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case a.mk
๐ : Type u_1
E : Type u_2
F : Type u_3
instโโด : IsROrC ๐
instโยณ : NormedAddCommGroup E
instโยฒ : InnerProductSpace ๐ E
instโยน : NormedAddCommGroup F
instโ : InnerProductSpace ๐ F
K : Submodule ๐ E
v : E
hv : v โ K
w : E
hw : w โ Kแฎ
โข w โ LinearMap.ker ((innerSL ๐) โ{ val := v, property := hv })
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sรฉbastien Gouรซzel, Frรฉdรฉric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
/-!
# Orthogonal complements of submodules
In this file, the `orthogonal` complement of a submodule `K` is defined, and basic API established.
Some of the more subtle results about the orthogonal complement are delayed to
`Analysis.InnerProductSpace.Projection`.
See also `BilinForm.orthogonal` for orthogonality with respect to a general bilinear form.
## Notation
The orthogonal complement of a submodule `K` is denoted by `Kแฎ`.
The proposition that two submodules are orthogonal, `Submodule.IsOrtho`, is denoted by `U โ V`.
Note this is not the same unicode symbol as `โฅ` (`Bot`).
-/
variable {๐ E F : Type*} [IsROrC ๐]
variable [NormedAddCommGroup E] [InnerProductSpace ๐ E]
variable [NormedAddCommGroup F] [InnerProductSpace ๐ F]
local notation "โช" x ", " y "โซ" => @inner ๐ _ _ x y
namespace Submodule
variable (K : Submodule ๐ E)
/-- The subspace of vectors orthogonal to a given subspace. -/
def orthogonal : Submodule ๐ E where
carrier := { v | โ u โ K, โชu, vโซ = 0 }
zero_mem' _ _ := inner_zero_right _
add_mem' hx hy u hu := by rw [inner_add_right, hx u hu, hy u hu, add_zero]
smul_mem' c x hx u hu := by rw [inner_smul_right, hx u hu, mul_zero]
#align submodule.orthogonal Submodule.orthogonal
@[inherit_doc]
notation:1200 K "แฎ" => orthogonal K
/-- When a vector is in `Kแฎ`. -/
theorem mem_orthogonal (v : E) : v โ Kแฎ โ โ u โ K, โชu, vโซ = 0 :=
Iff.rfl
#align submodule.mem_orthogonal Submodule.mem_orthogonal
/-- When a vector is in `Kแฎ`, with the inner product the
other way round. -/
theorem mem_orthogonal' (v : E) : v โ Kแฎ โ โ u โ K, โชv, uโซ = 0 := by
simp_rw [mem_orthogonal, inner_eq_zero_symm]
#align submodule.mem_orthogonal' Submodule.mem_orthogonal'
variable {K}
/-- A vector in `K` is orthogonal to one in `Kแฎ`. -/
theorem inner_right_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชu, vโซ = 0 :=
(K.mem_orthogonal v).1 hv u hu
#align submodule.inner_right_of_mem_orthogonal Submodule.inner_right_of_mem_orthogonal
/-- A vector in `Kแฎ` is orthogonal to one in `K`. -/
theorem inner_left_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชv, uโซ = 0 := by
rw [inner_eq_zero_symm]; exact inner_right_of_mem_orthogonal hu hv
#align submodule.inner_left_of_mem_orthogonal Submodule.inner_left_of_mem_orthogonal
/-- A vector is in `(๐ โ u)แฎ` iff it is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_right {u v : E} : v โ (๐ โ u)แฎ โ โชu, vโซ = 0 := by
refine' โจinner_right_of_mem_orthogonal (mem_span_singleton_self u), _โฉ
intro hv w hw
rw [mem_span_singleton] at hw
obtain โจc, rflโฉ := hw
simp [inner_smul_left, hv]
#align submodule.mem_orthogonal_singleton_iff_inner_right Submodule.mem_orthogonal_singleton_iff_inner_right
/-- A vector in `(๐ โ u)แฎ` is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_left {u v : E} : v โ (๐ โ u)แฎ โ โชv, uโซ = 0 := by
rw [mem_orthogonal_singleton_iff_inner_right, inner_eq_zero_symm]
#align submodule.mem_orthogonal_singleton_iff_inner_left Submodule.mem_orthogonal_singleton_iff_inner_left
theorem sub_mem_orthogonal_of_inner_left {x y : E} (h : โ v : K, โชx, vโซ = โชy, vโซ) : x - y โ Kแฎ := by
rw [mem_orthogonal']
intro u hu
rw [inner_sub_left, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_left Submodule.sub_mem_orthogonal_of_inner_left
theorem sub_mem_orthogonal_of_inner_right {x y : E} (h : โ v : K, โช(v : E), xโซ = โช(v : E), yโซ) :
x - y โ Kแฎ := by
intro u hu
rw [inner_sub_right, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_right Submodule.sub_mem_orthogonal_of_inner_right
variable (K)
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem inf_orthogonal_eq_bot : K โ Kแฎ = โฅ := by
rw [eq_bot_iff]
intro x
rw [mem_inf]
exact fun โจhx, hoโฉ => inner_self_eq_zero.1 (ho x hx)
#align submodule.inf_orthogonal_eq_bot Submodule.inf_orthogonal_eq_bot
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem orthogonal_disjoint : Disjoint K Kแฎ := by simp [disjoint_iff, K.inf_orthogonal_eq_bot]
#align submodule.orthogonal_disjoint Submodule.orthogonal_disjoint
/-- `Kแฎ` can be characterized as the intersection of the kernels of the operations of
inner product with each of the elements of `K`. -/
theorem orthogonal_eq_inter : Kแฎ = โจ
v : K, LinearMap.ker (innerSL ๐ (v : E)) := by
apply le_antisymm
ยท rw [le_iInf_iff]
rintro โจv, hvโฉ w hw
|
simpa using hw _ hv
|
/-- `Kแฎ` can be characterized as the intersection of the kernels of the operations of
inner product with each of the elements of `K`. -/
theorem orthogonal_eq_inter : Kแฎ = โจ
v : K, LinearMap.ker (innerSL ๐ (v : E)) := by
apply le_antisymm
ยท rw [le_iInf_iff]
rintro โจv, hvโฉ w hw
|
Mathlib.Analysis.InnerProductSpace.Orthogonal.116_0.QXx0GYqLoAbtfq1
|
/-- `Kแฎ` can be characterized as the intersection of the kernels of the operations of
inner product with each of the elements of `K`. -/
theorem orthogonal_eq_inter : Kแฎ = โจ
v : K, LinearMap.ker (innerSL ๐ (v : E))
|
Mathlib_Analysis_InnerProductSpace_Orthogonal
|
case a
๐ : Type u_1
E : Type u_2
F : Type u_3
instโโด : IsROrC ๐
instโยณ : NormedAddCommGroup E
instโยฒ : InnerProductSpace ๐ E
instโยน : NormedAddCommGroup F
instโ : InnerProductSpace ๐ F
K : Submodule ๐ E
โข โจ
v, LinearMap.ker ((innerSL ๐) โv) โค Kแฎ
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sรฉbastien Gouรซzel, Frรฉdรฉric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
/-!
# Orthogonal complements of submodules
In this file, the `orthogonal` complement of a submodule `K` is defined, and basic API established.
Some of the more subtle results about the orthogonal complement are delayed to
`Analysis.InnerProductSpace.Projection`.
See also `BilinForm.orthogonal` for orthogonality with respect to a general bilinear form.
## Notation
The orthogonal complement of a submodule `K` is denoted by `Kแฎ`.
The proposition that two submodules are orthogonal, `Submodule.IsOrtho`, is denoted by `U โ V`.
Note this is not the same unicode symbol as `โฅ` (`Bot`).
-/
variable {๐ E F : Type*} [IsROrC ๐]
variable [NormedAddCommGroup E] [InnerProductSpace ๐ E]
variable [NormedAddCommGroup F] [InnerProductSpace ๐ F]
local notation "โช" x ", " y "โซ" => @inner ๐ _ _ x y
namespace Submodule
variable (K : Submodule ๐ E)
/-- The subspace of vectors orthogonal to a given subspace. -/
def orthogonal : Submodule ๐ E where
carrier := { v | โ u โ K, โชu, vโซ = 0 }
zero_mem' _ _ := inner_zero_right _
add_mem' hx hy u hu := by rw [inner_add_right, hx u hu, hy u hu, add_zero]
smul_mem' c x hx u hu := by rw [inner_smul_right, hx u hu, mul_zero]
#align submodule.orthogonal Submodule.orthogonal
@[inherit_doc]
notation:1200 K "แฎ" => orthogonal K
/-- When a vector is in `Kแฎ`. -/
theorem mem_orthogonal (v : E) : v โ Kแฎ โ โ u โ K, โชu, vโซ = 0 :=
Iff.rfl
#align submodule.mem_orthogonal Submodule.mem_orthogonal
/-- When a vector is in `Kแฎ`, with the inner product the
other way round. -/
theorem mem_orthogonal' (v : E) : v โ Kแฎ โ โ u โ K, โชv, uโซ = 0 := by
simp_rw [mem_orthogonal, inner_eq_zero_symm]
#align submodule.mem_orthogonal' Submodule.mem_orthogonal'
variable {K}
/-- A vector in `K` is orthogonal to one in `Kแฎ`. -/
theorem inner_right_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชu, vโซ = 0 :=
(K.mem_orthogonal v).1 hv u hu
#align submodule.inner_right_of_mem_orthogonal Submodule.inner_right_of_mem_orthogonal
/-- A vector in `Kแฎ` is orthogonal to one in `K`. -/
theorem inner_left_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชv, uโซ = 0 := by
rw [inner_eq_zero_symm]; exact inner_right_of_mem_orthogonal hu hv
#align submodule.inner_left_of_mem_orthogonal Submodule.inner_left_of_mem_orthogonal
/-- A vector is in `(๐ โ u)แฎ` iff it is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_right {u v : E} : v โ (๐ โ u)แฎ โ โชu, vโซ = 0 := by
refine' โจinner_right_of_mem_orthogonal (mem_span_singleton_self u), _โฉ
intro hv w hw
rw [mem_span_singleton] at hw
obtain โจc, rflโฉ := hw
simp [inner_smul_left, hv]
#align submodule.mem_orthogonal_singleton_iff_inner_right Submodule.mem_orthogonal_singleton_iff_inner_right
/-- A vector in `(๐ โ u)แฎ` is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_left {u v : E} : v โ (๐ โ u)แฎ โ โชv, uโซ = 0 := by
rw [mem_orthogonal_singleton_iff_inner_right, inner_eq_zero_symm]
#align submodule.mem_orthogonal_singleton_iff_inner_left Submodule.mem_orthogonal_singleton_iff_inner_left
theorem sub_mem_orthogonal_of_inner_left {x y : E} (h : โ v : K, โชx, vโซ = โชy, vโซ) : x - y โ Kแฎ := by
rw [mem_orthogonal']
intro u hu
rw [inner_sub_left, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_left Submodule.sub_mem_orthogonal_of_inner_left
theorem sub_mem_orthogonal_of_inner_right {x y : E} (h : โ v : K, โช(v : E), xโซ = โช(v : E), yโซ) :
x - y โ Kแฎ := by
intro u hu
rw [inner_sub_right, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_right Submodule.sub_mem_orthogonal_of_inner_right
variable (K)
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem inf_orthogonal_eq_bot : K โ Kแฎ = โฅ := by
rw [eq_bot_iff]
intro x
rw [mem_inf]
exact fun โจhx, hoโฉ => inner_self_eq_zero.1 (ho x hx)
#align submodule.inf_orthogonal_eq_bot Submodule.inf_orthogonal_eq_bot
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem orthogonal_disjoint : Disjoint K Kแฎ := by simp [disjoint_iff, K.inf_orthogonal_eq_bot]
#align submodule.orthogonal_disjoint Submodule.orthogonal_disjoint
/-- `Kแฎ` can be characterized as the intersection of the kernels of the operations of
inner product with each of the elements of `K`. -/
theorem orthogonal_eq_inter : Kแฎ = โจ
v : K, LinearMap.ker (innerSL ๐ (v : E)) := by
apply le_antisymm
ยท rw [le_iInf_iff]
rintro โจv, hvโฉ w hw
simpa using hw _ hv
ยท
|
intro v hv w hw
|
/-- `Kแฎ` can be characterized as the intersection of the kernels of the operations of
inner product with each of the elements of `K`. -/
theorem orthogonal_eq_inter : Kแฎ = โจ
v : K, LinearMap.ker (innerSL ๐ (v : E)) := by
apply le_antisymm
ยท rw [le_iInf_iff]
rintro โจv, hvโฉ w hw
simpa using hw _ hv
ยท
|
Mathlib.Analysis.InnerProductSpace.Orthogonal.116_0.QXx0GYqLoAbtfq1
|
/-- `Kแฎ` can be characterized as the intersection of the kernels of the operations of
inner product with each of the elements of `K`. -/
theorem orthogonal_eq_inter : Kแฎ = โจ
v : K, LinearMap.ker (innerSL ๐ (v : E))
|
Mathlib_Analysis_InnerProductSpace_Orthogonal
|
case a
๐ : Type u_1
E : Type u_2
F : Type u_3
instโโด : IsROrC ๐
instโยณ : NormedAddCommGroup E
instโยฒ : InnerProductSpace ๐ E
instโยน : NormedAddCommGroup F
instโ : InnerProductSpace ๐ F
K : Submodule ๐ E
v : E
hv : v โ โจ
v, LinearMap.ker ((innerSL ๐) โv)
w : E
hw : w โ K
โข inner w v = 0
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sรฉbastien Gouรซzel, Frรฉdรฉric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
/-!
# Orthogonal complements of submodules
In this file, the `orthogonal` complement of a submodule `K` is defined, and basic API established.
Some of the more subtle results about the orthogonal complement are delayed to
`Analysis.InnerProductSpace.Projection`.
See also `BilinForm.orthogonal` for orthogonality with respect to a general bilinear form.
## Notation
The orthogonal complement of a submodule `K` is denoted by `Kแฎ`.
The proposition that two submodules are orthogonal, `Submodule.IsOrtho`, is denoted by `U โ V`.
Note this is not the same unicode symbol as `โฅ` (`Bot`).
-/
variable {๐ E F : Type*} [IsROrC ๐]
variable [NormedAddCommGroup E] [InnerProductSpace ๐ E]
variable [NormedAddCommGroup F] [InnerProductSpace ๐ F]
local notation "โช" x ", " y "โซ" => @inner ๐ _ _ x y
namespace Submodule
variable (K : Submodule ๐ E)
/-- The subspace of vectors orthogonal to a given subspace. -/
def orthogonal : Submodule ๐ E where
carrier := { v | โ u โ K, โชu, vโซ = 0 }
zero_mem' _ _ := inner_zero_right _
add_mem' hx hy u hu := by rw [inner_add_right, hx u hu, hy u hu, add_zero]
smul_mem' c x hx u hu := by rw [inner_smul_right, hx u hu, mul_zero]
#align submodule.orthogonal Submodule.orthogonal
@[inherit_doc]
notation:1200 K "แฎ" => orthogonal K
/-- When a vector is in `Kแฎ`. -/
theorem mem_orthogonal (v : E) : v โ Kแฎ โ โ u โ K, โชu, vโซ = 0 :=
Iff.rfl
#align submodule.mem_orthogonal Submodule.mem_orthogonal
/-- When a vector is in `Kแฎ`, with the inner product the
other way round. -/
theorem mem_orthogonal' (v : E) : v โ Kแฎ โ โ u โ K, โชv, uโซ = 0 := by
simp_rw [mem_orthogonal, inner_eq_zero_symm]
#align submodule.mem_orthogonal' Submodule.mem_orthogonal'
variable {K}
/-- A vector in `K` is orthogonal to one in `Kแฎ`. -/
theorem inner_right_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชu, vโซ = 0 :=
(K.mem_orthogonal v).1 hv u hu
#align submodule.inner_right_of_mem_orthogonal Submodule.inner_right_of_mem_orthogonal
/-- A vector in `Kแฎ` is orthogonal to one in `K`. -/
theorem inner_left_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชv, uโซ = 0 := by
rw [inner_eq_zero_symm]; exact inner_right_of_mem_orthogonal hu hv
#align submodule.inner_left_of_mem_orthogonal Submodule.inner_left_of_mem_orthogonal
/-- A vector is in `(๐ โ u)แฎ` iff it is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_right {u v : E} : v โ (๐ โ u)แฎ โ โชu, vโซ = 0 := by
refine' โจinner_right_of_mem_orthogonal (mem_span_singleton_self u), _โฉ
intro hv w hw
rw [mem_span_singleton] at hw
obtain โจc, rflโฉ := hw
simp [inner_smul_left, hv]
#align submodule.mem_orthogonal_singleton_iff_inner_right Submodule.mem_orthogonal_singleton_iff_inner_right
/-- A vector in `(๐ โ u)แฎ` is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_left {u v : E} : v โ (๐ โ u)แฎ โ โชv, uโซ = 0 := by
rw [mem_orthogonal_singleton_iff_inner_right, inner_eq_zero_symm]
#align submodule.mem_orthogonal_singleton_iff_inner_left Submodule.mem_orthogonal_singleton_iff_inner_left
theorem sub_mem_orthogonal_of_inner_left {x y : E} (h : โ v : K, โชx, vโซ = โชy, vโซ) : x - y โ Kแฎ := by
rw [mem_orthogonal']
intro u hu
rw [inner_sub_left, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_left Submodule.sub_mem_orthogonal_of_inner_left
theorem sub_mem_orthogonal_of_inner_right {x y : E} (h : โ v : K, โช(v : E), xโซ = โช(v : E), yโซ) :
x - y โ Kแฎ := by
intro u hu
rw [inner_sub_right, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_right Submodule.sub_mem_orthogonal_of_inner_right
variable (K)
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem inf_orthogonal_eq_bot : K โ Kแฎ = โฅ := by
rw [eq_bot_iff]
intro x
rw [mem_inf]
exact fun โจhx, hoโฉ => inner_self_eq_zero.1 (ho x hx)
#align submodule.inf_orthogonal_eq_bot Submodule.inf_orthogonal_eq_bot
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem orthogonal_disjoint : Disjoint K Kแฎ := by simp [disjoint_iff, K.inf_orthogonal_eq_bot]
#align submodule.orthogonal_disjoint Submodule.orthogonal_disjoint
/-- `Kแฎ` can be characterized as the intersection of the kernels of the operations of
inner product with each of the elements of `K`. -/
theorem orthogonal_eq_inter : Kแฎ = โจ
v : K, LinearMap.ker (innerSL ๐ (v : E)) := by
apply le_antisymm
ยท rw [le_iInf_iff]
rintro โจv, hvโฉ w hw
simpa using hw _ hv
ยท intro v hv w hw
|
simp only [mem_iInf] at hv
|
/-- `Kแฎ` can be characterized as the intersection of the kernels of the operations of
inner product with each of the elements of `K`. -/
theorem orthogonal_eq_inter : Kแฎ = โจ
v : K, LinearMap.ker (innerSL ๐ (v : E)) := by
apply le_antisymm
ยท rw [le_iInf_iff]
rintro โจv, hvโฉ w hw
simpa using hw _ hv
ยท intro v hv w hw
|
Mathlib.Analysis.InnerProductSpace.Orthogonal.116_0.QXx0GYqLoAbtfq1
|
/-- `Kแฎ` can be characterized as the intersection of the kernels of the operations of
inner product with each of the elements of `K`. -/
theorem orthogonal_eq_inter : Kแฎ = โจ
v : K, LinearMap.ker (innerSL ๐ (v : E))
|
Mathlib_Analysis_InnerProductSpace_Orthogonal
|
case a
๐ : Type u_1
E : Type u_2
F : Type u_3
instโโด : IsROrC ๐
instโยณ : NormedAddCommGroup E
instโยฒ : InnerProductSpace ๐ E
instโยน : NormedAddCommGroup F
instโ : InnerProductSpace ๐ F
K : Submodule ๐ E
v w : E
hw : w โ K
hv : โ (i : โฅK), v โ LinearMap.ker ((innerSL ๐) โi)
โข inner w v = 0
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sรฉbastien Gouรซzel, Frรฉdรฉric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
/-!
# Orthogonal complements of submodules
In this file, the `orthogonal` complement of a submodule `K` is defined, and basic API established.
Some of the more subtle results about the orthogonal complement are delayed to
`Analysis.InnerProductSpace.Projection`.
See also `BilinForm.orthogonal` for orthogonality with respect to a general bilinear form.
## Notation
The orthogonal complement of a submodule `K` is denoted by `Kแฎ`.
The proposition that two submodules are orthogonal, `Submodule.IsOrtho`, is denoted by `U โ V`.
Note this is not the same unicode symbol as `โฅ` (`Bot`).
-/
variable {๐ E F : Type*} [IsROrC ๐]
variable [NormedAddCommGroup E] [InnerProductSpace ๐ E]
variable [NormedAddCommGroup F] [InnerProductSpace ๐ F]
local notation "โช" x ", " y "โซ" => @inner ๐ _ _ x y
namespace Submodule
variable (K : Submodule ๐ E)
/-- The subspace of vectors orthogonal to a given subspace. -/
def orthogonal : Submodule ๐ E where
carrier := { v | โ u โ K, โชu, vโซ = 0 }
zero_mem' _ _ := inner_zero_right _
add_mem' hx hy u hu := by rw [inner_add_right, hx u hu, hy u hu, add_zero]
smul_mem' c x hx u hu := by rw [inner_smul_right, hx u hu, mul_zero]
#align submodule.orthogonal Submodule.orthogonal
@[inherit_doc]
notation:1200 K "แฎ" => orthogonal K
/-- When a vector is in `Kแฎ`. -/
theorem mem_orthogonal (v : E) : v โ Kแฎ โ โ u โ K, โชu, vโซ = 0 :=
Iff.rfl
#align submodule.mem_orthogonal Submodule.mem_orthogonal
/-- When a vector is in `Kแฎ`, with the inner product the
other way round. -/
theorem mem_orthogonal' (v : E) : v โ Kแฎ โ โ u โ K, โชv, uโซ = 0 := by
simp_rw [mem_orthogonal, inner_eq_zero_symm]
#align submodule.mem_orthogonal' Submodule.mem_orthogonal'
variable {K}
/-- A vector in `K` is orthogonal to one in `Kแฎ`. -/
theorem inner_right_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชu, vโซ = 0 :=
(K.mem_orthogonal v).1 hv u hu
#align submodule.inner_right_of_mem_orthogonal Submodule.inner_right_of_mem_orthogonal
/-- A vector in `Kแฎ` is orthogonal to one in `K`. -/
theorem inner_left_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชv, uโซ = 0 := by
rw [inner_eq_zero_symm]; exact inner_right_of_mem_orthogonal hu hv
#align submodule.inner_left_of_mem_orthogonal Submodule.inner_left_of_mem_orthogonal
/-- A vector is in `(๐ โ u)แฎ` iff it is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_right {u v : E} : v โ (๐ โ u)แฎ โ โชu, vโซ = 0 := by
refine' โจinner_right_of_mem_orthogonal (mem_span_singleton_self u), _โฉ
intro hv w hw
rw [mem_span_singleton] at hw
obtain โจc, rflโฉ := hw
simp [inner_smul_left, hv]
#align submodule.mem_orthogonal_singleton_iff_inner_right Submodule.mem_orthogonal_singleton_iff_inner_right
/-- A vector in `(๐ โ u)แฎ` is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_left {u v : E} : v โ (๐ โ u)แฎ โ โชv, uโซ = 0 := by
rw [mem_orthogonal_singleton_iff_inner_right, inner_eq_zero_symm]
#align submodule.mem_orthogonal_singleton_iff_inner_left Submodule.mem_orthogonal_singleton_iff_inner_left
theorem sub_mem_orthogonal_of_inner_left {x y : E} (h : โ v : K, โชx, vโซ = โชy, vโซ) : x - y โ Kแฎ := by
rw [mem_orthogonal']
intro u hu
rw [inner_sub_left, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_left Submodule.sub_mem_orthogonal_of_inner_left
theorem sub_mem_orthogonal_of_inner_right {x y : E} (h : โ v : K, โช(v : E), xโซ = โช(v : E), yโซ) :
x - y โ Kแฎ := by
intro u hu
rw [inner_sub_right, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_right Submodule.sub_mem_orthogonal_of_inner_right
variable (K)
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem inf_orthogonal_eq_bot : K โ Kแฎ = โฅ := by
rw [eq_bot_iff]
intro x
rw [mem_inf]
exact fun โจhx, hoโฉ => inner_self_eq_zero.1 (ho x hx)
#align submodule.inf_orthogonal_eq_bot Submodule.inf_orthogonal_eq_bot
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem orthogonal_disjoint : Disjoint K Kแฎ := by simp [disjoint_iff, K.inf_orthogonal_eq_bot]
#align submodule.orthogonal_disjoint Submodule.orthogonal_disjoint
/-- `Kแฎ` can be characterized as the intersection of the kernels of the operations of
inner product with each of the elements of `K`. -/
theorem orthogonal_eq_inter : Kแฎ = โจ
v : K, LinearMap.ker (innerSL ๐ (v : E)) := by
apply le_antisymm
ยท rw [le_iInf_iff]
rintro โจv, hvโฉ w hw
simpa using hw _ hv
ยท intro v hv w hw
simp only [mem_iInf] at hv
|
exact hv โจw, hwโฉ
|
/-- `Kแฎ` can be characterized as the intersection of the kernels of the operations of
inner product with each of the elements of `K`. -/
theorem orthogonal_eq_inter : Kแฎ = โจ
v : K, LinearMap.ker (innerSL ๐ (v : E)) := by
apply le_antisymm
ยท rw [le_iInf_iff]
rintro โจv, hvโฉ w hw
simpa using hw _ hv
ยท intro v hv w hw
simp only [mem_iInf] at hv
|
Mathlib.Analysis.InnerProductSpace.Orthogonal.116_0.QXx0GYqLoAbtfq1
|
/-- `Kแฎ` can be characterized as the intersection of the kernels of the operations of
inner product with each of the elements of `K`. -/
theorem orthogonal_eq_inter : Kแฎ = โจ
v : K, LinearMap.ker (innerSL ๐ (v : E))
|
Mathlib_Analysis_InnerProductSpace_Orthogonal
|
๐ : Type u_1
E : Type u_2
F : Type u_3
instโโด : IsROrC ๐
instโยณ : NormedAddCommGroup E
instโยฒ : InnerProductSpace ๐ E
instโยน : NormedAddCommGroup F
instโ : InnerProductSpace ๐ F
K : Submodule ๐ E
โข IsClosed โKแฎ
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sรฉbastien Gouรซzel, Frรฉdรฉric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
/-!
# Orthogonal complements of submodules
In this file, the `orthogonal` complement of a submodule `K` is defined, and basic API established.
Some of the more subtle results about the orthogonal complement are delayed to
`Analysis.InnerProductSpace.Projection`.
See also `BilinForm.orthogonal` for orthogonality with respect to a general bilinear form.
## Notation
The orthogonal complement of a submodule `K` is denoted by `Kแฎ`.
The proposition that two submodules are orthogonal, `Submodule.IsOrtho`, is denoted by `U โ V`.
Note this is not the same unicode symbol as `โฅ` (`Bot`).
-/
variable {๐ E F : Type*} [IsROrC ๐]
variable [NormedAddCommGroup E] [InnerProductSpace ๐ E]
variable [NormedAddCommGroup F] [InnerProductSpace ๐ F]
local notation "โช" x ", " y "โซ" => @inner ๐ _ _ x y
namespace Submodule
variable (K : Submodule ๐ E)
/-- The subspace of vectors orthogonal to a given subspace. -/
def orthogonal : Submodule ๐ E where
carrier := { v | โ u โ K, โชu, vโซ = 0 }
zero_mem' _ _ := inner_zero_right _
add_mem' hx hy u hu := by rw [inner_add_right, hx u hu, hy u hu, add_zero]
smul_mem' c x hx u hu := by rw [inner_smul_right, hx u hu, mul_zero]
#align submodule.orthogonal Submodule.orthogonal
@[inherit_doc]
notation:1200 K "แฎ" => orthogonal K
/-- When a vector is in `Kแฎ`. -/
theorem mem_orthogonal (v : E) : v โ Kแฎ โ โ u โ K, โชu, vโซ = 0 :=
Iff.rfl
#align submodule.mem_orthogonal Submodule.mem_orthogonal
/-- When a vector is in `Kแฎ`, with the inner product the
other way round. -/
theorem mem_orthogonal' (v : E) : v โ Kแฎ โ โ u โ K, โชv, uโซ = 0 := by
simp_rw [mem_orthogonal, inner_eq_zero_symm]
#align submodule.mem_orthogonal' Submodule.mem_orthogonal'
variable {K}
/-- A vector in `K` is orthogonal to one in `Kแฎ`. -/
theorem inner_right_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชu, vโซ = 0 :=
(K.mem_orthogonal v).1 hv u hu
#align submodule.inner_right_of_mem_orthogonal Submodule.inner_right_of_mem_orthogonal
/-- A vector in `Kแฎ` is orthogonal to one in `K`. -/
theorem inner_left_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชv, uโซ = 0 := by
rw [inner_eq_zero_symm]; exact inner_right_of_mem_orthogonal hu hv
#align submodule.inner_left_of_mem_orthogonal Submodule.inner_left_of_mem_orthogonal
/-- A vector is in `(๐ โ u)แฎ` iff it is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_right {u v : E} : v โ (๐ โ u)แฎ โ โชu, vโซ = 0 := by
refine' โจinner_right_of_mem_orthogonal (mem_span_singleton_self u), _โฉ
intro hv w hw
rw [mem_span_singleton] at hw
obtain โจc, rflโฉ := hw
simp [inner_smul_left, hv]
#align submodule.mem_orthogonal_singleton_iff_inner_right Submodule.mem_orthogonal_singleton_iff_inner_right
/-- A vector in `(๐ โ u)แฎ` is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_left {u v : E} : v โ (๐ โ u)แฎ โ โชv, uโซ = 0 := by
rw [mem_orthogonal_singleton_iff_inner_right, inner_eq_zero_symm]
#align submodule.mem_orthogonal_singleton_iff_inner_left Submodule.mem_orthogonal_singleton_iff_inner_left
theorem sub_mem_orthogonal_of_inner_left {x y : E} (h : โ v : K, โชx, vโซ = โชy, vโซ) : x - y โ Kแฎ := by
rw [mem_orthogonal']
intro u hu
rw [inner_sub_left, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_left Submodule.sub_mem_orthogonal_of_inner_left
theorem sub_mem_orthogonal_of_inner_right {x y : E} (h : โ v : K, โช(v : E), xโซ = โช(v : E), yโซ) :
x - y โ Kแฎ := by
intro u hu
rw [inner_sub_right, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_right Submodule.sub_mem_orthogonal_of_inner_right
variable (K)
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem inf_orthogonal_eq_bot : K โ Kแฎ = โฅ := by
rw [eq_bot_iff]
intro x
rw [mem_inf]
exact fun โจhx, hoโฉ => inner_self_eq_zero.1 (ho x hx)
#align submodule.inf_orthogonal_eq_bot Submodule.inf_orthogonal_eq_bot
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem orthogonal_disjoint : Disjoint K Kแฎ := by simp [disjoint_iff, K.inf_orthogonal_eq_bot]
#align submodule.orthogonal_disjoint Submodule.orthogonal_disjoint
/-- `Kแฎ` can be characterized as the intersection of the kernels of the operations of
inner product with each of the elements of `K`. -/
theorem orthogonal_eq_inter : Kแฎ = โจ
v : K, LinearMap.ker (innerSL ๐ (v : E)) := by
apply le_antisymm
ยท rw [le_iInf_iff]
rintro โจv, hvโฉ w hw
simpa using hw _ hv
ยท intro v hv w hw
simp only [mem_iInf] at hv
exact hv โจw, hwโฉ
#align submodule.orthogonal_eq_inter Submodule.orthogonal_eq_inter
/-- The orthogonal complement of any submodule `K` is closed. -/
theorem isClosed_orthogonal : IsClosed (Kแฎ : Set E) := by
|
rw [orthogonal_eq_inter K]
|
/-- The orthogonal complement of any submodule `K` is closed. -/
theorem isClosed_orthogonal : IsClosed (Kแฎ : Set E) := by
|
Mathlib.Analysis.InnerProductSpace.Orthogonal.128_0.QXx0GYqLoAbtfq1
|
/-- The orthogonal complement of any submodule `K` is closed. -/
theorem isClosed_orthogonal : IsClosed (Kแฎ : Set E)
|
Mathlib_Analysis_InnerProductSpace_Orthogonal
|
๐ : Type u_1
E : Type u_2
F : Type u_3
instโโด : IsROrC ๐
instโยณ : NormedAddCommGroup E
instโยฒ : InnerProductSpace ๐ E
instโยน : NormedAddCommGroup F
instโ : InnerProductSpace ๐ F
K : Submodule ๐ E
โข IsClosed โ(โจ
v, LinearMap.ker ((innerSL ๐) โv))
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sรฉbastien Gouรซzel, Frรฉdรฉric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
/-!
# Orthogonal complements of submodules
In this file, the `orthogonal` complement of a submodule `K` is defined, and basic API established.
Some of the more subtle results about the orthogonal complement are delayed to
`Analysis.InnerProductSpace.Projection`.
See also `BilinForm.orthogonal` for orthogonality with respect to a general bilinear form.
## Notation
The orthogonal complement of a submodule `K` is denoted by `Kแฎ`.
The proposition that two submodules are orthogonal, `Submodule.IsOrtho`, is denoted by `U โ V`.
Note this is not the same unicode symbol as `โฅ` (`Bot`).
