp_unitary selfAdjoint.expUnitary
open selfAdjoint
theorem Commute.expUnitary_add {a b : selfAdjoint A} (h : Commute (a : A) (b : A)) :
expUnitary (a + b) = expUnitary a * expUnitary b := by
ext
have hcomm : Commute (I • (a : A)) (I • (b : A)) := by
|
unfold Commute SemiconjBy
|
theorem Commute.expUnitary_add {a b : selfAdjoint A} (h : Commute (a : A) (b : A)) :
expUnitary (a + b) = expUnitary a * expUnitary b := by
ext
have hcomm : Commute (I • (a : A)) (I • (b : A)) := by
|
Mathlib.Analysis.NormedSpace.Star.Exponential.42_0.QAfFqeOSAt19Prd
|
theorem Commute.expUnitary_add {a b : selfAdjoint A} (h : Commute (a : A) (b : A)) :
expUnitary (a + b) = expUnitary a * expUnitary b
|
Mathlib_Analysis_NormedSpace_Star_Exponential
|
A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra ℂ A
inst✝³ : StarRing A
inst✝² : ContinuousStar A
inst✝¹ : CompleteSpace A
inst✝ : StarModule ℂ A
a b : ↥(selfAdjoint A)
h : Commute ↑a ↑b
⊢ I • ↑a * I • ↑b = I • ↑b * I • ↑a
|
/-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Analysis.NormedSpace.Exponential
#align_import analysis.normed_space.star.exponential from "leanprover-community/mathlib"@"1e3201306d4d9eb1fd54c60d7c4510ad5126f6f9"
/-! # The exponential map from selfadjoint to unitary
In this file, we establish various properties related to the map `λ a, exp ℂ A (I • a)` between the
subtypes `selfAdjoint A` and `unitary A`.
## TODO
* Show that any exponential unitary is path-connected in `unitary A` to `1 : unitary A`.
* Prove any unitary whose distance to `1 : unitary A` is less than `1` can be expressed as an
exponential unitary.
* A unitary is in the path component of `1` if and only if it is a finite product of exponential
unitaries.
-/
open NormedSpace -- For `exp`.
section Star
variable {A : Type*} [NormedRing A] [NormedAlgebra ℂ A] [StarRing A] [ContinuousStar A]
[CompleteSpace A] [StarModule ℂ A]
open Complex
/-- The map from the selfadjoint real subspace to the unitary group. This map only makes sense
over ℂ. -/
@[simps]
noncomputable def selfAdjoint.expUnitary (a : selfAdjoint A) : unitary A :=
⟨exp ℂ ((I • a.val) : A),
exp_mem_unitary_of_mem_skewAdjoint _ (a.prop.smul_mem_skewAdjoint conj_I)⟩
#align self_adjoint.exp_unitary selfAdjoint.expUnitary
open selfAdjoint
theorem Commute.expUnitary_add {a b : selfAdjoint A} (h : Commute (a : A) (b : A)) :
expUnitary (a + b) = expUnitary a * expUnitary b := by
ext
have hcomm : Commute (I • (a : A)) (I • (b : A)) := by
unfold Commute SemiconjBy
|
simp only [h.eq, Algebra.smul_mul_assoc, Algebra.mul_smul_comm]
|
theorem Commute.expUnitary_add {a b : selfAdjoint A} (h : Commute (a : A) (b : A)) :
expUnitary (a + b) = expUnitary a * expUnitary b := by
ext
have hcomm : Commute (I • (a : A)) (I • (b : A)) := by
unfold Commute SemiconjBy
|
Mathlib.Analysis.NormedSpace.Star.Exponential.42_0.QAfFqeOSAt19Prd
|
theorem Commute.expUnitary_add {a b : selfAdjoint A} (h : Commute (a : A) (b : A)) :
expUnitary (a + b) = expUnitary a * expUnitary b
|
Mathlib_Analysis_NormedSpace_Star_Exponential
|
case a
A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra ℂ A
inst✝³ : StarRing A
inst✝² : ContinuousStar A
inst✝¹ : CompleteSpace A
inst✝ : StarModule ℂ A
a b : ↥(selfAdjoint A)
h : Commute ↑a ↑b
hcomm : Commute (I • ↑a) (I • ↑b)
⊢ ↑(expUnitary (a + b)) = ↑(expUnitary a * expUnitary b)
|
/-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Analysis.NormedSpace.Exponential
#align_import analysis.normed_space.star.exponential from "leanprover-community/mathlib"@"1e3201306d4d9eb1fd54c60d7c4510ad5126f6f9"
/-! # The exponential map from selfadjoint to unitary
In this file, we establish various properties related to the map `λ a, exp ℂ A (I • a)` between the
subtypes `selfAdjoint A` and `unitary A`.
## TODO
* Show that any exponential unitary is path-connected in `unitary A` to `1 : unitary A`.
* Prove any unitary whose distance to `1 : unitary A` is less than `1` can be expressed as an
exponential unitary.
* A unitary is in the path component of `1` if and only if it is a finite product of exponential
unitaries.
-/
open NormedSpace -- For `exp`.
section Star
variable {A : Type*} [NormedRing A] [NormedAlgebra ℂ A] [StarRing A] [ContinuousStar A]
[CompleteSpace A] [StarModule ℂ A]
open Complex
/-- The map from the selfadjoint real subspace to the unitary group. This map only makes sense
over ℂ. -/
@[simps]
noncomputable def selfAdjoint.expUnitary (a : selfAdjoint A) : unitary A :=
⟨exp ℂ ((I • a.val) : A),
exp_mem_unitary_of_mem_skewAdjoint _ (a.prop.smul_mem_skewAdjoint conj_I)⟩
#align self_adjoint.exp_unitary selfAdjoint.expUnitary
open selfAdjoint
theorem Commute.expUnitary_add {a b : selfAdjoint A} (h : Commute (a : A) (b : A)) :
expUnitary (a + b) = expUnitary a * expUnitary b := by
ext
have hcomm : Commute (I • (a : A)) (I • (b : A)) := by
unfold Commute SemiconjBy
simp only [h.eq, Algebra.smul_mul_assoc, Algebra.mul_smul_comm]
|
simpa only [expUnitary_coe, AddSubgroup.coe_add, smul_add] using exp_add_of_commute hcomm
|
theorem Commute.expUnitary_add {a b : selfAdjoint A} (h : Commute (a : A) (b : A)) :
expUnitary (a + b) = expUnitary a * expUnitary b := by
ext
have hcomm : Commute (I • (a : A)) (I • (b : A)) := by
unfold Commute SemiconjBy
simp only [h.eq, Algebra.smul_mul_assoc, Algebra.mul_smul_comm]
|
Mathlib.Analysis.NormedSpace.Star.Exponential.42_0.QAfFqeOSAt19Prd
|
theorem Commute.expUnitary_add {a b : selfAdjoint A} (h : Commute (a : A) (b : A)) :
expUnitary (a + b) = expUnitary a * expUnitary b
|
Mathlib_Analysis_NormedSpace_Star_Exponential
|
A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra ℂ A
inst✝³ : StarRing A
inst✝² : ContinuousStar A
inst✝¹ : CompleteSpace A
inst✝ : StarModule ℂ A
a b : ↥(selfAdjoint A)
h : Commute ↑a ↑b
⊢ selfAdjoint.expUnitary a * selfAdjoint.expUnitary b = selfAdjoint.expUnitary b * selfAdjoint.expUnitary a
|
/-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Analysis.NormedSpace.Exponential
#align_import analysis.normed_space.star.exponential from "leanprover-community/mathlib"@"1e3201306d4d9eb1fd54c60d7c4510ad5126f6f9"
/-! # The exponential map from selfadjoint to unitary
In this file, we establish various properties related to the map `λ a, exp ℂ A (I • a)` between the
subtypes `selfAdjoint A` and `unitary A`.
## TODO
* Show that any exponential unitary is path-connected in `unitary A` to `1 : unitary A`.
* Prove any unitary whose distance to `1 : unitary A` is less than `1` can be expressed as an
exponential unitary.
* A unitary is in the path component of `1` if and only if it is a finite product of exponential
unitaries.
-/
open NormedSpace -- For `exp`.
section Star
variable {A : Type*} [NormedRing A] [NormedAlgebra ℂ A] [StarRing A] [ContinuousStar A]
[CompleteSpace A] [StarModule ℂ A]
open Complex
/-- The map from the selfadjoint real subspace to the unitary group. This map only makes sense
over ℂ. -/
@[simps]
noncomputable def selfAdjoint.expUnitary (a : selfAdjoint A) : unitary A :=
⟨exp ℂ ((I • a.val) : A),
exp_mem_unitary_of_mem_skewAdjoint _ (a.prop.smul_mem_skewAdjoint conj_I)⟩
#align self_adjoint.exp_unitary selfAdjoint.expUnitary
open selfAdjoint
theorem Commute.expUnitary_add {a b : selfAdjoint A} (h : Commute (a : A) (b : A)) :
expUnitary (a + b) = expUnitary a * expUnitary b := by
ext
have hcomm : Commute (I • (a : A)) (I • (b : A)) := by
unfold Commute SemiconjBy
simp only [h.eq, Algebra.smul_mul_assoc, Algebra.mul_smul_comm]
simpa only [expUnitary_coe, AddSubgroup.coe_add, smul_add] using exp_add_of_commute hcomm
#align commute.exp_unitary_add Commute.expUnitary_add
theorem Commute.expUnitary {a b : selfAdjoint A} (h : Commute (a : A) (b : A)) :
Commute (expUnitary a) (expUnitary b) :=
calc
selfAdjoint.expUnitary a * selfAdjoint.expUnitary b =
selfAdjoint.expUnitary b * selfAdjoint.expUnitary a := by
|
rw [← h.expUnitary_add, ← h.symm.expUnitary_add, add_comm]
|
theorem Commute.expUnitary {a b : selfAdjoint A} (h : Commute (a : A) (b : A)) :
Commute (expUnitary a) (expUnitary b) :=
calc
selfAdjoint.expUnitary a * selfAdjoint.expUnitary b =
selfAdjoint.expUnitary b * selfAdjoint.expUnitary a := by
|
Mathlib.Analysis.NormedSpace.Star.Exponential.51_0.QAfFqeOSAt19Prd
|
theorem Commute.expUnitary {a b : selfAdjoint A} (h : Commute (a : A) (b : A)) :
Commute (expUnitary a) (expUnitary b)
|
Mathlib_Analysis_NormedSpace_Star_Exponential
|
α : Type u_1
M₀ : Type u_2
G₀ : Type u_3
M₀' : Type u_4
G₀' : Type u_5
F : Type u_6
F' : Type u_7
inst✝¹ : MonoidWithZero M₀
inst✝ : GroupWithZero G₀
a✝ b c a : G₀
hc : c ≠ 0
⊢ a / c / (b / c) = a / b
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Hom.Basic
import Mathlib.Algebra.Group.Units.Hom
import Mathlib.Algebra.GroupWithZero.Commute
import Mathlib.Algebra.GroupWithZero.Units.Basic
import Mathlib.GroupTheory.GroupAction.Units
#align_import algebra.group_with_zero.units.lemmas from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
/-!
# Further lemmas about units in a `MonoidWithZero` or a `GroupWithZero`.
-/
variable {α M₀ G₀ M₀' G₀' F F' : Type*}
variable [MonoidWithZero M₀]
section GroupWithZero
variable [GroupWithZero G₀] {a b c : G₀}
@[simp]
theorem div_self (h : a ≠ 0) : a / a = 1 :=
IsUnit.div_self h.isUnit
#align div_self div_self
theorem eq_mul_inv_iff_mul_eq₀ (hc : c ≠ 0) : a = b * c⁻¹ ↔ a * c = b :=
IsUnit.eq_mul_inv_iff_mul_eq hc.isUnit
#align eq_mul_inv_iff_mul_eq₀ eq_mul_inv_iff_mul_eq₀
theorem eq_inv_mul_iff_mul_eq₀ (hb : b ≠ 0) : a = b⁻¹ * c ↔ b * a = c :=
IsUnit.eq_inv_mul_iff_mul_eq hb.isUnit
#align eq_inv_mul_iff_mul_eq₀ eq_inv_mul_iff_mul_eq₀
theorem inv_mul_eq_iff_eq_mul₀ (ha : a ≠ 0) : a⁻¹ * b = c ↔ b = a * c :=
IsUnit.inv_mul_eq_iff_eq_mul ha.isUnit
#align inv_mul_eq_iff_eq_mul₀ inv_mul_eq_iff_eq_mul₀
theorem mul_inv_eq_iff_eq_mul₀ (hb : b ≠ 0) : a * b⁻¹ = c ↔ a = c * b :=
IsUnit.mul_inv_eq_iff_eq_mul hb.isUnit
#align mul_inv_eq_iff_eq_mul₀ mul_inv_eq_iff_eq_mul₀
theorem mul_inv_eq_one₀ (hb : b ≠ 0) : a * b⁻¹ = 1 ↔ a = b :=
IsUnit.mul_inv_eq_one hb.isUnit
#align mul_inv_eq_one₀ mul_inv_eq_one₀
theorem inv_mul_eq_one₀ (ha : a ≠ 0) : a⁻¹ * b = 1 ↔ a = b :=
IsUnit.inv_mul_eq_one ha.isUnit
#align inv_mul_eq_one₀ inv_mul_eq_one₀
theorem mul_eq_one_iff_eq_inv₀ (hb : b ≠ 0) : a * b = 1 ↔ a = b⁻¹ :=
IsUnit.mul_eq_one_iff_eq_inv hb.isUnit
#align mul_eq_one_iff_eq_inv₀ mul_eq_one_iff_eq_inv₀
theorem mul_eq_one_iff_inv_eq₀ (ha : a ≠ 0) : a * b = 1 ↔ a⁻¹ = b :=
IsUnit.mul_eq_one_iff_inv_eq ha.isUnit
#align mul_eq_one_iff_inv_eq₀ mul_eq_one_iff_inv_eq₀
@[simp]
theorem div_mul_cancel (a : G₀) (h : b ≠ 0) : a / b * b = a :=
IsUnit.div_mul_cancel h.isUnit _
#align div_mul_cancel div_mul_cancel
@[simp]
theorem mul_div_cancel (a : G₀) (h : b ≠ 0) : a * b / b = a :=
IsUnit.mul_div_cancel h.isUnit _
#align mul_div_cancel mul_div_cancel
theorem mul_one_div_cancel (h : a ≠ 0) : a * (1 / a) = 1 :=
IsUnit.mul_one_div_cancel h.isUnit
#align mul_one_div_cancel mul_one_div_cancel
theorem one_div_mul_cancel (h : a ≠ 0) : 1 / a * a = 1 :=
IsUnit.one_div_mul_cancel h.isUnit
#align one_div_mul_cancel one_div_mul_cancel
theorem div_left_inj' (hc : c ≠ 0) : a / c = b / c ↔ a = b :=
IsUnit.div_left_inj hc.isUnit
#align div_left_inj' div_left_inj'
@[field_simps]
theorem div_eq_iff (hb : b ≠ 0) : a / b = c ↔ a = c * b :=
IsUnit.div_eq_iff hb.isUnit
#align div_eq_iff div_eq_iff
@[field_simps]
theorem eq_div_iff (hb : b ≠ 0) : c = a / b ↔ c * b = a :=
IsUnit.eq_div_iff hb.isUnit
#align eq_div_iff eq_div_iff
theorem div_eq_iff_mul_eq (hb : b ≠ 0) : a / b = c ↔ c * b = a :=
(IsUnit.div_eq_iff hb.isUnit).trans eq_comm
#align div_eq_iff_mul_eq div_eq_iff_mul_eq
theorem eq_div_iff_mul_eq (hc : c ≠ 0) : a = b / c ↔ a * c = b :=
IsUnit.eq_div_iff hc.isUnit
#align eq_div_iff_mul_eq eq_div_iff_mul_eq
theorem div_eq_of_eq_mul (hb : b ≠ 0) : a = c * b → a / b = c :=
IsUnit.div_eq_of_eq_mul hb.isUnit
#align div_eq_of_eq_mul div_eq_of_eq_mul
theorem eq_div_of_mul_eq (hc : c ≠ 0) : a * c = b → a = b / c :=
IsUnit.eq_div_of_mul_eq hc.isUnit
#align eq_div_of_mul_eq eq_div_of_mul_eq
theorem div_eq_one_iff_eq (hb : b ≠ 0) : a / b = 1 ↔ a = b :=
IsUnit.div_eq_one_iff_eq hb.isUnit
#align div_eq_one_iff_eq div_eq_one_iff_eq
theorem div_mul_left (hb : b ≠ 0) : b / (a * b) = 1 / a :=
IsUnit.div_mul_left hb.isUnit
#align div_mul_left div_mul_left
theorem mul_div_mul_right (a b : G₀) (hc : c ≠ 0) : a * c / (b * c) = a / b :=
IsUnit.mul_div_mul_right hc.isUnit _ _
#align mul_div_mul_right mul_div_mul_right
theorem mul_mul_div (a : G₀) (hb : b ≠ 0) : a = a * b * (1 / b) :=
(IsUnit.mul_mul_div _ hb.isUnit).symm
#align mul_mul_div mul_mul_div
theorem div_div_div_cancel_right (a : G₀) (hc : c ≠ 0) : a / c / (b / c) = a / b := by
|
rw [div_div_eq_mul_div, div_mul_cancel _ hc]
|
theorem div_div_div_cancel_right (a : G₀) (hc : c ≠ 0) : a / c / (b / c) = a / b := by
|
Mathlib.Algebra.GroupWithZero.Units.Lemmas.129_0.ICkvbDoLYHVogsB
|
theorem div_div_div_cancel_right (a : G₀) (hc : c ≠ 0) : a / c / (b / c) = a / b
|
Mathlib_Algebra_GroupWithZero_Units_Lemmas
|
α : Type u_1
M₀ : Type u_2
G₀ : Type u_3
M₀' : Type u_4
G₀' : Type u_5
F : Type u_6
F' : Type u_7
inst✝¹ : MonoidWithZero M₀
inst✝ : GroupWithZero G₀
a✝ b c a : G₀
hc : c ≠ 0
⊢ a / c * (c / b) = a / b
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Hom.Basic
import Mathlib.Algebra.Group.Units.Hom
import Mathlib.Algebra.GroupWithZero.Commute
import Mathlib.Algebra.GroupWithZero.Units.Basic
import Mathlib.GroupTheory.GroupAction.Units
#align_import algebra.group_with_zero.units.lemmas from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
/-!