-/
variable {๐ E F : Type*} [IsROrC ๐]
variable [NormedAddCommGroup E] [InnerProductSpace ๐ E]
variable [NormedAddCommGroup F] [InnerProductSpace ๐ F]
local notation "โช" x ", " y "โซ" => @inner ๐ _ _ x y
namespace Submodule
variable (K : Submodule ๐ E)
/-- The subspace of vectors orthogonal to a given subspace. -/
def orthogonal : Submodule ๐ E where
carrier := { v | โ u โ K, โชu, vโซ = 0 }
zero_mem' _ _ := inner_zero_right _
add_mem' hx hy u hu := by rw [inner_add_right, hx u hu, hy u hu, add_zero]
smul_mem' c x hx u hu := by rw [inner_smul_right, hx u hu, mul_zero]
#align submodule.orthogonal Submodule.orthogonal
@[inherit_doc]
notation:1200 K "แฎ" => orthogonal K
/-- When a vector is in `Kแฎ`. -/
theorem mem_orthogonal (v : E) : v โ Kแฎ โ โ u โ K, โชu, vโซ = 0 :=
Iff.rfl
#align submodule.mem_orthogonal Submodule.mem_orthogonal
/-- When a vector is in `Kแฎ`, with the inner product the
other way round. -/
theorem mem_orthogonal' (v : E) : v โ Kแฎ โ โ u โ K, โชv, uโซ = 0 := by
simp_rw [mem_orthogonal, inner_eq_zero_symm]
#align submodule.mem_orthogonal' Submodule.mem_orthogonal'
variable {K}
/-- A vector in `K` is orthogonal to one in `Kแฎ`. -/
theorem inner_right_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชu, vโซ = 0 :=
(K.mem_orthogonal v).1 hv u hu
#align submodule.inner_right_of_mem_orthogonal Submodule.inner_right_of_mem_orthogonal
/-- A vector in `Kแฎ` is orthogonal to one in `K`. -/
theorem inner_left_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชv, uโซ = 0 := by
rw [inner_eq_zero_symm]; exact inner_right_of_mem_orthogonal hu hv
#align submodule.inner_left_of_mem_orthogonal Submodule.inner_left_of_mem_orthogonal
/-- A vector is in `(๐ โ u)แฎ` iff it is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_right {u v : E} : v โ (๐ โ u)แฎ โ โชu, vโซ = 0 := by
refine' โจinner_right_of_mem_orthogonal (mem_span_singleton_self u), _โฉ
intro hv w hw
rw [mem_span_singleton] at hw
obtain โจc, rflโฉ := hw
simp [inner_smul_left, hv]
#align submodule.mem_orthogonal_singleton_iff_inner_right Submodule.mem_orthogonal_singleton_iff_inner_right
/-- A vector in `(๐ โ u)แฎ` is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_left {u v : E} : v โ (๐ โ u)แฎ โ โชv, uโซ = 0 := by
rw [mem_orthogonal_singleton_iff_inner_right, inner_eq_zero_symm]
#align submodule.mem_orthogonal_singleton_iff_inner_left Submodule.mem_orthogonal_singleton_iff_inner_left
theorem sub_mem_orthogonal_of_inner_left {x y : E} (h : โ v : K, โชx, vโซ = โชy, vโซ) : x - y โ Kแฎ := by
rw [mem_orthogonal']
intro u hu
rw [inner_sub_left, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_left Submodule.sub_mem_orthogonal_of_inner_left
theorem sub_mem_orthogonal_of_inner_right {x y : E} (h : โ v : K, โช(v : E), xโซ = โช(v : E), yโซ) :
x - y โ Kแฎ := by
intro u hu
rw [inner_sub_right, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_right Submodule.sub_mem_orthogonal_of_inner_right
variable (K)
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem inf_orthogonal_eq_bot : K โ Kแฎ = โฅ := by
rw [eq_bot_iff]
intro x
rw [mem_inf]
exact fun โจhx, hoโฉ => inner_self_eq_zero.1 (ho x hx)
#align submodule.inf_orthogonal_eq_bot Submodule.inf_orthogonal_eq_bot
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem orthogonal_disjoint : Disjoint K Kแฎ := by simp [disjoint_iff, K.inf_orthogonal_eq_bot]
#align submodule.orthogonal_disjoint Submodule.orthogonal_disjoint
/-- `Kแฎ` can be characterized as the intersection of the kernels of the operations of
inner product with each of the elements of `K`. -/
theorem orthogonal_eq_inter : Kแฎ = โจ
v : K, LinearMap.ker (innerSL ๐ (v : E)) := by
apply le_antisymm
ยท rw [le_iInf_iff]
rintro โจv, hvโฉ w hw
simpa using hw _ hv
ยท intro v hv w hw
simp only [mem_iInf] at hv
exact hv โจw, hwโฉ
#align submodule.orthogonal_eq_inter Submodule.orthogonal_eq_inter
/-- The orthogonal complement of any submodule `K` is closed. -/
theorem isClosed_orthogonal : IsClosed (Kแฎ : Set E) := by
rw [orthogonal_eq_inter K]
|
have := fun v : K => ContinuousLinearMap.isClosed_ker (innerSL ๐ (v : E))
|
/-- The orthogonal complement of any submodule `K` is closed. -/
theorem isClosed_orthogonal : IsClosed (Kแฎ : Set E) := by
rw [orthogonal_eq_inter K]
|
Mathlib.Analysis.InnerProductSpace.Orthogonal.128_0.QXx0GYqLoAbtfq1
|
/-- The orthogonal complement of any submodule `K` is closed. -/
theorem isClosed_orthogonal : IsClosed (Kแฎ : Set E)
|
Mathlib_Analysis_InnerProductSpace_Orthogonal
|
๐ : Type u_1
E : Type u_2
F : Type u_3
instโโด : IsROrC ๐
instโยณ : NormedAddCommGroup E
instโยฒ : InnerProductSpace ๐ E
instโยน : NormedAddCommGroup F
instโ : InnerProductSpace ๐ F
K : Submodule ๐ E
this : โ (v : โฅK), IsClosed โ(LinearMap.ker ((innerSL ๐) โv))
โข IsClosed โ(โจ
v, LinearMap.ker ((innerSL ๐) โv))
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sรฉbastien Gouรซzel, Frรฉdรฉric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
/-!
# Orthogonal complements of submodules
In this file, the `orthogonal` complement of a submodule `K` is defined, and basic API established.
Some of the more subtle results about the orthogonal complement are delayed to
`Analysis.InnerProductSpace.Projection`.
See also `BilinForm.orthogonal` for orthogonality with respect to a general bilinear form.
## Notation
The orthogonal complement of a submodule `K` is denoted by `Kแฎ`.
The proposition that two submodules are orthogonal, `Submodule.IsOrtho`, is denoted by `U โ V`.
Note this is not the same unicode symbol as `โฅ` (`Bot`).
-/
variable {๐ E F : Type*} [IsROrC ๐]
variable [NormedAddCommGroup E] [InnerProductSpace ๐ E]
variable [NormedAddCommGroup F] [InnerProductSpace ๐ F]
local notation "โช" x ", " y "โซ" => @inner ๐ _ _ x y
namespace Submodule
variable (K : Submodule ๐ E)
/-- The subspace of vectors orthogonal to a given subspace. -/
def orthogonal : Submodule ๐ E where
carrier := { v | โ u โ K, โชu, vโซ = 0 }
zero_mem' _ _ := inner_zero_right _
add_mem' hx hy u hu := by rw [inner_add_right, hx u hu, hy u hu, add_zero]
smul_mem' c x hx u hu := by rw [inner_smul_right, hx u hu, mul_zero]
#align submodule.orthogonal Submodule.orthogonal
@[inherit_doc]
notation:1200 K "แฎ" => orthogonal K
/-- When a vector is in `Kแฎ`. -/
theorem mem_orthogonal (v : E) : v โ Kแฎ โ โ u โ K, โชu, vโซ = 0 :=
Iff.rfl
#align submodule.mem_orthogonal Submodule.mem_orthogonal
/-- When a vector is in `Kแฎ`, with the inner product the
other way round. -/
theorem mem_orthogonal' (v : E) : v โ Kแฎ โ โ u โ K, โชv, uโซ = 0 := by
simp_rw [mem_orthogonal, inner_eq_zero_symm]
#align submodule.mem_orthogonal' Submodule.mem_orthogonal'
variable {K}
/-- A vector in `K` is orthogonal to one in `Kแฎ`. -/
theorem inner_right_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชu, vโซ = 0 :=
(K.mem_orthogonal v).1 hv u hu
#align submodule.inner_right_of_mem_orthogonal Submodule.inner_right_of_mem_orthogonal
/-- A vector in `Kแฎ` is orthogonal to one in `K`. -/
theorem inner_left_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชv, uโซ = 0 := by
rw [inner_eq_zero_symm]; exact inner_right_of_mem_orthogonal hu hv
#align submodule.inner_left_of_mem_orthogonal Submodule.inner_left_of_mem_orthogonal
/-- A vector is in `(๐ โ u)แฎ` iff it is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_right {u v : E} : v โ (๐ โ u)แฎ โ โชu, vโซ = 0 := by
refine' โจinner_right_of_mem_orthogonal (mem_span_singleton_self u), _โฉ
intro hv w hw
rw [mem_span_singleton] at hw
obtain โจc, rflโฉ := hw
simp [inner_smul_left, hv]
#align submodule.mem_orthogonal_singleton_iff_inner_right Submodule.mem_orthogonal_singleton_iff_inner_right
/-- A vector in `(๐ โ u)แฎ` is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_left {u v : E} : v โ (๐ โ u)แฎ โ โชv, uโซ = 0 := by
rw [mem_orthogonal_singleton_iff_inner_right, inner_eq_zero_symm]
#align submodule.mem_orthogonal_singleton_iff_inner_left Submodule.mem_orthogonal_singleton_iff_inner_left
theorem sub_mem_orthogonal_of_inner_left {x y : E} (h : โ v : K, โชx, vโซ = โชy, vโซ) : x - y โ Kแฎ := by
rw [mem_orthogonal']
intro u hu
rw [inner_sub_left, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_left Submodule.sub_mem_orthogonal_of_inner_left
theorem sub_mem_orthogonal_of_inner_right {x y : E} (h : โ v : K, โช(v : E), xโซ = โช(v : E), yโซ) :
x - y โ Kแฎ := by
intro u hu
rw [inner_sub_right, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_right Submodule.sub_mem_orthogonal_of_inner_right
variable (K)
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem inf_orthogonal_eq_bot : K โ Kแฎ = โฅ := by
rw [eq_bot_iff]
intro x
rw [mem_inf]
exact fun โจhx, hoโฉ => inner_self_eq_zero.1 (ho x hx)
#align submodule.inf_orthogonal_eq_bot Submodule.inf_orthogonal_eq_bot
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem orthogonal_disjoint : Disjoint K Kแฎ := by simp [disjoint_iff, K.inf_orthogonal_eq_bot]
#align submodule.orthogonal_disjoint Submodule.orthogonal_disjoint
/-- `Kแฎ` can be characterized as the intersection of the kernels of the operations of
inner product with each of the elements of `K`. -/
theorem orthogonal_eq_inter : Kแฎ = โจ
v : K, LinearMap.ker (innerSL ๐ (v : E)) := by
apply le_antisymm
ยท rw [le_iInf_iff]
rintro โจv, hvโฉ w hw
simpa using hw _ hv
ยท intro v hv w hw
simp only [mem_iInf] at hv
exact hv โจw, hwโฉ
#align submodule.orthogonal_eq_inter Submodule.orthogonal_eq_inter
/-- The orthogonal complement of any submodule `K` is closed. -/
theorem isClosed_orthogonal : IsClosed (Kแฎ : Set E) := by
rw [orthogonal_eq_inter K]
have := fun v : K => ContinuousLinearMap.isClosed_ker (innerSL ๐ (v : E))
|
convert isClosed_iInter this
|
/-- The orthogonal complement of any submodule `K` is closed. -/
theorem isClosed_orthogonal : IsClosed (Kแฎ : Set E) := by
rw [orthogonal_eq_inter K]
have := fun v : K => ContinuousLinearMap.isClosed_ker (innerSL ๐ (v : E))
|
Mathlib.Analysis.InnerProductSpace.Orthogonal.128_0.QXx0GYqLoAbtfq1
|
/-- The orthogonal complement of any submodule `K` is closed. -/
theorem isClosed_orthogonal : IsClosed (Kแฎ : Set E)
|
Mathlib_Analysis_InnerProductSpace_Orthogonal
|
case h.e'_3
๐ : Type u_1
E : Type u_2
F : Type u_3
instโโด : IsROrC ๐
instโยณ : NormedAddCommGroup E
instโยฒ : InnerProductSpace ๐ E
instโยน : NormedAddCommGroup F
instโ : InnerProductSpace ๐ F
K : Submodule ๐ E
this : โ (v : โฅK), IsClosed โ(LinearMap.ker ((innerSL ๐) โv))
โข โ(โจ
v, LinearMap.ker ((innerSL ๐) โv)) = โ i, โ(LinearMap.ker ((innerSL ๐) โi))
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sรฉbastien Gouรซzel, Frรฉdรฉric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
/-!
# Orthogonal complements of submodules
In this file, the `orthogonal` complement of a submodule `K` is defined, and basic API established.
Some of the more subtle results about the orthogonal complement are delayed to
`Analysis.InnerProductSpace.Projection`.
See also `BilinForm.orthogonal` for orthogonality with respect to a general bilinear form.
## Notation
The orthogonal complement of a submodule `K` is denoted by `Kแฎ`.
The proposition that two submodules are orthogonal, `Submodule.IsOrtho`, is denoted by `U โ V`.
Note this is not the same unicode symbol as `โฅ` (`Bot`).
-/
variable {๐ E F : Type*} [IsROrC ๐]
variable [NormedAddCommGroup E] [InnerProductSpace ๐ E]
variable [NormedAddCommGroup F] [InnerProductSpace ๐ F]
local notation "โช" x ", " y "โซ" => @inner ๐ _ _ x y
namespace Submodule
variable (K : Submodule ๐ E)
/-- The subspace of vectors orthogonal to a given subspace. -/
def orthogonal : Submodule ๐ E where
carrier := { v | โ u โ K, โชu, vโซ = 0 }
zero_mem' _ _ := inner_zero_right _
add_mem' hx hy u hu := by rw [inner_add_right, hx u hu, hy u hu, add_zero]
smul_mem' c x hx u hu := by rw [inner_smul_right, hx u hu, mul_zero]
#align submodule.orthogonal Submodule.orthogonal
@[inherit_doc]
notation:1200 K "แฎ" => orthogonal K
/-- When a vector is in `Kแฎ`. -/
theorem mem_orthogonal (v : E) : v โ Kแฎ โ โ u โ K, โชu, vโซ = 0 :=
Iff.rfl
#align submodule.mem_orthogonal Submodule.mem_orthogonal
/-- When a vector is in `Kแฎ`, with the inner product the
other way round. -/
theorem mem_orthogonal' (v : E) : v โ Kแฎ โ โ u โ K, โชv, uโซ = 0 := by
simp_rw [mem_orthogonal, inner_eq_zero_symm]
#align submodule.mem_orthogonal' Submodule.mem_orthogonal'
variable {K}
/-- A vector in `K` is orthogonal to one in `Kแฎ`. -/
theorem inner_right_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชu, vโซ = 0 :=
(K.mem_orthogonal v).1 hv u hu
#align submodule.inner_right_of_mem_orthogonal Submodule.inner_right_of_mem_orthogonal
/-- A vector in `Kแฎ` is orthogonal to one in `K`. -/
theorem inner_left_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชv, uโซ = 0 := by
rw [inner_eq_zero_symm]; exact inner_right_of_mem_orthogonal hu hv
#align submodule.inner_left_of_mem_orthogonal Submodule.inner_left_of_mem_orthogonal
/-- A vector is in `(๐ โ u)แฎ` iff it is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_right {u v : E} : v โ (๐ โ u)แฎ โ โชu, vโซ = 0 := by
refine' โจinner_right_of_mem_orthogonal (mem_span_singleton_self u), _โฉ
intro hv w hw
rw [mem_span_singleton] at hw
obtain โจc, rflโฉ := hw
simp [inner_smul_left, hv]
#align submodule.mem_orthogonal_singleton_iff_inner_right Submodule.mem_orthogonal_singleton_iff_inner_right
/-- A vector in `(๐ โ u)แฎ` is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_left {u v : E} : v โ (๐ โ u)แฎ โ โชv, uโซ = 0 := by
rw [mem_orthogonal_singleton_iff_inner_right, inner_eq_zero_symm]
#align submodule.mem_orthogonal_singleton_iff_inner_left Submodule.mem_orthogonal_singleton_iff_inner_left
theorem sub_mem_orthogonal_of_inner_left {x y : E} (h : โ v : K, โชx, vโซ = โชy, vโซ) : x - y โ Kแฎ := by
rw [mem_orthogonal']
intro u hu
rw [inner_sub_left, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_left Submodule.sub_mem_orthogonal_of_inner_left
theorem sub_mem_orthogonal_of_inner_right {x y : E} (h : โ v : K, โช(v : E), xโซ = โช(v : E), yโซ) :
x - y โ Kแฎ := by
intro u hu
rw [inner_sub_right, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_right Submodule.sub_mem_orthogonal_of_inner_right
variable (K)
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem inf_orthogonal_eq_bot : K โ Kแฎ = โฅ := by
rw [eq_bot_iff]
intro x
rw [mem_inf]
exact fun โจhx, hoโฉ => inner_self_eq_zero.1 (ho x hx)
#align submodule.inf_orthogonal_eq_bot Submodule.inf_orthogonal_eq_bot
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem orthogonal_disjoint : Disjoint K Kแฎ := by simp [disjoint_iff, K.inf_orthogonal_eq_bot]
#align submodule.orthogonal_disjoint Submodule.orthogonal_disjoint
/-- `Kแฎ` can be characterized as the intersection of the kernels of the operations of
inner product with each of the elements of `K`. -/
theorem orthogonal_eq_inter : Kแฎ = โจ
v : K, LinearMap.ker (innerSL ๐ (v : E)) := by
apply le_antisymm
ยท rw [le_iInf_iff]
rintro โจv, hvโฉ w hw
simpa using hw _ hv
ยท intro v hv w hw
simp only [mem_iInf] at hv
exact hv โจw, hwโฉ
#align submodule.orthogonal_eq_inter Submodule.orthogonal_eq_inter
/-- The orthogonal complement of any submodule `K` is closed. -/
theorem isClosed_orthogonal : IsClosed (Kแฎ : Set E) := by
rw [orthogonal_eq_inter K]
have := fun v : K => ContinuousLinearMap.isClosed_ker (innerSL ๐ (v : E))
convert isClosed_iInter this
|
simp only [iInf_coe]
|
/-- The orthogonal complement of any submodule `K` is closed. -/
theorem isClosed_orthogonal : IsClosed (Kแฎ : Set E) := by
rw [orthogonal_eq_inter K]
have := fun v : K => ContinuousLinearMap.isClosed_ker (innerSL ๐ (v : E))
convert isClosed_iInter this
|
Mathlib.Analysis.InnerProductSpace.Orthogonal.128_0.QXx0GYqLoAbtfq1
|
/-- The orthogonal complement of any submodule `K` is closed. -/
theorem isClosed_orthogonal : IsClosed (Kแฎ : Set E)
|
Mathlib_Analysis_InnerProductSpace_Orthogonal
|
๐ : Type u_1
E : Type u_2
F : Type u_3
instโโด : IsROrC ๐
instโยณ : NormedAddCommGroup E
instโยฒ : InnerProductSpace ๐ E
instโยน : NormedAddCommGroup F
instโ : InnerProductSpace ๐ F
K : Submodule ๐ E
โข โคแฎ = โฅ
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sรฉbastien Gouรซzel, Frรฉdรฉric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
/-!
# Orthogonal complements of submodules
In this file, the `orthogonal` complement of a submodule `K` is defined, and basic API established.
Some of the more subtle results about the orthogonal complement are delayed to
`Analysis.InnerProductSpace.Projection`.
See also `BilinForm.orthogonal` for orthogonality with respect to a general bilinear form.
## Notation
The orthogonal complement of a submodule `K` is denoted by `Kแฎ`.
The proposition that two submodules are orthogonal, `Submodule.IsOrtho`, is denoted by `U โ V`.
Note this is not the same unicode symbol as `โฅ` (`Bot`).
-/
variable {๐ E F : Type*} [IsROrC ๐]
variable [NormedAddCommGroup E] [InnerProductSpace ๐ E]
variable [NormedAddCommGroup F] [InnerProductSpace ๐ F]
local notation "โช" x ", " y "โซ" => @inner ๐ _ _ x y
namespace Submodule
variable (K : Submodule ๐ E)
/-- The subspace of vectors orthogonal to a given subspace. -/
def orthogonal : Submodule ๐ E where
carrier := { v | โ u โ K, โชu, vโซ = 0 }
zero_mem' _ _ := inner_zero_right _
add_mem' hx hy u hu := by rw [inner_add_right, hx u hu, hy u hu, add_zero]
smul_mem' c x hx u hu := by rw [inner_smul_right, hx u hu, mul_zero]
#align submodule.orthogonal Submodule.orthogonal
@[inherit_doc]
notation:1200 K "แฎ" => orthogonal K
/-- When a vector is in `Kแฎ`. -/
theorem mem_orthogonal (v : E) : v โ Kแฎ โ โ u โ K, โชu, vโซ = 0 :=
Iff.rfl
#align submodule.mem_orthogonal Submodule.mem_orthogonal
/-- When a vector is in `Kแฎ`, with the inner product the
other way round. -/
theorem mem_orthogonal' (v : E) : v โ Kแฎ โ โ u โ K, โชv, uโซ = 0 := by
simp_rw [mem_orthogonal, inner_eq_zero_symm]
#align submodule.mem_orthogonal' Submodule.mem_orthogonal'
variable {K}
/-- A vector in `K` is orthogonal to one in `Kแฎ`. -/
theorem inner_right_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชu, vโซ = 0 :=
(K.mem_orthogonal v).1 hv u hu
#align submodule.inner_right_of_mem_orthogonal Submodule.inner_right_of_mem_orthogonal
/-- A vector in `Kแฎ` is orthogonal to one in `K`. -/
theorem inner_left_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชv, uโซ = 0 := by
rw [inner_eq_zero_symm]; exact inner_right_of_mem_orthogonal hu hv
#align submodule.inner_left_of_mem_orthogonal Submodule.inner_left_of_mem_orthogonal
/-- A vector is in `(๐ โ u)แฎ` iff it is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_right {u v : E} : v โ (๐ โ u)แฎ โ โชu, vโซ = 0 := by
refine' โจinner_right_of_mem_orthogonal (mem_span_singleton_self u), _โฉ
intro hv w hw
rw [mem_span_singleton] at hw
obtain โจc, rflโฉ := hw
simp [inner_smul_left, hv]
#align submodule.mem_orthogonal_singleton_iff_inner_right Submodule.mem_orthogonal_singleton_iff_inner_right
/-- A vector in `(๐ โ u)แฎ` is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_left {u v : E} : v โ (๐ โ u)แฎ โ โชv, uโซ = 0 := by
rw [mem_orthogonal_singleton_iff_inner_right, inner_eq_zero_symm]
#align submodule.mem_orthogonal_singleton_iff_inner_left Submodule.mem_orthogonal_singleton_iff_inner_left
theorem sub_mem_orthogonal_of_inner_left {x y : E} (h : โ v : K, โชx, vโซ = โชy, vโซ) : x - y โ Kแฎ := by
rw [mem_orthogonal']
intro u hu
rw [inner_sub_left, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_left Submodule.sub_mem_orthogonal_of_inner_left
theorem sub_mem_orthogonal_of_inner_right {x y : E} (h : โ v : K, โช(v : E), xโซ = โช(v : E), yโซ) :
x - y โ Kแฎ := by
intro u hu
rw [inner_sub_right, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_right Submodule.sub_mem_orthogonal_of_inner_right
variable (K)
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem inf_orthogonal_eq_bot : K โ Kแฎ = โฅ := by
rw [eq_bot_iff]
intro x
rw [mem_inf]
exact fun โจhx, hoโฉ => inner_self_eq_zero.1 (ho x hx)
#align submodule.inf_orthogonal_eq_bot Submodule.inf_orthogonal_eq_bot
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem orthogonal_disjoint : Disjoint K Kแฎ := by simp [disjoint_iff, K.inf_orthogonal_eq_bot]
#align submodule.orthogonal_disjoint Submodule.orthogonal_disjoint
/-- `Kแฎ` can be characterized as the intersection of the kernels of the operations of
inner product with each of the elements of `K`. -/
theorem orthogonal_eq_inter : Kแฎ = โจ
v : K, LinearMap.ker (innerSL ๐ (v : E)) := by
apply le_antisymm
ยท rw [le_iInf_iff]
rintro โจv, hvโฉ w hw
simpa using hw _ hv
ยท intro v hv w hw
simp only [mem_iInf] at hv
exact hv โจw, hwโฉ
#align submodule.orthogonal_eq_inter Submodule.orthogonal_eq_inter
/-- The orthogonal complement of any submodule `K` is closed. -/
theorem isClosed_orthogonal : IsClosed (Kแฎ : Set E) := by
rw [orthogonal_eq_inter K]
have := fun v : K => ContinuousLinearMap.isClosed_ker (innerSL ๐ (v : E))
convert isClosed_iInter this
simp only [iInf_coe]
#align submodule.is_closed_orthogonal Submodule.isClosed_orthogonal
/-- In a complete space, the orthogonal complement of any submodule `K` is complete. -/
instance instOrthogonalCompleteSpace [CompleteSpace E] : CompleteSpace Kแฎ :=
K.isClosed_orthogonal.completeSpace_coe
variable (๐ E)
/-- `orthogonal` gives a `GaloisConnection` between
`Submodule ๐ E` and its `OrderDual`. -/
theorem orthogonal_gc :
@GaloisConnection (Submodule ๐ E) (Submodule ๐ E)แตแต _ _ orthogonal orthogonal := fun _Kโ _Kโ =>
โจfun h _v hv _u hu => inner_left_of_mem_orthogonal hv (h hu), fun h _v hv _u hu =>
inner_left_of_mem_orthogonal hv (h hu)โฉ
#align submodule.orthogonal_gc Submodule.orthogonal_gc
variable {๐ E}
/-- `orthogonal` reverses the `โค` ordering of two
subspaces. -/
theorem orthogonal_le {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎ โค Kโแฎ :=
(orthogonal_gc ๐ E).monotone_l h
#align submodule.orthogonal_le Submodule.orthogonal_le
/-- `orthogonal.orthogonal` preserves the `โค` ordering of two
subspaces. -/
theorem orthogonal_orthogonal_monotone {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎแฎ โค Kโแฎแฎ :=
orthogonal_le (orthogonal_le h)
#align submodule.orthogonal_orthogonal_monotone Submodule.orthogonal_orthogonal_monotone
/-- `K` is contained in `Kแฎแฎ`. -/
theorem le_orthogonal_orthogonal : K โค Kแฎแฎ :=
(orthogonal_gc ๐ E).le_u_l _
#align submodule.le_orthogonal_orthogonal Submodule.le_orthogonal_orthogonal
/-- The inf of two orthogonal subspaces equals the subspace orthogonal
to the sup. -/
theorem inf_orthogonal (Kโ Kโ : Submodule ๐ E) : Kโแฎ โ Kโแฎ = (Kโ โ Kโ)แฎ :=
(orthogonal_gc ๐ E).l_sup.symm
#align submodule.inf_orthogonal Submodule.inf_orthogonal
/-- The inf of an indexed family of orthogonal subspaces equals the
subspace orthogonal to the sup. -/
theorem iInf_orthogonal {ฮน : Type*} (K : ฮน โ Submodule ๐ E) : โจ
i, (K i)แฎ = (iSup K)แฎ :=
(orthogonal_gc ๐ E).l_iSup.symm
#align submodule.infi_orthogonal Submodule.iInf_orthogonal
/-- The inf of a set of orthogonal subspaces equals the subspace orthogonal to the sup. -/
theorem sInf_orthogonal (s : Set <| Submodule ๐ E) : โจ
K โ s, Kแฎ = (sSup s)แฎ :=
(orthogonal_gc ๐ E).l_sSup.symm
#align submodule.Inf_orthogonal Submodule.sInf_orthogonal
@[simp]
theorem top_orthogonal_eq_bot : (โค : Submodule ๐ E)แฎ = โฅ := by
|
ext x
|
@[simp]
theorem top_orthogonal_eq_bot : (โค : Submodule ๐ E)แฎ = โฅ := by
|
Mathlib.Analysis.InnerProductSpace.Orthogonal.186_0.QXx0GYqLoAbtfq1
|
@[simp]
theorem top_orthogonal_eq_bot : (โค : Submodule ๐ E)แฎ = โฅ
|
Mathlib_Analysis_InnerProductSpace_Orthogonal
|
case h
๐ : Type u_1
E : Type u_2
F : Type u_3
instโโด : IsROrC ๐
instโยณ : NormedAddCommGroup E
instโยฒ : InnerProductSpace ๐ E
instโยน : NormedAddCommGroup F
instโ : InnerProductSpace ๐ F
K : Submodule ๐ E
x : E
โข x โ โคแฎ โ x โ โฅ
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sรฉbastien Gouรซzel, Frรฉdรฉric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
/-!
# Orthogonal complements of submodules
In this file, the `orthogonal` complement of a submodule `K` is defined, and basic API established.
Some of the more subtle results about the orthogonal complement are delayed to
`Analysis.InnerProductSpace.Projection`.
See also `BilinForm.orthogonal` for orthogonality with respect to a general bilinear form.
## Notation
The orthogonal complement of a submodule `K` is denoted by `Kแฎ`.
The proposition that two submodules are orthogonal, `Submodule.IsOrtho`, is denoted by `U โ V`.
Note this is not the same unicode symbol as `โฅ` (`Bot`).
-/
variable {๐ E F : Type*} [IsROrC ๐]
variable [NormedAddCommGroup E] [InnerProductSpace ๐ E]
variable [NormedAddCommGroup F] [InnerProductSpace ๐ F]
local notation "โช" x ", " y "โซ" => @inner ๐ _ _ x y
namespace Submodule
variable (K : Submodule ๐ E)
/-- The subspace of vectors orthogonal to a given subspace. -/
def orthogonal : Submodule ๐ E where
carrier := { v | โ u โ K, โชu, vโซ = 0 }
zero_mem' _ _ := inner_zero_right _
add_mem' hx hy u hu := by rw [inner_add_right, hx u hu, hy u hu, add_zero]
smul_mem' c x hx u hu := by rw [inner_smul_right, hx u hu, mul_zero]
#align submodule.orthogonal Submodule.orthogonal
@[inherit_doc]
notation:1200 K "แฎ" => orthogonal K
/-- When a vector is in `Kแฎ`. -/
theorem mem_orthogonal (v : E) : v โ Kแฎ โ โ u โ K, โชu, vโซ = 0 :=
Iff.rfl
#align submodule.mem_orthogonal Submodule.mem_orthogonal
/-- When a vector is in `Kแฎ`, with the inner product the
other way round. -/
theorem mem_orthogonal' (v : E) : v โ Kแฎ โ โ u โ K, โชv, uโซ = 0 := by
simp_rw [mem_orthogonal, inner_eq_zero_symm]
#align submodule.mem_orthogonal' Submodule.mem_orthogonal'
variable {K}
/-- A vector in `K` is orthogonal to one in `Kแฎ`. -/
theorem inner_right_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชu, vโซ = 0 :=
(K.mem_orthogonal v).1 hv u hu
#align submodule.inner_right_of_mem_orthogonal Submodule.inner_right_of_mem_orthogonal
/-- A vector in `Kแฎ` is orthogonal to one in `K`. -/
theorem inner_left_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชv, uโซ = 0 := by
rw [inner_eq_zero_symm]; exact inner_right_of_mem_orthogonal hu hv
#align submodule.inner_left_of_mem_orthogonal Submodule.inner_left_of_mem_orthogonal
/-- A vector is in `(๐ โ u)แฎ` iff it is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_right {u v : E} : v โ (๐ โ u)แฎ โ โชu, vโซ = 0 := by
refine' โจinner_right_of_mem_orthogonal (mem_span_singleton_self u), _โฉ
intro hv w hw
rw [mem_span_singleton] at hw
obtain โจc, rflโฉ := hw
simp [inner_smul_left, hv]
#align submodule.mem_orthogonal_singleton_iff_inner_right Submodule.mem_orthogonal_singleton_iff_inner_right
/-- A vector in `(๐ โ u)แฎ` is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_left {u v : E} : v โ (๐ โ u)แฎ โ โชv, uโซ = 0 := by
rw [mem_orthogonal_singleton_iff_inner_right, inner_eq_zero_symm]
#align submodule.mem_orthogonal_singleton_iff_inner_left Submodule.mem_orthogonal_singleton_iff_inner_left
theorem sub_mem_orthogonal_of_inner_left {x y : E} (h : โ v : K, โชx, vโซ = โชy, vโซ) : x - y โ Kแฎ := by
rw [mem_orthogonal']
intro u hu
rw [inner_sub_left, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_left Submodule.sub_mem_orthogonal_of_inner_left
theorem sub_mem_orthogonal_of_inner_right {x y : E} (h : โ v : K, โช(v : E), xโซ = โช(v : E), yโซ) :
x - y โ Kแฎ := by
intro u hu
rw [inner_sub_right, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_right Submodule.sub_mem_orthogonal_of_inner_right
variable (K)
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem inf_orthogonal_eq_bot : K โ Kแฎ = โฅ := by
rw [eq_bot_iff]
intro x
rw [mem_inf]
exact fun โจhx, hoโฉ => inner_self_eq_zero.1 (ho x hx)
#align submodule.inf_orthogonal_eq_bot Submodule.inf_orthogonal_eq_bot
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem orthogonal_disjoint : Disjoint K Kแฎ := by simp [disjoint_iff, K.inf_orthogonal_eq_bot]
#align submodule.orthogonal_disjoint Submodule.orthogonal_disjoint
/-- `Kแฎ` can be characterized as the intersection of the kernels of the operations of
inner product with each of the elements of `K`. -/
theorem orthogonal_eq_inter : Kแฎ = โจ
v : K, LinearMap.ker (innerSL ๐ (v : E)) := by
apply le_antisymm
ยท rw [le_iInf_iff]
rintro โจv, hvโฉ w hw
simpa using hw _ hv
ยท intro v hv w hw
simp only [mem_iInf] at hv
exact hv โจw, hwโฉ
#align submodule.orthogonal_eq_inter Submodule.orthogonal_eq_inter
/-- The orthogonal complement of any submodule `K` is closed. -/
theorem isClosed_orthogonal : IsClosed (Kแฎ : Set E) := by
rw [orthogonal_eq_inter K]
have := fun v : K => ContinuousLinearMap.isClosed_ker (innerSL ๐ (v : E))
convert isClosed_iInter this
simp only [iInf_coe]
#align submodule.is_closed_orthogonal Submodule.isClosed_orthogonal
/-- In a complete space, the orthogonal complement of any submodule `K` is complete. -/
instance instOrthogonalCompleteSpace [CompleteSpace E] : CompleteSpace Kแฎ :=
K.isClosed_orthogonal.completeSpace_coe
variable (๐ E)
/-- `orthogonal` gives a `GaloisConnection` between
`Submodule ๐ E` and its `OrderDual`. -/
theorem orthogonal_gc :
@GaloisConnection (Submodule ๐ E) (Submodule ๐ E)แตแต _ _ orthogonal orthogonal := fun _Kโ _Kโ =>
โจfun h _v hv _u hu => inner_left_of_mem_orthogonal hv (h hu), fun h _v hv _u hu =>
inner_left_of_mem_orthogonal hv (h hu)โฉ
#align submodule.orthogonal_gc Submodule.orthogonal_gc
variable {๐ E}
/-- `orthogonal` reverses the `โค` ordering of two
subspaces. -/
theorem orthogonal_le {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎ โค Kโแฎ :=
(orthogonal_gc ๐ E).monotone_l h
#align submodule.orthogonal_le Submodule.orthogonal_le
/-- `orthogonal.orthogonal` preserves the `โค` ordering of two
subspaces. -/
theorem orthogonal_orthogonal_monotone {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎแฎ โค Kโแฎแฎ :=
orthogonal_le (orthogonal_le h)
#align submodule.orthogonal_orthogonal_monotone Submodule.orthogonal_orthogonal_monotone
/-- `K` is contained in `Kแฎแฎ`. -/
theorem le_orthogonal_orthogonal : K โค Kแฎแฎ :=
(orthogonal_gc ๐ E).le_u_l _
#align submodule.le_orthogonal_orthogonal Submodule.le_orthogonal_orthogonal
/-- The inf of two orthogonal subspaces equals the subspace orthogonal
to the sup. -/
theorem inf_orthogonal (Kโ Kโ : Submodule ๐ E) : Kโแฎ โ Kโแฎ = (Kโ โ Kโ)แฎ :=
(orthogonal_gc ๐ E).l_sup.symm
#align submodule.inf_orthogonal Submodule.inf_orthogonal
/-- The inf of an indexed family of orthogonal subspaces equals the
subspace orthogonal to the sup. -/
theorem iInf_orthogonal {ฮน : Type*} (K : ฮน โ Submodule ๐ E) : โจ
i, (K i)แฎ = (iSup K)แฎ :=
(orthogonal_gc ๐ E).l_iSup.symm
#align submodule.infi_orthogonal Submodule.iInf_orthogonal
/-- The inf of a set of orthogonal subspaces equals the subspace orthogonal to the sup. -/
theorem sInf_orthogonal (s : Set <| Submodule ๐ E) : โจ
K โ s, Kแฎ = (sSup s)แฎ :=
(orthogonal_gc ๐ E).l_sSup.symm
#align submodule.Inf_orthogonal Submodule.sInf_orthogonal
@[simp]
theorem top_orthogonal_eq_bot : (โค : Submodule ๐ E)แฎ = โฅ := by
ext x
|
rw [mem_bot, mem_orthogonal]
|
@[simp]
theorem top_orthogonal_eq_bot : (โค : Submodule ๐ E)แฎ = โฅ := by
ext x
|
Mathlib.Analysis.InnerProductSpace.Orthogonal.186_0.QXx0GYqLoAbtfq1
|
@[simp]
theorem top_orthogonal_eq_bot : (โค : Submodule ๐ E)แฎ = โฅ
|
Mathlib_Analysis_InnerProductSpace_Orthogonal
|
case h
๐ : Type u_1
E : Type u_2
F : Type u_3
instโโด : IsROrC ๐
instโยณ : NormedAddCommGroup E
instโยฒ : InnerProductSpace ๐ E
instโยน : NormedAddCommGroup F
instโ : InnerProductSpace ๐ F
K : Submodule ๐ E
x : E
โข (โ u โ โค, inner u x = 0) โ x = 0
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sรฉbastien Gouรซzel, Frรฉdรฉric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
/-!
# Orthogonal complements of submodules
In this file, the `orthogonal` complement of a submodule `K` is defined, and basic API established.
Some of the more subtle results about the orthogonal complement are delayed to
`Analysis.InnerProductSpace.Projection`.
See also `BilinForm.orthogonal` for orthogonality with respect to a general bilinear form.
## Notation
The orthogonal complement of a submodule `K` is denoted by `Kแฎ`.
The proposition that two submodules are orthogonal, `Submodule.IsOrtho`, is denoted by `U โ V`.
Note this is not the same unicode symbol as `โฅ` (`Bot`).
-/
variable {๐ E F : Type*} [IsROrC ๐]
variable [NormedAddCommGroup E] [InnerProductSpace ๐ E]
variable [NormedAddCommGroup F] [InnerProductSpace ๐ F]
local notation "โช" x ", " y "โซ" => @inner ๐ _ _ x y
namespace Submodule
variable (K : Submodule ๐ E)
/-- The subspace of vectors orthogonal to a given subspace. -/
def orthogonal : Submodule ๐ E where
carrier := { v | โ u โ K, โชu, vโซ = 0 }
zero_mem' _ _ := inner_zero_right _
add_mem' hx hy u hu := by rw [inner_add_right, hx u hu, hy u hu, add_zero]
smul_mem' c x hx u hu := by rw [inner_smul_right, hx u hu, mul_zero]
#align submodule.orthogonal Submodule.orthogonal
@[inherit_doc]
notation:1200 K "แฎ" => orthogonal K
/-- When a vector is in `Kแฎ`. -/
theorem mem_orthogonal (v : E) : v โ Kแฎ โ โ u โ K, โชu, vโซ = 0 :=
Iff.rfl
#align submodule.mem_orthogonal Submodule.mem_orthogonal
/-- When a vector is in `Kแฎ`, with the inner product the
other way round. -/
theorem mem_orthogonal' (v : E) : v โ Kแฎ โ โ u โ K, โชv, uโซ = 0 := by
simp_rw [mem_orthogonal, inner_eq_zero_symm]
#align submodule.mem_orthogonal' Submodule.mem_orthogonal'
variable {K}
/-- A vector in `K` is orthogonal to one in `Kแฎ`. -/
theorem inner_right_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชu, vโซ = 0 :=
(K.mem_orthogonal v).1 hv u hu
#align submodule.inner_right_of_mem_orthogonal Submodule.inner_right_of_mem_orthogonal
/-- A vector in `Kแฎ` is orthogonal to one in `K`. -/
theorem inner_left_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชv, uโซ = 0 := by
rw [inner_eq_zero_symm]; exact inner_right_of_mem_orthogonal hu hv
#align submodule.inner_left_of_mem_orthogonal Submodule.inner_left_of_mem_orthogonal
/-- A vector is in `(๐ โ u)แฎ` iff it is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_right {u v : E} : v โ (๐ โ u)แฎ โ โชu, vโซ = 0 := by
refine' โจinner_right_of_mem_orthogonal (mem_span_singleton_self u), _โฉ
intro hv w hw
rw [mem_span_singleton] at hw
obtain โจc, rflโฉ := hw
simp [inner_smul_left, hv]
#align submodule.mem_orthogonal_singleton_iff_inner_right Submodule.mem_orthogonal_singleton_iff_inner_right
/-- A vector in `(๐ โ u)แฎ` is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_left {u v : E} : v โ (๐ โ u)แฎ โ โชv, uโซ = 0 := by
rw [mem_orthogonal_singleton_iff_inner_right, inner_eq_zero_symm]
#align submodule.mem_orthogonal_singleton_iff_inner_left Submodule.mem_orthogonal_singleton_iff_inner_left
theorem sub_mem_orthogonal_of_inner_left {x y : E} (h : โ v : K, โชx, vโซ = โชy, vโซ) : x - y โ Kแฎ := by
rw [mem_orthogonal']
intro u hu
rw [inner_sub_left, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_left Submodule.sub_mem_orthogonal_of_inner_left
theorem sub_mem_orthogonal_of_inner_right {x y : E} (h : โ v : K, โช(v : E), xโซ = โช(v : E), yโซ) :
x - y โ Kแฎ := by
intro u hu
rw [inner_sub_right, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_right Submodule.sub_mem_orthogonal_of_inner_right
variable (K)
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem inf_orthogonal_eq_bot : K โ Kแฎ = โฅ := by
rw [eq_bot_iff]
intro x
rw [mem_inf]
exact fun โจhx, hoโฉ => inner_self_eq_zero.1 (ho x hx)
#align submodule.inf_orthogonal_eq_bot Submodule.inf_orthogonal_eq_bot
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem orthogonal_disjoint : Disjoint K Kแฎ := by simp [disjoint_iff, K.inf_orthogonal_eq_bot]
#align submodule.orthogonal_disjoint Submodule.orthogonal_disjoint
/-- `Kแฎ` can be characterized as the intersection of the kernels of the operations of
inner product with each of the elements of `K`. -/
theorem orthogonal_eq_inter : Kแฎ = โจ
v : K, LinearMap.ker (innerSL ๐ (v : E)) := by
apply le_antisymm
ยท rw [le_iInf_iff]
rintro โจv, hvโฉ w hw
simpa using hw _ hv
ยท intro v hv w hw
simp only [mem_iInf] at hv
exact hv โจw, hwโฉ
#align submodule.orthogonal_eq_inter Submodule.orthogonal_eq_inter
/-- The orthogonal complement of any submodule `K` is closed. -/
theorem isClosed_orthogonal : IsClosed (Kแฎ : Set E) := by
rw [orthogonal_eq_inter K]
have := fun v : K => ContinuousLinearMap.isClosed_ker (innerSL ๐ (v : E))
convert isClosed_iInter this
simp only [iInf_coe]
#align submodule.is_closed_orthogonal Submodule.isClosed_orthogonal
/-- In a complete space, the orthogonal complement of any submodule `K` is complete. -/
instance instOrthogonalCompleteSpace [CompleteSpace E] : CompleteSpace Kแฎ :=
K.isClosed_orthogonal.completeSpace_coe
variable (๐ E)
/-- `orthogonal` gives a `GaloisConnection` between
`Submodule ๐ E` and its `OrderDual`. -/
theorem orthogonal_gc :
@GaloisConnection (Submodule ๐ E) (Submodule ๐ E)แตแต _ _ orthogonal orthogonal := fun _Kโ _Kโ =>
โจfun h _v hv _u hu => inner_left_of_mem_orthogonal hv (h hu), fun h _v hv _u hu =>
inner_left_of_mem_orthogonal hv (h hu)โฉ
#align submodule.orthogonal_gc Submodule.orthogonal_gc
variable {๐ E}
/-- `orthogonal` reverses the `โค` ordering of two
subspaces. -/
theorem orthogonal_le {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎ โค Kโแฎ :=
(orthogonal_gc ๐ E).monotone_l h
#align submodule.orthogonal_le Submodule.orthogonal_le
/-- `orthogonal.orthogonal` preserves the `โค` ordering of two
subspaces. -/
theorem orthogonal_orthogonal_monotone {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎแฎ โค Kโแฎแฎ :=
orthogonal_le (orthogonal_le h)
#align submodule.orthogonal_orthogonal_monotone Submodule.orthogonal_orthogonal_monotone
/-- `K` is contained in `Kแฎแฎ`. -/
theorem le_orthogonal_orthogonal : K โค Kแฎแฎ :=
(orthogonal_gc ๐ E).le_u_l _
#align submodule.le_orthogonal_orthogonal Submodule.le_orthogonal_orthogonal
/-- The inf of two orthogonal subspaces equals the subspace orthogonal
to the sup. -/
theorem inf_orthogonal (Kโ Kโ : Submodule ๐ E) : Kโแฎ โ Kโแฎ = (Kโ โ Kโ)แฎ :=
(orthogonal_gc ๐ E).l_sup.symm
#align submodule.inf_orthogonal Submodule.inf_orthogonal
/-- The inf of an indexed family of orthogonal subspaces equals the
subspace orthogonal to the sup. -/
theorem iInf_orthogonal {ฮน : Type*} (K : ฮน โ Submodule ๐ E) : โจ
i, (K i)แฎ = (iSup K)แฎ :=
(orthogonal_gc ๐ E).l_iSup.symm
#align submodule.infi_orthogonal Submodule.iInf_orthogonal
/-- The inf of a set of orthogonal subspaces equals the subspace orthogonal to the sup. -/
theorem sInf_orthogonal (s : Set <| Submodule ๐ E) : โจ
K โ s, Kแฎ = (sSup s)แฎ :=
(orthogonal_gc ๐ E).l_sSup.symm
#align submodule.Inf_orthogonal Submodule.sInf_orthogonal
@[simp]
theorem top_orthogonal_eq_bot : (โค : Submodule ๐ E)แฎ = โฅ := by
ext x
rw [mem_bot, mem_orthogonal]
|
exact
โจfun h => inner_self_eq_zero.mp (h x mem_top), by
rintro rfl
simpโฉ
|
@[simp]
theorem top_orthogonal_eq_bot : (โค : Submodule ๐ E)แฎ = โฅ := by
ext x
rw [mem_bot, mem_orthogonal]
|
Mathlib.Analysis.InnerProductSpace.Orthogonal.186_0.QXx0GYqLoAbtfq1
|
@[simp]
theorem top_orthogonal_eq_bot : (โค : Submodule ๐ E)แฎ = โฅ
|
Mathlib_Analysis_InnerProductSpace_Orthogonal
|
๐ : Type u_1
E : Type u_2
F : Type u_3
instโโด : IsROrC ๐
instโยณ : NormedAddCommGroup E
instโยฒ : InnerProductSpace ๐ E
instโยน : NormedAddCommGroup F
instโ : InnerProductSpace ๐ F
K : Submodule ๐ E
x : E
โข x = 0 โ โ u โ โค, inner u x = 0
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sรฉbastien Gouรซzel, Frรฉdรฉric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
/-!
# Orthogonal complements of submodules
In this file, the `orthogonal` complement of a submodule `K` is defined, and basic API established.
Some of the more subtle results about the orthogonal complement are delayed to
`Analysis.InnerProductSpace.Projection`.
See also `BilinForm.orthogonal` for orthogonality with respect to a general bilinear form.
## Notation
The orthogonal complement of a submodule `K` is denoted by `Kแฎ`.
The proposition that two submodules are orthogonal, `Submodule.IsOrtho`, is denoted by `U โ V`.
Note this is not the same unicode symbol as `โฅ` (`Bot`).
-/
variable {๐ E F : Type*} [IsROrC ๐]
variable [NormedAddCommGroup E] [InnerProductSpace ๐ E]
variable [NormedAddCommGroup F] [InnerProductSpace ๐ F]
local notation "โช" x ", " y "โซ" => @inner ๐ _ _ x y
namespace Submodule
variable (K : Submodule ๐ E)
/-- The subspace of vectors orthogonal to a given subspace. -/
def orthogonal : Submodule ๐ E where
carrier := { v | โ u โ K, โชu, vโซ = 0 }
zero_mem' _ _ := inner_zero_right _
add_mem' hx hy u hu := by rw [inner_add_right, hx u hu, hy u hu, add_zero]
smul_mem' c x hx u hu := by rw [inner_smul_right, hx u hu, mul_zero]
#align submodule.orthogonal Submodule.orthogonal
@[inherit_doc]
notation:1200 K "แฎ" => orthogonal K
/-- When a vector is in `Kแฎ`. -/
theorem mem_orthogonal (v : E) : v โ Kแฎ โ โ u โ K, โชu, vโซ = 0 :=
Iff.rfl
#align submodule.mem_orthogonal Submodule.mem_orthogonal
/-- When a vector is in `Kแฎ`, with the inner product the
other way round. -/
theorem mem_orthogonal' (v : E) : v โ Kแฎ โ โ u โ K, โชv, uโซ = 0 := by
simp_rw [mem_orthogonal, inner_eq_zero_symm]
#align submodule.mem_orthogonal' Submodule.mem_orthogonal'
variable {K}
/-- A vector in `K` is orthogonal to one in `Kแฎ`. -/
theorem inner_right_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชu, vโซ = 0 :=
(K.mem_orthogonal v).1 hv u hu
#align submodule.inner_right_of_mem_orthogonal Submodule.inner_right_of_mem_orthogonal
/-- A vector in `Kแฎ` is orthogonal to one in `K`. -/
theorem inner_left_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชv, uโซ = 0 := by
rw [inner_eq_zero_symm]; exact inner_right_of_mem_orthogonal hu hv
#align submodule.inner_left_of_mem_orthogonal Submodule.inner_left_of_mem_orthogonal
/-- A vector is in `(๐ โ u)แฎ` iff it is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_right {u v : E} : v โ (๐ โ u)แฎ โ โชu, vโซ = 0 := by
refine' โจinner_right_of_mem_orthogonal (mem_span_singleton_self u), _โฉ
intro hv w hw
rw [mem_span_singleton] at hw
obtain โจc, rflโฉ := hw
simp [inner_smul_left, hv]
#align submodule.mem_orthogonal_singleton_iff_inner_right Submodule.mem_orthogonal_singleton_iff_inner_right
/-- A vector in `(๐ โ u)แฎ` is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_left {u v : E} : v โ (๐ โ u)แฎ โ โชv, uโซ = 0 := by
rw [mem_orthogonal_singleton_iff_inner_right, inner_eq_zero_symm]
#align submodule.mem_orthogonal_singleton_iff_inner_left Submodule.mem_orthogonal_singleton_iff_inner_left
theorem sub_mem_orthogonal_of_inner_left {x y : E} (h : โ v : K, โชx, vโซ = โชy, vโซ) : x - y โ Kแฎ := by
rw [mem_orthogonal']
intro u hu
rw [inner_sub_left, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_left Submodule.sub_mem_orthogonal_of_inner_left
theorem sub_mem_orthogonal_of_inner_right {x y : E} (h : โ v : K, โช(v : E), xโซ = โช(v : E), yโซ) :
x - y โ Kแฎ := by
intro u hu
rw [inner_sub_right, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_right Submodule.sub_mem_orthogonal_of_inner_right
variable (K)
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem inf_orthogonal_eq_bot : K โ Kแฎ = โฅ := by
rw [eq_bot_iff]
intro x
rw [mem_inf]
exact fun โจhx, hoโฉ => inner_self_eq_zero.1 (ho x hx)
#align submodule.inf_orthogonal_eq_bot Submodule.inf_orthogonal_eq_bot
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem orthogonal_disjoint : Disjoint K Kแฎ := by simp [disjoint_iff, K.inf_orthogonal_eq_bot]
#align submodule.orthogonal_disjoint Submodule.orthogonal_disjoint
/-- `Kแฎ` can be characterized as the intersection of the kernels of the operations of
inner product with each of the elements of `K`. -/
theorem orthogonal_eq_inter : Kแฎ = โจ
v : K, LinearMap.ker (innerSL ๐ (v : E)) := by
apply le_antisymm
ยท rw [le_iInf_iff]
rintro โจv, hvโฉ w hw
simpa using hw _ hv
ยท intro v hv w hw
simp only [mem_iInf] at hv
exact hv โจw, hwโฉ
#align submodule.orthogonal_eq_inter Submodule.orthogonal_eq_inter
/-- The orthogonal complement of any submodule `K` is closed. -/
theorem isClosed_orthogonal : IsClosed (Kแฎ : Set E) := by
rw [orthogonal_eq_inter K]
have := fun v : K => ContinuousLinearMap.isClosed_ker (innerSL ๐ (v : E))
convert isClosed_iInter this
simp only [iInf_coe]
#align submodule.is_closed_orthogonal Submodule.isClosed_orthogonal
/-- In a complete space, the orthogonal complement of any submodule `K` is complete. -/
instance instOrthogonalCompleteSpace [CompleteSpace E] : CompleteSpace Kแฎ :=
K.isClosed_orthogonal.completeSpace_coe
variable (๐ E)
/-- `orthogonal` gives a `GaloisConnection` between
`Submodule ๐ E` and its `OrderDual`. -/
theorem orthogonal_gc :
@GaloisConnection (Submodule ๐ E) (Submodule ๐ E)แตแต _ _ orthogonal orthogonal := fun _Kโ _Kโ =>
โจfun h _v hv _u hu => inner_left_of_mem_orthogonal hv (h hu), fun h _v hv _u hu =>
inner_left_of_mem_orthogonal hv (h hu)โฉ
#align submodule.orthogonal_gc Submodule.orthogonal_gc
variable {๐ E}
/-- `orthogonal` reverses the `โค` ordering of two
subspaces. -/
theorem orthogonal_le {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎ โค Kโแฎ :=
(orthogonal_gc ๐ E).monotone_l h
#align submodule.orthogonal_le Submodule.orthogonal_le
/-- `orthogonal.orthogonal` preserves the `โค` ordering of two
subspaces. -/
theorem orthogonal_orthogonal_monotone {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎแฎ โค Kโแฎแฎ :=
orthogonal_le (orthogonal_le h)
#align submodule.orthogonal_orthogonal_monotone Submodule.orthogonal_orthogonal_monotone
/-- `K` is contained in `Kแฎแฎ`. -/
theorem le_orthogonal_orthogonal : K โค Kแฎแฎ :=
(orthogonal_gc ๐ E).le_u_l _
#align submodule.le_orthogonal_orthogonal Submodule.le_orthogonal_orthogonal
/-- The inf of two orthogonal subspaces equals the subspace orthogonal
to the sup. -/
theorem inf_orthogonal (Kโ Kโ : Submodule ๐ E) : Kโแฎ โ Kโแฎ = (Kโ โ Kโ)แฎ :=
(orthogonal_gc ๐ E).l_sup.symm
#align submodule.inf_orthogonal Submodule.inf_orthogonal
/-- The inf of an indexed family of orthogonal subspaces equals the
subspace orthogonal to the sup. -/
theorem iInf_orthogonal {ฮน : Type*} (K : ฮน โ Submodule ๐ E) : โจ
i, (K i)แฎ = (iSup K)แฎ :=
(orthogonal_gc ๐ E).l_iSup.symm
#align submodule.infi_orthogonal Submodule.iInf_orthogonal
/-- The inf of a set of orthogonal subspaces equals the subspace orthogonal to the sup. -/
theorem sInf_orthogonal (s : Set <| Submodule ๐ E) : โจ
K โ s, Kแฎ = (sSup s)แฎ :=
(orthogonal_gc ๐ E).l_sSup.symm
#align submodule.Inf_orthogonal Submodule.sInf_orthogonal
@[simp]
theorem top_orthogonal_eq_bot : (โค : Submodule ๐ E)แฎ = โฅ := by
ext x
rw [mem_bot, mem_orthogonal]
exact
โจfun h => inner_self_eq_zero.mp (h x mem_top), by
|
rintro rfl
|
@[simp]
theorem top_orthogonal_eq_bot : (โค : Submodule ๐ E)แฎ = โฅ := by
ext x
rw [mem_bot, mem_orthogonal]
exact
โจfun h => inner_self_eq_zero.mp (h x mem_top), by
|
Mathlib.Analysis.InnerProductSpace.Orthogonal.186_0.QXx0GYqLoAbtfq1
|
@[simp]
theorem top_orthogonal_eq_bot : (โค : Submodule ๐ E)แฎ = โฅ
|
Mathlib_Analysis_InnerProductSpace_Orthogonal
|
๐ : Type u_1
E : Type u_2
F : Type u_3
instโโด : IsROrC ๐
instโยณ : NormedAddCommGroup E
instโยฒ : InnerProductSpace ๐ E
instโยน : NormedAddCommGroup F
instโ : InnerProductSpace ๐ F
K : Submodule ๐ E
โข โ u โ โค, inner u 0 = 0
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sรฉbastien Gouรซzel, Frรฉdรฉric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
/-!
# Orthogonal complements of submodules
In this file, the `orthogonal` complement of a submodule `K` is defined, and basic API established.
Some of the more subtle results about the orthogonal complement are delayed to
`Analysis.InnerProductSpace.Projection`.
See also `BilinForm.orthogonal` for orthogonality with respect to a general bilinear form.
## Notation
The orthogonal complement of a submodule `K` is denoted by `Kแฎ`.
The proposition that two submodules are orthogonal, `Submodule.IsOrtho`, is denoted by `U โ V`.
Note this is not the same unicode symbol as `โฅ` (`Bot`).
-/
variable {๐ E F : Type*} [IsROrC ๐]
variable [NormedAddCommGroup E] [InnerProductSpace ๐ E]
variable [NormedAddCommGroup F] [InnerProductSpace ๐ F]
local notation "โช" x ", " y "โซ" => @inner ๐ _ _ x y
namespace Submodule
variable (K : Submodule ๐ E)
/-- The subspace of vectors orthogonal to a given subspace. -/
def orthogonal : Submodule ๐ E where
carrier := { v | โ u โ K, โชu, vโซ = 0 }
zero_mem' _ _ := inner_zero_right _
add_mem' hx hy u hu := by rw [inner_add_right, hx u hu, hy u hu, add_zero]
smul_mem' c x hx u hu := by rw [inner_smul_right, hx u hu, mul_zero]
#align submodule.orthogonal Submodule.orthogonal
@[inherit_doc]
notation:1200 K "แฎ" => orthogonal K
/-- When a vector is in `Kแฎ`. -/
theorem mem_orthogonal (v : E) : v โ Kแฎ โ โ u โ K, โชu, vโซ = 0 :=
Iff.rfl
#align submodule.mem_orthogonal Submodule.mem_orthogonal
/-- When a vector is in `Kแฎ`, with the inner product the
other way round. -/
theorem mem_orthogonal' (v : E) : v โ Kแฎ โ โ u โ K, โชv, uโซ = 0 := by
simp_rw [mem_orthogonal, inner_eq_zero_symm]
#align submodule.mem_orthogonal' Submodule.mem_orthogonal'
variable {K}
/-- A vector in `K` is orthogonal to one in `Kแฎ`. -/
theorem inner_right_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชu, vโซ = 0 :=
(K.mem_orthogonal v).1 hv u hu
#align submodule.inner_right_of_mem_orthogonal Submodule.inner_right_of_mem_orthogonal
/-- A vector in `Kแฎ` is orthogonal to one in `K`. -/
theorem inner_left_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชv, uโซ = 0 := by
rw [inner_eq_zero_symm]; exact inner_right_of_mem_orthogonal hu hv
#align submodule.inner_left_of_mem_orthogonal Submodule.inner_left_of_mem_orthogonal
/-- A vector is in `(๐ โ u)แฎ` iff it is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_right {u v : E} : v โ (๐ โ u)แฎ โ โชu, vโซ = 0 := by
refine' โจinner_right_of_mem_orthogonal (mem_span_singleton_self u), _โฉ
intro hv w hw
rw [mem_span_singleton] at hw
obtain โจc, rflโฉ := hw
simp [inner_smul_left, hv]
#align submodule.mem_orthogonal_singleton_iff_inner_right Submodule.mem_orthogonal_singleton_iff_inner_right
/-- A vector in `(๐ โ u)แฎ` is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_left {u v : E} : v โ (๐ โ u)แฎ โ โชv, uโซ = 0 := by
rw [mem_orthogonal_singleton_iff_inner_right, inner_eq_zero_symm]
#align submodule.mem_orthogonal_singleton_iff_inner_left Submodule.mem_orthogonal_singleton_iff_inner_left
theorem sub_mem_orthogonal_of_inner_left {x y : E} (h : โ v : K, โชx, vโซ = โชy, vโซ) : x - y โ Kแฎ := by
rw [mem_orthogonal']
intro u hu
rw [inner_sub_left, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_left Submodule.sub_mem_orthogonal_of_inner_left
theorem sub_mem_orthogonal_of_inner_right {x y : E} (h : โ v : K, โช(v : E), xโซ = โช(v : E), yโซ) :
x - y โ Kแฎ := by
intro u hu
rw [inner_sub_right, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_right Submodule.sub_mem_orthogonal_of_inner_right
variable (K)
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem inf_orthogonal_eq_bot : K โ Kแฎ = โฅ := by
rw [eq_bot_iff]
intro x
rw [mem_inf]
exact fun โจhx, hoโฉ => inner_self_eq_zero.1 (ho x hx)
#align submodule.inf_orthogonal_eq_bot Submodule.inf_orthogonal_eq_bot
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem orthogonal_disjoint : Disjoint K Kแฎ := by simp [disjoint_iff, K.inf_orthogonal_eq_bot]
#align submodule.orthogonal_disjoint Submodule.orthogonal_disjoint
/-- `Kแฎ` can be characterized as the intersection of the kernels of the operations of
inner product with each of the elements of `K`. -/
theorem orthogonal_eq_inter : Kแฎ = โจ
v : K, LinearMap.ker (innerSL ๐ (v : E)) := by
apply le_antisymm
ยท rw [le_iInf_iff]
rintro โจv, hvโฉ w hw
simpa using hw _ hv
ยท intro v hv w hw
simp only [mem_iInf] at hv
exact hv โจw, hwโฉ
#align submodule.orthogonal_eq_inter Submodule.orthogonal_eq_inter
/-- The orthogonal complement of any submodule `K` is closed. -/
theorem isClosed_orthogonal : IsClosed (Kแฎ : Set E) := by
rw [orthogonal_eq_inter K]
have := fun v : K => ContinuousLinearMap.isClosed_ker (innerSL ๐ (v : E))
convert isClosed_iInter this
simp only [iInf_coe]
#align submodule.is_closed_orthogonal Submodule.isClosed_orthogonal
/-- In a complete space, the orthogonal complement of any submodule `K` is complete. -/
instance instOrthogonalCompleteSpace [CompleteSpace E] : CompleteSpace Kแฎ :=
K.isClosed_orthogonal.completeSpace_coe
variable (๐ E)
/-- `orthogonal` gives a `GaloisConnection` between
`Submodule ๐ E` and its `OrderDual`. -/
theorem orthogonal_gc :
@GaloisConnection (Submodule ๐ E) (Submodule ๐ E)แตแต _ _ orthogonal orthogonal := fun _Kโ _Kโ =>
โจfun h _v hv _u hu => inner_left_of_mem_orthogonal hv (h hu), fun h _v hv _u hu =>
inner_left_of_mem_orthogonal hv (h hu)โฉ
#align submodule.orthogonal_gc Submodule.orthogonal_gc
variable {๐ E}
/-- `orthogonal` reverses the `โค` ordering of two
subspaces. -/
theorem orthogonal_le {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎ โค Kโแฎ :=
(orthogonal_gc ๐ E).monotone_l h
#align submodule.orthogonal_le Submodule.orthogonal_le
/-- `orthogonal.orthogonal` preserves the `โค` ordering of two
subspaces. -/
theorem orthogonal_orthogonal_monotone {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎแฎ โค Kโแฎแฎ :=
orthogonal_le (orthogonal_le h)
#align submodule.orthogonal_orthogonal_monotone Submodule.orthogonal_orthogonal_monotone
/-- `K` is contained in `Kแฎแฎ`. -/
theorem le_orthogonal_orthogonal : K โค Kแฎแฎ :=
(orthogonal_gc ๐ E).le_u_l _
#align submodule.le_orthogonal_orthogonal Submodule.le_orthogonal_orthogonal
/-- The inf of two orthogonal subspaces equals the subspace orthogonal
to the sup. -/
theorem inf_orthogonal (Kโ Kโ : Submodule ๐ E) : Kโแฎ โ Kโแฎ = (Kโ โ Kโ)แฎ :=
(orthogonal_gc ๐ E).l_sup.symm
#align submodule.inf_orthogonal Submodule.inf_orthogonal
/-- The inf of an indexed family of orthogonal subspaces equals the
subspace orthogonal to the sup. -/
theorem iInf_orthogonal {ฮน : Type*} (K : ฮน โ Submodule ๐ E) : โจ
i, (K i)แฎ = (iSup K)แฎ :=
(orthogonal_gc ๐ E).l_iSup.symm
#align submodule.infi_orthogonal Submodule.iInf_orthogonal
/-- The inf of a set of orthogonal subspaces equals the subspace orthogonal to the sup. -/
theorem sInf_orthogonal (s : Set <| Submodule ๐ E) : โจ
K โ s, Kแฎ = (sSup s)แฎ :=
(orthogonal_gc ๐ E).l_sSup.symm
#align submodule.Inf_orthogonal Submodule.sInf_orthogonal
@[simp]
theorem top_orthogonal_eq_bot : (โค : Submodule ๐ E)แฎ = โฅ := by
ext x
rw [mem_bot, mem_orthogonal]
exact
โจfun h => inner_self_eq_zero.mp (h x mem_top), by
rintro rfl
|
simp
|
@[simp]
theorem top_orthogonal_eq_bot : (โค : Submodule ๐ E)แฎ = โฅ := by
ext x
rw [mem_bot, mem_orthogonal]
exact
โจfun h => inner_self_eq_zero.mp (h x mem_top), by
rintro rfl
|
Mathlib.Analysis.InnerProductSpace.Orthogonal.186_0.QXx0GYqLoAbtfq1
|
@[simp]
theorem top_orthogonal_eq_bot : (โค : Submodule ๐ E)แฎ = โฅ
|
Mathlib_Analysis_InnerProductSpace_Orthogonal
|
๐ : Type u_1
E : Type u_2
F : Type u_3
instโโด : IsROrC ๐
instโยณ : NormedAddCommGroup E
instโยฒ : InnerProductSpace ๐ E
instโยน : NormedAddCommGroup F
instโ : InnerProductSpace ๐ F
K : Submodule ๐ E
โข โฅแฎ = โค
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sรฉbastien Gouรซzel, Frรฉdรฉric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
/-!