# Further lemmas about units in a `MonoidWithZero` or a `GroupWithZero`.
-/
variable {α M₀ G₀ M₀' G₀' F F' : Type*}
variable [MonoidWithZero M₀]
section GroupWithZero
variable [GroupWithZero G₀] {a b c : G₀}
@[simp]
theorem div_self (h : a ≠ 0) : a / a = 1 :=
IsUnit.div_self h.isUnit
#align div_self div_self
theorem eq_mul_inv_iff_mul_eq₀ (hc : c ≠ 0) : a = b * c⁻¹ ↔ a * c = b :=
IsUnit.eq_mul_inv_iff_mul_eq hc.isUnit
#align eq_mul_inv_iff_mul_eq₀ eq_mul_inv_iff_mul_eq₀
theorem eq_inv_mul_iff_mul_eq₀ (hb : b ≠ 0) : a = b⁻¹ * c ↔ b * a = c :=
IsUnit.eq_inv_mul_iff_mul_eq hb.isUnit
#align eq_inv_mul_iff_mul_eq₀ eq_inv_mul_iff_mul_eq₀
theorem inv_mul_eq_iff_eq_mul₀ (ha : a ≠ 0) : a⁻¹ * b = c ↔ b = a * c :=
IsUnit.inv_mul_eq_iff_eq_mul ha.isUnit
#align inv_mul_eq_iff_eq_mul₀ inv_mul_eq_iff_eq_mul₀
theorem mul_inv_eq_iff_eq_mul₀ (hb : b ≠ 0) : a * b⁻¹ = c ↔ a = c * b :=
IsUnit.mul_inv_eq_iff_eq_mul hb.isUnit
#align mul_inv_eq_iff_eq_mul₀ mul_inv_eq_iff_eq_mul₀
theorem mul_inv_eq_one₀ (hb : b ≠ 0) : a * b⁻¹ = 1 ↔ a = b :=
IsUnit.mul_inv_eq_one hb.isUnit
#align mul_inv_eq_one₀ mul_inv_eq_one₀
theorem inv_mul_eq_one₀ (ha : a ≠ 0) : a⁻¹ * b = 1 ↔ a = b :=
IsUnit.inv_mul_eq_one ha.isUnit
#align inv_mul_eq_one₀ inv_mul_eq_one₀
theorem mul_eq_one_iff_eq_inv₀ (hb : b ≠ 0) : a * b = 1 ↔ a = b⁻¹ :=
IsUnit.mul_eq_one_iff_eq_inv hb.isUnit
#align mul_eq_one_iff_eq_inv₀ mul_eq_one_iff_eq_inv₀
theorem mul_eq_one_iff_inv_eq₀ (ha : a ≠ 0) : a * b = 1 ↔ a⁻¹ = b :=
IsUnit.mul_eq_one_iff_inv_eq ha.isUnit
#align mul_eq_one_iff_inv_eq₀ mul_eq_one_iff_inv_eq₀
@[simp]
theorem div_mul_cancel (a : G₀) (h : b ≠ 0) : a / b * b = a :=
IsUnit.div_mul_cancel h.isUnit _
#align div_mul_cancel div_mul_cancel
@[simp]
theorem mul_div_cancel (a : G₀) (h : b ≠ 0) : a * b / b = a :=
IsUnit.mul_div_cancel h.isUnit _
#align mul_div_cancel mul_div_cancel
theorem mul_one_div_cancel (h : a ≠ 0) : a * (1 / a) = 1 :=
IsUnit.mul_one_div_cancel h.isUnit
#align mul_one_div_cancel mul_one_div_cancel
theorem one_div_mul_cancel (h : a ≠ 0) : 1 / a * a = 1 :=
IsUnit.one_div_mul_cancel h.isUnit
#align one_div_mul_cancel one_div_mul_cancel
theorem div_left_inj' (hc : c ≠ 0) : a / c = b / c ↔ a = b :=
IsUnit.div_left_inj hc.isUnit
#align div_left_inj' div_left_inj'
@[field_simps]
theorem div_eq_iff (hb : b ≠ 0) : a / b = c ↔ a = c * b :=
IsUnit.div_eq_iff hb.isUnit
#align div_eq_iff div_eq_iff
@[field_simps]
theorem eq_div_iff (hb : b ≠ 0) : c = a / b ↔ c * b = a :=
IsUnit.eq_div_iff hb.isUnit
#align eq_div_iff eq_div_iff
theorem div_eq_iff_mul_eq (hb : b ≠ 0) : a / b = c ↔ c * b = a :=
(IsUnit.div_eq_iff hb.isUnit).trans eq_comm
#align div_eq_iff_mul_eq div_eq_iff_mul_eq
theorem eq_div_iff_mul_eq (hc : c ≠ 0) : a = b / c ↔ a * c = b :=
IsUnit.eq_div_iff hc.isUnit
#align eq_div_iff_mul_eq eq_div_iff_mul_eq
theorem div_eq_of_eq_mul (hb : b ≠ 0) : a = c * b → a / b = c :=
IsUnit.div_eq_of_eq_mul hb.isUnit
#align div_eq_of_eq_mul div_eq_of_eq_mul
theorem eq_div_of_mul_eq (hc : c ≠ 0) : a * c = b → a = b / c :=
IsUnit.eq_div_of_mul_eq hc.isUnit
#align eq_div_of_mul_eq eq_div_of_mul_eq
theorem div_eq_one_iff_eq (hb : b ≠ 0) : a / b = 1 ↔ a = b :=
IsUnit.div_eq_one_iff_eq hb.isUnit
#align div_eq_one_iff_eq div_eq_one_iff_eq
theorem div_mul_left (hb : b ≠ 0) : b / (a * b) = 1 / a :=
IsUnit.div_mul_left hb.isUnit
#align div_mul_left div_mul_left
theorem mul_div_mul_right (a b : G₀) (hc : c ≠ 0) : a * c / (b * c) = a / b :=
IsUnit.mul_div_mul_right hc.isUnit _ _
#align mul_div_mul_right mul_div_mul_right
theorem mul_mul_div (a : G₀) (hb : b ≠ 0) : a = a * b * (1 / b) :=
(IsUnit.mul_mul_div _ hb.isUnit).symm
#align mul_mul_div mul_mul_div
theorem div_div_div_cancel_right (a : G₀) (hc : c ≠ 0) : a / c / (b / c) = a / b := by
rw [div_div_eq_mul_div, div_mul_cancel _ hc]
#align div_div_div_cancel_right div_div_div_cancel_right
theorem div_mul_div_cancel (a : G₀) (hc : c ≠ 0) : a / c * (c / b) = a / b := by
|
rw [← mul_div_assoc, div_mul_cancel _ hc]
|
theorem div_mul_div_cancel (a : G₀) (hc : c ≠ 0) : a / c * (c / b) = a / b := by
|
Mathlib.Algebra.GroupWithZero.Units.Lemmas.133_0.ICkvbDoLYHVogsB
|
theorem div_mul_div_cancel (a : G₀) (hc : c ≠ 0) : a / c * (c / b) = a / b
|
Mathlib_Algebra_GroupWithZero_Units_Lemmas
|
α : Type u_1
M₀ : Type u_2
G₀ : Type u_3
M₀' : Type u_4
G₀' : Type u_5
F : Type u_6
F' : Type u_7
inst✝¹ : MonoidWithZero M₀
inst✝ : GroupWithZero G₀
a✝ b✝ c a b : G₀
h : b = 0 → a = 0
hb : b = 0
⊢ a / b * b = a
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Hom.Basic
import Mathlib.Algebra.Group.Units.Hom
import Mathlib.Algebra.GroupWithZero.Commute
import Mathlib.Algebra.GroupWithZero.Units.Basic
import Mathlib.GroupTheory.GroupAction.Units
#align_import algebra.group_with_zero.units.lemmas from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
/-!
# Further lemmas about units in a `MonoidWithZero` or a `GroupWithZero`.
-/
variable {α M₀ G₀ M₀' G₀' F F' : Type*}
variable [MonoidWithZero M₀]
section GroupWithZero
variable [GroupWithZero G₀] {a b c : G₀}
@[simp]
theorem div_self (h : a ≠ 0) : a / a = 1 :=
IsUnit.div_self h.isUnit
#align div_self div_self
theorem eq_mul_inv_iff_mul_eq₀ (hc : c ≠ 0) : a = b * c⁻¹ ↔ a * c = b :=
IsUnit.eq_mul_inv_iff_mul_eq hc.isUnit
#align eq_mul_inv_iff_mul_eq₀ eq_mul_inv_iff_mul_eq₀
theorem eq_inv_mul_iff_mul_eq₀ (hb : b ≠ 0) : a = b⁻¹ * c ↔ b * a = c :=
IsUnit.eq_inv_mul_iff_mul_eq hb.isUnit
#align eq_inv_mul_iff_mul_eq₀ eq_inv_mul_iff_mul_eq₀
theorem inv_mul_eq_iff_eq_mul₀ (ha : a ≠ 0) : a⁻¹ * b = c ↔ b = a * c :=
IsUnit.inv_mul_eq_iff_eq_mul ha.isUnit
#align inv_mul_eq_iff_eq_mul₀ inv_mul_eq_iff_eq_mul₀
theorem mul_inv_eq_iff_eq_mul₀ (hb : b ≠ 0) : a * b⁻¹ = c ↔ a = c * b :=
IsUnit.mul_inv_eq_iff_eq_mul hb.isUnit
#align mul_inv_eq_iff_eq_mul₀ mul_inv_eq_iff_eq_mul₀
theorem mul_inv_eq_one₀ (hb : b ≠ 0) : a * b⁻¹ = 1 ↔ a = b :=
IsUnit.mul_inv_eq_one hb.isUnit
#align mul_inv_eq_one₀ mul_inv_eq_one₀
theorem inv_mul_eq_one₀ (ha : a ≠ 0) : a⁻¹ * b = 1 ↔ a = b :=
IsUnit.inv_mul_eq_one ha.isUnit
#align inv_mul_eq_one₀ inv_mul_eq_one₀
theorem mul_eq_one_iff_eq_inv₀ (hb : b ≠ 0) : a * b = 1 ↔ a = b⁻¹ :=
IsUnit.mul_eq_one_iff_eq_inv hb.isUnit
#align mul_eq_one_iff_eq_inv₀ mul_eq_one_iff_eq_inv₀
theorem mul_eq_one_iff_inv_eq₀ (ha : a ≠ 0) : a * b = 1 ↔ a⁻¹ = b :=
IsUnit.mul_eq_one_iff_inv_eq ha.isUnit
#align mul_eq_one_iff_inv_eq₀ mul_eq_one_iff_inv_eq₀
@[simp]
theorem div_mul_cancel (a : G₀) (h : b ≠ 0) : a / b * b = a :=
IsUnit.div_mul_cancel h.isUnit _
#align div_mul_cancel div_mul_cancel
@[simp]
theorem mul_div_cancel (a : G₀) (h : b ≠ 0) : a * b / b = a :=
IsUnit.mul_div_cancel h.isUnit _
#align mul_div_cancel mul_div_cancel
theorem mul_one_div_cancel (h : a ≠ 0) : a * (1 / a) = 1 :=
IsUnit.mul_one_div_cancel h.isUnit
#align mul_one_div_cancel mul_one_div_cancel
theorem one_div_mul_cancel (h : a ≠ 0) : 1 / a * a = 1 :=
IsUnit.one_div_mul_cancel h.isUnit
#align one_div_mul_cancel one_div_mul_cancel
theorem div_left_inj' (hc : c ≠ 0) : a / c = b / c ↔ a = b :=
IsUnit.div_left_inj hc.isUnit
#align div_left_inj' div_left_inj'
@[field_simps]
theorem div_eq_iff (hb : b ≠ 0) : a / b = c ↔ a = c * b :=
IsUnit.div_eq_iff hb.isUnit
#align div_eq_iff div_eq_iff
@[field_simps]
theorem eq_div_iff (hb : b ≠ 0) : c = a / b ↔ c * b = a :=
IsUnit.eq_div_iff hb.isUnit
#align eq_div_iff eq_div_iff
theorem div_eq_iff_mul_eq (hb : b ≠ 0) : a / b = c ↔ c * b = a :=
(IsUnit.div_eq_iff hb.isUnit).trans eq_comm
#align div_eq_iff_mul_eq div_eq_iff_mul_eq
theorem eq_div_iff_mul_eq (hc : c ≠ 0) : a = b / c ↔ a * c = b :=