# Orthogonal complements of submodules
In this file, the `orthogonal` complement of a submodule `K` is defined, and basic API established.
Some of the more subtle results about the orthogonal complement are delayed to
`Analysis.InnerProductSpace.Projection`.
See also `BilinForm.orthogonal` for orthogonality with respect to a general bilinear form.
## Notation
The orthogonal complement of a submodule `K` is denoted by `Kแฎ`.
The proposition that two submodules are orthogonal, `Submodule.IsOrtho`, is denoted by `U โ V`.
Note this is not the same unicode symbol as `โฅ` (`Bot`).
-/
variable {๐ E F : Type*} [IsROrC ๐]
variable [NormedAddCommGroup E] [InnerProductSpace ๐ E]
variable [NormedAddCommGroup F] [InnerProductSpace ๐ F]
local notation "โช" x ", " y "โซ" => @inner ๐ _ _ x y
namespace Submodule
variable (K : Submodule ๐ E)
/-- The subspace of vectors orthogonal to a given subspace. -/
def orthogonal : Submodule ๐ E where
carrier := { v | โ u โ K, โชu, vโซ = 0 }
zero_mem' _ _ := inner_zero_right _
add_mem' hx hy u hu := by rw [inner_add_right, hx u hu, hy u hu, add_zero]
smul_mem' c x hx u hu := by rw [inner_smul_right, hx u hu, mul_zero]
#align submodule.orthogonal Submodule.orthogonal
@[inherit_doc]
notation:1200 K "แฎ" => orthogonal K
/-- When a vector is in `Kแฎ`. -/
theorem mem_orthogonal (v : E) : v โ Kแฎ โ โ u โ K, โชu, vโซ = 0 :=
Iff.rfl
#align submodule.mem_orthogonal Submodule.mem_orthogonal
/-- When a vector is in `Kแฎ`, with the inner product the
other way round. -/
theorem mem_orthogonal' (v : E) : v โ Kแฎ โ โ u โ K, โชv, uโซ = 0 := by
simp_rw [mem_orthogonal, inner_eq_zero_symm]
#align submodule.mem_orthogonal' Submodule.mem_orthogonal'
variable {K}
/-- A vector in `K` is orthogonal to one in `Kแฎ`. -/
theorem inner_right_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชu, vโซ = 0 :=
(K.mem_orthogonal v).1 hv u hu
#align submodule.inner_right_of_mem_orthogonal Submodule.inner_right_of_mem_orthogonal
/-- A vector in `Kแฎ` is orthogonal to one in `K`. -/
theorem inner_left_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชv, uโซ = 0 := by
rw [inner_eq_zero_symm]; exact inner_right_of_mem_orthogonal hu hv
#align submodule.inner_left_of_mem_orthogonal Submodule.inner_left_of_mem_orthogonal
/-- A vector is in `(๐ โ u)แฎ` iff it is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_right {u v : E} : v โ (๐ โ u)แฎ โ โชu, vโซ = 0 := by
refine' โจinner_right_of_mem_orthogonal (mem_span_singleton_self u), _โฉ
intro hv w hw
rw [mem_span_singleton] at hw
obtain โจc, rflโฉ := hw
simp [inner_smul_left, hv]
#align submodule.mem_orthogonal_singleton_iff_inner_right Submodule.mem_orthogonal_singleton_iff_inner_right
/-- A vector in `(๐ โ u)แฎ` is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_left {u v : E} : v โ (๐ โ u)แฎ โ โชv, uโซ = 0 := by
rw [mem_orthogonal_singleton_iff_inner_right, inner_eq_zero_symm]
#align submodule.mem_orthogonal_singleton_iff_inner_left Submodule.mem_orthogonal_singleton_iff_inner_left
theorem sub_mem_orthogonal_of_inner_left {x y : E} (h : โ v : K, โชx, vโซ = โชy, vโซ) : x - y โ Kแฎ := by
rw [mem_orthogonal']
intro u hu
rw [inner_sub_left, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_left Submodule.sub_mem_orthogonal_of_inner_left
theorem sub_mem_orthogonal_of_inner_right {x y : E} (h : โ v : K, โช(v : E), xโซ = โช(v : E), yโซ) :
x - y โ Kแฎ := by
intro u hu
rw [inner_sub_right, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_right Submodule.sub_mem_orthogonal_of_inner_right
variable (K)
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem inf_orthogonal_eq_bot : K โ Kแฎ = โฅ := by
rw [eq_bot_iff]
intro x
rw [mem_inf]
exact fun โจhx, hoโฉ => inner_self_eq_zero.1 (ho x hx)
#align submodule.inf_orthogonal_eq_bot Submodule.inf_orthogonal_eq_bot
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem orthogonal_disjoint : Disjoint K Kแฎ := by simp [disjoint_iff, K.inf_orthogonal_eq_bot]
#align submodule.orthogonal_disjoint Submodule.orthogonal_disjoint
/-- `Kแฎ` can be characterized as the intersection of the kernels of the operations of
inner product with each of the elements of `K`. -/
theorem orthogonal_eq_inter : Kแฎ = โจ
v : K, LinearMap.ker (innerSL ๐ (v : E)) := by
apply le_antisymm
ยท rw [le_iInf_iff]
rintro โจv, hvโฉ w hw
simpa using hw _ hv
ยท intro v hv w hw
simp only [mem_iInf] at hv
exact hv โจw, hwโฉ
#align submodule.orthogonal_eq_inter Submodule.orthogonal_eq_inter
/-- The orthogonal complement of any submodule `K` is closed. -/
theorem isClosed_orthogonal : IsClosed (Kแฎ : Set E) := by
rw [orthogonal_eq_inter K]
have := fun v : K => ContinuousLinearMap.isClosed_ker (innerSL ๐ (v : E))
convert isClosed_iInter this
simp only [iInf_coe]
#align submodule.is_closed_orthogonal Submodule.isClosed_orthogonal
/-- In a complete space, the orthogonal complement of any submodule `K` is complete. -/
instance instOrthogonalCompleteSpace [CompleteSpace E] : CompleteSpace Kแฎ :=
K.isClosed_orthogonal.completeSpace_coe
variable (๐ E)
/-- `orthogonal` gives a `GaloisConnection` between
`Submodule ๐ E` and its `OrderDual`. -/
theorem orthogonal_gc :
@GaloisConnection (Submodule ๐ E) (Submodule ๐ E)แตแต _ _ orthogonal orthogonal := fun _Kโ _Kโ =>
โจfun h _v hv _u hu => inner_left_of_mem_orthogonal hv (h hu), fun h _v hv _u hu =>
inner_left_of_mem_orthogonal hv (h hu)โฉ
#align submodule.orthogonal_gc Submodule.orthogonal_gc
variable {๐ E}
/-- `orthogonal` reverses the `โค` ordering of two
subspaces. -/
theorem orthogonal_le {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎ โค Kโแฎ :=
(orthogonal_gc ๐ E).monotone_l h
#align submodule.orthogonal_le Submodule.orthogonal_le
/-- `orthogonal.orthogonal` preserves the `โค` ordering of two
subspaces. -/
theorem orthogonal_orthogonal_monotone {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎแฎ โค Kโแฎแฎ :=
orthogonal_le (orthogonal_le h)
#align submodule.orthogonal_orthogonal_monotone Submodule.orthogonal_orthogonal_monotone
/-- `K` is contained in `Kแฎแฎ`. -/
theorem le_orthogonal_orthogonal : K โค Kแฎแฎ :=
(orthogonal_gc ๐ E).le_u_l _
#align submodule.le_orthogonal_orthogonal Submodule.le_orthogonal_orthogonal
/-- The inf of two orthogonal subspaces equals the subspace orthogonal
to the sup. -/
theorem inf_orthogonal (Kโ Kโ : Submodule ๐ E) : Kโแฎ โ Kโแฎ = (Kโ โ Kโ)แฎ :=
(orthogonal_gc ๐ E).l_sup.symm
#align submodule.inf_orthogonal Submodule.inf_orthogonal
/-- The inf of an indexed family of orthogonal subspaces equals the
subspace orthogonal to the sup. -/
theorem iInf_orthogonal {ฮน : Type*} (K : ฮน โ Submodule ๐ E) : โจ
i, (K i)แฎ = (iSup K)แฎ :=
(orthogonal_gc ๐ E).l_iSup.symm
#align submodule.infi_orthogonal Submodule.iInf_orthogonal
/-- The inf of a set of orthogonal subspaces equals the subspace orthogonal to the sup. -/
theorem sInf_orthogonal (s : Set <| Submodule ๐ E) : โจ
K โ s, Kแฎ = (sSup s)แฎ :=
(orthogonal_gc ๐ E).l_sSup.symm
#align submodule.Inf_orthogonal Submodule.sInf_orthogonal
@[simp]
theorem top_orthogonal_eq_bot : (โค : Submodule ๐ E)แฎ = โฅ := by
ext x
rw [mem_bot, mem_orthogonal]
exact
โจfun h => inner_self_eq_zero.mp (h x mem_top), by
rintro rfl
simpโฉ
#align submodule.top_orthogonal_eq_bot Submodule.top_orthogonal_eq_bot
@[simp]
theorem bot_orthogonal_eq_top : (โฅ : Submodule ๐ E)แฎ = โค := by
|
rw [โ top_orthogonal_eq_bot, eq_top_iff]
|
@[simp]
theorem bot_orthogonal_eq_top : (โฅ : Submodule ๐ E)แฎ = โค := by
|
Mathlib.Analysis.InnerProductSpace.Orthogonal.196_0.QXx0GYqLoAbtfq1
|
@[simp]
theorem bot_orthogonal_eq_top : (โฅ : Submodule ๐ E)แฎ = โค
|
Mathlib_Analysis_InnerProductSpace_Orthogonal
|
๐ : Type u_1
E : Type u_2
F : Type u_3
instโโด : IsROrC ๐
instโยณ : NormedAddCommGroup E
instโยฒ : InnerProductSpace ๐ E
instโยน : NormedAddCommGroup F
instโ : InnerProductSpace ๐ F
K : Submodule ๐ E
โข โค โค โคแฎแฎ
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sรฉbastien Gouรซzel, Frรฉdรฉric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
/-!
# Orthogonal complements of submodules
In this file, the `orthogonal` complement of a submodule `K` is defined, and basic API established.
Some of the more subtle results about the orthogonal complement are delayed to
`Analysis.InnerProductSpace.Projection`.
See also `BilinForm.orthogonal` for orthogonality with respect to a general bilinear form.
## Notation
The orthogonal complement of a submodule `K` is denoted by `Kแฎ`.
The proposition that two submodules are orthogonal, `Submodule.IsOrtho`, is denoted by `U โ V`.
Note this is not the same unicode symbol as `โฅ` (`Bot`).
-/
variable {๐ E F : Type*} [IsROrC ๐]
variable [NormedAddCommGroup E] [InnerProductSpace ๐ E]
variable [NormedAddCommGroup F] [InnerProductSpace ๐ F]
local notation "โช" x ", " y "โซ" => @inner ๐ _ _ x y
namespace Submodule
variable (K : Submodule ๐ E)
/-- The subspace of vectors orthogonal to a given subspace. -/
def orthogonal : Submodule ๐ E where
carrier := { v | โ u โ K, โชu, vโซ = 0 }
zero_mem' _ _ := inner_zero_right _
add_mem' hx hy u hu := by rw [inner_add_right, hx u hu, hy u hu, add_zero]
smul_mem' c x hx u hu := by rw [inner_smul_right, hx u hu, mul_zero]
#align submodule.orthogonal Submodule.orthogonal
@[inherit_doc]
notation:1200 K "แฎ" => orthogonal K
/-- When a vector is in `Kแฎ`. -/
theorem mem_orthogonal (v : E) : v โ Kแฎ โ โ u โ K, โชu, vโซ = 0 :=
Iff.rfl
#align submodule.mem_orthogonal Submodule.mem_orthogonal
/-- When a vector is in `Kแฎ`, with the inner product the
other way round. -/
theorem mem_orthogonal' (v : E) : v โ Kแฎ โ โ u โ K, โชv, uโซ = 0 := by
simp_rw [mem_orthogonal, inner_eq_zero_symm]
#align submodule.mem_orthogonal' Submodule.mem_orthogonal'
variable {K}
/-- A vector in `K` is orthogonal to one in `Kแฎ`. -/
theorem inner_right_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชu, vโซ = 0 :=
(K.mem_orthogonal v).1 hv u hu
#align submodule.inner_right_of_mem_orthogonal Submodule.inner_right_of_mem_orthogonal
/-- A vector in `Kแฎ` is orthogonal to one in `K`. -/
theorem inner_left_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชv, uโซ = 0 := by
rw [inner_eq_zero_symm]; exact inner_right_of_mem_orthogonal hu hv
#align submodule.inner_left_of_mem_orthogonal Submodule.inner_left_of_mem_orthogonal
/-- A vector is in `(๐ โ u)แฎ` iff it is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_right {u v : E} : v โ (๐ โ u)แฎ โ โชu, vโซ = 0 := by
refine' โจinner_right_of_mem_orthogonal (mem_span_singleton_self u), _โฉ
intro hv w hw
rw [mem_span_singleton] at hw
obtain โจc, rflโฉ := hw
simp [inner_smul_left, hv]
#align submodule.mem_orthogonal_singleton_iff_inner_right Submodule.mem_orthogonal_singleton_iff_inner_right
/-- A vector in `(๐ โ u)แฎ` is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_left {u v : E} : v โ (๐ โ u)แฎ โ โชv, uโซ = 0 := by
rw [mem_orthogonal_singleton_iff_inner_right, inner_eq_zero_symm]
#align submodule.mem_orthogonal_singleton_iff_inner_left Submodule.mem_orthogonal_singleton_iff_inner_left
theorem sub_mem_orthogonal_of_inner_left {x y : E} (h : โ v : K, โชx, vโซ = โชy, vโซ) : x - y โ Kแฎ := by
rw [mem_orthogonal']
intro u hu
rw [inner_sub_left, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_left Submodule.sub_mem_orthogonal_of_inner_left
theorem sub_mem_orthogonal_of_inner_right {x y : E} (h : โ v : K, โช(v : E), xโซ = โช(v : E), yโซ) :
x - y โ Kแฎ := by
intro u hu
rw [inner_sub_right, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_right Submodule.sub_mem_orthogonal_of_inner_right
variable (K)
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem inf_orthogonal_eq_bot : K โ Kแฎ = โฅ := by
rw [eq_bot_iff]
intro x
rw [mem_inf]
exact fun โจhx, hoโฉ => inner_self_eq_zero.1 (ho x hx)
#align submodule.inf_orthogonal_eq_bot Submodule.inf_orthogonal_eq_bot
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem orthogonal_disjoint : Disjoint K Kแฎ := by simp [disjoint_iff, K.inf_orthogonal_eq_bot]
#align submodule.orthogonal_disjoint Submodule.orthogonal_disjoint
/-- `Kแฎ` can be characterized as the intersection of the kernels of the operations of
inner product with each of the elements of `K`. -/
theorem orthogonal_eq_inter : Kแฎ = โจ
v : K, LinearMap.ker (innerSL ๐ (v : E)) := by
apply le_antisymm
ยท rw [le_iInf_iff]
rintro โจv, hvโฉ w hw
simpa using hw _ hv
ยท intro v hv w hw
simp only [mem_iInf] at hv
exact hv โจw, hwโฉ
#align submodule.orthogonal_eq_inter Submodule.orthogonal_eq_inter
/-- The orthogonal complement of any submodule `K` is closed. -/
theorem isClosed_orthogonal : IsClosed (Kแฎ : Set E) := by
rw [orthogonal_eq_inter K]
have := fun v : K => ContinuousLinearMap.isClosed_ker (innerSL ๐ (v : E))
convert isClosed_iInter this
simp only [iInf_coe]
#align submodule.is_closed_orthogonal Submodule.isClosed_orthogonal
/-- In a complete space, the orthogonal complement of any submodule `K` is complete. -/
instance instOrthogonalCompleteSpace [CompleteSpace E] : CompleteSpace Kแฎ :=
K.isClosed_orthogonal.completeSpace_coe
variable (๐ E)
/-- `orthogonal` gives a `GaloisConnection` between
`Submodule ๐ E` and its `OrderDual`. -/
theorem orthogonal_gc :
@GaloisConnection (Submodule ๐ E) (Submodule ๐ E)แตแต _ _ orthogonal orthogonal := fun _Kโ _Kโ =>
โจfun h _v hv _u hu => inner_left_of_mem_orthogonal hv (h hu), fun h _v hv _u hu =>
inner_left_of_mem_orthogonal hv (h hu)โฉ
#align submodule.orthogonal_gc Submodule.orthogonal_gc
variable {๐ E}
/-- `orthogonal` reverses the `โค` ordering of two
subspaces. -/
theorem orthogonal_le {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎ โค Kโแฎ :=
(orthogonal_gc ๐ E).monotone_l h
#align submodule.orthogonal_le Submodule.orthogonal_le
/-- `orthogonal.orthogonal` preserves the `โค` ordering of two
subspaces. -/
theorem orthogonal_orthogonal_monotone {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎแฎ โค Kโแฎแฎ :=
orthogonal_le (orthogonal_le h)
#align submodule.orthogonal_orthogonal_monotone Submodule.orthogonal_orthogonal_monotone
/-- `K` is contained in `Kแฎแฎ`. -/
theorem le_orthogonal_orthogonal : K โค Kแฎแฎ :=
(orthogonal_gc ๐ E).le_u_l _
#align submodule.le_orthogonal_orthogonal Submodule.le_orthogonal_orthogonal
/-- The inf of two orthogonal subspaces equals the subspace orthogonal
to the sup. -/
theorem inf_orthogonal (Kโ Kโ : Submodule ๐ E) : Kโแฎ โ Kโแฎ = (Kโ โ Kโ)แฎ :=
(orthogonal_gc ๐ E).l_sup.symm
#align submodule.inf_orthogonal Submodule.inf_orthogonal
/-- The inf of an indexed family of orthogonal subspaces equals the
subspace orthogonal to the sup. -/
theorem iInf_orthogonal {ฮน : Type*} (K : ฮน โ Submodule ๐ E) : โจ
i, (K i)แฎ = (iSup K)แฎ :=
(orthogonal_gc ๐ E).l_iSup.symm
#align submodule.infi_orthogonal Submodule.iInf_orthogonal
/-- The inf of a set of orthogonal subspaces equals the subspace orthogonal to the sup. -/
theorem sInf_orthogonal (s : Set <| Submodule ๐ E) : โจ
K โ s, Kแฎ = (sSup s)แฎ :=
(orthogonal_gc ๐ E).l_sSup.symm
#align submodule.Inf_orthogonal Submodule.sInf_orthogonal
@[simp]
theorem top_orthogonal_eq_bot : (โค : Submodule ๐ E)แฎ = โฅ := by
ext x
rw [mem_bot, mem_orthogonal]
exact
โจfun h => inner_self_eq_zero.mp (h x mem_top), by
rintro rfl
simpโฉ
#align submodule.top_orthogonal_eq_bot Submodule.top_orthogonal_eq_bot
@[simp]
theorem bot_orthogonal_eq_top : (โฅ : Submodule ๐ E)แฎ = โค := by
rw [โ top_orthogonal_eq_bot, eq_top_iff]
|
exact le_orthogonal_orthogonal โค
|
@[simp]
theorem bot_orthogonal_eq_top : (โฅ : Submodule ๐ E)แฎ = โค := by
rw [โ top_orthogonal_eq_bot, eq_top_iff]
|
Mathlib.Analysis.InnerProductSpace.Orthogonal.196_0.QXx0GYqLoAbtfq1
|
@[simp]
theorem bot_orthogonal_eq_top : (โฅ : Submodule ๐ E)แฎ = โค
|
Mathlib_Analysis_InnerProductSpace_Orthogonal
|
๐ : Type u_1
E : Type u_2
F : Type u_3
instโโด : IsROrC ๐
instโยณ : NormedAddCommGroup E
instโยฒ : InnerProductSpace ๐ E
instโยน : NormedAddCommGroup F
instโ : InnerProductSpace ๐ F
K : Submodule ๐ E
โข Kแฎ = โค โ K = โฅ
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sรฉbastien Gouรซzel, Frรฉdรฉric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
/-!
# Orthogonal complements of submodules
In this file, the `orthogonal` complement of a submodule `K` is defined, and basic API established.
Some of the more subtle results about the orthogonal complement are delayed to
`Analysis.InnerProductSpace.Projection`.
See also `BilinForm.orthogonal` for orthogonality with respect to a general bilinear form.
## Notation
The orthogonal complement of a submodule `K` is denoted by `Kแฎ`.
The proposition that two submodules are orthogonal, `Submodule.IsOrtho`, is denoted by `U โ V`.
Note this is not the same unicode symbol as `โฅ` (`Bot`).
-/
variable {๐ E F : Type*} [IsROrC ๐]
variable [NormedAddCommGroup E] [InnerProductSpace ๐ E]
variable [NormedAddCommGroup F] [InnerProductSpace ๐ F]
local notation "โช" x ", " y "โซ" => @inner ๐ _ _ x y
namespace Submodule
variable (K : Submodule ๐ E)
/-- The subspace of vectors orthogonal to a given subspace. -/
def orthogonal : Submodule ๐ E where
carrier := { v | โ u โ K, โชu, vโซ = 0 }
zero_mem' _ _ := inner_zero_right _
add_mem' hx hy u hu := by rw [inner_add_right, hx u hu, hy u hu, add_zero]
smul_mem' c x hx u hu := by rw [inner_smul_right, hx u hu, mul_zero]
#align submodule.orthogonal Submodule.orthogonal
@[inherit_doc]
notation:1200 K "แฎ" => orthogonal K
/-- When a vector is in `Kแฎ`. -/
theorem mem_orthogonal (v : E) : v โ Kแฎ โ โ u โ K, โชu, vโซ = 0 :=
Iff.rfl
#align submodule.mem_orthogonal Submodule.mem_orthogonal
/-- When a vector is in `Kแฎ`, with the inner product the
other way round. -/
theorem mem_orthogonal' (v : E) : v โ Kแฎ โ โ u โ K, โชv, uโซ = 0 := by
simp_rw [mem_orthogonal, inner_eq_zero_symm]
#align submodule.mem_orthogonal' Submodule.mem_orthogonal'
variable {K}
/-- A vector in `K` is orthogonal to one in `Kแฎ`. -/
theorem inner_right_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชu, vโซ = 0 :=
(K.mem_orthogonal v).1 hv u hu
#align submodule.inner_right_of_mem_orthogonal Submodule.inner_right_of_mem_orthogonal
/-- A vector in `Kแฎ` is orthogonal to one in `K`. -/
theorem inner_left_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชv, uโซ = 0 := by
rw [inner_eq_zero_symm]; exact inner_right_of_mem_orthogonal hu hv
#align submodule.inner_left_of_mem_orthogonal Submodule.inner_left_of_mem_orthogonal
/-- A vector is in `(๐ โ u)แฎ` iff it is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_right {u v : E} : v โ (๐ โ u)แฎ โ โชu, vโซ = 0 := by
refine' โจinner_right_of_mem_orthogonal (mem_span_singleton_self u), _โฉ
intro hv w hw
rw [mem_span_singleton] at hw
obtain โจc, rflโฉ := hw
simp [inner_smul_left, hv]
#align submodule.mem_orthogonal_singleton_iff_inner_right Submodule.mem_orthogonal_singleton_iff_inner_right
/-- A vector in `(๐ โ u)แฎ` is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_left {u v : E} : v โ (๐ โ u)แฎ โ โชv, uโซ = 0 := by
rw [mem_orthogonal_singleton_iff_inner_right, inner_eq_zero_symm]
#align submodule.mem_orthogonal_singleton_iff_inner_left Submodule.mem_orthogonal_singleton_iff_inner_left
theorem sub_mem_orthogonal_of_inner_left {x y : E} (h : โ v : K, โชx, vโซ = โชy, vโซ) : x - y โ Kแฎ := by
rw [mem_orthogonal']
intro u hu
rw [inner_sub_left, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_left Submodule.sub_mem_orthogonal_of_inner_left
theorem sub_mem_orthogonal_of_inner_right {x y : E} (h : โ v : K, โช(v : E), xโซ = โช(v : E), yโซ) :
x - y โ Kแฎ := by
intro u hu
rw [inner_sub_right, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_right Submodule.sub_mem_orthogonal_of_inner_right
variable (K)
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem inf_orthogonal_eq_bot : K โ Kแฎ = โฅ := by
rw [eq_bot_iff]
intro x
rw [mem_inf]
exact fun โจhx, hoโฉ => inner_self_eq_zero.1 (ho x hx)
#align submodule.inf_orthogonal_eq_bot Submodule.inf_orthogonal_eq_bot
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem orthogonal_disjoint : Disjoint K Kแฎ := by simp [disjoint_iff, K.inf_orthogonal_eq_bot]
#align submodule.orthogonal_disjoint Submodule.orthogonal_disjoint
/-- `Kแฎ` can be characterized as the intersection of the kernels of the operations of
inner product with each of the elements of `K`. -/
theorem orthogonal_eq_inter : Kแฎ = โจ
v : K, LinearMap.ker (innerSL ๐ (v : E)) := by
apply le_antisymm
ยท rw [le_iInf_iff]
rintro โจv, hvโฉ w hw
simpa using hw _ hv
ยท intro v hv w hw
simp only [mem_iInf] at hv
exact hv โจw, hwโฉ
#align submodule.orthogonal_eq_inter Submodule.orthogonal_eq_inter
/-- The orthogonal complement of any submodule `K` is closed. -/
theorem isClosed_orthogonal : IsClosed (Kแฎ : Set E) := by
rw [orthogonal_eq_inter K]
have := fun v : K => ContinuousLinearMap.isClosed_ker (innerSL ๐ (v : E))
convert isClosed_iInter this
simp only [iInf_coe]
#align submodule.is_closed_orthogonal Submodule.isClosed_orthogonal
/-- In a complete space, the orthogonal complement of any submodule `K` is complete. -/
instance instOrthogonalCompleteSpace [CompleteSpace E] : CompleteSpace Kแฎ :=
K.isClosed_orthogonal.completeSpace_coe
variable (๐ E)
/-- `orthogonal` gives a `GaloisConnection` between
`Submodule ๐ E` and its `OrderDual`. -/
theorem orthogonal_gc :
@GaloisConnection (Submodule ๐ E) (Submodule ๐ E)แตแต _ _ orthogonal orthogonal := fun _Kโ _Kโ =>
โจfun h _v hv _u hu => inner_left_of_mem_orthogonal hv (h hu), fun h _v hv _u hu =>
inner_left_of_mem_orthogonal hv (h hu)โฉ
#align submodule.orthogonal_gc Submodule.orthogonal_gc
variable {๐ E}
/-- `orthogonal` reverses the `โค` ordering of two
subspaces. -/
theorem orthogonal_le {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎ โค Kโแฎ :=
(orthogonal_gc ๐ E).monotone_l h
#align submodule.orthogonal_le Submodule.orthogonal_le
/-- `orthogonal.orthogonal` preserves the `โค` ordering of two
subspaces. -/
theorem orthogonal_orthogonal_monotone {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎแฎ โค Kโแฎแฎ :=
orthogonal_le (orthogonal_le h)
#align submodule.orthogonal_orthogonal_monotone Submodule.orthogonal_orthogonal_monotone
/-- `K` is contained in `Kแฎแฎ`. -/
theorem le_orthogonal_orthogonal : K โค Kแฎแฎ :=
(orthogonal_gc ๐ E).le_u_l _
#align submodule.le_orthogonal_orthogonal Submodule.le_orthogonal_orthogonal
/-- The inf of two orthogonal subspaces equals the subspace orthogonal
to the sup. -/
theorem inf_orthogonal (Kโ Kโ : Submodule ๐ E) : Kโแฎ โ Kโแฎ = (Kโ โ Kโ)แฎ :=
(orthogonal_gc ๐ E).l_sup.symm
#align submodule.inf_orthogonal Submodule.inf_orthogonal
/-- The inf of an indexed family of orthogonal subspaces equals the
subspace orthogonal to the sup. -/
theorem iInf_orthogonal {ฮน : Type*} (K : ฮน โ Submodule ๐ E) : โจ
i, (K i)แฎ = (iSup K)แฎ :=
(orthogonal_gc ๐ E).l_iSup.symm
#align submodule.infi_orthogonal Submodule.iInf_orthogonal
/-- The inf of a set of orthogonal subspaces equals the subspace orthogonal to the sup. -/
theorem sInf_orthogonal (s : Set <| Submodule ๐ E) : โจ
K โ s, Kแฎ = (sSup s)แฎ :=
(orthogonal_gc ๐ E).l_sSup.symm
#align submodule.Inf_orthogonal Submodule.sInf_orthogonal
@[simp]
theorem top_orthogonal_eq_bot : (โค : Submodule ๐ E)แฎ = โฅ := by
ext x
rw [mem_bot, mem_orthogonal]
exact
โจfun h => inner_self_eq_zero.mp (h x mem_top), by
rintro rfl
simpโฉ
#align submodule.top_orthogonal_eq_bot Submodule.top_orthogonal_eq_bot
@[simp]
theorem bot_orthogonal_eq_top : (โฅ : Submodule ๐ E)แฎ = โค := by
rw [โ top_orthogonal_eq_bot, eq_top_iff]
exact le_orthogonal_orthogonal โค
#align submodule.bot_orthogonal_eq_top Submodule.bot_orthogonal_eq_top
@[simp]
theorem orthogonal_eq_top_iff : Kแฎ = โค โ K = โฅ := by
|
refine'
โจ_, by
rintro rfl
exact bot_orthogonal_eq_topโฉ
|
@[simp]
theorem orthogonal_eq_top_iff : Kแฎ = โค โ K = โฅ := by
|
Mathlib.Analysis.InnerProductSpace.Orthogonal.202_0.QXx0GYqLoAbtfq1
|
@[simp]
theorem orthogonal_eq_top_iff : Kแฎ = โค โ K = โฅ
|
Mathlib_Analysis_InnerProductSpace_Orthogonal
|
๐ : Type u_1
E : Type u_2
F : Type u_3
instโโด : IsROrC ๐
instโยณ : NormedAddCommGroup E
instโยฒ : InnerProductSpace ๐ E
instโยน : NormedAddCommGroup F
instโ : InnerProductSpace ๐ F
K : Submodule ๐ E
โข K = โฅ โ Kแฎ = โค
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sรฉbastien Gouรซzel, Frรฉdรฉric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
/-!
# Orthogonal complements of submodules
In this file, the `orthogonal` complement of a submodule `K` is defined, and basic API established.
Some of the more subtle results about the orthogonal complement are delayed to
`Analysis.InnerProductSpace.Projection`.
See also `BilinForm.orthogonal` for orthogonality with respect to a general bilinear form.
## Notation
The orthogonal complement of a submodule `K` is denoted by `Kแฎ`.
The proposition that two submodules are orthogonal, `Submodule.IsOrtho`, is denoted by `U โ V`.
Note this is not the same unicode symbol as `โฅ` (`Bot`).