IsUnit.eq_div_iff hc.isUnit
#align eq_div_iff_mul_eq eq_div_iff_mul_eq
theorem div_eq_of_eq_mul (hb : b ≠ 0) : a = c * b → a / b = c :=
IsUnit.div_eq_of_eq_mul hb.isUnit
#align div_eq_of_eq_mul div_eq_of_eq_mul
theorem eq_div_of_mul_eq (hc : c ≠ 0) : a * c = b → a = b / c :=
IsUnit.eq_div_of_mul_eq hc.isUnit
#align eq_div_of_mul_eq eq_div_of_mul_eq
theorem div_eq_one_iff_eq (hb : b ≠ 0) : a / b = 1 ↔ a = b :=
IsUnit.div_eq_one_iff_eq hb.isUnit
#align div_eq_one_iff_eq div_eq_one_iff_eq
theorem div_mul_left (hb : b ≠ 0) : b / (a * b) = 1 / a :=
IsUnit.div_mul_left hb.isUnit
#align div_mul_left div_mul_left
theorem mul_div_mul_right (a b : G₀) (hc : c ≠ 0) : a * c / (b * c) = a / b :=
IsUnit.mul_div_mul_right hc.isUnit _ _
#align mul_div_mul_right mul_div_mul_right
theorem mul_mul_div (a : G₀) (hb : b ≠ 0) : a = a * b * (1 / b) :=
(IsUnit.mul_mul_div _ hb.isUnit).symm
#align mul_mul_div mul_mul_div
theorem div_div_div_cancel_right (a : G₀) (hc : c ≠ 0) : a / c / (b / c) = a / b := by
rw [div_div_eq_mul_div, div_mul_cancel _ hc]
#align div_div_div_cancel_right div_div_div_cancel_right
theorem div_mul_div_cancel (a : G₀) (hc : c ≠ 0) : a / c * (c / b) = a / b := by
rw [← mul_div_assoc, div_mul_cancel _ hc]
#align div_mul_div_cancel div_mul_div_cancel
theorem div_mul_cancel_of_imp {a b : G₀} (h : b = 0 → a = 0) : a / b * b = a :=
Classical.by_cases (fun hb : b = 0 => by
|
simp [*]
|
theorem div_mul_cancel_of_imp {a b : G₀} (h : b = 0 → a = 0) : a / b * b = a :=
Classical.by_cases (fun hb : b = 0 => by
|
Mathlib.Algebra.GroupWithZero.Units.Lemmas.137_0.ICkvbDoLYHVogsB
|
theorem div_mul_cancel_of_imp {a b : G₀} (h : b = 0 → a = 0) : a / b * b = a
|
Mathlib_Algebra_GroupWithZero_Units_Lemmas
|
α : Type u_1
M₀ : Type u_2
G₀ : Type u_3
M₀' : Type u_4
G₀' : Type u_5
F : Type u_6
F' : Type u_7
inst✝¹ : MonoidWithZero M₀
inst✝ : GroupWithZero G₀
a✝ b✝ c a b : G₀
h : b = 0 → a = 0
hb : b = 0
⊢ a * b / b = a
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Hom.Basic
import Mathlib.Algebra.Group.Units.Hom
import Mathlib.Algebra.GroupWithZero.Commute
import Mathlib.Algebra.GroupWithZero.Units.Basic
import Mathlib.GroupTheory.GroupAction.Units
#align_import algebra.group_with_zero.units.lemmas from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
/-!
# Further lemmas about units in a `MonoidWithZero` or a `GroupWithZero`.
-/
variable {α M₀ G₀ M₀' G₀' F F' : Type*}
variable [MonoidWithZero M₀]
section GroupWithZero
variable [GroupWithZero G₀] {a b c : G₀}
@[simp]
theorem div_self (h : a ≠ 0) : a / a = 1 :=
IsUnit.div_self h.isUnit
#align div_self div_self
theorem eq_mul_inv_iff_mul_eq₀ (hc : c ≠ 0) : a = b * c⁻¹ ↔ a * c = b :=
IsUnit.eq_mul_inv_iff_mul_eq hc.isUnit
#align eq_mul_inv_iff_mul_eq₀ eq_mul_inv_iff_mul_eq₀
theorem eq_inv_mul_iff_mul_eq₀ (hb : b ≠ 0) : a = b⁻¹ * c ↔ b * a = c :=
IsUnit.eq_inv_mul_iff_mul_eq hb.isUnit
#align eq_inv_mul_iff_mul_eq₀ eq_inv_mul_iff_mul_eq₀
theorem inv_mul_eq_iff_eq_mul₀ (ha : a ≠ 0) : a⁻¹ * b = c ↔ b = a * c :=
IsUnit.inv_mul_eq_iff_eq_mul ha.isUnit
#align inv_mul_eq_iff_eq_mul₀ inv_mul_eq_iff_eq_mul₀
theorem mul_inv_eq_iff_eq_mul₀ (hb : b ≠ 0) : a * b⁻¹ = c ↔ a = c * b :=
IsUnit.mul_inv_eq_iff_eq_mul hb.isUnit
#align mul_inv_eq_iff_eq_mul₀ mul_inv_eq_iff_eq_mul₀
theorem mul_inv_eq_one₀ (hb : b ≠ 0) : a * b⁻¹ = 1 ↔ a = b :=
IsUnit.mul_inv_eq_one hb.isUnit
#align mul_inv_eq_one₀ mul_inv_eq_one₀
theorem inv_mul_eq_one₀ (ha : a ≠ 0) : a⁻¹ * b = 1 ↔ a = b :=
IsUnit.inv_mul_eq_one ha.isUnit
#align inv_mul_eq_one₀ inv_mul_eq_one₀
theorem mul_eq_one_iff_eq_inv₀ (hb : b ≠ 0) : a * b = 1 ↔ a = b⁻¹ :=
IsUnit.mul_eq_one_iff_eq_inv hb.isUnit
#align mul_eq_one_iff_eq_inv₀ mul_eq_one_iff_eq_inv₀
theorem mul_eq_one_iff_inv_eq₀ (ha : a ≠ 0) : a * b = 1 ↔ a⁻¹ = b :=
IsUnit.mul_eq_one_iff_inv_eq ha.isUnit
#align mul_eq_one_iff_inv_eq₀ mul_eq_one_iff_inv_eq₀
@[simp]
theorem div_mul_cancel (a : G₀) (h : b ≠ 0) : a / b * b = a :=
IsUnit.div_mul_cancel h.isUnit _
#align div_mul_cancel div_mul_cancel
@[simp]
theorem mul_div_cancel (a : G₀) (h : b ≠ 0) : a * b / b = a :=
IsUnit.mul_div_cancel h.isUnit _
#align mul_div_cancel mul_div_cancel
theorem mul_one_div_cancel (h : a ≠ 0) : a * (1 / a) = 1 :=
IsUnit.mul_one_div_cancel h.isUnit
#align mul_one_div_cancel mul_one_div_cancel
theorem one_div_mul_cancel (h : a ≠ 0) : 1 / a * a = 1 :=
IsUnit.one_div_mul_cancel h.isUnit
#align one_div_mul_cancel one_div_mul_cancel
theorem div_left_inj' (hc : c ≠ 0) : a / c = b / c ↔ a = b :=
IsUnit.div_left_inj hc.isUnit
#align div_left_inj' div_left_inj'
@[field_simps]
theorem div_eq_iff (hb : b ≠ 0) : a / b = c ↔ a = c * b :=
IsUnit.div_eq_iff hb.isUnit
#align div_eq_iff div_eq_iff
@[field_simps]
theorem eq_div_iff (hb : b ≠ 0) : c = a / b ↔ c * b = a :=
IsUnit.eq_div_iff hb.isUnit
#align eq_div_iff eq_div_iff
theorem div_eq_iff_mul_eq (hb : b ≠ 0) : a / b = c ↔ c * b = a :=
(IsUnit.div_eq_iff hb.isUnit).trans eq_comm
#align div_eq_iff_mul_eq div_eq_iff_mul_eq
theorem eq_div_iff_mul_eq (hc : c ≠ 0) : a = b / c ↔ a * c = b :=
IsUnit.eq_div_iff hc.isUnit
#align eq_div_iff_mul_eq eq_div_iff_mul_eq
theorem div_eq_of_eq_mul (hb : b ≠ 0) : a = c * b → a / b = c :=
IsUnit.div_eq_of_eq_mul hb.isUnit
#align div_eq_of_eq_mul div_eq_of_eq_mul
theorem eq_div_of_mul_eq (hc : c ≠ 0) : a * c = b → a = b / c :=
IsUnit.eq_div_of_mul_eq hc.isUnit
#align eq_div_of_mul_eq eq_div_of_mul_eq
theorem div_eq_one_iff_eq (hb : b ≠ 0) : a / b = 1 ↔ a = b :=
IsUnit.div_eq_one_iff_eq hb.isUnit
#align div_eq_one_iff_eq div_eq_one_iff_eq
theorem div_mul_left (hb : b ≠ 0) : b / (a * b) = 1 / a :=
IsUnit.div_mul_left hb.isUnit
#align div_mul_left div_mul_left
theorem mul_div_mul_right (a b : G₀) (hc : c ≠ 0) : a * c / (b * c) = a / b :=
IsUnit.mul_div_mul_right hc.isUnit _ _
#align mul_div_mul_right mul_div_mul_right
theorem mul_mul_div (a : G₀) (hb : b ≠ 0) : a = a * b * (1 / b) :=
(IsUnit.mul_mul_div _ hb.isUnit).symm
#align mul_mul_div mul_mul_div
theorem div_div_div_cancel_right (a : G₀) (hc : c ≠ 0) : a / c / (b / c) = a / b := by
rw [div_div_eq_mul_div, div_mul_cancel _ hc]
#align div_div_div_cancel_right div_div_div_cancel_right
theorem div_mul_div_cancel (a : G₀) (hc : c ≠ 0) : a / c * (c / b) = a / b := by
rw [← mul_div_assoc, div_mul_cancel _ hc]
#align div_mul_div_cancel div_mul_div_cancel
theorem div_mul_cancel_of_imp {a b : G₀} (h : b = 0 → a = 0) : a / b * b = a :=
Classical.by_cases (fun hb : b = 0 => by simp [*]) (div_mul_cancel a)
#align div_mul_cancel_of_imp div_mul_cancel_of_imp
theorem mul_div_cancel_of_imp {a b : G₀} (h : b = 0 → a = 0) : a * b / b = a :=
Classical.by_cases (fun hb : b = 0 => by
|
simp [*]
|
theorem mul_div_cancel_of_imp {a b : G₀} (h : b = 0 → a = 0) : a * b / b = a :=
Classical.by_cases (fun hb : b = 0 => by
|
Mathlib.Algebra.GroupWithZero.Units.Lemmas.141_0.ICkvbDoLYHVogsB
|
theorem mul_div_cancel_of_imp {a b : G₀} (h : b = 0 → a = 0) : a * b / b = a
|
Mathlib_Algebra_GroupWithZero_Units_Lemmas
|
α : Type u_1
M₀ : Type u_2
G₀ : Type u_3
M₀' : Type u_4
G₀' : Type u_5
F : Type u_6
F' : Type u_7
inst✝¹ : MonoidWithZero M₀
inst✝ : CommGroupWithZero G₀
a✝ b✝ c d a b : G₀
h : a = 0 → b = 0
⊢ a * b / a = b
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Hom.Basic
import Mathlib.Algebra.Group.Units.Hom
import Mathlib.Algebra.GroupWithZero.Commute
import Mathlib.Algebra.GroupWithZero.Units.Basic
import Mathlib.GroupTheory.GroupAction.Units
#align_import algebra.group_with_zero.units.lemmas from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
/-!