-/
variable {๐ E F : Type*} [IsROrC ๐]
variable [NormedAddCommGroup E] [InnerProductSpace ๐ E]
variable [NormedAddCommGroup F] [InnerProductSpace ๐ F]
local notation "โช" x ", " y "โซ" => @inner ๐ _ _ x y
namespace Submodule
variable (K : Submodule ๐ E)
/-- The subspace of vectors orthogonal to a given subspace. -/
def orthogonal : Submodule ๐ E where
carrier := { v | โ u โ K, โชu, vโซ = 0 }
zero_mem' _ _ := inner_zero_right _
add_mem' hx hy u hu := by rw [inner_add_right, hx u hu, hy u hu, add_zero]
smul_mem' c x hx u hu := by rw [inner_smul_right, hx u hu, mul_zero]
#align submodule.orthogonal Submodule.orthogonal
@[inherit_doc]
notation:1200 K "แฎ" => orthogonal K
/-- When a vector is in `Kแฎ`. -/
theorem mem_orthogonal (v : E) : v โ Kแฎ โ โ u โ K, โชu, vโซ = 0 :=
Iff.rfl
#align submodule.mem_orthogonal Submodule.mem_orthogonal
/-- When a vector is in `Kแฎ`, with the inner product the
other way round. -/
theorem mem_orthogonal' (v : E) : v โ Kแฎ โ โ u โ K, โชv, uโซ = 0 := by
simp_rw [mem_orthogonal, inner_eq_zero_symm]
#align submodule.mem_orthogonal' Submodule.mem_orthogonal'
variable {K}
/-- A vector in `K` is orthogonal to one in `Kแฎ`. -/
theorem inner_right_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชu, vโซ = 0 :=
(K.mem_orthogonal v).1 hv u hu
#align submodule.inner_right_of_mem_orthogonal Submodule.inner_right_of_mem_orthogonal
/-- A vector in `Kแฎ` is orthogonal to one in `K`. -/
theorem inner_left_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชv, uโซ = 0 := by
rw [inner_eq_zero_symm]; exact inner_right_of_mem_orthogonal hu hv
#align submodule.inner_left_of_mem_orthogonal Submodule.inner_left_of_mem_orthogonal
/-- A vector is in `(๐ โ u)แฎ` iff it is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_right {u v : E} : v โ (๐ โ u)แฎ โ โชu, vโซ = 0 := by
refine' โจinner_right_of_mem_orthogonal (mem_span_singleton_self u), _โฉ
intro hv w hw
rw [mem_span_singleton] at hw
obtain โจc, rflโฉ := hw
simp [inner_smul_left, hv]
#align submodule.mem_orthogonal_singleton_iff_inner_right Submodule.mem_orthogonal_singleton_iff_inner_right
/-- A vector in `(๐ โ u)แฎ` is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_left {u v : E} : v โ (๐ โ u)แฎ โ โชv, uโซ = 0 := by
rw [mem_orthogonal_singleton_iff_inner_right, inner_eq_zero_symm]
#align submodule.mem_orthogonal_singleton_iff_inner_left Submodule.mem_orthogonal_singleton_iff_inner_left
theorem sub_mem_orthogonal_of_inner_left {x y : E} (h : โ v : K, โชx, vโซ = โชy, vโซ) : x - y โ Kแฎ := by
rw [mem_orthogonal']
intro u hu
rw [inner_sub_left, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_left Submodule.sub_mem_orthogonal_of_inner_left
theorem sub_mem_orthogonal_of_inner_right {x y : E} (h : โ v : K, โช(v : E), xโซ = โช(v : E), yโซ) :
x - y โ Kแฎ := by
intro u hu
rw [inner_sub_right, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_right Submodule.sub_mem_orthogonal_of_inner_right
variable (K)
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem inf_orthogonal_eq_bot : K โ Kแฎ = โฅ := by
rw [eq_bot_iff]
intro x
rw [mem_inf]
exact fun โจhx, hoโฉ => inner_self_eq_zero.1 (ho x hx)
#align submodule.inf_orthogonal_eq_bot Submodule.inf_orthogonal_eq_bot
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem orthogonal_disjoint : Disjoint K Kแฎ := by simp [disjoint_iff, K.inf_orthogonal_eq_bot]
#align submodule.orthogonal_disjoint Submodule.orthogonal_disjoint
/-- `Kแฎ` can be characterized as the intersection of the kernels of the operations of
inner product with each of the elements of `K`. -/
theorem orthogonal_eq_inter : Kแฎ = โจ
v : K, LinearMap.ker (innerSL ๐ (v : E)) := by
apply le_antisymm
ยท rw [le_iInf_iff]
rintro โจv, hvโฉ w hw
simpa using hw _ hv
ยท intro v hv w hw
simp only [mem_iInf] at hv
exact hv โจw, hwโฉ
#align submodule.orthogonal_eq_inter Submodule.orthogonal_eq_inter
/-- The orthogonal complement of any submodule `K` is closed. -/
theorem isClosed_orthogonal : IsClosed (Kแฎ : Set E) := by
rw [orthogonal_eq_inter K]
have := fun v : K => ContinuousLinearMap.isClosed_ker (innerSL ๐ (v : E))
convert isClosed_iInter this
simp only [iInf_coe]
#align submodule.is_closed_orthogonal Submodule.isClosed_orthogonal
/-- In a complete space, the orthogonal complement of any submodule `K` is complete. -/
instance instOrthogonalCompleteSpace [CompleteSpace E] : CompleteSpace Kแฎ :=
K.isClosed_orthogonal.completeSpace_coe
variable (๐ E)
/-- `orthogonal` gives a `GaloisConnection` between
`Submodule ๐ E` and its `OrderDual`. -/
theorem orthogonal_gc :
@GaloisConnection (Submodule ๐ E) (Submodule ๐ E)แตแต _ _ orthogonal orthogonal := fun _Kโ _Kโ =>
โจfun h _v hv _u hu => inner_left_of_mem_orthogonal hv (h hu), fun h _v hv _u hu =>
inner_left_of_mem_orthogonal hv (h hu)โฉ
#align submodule.orthogonal_gc Submodule.orthogonal_gc
variable {๐ E}
/-- `orthogonal` reverses the `โค` ordering of two
subspaces. -/
theorem orthogonal_le {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎ โค Kโแฎ :=
(orthogonal_gc ๐ E).monotone_l h
#align submodule.orthogonal_le Submodule.orthogonal_le
/-- `orthogonal.orthogonal` preserves the `โค` ordering of two
subspaces. -/
theorem orthogonal_orthogonal_monotone {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎแฎ โค Kโแฎแฎ :=
orthogonal_le (orthogonal_le h)
#align submodule.orthogonal_orthogonal_monotone Submodule.orthogonal_orthogonal_monotone
/-- `K` is contained in `Kแฎแฎ`. -/
theorem le_orthogonal_orthogonal : K โค Kแฎแฎ :=
(orthogonal_gc ๐ E).le_u_l _
#align submodule.le_orthogonal_orthogonal Submodule.le_orthogonal_orthogonal
/-- The inf of two orthogonal subspaces equals the subspace orthogonal
to the sup. -/
theorem inf_orthogonal (Kโ Kโ : Submodule ๐ E) : Kโแฎ โ Kโแฎ = (Kโ โ Kโ)แฎ :=
(orthogonal_gc ๐ E).l_sup.symm
#align submodule.inf_orthogonal Submodule.inf_orthogonal
/-- The inf of an indexed family of orthogonal subspaces equals the
subspace orthogonal to the sup. -/
theorem iInf_orthogonal {ฮน : Type*} (K : ฮน โ Submodule ๐ E) : โจ
i, (K i)แฎ = (iSup K)แฎ :=
(orthogonal_gc ๐ E).l_iSup.symm
#align submodule.infi_orthogonal Submodule.iInf_orthogonal
/-- The inf of a set of orthogonal subspaces equals the subspace orthogonal to the sup. -/
theorem sInf_orthogonal (s : Set <| Submodule ๐ E) : โจ
K โ s, Kแฎ = (sSup s)แฎ :=
(orthogonal_gc ๐ E).l_sSup.symm
#align submodule.Inf_orthogonal Submodule.sInf_orthogonal
@[simp]
theorem top_orthogonal_eq_bot : (โค : Submodule ๐ E)แฎ = โฅ := by
ext x
rw [mem_bot, mem_orthogonal]
exact
โจfun h => inner_self_eq_zero.mp (h x mem_top), by
rintro rfl
simpโฉ
#align submodule.top_orthogonal_eq_bot Submodule.top_orthogonal_eq_bot
@[simp]
theorem bot_orthogonal_eq_top : (โฅ : Submodule ๐ E)แฎ = โค := by
rw [โ top_orthogonal_eq_bot, eq_top_iff]
exact le_orthogonal_orthogonal โค
#align submodule.bot_orthogonal_eq_top Submodule.bot_orthogonal_eq_top
@[simp]
theorem orthogonal_eq_top_iff : Kแฎ = โค โ K = โฅ := by
refine'
โจ_, by
|
rintro rfl
|
@[simp]
theorem orthogonal_eq_top_iff : Kแฎ = โค โ K = โฅ := by
refine'
โจ_, by
|
Mathlib.Analysis.InnerProductSpace.Orthogonal.202_0.QXx0GYqLoAbtfq1
|
@[simp]
theorem orthogonal_eq_top_iff : Kแฎ = โค โ K = โฅ
|
Mathlib_Analysis_InnerProductSpace_Orthogonal
|
๐ : Type u_1
E : Type u_2
F : Type u_3
instโโด : IsROrC ๐
instโยณ : NormedAddCommGroup E
instโยฒ : InnerProductSpace ๐ E
instโยน : NormedAddCommGroup F
instโ : InnerProductSpace ๐ F
โข โฅแฎ = โค
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sรฉbastien Gouรซzel, Frรฉdรฉric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
/-!
# Orthogonal complements of submodules
In this file, the `orthogonal` complement of a submodule `K` is defined, and basic API established.
Some of the more subtle results about the orthogonal complement are delayed to
`Analysis.InnerProductSpace.Projection`.
See also `BilinForm.orthogonal` for orthogonality with respect to a general bilinear form.
## Notation
The orthogonal complement of a submodule `K` is denoted by `Kแฎ`.
The proposition that two submodules are orthogonal, `Submodule.IsOrtho`, is denoted by `U โ V`.
Note this is not the same unicode symbol as `โฅ` (`Bot`).
-/
variable {๐ E F : Type*} [IsROrC ๐]
variable [NormedAddCommGroup E] [InnerProductSpace ๐ E]
variable [NormedAddCommGroup F] [InnerProductSpace ๐ F]
local notation "โช" x ", " y "โซ" => @inner ๐ _ _ x y
namespace Submodule
variable (K : Submodule ๐ E)
/-- The subspace of vectors orthogonal to a given subspace. -/
def orthogonal : Submodule ๐ E where
carrier := { v | โ u โ K, โชu, vโซ = 0 }
zero_mem' _ _ := inner_zero_right _
add_mem' hx hy u hu := by rw [inner_add_right, hx u hu, hy u hu, add_zero]
smul_mem' c x hx u hu := by rw [inner_smul_right, hx u hu, mul_zero]
#align submodule.orthogonal Submodule.orthogonal
@[inherit_doc]
notation:1200 K "แฎ" => orthogonal K
/-- When a vector is in `Kแฎ`. -/
theorem mem_orthogonal (v : E) : v โ Kแฎ โ โ u โ K, โชu, vโซ = 0 :=
Iff.rfl
#align submodule.mem_orthogonal Submodule.mem_orthogonal
/-- When a vector is in `Kแฎ`, with the inner product the
other way round. -/
theorem mem_orthogonal' (v : E) : v โ Kแฎ โ โ u โ K, โชv, uโซ = 0 := by
simp_rw [mem_orthogonal, inner_eq_zero_symm]
#align submodule.mem_orthogonal' Submodule.mem_orthogonal'
variable {K}
/-- A vector in `K` is orthogonal to one in `Kแฎ`. -/
theorem inner_right_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชu, vโซ = 0 :=
(K.mem_orthogonal v).1 hv u hu
#align submodule.inner_right_of_mem_orthogonal Submodule.inner_right_of_mem_orthogonal
/-- A vector in `Kแฎ` is orthogonal to one in `K`. -/
theorem inner_left_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชv, uโซ = 0 := by
rw [inner_eq_zero_symm]; exact inner_right_of_mem_orthogonal hu hv
#align submodule.inner_left_of_mem_orthogonal Submodule.inner_left_of_mem_orthogonal
/-- A vector is in `(๐ โ u)แฎ` iff it is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_right {u v : E} : v โ (๐ โ u)แฎ โ โชu, vโซ = 0 := by
refine' โจinner_right_of_mem_orthogonal (mem_span_singleton_self u), _โฉ
intro hv w hw
rw [mem_span_singleton] at hw
obtain โจc, rflโฉ := hw
simp [inner_smul_left, hv]
#align submodule.mem_orthogonal_singleton_iff_inner_right Submodule.mem_orthogonal_singleton_iff_inner_right
/-- A vector in `(๐ โ u)แฎ` is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_left {u v : E} : v โ (๐ โ u)แฎ โ โชv, uโซ = 0 := by
rw [mem_orthogonal_singleton_iff_inner_right, inner_eq_zero_symm]
#align submodule.mem_orthogonal_singleton_iff_inner_left Submodule.mem_orthogonal_singleton_iff_inner_left
theorem sub_mem_orthogonal_of_inner_left {x y : E} (h : โ v : K, โชx, vโซ = โชy, vโซ) : x - y โ Kแฎ := by
rw [mem_orthogonal']
intro u hu
rw [inner_sub_left, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_left Submodule.sub_mem_orthogonal_of_inner_left
theorem sub_mem_orthogonal_of_inner_right {x y : E} (h : โ v : K, โช(v : E), xโซ = โช(v : E), yโซ) :
x - y โ Kแฎ := by
intro u hu
rw [inner_sub_right, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_right Submodule.sub_mem_orthogonal_of_inner_right
variable (K)
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem inf_orthogonal_eq_bot : K โ Kแฎ = โฅ := by
rw [eq_bot_iff]
intro x
rw [mem_inf]
exact fun โจhx, hoโฉ => inner_self_eq_zero.1 (ho x hx)
#align submodule.inf_orthogonal_eq_bot Submodule.inf_orthogonal_eq_bot
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem orthogonal_disjoint : Disjoint K Kแฎ := by simp [disjoint_iff, K.inf_orthogonal_eq_bot]
#align submodule.orthogonal_disjoint Submodule.orthogonal_disjoint
/-- `Kแฎ` can be characterized as the intersection of the kernels of the operations of
inner product with each of the elements of `K`. -/
theorem orthogonal_eq_inter : Kแฎ = โจ
v : K, LinearMap.ker (innerSL ๐ (v : E)) := by
apply le_antisymm
ยท rw [le_iInf_iff]
rintro โจv, hvโฉ w hw
simpa using hw _ hv
ยท intro v hv w hw
simp only [mem_iInf] at hv
exact hv โจw, hwโฉ
#align submodule.orthogonal_eq_inter Submodule.orthogonal_eq_inter
/-- The orthogonal complement of any submodule `K` is closed. -/
theorem isClosed_orthogonal : IsClosed (Kแฎ : Set E) := by
rw [orthogonal_eq_inter K]
have := fun v : K => ContinuousLinearMap.isClosed_ker (innerSL ๐ (v : E))
convert isClosed_iInter this
simp only [iInf_coe]
#align submodule.is_closed_orthogonal Submodule.isClosed_orthogonal
/-- In a complete space, the orthogonal complement of any submodule `K` is complete. -/
instance instOrthogonalCompleteSpace [CompleteSpace E] : CompleteSpace Kแฎ :=
K.isClosed_orthogonal.completeSpace_coe
variable (๐ E)
/-- `orthogonal` gives a `GaloisConnection` between
`Submodule ๐ E` and its `OrderDual`. -/
theorem orthogonal_gc :
@GaloisConnection (Submodule ๐ E) (Submodule ๐ E)แตแต _ _ orthogonal orthogonal := fun _Kโ _Kโ =>
โจfun h _v hv _u hu => inner_left_of_mem_orthogonal hv (h hu), fun h _v hv _u hu =>
inner_left_of_mem_orthogonal hv (h hu)โฉ
#align submodule.orthogonal_gc Submodule.orthogonal_gc
variable {๐ E}
/-- `orthogonal` reverses the `โค` ordering of two
subspaces. -/
theorem orthogonal_le {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎ โค Kโแฎ :=
(orthogonal_gc ๐ E).monotone_l h
#align submodule.orthogonal_le Submodule.orthogonal_le
/-- `orthogonal.orthogonal` preserves the `โค` ordering of two
subspaces. -/
theorem orthogonal_orthogonal_monotone {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎแฎ โค Kโแฎแฎ :=
orthogonal_le (orthogonal_le h)
#align submodule.orthogonal_orthogonal_monotone Submodule.orthogonal_orthogonal_monotone
/-- `K` is contained in `Kแฎแฎ`. -/
theorem le_orthogonal_orthogonal : K โค Kแฎแฎ :=
(orthogonal_gc ๐ E).le_u_l _
#align submodule.le_orthogonal_orthogonal Submodule.le_orthogonal_orthogonal
/-- The inf of two orthogonal subspaces equals the subspace orthogonal
to the sup. -/
theorem inf_orthogonal (Kโ Kโ : Submodule ๐ E) : Kโแฎ โ Kโแฎ = (Kโ โ Kโ)แฎ :=
(orthogonal_gc ๐ E).l_sup.symm
#align submodule.inf_orthogonal Submodule.inf_orthogonal
/-- The inf of an indexed family of orthogonal subspaces equals the
subspace orthogonal to the sup. -/
theorem iInf_orthogonal {ฮน : Type*} (K : ฮน โ Submodule ๐ E) : โจ
i, (K i)แฎ = (iSup K)แฎ :=
(orthogonal_gc ๐ E).l_iSup.symm
#align submodule.infi_orthogonal Submodule.iInf_orthogonal
/-- The inf of a set of orthogonal subspaces equals the subspace orthogonal to the sup. -/
theorem sInf_orthogonal (s : Set <| Submodule ๐ E) : โจ
K โ s, Kแฎ = (sSup s)แฎ :=
(orthogonal_gc ๐ E).l_sSup.symm
#align submodule.Inf_orthogonal Submodule.sInf_orthogonal
@[simp]
theorem top_orthogonal_eq_bot : (โค : Submodule ๐ E)แฎ = โฅ := by
ext x
rw [mem_bot, mem_orthogonal]
exact
โจfun h => inner_self_eq_zero.mp (h x mem_top), by
rintro rfl
simpโฉ
#align submodule.top_orthogonal_eq_bot Submodule.top_orthogonal_eq_bot
@[simp]
theorem bot_orthogonal_eq_top : (โฅ : Submodule ๐ E)แฎ = โค := by
rw [โ top_orthogonal_eq_bot, eq_top_iff]
exact le_orthogonal_orthogonal โค
#align submodule.bot_orthogonal_eq_top Submodule.bot_orthogonal_eq_top
@[simp]
theorem orthogonal_eq_top_iff : Kแฎ = โค โ K = โฅ := by
refine'
โจ_, by
rintro rfl
|
exact bot_orthogonal_eq_top
|
@[simp]
theorem orthogonal_eq_top_iff : Kแฎ = โค โ K = โฅ := by
refine'
โจ_, by
rintro rfl
|
Mathlib.Analysis.InnerProductSpace.Orthogonal.202_0.QXx0GYqLoAbtfq1
|
@[simp]
theorem orthogonal_eq_top_iff : Kแฎ = โค โ K = โฅ
|
Mathlib_Analysis_InnerProductSpace_Orthogonal
|
๐ : Type u_1
E : Type u_2
F : Type u_3
instโโด : IsROrC ๐
instโยณ : NormedAddCommGroup E
instโยฒ : InnerProductSpace ๐ E
instโยน : NormedAddCommGroup F
instโ : InnerProductSpace ๐ F
K : Submodule ๐ E
โข Kแฎ = โค โ K = โฅ
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sรฉbastien Gouรซzel, Frรฉdรฉric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
/-!
# Orthogonal complements of submodules
In this file, the `orthogonal` complement of a submodule `K` is defined, and basic API established.
Some of the more subtle results about the orthogonal complement are delayed to
`Analysis.InnerProductSpace.Projection`.
See also `BilinForm.orthogonal` for orthogonality with respect to a general bilinear form.
## Notation
The orthogonal complement of a submodule `K` is denoted by `Kแฎ`.
The proposition that two submodules are orthogonal, `Submodule.IsOrtho`, is denoted by `U โ V`.
Note this is not the same unicode symbol as `โฅ` (`Bot`).
-/
variable {๐ E F : Type*} [IsROrC ๐]
variable [NormedAddCommGroup E] [InnerProductSpace ๐ E]
variable [NormedAddCommGroup F] [InnerProductSpace ๐ F]
local notation "โช" x ", " y "โซ" => @inner ๐ _ _ x y
namespace Submodule
variable (K : Submodule ๐ E)
/-- The subspace of vectors orthogonal to a given subspace. -/
def orthogonal : Submodule ๐ E where
carrier := { v | โ u โ K, โชu, vโซ = 0 }
zero_mem' _ _ := inner_zero_right _
add_mem' hx hy u hu := by rw [inner_add_right, hx u hu, hy u hu, add_zero]
smul_mem' c x hx u hu := by rw [inner_smul_right, hx u hu, mul_zero]
#align submodule.orthogonal Submodule.orthogonal
@[inherit_doc]
notation:1200 K "แฎ" => orthogonal K
/-- When a vector is in `Kแฎ`. -/
theorem mem_orthogonal (v : E) : v โ Kแฎ โ โ u โ K, โชu, vโซ = 0 :=
Iff.rfl
#align submodule.mem_orthogonal Submodule.mem_orthogonal
/-- When a vector is in `Kแฎ`, with the inner product the
other way round. -/
theorem mem_orthogonal' (v : E) : v โ Kแฎ โ โ u โ K, โชv, uโซ = 0 := by
simp_rw [mem_orthogonal, inner_eq_zero_symm]
#align submodule.mem_orthogonal' Submodule.mem_orthogonal'
variable {K}
/-- A vector in `K` is orthogonal to one in `Kแฎ`. -/
theorem inner_right_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชu, vโซ = 0 :=
(K.mem_orthogonal v).1 hv u hu
#align submodule.inner_right_of_mem_orthogonal Submodule.inner_right_of_mem_orthogonal
/-- A vector in `Kแฎ` is orthogonal to one in `K`. -/
theorem inner_left_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชv, uโซ = 0 := by
rw [inner_eq_zero_symm]; exact inner_right_of_mem_orthogonal hu hv
#align submodule.inner_left_of_mem_orthogonal Submodule.inner_left_of_mem_orthogonal
/-- A vector is in `(๐ โ u)แฎ` iff it is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_right {u v : E} : v โ (๐ โ u)แฎ โ โชu, vโซ = 0 := by
refine' โจinner_right_of_mem_orthogonal (mem_span_singleton_self u), _โฉ
intro hv w hw
rw [mem_span_singleton] at hw
obtain โจc, rflโฉ := hw
simp [inner_smul_left, hv]
#align submodule.mem_orthogonal_singleton_iff_inner_right Submodule.mem_orthogonal_singleton_iff_inner_right
/-- A vector in `(๐ โ u)แฎ` is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_left {u v : E} : v โ (๐ โ u)แฎ โ โชv, uโซ = 0 := by
rw [mem_orthogonal_singleton_iff_inner_right, inner_eq_zero_symm]
#align submodule.mem_orthogonal_singleton_iff_inner_left Submodule.mem_orthogonal_singleton_iff_inner_left
theorem sub_mem_orthogonal_of_inner_left {x y : E} (h : โ v : K, โชx, vโซ = โชy, vโซ) : x - y โ Kแฎ := by
rw [mem_orthogonal']
intro u hu
rw [inner_sub_left, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_left Submodule.sub_mem_orthogonal_of_inner_left
theorem sub_mem_orthogonal_of_inner_right {x y : E} (h : โ v : K, โช(v : E), xโซ = โช(v : E), yโซ) :
x - y โ Kแฎ := by
intro u hu
rw [inner_sub_right, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_right Submodule.sub_mem_orthogonal_of_inner_right
variable (K)
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem inf_orthogonal_eq_bot : K โ Kแฎ = โฅ := by
rw [eq_bot_iff]
intro x
rw [mem_inf]
exact fun โจhx, hoโฉ => inner_self_eq_zero.1 (ho x hx)
#align submodule.inf_orthogonal_eq_bot Submodule.inf_orthogonal_eq_bot
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem orthogonal_disjoint : Disjoint K Kแฎ := by simp [disjoint_iff, K.inf_orthogonal_eq_bot]
#align submodule.orthogonal_disjoint Submodule.orthogonal_disjoint
/-- `Kแฎ` can be characterized as the intersection of the kernels of the operations of
inner product with each of the elements of `K`. -/
theorem orthogonal_eq_inter : Kแฎ = โจ
v : K, LinearMap.ker (innerSL ๐ (v : E)) := by
apply le_antisymm
ยท rw [le_iInf_iff]
rintro โจv, hvโฉ w hw
simpa using hw _ hv
ยท intro v hv w hw
simp only [mem_iInf] at hv
exact hv โจw, hwโฉ
#align submodule.orthogonal_eq_inter Submodule.orthogonal_eq_inter
/-- The orthogonal complement of any submodule `K` is closed. -/
theorem isClosed_orthogonal : IsClosed (Kแฎ : Set E) := by
rw [orthogonal_eq_inter K]
have := fun v : K => ContinuousLinearMap.isClosed_ker (innerSL ๐ (v : E))
convert isClosed_iInter this
simp only [iInf_coe]
#align submodule.is_closed_orthogonal Submodule.isClosed_orthogonal
/-- In a complete space, the orthogonal complement of any submodule `K` is complete. -/
instance instOrthogonalCompleteSpace [CompleteSpace E] : CompleteSpace Kแฎ :=
K.isClosed_orthogonal.completeSpace_coe
variable (๐ E)
/-- `orthogonal` gives a `GaloisConnection` between
`Submodule ๐ E` and its `OrderDual`. -/
theorem orthogonal_gc :
@GaloisConnection (Submodule ๐ E) (Submodule ๐ E)แตแต _ _ orthogonal orthogonal := fun _Kโ _Kโ =>
โจfun h _v hv _u hu => inner_left_of_mem_orthogonal hv (h hu), fun h _v hv _u hu =>
inner_left_of_mem_orthogonal hv (h hu)โฉ
#align submodule.orthogonal_gc Submodule.orthogonal_gc
variable {๐ E}
/-- `orthogonal` reverses the `โค` ordering of two
subspaces. -/
theorem orthogonal_le {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎ โค Kโแฎ :=
(orthogonal_gc ๐ E).monotone_l h
#align submodule.orthogonal_le Submodule.orthogonal_le
/-- `orthogonal.orthogonal` preserves the `โค` ordering of two
subspaces. -/
theorem orthogonal_orthogonal_monotone {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎแฎ โค Kโแฎแฎ :=
orthogonal_le (orthogonal_le h)
#align submodule.orthogonal_orthogonal_monotone Submodule.orthogonal_orthogonal_monotone
/-- `K` is contained in `Kแฎแฎ`. -/
theorem le_orthogonal_orthogonal : K โค Kแฎแฎ :=
(orthogonal_gc ๐ E).le_u_l _
#align submodule.le_orthogonal_orthogonal Submodule.le_orthogonal_orthogonal
/-- The inf of two orthogonal subspaces equals the subspace orthogonal
to the sup. -/
theorem inf_orthogonal (Kโ Kโ : Submodule ๐ E) : Kโแฎ โ Kโแฎ = (Kโ โ Kโ)แฎ :=
(orthogonal_gc ๐ E).l_sup.symm
#align submodule.inf_orthogonal Submodule.inf_orthogonal
/-- The inf of an indexed family of orthogonal subspaces equals the
subspace orthogonal to the sup. -/
theorem iInf_orthogonal {ฮน : Type*} (K : ฮน โ Submodule ๐ E) : โจ
i, (K i)แฎ = (iSup K)แฎ :=
(orthogonal_gc ๐ E).l_iSup.symm
#align submodule.infi_orthogonal Submodule.iInf_orthogonal
/-- The inf of a set of orthogonal subspaces equals the subspace orthogonal to the sup. -/
theorem sInf_orthogonal (s : Set <| Submodule ๐ E) : โจ
K โ s, Kแฎ = (sSup s)แฎ :=
(orthogonal_gc ๐ E).l_sSup.symm
#align submodule.Inf_orthogonal Submodule.sInf_orthogonal
@[simp]
theorem top_orthogonal_eq_bot : (โค : Submodule ๐ E)แฎ = โฅ := by
ext x
rw [mem_bot, mem_orthogonal]
exact
โจfun h => inner_self_eq_zero.mp (h x mem_top), by
rintro rfl
simpโฉ
#align submodule.top_orthogonal_eq_bot Submodule.top_orthogonal_eq_bot
@[simp]
theorem bot_orthogonal_eq_top : (โฅ : Submodule ๐ E)แฎ = โค := by
rw [โ top_orthogonal_eq_bot, eq_top_iff]
exact le_orthogonal_orthogonal โค
#align submodule.bot_orthogonal_eq_top Submodule.bot_orthogonal_eq_top
@[simp]
theorem orthogonal_eq_top_iff : Kแฎ = โค โ K = โฅ := by
refine'
โจ_, by
rintro rfl
exact bot_orthogonal_eq_topโฉ
|
intro h
|
@[simp]
theorem orthogonal_eq_top_iff : Kแฎ = โค โ K = โฅ := by
refine'
โจ_, by
rintro rfl
exact bot_orthogonal_eq_topโฉ
|
Mathlib.Analysis.InnerProductSpace.Orthogonal.202_0.QXx0GYqLoAbtfq1
|
@[simp]
theorem orthogonal_eq_top_iff : Kแฎ = โค โ K = โฅ
|
Mathlib_Analysis_InnerProductSpace_Orthogonal
|
๐ : Type u_1
E : Type u_2
F : Type u_3
instโโด : IsROrC ๐
instโยณ : NormedAddCommGroup E
instโยฒ : InnerProductSpace ๐ E
instโยน : NormedAddCommGroup F
instโ : InnerProductSpace ๐ F
K : Submodule ๐ E
h : Kแฎ = โค
โข K = โฅ
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sรฉbastien Gouรซzel, Frรฉdรฉric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
/-!
# Orthogonal complements of submodules
In this file, the `orthogonal` complement of a submodule `K` is defined, and basic API established.
Some of the more subtle results about the orthogonal complement are delayed to
`Analysis.InnerProductSpace.Projection`.
See also `BilinForm.orthogonal` for orthogonality with respect to a general bilinear form.
## Notation
The orthogonal complement of a submodule `K` is denoted by `Kแฎ`.
The proposition that two submodules are orthogonal, `Submodule.IsOrtho`, is denoted by `U โ V`.
Note this is not the same unicode symbol as `โฅ` (`Bot`).
-/
variable {๐ E F : Type*} [IsROrC ๐]
variable [NormedAddCommGroup E] [InnerProductSpace ๐ E]
variable [NormedAddCommGroup F] [InnerProductSpace ๐ F]
local notation "โช" x ", " y "โซ" => @inner ๐ _ _ x y
namespace Submodule
variable (K : Submodule ๐ E)
/-- The subspace of vectors orthogonal to a given subspace. -/
def orthogonal : Submodule ๐ E where
carrier := { v | โ u โ K, โชu, vโซ = 0 }
zero_mem' _ _ := inner_zero_right _
add_mem' hx hy u hu := by rw [inner_add_right, hx u hu, hy u hu, add_zero]
smul_mem' c x hx u hu := by rw [inner_smul_right, hx u hu, mul_zero]
#align submodule.orthogonal Submodule.orthogonal
@[inherit_doc]
notation:1200 K "แฎ" => orthogonal K
/-- When a vector is in `Kแฎ`. -/
theorem mem_orthogonal (v : E) : v โ Kแฎ โ โ u โ K, โชu, vโซ = 0 :=
Iff.rfl
#align submodule.mem_orthogonal Submodule.mem_orthogonal
/-- When a vector is in `Kแฎ`, with the inner product the
other way round. -/
theorem mem_orthogonal' (v : E) : v โ Kแฎ โ โ u โ K, โชv, uโซ = 0 := by
simp_rw [mem_orthogonal, inner_eq_zero_symm]
#align submodule.mem_orthogonal' Submodule.mem_orthogonal'
variable {K}
/-- A vector in `K` is orthogonal to one in `Kแฎ`. -/
theorem inner_right_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชu, vโซ = 0 :=
(K.mem_orthogonal v).1 hv u hu
#align submodule.inner_right_of_mem_orthogonal Submodule.inner_right_of_mem_orthogonal
/-- A vector in `Kแฎ` is orthogonal to one in `K`. -/
theorem inner_left_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชv, uโซ = 0 := by
rw [inner_eq_zero_symm]; exact inner_right_of_mem_orthogonal hu hv
#align submodule.inner_left_of_mem_orthogonal Submodule.inner_left_of_mem_orthogonal
/-- A vector is in `(๐ โ u)แฎ` iff it is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_right {u v : E} : v โ (๐ โ u)แฎ โ โชu, vโซ = 0 := by
refine' โจinner_right_of_mem_orthogonal (mem_span_singleton_self u), _โฉ
intro hv w hw
rw [mem_span_singleton] at hw
obtain โจc, rflโฉ := hw
simp [inner_smul_left, hv]
#align submodule.mem_orthogonal_singleton_iff_inner_right Submodule.mem_orthogonal_singleton_iff_inner_right
/-- A vector in `(๐ โ u)แฎ` is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_left {u v : E} : v โ (๐ โ u)แฎ โ โชv, uโซ = 0 := by
rw [mem_orthogonal_singleton_iff_inner_right, inner_eq_zero_symm]
#align submodule.mem_orthogonal_singleton_iff_inner_left Submodule.mem_orthogonal_singleton_iff_inner_left
theorem sub_mem_orthogonal_of_inner_left {x y : E} (h : โ v : K, โชx, vโซ = โชy, vโซ) : x - y โ Kแฎ := by
rw [mem_orthogonal']
intro u hu
rw [inner_sub_left, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_left Submodule.sub_mem_orthogonal_of_inner_left
theorem sub_mem_orthogonal_of_inner_right {x y : E} (h : โ v : K, โช(v : E), xโซ = โช(v : E), yโซ) :
x - y โ Kแฎ := by
intro u hu
rw [inner_sub_right, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_right Submodule.sub_mem_orthogonal_of_inner_right
variable (K)
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem inf_orthogonal_eq_bot : K โ Kแฎ = โฅ := by
rw [eq_bot_iff]
intro x
rw [mem_inf]
exact fun โจhx, hoโฉ => inner_self_eq_zero.1 (ho x hx)
#align submodule.inf_orthogonal_eq_bot Submodule.inf_orthogonal_eq_bot
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem orthogonal_disjoint : Disjoint K Kแฎ := by simp [disjoint_iff, K.inf_orthogonal_eq_bot]
#align submodule.orthogonal_disjoint Submodule.orthogonal_disjoint
/-- `Kแฎ` can be characterized as the intersection of the kernels of the operations of
inner product with each of the elements of `K`. -/
theorem orthogonal_eq_inter : Kแฎ = โจ
v : K, LinearMap.ker (innerSL ๐ (v : E)) := by
apply le_antisymm
ยท rw [le_iInf_iff]
rintro โจv, hvโฉ w hw
simpa using hw _ hv
ยท intro v hv w hw
simp only [mem_iInf] at hv
exact hv โจw, hwโฉ
#align submodule.orthogonal_eq_inter Submodule.orthogonal_eq_inter
/-- The orthogonal complement of any submodule `K` is closed. -/
theorem isClosed_orthogonal : IsClosed (Kแฎ : Set E) := by
rw [orthogonal_eq_inter K]
have := fun v : K => ContinuousLinearMap.isClosed_ker (innerSL ๐ (v : E))
convert isClosed_iInter this
simp only [iInf_coe]
#align submodule.is_closed_orthogonal Submodule.isClosed_orthogonal
/-- In a complete space, the orthogonal complement of any submodule `K` is complete. -/
instance instOrthogonalCompleteSpace [CompleteSpace E] : CompleteSpace Kแฎ :=
K.isClosed_orthogonal.completeSpace_coe
variable (๐ E)
/-- `orthogonal` gives a `GaloisConnection` between
`Submodule ๐ E` and its `OrderDual`. -/
theorem orthogonal_gc :
@GaloisConnection (Submodule ๐ E) (Submodule ๐ E)แตแต _ _ orthogonal orthogonal := fun _Kโ _Kโ =>
โจfun h _v hv _u hu => inner_left_of_mem_orthogonal hv (h hu), fun h _v hv _u hu =>
inner_left_of_mem_orthogonal hv (h hu)โฉ
#align submodule.orthogonal_gc Submodule.orthogonal_gc
variable {๐ E}
/-- `orthogonal` reverses the `โค` ordering of two
subspaces. -/
theorem orthogonal_le {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎ โค Kโแฎ :=
(orthogonal_gc ๐ E).monotone_l h
#align submodule.orthogonal_le Submodule.orthogonal_le
/-- `orthogonal.orthogonal` preserves the `โค` ordering of two
subspaces. -/
theorem orthogonal_orthogonal_monotone {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎแฎ โค Kโแฎแฎ :=
orthogonal_le (orthogonal_le h)
#align submodule.orthogonal_orthogonal_monotone Submodule.orthogonal_orthogonal_monotone
/-- `K` is contained in `Kแฎแฎ`. -/
theorem le_orthogonal_orthogonal : K โค Kแฎแฎ :=
(orthogonal_gc ๐ E).le_u_l _
#align submodule.le_orthogonal_orthogonal Submodule.le_orthogonal_orthogonal
/-- The inf of two orthogonal subspaces equals the subspace orthogonal
to the sup. -/
theorem inf_orthogonal (Kโ Kโ : Submodule ๐ E) : Kโแฎ โ Kโแฎ = (Kโ โ Kโ)แฎ :=
(orthogonal_gc ๐ E).l_sup.symm
#align submodule.inf_orthogonal Submodule.inf_orthogonal
/-- The inf of an indexed family of orthogonal subspaces equals the
subspace orthogonal to the sup. -/
theorem iInf_orthogonal {ฮน : Type*} (K : ฮน โ Submodule ๐ E) : โจ
i, (K i)แฎ = (iSup K)แฎ :=
(orthogonal_gc ๐ E).l_iSup.symm
#align submodule.infi_orthogonal Submodule.iInf_orthogonal
/-- The inf of a set of orthogonal subspaces equals the subspace orthogonal to the sup. -/
theorem sInf_orthogonal (s : Set <| Submodule ๐ E) : โจ
K โ s, Kแฎ = (sSup s)แฎ :=
(orthogonal_gc ๐ E).l_sSup.symm
#align submodule.Inf_orthogonal Submodule.sInf_orthogonal
@[simp]
theorem top_orthogonal_eq_bot : (โค : Submodule ๐ E)แฎ = โฅ := by
ext x
rw [mem_bot, mem_orthogonal]
exact
โจfun h => inner_self_eq_zero.mp (h x mem_top), by
rintro rfl
simpโฉ
#align submodule.top_orthogonal_eq_bot Submodule.top_orthogonal_eq_bot
@[simp]
theorem bot_orthogonal_eq_top : (โฅ : Submodule ๐ E)แฎ = โค := by
rw [โ top_orthogonal_eq_bot, eq_top_iff]
exact le_orthogonal_orthogonal โค
#align submodule.bot_orthogonal_eq_top Submodule.bot_orthogonal_eq_top
@[simp]
theorem orthogonal_eq_top_iff : Kแฎ = โค โ K = โฅ := by
refine'
โจ_, by
rintro rfl
exact bot_orthogonal_eq_topโฉ
intro h
|
have : K โ Kแฎ = โฅ := K.orthogonal_disjoint.eq_bot
|
@[simp]
theorem orthogonal_eq_top_iff : Kแฎ = โค โ K = โฅ := by
refine'
โจ_, by
rintro rfl
exact bot_orthogonal_eq_topโฉ
intro h
|
Mathlib.Analysis.InnerProductSpace.Orthogonal.202_0.QXx0GYqLoAbtfq1
|
@[simp]
theorem orthogonal_eq_top_iff : Kแฎ = โค โ K = โฅ
|
Mathlib_Analysis_InnerProductSpace_Orthogonal
|
๐ : Type u_1
E : Type u_2
F : Type u_3
instโโด : IsROrC ๐
instโยณ : NormedAddCommGroup E
instโยฒ : InnerProductSpace ๐ E
instโยน : NormedAddCommGroup F
instโ : InnerProductSpace ๐ F
K : Submodule ๐ E
h : Kแฎ = โค
this : K โ Kแฎ = โฅ
โข K = โฅ
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sรฉbastien Gouรซzel, Frรฉdรฉric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
/-!
# Orthogonal complements of submodules
In this file, the `orthogonal` complement of a submodule `K` is defined, and basic API established.
Some of the more subtle results about the orthogonal complement are delayed to
`Analysis.InnerProductSpace.Projection`.
See also `BilinForm.orthogonal` for orthogonality with respect to a general bilinear form.
## Notation
The orthogonal complement of a submodule `K` is denoted by `Kแฎ`.
The proposition that two submodules are orthogonal, `Submodule.IsOrtho`, is denoted by `U โ V`.
Note this is not the same unicode symbol as `โฅ` (`Bot`).
-/
variable {๐ E F : Type*} [IsROrC ๐]
variable [NormedAddCommGroup E] [InnerProductSpace ๐ E]
variable [NormedAddCommGroup F] [InnerProductSpace ๐ F]
local notation "โช" x ", " y "โซ" => @inner ๐ _ _ x y
namespace Submodule
variable (K : Submodule ๐ E)
/-- The subspace of vectors orthogonal to a given subspace. -/
def orthogonal : Submodule ๐ E where
carrier := { v | โ u โ K, โชu, vโซ = 0 }
zero_mem' _ _ := inner_zero_right _
add_mem' hx hy u hu := by rw [inner_add_right, hx u hu, hy u hu, add_zero]
smul_mem' c x hx u hu := by rw [inner_smul_right, hx u hu, mul_zero]
#align submodule.orthogonal Submodule.orthogonal
@[inherit_doc]
notation:1200 K "แฎ" => orthogonal K
/-- When a vector is in `Kแฎ`. -/
theorem mem_orthogonal (v : E) : v โ Kแฎ โ โ u โ K, โชu, vโซ = 0 :=
Iff.rfl
#align submodule.mem_orthogonal Submodule.mem_orthogonal
/-- When a vector is in `Kแฎ`, with the inner product the
other way round. -/
theorem mem_orthogonal' (v : E) : v โ Kแฎ โ โ u โ K, โชv, uโซ = 0 := by
simp_rw [mem_orthogonal, inner_eq_zero_symm]
#align submodule.mem_orthogonal' Submodule.mem_orthogonal'
variable {K}
/-- A vector in `K` is orthogonal to one in `Kแฎ`. -/
theorem inner_right_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชu, vโซ = 0 :=
(K.mem_orthogonal v).1 hv u hu
#align submodule.inner_right_of_mem_orthogonal Submodule.inner_right_of_mem_orthogonal
/-- A vector in `Kแฎ` is orthogonal to one in `K`. -/
theorem inner_left_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชv, uโซ = 0 := by
rw [inner_eq_zero_symm]; exact inner_right_of_mem_orthogonal hu hv
#align submodule.inner_left_of_mem_orthogonal Submodule.inner_left_of_mem_orthogonal
/-- A vector is in `(๐ โ u)แฎ` iff it is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_right {u v : E} : v โ (๐ โ u)แฎ โ โชu, vโซ = 0 := by
refine' โจinner_right_of_mem_orthogonal (mem_span_singleton_self u), _โฉ
intro hv w hw
rw [mem_span_singleton] at hw
obtain โจc, rflโฉ := hw
simp [inner_smul_left, hv]
#align submodule.mem_orthogonal_singleton_iff_inner_right Submodule.mem_orthogonal_singleton_iff_inner_right
/-- A vector in `(๐ โ u)แฎ` is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_left {u v : E} : v โ (๐ โ u)แฎ โ โชv, uโซ = 0 := by
rw [mem_orthogonal_singleton_iff_inner_right, inner_eq_zero_symm]
#align submodule.mem_orthogonal_singleton_iff_inner_left Submodule.mem_orthogonal_singleton_iff_inner_left
theorem sub_mem_orthogonal_of_inner_left {x y : E} (h : โ v : K, โชx, vโซ = โชy, vโซ) : x - y โ Kแฎ := by
rw [mem_orthogonal']
intro u hu
rw [inner_sub_left, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_left Submodule.sub_mem_orthogonal_of_inner_left
theorem sub_mem_orthogonal_of_inner_right {x y : E} (h : โ v : K, โช(v : E), xโซ = โช(v : E), yโซ) :
x - y โ Kแฎ := by
intro u hu
rw [inner_sub_right, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_right Submodule.sub_mem_orthogonal_of_inner_right
variable (K)
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem inf_orthogonal_eq_bot : K โ Kแฎ = โฅ := by
rw [eq_bot_iff]
intro x
rw [mem_inf]
exact fun โจhx, hoโฉ => inner_self_eq_zero.1 (ho x hx)
#align submodule.inf_orthogonal_eq_bot Submodule.inf_orthogonal_eq_bot
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem orthogonal_disjoint : Disjoint K Kแฎ := by simp [disjoint_iff, K.inf_orthogonal_eq_bot]
#align submodule.orthogonal_disjoint Submodule.orthogonal_disjoint
/-- `Kแฎ` can be characterized as the intersection of the kernels of the operations of
inner product with each of the elements of `K`. -/
theorem orthogonal_eq_inter : Kแฎ = โจ
v : K, LinearMap.ker (innerSL ๐ (v : E)) := by
apply le_antisymm
ยท rw [le_iInf_iff]
rintro โจv, hvโฉ w hw
simpa using hw _ hv
ยท intro v hv w hw
simp only [mem_iInf] at hv
exact hv โจw, hwโฉ
#align submodule.orthogonal_eq_inter Submodule.orthogonal_eq_inter
/-- The orthogonal complement of any submodule `K` is closed. -/
theorem isClosed_orthogonal : IsClosed (Kแฎ : Set E) := by
rw [orthogonal_eq_inter K]
have := fun v : K => ContinuousLinearMap.isClosed_ker (innerSL ๐ (v : E))
convert isClosed_iInter this
simp only [iInf_coe]
#align submodule.is_closed_orthogonal Submodule.isClosed_orthogonal
/-- In a complete space, the orthogonal complement of any submodule `K` is complete. -/
instance instOrthogonalCompleteSpace [CompleteSpace E] : CompleteSpace Kแฎ :=
K.isClosed_orthogonal.completeSpace_coe
variable (๐ E)
/-- `orthogonal` gives a `GaloisConnection` between
`Submodule ๐ E` and its `OrderDual`. -/
theorem orthogonal_gc :
@GaloisConnection (Submodule ๐ E) (Submodule ๐ E)แตแต _ _ orthogonal orthogonal := fun _Kโ _Kโ =>
โจfun h _v hv _u hu => inner_left_of_mem_orthogonal hv (h hu), fun h _v hv _u hu =>
inner_left_of_mem_orthogonal hv (h hu)โฉ
#align submodule.orthogonal_gc Submodule.orthogonal_gc
variable {๐ E}
/-- `orthogonal` reverses the `โค` ordering of two
subspaces. -/
theorem orthogonal_le {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎ โค Kโแฎ :=
(orthogonal_gc ๐ E).monotone_l h
#align submodule.orthogonal_le Submodule.orthogonal_le
/-- `orthogonal.orthogonal` preserves the `โค` ordering of two
subspaces. -/
theorem orthogonal_orthogonal_monotone {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎแฎ โค Kโแฎแฎ :=
orthogonal_le (orthogonal_le h)
#align submodule.orthogonal_orthogonal_monotone Submodule.orthogonal_orthogonal_monotone
/-- `K` is contained in `Kแฎแฎ`. -/
theorem le_orthogonal_orthogonal : K โค Kแฎแฎ :=
(orthogonal_gc ๐ E).le_u_l _
#align submodule.le_orthogonal_orthogonal Submodule.le_orthogonal_orthogonal
/-- The inf of two orthogonal subspaces equals the subspace orthogonal
to the sup. -/
theorem inf_orthogonal (Kโ Kโ : Submodule ๐ E) : Kโแฎ โ Kโแฎ = (Kโ โ Kโ)แฎ :=
(orthogonal_gc ๐ E).l_sup.symm
#align submodule.inf_orthogonal Submodule.inf_orthogonal
/-- The inf of an indexed family of orthogonal subspaces equals the
subspace orthogonal to the sup. -/
theorem iInf_orthogonal {ฮน : Type*} (K : ฮน โ Submodule ๐ E) : โจ
i, (K i)แฎ = (iSup K)แฎ :=
(orthogonal_gc ๐ E).l_iSup.symm
#align submodule.infi_orthogonal Submodule.iInf_orthogonal
/-- The inf of a set of orthogonal subspaces equals the subspace orthogonal to the sup. -/
theorem sInf_orthogonal (s : Set <| Submodule ๐ E) : โจ
K โ s, Kแฎ = (sSup s)แฎ :=
(orthogonal_gc ๐ E).l_sSup.symm
#align submodule.Inf_orthogonal Submodule.sInf_orthogonal
@[simp]
theorem top_orthogonal_eq_bot : (โค : Submodule ๐ E)แฎ = โฅ := by
ext x
rw [mem_bot, mem_orthogonal]
exact
โจfun h => inner_self_eq_zero.mp (h x mem_top), by
rintro rfl
simpโฉ
#align submodule.top_orthogonal_eq_bot Submodule.top_orthogonal_eq_bot
@[simp]
theorem bot_orthogonal_eq_top : (โฅ : Submodule ๐ E)แฎ = โค := by
rw [โ top_orthogonal_eq_bot, eq_top_iff]
exact le_orthogonal_orthogonal โค
#align submodule.bot_orthogonal_eq_top Submodule.bot_orthogonal_eq_top
@[simp]
theorem orthogonal_eq_top_iff : Kแฎ = โค โ K = โฅ := by
refine'
โจ_, by
rintro rfl
exact bot_orthogonal_eq_topโฉ
intro h
have : K โ Kแฎ = โฅ := K.orthogonal_disjoint.eq_bot
|
rwa [h, inf_comm, top_inf_eq] at this
|
@[simp]
theorem orthogonal_eq_top_iff : Kแฎ = โค โ K = โฅ := by
refine'
โจ_, by
rintro rfl
exact bot_orthogonal_eq_topโฉ
intro h
have : K โ Kแฎ = โฅ := K.orthogonal_disjoint.eq_bot
|
Mathlib.Analysis.InnerProductSpace.Orthogonal.202_0.QXx0GYqLoAbtfq1
|
@[simp]
theorem orthogonal_eq_top_iff : Kแฎ = โค โ K = โฅ
|
Mathlib_Analysis_InnerProductSpace_Orthogonal
|
๐ : Type u_1
E : Type u_2
F : Type u_3
instโโด : IsROrC ๐
instโยณ : NormedAddCommGroup E
instโยฒ : InnerProductSpace ๐ E
instโยน : NormedAddCommGroup F
instโ : InnerProductSpace ๐ F
U : Submodule ๐ E
V : Set (Submodule ๐ E)
โข (โ Uแตข โ V, Uแตข โ U) โ โ Vแตข โ V, U โ Vแตข
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sรฉbastien Gouรซzel, Frรฉdรฉric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
/-!
# Orthogonal complements of submodules
In this file, the `orthogonal` complement of a submodule `K` is defined, and basic API established.
Some of the more subtle results about the orthogonal complement are delayed to
`Analysis.InnerProductSpace.Projection`.
See also `BilinForm.orthogonal` for orthogonality with respect to a general bilinear form.
## Notation
The orthogonal complement of a submodule `K` is denoted by `Kแฎ`.
The proposition that two submodules are orthogonal, `Submodule.IsOrtho`, is denoted by `U โ V`.
Note this is not the same unicode symbol as `โฅ` (`Bot`).
-/
variable {๐ E F : Type*} [IsROrC ๐]
variable [NormedAddCommGroup E] [InnerProductSpace ๐ E]
variable [NormedAddCommGroup F] [InnerProductSpace ๐ F]
local notation "โช" x ", " y "โซ" => @inner ๐ _ _ x y
namespace Submodule
variable (K : Submodule ๐ E)
/-- The subspace of vectors orthogonal to a given subspace. -/
def orthogonal : Submodule ๐ E where
carrier := { v | โ u โ K, โชu, vโซ = 0 }
zero_mem' _ _ := inner_zero_right _
add_mem' hx hy u hu := by rw [inner_add_right, hx u hu, hy u hu, add_zero]
smul_mem' c x hx u hu := by rw [inner_smul_right, hx u hu, mul_zero]
#align submodule.orthogonal Submodule.orthogonal
@[inherit_doc]
notation:1200 K "แฎ" => orthogonal K
/-- When a vector is in `Kแฎ`. -/
theorem mem_orthogonal (v : E) : v โ Kแฎ โ โ u โ K, โชu, vโซ = 0 :=
Iff.rfl
#align submodule.mem_orthogonal Submodule.mem_orthogonal
/-- When a vector is in `Kแฎ`, with the inner product the
other way round. -/
theorem mem_orthogonal' (v : E) : v โ Kแฎ โ โ u โ K, โชv, uโซ = 0 := by
simp_rw [mem_orthogonal, inner_eq_zero_symm]
#align submodule.mem_orthogonal' Submodule.mem_orthogonal'
variable {K}
/-- A vector in `K` is orthogonal to one in `Kแฎ`. -/
theorem inner_right_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชu, vโซ = 0 :=
(K.mem_orthogonal v).1 hv u hu
#align submodule.inner_right_of_mem_orthogonal Submodule.inner_right_of_mem_orthogonal
/-- A vector in `Kแฎ` is orthogonal to one in `K`. -/
theorem inner_left_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชv, uโซ = 0 := by
rw [inner_eq_zero_symm]; exact inner_right_of_mem_orthogonal hu hv
#align submodule.inner_left_of_mem_orthogonal Submodule.inner_left_of_mem_orthogonal
/-- A vector is in `(๐ โ u)แฎ` iff it is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_right {u v : E} : v โ (๐ โ u)แฎ โ โชu, vโซ = 0 := by
refine' โจinner_right_of_mem_orthogonal (mem_span_singleton_self u), _โฉ
intro hv w hw
rw [mem_span_singleton] at hw
obtain โจc, rflโฉ := hw
simp [inner_smul_left, hv]
#align submodule.mem_orthogonal_singleton_iff_inner_right Submodule.mem_orthogonal_singleton_iff_inner_right
/-- A vector in `(๐ โ u)แฎ` is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_left {u v : E} : v โ (๐ โ u)แฎ โ โชv, uโซ = 0 := by
rw [mem_orthogonal_singleton_iff_inner_right, inner_eq_zero_symm]
#align submodule.mem_orthogonal_singleton_iff_inner_left Submodule.mem_orthogonal_singleton_iff_inner_left
theorem sub_mem_orthogonal_of_inner_left {x y : E} (h : โ v : K, โชx, vโซ = โชy, vโซ) : x - y โ Kแฎ := by
rw [mem_orthogonal']
intro u hu
rw [inner_sub_left, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_left Submodule.sub_mem_orthogonal_of_inner_left
theorem sub_mem_orthogonal_of_inner_right {x y : E} (h : โ v : K, โช(v : E), xโซ = โช(v : E), yโซ) :
x - y โ Kแฎ := by
intro u hu
rw [inner_sub_right, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_right Submodule.sub_mem_orthogonal_of_inner_right
variable (K)
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem inf_orthogonal_eq_bot : K โ Kแฎ = โฅ := by
rw [eq_bot_iff]
intro x
rw [mem_inf]
exact fun โจhx, hoโฉ => inner_self_eq_zero.1 (ho x hx)
#align submodule.inf_orthogonal_eq_bot Submodule.inf_orthogonal_eq_bot
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem orthogonal_disjoint : Disjoint K Kแฎ := by simp [disjoint_iff, K.inf_orthogonal_eq_bot]
#align submodule.orthogonal_disjoint Submodule.orthogonal_disjoint
/-- `Kแฎ` can be characterized as the intersection of the kernels of the operations of
inner product with each of the elements of `K`. -/
theorem orthogonal_eq_inter : Kแฎ = โจ
v : K, LinearMap.ker (innerSL ๐ (v : E)) := by
apply le_antisymm
ยท rw [le_iInf_iff]
rintro โจv, hvโฉ w hw
simpa using hw _ hv
ยท intro v hv w hw
simp only [mem_iInf] at hv
exact hv โจw, hwโฉ
#align submodule.orthogonal_eq_inter Submodule.orthogonal_eq_inter
/-- The orthogonal complement of any submodule `K` is closed. -/
theorem isClosed_orthogonal : IsClosed (Kแฎ : Set E) := by
rw [orthogonal_eq_inter K]
have := fun v : K => ContinuousLinearMap.isClosed_ker (innerSL ๐ (v : E))
convert isClosed_iInter this
simp only [iInf_coe]
#align submodule.is_closed_orthogonal Submodule.isClosed_orthogonal
/-- In a complete space, the orthogonal complement of any submodule `K` is complete. -/
instance instOrthogonalCompleteSpace [CompleteSpace E] : CompleteSpace Kแฎ :=
K.isClosed_orthogonal.completeSpace_coe
variable (๐ E)
/-- `orthogonal` gives a `GaloisConnection` between
`Submodule ๐ E` and its `OrderDual`. -/
theorem orthogonal_gc :
@GaloisConnection (Submodule ๐ E) (Submodule ๐ E)แตแต _ _ orthogonal orthogonal := fun _Kโ _Kโ =>
โจfun h _v hv _u hu => inner_left_of_mem_orthogonal hv (h hu), fun h _v hv _u hu =>
inner_left_of_mem_orthogonal hv (h hu)โฉ
#align submodule.orthogonal_gc Submodule.orthogonal_gc
variable {๐ E}
/-- `orthogonal` reverses the `โค` ordering of two
subspaces. -/
theorem orthogonal_le {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎ โค Kโแฎ :=
(orthogonal_gc ๐ E).monotone_l h
#align submodule.orthogonal_le Submodule.orthogonal_le
/-- `orthogonal.orthogonal` preserves the `โค` ordering of two
subspaces. -/
theorem orthogonal_orthogonal_monotone {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎแฎ โค Kโแฎแฎ :=
orthogonal_le (orthogonal_le h)
#align submodule.orthogonal_orthogonal_monotone Submodule.orthogonal_orthogonal_monotone
/-- `K` is contained in `Kแฎแฎ`. -/
theorem le_orthogonal_orthogonal : K โค Kแฎแฎ :=
(orthogonal_gc ๐ E).le_u_l _
#align submodule.le_orthogonal_orthogonal Submodule.le_orthogonal_orthogonal
/-- The inf of two orthogonal subspaces equals the subspace orthogonal
to the sup. -/
theorem inf_orthogonal (Kโ Kโ : Submodule ๐ E) : Kโแฎ โ Kโแฎ = (Kโ โ Kโ)แฎ :=
(orthogonal_gc ๐ E).l_sup.symm
#align submodule.inf_orthogonal Submodule.inf_orthogonal
/-- The inf of an indexed family of orthogonal subspaces equals the
subspace orthogonal to the sup. -/
theorem iInf_orthogonal {ฮน : Type*} (K : ฮน โ Submodule ๐ E) : โจ
i, (K i)แฎ = (iSup K)แฎ :=
(orthogonal_gc ๐ E).l_iSup.symm
#align submodule.infi_orthogonal Submodule.iInf_orthogonal
/-- The inf of a set of orthogonal subspaces equals the subspace orthogonal to the sup. -/
theorem sInf_orthogonal (s : Set <| Submodule ๐ E) : โจ
K โ s, Kแฎ = (sSup s)แฎ :=
(orthogonal_gc ๐ E).l_sSup.symm
#align submodule.Inf_orthogonal Submodule.sInf_orthogonal
@[simp]
theorem top_orthogonal_eq_bot : (โค : Submodule ๐ E)แฎ = โฅ := by
ext x
rw [mem_bot, mem_orthogonal]
exact
โจfun h => inner_self_eq_zero.mp (h x mem_top), by
rintro rfl
simpโฉ
#align submodule.top_orthogonal_eq_bot Submodule.top_orthogonal_eq_bot
@[simp]
theorem bot_orthogonal_eq_top : (โฅ : Submodule ๐ E)แฎ = โค := by
rw [โ top_orthogonal_eq_bot, eq_top_iff]
exact le_orthogonal_orthogonal โค
#align submodule.bot_orthogonal_eq_top Submodule.bot_orthogonal_eq_top
@[simp]
theorem orthogonal_eq_top_iff : Kแฎ = โค โ K = โฅ := by
refine'
โจ_, by
rintro rfl
exact bot_orthogonal_eq_topโฉ
intro h
have : K โ Kแฎ = โฅ := K.orthogonal_disjoint.eq_bot
rwa [h, inf_comm, top_inf_eq] at this
#align submodule.orthogonal_eq_top_iff Submodule.orthogonal_eq_top_iff
theorem orthogonalFamily_self :
OrthogonalFamily ๐ (fun b => โฅ(cond b K Kแฎ)) fun b => (cond b K Kแฎ).subtypeโแตข
| true, true => absurd rfl
| true, false => fun _ x y => inner_right_of_mem_orthogonal x.prop y.prop
| false, true => fun _ x y => inner_left_of_mem_orthogonal y.prop x.prop
| false, false => absurd rfl
#align submodule.orthogonal_family_self Submodule.orthogonalFamily_self
end Submodule
@[simp]
theorem bilinFormOfRealInner_orthogonal {E} [NormedAddCommGroup E] [InnerProductSpace โ E]
(K : Submodule โ E) : bilinFormOfRealInner.orthogonal K = Kแฎ :=
rfl
#align bilin_form_of_real_inner_orthogonal bilinFormOfRealInner_orthogonal
/-!
### Orthogonality of submodules
In this section we define `Submodule.IsOrtho U V`, with notation `U โ V`.
The API roughly matches that of `Disjoint`.
-/
namespace Submodule
/-- The proposition that two submodules are orthogonal. Has notation `U โ V`. -/
def IsOrtho (U V : Submodule ๐ E) : Prop :=
U โค Vแฎ
#align submodule.is_ortho Submodule.IsOrtho
@[inherit_doc]
infixl:50 " โ " => Submodule.IsOrtho
theorem isOrtho_iff_le {U V : Submodule ๐ E} : U โ V โ U โค Vแฎ :=
Iff.rfl
#align submodule.is_ortho_iff_le Submodule.isOrtho_iff_le
@[symm]
theorem IsOrtho.symm {U V : Submodule ๐ E} (h : U โ V) : V โ U :=
(le_orthogonal_orthogonal _).trans (orthogonal_le h)
#align submodule.is_ortho.symm Submodule.IsOrtho.symm
theorem isOrtho_comm {U V : Submodule ๐ E} : U โ V โ V โ U :=
โจIsOrtho.symm, IsOrtho.symmโฉ
#align submodule.is_ortho_comm Submodule.isOrtho_comm
theorem symmetric_isOrtho : Symmetric (IsOrtho : Submodule ๐ E โ Submodule ๐ E โ Prop) := fun _ _ =>
IsOrtho.symm
#align submodule.symmetric_is_ortho Submodule.symmetric_isOrtho
theorem IsOrtho.inner_eq {U V : Submodule ๐ E} (h : U โ V) {u v : E} (hu : u โ U) (hv : v โ V) :
โชu, vโซ = 0 :=
h.symm hv _ hu
#align submodule.is_ortho.inner_eq Submodule.IsOrtho.inner_eq
theorem isOrtho_iff_inner_eq {U V : Submodule ๐ E} : U โ V โ โ u โ U, โ v โ V, โชu, vโซ = 0 :=
forallโ_congr fun _u _hu _v _hv => inner_eq_zero_symm
#align submodule.is_ortho_iff_inner_eq Submodule.isOrtho_iff_inner_eq
/- TODO: generalize `Submodule.mapโ` to semilinear maps, so that we can state
`U โ V โ Submodule.mapโ (innerโโ ๐) U V โค โฅ`. -/
@[simp]
theorem isOrtho_bot_left {V : Submodule ๐ E} : โฅ โ V :=
bot_le
#align submodule.is_ortho_bot_left Submodule.isOrtho_bot_left
@[simp]
theorem isOrtho_bot_right {U : Submodule ๐ E} : U โ โฅ :=
isOrtho_bot_left.symm
#align submodule.is_ortho_bot_right Submodule.isOrtho_bot_right
theorem IsOrtho.mono_left {Uโ Uโ V : Submodule ๐ E} (hU : Uโ โค Uโ) (h : Uโ โ V) : Uโ โ V :=
hU.trans h
#align submodule.is_ortho.mono_left Submodule.IsOrtho.mono_left
theorem IsOrtho.mono_right {U Vโ Vโ : Submodule ๐ E} (hV : Vโ โค Vโ) (h : U โ Vโ) : U โ Vโ :=
(h.symm.mono_left hV).symm
#align submodule.is_ortho.mono_right Submodule.IsOrtho.mono_right
theorem IsOrtho.mono {Uโ Vโ Uโ Vโ : Submodule ๐ E} (hU : Uโ โค Uโ) (hV : Vโ โค Vโ) (h : Uโ โ Vโ) :
Uโ โ Vโ :=
(h.mono_right hV).mono_left hU
#align submodule.is_ortho.mono Submodule.IsOrtho.mono
@[simp]
theorem isOrtho_self {U : Submodule ๐ E} : U โ U โ U = โฅ :=
โจfun h => eq_bot_iff.mpr fun x hx => inner_self_eq_zero.mp (h hx x hx), fun h =>
h.symm โธ isOrtho_bot_leftโฉ
#align submodule.is_ortho_self Submodule.isOrtho_self
@[simp]
theorem isOrtho_orthogonal_right (U : Submodule ๐ E) : U โ Uแฎ :=
le_orthogonal_orthogonal _
#align submodule.is_ortho_orthogonal_right Submodule.isOrtho_orthogonal_right
@[simp]
theorem isOrtho_orthogonal_left (U : Submodule ๐ E) : Uแฎ โ U :=
(isOrtho_orthogonal_right U).symm
#align submodule.is_ortho_orthogonal_left Submodule.isOrtho_orthogonal_left
theorem IsOrtho.le {U V : Submodule ๐ E} (h : U โ V) : U โค Vแฎ :=
h
#align submodule.is_ortho.le Submodule.IsOrtho.le
theorem IsOrtho.ge {U V : Submodule ๐ E} (h : U โ V) : V โค Uแฎ :=
h.symm
#align submodule.is_ortho.ge Submodule.IsOrtho.ge
@[simp]
theorem isOrtho_top_right {U : Submodule ๐ E} : U โ โค โ U = โฅ :=
โจfun h => eq_bot_iff.mpr fun _x hx => inner_self_eq_zero.mp (h hx _ mem_top), fun h =>
h.symm โธ isOrtho_bot_leftโฉ
#align submodule.is_ortho_top_right Submodule.isOrtho_top_right
@[simp]
theorem isOrtho_top_left {V : Submodule ๐ E} : โค โ V โ V = โฅ :=
isOrtho_comm.trans isOrtho_top_right
#align submodule.is_ortho_top_left Submodule.isOrtho_top_left
/-- Orthogonal submodules are disjoint. -/
theorem IsOrtho.disjoint {U V : Submodule ๐ E} (h : U โ V) : Disjoint U V :=
(Submodule.orthogonal_disjoint _).mono_right h.symm
#align submodule.is_ortho.disjoint Submodule.IsOrtho.disjoint
@[simp]
theorem isOrtho_sup_left {Uโ Uโ V : Submodule ๐ E} : Uโ โ Uโ โ V โ Uโ โ V โง Uโ โ V :=
sup_le_iff
#align submodule.is_ortho_sup_left Submodule.isOrtho_sup_left
@[simp]
theorem isOrtho_sup_right {U Vโ Vโ : Submodule ๐ E} : U โ Vโ โ Vโ โ U โ Vโ โง U โ Vโ :=
isOrtho_comm.trans <| isOrtho_sup_left.trans <| isOrtho_comm.and isOrtho_comm
#align submodule.is_ortho_sup_right Submodule.isOrtho_sup_right
@[simp]
theorem isOrtho_sSup_left {U : Set (Submodule ๐ E)} {V : Submodule ๐ E} :
sSup U โ V โ โ Uแตข โ U, Uแตข โ V :=
sSup_le_iff
#align submodule.is_ortho_Sup_left Submodule.isOrtho_sSup_left
@[simp]
theorem isOrtho_sSup_right {U : Submodule ๐ E} {V : Set (Submodule ๐ E)} :
U โ sSup V โ โ Vแตข โ V, U โ Vแตข :=
isOrtho_comm.trans <| isOrtho_sSup_left.trans <| by
|
simp_rw [isOrtho_comm]
|
@[simp]
theorem isOrtho_sSup_right {U : Submodule ๐ E} {V : Set (Submodule ๐ E)} :
U โ sSup V โ โ Vแตข โ V, U โ Vแตข :=
isOrtho_comm.trans <| isOrtho_sSup_left.trans <| by
|
Mathlib.Analysis.InnerProductSpace.Orthogonal.355_0.QXx0GYqLoAbtfq1
|
@[simp]
theorem isOrtho_sSup_right {U : Submodule ๐ E} {V : Set (Submodule ๐ E)} :
U โ sSup V โ โ Vแตข โ V, U โ Vแตข
|
Mathlib_Analysis_InnerProductSpace_Orthogonal
|
๐ : Type u_1
E : Type u_2
F : Type u_3
instโโด : IsROrC ๐
instโยณ : NormedAddCommGroup E
instโยฒ : InnerProductSpace ๐ E
instโยน : NormedAddCommGroup F
instโ : InnerProductSpace ๐ F
ฮน : Sort u_4
U : Submodule ๐ E
V : ฮน โ Submodule ๐ E
โข (โ (i : ฮน), V i โ U) โ โ (i : ฮน), U โ V i
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sรฉbastien Gouรซzel, Frรฉdรฉric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
/-!