# Further lemmas about units in a `MonoidWithZero` or a `GroupWithZero`.
-/
variable {α M₀ G₀ M₀' G₀' F F' : Type*}
variable [MonoidWithZero M₀]
section GroupWithZero
variable [GroupWithZero G₀] {a b c : G₀}
@[simp]
theorem div_self (h : a ≠ 0) : a / a = 1 :=
IsUnit.div_self h.isUnit
#align div_self div_self
theorem eq_mul_inv_iff_mul_eq₀ (hc : c ≠ 0) : a = b * c⁻¹ ↔ a * c = b :=
IsUnit.eq_mul_inv_iff_mul_eq hc.isUnit
#align eq_mul_inv_iff_mul_eq₀ eq_mul_inv_iff_mul_eq₀
theorem eq_inv_mul_iff_mul_eq₀ (hb : b ≠ 0) : a = b⁻¹ * c ↔ b * a = c :=
IsUnit.eq_inv_mul_iff_mul_eq hb.isUnit
#align eq_inv_mul_iff_mul_eq₀ eq_inv_mul_iff_mul_eq₀
theorem inv_mul_eq_iff_eq_mul₀ (ha : a ≠ 0) : a⁻¹ * b = c ↔ b = a * c :=
IsUnit.inv_mul_eq_iff_eq_mul ha.isUnit
#align inv_mul_eq_iff_eq_mul₀ inv_mul_eq_iff_eq_mul₀
theorem mul_inv_eq_iff_eq_mul₀ (hb : b ≠ 0) : a * b⁻¹ = c ↔ a = c * b :=
IsUnit.mul_inv_eq_iff_eq_mul hb.isUnit
#align mul_inv_eq_iff_eq_mul₀ mul_inv_eq_iff_eq_mul₀
theorem mul_inv_eq_one₀ (hb : b ≠ 0) : a * b⁻¹ = 1 ↔ a = b :=
IsUnit.mul_inv_eq_one hb.isUnit
#align mul_inv_eq_one₀ mul_inv_eq_one₀
theorem inv_mul_eq_one₀ (ha : a ≠ 0) : a⁻¹ * b = 1 ↔ a = b :=
IsUnit.inv_mul_eq_one ha.isUnit
#align inv_mul_eq_one₀ inv_mul_eq_one₀
theorem mul_eq_one_iff_eq_inv₀ (hb : b ≠ 0) : a * b = 1 ↔ a = b⁻¹ :=
IsUnit.mul_eq_one_iff_eq_inv hb.isUnit
#align mul_eq_one_iff_eq_inv₀ mul_eq_one_iff_eq_inv₀
theorem mul_eq_one_iff_inv_eq₀ (ha : a ≠ 0) : a * b = 1 ↔ a⁻¹ = b :=
IsUnit.mul_eq_one_iff_inv_eq ha.isUnit
#align mul_eq_one_iff_inv_eq₀ mul_eq_one_iff_inv_eq₀
@[simp]
theorem div_mul_cancel (a : G₀) (h : b ≠ 0) : a / b * b = a :=
IsUnit.div_mul_cancel h.isUnit _
#align div_mul_cancel div_mul_cancel
@[simp]
theorem mul_div_cancel (a : G₀) (h : b ≠ 0) : a * b / b = a :=
IsUnit.mul_div_cancel h.isUnit _
#align mul_div_cancel mul_div_cancel
theorem mul_one_div_cancel (h : a ≠ 0) : a * (1 / a) = 1 :=
IsUnit.mul_one_div_cancel h.isUnit
#align mul_one_div_cancel mul_one_div_cancel
theorem one_div_mul_cancel (h : a ≠ 0) : 1 / a * a = 1 :=
IsUnit.one_div_mul_cancel h.isUnit
#align one_div_mul_cancel one_div_mul_cancel
theorem div_left_inj' (hc : c ≠ 0) : a / c = b / c ↔ a = b :=
IsUnit.div_left_inj hc.isUnit
#align div_left_inj' div_left_inj'
@[field_simps]
theorem div_eq_iff (hb : b ≠ 0) : a / b = c ↔ a = c * b :=
IsUnit.div_eq_iff hb.isUnit
#align div_eq_iff div_eq_iff
@[field_simps]
theorem eq_div_iff (hb : b ≠ 0) : c = a / b ↔ c * b = a :=
IsUnit.eq_div_iff hb.isUnit
#align eq_div_iff eq_div_iff
theorem div_eq_iff_mul_eq (hb : b ≠ 0) : a / b = c ↔ c * b = a :=
(IsUnit.div_eq_iff hb.isUnit).trans eq_comm
#align div_eq_iff_mul_eq div_eq_iff_mul_eq
theorem eq_div_iff_mul_eq (hc : c ≠ 0) : a = b / c ↔ a * c = b :=
IsUnit.eq_div_iff hc.isUnit
#align eq_div_iff_mul_eq eq_div_iff_mul_eq
theorem div_eq_of_eq_mul (hb : b ≠ 0) : a = c * b → a / b = c :=
IsUnit.div_eq_of_eq_mul hb.isUnit
#align div_eq_of_eq_mul div_eq_of_eq_mul
theorem eq_div_of_mul_eq (hc : c ≠ 0) : a * c = b → a = b / c :=
IsUnit.eq_div_of_mul_eq hc.isUnit
#align eq_div_of_mul_eq eq_div_of_mul_eq
theorem div_eq_one_iff_eq (hb : b ≠ 0) : a / b = 1 ↔ a = b :=
IsUnit.div_eq_one_iff_eq hb.isUnit
#align div_eq_one_iff_eq div_eq_one_iff_eq
theorem div_mul_left (hb : b ≠ 0) : b / (a * b) = 1 / a :=
IsUnit.div_mul_left hb.isUnit
#align div_mul_left div_mul_left
theorem mul_div_mul_right (a b : G₀) (hc : c ≠ 0) : a * c / (b * c) = a / b :=
IsUnit.mul_div_mul_right hc.isUnit _ _
#align mul_div_mul_right mul_div_mul_right
theorem mul_mul_div (a : G₀) (hb : b ≠ 0) : a = a * b * (1 / b) :=
(IsUnit.mul_mul_div _ hb.isUnit).symm
#align mul_mul_div mul_mul_div
theorem div_div_div_cancel_right (a : G₀) (hc : c ≠ 0) : a / c / (b / c) = a / b := by
rw [div_div_eq_mul_div, div_mul_cancel _ hc]
#align div_div_div_cancel_right div_div_div_cancel_right
theorem div_mul_div_cancel (a : G₀) (hc : c ≠ 0) : a / c * (c / b) = a / b := by
rw [← mul_div_assoc, div_mul_cancel _ hc]
#align div_mul_div_cancel div_mul_div_cancel
theorem div_mul_cancel_of_imp {a b : G₀} (h : b = 0 → a = 0) : a / b * b = a :=
Classical.by_cases (fun hb : b = 0 => by simp [*]) (div_mul_cancel a)
#align div_mul_cancel_of_imp div_mul_cancel_of_imp
theorem mul_div_cancel_of_imp {a b : G₀} (h : b = 0 → a = 0) : a * b / b = a :=
Classical.by_cases (fun hb : b = 0 => by simp [*]) (mul_div_cancel a)
#align mul_div_cancel_of_imp mul_div_cancel_of_imp
@[simp]
theorem divp_mk0 (a : G₀) {b : G₀} (hb : b ≠ 0) : a /ₚ Units.mk0 b hb = a / b :=
divp_eq_div _ _
#align divp_mk0 divp_mk0
end GroupWithZero
section CommGroupWithZero
-- comm
variable [CommGroupWithZero G₀] {a b c d : G₀}
theorem div_mul_right (b : G₀) (ha : a ≠ 0) : a / (a * b) = 1 / b :=
IsUnit.div_mul_right ha.isUnit _
#align div_mul_right div_mul_right
theorem mul_div_cancel_left_of_imp {a b : G₀} (h : a = 0 → b = 0) : a * b / a = b := by
|
rw [mul_comm, mul_div_cancel_of_imp h]
|
theorem mul_div_cancel_left_of_imp {a b : G₀} (h : a = 0 → b = 0) : a * b / a = b := by
|
Mathlib.Algebra.GroupWithZero.Units.Lemmas.161_0.ICkvbDoLYHVogsB
|
theorem mul_div_cancel_left_of_imp {a b : G₀} (h : a = 0 → b = 0) : a * b / a = b
|
Mathlib_Algebra_GroupWithZero_Units_Lemmas
|
α : Type u_1
M₀ : Type u_2
G₀ : Type u_3
M₀' : Type u_4
G₀' : Type u_5
F : Type u_6
F' : Type u_7
inst✝¹ : MonoidWithZero M₀
inst✝ : CommGroupWithZero G₀
a✝ b✝ c d a b : G₀
h : b = 0 → a = 0
⊢ b * (a / b) = a
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Hom.Basic
import Mathlib.Algebra.Group.Units.Hom
import Mathlib.Algebra.GroupWithZero.Commute
import Mathlib.Algebra.GroupWithZero.Units.Basic
import Mathlib.GroupTheory.GroupAction.Units
#align_import algebra.group_with_zero.units.lemmas from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
/-!
# Further lemmas about units in a `MonoidWithZero` or a `GroupWithZero`.
-/
variable {α M₀ G₀ M₀' G₀' F F' : Type*}
variable [MonoidWithZero M₀]
section GroupWithZero
variable [GroupWithZero G₀] {a b c : G₀}
@[simp]
theorem div_self (h : a ≠ 0) : a / a = 1 :=
IsUnit.div_self h.isUnit
#align div_self div_self
theorem eq_mul_inv_iff_mul_eq₀ (hc : c ≠ 0) : a = b * c⁻¹ ↔ a * c = b :=
IsUnit.eq_mul_inv_iff_mul_eq hc.isUnit
#align eq_mul_inv_iff_mul_eq₀ eq_mul_inv_iff_mul_eq₀
theorem eq_inv_mul_iff_mul_eq₀ (hb : b ≠ 0) : a = b⁻¹ * c ↔ b * a = c :=
IsUnit.eq_inv_mul_iff_mul_eq hb.isUnit
#align eq_inv_mul_iff_mul_eq₀ eq_inv_mul_iff_mul_eq₀
theorem inv_mul_eq_iff_eq_mul₀ (ha : a ≠ 0) : a⁻¹ * b = c ↔ b = a * c :=
IsUnit.inv_mul_eq_iff_eq_mul ha.isUnit
#align inv_mul_eq_iff_eq_mul₀ inv_mul_eq_iff_eq_mul₀
theorem mul_inv_eq_iff_eq_mul₀ (hb : b ≠ 0) : a * b⁻¹ = c ↔ a = c * b :=
IsUnit.mul_inv_eq_iff_eq_mul hb.isUnit
#align mul_inv_eq_iff_eq_mul₀ mul_inv_eq_iff_eq_mul₀
theorem mul_inv_eq_one₀ (hb : b ≠ 0) : a * b⁻¹ = 1 ↔ a = b :=
IsUnit.mul_inv_eq_one hb.isUnit
#align mul_inv_eq_one₀ mul_inv_eq_one₀
theorem inv_mul_eq_one₀ (ha : a ≠ 0) : a⁻¹ * b = 1 ↔ a = b :=
IsUnit.inv_mul_eq_one ha.isUnit
#align inv_mul_eq_one₀ inv_mul_eq_one₀
theorem mul_eq_one_iff_eq_inv₀ (hb : b ≠ 0) : a * b = 1 ↔ a = b⁻¹ :=
IsUnit.mul_eq_one_iff_eq_inv hb.isUnit
#align mul_eq_one_iff_eq_inv₀ mul_eq_one_iff_eq_inv₀
theorem mul_eq_one_iff_inv_eq₀ (ha : a ≠ 0) : a * b = 1 ↔ a⁻¹ = b :=
IsUnit.mul_eq_one_iff_inv_eq ha.isUnit
#align mul_eq_one_iff_inv_eq₀ mul_eq_one_iff_inv_eq₀
@[simp]
theorem div_mul_cancel (a : G₀) (h : b ≠ 0) : a / b * b = a :=
IsUnit.div_mul_cancel h.isUnit _
#align div_mul_cancel div_mul_cancel
@[simp]
theorem mul_div_cancel (a : G₀) (h : b ≠ 0) : a * b / b = a :=
IsUnit.mul_div_cancel h.isUnit _
#align mul_div_cancel mul_div_cancel
theorem mul_one_div_cancel (h : a ≠ 0) : a * (1 / a) = 1 :=
IsUnit.mul_one_div_cancel h.isUnit
#align mul_one_div_cancel mul_one_div_cancel
theorem one_div_mul_cancel (h : a ≠ 0) : 1 / a * a = 1 :=
IsUnit.one_div_mul_cancel h.isUnit
#align one_div_mul_cancel one_div_mul_cancel
theorem div_left_inj' (hc : c ≠ 0) : a / c = b / c ↔ a = b :=
IsUnit.div_left_inj hc.isUnit
#align div_left_inj' div_left_inj'
@[field_simps]
theorem div_eq_iff (hb : b ≠ 0) : a / b = c ↔ a = c * b :=
IsUnit.div_eq_iff hb.isUnit
#align div_eq_iff div_eq_iff
@[field_simps]
theorem eq_div_iff (hb : b ≠ 0) : c = a / b ↔ c * b = a :=
IsUnit.eq_div_iff hb.isUnit
#align eq_div_iff eq_div_iff
theorem div_eq_iff_mul_eq (hb : b ≠ 0) : a / b = c ↔ c * b = a :=
(IsUnit.div_eq_iff hb.isUnit).trans eq_comm
#align div_eq_iff_mul_eq div_eq_iff_mul_eq
theorem eq_div_iff_mul_eq (hc : c ≠ 0) : a = b / c ↔ a * c = b :=
IsUnit.eq_div_iff hc.isUnit
#align eq_div_iff_mul_eq eq_div_iff_mul_eq
theorem div_eq_of_eq_mul (hb : b ≠ 0) : a = c * b → a / b = c :=
IsUnit.div_eq_of_eq_mul hb.isUnit
#align div_eq_of_eq_mul div_eq_of_eq_mul
theorem eq_div_of_mul_eq (hc : c ≠ 0) : a * c = b → a = b / c :=
IsUnit.eq_div_of_mul_eq hc.isUnit
#align eq_div_of_mul_eq eq_div_of_mul_eq
theorem div_eq_one_iff_eq (hb : b ≠ 0) : a / b = 1 ↔ a = b :=
IsUnit.div_eq_one_iff_eq hb.isUnit
#align div_eq_one_iff_eq div_eq_one_iff_eq
theorem div_mul_left (hb : b ≠ 0) : b / (a * b) = 1 / a :=
IsUnit.div_mul_left hb.isUnit
#align div_mul_left div_mul_left
theorem mul_div_mul_right (a b : G₀) (hc : c ≠ 0) : a * c / (b * c) = a / b :=
IsUnit.mul_div_mul_right hc.isUnit _ _
#align mul_div_mul_right mul_div_mul_right
theorem mul_mul_div (a : G₀) (hb : b ≠ 0) : a = a * b * (1 / b) :=
(IsUnit.mul_mul_div _ hb.isUnit).symm
#align mul_mul_div mul_mul_div
theorem div_div_div_cancel_right (a : G₀) (hc : c ≠ 0) : a / c / (b / c) = a / b := by
rw [div_div_eq_mul_div, div_mul_cancel _ hc]
#align div_div_div_cancel_right div_div_div_cancel_right
theorem div_mul_div_cancel (a : G₀) (hc : c ≠ 0) : a / c * (c / b) = a / b := by
rw [← mul_div_assoc, div_mul_cancel _ hc]
#align div_mul_div_cancel div_mul_div_cancel
theorem div_mul_cancel_of_imp {a b : G₀} (h : b = 0 → a = 0) : a / b * b = a :=
Classical.by_cases (fun hb : b = 0 => by simp [*]) (div_mul_cancel a)
#align div_mul_cancel_of_imp div_mul_cancel_of_imp
theorem mul_div_cancel_of_imp {a b : G₀} (h : b = 0 → a = 0) : a * b / b = a :=
Classical.by_cases (fun hb : b = 0 => by simp [*]) (mul_div_cancel a)
#align mul_div_cancel_of_imp mul_div_cancel_of_imp
@[simp]
theorem divp_mk0 (a : G₀) {b : G₀} (hb : b ≠ 0) : a /ₚ Units.mk0 b hb = a / b :=
divp_eq_div _ _
#align divp_mk0 divp_mk0
end GroupWithZero
section CommGroupWithZero
-- comm
variable [CommGroupWithZero G₀] {a b c d : G₀}
theorem div_mul_right (b : G₀) (ha : a ≠ 0) : a / (a * b) = 1 / b :=
IsUnit.div_mul_right ha.isUnit _
#align div_mul_right div_mul_right
theorem mul_div_cancel_left_of_imp {a b : G₀} (h : a = 0 → b = 0) : a * b / a = b := by
rw [mul_comm, mul_div_cancel_of_imp h]
#align mul_div_cancel_left_of_imp mul_div_cancel_left_of_imp
theorem mul_div_cancel_left (b : G₀) (ha : a ≠ 0) : a * b / a = b :=
IsUnit.mul_div_cancel_left ha.isUnit _
#align mul_div_cancel_left mul_div_cancel_left
theorem mul_div_cancel_of_imp' {a b : G₀} (h : b = 0 → a = 0) : b * (a / b) = a := by
|
rw [mul_comm, div_mul_cancel_of_imp h]
|
theorem mul_div_cancel_of_imp' {a b : G₀} (h : b = 0 → a = 0) : b * (a / b) = a := by
|
Mathlib.Algebra.GroupWithZero.Units.Lemmas.169_0.ICkvbDoLYHVogsB
|
theorem mul_div_cancel_of_imp' {a b : G₀} (h : b = 0 → a = 0) : b * (a / b) = a
|
Mathlib_Algebra_GroupWithZero_Units_Lemmas
|
α : Type u_1
M₀ : Type u_2
G₀ : Type u_3
M₀' : Type u_4
G₀' : Type u_5
F : Type u_6
F' : Type u_7
inst✝¹ : MonoidWithZero M₀
inst✝ : CommGroupWithZero G₀
a✝ b✝ c✝ d✝ a b c d : G₀
hb : b ≠ 0
hd : d ≠ 0
h : a / b = c / d
⊢ a * d = c * b
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Hom.Basic
import Mathlib.Algebra.Group.Units.Hom
import Mathlib.Algebra.GroupWithZero.Commute
import Mathlib.Algebra.GroupWithZero.Units.Basic
import Mathlib.GroupTheory.GroupAction.Units
#align_import algebra.group_with_zero.units.lemmas from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
/-!