# Orthogonal complements of submodules
In this file, the `orthogonal` complement of a submodule `K` is defined, and basic API established.
Some of the more subtle results about the orthogonal complement are delayed to
`Analysis.InnerProductSpace.Projection`.
See also `BilinForm.orthogonal` for orthogonality with respect to a general bilinear form.
## Notation
The orthogonal complement of a submodule `K` is denoted by `Kแฎ`.
The proposition that two submodules are orthogonal, `Submodule.IsOrtho`, is denoted by `U โ V`.
Note this is not the same unicode symbol as `โฅ` (`Bot`).
-/
variable {๐ E F : Type*} [IsROrC ๐]
variable [NormedAddCommGroup E] [InnerProductSpace ๐ E]
variable [NormedAddCommGroup F] [InnerProductSpace ๐ F]
local notation "โช" x ", " y "โซ" => @inner ๐ _ _ x y
namespace Submodule
variable (K : Submodule ๐ E)
/-- The subspace of vectors orthogonal to a given subspace. -/
def orthogonal : Submodule ๐ E where
carrier := { v | โ u โ K, โชu, vโซ = 0 }
zero_mem' _ _ := inner_zero_right _
add_mem' hx hy u hu := by rw [inner_add_right, hx u hu, hy u hu, add_zero]
smul_mem' c x hx u hu := by rw [inner_smul_right, hx u hu, mul_zero]
#align submodule.orthogonal Submodule.orthogonal
@[inherit_doc]
notation:1200 K "แฎ" => orthogonal K
/-- When a vector is in `Kแฎ`. -/
theorem mem_orthogonal (v : E) : v โ Kแฎ โ โ u โ K, โชu, vโซ = 0 :=
Iff.rfl
#align submodule.mem_orthogonal Submodule.mem_orthogonal
/-- When a vector is in `Kแฎ`, with the inner product the
other way round. -/
theorem mem_orthogonal' (v : E) : v โ Kแฎ โ โ u โ K, โชv, uโซ = 0 := by
simp_rw [mem_orthogonal, inner_eq_zero_symm]
#align submodule.mem_orthogonal' Submodule.mem_orthogonal'
variable {K}
/-- A vector in `K` is orthogonal to one in `Kแฎ`. -/
theorem inner_right_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชu, vโซ = 0 :=
(K.mem_orthogonal v).1 hv u hu
#align submodule.inner_right_of_mem_orthogonal Submodule.inner_right_of_mem_orthogonal
/-- A vector in `Kแฎ` is orthogonal to one in `K`. -/
theorem inner_left_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชv, uโซ = 0 := by
rw [inner_eq_zero_symm]; exact inner_right_of_mem_orthogonal hu hv
#align submodule.inner_left_of_mem_orthogonal Submodule.inner_left_of_mem_orthogonal
/-- A vector is in `(๐ โ u)แฎ` iff it is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_right {u v : E} : v โ (๐ โ u)แฎ โ โชu, vโซ = 0 := by
refine' โจinner_right_of_mem_orthogonal (mem_span_singleton_self u), _โฉ
intro hv w hw
rw [mem_span_singleton] at hw
obtain โจc, rflโฉ := hw
simp [inner_smul_left, hv]
#align submodule.mem_orthogonal_singleton_iff_inner_right Submodule.mem_orthogonal_singleton_iff_inner_right
/-- A vector in `(๐ โ u)แฎ` is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_left {u v : E} : v โ (๐ โ u)แฎ โ โชv, uโซ = 0 := by
rw [mem_orthogonal_singleton_iff_inner_right, inner_eq_zero_symm]
#align submodule.mem_orthogonal_singleton_iff_inner_left Submodule.mem_orthogonal_singleton_iff_inner_left
theorem sub_mem_orthogonal_of_inner_left {x y : E} (h : โ v : K, โชx, vโซ = โชy, vโซ) : x - y โ Kแฎ := by
rw [mem_orthogonal']
intro u hu
rw [inner_sub_left, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_left Submodule.sub_mem_orthogonal_of_inner_left
theorem sub_mem_orthogonal_of_inner_right {x y : E} (h : โ v : K, โช(v : E), xโซ = โช(v : E), yโซ) :
x - y โ Kแฎ := by
intro u hu
rw [inner_sub_right, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_right Submodule.sub_mem_orthogonal_of_inner_right
variable (K)
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem inf_orthogonal_eq_bot : K โ Kแฎ = โฅ := by
rw [eq_bot_iff]
intro x
rw [mem_inf]
exact fun โจhx, hoโฉ => inner_self_eq_zero.1 (ho x hx)
#align submodule.inf_orthogonal_eq_bot Submodule.inf_orthogonal_eq_bot
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem orthogonal_disjoint : Disjoint K Kแฎ := by simp [disjoint_iff, K.inf_orthogonal_eq_bot]
#align submodule.orthogonal_disjoint Submodule.orthogonal_disjoint
/-- `Kแฎ` can be characterized as the intersection of the kernels of the operations of
inner product with each of the elements of `K`. -/
theorem orthogonal_eq_inter : Kแฎ = โจ
v : K, LinearMap.ker (innerSL ๐ (v : E)) := by
apply le_antisymm
ยท rw [le_iInf_iff]
rintro โจv, hvโฉ w hw
simpa using hw _ hv
ยท intro v hv w hw
simp only [mem_iInf] at hv
exact hv โจw, hwโฉ
#align submodule.orthogonal_eq_inter Submodule.orthogonal_eq_inter
/-- The orthogonal complement of any submodule `K` is closed. -/
theorem isClosed_orthogonal : IsClosed (Kแฎ : Set E) := by
rw [orthogonal_eq_inter K]
have := fun v : K => ContinuousLinearMap.isClosed_ker (innerSL ๐ (v : E))
convert isClosed_iInter this
simp only [iInf_coe]
#align submodule.is_closed_orthogonal Submodule.isClosed_orthogonal
/-- In a complete space, the orthogonal complement of any submodule `K` is complete. -/
instance instOrthogonalCompleteSpace [CompleteSpace E] : CompleteSpace Kแฎ :=
K.isClosed_orthogonal.completeSpace_coe
variable (๐ E)
/-- `orthogonal` gives a `GaloisConnection` between
`Submodule ๐ E` and its `OrderDual`. -/
theorem orthogonal_gc :
@GaloisConnection (Submodule ๐ E) (Submodule ๐ E)แตแต _ _ orthogonal orthogonal := fun _Kโ _Kโ =>
โจfun h _v hv _u hu => inner_left_of_mem_orthogonal hv (h hu), fun h _v hv _u hu =>
inner_left_of_mem_orthogonal hv (h hu)โฉ
#align submodule.orthogonal_gc Submodule.orthogonal_gc
variable {๐ E}
/-- `orthogonal` reverses the `โค` ordering of two
subspaces. -/
theorem orthogonal_le {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎ โค Kโแฎ :=
(orthogonal_gc ๐ E).monotone_l h
#align submodule.orthogonal_le Submodule.orthogonal_le
/-- `orthogonal.orthogonal` preserves the `โค` ordering of two
subspaces. -/
theorem orthogonal_orthogonal_monotone {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎแฎ โค Kโแฎแฎ :=
orthogonal_le (orthogonal_le h)
#align submodule.orthogonal_orthogonal_monotone Submodule.orthogonal_orthogonal_monotone
/-- `K` is contained in `Kแฎแฎ`. -/
theorem le_orthogonal_orthogonal : K โค Kแฎแฎ :=
(orthogonal_gc ๐ E).le_u_l _
#align submodule.le_orthogonal_orthogonal Submodule.le_orthogonal_orthogonal
/-- The inf of two orthogonal subspaces equals the subspace orthogonal
to the sup. -/
theorem inf_orthogonal (Kโ Kโ : Submodule ๐ E) : Kโแฎ โ Kโแฎ = (Kโ โ Kโ)แฎ :=
(orthogonal_gc ๐ E).l_sup.symm
#align submodule.inf_orthogonal Submodule.inf_orthogonal
/-- The inf of an indexed family of orthogonal subspaces equals the
subspace orthogonal to the sup. -/
theorem iInf_orthogonal {ฮน : Type*} (K : ฮน โ Submodule ๐ E) : โจ
i, (K i)แฎ = (iSup K)แฎ :=
(orthogonal_gc ๐ E).l_iSup.symm
#align submodule.infi_orthogonal Submodule.iInf_orthogonal
/-- The inf of a set of orthogonal subspaces equals the subspace orthogonal to the sup. -/
theorem sInf_orthogonal (s : Set <| Submodule ๐ E) : โจ
K โ s, Kแฎ = (sSup s)แฎ :=
(orthogonal_gc ๐ E).l_sSup.symm
#align submodule.Inf_orthogonal Submodule.sInf_orthogonal
@[simp]
theorem top_orthogonal_eq_bot : (โค : Submodule ๐ E)แฎ = โฅ := by
ext x
rw [mem_bot, mem_orthogonal]
exact
โจfun h => inner_self_eq_zero.mp (h x mem_top), by
rintro rfl
simpโฉ
#align submodule.top_orthogonal_eq_bot Submodule.top_orthogonal_eq_bot
@[simp]
theorem bot_orthogonal_eq_top : (โฅ : Submodule ๐ E)แฎ = โค := by
rw [โ top_orthogonal_eq_bot, eq_top_iff]
exact le_orthogonal_orthogonal โค
#align submodule.bot_orthogonal_eq_top Submodule.bot_orthogonal_eq_top
@[simp]
theorem orthogonal_eq_top_iff : Kแฎ = โค โ K = โฅ := by
refine'
โจ_, by
rintro rfl
exact bot_orthogonal_eq_topโฉ
intro h
have : K โ Kแฎ = โฅ := K.orthogonal_disjoint.eq_bot
rwa [h, inf_comm, top_inf_eq] at this
#align submodule.orthogonal_eq_top_iff Submodule.orthogonal_eq_top_iff
theorem orthogonalFamily_self :
OrthogonalFamily ๐ (fun b => โฅ(cond b K Kแฎ)) fun b => (cond b K Kแฎ).subtypeโแตข
| true, true => absurd rfl
| true, false => fun _ x y => inner_right_of_mem_orthogonal x.prop y.prop
| false, true => fun _ x y => inner_left_of_mem_orthogonal y.prop x.prop
| false, false => absurd rfl
#align submodule.orthogonal_family_self Submodule.orthogonalFamily_self
end Submodule
@[simp]
theorem bilinFormOfRealInner_orthogonal {E} [NormedAddCommGroup E] [InnerProductSpace โ E]
(K : Submodule โ E) : bilinFormOfRealInner.orthogonal K = Kแฎ :=
rfl
#align bilin_form_of_real_inner_orthogonal bilinFormOfRealInner_orthogonal
/-!
### Orthogonality of submodules
In this section we define `Submodule.IsOrtho U V`, with notation `U โ V`.
The API roughly matches that of `Disjoint`.
-/
namespace Submodule
/-- The proposition that two submodules are orthogonal. Has notation `U โ V`. -/
def IsOrtho (U V : Submodule ๐ E) : Prop :=
U โค Vแฎ
#align submodule.is_ortho Submodule.IsOrtho
@[inherit_doc]
infixl:50 " โ " => Submodule.IsOrtho
theorem isOrtho_iff_le {U V : Submodule ๐ E} : U โ V โ U โค Vแฎ :=
Iff.rfl
#align submodule.is_ortho_iff_le Submodule.isOrtho_iff_le
@[symm]
theorem IsOrtho.symm {U V : Submodule ๐ E} (h : U โ V) : V โ U :=
(le_orthogonal_orthogonal _).trans (orthogonal_le h)
#align submodule.is_ortho.symm Submodule.IsOrtho.symm
theorem isOrtho_comm {U V : Submodule ๐ E} : U โ V โ V โ U :=
โจIsOrtho.symm, IsOrtho.symmโฉ
#align submodule.is_ortho_comm Submodule.isOrtho_comm
theorem symmetric_isOrtho : Symmetric (IsOrtho : Submodule ๐ E โ Submodule ๐ E โ Prop) := fun _ _ =>
IsOrtho.symm
#align submodule.symmetric_is_ortho Submodule.symmetric_isOrtho
theorem IsOrtho.inner_eq {U V : Submodule ๐ E} (h : U โ V) {u v : E} (hu : u โ U) (hv : v โ V) :
โชu, vโซ = 0 :=
h.symm hv _ hu
#align submodule.is_ortho.inner_eq Submodule.IsOrtho.inner_eq
theorem isOrtho_iff_inner_eq {U V : Submodule ๐ E} : U โ V โ โ u โ U, โ v โ V, โชu, vโซ = 0 :=
forallโ_congr fun _u _hu _v _hv => inner_eq_zero_symm
#align submodule.is_ortho_iff_inner_eq Submodule.isOrtho_iff_inner_eq
/- TODO: generalize `Submodule.mapโ` to semilinear maps, so that we can state
`U โ V โ Submodule.mapโ (innerโโ ๐) U V โค โฅ`. -/
@[simp]
theorem isOrtho_bot_left {V : Submodule ๐ E} : โฅ โ V :=
bot_le
#align submodule.is_ortho_bot_left Submodule.isOrtho_bot_left
@[simp]
theorem isOrtho_bot_right {U : Submodule ๐ E} : U โ โฅ :=
isOrtho_bot_left.symm
#align submodule.is_ortho_bot_right Submodule.isOrtho_bot_right
theorem IsOrtho.mono_left {Uโ Uโ V : Submodule ๐ E} (hU : Uโ โค Uโ) (h : Uโ โ V) : Uโ โ V :=
hU.trans h
#align submodule.is_ortho.mono_left Submodule.IsOrtho.mono_left
theorem IsOrtho.mono_right {U Vโ Vโ : Submodule ๐ E} (hV : Vโ โค Vโ) (h : U โ Vโ) : U โ Vโ :=
(h.symm.mono_left hV).symm
#align submodule.is_ortho.mono_right Submodule.IsOrtho.mono_right
theorem IsOrtho.mono {Uโ Vโ Uโ Vโ : Submodule ๐ E} (hU : Uโ โค Uโ) (hV : Vโ โค Vโ) (h : Uโ โ Vโ) :
Uโ โ Vโ :=
(h.mono_right hV).mono_left hU
#align submodule.is_ortho.mono Submodule.IsOrtho.mono
@[simp]
theorem isOrtho_self {U : Submodule ๐ E} : U โ U โ U = โฅ :=
โจfun h => eq_bot_iff.mpr fun x hx => inner_self_eq_zero.mp (h hx x hx), fun h =>
h.symm โธ isOrtho_bot_leftโฉ
#align submodule.is_ortho_self Submodule.isOrtho_self
@[simp]
theorem isOrtho_orthogonal_right (U : Submodule ๐ E) : U โ Uแฎ :=
le_orthogonal_orthogonal _
#align submodule.is_ortho_orthogonal_right Submodule.isOrtho_orthogonal_right
@[simp]
theorem isOrtho_orthogonal_left (U : Submodule ๐ E) : Uแฎ โ U :=
(isOrtho_orthogonal_right U).symm
#align submodule.is_ortho_orthogonal_left Submodule.isOrtho_orthogonal_left
theorem IsOrtho.le {U V : Submodule ๐ E} (h : U โ V) : U โค Vแฎ :=
h
#align submodule.is_ortho.le Submodule.IsOrtho.le
theorem IsOrtho.ge {U V : Submodule ๐ E} (h : U โ V) : V โค Uแฎ :=
h.symm
#align submodule.is_ortho.ge Submodule.IsOrtho.ge
@[simp]
theorem isOrtho_top_right {U : Submodule ๐ E} : U โ โค โ U = โฅ :=
โจfun h => eq_bot_iff.mpr fun _x hx => inner_self_eq_zero.mp (h hx _ mem_top), fun h =>
h.symm โธ isOrtho_bot_leftโฉ
#align submodule.is_ortho_top_right Submodule.isOrtho_top_right
@[simp]
theorem isOrtho_top_left {V : Submodule ๐ E} : โค โ V โ V = โฅ :=
isOrtho_comm.trans isOrtho_top_right
#align submodule.is_ortho_top_left Submodule.isOrtho_top_left
/-- Orthogonal submodules are disjoint. -/
theorem IsOrtho.disjoint {U V : Submodule ๐ E} (h : U โ V) : Disjoint U V :=
(Submodule.orthogonal_disjoint _).mono_right h.symm
#align submodule.is_ortho.disjoint Submodule.IsOrtho.disjoint
@[simp]
theorem isOrtho_sup_left {Uโ Uโ V : Submodule ๐ E} : Uโ โ Uโ โ V โ Uโ โ V โง Uโ โ V :=
sup_le_iff
#align submodule.is_ortho_sup_left Submodule.isOrtho_sup_left
@[simp]
theorem isOrtho_sup_right {U Vโ Vโ : Submodule ๐ E} : U โ Vโ โ Vโ โ U โ Vโ โง U โ Vโ :=
isOrtho_comm.trans <| isOrtho_sup_left.trans <| isOrtho_comm.and isOrtho_comm
#align submodule.is_ortho_sup_right Submodule.isOrtho_sup_right
@[simp]
theorem isOrtho_sSup_left {U : Set (Submodule ๐ E)} {V : Submodule ๐ E} :
sSup U โ V โ โ Uแตข โ U, Uแตข โ V :=
sSup_le_iff
#align submodule.is_ortho_Sup_left Submodule.isOrtho_sSup_left
@[simp]
theorem isOrtho_sSup_right {U : Submodule ๐ E} {V : Set (Submodule ๐ E)} :
U โ sSup V โ โ Vแตข โ V, U โ Vแตข :=
isOrtho_comm.trans <| isOrtho_sSup_left.trans <| by simp_rw [isOrtho_comm]
#align submodule.is_ortho_Sup_right Submodule.isOrtho_sSup_right
@[simp]
theorem isOrtho_iSup_left {ฮน : Sort*} {U : ฮน โ Submodule ๐ E} {V : Submodule ๐ E} :
iSup U โ V โ โ i, U i โ V :=
iSup_le_iff
#align submodule.is_ortho_supr_left Submodule.isOrtho_iSup_left
@[simp]
theorem isOrtho_iSup_right {ฮน : Sort*} {U : Submodule ๐ E} {V : ฮน โ Submodule ๐ E} :
U โ iSup V โ โ i, U โ V i :=
isOrtho_comm.trans <| isOrtho_iSup_left.trans <| by
|
simp_rw [isOrtho_comm]
|
@[simp]
theorem isOrtho_iSup_right {ฮน : Sort*} {U : Submodule ๐ E} {V : ฮน โ Submodule ๐ E} :
U โ iSup V โ โ i, U โ V i :=
isOrtho_comm.trans <| isOrtho_iSup_left.trans <| by
|
Mathlib.Analysis.InnerProductSpace.Orthogonal.367_0.QXx0GYqLoAbtfq1
|
@[simp]
theorem isOrtho_iSup_right {ฮน : Sort*} {U : Submodule ๐ E} {V : ฮน โ Submodule ๐ E} :
U โ iSup V โ โ i, U โ V i
|
Mathlib_Analysis_InnerProductSpace_Orthogonal
|
๐ : Type u_1
E : Type u_2
F : Type u_3
instโโด : IsROrC ๐
instโยณ : NormedAddCommGroup E
instโยฒ : InnerProductSpace ๐ E
instโยน : NormedAddCommGroup F
instโ : InnerProductSpace ๐ F
s t : Set E
โข span ๐ s โ span ๐ t โ โ โฆu : Eโฆ, u โ s โ โ โฆv : Eโฆ, v โ t โ inner u v = 0
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sรฉbastien Gouรซzel, Frรฉdรฉric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
/-!
# Orthogonal complements of submodules
In this file, the `orthogonal` complement of a submodule `K` is defined, and basic API established.
Some of the more subtle results about the orthogonal complement are delayed to
`Analysis.InnerProductSpace.Projection`.
See also `BilinForm.orthogonal` for orthogonality with respect to a general bilinear form.
## Notation
The orthogonal complement of a submodule `K` is denoted by `Kแฎ`.
The proposition that two submodules are orthogonal, `Submodule.IsOrtho`, is denoted by `U โ V`.
Note this is not the same unicode symbol as `โฅ` (`Bot`).
-/
variable {๐ E F : Type*} [IsROrC ๐]
variable [NormedAddCommGroup E] [InnerProductSpace ๐ E]
variable [NormedAddCommGroup F] [InnerProductSpace ๐ F]
local notation "โช" x ", " y "โซ" => @inner ๐ _ _ x y
namespace Submodule
variable (K : Submodule ๐ E)
/-- The subspace of vectors orthogonal to a given subspace. -/
def orthogonal : Submodule ๐ E where
carrier := { v | โ u โ K, โชu, vโซ = 0 }
zero_mem' _ _ := inner_zero_right _
add_mem' hx hy u hu := by rw [inner_add_right, hx u hu, hy u hu, add_zero]
smul_mem' c x hx u hu := by rw [inner_smul_right, hx u hu, mul_zero]
#align submodule.orthogonal Submodule.orthogonal
@[inherit_doc]
notation:1200 K "แฎ" => orthogonal K
/-- When a vector is in `Kแฎ`. -/
theorem mem_orthogonal (v : E) : v โ Kแฎ โ โ u โ K, โชu, vโซ = 0 :=
Iff.rfl
#align submodule.mem_orthogonal Submodule.mem_orthogonal
/-- When a vector is in `Kแฎ`, with the inner product the
other way round. -/
theorem mem_orthogonal' (v : E) : v โ Kแฎ โ โ u โ K, โชv, uโซ = 0 := by
simp_rw [mem_orthogonal, inner_eq_zero_symm]
#align submodule.mem_orthogonal' Submodule.mem_orthogonal'
variable {K}
/-- A vector in `K` is orthogonal to one in `Kแฎ`. -/
theorem inner_right_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชu, vโซ = 0 :=
(K.mem_orthogonal v).1 hv u hu
#align submodule.inner_right_of_mem_orthogonal Submodule.inner_right_of_mem_orthogonal
/-- A vector in `Kแฎ` is orthogonal to one in `K`. -/
theorem inner_left_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชv, uโซ = 0 := by
rw [inner_eq_zero_symm]; exact inner_right_of_mem_orthogonal hu hv
#align submodule.inner_left_of_mem_orthogonal Submodule.inner_left_of_mem_orthogonal
/-- A vector is in `(๐ โ u)แฎ` iff it is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_right {u v : E} : v โ (๐ โ u)แฎ โ โชu, vโซ = 0 := by
refine' โจinner_right_of_mem_orthogonal (mem_span_singleton_self u), _โฉ
intro hv w hw
rw [mem_span_singleton] at hw
obtain โจc, rflโฉ := hw
simp [inner_smul_left, hv]
#align submodule.mem_orthogonal_singleton_iff_inner_right Submodule.mem_orthogonal_singleton_iff_inner_right
/-- A vector in `(๐ โ u)แฎ` is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_left {u v : E} : v โ (๐ โ u)แฎ โ โชv, uโซ = 0 := by
rw [mem_orthogonal_singleton_iff_inner_right, inner_eq_zero_symm]
#align submodule.mem_orthogonal_singleton_iff_inner_left Submodule.mem_orthogonal_singleton_iff_inner_left
theorem sub_mem_orthogonal_of_inner_left {x y : E} (h : โ v : K, โชx, vโซ = โชy, vโซ) : x - y โ Kแฎ := by
rw [mem_orthogonal']
intro u hu
rw [inner_sub_left, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_left Submodule.sub_mem_orthogonal_of_inner_left
theorem sub_mem_orthogonal_of_inner_right {x y : E} (h : โ v : K, โช(v : E), xโซ = โช(v : E), yโซ) :
x - y โ Kแฎ := by
intro u hu
rw [inner_sub_right, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_right Submodule.sub_mem_orthogonal_of_inner_right
variable (K)
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem inf_orthogonal_eq_bot : K โ Kแฎ = โฅ := by
rw [eq_bot_iff]
intro x
rw [mem_inf]
exact fun โจhx, hoโฉ => inner_self_eq_zero.1 (ho x hx)
#align submodule.inf_orthogonal_eq_bot Submodule.inf_orthogonal_eq_bot
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem orthogonal_disjoint : Disjoint K Kแฎ := by simp [disjoint_iff, K.inf_orthogonal_eq_bot]
#align submodule.orthogonal_disjoint Submodule.orthogonal_disjoint
/-- `Kแฎ` can be characterized as the intersection of the kernels of the operations of
inner product with each of the elements of `K`. -/
theorem orthogonal_eq_inter : Kแฎ = โจ
v : K, LinearMap.ker (innerSL ๐ (v : E)) := by
apply le_antisymm
ยท rw [le_iInf_iff]
rintro โจv, hvโฉ w hw
simpa using hw _ hv
ยท intro v hv w hw
simp only [mem_iInf] at hv
exact hv โจw, hwโฉ
#align submodule.orthogonal_eq_inter Submodule.orthogonal_eq_inter
/-- The orthogonal complement of any submodule `K` is closed. -/
theorem isClosed_orthogonal : IsClosed (Kแฎ : Set E) := by
rw [orthogonal_eq_inter K]
have := fun v : K => ContinuousLinearMap.isClosed_ker (innerSL ๐ (v : E))
convert isClosed_iInter this
simp only [iInf_coe]
#align submodule.is_closed_orthogonal Submodule.isClosed_orthogonal
/-- In a complete space, the orthogonal complement of any submodule `K` is complete. -/
instance instOrthogonalCompleteSpace [CompleteSpace E] : CompleteSpace Kแฎ :=
K.isClosed_orthogonal.completeSpace_coe
variable (๐ E)
/-- `orthogonal` gives a `GaloisConnection` between
`Submodule ๐ E` and its `OrderDual`. -/
theorem orthogonal_gc :
@GaloisConnection (Submodule ๐ E) (Submodule ๐ E)แตแต _ _ orthogonal orthogonal := fun _Kโ _Kโ =>
โจfun h _v hv _u hu => inner_left_of_mem_orthogonal hv (h hu), fun h _v hv _u hu =>
inner_left_of_mem_orthogonal hv (h hu)โฉ
#align submodule.orthogonal_gc Submodule.orthogonal_gc
variable {๐ E}
/-- `orthogonal` reverses the `โค` ordering of two
subspaces. -/
theorem orthogonal_le {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎ โค Kโแฎ :=
(orthogonal_gc ๐ E).monotone_l h
#align submodule.orthogonal_le Submodule.orthogonal_le
/-- `orthogonal.orthogonal` preserves the `โค` ordering of two
subspaces. -/
theorem orthogonal_orthogonal_monotone {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎแฎ โค Kโแฎแฎ :=
orthogonal_le (orthogonal_le h)
#align submodule.orthogonal_orthogonal_monotone Submodule.orthogonal_orthogonal_monotone
/-- `K` is contained in `Kแฎแฎ`. -/
theorem le_orthogonal_orthogonal : K โค Kแฎแฎ :=
(orthogonal_gc ๐ E).le_u_l _
#align submodule.le_orthogonal_orthogonal Submodule.le_orthogonal_orthogonal
/-- The inf of two orthogonal subspaces equals the subspace orthogonal
to the sup. -/
theorem inf_orthogonal (Kโ Kโ : Submodule ๐ E) : Kโแฎ โ Kโแฎ = (Kโ โ Kโ)แฎ :=
(orthogonal_gc ๐ E).l_sup.symm
#align submodule.inf_orthogonal Submodule.inf_orthogonal
/-- The inf of an indexed family of orthogonal subspaces equals the
subspace orthogonal to the sup. -/
theorem iInf_orthogonal {ฮน : Type*} (K : ฮน โ Submodule ๐ E) : โจ
i, (K i)แฎ = (iSup K)แฎ :=
(orthogonal_gc ๐ E).l_iSup.symm
#align submodule.infi_orthogonal Submodule.iInf_orthogonal
/-- The inf of a set of orthogonal subspaces equals the subspace orthogonal to the sup. -/
theorem sInf_orthogonal (s : Set <| Submodule ๐ E) : โจ
K โ s, Kแฎ = (sSup s)แฎ :=
(orthogonal_gc ๐ E).l_sSup.symm
#align submodule.Inf_orthogonal Submodule.sInf_orthogonal
@[simp]
theorem top_orthogonal_eq_bot : (โค : Submodule ๐ E)แฎ = โฅ := by
ext x
rw [mem_bot, mem_orthogonal]
exact
โจfun h => inner_self_eq_zero.mp (h x mem_top), by
rintro rfl
simpโฉ
#align submodule.top_orthogonal_eq_bot Submodule.top_orthogonal_eq_bot
@[simp]
theorem bot_orthogonal_eq_top : (โฅ : Submodule ๐ E)แฎ = โค := by
rw [โ top_orthogonal_eq_bot, eq_top_iff]
exact le_orthogonal_orthogonal โค
#align submodule.bot_orthogonal_eq_top Submodule.bot_orthogonal_eq_top
@[simp]
theorem orthogonal_eq_top_iff : Kแฎ = โค โ K = โฅ := by
refine'
โจ_, by
rintro rfl
exact bot_orthogonal_eq_topโฉ
intro h
have : K โ Kแฎ = โฅ := K.orthogonal_disjoint.eq_bot
rwa [h, inf_comm, top_inf_eq] at this
#align submodule.orthogonal_eq_top_iff Submodule.orthogonal_eq_top_iff
theorem orthogonalFamily_self :
OrthogonalFamily ๐ (fun b => โฅ(cond b K Kแฎ)) fun b => (cond b K Kแฎ).subtypeโแตข
| true, true => absurd rfl
| true, false => fun _ x y => inner_right_of_mem_orthogonal x.prop y.prop
| false, true => fun _ x y => inner_left_of_mem_orthogonal y.prop x.prop
| false, false => absurd rfl
#align submodule.orthogonal_family_self Submodule.orthogonalFamily_self
end Submodule
@[simp]
theorem bilinFormOfRealInner_orthogonal {E} [NormedAddCommGroup E] [InnerProductSpace โ E]
(K : Submodule โ E) : bilinFormOfRealInner.orthogonal K = Kแฎ :=
rfl
#align bilin_form_of_real_inner_orthogonal bilinFormOfRealInner_orthogonal
/-!
### Orthogonality of submodules
In this section we define `Submodule.IsOrtho U V`, with notation `U โ V`.
The API roughly matches that of `Disjoint`.