# Further lemmas about units in a `MonoidWithZero` or a `GroupWithZero`.
-/
variable {α M₀ G₀ M₀' G₀' F F' : Type*}
variable [MonoidWithZero M₀]
section GroupWithZero
variable [GroupWithZero G₀] {a b c : G₀}
@[simp]
theorem div_self (h : a ≠ 0) : a / a = 1 :=
IsUnit.div_self h.isUnit
#align div_self div_self
theorem eq_mul_inv_iff_mul_eq₀ (hc : c ≠ 0) : a = b * c⁻¹ ↔ a * c = b :=
IsUnit.eq_mul_inv_iff_mul_eq hc.isUnit
#align eq_mul_inv_iff_mul_eq₀ eq_mul_inv_iff_mul_eq₀
theorem eq_inv_mul_iff_mul_eq₀ (hb : b ≠ 0) : a = b⁻¹ * c ↔ b * a = c :=
IsUnit.eq_inv_mul_iff_mul_eq hb.isUnit
#align eq_inv_mul_iff_mul_eq₀ eq_inv_mul_iff_mul_eq₀
theorem inv_mul_eq_iff_eq_mul₀ (ha : a ≠ 0) : a⁻¹ * b = c ↔ b = a * c :=
IsUnit.inv_mul_eq_iff_eq_mul ha.isUnit
#align inv_mul_eq_iff_eq_mul₀ inv_mul_eq_iff_eq_mul₀
theorem mul_inv_eq_iff_eq_mul₀ (hb : b ≠ 0) : a * b⁻¹ = c ↔ a = c * b :=
IsUnit.mul_inv_eq_iff_eq_mul hb.isUnit
#align mul_inv_eq_iff_eq_mul₀ mul_inv_eq_iff_eq_mul₀
theorem mul_inv_eq_one₀ (hb : b ≠ 0) : a * b⁻¹ = 1 ↔ a = b :=
IsUnit.mul_inv_eq_one hb.isUnit
#align mul_inv_eq_one₀ mul_inv_eq_one₀
theorem inv_mul_eq_one₀ (ha : a ≠ 0) : a⁻¹ * b = 1 ↔ a = b :=
IsUnit.inv_mul_eq_one ha.isUnit
#align inv_mul_eq_one₀ inv_mul_eq_one₀
theorem mul_eq_one_iff_eq_inv₀ (hb : b ≠ 0) : a * b = 1 ↔ a = b⁻¹ :=
IsUnit.mul_eq_one_iff_eq_inv hb.isUnit
#align mul_eq_one_iff_eq_inv₀ mul_eq_one_iff_eq_inv₀
theorem mul_eq_one_iff_inv_eq₀ (ha : a ≠ 0) : a * b = 1 ↔ a⁻¹ = b :=
IsUnit.mul_eq_one_iff_inv_eq ha.isUnit
#align mul_eq_one_iff_inv_eq₀ mul_eq_one_iff_inv_eq₀
@[simp]
theorem div_mul_cancel (a : G₀) (h : b ≠ 0) : a / b * b = a :=
IsUnit.div_mul_cancel h.isUnit _
#align div_mul_cancel div_mul_cancel
@[simp]
theorem mul_div_cancel (a : G₀) (h : b ≠ 0) : a * b / b = a :=
IsUnit.mul_div_cancel h.isUnit _
#align mul_div_cancel mul_div_cancel
theorem mul_one_div_cancel (h : a ≠ 0) : a * (1 / a) = 1 :=
IsUnit.mul_one_div_cancel h.isUnit
#align mul_one_div_cancel mul_one_div_cancel
theorem one_div_mul_cancel (h : a ≠ 0) : 1 / a * a = 1 :=
IsUnit.one_div_mul_cancel h.isUnit
#align one_div_mul_cancel one_div_mul_cancel
theorem div_left_inj' (hc : c ≠ 0) : a / c = b / c ↔ a = b :=
IsUnit.div_left_inj hc.isUnit
#align div_left_inj' div_left_inj'
@[field_simps]
theorem div_eq_iff (hb : b ≠ 0) : a / b = c ↔ a = c * b :=
IsUnit.div_eq_iff hb.isUnit
#align div_eq_iff div_eq_iff
@[field_simps]
theorem eq_div_iff (hb : b ≠ 0) : c = a / b ↔ c * b = a :=
IsUnit.eq_div_iff hb.isUnit
#align eq_div_iff eq_div_iff
theorem div_eq_iff_mul_eq (hb : b ≠ 0) : a / b = c ↔ c * b = a :=
(IsUnit.div_eq_iff hb.isUnit).trans eq_comm
#align div_eq_iff_mul_eq div_eq_iff_mul_eq
theorem eq_div_iff_mul_eq (hc : c ≠ 0) : a = b / c ↔ a * c = b :=
IsUnit.eq_div_iff hc.isUnit
#align eq_div_iff_mul_eq eq_div_iff_mul_eq
theorem div_eq_of_eq_mul (hb : b ≠ 0) : a = c * b → a / b = c :=
IsUnit.div_eq_of_eq_mul hb.isUnit
#align div_eq_of_eq_mul div_eq_of_eq_mul
theorem eq_div_of_mul_eq (hc : c ≠ 0) : a * c = b → a = b / c :=
IsUnit.eq_div_of_mul_eq hc.isUnit
#align eq_div_of_mul_eq eq_div_of_mul_eq
theorem div_eq_one_iff_eq (hb : b ≠ 0) : a / b = 1 ↔ a = b :=
IsUnit.div_eq_one_iff_eq hb.isUnit
#align div_eq_one_iff_eq div_eq_one_iff_eq
theorem div_mul_left (hb : b ≠ 0) : b / (a * b) = 1 / a :=
IsUnit.div_mul_left hb.isUnit
#align div_mul_left div_mul_left
theorem mul_div_mul_right (a b : G₀) (hc : c ≠ 0) : a * c / (b * c) = a / b :=
IsUnit.mul_div_mul_right hc.isUnit _ _
#align mul_div_mul_right mul_div_mul_right
theorem mul_mul_div (a : G₀) (hb : b ≠ 0) : a = a * b * (1 / b) :=
(IsUnit.mul_mul_div _ hb.isUnit).symm
#align mul_mul_div mul_mul_div
theorem div_div_div_cancel_right (a : G₀) (hc : c ≠ 0) : a / c / (b / c) = a / b := by
rw [div_div_eq_mul_div, div_mul_cancel _ hc]
#align div_div_div_cancel_right div_div_div_cancel_right
theorem div_mul_div_cancel (a : G₀) (hc : c ≠ 0) : a / c * (c / b) = a / b := by
rw [← mul_div_assoc, div_mul_cancel _ hc]
#align div_mul_div_cancel div_mul_div_cancel
theorem div_mul_cancel_of_imp {a b : G₀} (h : b = 0 → a = 0) : a / b * b = a :=
Classical.by_cases (fun hb : b = 0 => by simp [*]) (div_mul_cancel a)
#align div_mul_cancel_of_imp div_mul_cancel_of_imp
theorem mul_div_cancel_of_imp {a b : G₀} (h : b = 0 → a = 0) : a * b / b = a :=
Classical.by_cases (fun hb : b = 0 => by simp [*]) (mul_div_cancel a)
#align mul_div_cancel_of_imp mul_div_cancel_of_imp
@[simp]
theorem divp_mk0 (a : G₀) {b : G₀} (hb : b ≠ 0) : a /ₚ Units.mk0 b hb = a / b :=
divp_eq_div _ _
#align divp_mk0 divp_mk0
end GroupWithZero
section CommGroupWithZero
-- comm
variable [CommGroupWithZero G₀] {a b c d : G₀}
theorem div_mul_right (b : G₀) (ha : a ≠ 0) : a / (a * b) = 1 / b :=
IsUnit.div_mul_right ha.isUnit _
#align div_mul_right div_mul_right
theorem mul_div_cancel_left_of_imp {a b : G₀} (h : a = 0 → b = 0) : a * b / a = b := by
rw [mul_comm, mul_div_cancel_of_imp h]
#align mul_div_cancel_left_of_imp mul_div_cancel_left_of_imp
theorem mul_div_cancel_left (b : G₀) (ha : a ≠ 0) : a * b / a = b :=
IsUnit.mul_div_cancel_left ha.isUnit _
#align mul_div_cancel_left mul_div_cancel_left
theorem mul_div_cancel_of_imp' {a b : G₀} (h : b = 0 → a = 0) : b * (a / b) = a := by
rw [mul_comm, div_mul_cancel_of_imp h]
#align mul_div_cancel_of_imp' mul_div_cancel_of_imp'
theorem mul_div_cancel' (a : G₀) (hb : b ≠ 0) : b * (a / b) = a :=
IsUnit.mul_div_cancel' hb.isUnit _
#align mul_div_cancel' mul_div_cancel'
theorem mul_div_mul_left (a b : G₀) (hc : c ≠ 0) : c * a / (c * b) = a / b :=
IsUnit.mul_div_mul_left hc.isUnit _ _
#align mul_div_mul_left mul_div_mul_left
theorem mul_eq_mul_of_div_eq_div (a : G₀) {b : G₀} (c : G₀) {d : G₀} (hb : b ≠ 0) (hd : d ≠ 0)
(h : a / b = c / d) : a * d = c * b := by
|
rw [← mul_one a, ← div_self hb, ← mul_comm_div, h, div_mul_eq_mul_div, div_mul_cancel _ hd]
|
theorem mul_eq_mul_of_div_eq_div (a : G₀) {b : G₀} (c : G₀) {d : G₀} (hb : b ≠ 0) (hd : d ≠ 0)
(h : a / b = c / d) : a * d = c * b := by
|
Mathlib.Algebra.GroupWithZero.Units.Lemmas.181_0.ICkvbDoLYHVogsB
|
theorem mul_eq_mul_of_div_eq_div (a : G₀) {b : G₀} (c : G₀) {d : G₀} (hb : b ≠ 0) (hd : d ≠ 0)
(h : a / b = c / d) : a * d = c * b
|
Mathlib_Algebra_GroupWithZero_Units_Lemmas
|
α : Type u_1
M₀ : Type u_2
G₀ : Type u_3
M₀' : Type u_4
G₀' : Type u_5
F : Type u_6
F' : Type u_7
inst✝¹ : MonoidWithZero M₀
inst✝ : CommGroupWithZero G₀
a b✝ c d b : G₀
h : a ≠ 0
⊢ 1 / (a * b) * a = 1 / b
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Hom.Basic
import Mathlib.Algebra.Group.Units.Hom
import Mathlib.Algebra.GroupWithZero.Commute
import Mathlib.Algebra.GroupWithZero.Units.Basic
import Mathlib.GroupTheory.GroupAction.Units
#align_import algebra.group_with_zero.units.lemmas from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
/-!
# Further lemmas about units in a `MonoidWithZero` or a `GroupWithZero`.
-/
variable {α M₀ G₀ M₀' G₀' F F' : Type*}
variable [MonoidWithZero M₀]
section GroupWithZero
variable [GroupWithZero G₀] {a b c : G₀}
@[simp]
theorem div_self (h : a ≠ 0) : a / a = 1 :=
IsUnit.div_self h.isUnit
#align div_self div_self
theorem eq_mul_inv_iff_mul_eq₀ (hc : c ≠ 0) : a = b * c⁻¹ ↔ a * c = b :=
IsUnit.eq_mul_inv_iff_mul_eq hc.isUnit
#align eq_mul_inv_iff_mul_eq₀ eq_mul_inv_iff_mul_eq₀
theorem eq_inv_mul_iff_mul_eq₀ (hb : b ≠ 0) : a = b⁻¹ * c ↔ b * a = c :=
IsUnit.eq_inv_mul_iff_mul_eq hb.isUnit
#align eq_inv_mul_iff_mul_eq₀ eq_inv_mul_iff_mul_eq₀
theorem inv_mul_eq_iff_eq_mul₀ (ha : a ≠ 0) : a⁻¹ * b = c ↔ b = a * c :=
IsUnit.inv_mul_eq_iff_eq_mul ha.isUnit
#align inv_mul_eq_iff_eq_mul₀ inv_mul_eq_iff_eq_mul₀
theorem mul_inv_eq_iff_eq_mul₀ (hb : b ≠ 0) : a * b⁻¹ = c ↔ a = c * b :=
IsUnit.mul_inv_eq_iff_eq_mul hb.isUnit
#align mul_inv_eq_iff_eq_mul₀ mul_inv_eq_iff_eq_mul₀
theorem mul_inv_eq_one₀ (hb : b ≠ 0) : a * b⁻¹ = 1 ↔ a = b :=
IsUnit.mul_inv_eq_one hb.isUnit
#align mul_inv_eq_one₀ mul_inv_eq_one₀
theorem inv_mul_eq_one₀ (ha : a ≠ 0) : a⁻¹ * b = 1 ↔ a = b :=
IsUnit.inv_mul_eq_one ha.isUnit
#align inv_mul_eq_one₀ inv_mul_eq_one₀
theorem mul_eq_one_iff_eq_inv₀ (hb : b ≠ 0) : a * b = 1 ↔ a = b⁻¹ :=
IsUnit.mul_eq_one_iff_eq_inv hb.isUnit
#align mul_eq_one_iff_eq_inv₀ mul_eq_one_iff_eq_inv₀
theorem mul_eq_one_iff_inv_eq₀ (ha : a ≠ 0) : a * b = 1 ↔ a⁻¹ = b :=
IsUnit.mul_eq_one_iff_inv_eq ha.isUnit
#align mul_eq_one_iff_inv_eq₀ mul_eq_one_iff_inv_eq₀
@[simp]
theorem div_mul_cancel (a : G₀) (h : b ≠ 0) : a / b * b = a :=
IsUnit.div_mul_cancel h.isUnit _
#align div_mul_cancel div_mul_cancel
@[simp]
theorem mul_div_cancel (a : G₀) (h : b ≠ 0) : a * b / b = a :=
IsUnit.mul_div_cancel h.isUnit _
#align mul_div_cancel mul_div_cancel
theorem mul_one_div_cancel (h : a ≠ 0) : a * (1 / a) = 1 :=
IsUnit.mul_one_div_cancel h.isUnit
#align mul_one_div_cancel mul_one_div_cancel
theorem one_div_mul_cancel (h : a ≠ 0) : 1 / a * a = 1 :=
IsUnit.one_div_mul_cancel h.isUnit
#align one_div_mul_cancel one_div_mul_cancel
theorem div_left_inj' (hc : c ≠ 0) : a / c = b / c ↔ a = b :=
IsUnit.div_left_inj hc.isUnit
#align div_left_inj' div_left_inj'
@[field_simps]
theorem div_eq_iff (hb : b ≠ 0) : a / b = c ↔ a = c * b :=
IsUnit.div_eq_iff hb.isUnit
#align div_eq_iff div_eq_iff
@[field_simps]
theorem eq_div_iff (hb : b ≠ 0) : c = a / b ↔ c * b = a :=
IsUnit.eq_div_iff hb.isUnit
#align eq_div_iff eq_div_iff
theorem div_eq_iff_mul_eq (hb : b ≠ 0) : a / b = c ↔ c * b = a :=
(IsUnit.div_eq_iff hb.isUnit).trans eq_comm
#align div_eq_iff_mul_eq div_eq_iff_mul_eq
theorem eq_div_iff_mul_eq (hc : c ≠ 0) : a = b / c ↔ a * c = b :=
IsUnit.eq_div_iff hc.isUnit
#align eq_div_iff_mul_eq eq_div_iff_mul_eq
theorem div_eq_of_eq_mul (hb : b ≠ 0) : a = c * b → a / b = c :=
IsUnit.div_eq_of_eq_mul hb.isUnit
#align div_eq_of_eq_mul div_eq_of_eq_mul
theorem eq_div_of_mul_eq (hc : c ≠ 0) : a * c = b → a = b / c :=
IsUnit.eq_div_of_mul_eq hc.isUnit
#align eq_div_of_mul_eq eq_div_of_mul_eq
theorem div_eq_one_iff_eq (hb : b ≠ 0) : a / b = 1 ↔ a = b :=
IsUnit.div_eq_one_iff_eq hb.isUnit
#align div_eq_one_iff_eq div_eq_one_iff_eq
theorem div_mul_left (hb : b ≠ 0) : b / (a * b) = 1 / a :=
IsUnit.div_mul_left hb.isUnit
#align div_mul_left div_mul_left
theorem mul_div_mul_right (a b : G₀) (hc : c ≠ 0) : a * c / (b * c) = a / b :=
IsUnit.mul_div_mul_right hc.isUnit _ _
#align mul_div_mul_right mul_div_mul_right
theorem mul_mul_div (a : G₀) (hb : b ≠ 0) : a = a * b * (1 / b) :=
(IsUnit.mul_mul_div _ hb.isUnit).symm
#align mul_mul_div mul_mul_div
theorem div_div_div_cancel_right (a : G₀) (hc : c ≠ 0) : a / c / (b / c) = a / b := by
rw [div_div_eq_mul_div, div_mul_cancel _ hc]
#align div_div_div_cancel_right div_div_div_cancel_right
theorem div_mul_div_cancel (a : G₀) (hc : c ≠ 0) : a / c * (c / b) = a / b := by
rw [← mul_div_assoc, div_mul_cancel _ hc]
#align div_mul_div_cancel div_mul_div_cancel
theorem div_mul_cancel_of_imp {a b : G₀} (h : b = 0 → a = 0) : a / b * b = a :=
Classical.by_cases (fun hb : b = 0 => by simp [*]) (div_mul_cancel a)
#align div_mul_cancel_of_imp div_mul_cancel_of_imp
theorem mul_div_cancel_of_imp {a b : G₀} (h : b = 0 → a = 0) : a * b / b = a :=
Classical.by_cases (fun hb : b = 0 => by simp [*]) (mul_div_cancel a)
#align mul_div_cancel_of_imp mul_div_cancel_of_imp
@[simp]
theorem divp_mk0 (a : G₀) {b : G₀} (hb : b ≠ 0) : a /ₚ Units.mk0 b hb = a / b :=
divp_eq_div _ _
#align divp_mk0 divp_mk0
end GroupWithZero
section CommGroupWithZero
-- comm
variable [CommGroupWithZero G₀] {a b c d : G₀}
theorem div_mul_right (b : G₀) (ha : a ≠ 0) : a / (a * b) = 1 / b :=
IsUnit.div_mul_right ha.isUnit _
#align div_mul_right div_mul_right
theorem mul_div_cancel_left_of_imp {a b : G₀} (h : a = 0 → b = 0) : a * b / a = b := by
rw [mul_comm, mul_div_cancel_of_imp h]
#align mul_div_cancel_left_of_imp mul_div_cancel_left_of_imp
theorem mul_div_cancel_left (b : G₀) (ha : a ≠ 0) : a * b / a = b :=
IsUnit.mul_div_cancel_left ha.isUnit _
#align mul_div_cancel_left mul_div_cancel_left
theorem mul_div_cancel_of_imp' {a b : G₀} (h : b = 0 → a = 0) : b * (a / b) = a := by
rw [mul_comm, div_mul_cancel_of_imp h]
#align mul_div_cancel_of_imp' mul_div_cancel_of_imp'
theorem mul_div_cancel' (a : G₀) (hb : b ≠ 0) : b * (a / b) = a :=
IsUnit.mul_div_cancel' hb.isUnit _
#align mul_div_cancel' mul_div_cancel'
theorem mul_div_mul_left (a b : G₀) (hc : c ≠ 0) : c * a / (c * b) = a / b :=
IsUnit.mul_div_mul_left hc.isUnit _ _
#align mul_div_mul_left mul_div_mul_left
theorem mul_eq_mul_of_div_eq_div (a : G₀) {b : G₀} (c : G₀) {d : G₀} (hb : b ≠ 0) (hd : d ≠ 0)
(h : a / b = c / d) : a * d = c * b := by
rw [← mul_one a, ← div_self hb, ← mul_comm_div, h, div_mul_eq_mul_div, div_mul_cancel _ hd]
#align mul_eq_mul_of_div_eq_div mul_eq_mul_of_div_eq_div
@[field_simps]
theorem div_eq_div_iff (hb : b ≠ 0) (hd : d ≠ 0) : a / b = c / d ↔ a * d = c * b :=
IsUnit.div_eq_div_iff hb.isUnit hd.isUnit
#align div_eq_div_iff div_eq_div_iff
theorem div_div_cancel' (ha : a ≠ 0) : a / (a / b) = b :=
IsUnit.div_div_cancel ha.isUnit
#align div_div_cancel' div_div_cancel'
theorem div_div_cancel_left' (ha : a ≠ 0) : a / b / a = b⁻¹ :=
ha.isUnit.div_div_cancel_left
#align div_div_cancel_left' div_div_cancel_left'
theorem div_helper (b : G₀) (h : a ≠ 0) : 1 / (a * b) * a = 1 / b := by
|
rw [div_mul_eq_mul_div, one_mul, div_mul_right _ h]
|
theorem div_helper (b : G₀) (h : a ≠ 0) : 1 / (a * b) * a = 1 / b := by
|
Mathlib.Algebra.GroupWithZero.Units.Lemmas.199_0.ICkvbDoLYHVogsB
|
theorem div_helper (b : G₀) (h : a ≠ 0) : 1 / (a * b) * a = 1 / b
|
Mathlib_Algebra_GroupWithZero_Units_Lemmas
|
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