-/
namespace Submodule
/-- The proposition that two submodules are orthogonal. Has notation `U โ V`. -/
def IsOrtho (U V : Submodule ๐ E) : Prop :=
U โค Vแฎ
#align submodule.is_ortho Submodule.IsOrtho
@[inherit_doc]
infixl:50 " โ " => Submodule.IsOrtho
theorem isOrtho_iff_le {U V : Submodule ๐ E} : U โ V โ U โค Vแฎ :=
Iff.rfl
#align submodule.is_ortho_iff_le Submodule.isOrtho_iff_le
@[symm]
theorem IsOrtho.symm {U V : Submodule ๐ E} (h : U โ V) : V โ U :=
(le_orthogonal_orthogonal _).trans (orthogonal_le h)
#align submodule.is_ortho.symm Submodule.IsOrtho.symm
theorem isOrtho_comm {U V : Submodule ๐ E} : U โ V โ V โ U :=
โจIsOrtho.symm, IsOrtho.symmโฉ
#align submodule.is_ortho_comm Submodule.isOrtho_comm
theorem symmetric_isOrtho : Symmetric (IsOrtho : Submodule ๐ E โ Submodule ๐ E โ Prop) := fun _ _ =>
IsOrtho.symm
#align submodule.symmetric_is_ortho Submodule.symmetric_isOrtho
theorem IsOrtho.inner_eq {U V : Submodule ๐ E} (h : U โ V) {u v : E} (hu : u โ U) (hv : v โ V) :
โชu, vโซ = 0 :=
h.symm hv _ hu
#align submodule.is_ortho.inner_eq Submodule.IsOrtho.inner_eq
theorem isOrtho_iff_inner_eq {U V : Submodule ๐ E} : U โ V โ โ u โ U, โ v โ V, โชu, vโซ = 0 :=
forallโ_congr fun _u _hu _v _hv => inner_eq_zero_symm
#align submodule.is_ortho_iff_inner_eq Submodule.isOrtho_iff_inner_eq
/- TODO: generalize `Submodule.mapโ` to semilinear maps, so that we can state
`U โ V โ Submodule.mapโ (innerโโ ๐) U V โค โฅ`. -/
@[simp]
theorem isOrtho_bot_left {V : Submodule ๐ E} : โฅ โ V :=
bot_le
#align submodule.is_ortho_bot_left Submodule.isOrtho_bot_left
@[simp]
theorem isOrtho_bot_right {U : Submodule ๐ E} : U โ โฅ :=
isOrtho_bot_left.symm
#align submodule.is_ortho_bot_right Submodule.isOrtho_bot_right
theorem IsOrtho.mono_left {Uโ Uโ V : Submodule ๐ E} (hU : Uโ โค Uโ) (h : Uโ โ V) : Uโ โ V :=
hU.trans h
#align submodule.is_ortho.mono_left Submodule.IsOrtho.mono_left
theorem IsOrtho.mono_right {U Vโ Vโ : Submodule ๐ E} (hV : Vโ โค Vโ) (h : U โ Vโ) : U โ Vโ :=
(h.symm.mono_left hV).symm
#align submodule.is_ortho.mono_right Submodule.IsOrtho.mono_right
theorem IsOrtho.mono {Uโ Vโ Uโ Vโ : Submodule ๐ E} (hU : Uโ โค Uโ) (hV : Vโ โค Vโ) (h : Uโ โ Vโ) :
Uโ โ Vโ :=
(h.mono_right hV).mono_left hU
#align submodule.is_ortho.mono Submodule.IsOrtho.mono
@[simp]
theorem isOrtho_self {U : Submodule ๐ E} : U โ U โ U = โฅ :=
โจfun h => eq_bot_iff.mpr fun x hx => inner_self_eq_zero.mp (h hx x hx), fun h =>
h.symm โธ isOrtho_bot_leftโฉ
#align submodule.is_ortho_self Submodule.isOrtho_self
@[simp]
theorem isOrtho_orthogonal_right (U : Submodule ๐ E) : U โ Uแฎ :=
le_orthogonal_orthogonal _
#align submodule.is_ortho_orthogonal_right Submodule.isOrtho_orthogonal_right
@[simp]
theorem isOrtho_orthogonal_left (U : Submodule ๐ E) : Uแฎ โ U :=
(isOrtho_orthogonal_right U).symm
#align submodule.is_ortho_orthogonal_left Submodule.isOrtho_orthogonal_left
theorem IsOrtho.le {U V : Submodule ๐ E} (h : U โ V) : U โค Vแฎ :=
h
#align submodule.is_ortho.le Submodule.IsOrtho.le
theorem IsOrtho.ge {U V : Submodule ๐ E} (h : U โ V) : V โค Uแฎ :=
h.symm
#align submodule.is_ortho.ge Submodule.IsOrtho.ge
@[simp]
theorem isOrtho_top_right {U : Submodule ๐ E} : U โ โค โ U = โฅ :=
โจfun h => eq_bot_iff.mpr fun _x hx => inner_self_eq_zero.mp (h hx _ mem_top), fun h =>
h.symm โธ isOrtho_bot_leftโฉ
#align submodule.is_ortho_top_right Submodule.isOrtho_top_right
@[simp]
theorem isOrtho_top_left {V : Submodule ๐ E} : โค โ V โ V = โฅ :=
isOrtho_comm.trans isOrtho_top_right
#align submodule.is_ortho_top_left Submodule.isOrtho_top_left
/-- Orthogonal submodules are disjoint. -/
theorem IsOrtho.disjoint {U V : Submodule ๐ E} (h : U โ V) : Disjoint U V :=
(Submodule.orthogonal_disjoint _).mono_right h.symm
#align submodule.is_ortho.disjoint Submodule.IsOrtho.disjoint
@[simp]
theorem isOrtho_sup_left {Uโ Uโ V : Submodule ๐ E} : Uโ โ Uโ โ V โ Uโ โ V โง Uโ โ V :=
sup_le_iff
#align submodule.is_ortho_sup_left Submodule.isOrtho_sup_left
@[simp]
theorem isOrtho_sup_right {U Vโ Vโ : Submodule ๐ E} : U โ Vโ โ Vโ โ U โ Vโ โง U โ Vโ :=
isOrtho_comm.trans <| isOrtho_sup_left.trans <| isOrtho_comm.and isOrtho_comm
#align submodule.is_ortho_sup_right Submodule.isOrtho_sup_right
@[simp]
theorem isOrtho_sSup_left {U : Set (Submodule ๐ E)} {V : Submodule ๐ E} :
sSup U โ V โ โ Uแตข โ U, Uแตข โ V :=
sSup_le_iff
#align submodule.is_ortho_Sup_left Submodule.isOrtho_sSup_left
@[simp]
theorem isOrtho_sSup_right {U : Submodule ๐ E} {V : Set (Submodule ๐ E)} :
U โ sSup V โ โ Vแตข โ V, U โ Vแตข :=
isOrtho_comm.trans <| isOrtho_sSup_left.trans <| by simp_rw [isOrtho_comm]
#align submodule.is_ortho_Sup_right Submodule.isOrtho_sSup_right
@[simp]
theorem isOrtho_iSup_left {ฮน : Sort*} {U : ฮน โ Submodule ๐ E} {V : Submodule ๐ E} :
iSup U โ V โ โ i, U i โ V :=
iSup_le_iff
#align submodule.is_ortho_supr_left Submodule.isOrtho_iSup_left
@[simp]
theorem isOrtho_iSup_right {ฮน : Sort*} {U : Submodule ๐ E} {V : ฮน โ Submodule ๐ E} :
U โ iSup V โ โ i, U โ V i :=
isOrtho_comm.trans <| isOrtho_iSup_left.trans <| by simp_rw [isOrtho_comm]
#align submodule.is_ortho_supr_right Submodule.isOrtho_iSup_right
@[simp]
theorem isOrtho_span {s t : Set E} :
span ๐ s โ span ๐ t โ โ โฆuโฆ, u โ s โ โ โฆvโฆ, v โ t โ โชu, vโซ = 0 := by
|
simp_rw [span_eq_iSup_of_singleton_spans s, span_eq_iSup_of_singleton_spans t, isOrtho_iSup_left,
isOrtho_iSup_right, isOrtho_iff_le, span_le, Set.subset_def, SetLike.mem_coe,
mem_orthogonal_singleton_iff_inner_left, Set.mem_singleton_iff, forall_eq]
|
@[simp]
theorem isOrtho_span {s t : Set E} :
span ๐ s โ span ๐ t โ โ โฆuโฆ, u โ s โ โ โฆvโฆ, v โ t โ โชu, vโซ = 0 := by
|
Mathlib.Analysis.InnerProductSpace.Orthogonal.373_0.QXx0GYqLoAbtfq1
|
@[simp]
theorem isOrtho_span {s t : Set E} :
span ๐ s โ span ๐ t โ โ โฆuโฆ, u โ s โ โ โฆvโฆ, v โ t โ โชu, vโซ = 0
|
Mathlib_Analysis_InnerProductSpace_Orthogonal
|
๐ : Type u_1
E : Type u_2
F : Type u_3
instโโด : IsROrC ๐
instโยณ : NormedAddCommGroup E
instโยฒ : InnerProductSpace ๐ E
instโยน : NormedAddCommGroup F
instโ : InnerProductSpace ๐ F
f : E โโแตข[๐] F
U V : Submodule ๐ E
h : U โ V
โข Submodule.map f U โ Submodule.map f V
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sรฉbastien Gouรซzel, Frรฉdรฉric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
/-!
# Orthogonal complements of submodules
In this file, the `orthogonal` complement of a submodule `K` is defined, and basic API established.
Some of the more subtle results about the orthogonal complement are delayed to
`Analysis.InnerProductSpace.Projection`.
See also `BilinForm.orthogonal` for orthogonality with respect to a general bilinear form.
## Notation
The orthogonal complement of a submodule `K` is denoted by `Kแฎ`.
The proposition that two submodules are orthogonal, `Submodule.IsOrtho`, is denoted by `U โ V`.
Note this is not the same unicode symbol as `โฅ` (`Bot`).
-/
variable {๐ E F : Type*} [IsROrC ๐]
variable [NormedAddCommGroup E] [InnerProductSpace ๐ E]
variable [NormedAddCommGroup F] [InnerProductSpace ๐ F]
local notation "โช" x ", " y "โซ" => @inner ๐ _ _ x y
namespace Submodule
variable (K : Submodule ๐ E)
/-- The subspace of vectors orthogonal to a given subspace. -/
def orthogonal : Submodule ๐ E where
carrier := { v | โ u โ K, โชu, vโซ = 0 }
zero_mem' _ _ := inner_zero_right _
add_mem' hx hy u hu := by rw [inner_add_right, hx u hu, hy u hu, add_zero]
smul_mem' c x hx u hu := by rw [inner_smul_right, hx u hu, mul_zero]
#align submodule.orthogonal Submodule.orthogonal
@[inherit_doc]
notation:1200 K "แฎ" => orthogonal K
/-- When a vector is in `Kแฎ`. -/
theorem mem_orthogonal (v : E) : v โ Kแฎ โ โ u โ K, โชu, vโซ = 0 :=
Iff.rfl
#align submodule.mem_orthogonal Submodule.mem_orthogonal
/-- When a vector is in `Kแฎ`, with the inner product the
other way round. -/
theorem mem_orthogonal' (v : E) : v โ Kแฎ โ โ u โ K, โชv, uโซ = 0 := by
simp_rw [mem_orthogonal, inner_eq_zero_symm]
#align submodule.mem_orthogonal' Submodule.mem_orthogonal'
variable {K}
/-- A vector in `K` is orthogonal to one in `Kแฎ`. -/
theorem inner_right_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชu, vโซ = 0 :=
(K.mem_orthogonal v).1 hv u hu
#align submodule.inner_right_of_mem_orthogonal Submodule.inner_right_of_mem_orthogonal
/-- A vector in `Kแฎ` is orthogonal to one in `K`. -/
theorem inner_left_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชv, uโซ = 0 := by
rw [inner_eq_zero_symm]; exact inner_right_of_mem_orthogonal hu hv
#align submodule.inner_left_of_mem_orthogonal Submodule.inner_left_of_mem_orthogonal
/-- A vector is in `(๐ โ u)แฎ` iff it is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_right {u v : E} : v โ (๐ โ u)แฎ โ โชu, vโซ = 0 := by
refine' โจinner_right_of_mem_orthogonal (mem_span_singleton_self u), _โฉ
intro hv w hw
rw [mem_span_singleton] at hw
obtain โจc, rflโฉ := hw
simp [inner_smul_left, hv]
#align submodule.mem_orthogonal_singleton_iff_inner_right Submodule.mem_orthogonal_singleton_iff_inner_right
/-- A vector in `(๐ โ u)แฎ` is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_left {u v : E} : v โ (๐ โ u)แฎ โ โชv, uโซ = 0 := by
rw [mem_orthogonal_singleton_iff_inner_right, inner_eq_zero_symm]
#align submodule.mem_orthogonal_singleton_iff_inner_left Submodule.mem_orthogonal_singleton_iff_inner_left
theorem sub_mem_orthogonal_of_inner_left {x y : E} (h : โ v : K, โชx, vโซ = โชy, vโซ) : x - y โ Kแฎ := by
rw [mem_orthogonal']
intro u hu
rw [inner_sub_left, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_left Submodule.sub_mem_orthogonal_of_inner_left
theorem sub_mem_orthogonal_of_inner_right {x y : E} (h : โ v : K, โช(v : E), xโซ = โช(v : E), yโซ) :
x - y โ Kแฎ := by
intro u hu
rw [inner_sub_right, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_right Submodule.sub_mem_orthogonal_of_inner_right
variable (K)
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem inf_orthogonal_eq_bot : K โ Kแฎ = โฅ := by
rw [eq_bot_iff]
intro x
rw [mem_inf]
exact fun โจhx, hoโฉ => inner_self_eq_zero.1 (ho x hx)
#align submodule.inf_orthogonal_eq_bot Submodule.inf_orthogonal_eq_bot
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem orthogonal_disjoint : Disjoint K Kแฎ := by simp [disjoint_iff, K.inf_orthogonal_eq_bot]
#align submodule.orthogonal_disjoint Submodule.orthogonal_disjoint
/-- `Kแฎ` can be characterized as the intersection of the kernels of the operations of
inner product with each of the elements of `K`. -/
theorem orthogonal_eq_inter : Kแฎ = โจ
v : K, LinearMap.ker (innerSL ๐ (v : E)) := by
apply le_antisymm
ยท rw [le_iInf_iff]
rintro โจv, hvโฉ w hw
simpa using hw _ hv
ยท intro v hv w hw
simp only [mem_iInf] at hv
exact hv โจw, hwโฉ
#align submodule.orthogonal_eq_inter Submodule.orthogonal_eq_inter
/-- The orthogonal complement of any submodule `K` is closed. -/
theorem isClosed_orthogonal : IsClosed (Kแฎ : Set E) := by
rw [orthogonal_eq_inter K]
have := fun v : K => ContinuousLinearMap.isClosed_ker (innerSL ๐ (v : E))
convert isClosed_iInter this
simp only [iInf_coe]
#align submodule.is_closed_orthogonal Submodule.isClosed_orthogonal
/-- In a complete space, the orthogonal complement of any submodule `K` is complete. -/
instance instOrthogonalCompleteSpace [CompleteSpace E] : CompleteSpace Kแฎ :=
K.isClosed_orthogonal.completeSpace_coe
variable (๐ E)
/-- `orthogonal` gives a `GaloisConnection` between
`Submodule ๐ E` and its `OrderDual`. -/
theorem orthogonal_gc :
@GaloisConnection (Submodule ๐ E) (Submodule ๐ E)แตแต _ _ orthogonal orthogonal := fun _Kโ _Kโ =>
โจfun h _v hv _u hu => inner_left_of_mem_orthogonal hv (h hu), fun h _v hv _u hu =>
inner_left_of_mem_orthogonal hv (h hu)โฉ
#align submodule.orthogonal_gc Submodule.orthogonal_gc
variable {๐ E}
/-- `orthogonal` reverses the `โค` ordering of two
subspaces. -/
theorem orthogonal_le {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎ โค Kโแฎ :=
(orthogonal_gc ๐ E).monotone_l h
#align submodule.orthogonal_le Submodule.orthogonal_le
/-- `orthogonal.orthogonal` preserves the `โค` ordering of two
subspaces. -/
theorem orthogonal_orthogonal_monotone {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎแฎ โค Kโแฎแฎ :=
orthogonal_le (orthogonal_le h)
#align submodule.orthogonal_orthogonal_monotone Submodule.orthogonal_orthogonal_monotone
/-- `K` is contained in `Kแฎแฎ`. -/
theorem le_orthogonal_orthogonal : K โค Kแฎแฎ :=
(orthogonal_gc ๐ E).le_u_l _
#align submodule.le_orthogonal_orthogonal Submodule.le_orthogonal_orthogonal
/-- The inf of two orthogonal subspaces equals the subspace orthogonal
to the sup. -/
theorem inf_orthogonal (Kโ Kโ : Submodule ๐ E) : Kโแฎ โ Kโแฎ = (Kโ โ Kโ)แฎ :=
(orthogonal_gc ๐ E).l_sup.symm
#align submodule.inf_orthogonal Submodule.inf_orthogonal
/-- The inf of an indexed family of orthogonal subspaces equals the
subspace orthogonal to the sup. -/
theorem iInf_orthogonal {ฮน : Type*} (K : ฮน โ Submodule ๐ E) : โจ
i, (K i)แฎ = (iSup K)แฎ :=
(orthogonal_gc ๐ E).l_iSup.symm
#align submodule.infi_orthogonal Submodule.iInf_orthogonal
/-- The inf of a set of orthogonal subspaces equals the subspace orthogonal to the sup. -/
theorem sInf_orthogonal (s : Set <| Submodule ๐ E) : โจ
K โ s, Kแฎ = (sSup s)แฎ :=
(orthogonal_gc ๐ E).l_sSup.symm
#align submodule.Inf_orthogonal Submodule.sInf_orthogonal
@[simp]
theorem top_orthogonal_eq_bot : (โค : Submodule ๐ E)แฎ = โฅ := by
ext x
rw [mem_bot, mem_orthogonal]
exact
โจfun h => inner_self_eq_zero.mp (h x mem_top), by
rintro rfl
simpโฉ
#align submodule.top_orthogonal_eq_bot Submodule.top_orthogonal_eq_bot
@[simp]
theorem bot_orthogonal_eq_top : (โฅ : Submodule ๐ E)แฎ = โค := by
rw [โ top_orthogonal_eq_bot, eq_top_iff]
exact le_orthogonal_orthogonal โค
#align submodule.bot_orthogonal_eq_top Submodule.bot_orthogonal_eq_top
@[simp]
theorem orthogonal_eq_top_iff : Kแฎ = โค โ K = โฅ := by
refine'
โจ_, by
rintro rfl
exact bot_orthogonal_eq_topโฉ
intro h
have : K โ Kแฎ = โฅ := K.orthogonal_disjoint.eq_bot
rwa [h, inf_comm, top_inf_eq] at this
#align submodule.orthogonal_eq_top_iff Submodule.orthogonal_eq_top_iff
theorem orthogonalFamily_self :
OrthogonalFamily ๐ (fun b => โฅ(cond b K Kแฎ)) fun b => (cond b K Kแฎ).subtypeโแตข
| true, true => absurd rfl
| true, false => fun _ x y => inner_right_of_mem_orthogonal x.prop y.prop
| false, true => fun _ x y => inner_left_of_mem_orthogonal y.prop x.prop
| false, false => absurd rfl
#align submodule.orthogonal_family_self Submodule.orthogonalFamily_self
end Submodule
@[simp]
theorem bilinFormOfRealInner_orthogonal {E} [NormedAddCommGroup E] [InnerProductSpace โ E]
(K : Submodule โ E) : bilinFormOfRealInner.orthogonal K = Kแฎ :=
rfl
#align bilin_form_of_real_inner_orthogonal bilinFormOfRealInner_orthogonal
/-!
### Orthogonality of submodules
In this section we define `Submodule.IsOrtho U V`, with notation `U โ V`.
The API roughly matches that of `Disjoint`.
-/
namespace Submodule
/-- The proposition that two submodules are orthogonal. Has notation `U โ V`. -/
def IsOrtho (U V : Submodule ๐ E) : Prop :=
U โค Vแฎ
#align submodule.is_ortho Submodule.IsOrtho
@[inherit_doc]
infixl:50 " โ " => Submodule.IsOrtho
theorem isOrtho_iff_le {U V : Submodule ๐ E} : U โ V โ U โค Vแฎ :=
Iff.rfl
#align submodule.is_ortho_iff_le Submodule.isOrtho_iff_le
@[symm]
theorem IsOrtho.symm {U V : Submodule ๐ E} (h : U โ V) : V โ U :=
(le_orthogonal_orthogonal _).trans (orthogonal_le h)
#align submodule.is_ortho.symm Submodule.IsOrtho.symm
theorem isOrtho_comm {U V : Submodule ๐ E} : U โ V โ V โ U :=
โจIsOrtho.symm, IsOrtho.symmโฉ
#align submodule.is_ortho_comm Submodule.isOrtho_comm
theorem symmetric_isOrtho : Symmetric (IsOrtho : Submodule ๐ E โ Submodule ๐ E โ Prop) := fun _ _ =>
IsOrtho.symm
#align submodule.symmetric_is_ortho Submodule.symmetric_isOrtho
theorem IsOrtho.inner_eq {U V : Submodule ๐ E} (h : U โ V) {u v : E} (hu : u โ U) (hv : v โ V) :
โชu, vโซ = 0 :=
h.symm hv _ hu
#align submodule.is_ortho.inner_eq Submodule.IsOrtho.inner_eq
theorem isOrtho_iff_inner_eq {U V : Submodule ๐ E} : U โ V โ โ u โ U, โ v โ V, โชu, vโซ = 0 :=
forallโ_congr fun _u _hu _v _hv => inner_eq_zero_symm
#align submodule.is_ortho_iff_inner_eq Submodule.isOrtho_iff_inner_eq
/- TODO: generalize `Submodule.mapโ` to semilinear maps, so that we can state
`U โ V โ Submodule.mapโ (innerโโ ๐) U V โค โฅ`. -/
@[simp]
theorem isOrtho_bot_left {V : Submodule ๐ E} : โฅ โ V :=
bot_le
#align submodule.is_ortho_bot_left Submodule.isOrtho_bot_left
@[simp]
theorem isOrtho_bot_right {U : Submodule ๐ E} : U โ โฅ :=
isOrtho_bot_left.symm
#align submodule.is_ortho_bot_right Submodule.isOrtho_bot_right
theorem IsOrtho.mono_left {Uโ Uโ V : Submodule ๐ E} (hU : Uโ โค Uโ) (h : Uโ โ V) : Uโ โ V :=
hU.trans h
#align submodule.is_ortho.mono_left Submodule.IsOrtho.mono_left
theorem IsOrtho.mono_right {U Vโ Vโ : Submodule ๐ E} (hV : Vโ โค Vโ) (h : U โ Vโ) : U โ Vโ :=
(h.symm.mono_left hV).symm
#align submodule.is_ortho.mono_right Submodule.IsOrtho.mono_right
theorem IsOrtho.mono {Uโ Vโ Uโ Vโ : Submodule ๐ E} (hU : Uโ โค Uโ) (hV : Vโ โค Vโ) (h : Uโ โ Vโ) :
Uโ โ Vโ :=
(h.mono_right hV).mono_left hU
#align submodule.is_ortho.mono Submodule.IsOrtho.mono
@[simp]
theorem isOrtho_self {U : Submodule ๐ E} : U โ U โ U = โฅ :=
โจfun h => eq_bot_iff.mpr fun x hx => inner_self_eq_zero.mp (h hx x hx), fun h =>
h.symm โธ isOrtho_bot_leftโฉ
#align submodule.is_ortho_self Submodule.isOrtho_self
@[simp]
theorem isOrtho_orthogonal_right (U : Submodule ๐ E) : U โ Uแฎ :=
le_orthogonal_orthogonal _
#align submodule.is_ortho_orthogonal_right Submodule.isOrtho_orthogonal_right
@[simp]
theorem isOrtho_orthogonal_left (U : Submodule ๐ E) : Uแฎ โ U :=
(isOrtho_orthogonal_right U).symm
#align submodule.is_ortho_orthogonal_left Submodule.isOrtho_orthogonal_left
theorem IsOrtho.le {U V : Submodule ๐ E} (h : U โ V) : U โค Vแฎ :=
h
#align submodule.is_ortho.le Submodule.IsOrtho.le
theorem IsOrtho.ge {U V : Submodule ๐ E} (h : U โ V) : V โค Uแฎ :=
h.symm
#align submodule.is_ortho.ge Submodule.IsOrtho.ge
@[simp]
theorem isOrtho_top_right {U : Submodule ๐ E} : U โ โค โ U = โฅ :=
โจfun h => eq_bot_iff.mpr fun _x hx => inner_self_eq_zero.mp (h hx _ mem_top), fun h =>
h.symm โธ isOrtho_bot_leftโฉ
#align submodule.is_ortho_top_right Submodule.isOrtho_top_right
@[simp]
theorem isOrtho_top_left {V : Submodule ๐ E} : โค โ V โ V = โฅ :=
isOrtho_comm.trans isOrtho_top_right
#align submodule.is_ortho_top_left Submodule.isOrtho_top_left
/-- Orthogonal submodules are disjoint. -/
theorem IsOrtho.disjoint {U V : Submodule ๐ E} (h : U โ V) : Disjoint U V :=
(Submodule.orthogonal_disjoint _).mono_right h.symm
#align submodule.is_ortho.disjoint Submodule.IsOrtho.disjoint
@[simp]
theorem isOrtho_sup_left {Uโ Uโ V : Submodule ๐ E} : Uโ โ Uโ โ V โ Uโ โ V โง Uโ โ V :=
sup_le_iff
#align submodule.is_ortho_sup_left Submodule.isOrtho_sup_left
@[simp]
theorem isOrtho_sup_right {U Vโ Vโ : Submodule ๐ E} : U โ Vโ โ Vโ โ U โ Vโ โง U โ Vโ :=
isOrtho_comm.trans <| isOrtho_sup_left.trans <| isOrtho_comm.and isOrtho_comm
#align submodule.is_ortho_sup_right Submodule.isOrtho_sup_right
@[simp]
theorem isOrtho_sSup_left {U : Set (Submodule ๐ E)} {V : Submodule ๐ E} :
sSup U โ V โ โ Uแตข โ U, Uแตข โ V :=
sSup_le_iff
#align submodule.is_ortho_Sup_left Submodule.isOrtho_sSup_left
@[simp]
theorem isOrtho_sSup_right {U : Submodule ๐ E} {V : Set (Submodule ๐ E)} :
U โ sSup V โ โ Vแตข โ V, U โ Vแตข :=
isOrtho_comm.trans <| isOrtho_sSup_left.trans <| by simp_rw [isOrtho_comm]
#align submodule.is_ortho_Sup_right Submodule.isOrtho_sSup_right
@[simp]
theorem isOrtho_iSup_left {ฮน : Sort*} {U : ฮน โ Submodule ๐ E} {V : Submodule ๐ E} :
iSup U โ V โ โ i, U i โ V :=
iSup_le_iff
#align submodule.is_ortho_supr_left Submodule.isOrtho_iSup_left
@[simp]
theorem isOrtho_iSup_right {ฮน : Sort*} {U : Submodule ๐ E} {V : ฮน โ Submodule ๐ E} :
U โ iSup V โ โ i, U โ V i :=
isOrtho_comm.trans <| isOrtho_iSup_left.trans <| by simp_rw [isOrtho_comm]
#align submodule.is_ortho_supr_right Submodule.isOrtho_iSup_right
@[simp]
theorem isOrtho_span {s t : Set E} :
span ๐ s โ span ๐ t โ โ โฆuโฆ, u โ s โ โ โฆvโฆ, v โ t โ โชu, vโซ = 0 := by
simp_rw [span_eq_iSup_of_singleton_spans s, span_eq_iSup_of_singleton_spans t, isOrtho_iSup_left,
isOrtho_iSup_right, isOrtho_iff_le, span_le, Set.subset_def, SetLike.mem_coe,
mem_orthogonal_singleton_iff_inner_left, Set.mem_singleton_iff, forall_eq]
#align submodule.is_ortho_span Submodule.isOrtho_span
theorem IsOrtho.map (f : E โโแตข[๐] F) {U V : Submodule ๐ E} (h : U โ V) : U.map f โ V.map f := by
|
rw [isOrtho_iff_inner_eq] at *
|
theorem IsOrtho.map (f : E โโแตข[๐] F) {U V : Submodule ๐ E} (h : U โ V) : U.map f โ V.map f := by
|
Mathlib.Analysis.InnerProductSpace.Orthogonal.381_0.QXx0GYqLoAbtfq1
|
theorem IsOrtho.map (f : E โโแตข[๐] F) {U V : Submodule ๐ E} (h : U โ V) : U.map f โ V.map f
|
Mathlib_Analysis_InnerProductSpace_Orthogonal
|
๐ : Type u_1
E : Type u_2
F : Type u_3
instโโด : IsROrC ๐
instโยณ : NormedAddCommGroup E
instโยฒ : InnerProductSpace ๐ E
instโยน : NormedAddCommGroup F
instโ : InnerProductSpace ๐ F
f : E โโแตข[๐] F
U V : Submodule ๐ E
h : โ u โ U, โ v โ V, inner u v = 0
โข โ u โ Submodule.map f U, โ v โ Submodule.map f V, inner u v = 0
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sรฉbastien Gouรซzel, Frรฉdรฉric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
/-!
# Orthogonal complements of submodules
In this file, the `orthogonal` complement of a submodule `K` is defined, and basic API established.
Some of the more subtle results about the orthogonal complement are delayed to
`Analysis.InnerProductSpace.Projection`.
See also `BilinForm.orthogonal` for orthogonality with respect to a general bilinear form.
## Notation
The orthogonal complement of a submodule `K` is denoted by `Kแฎ`.
The proposition that two submodules are orthogonal, `Submodule.IsOrtho`, is denoted by `U โ V`.
Note this is not the same unicode symbol as `โฅ` (`Bot`).
-/
variable {๐ E F : Type*} [IsROrC ๐]
variable [NormedAddCommGroup E] [InnerProductSpace ๐ E]
variable [NormedAddCommGroup F] [InnerProductSpace ๐ F]
local notation "โช" x ", " y "โซ" => @inner ๐ _ _ x y
namespace Submodule
variable (K : Submodule ๐ E)
/-- The subspace of vectors orthogonal to a given subspace. -/
def orthogonal : Submodule ๐ E where
carrier := { v | โ u โ K, โชu, vโซ = 0 }
zero_mem' _ _ := inner_zero_right _
add_mem' hx hy u hu := by rw [inner_add_right, hx u hu, hy u hu, add_zero]
smul_mem' c x hx u hu := by rw [inner_smul_right, hx u hu, mul_zero]
#align submodule.orthogonal Submodule.orthogonal
@[inherit_doc]
notation:1200 K "แฎ" => orthogonal K
/-- When a vector is in `Kแฎ`. -/
theorem mem_orthogonal (v : E) : v โ Kแฎ โ โ u โ K, โชu, vโซ = 0 :=
Iff.rfl
#align submodule.mem_orthogonal Submodule.mem_orthogonal
/-- When a vector is in `Kแฎ`, with the inner product the
other way round. -/
theorem mem_orthogonal' (v : E) : v โ Kแฎ โ โ u โ K, โชv, uโซ = 0 := by
simp_rw [mem_orthogonal, inner_eq_zero_symm]
#align submodule.mem_orthogonal' Submodule.mem_orthogonal'
variable {K}
/-- A vector in `K` is orthogonal to one in `Kแฎ`. -/
theorem inner_right_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชu, vโซ = 0 :=
(K.mem_orthogonal v).1 hv u hu
#align submodule.inner_right_of_mem_orthogonal Submodule.inner_right_of_mem_orthogonal
/-- A vector in `Kแฎ` is orthogonal to one in `K`. -/
theorem inner_left_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชv, uโซ = 0 := by
rw [inner_eq_zero_symm]; exact inner_right_of_mem_orthogonal hu hv
#align submodule.inner_left_of_mem_orthogonal Submodule.inner_left_of_mem_orthogonal
/-- A vector is in `(๐ โ u)แฎ` iff it is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_right {u v : E} : v โ (๐ โ u)แฎ โ โชu, vโซ = 0 := by
refine' โจinner_right_of_mem_orthogonal (mem_span_singleton_self u), _โฉ
intro hv w hw
rw [mem_span_singleton] at hw
obtain โจc, rflโฉ := hw
simp [inner_smul_left, hv]
#align submodule.mem_orthogonal_singleton_iff_inner_right Submodule.mem_orthogonal_singleton_iff_inner_right
/-- A vector in `(๐ โ u)แฎ` is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_left {u v : E} : v โ (๐ โ u)แฎ โ โชv, uโซ = 0 := by
rw [mem_orthogonal_singleton_iff_inner_right, inner_eq_zero_symm]
#align submodule.mem_orthogonal_singleton_iff_inner_left Submodule.mem_orthogonal_singleton_iff_inner_left
theorem sub_mem_orthogonal_of_inner_left {x y : E} (h : โ v : K, โชx, vโซ = โชy, vโซ) : x - y โ Kแฎ := by
rw [mem_orthogonal']
intro u hu
rw [inner_sub_left, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_left Submodule.sub_mem_orthogonal_of_inner_left
theorem sub_mem_orthogonal_of_inner_right {x y : E} (h : โ v : K, โช(v : E), xโซ = โช(v : E), yโซ) :
x - y โ Kแฎ := by
intro u hu
rw [inner_sub_right, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_right Submodule.sub_mem_orthogonal_of_inner_right
variable (K)
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem inf_orthogonal_eq_bot : K โ Kแฎ = โฅ := by
rw [eq_bot_iff]
intro x
rw [mem_inf]
exact fun โจhx, hoโฉ => inner_self_eq_zero.1 (ho x hx)
#align submodule.inf_orthogonal_eq_bot Submodule.inf_orthogonal_eq_bot
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem orthogonal_disjoint : Disjoint K Kแฎ := by simp [disjoint_iff, K.inf_orthogonal_eq_bot]
#align submodule.orthogonal_disjoint Submodule.orthogonal_disjoint
/-- `Kแฎ` can be characterized as the intersection of the kernels of the operations of
inner product with each of the elements of `K`. -/
theorem orthogonal_eq_inter : Kแฎ = โจ
v : K, LinearMap.ker (innerSL ๐ (v : E)) := by
apply le_antisymm
ยท rw [le_iInf_iff]
rintro โจv, hvโฉ w hw
simpa using hw _ hv
ยท intro v hv w hw
simp only [mem_iInf] at hv
exact hv โจw, hwโฉ
#align submodule.orthogonal_eq_inter Submodule.orthogonal_eq_inter
/-- The orthogonal complement of any submodule `K` is closed. -/
theorem isClosed_orthogonal : IsClosed (Kแฎ : Set E) := by
rw [orthogonal_eq_inter K]
have := fun v : K => ContinuousLinearMap.isClosed_ker (innerSL ๐ (v : E))
convert isClosed_iInter this
simp only [iInf_coe]
#align submodule.is_closed_orthogonal Submodule.isClosed_orthogonal
/-- In a complete space, the orthogonal complement of any submodule `K` is complete. -/
instance instOrthogonalCompleteSpace [CompleteSpace E] : CompleteSpace Kแฎ :=
K.isClosed_orthogonal.completeSpace_coe
variable (๐ E)
/-- `orthogonal` gives a `GaloisConnection` between
`Submodule ๐ E` and its `OrderDual`. -/
theorem orthogonal_gc :
@GaloisConnection (Submodule ๐ E) (Submodule ๐ E)แตแต _ _ orthogonal orthogonal := fun _Kโ _Kโ =>
โจfun h _v hv _u hu => inner_left_of_mem_orthogonal hv (h hu), fun h _v hv _u hu =>
inner_left_of_mem_orthogonal hv (h hu)โฉ
#align submodule.orthogonal_gc Submodule.orthogonal_gc
variable {๐ E}
/-- `orthogonal` reverses the `โค` ordering of two
subspaces. -/
theorem orthogonal_le {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎ โค Kโแฎ :=
(orthogonal_gc ๐ E).monotone_l h
#align submodule.orthogonal_le Submodule.orthogonal_le
/-- `orthogonal.orthogonal` preserves the `โค` ordering of two
subspaces. -/
theorem orthogonal_orthogonal_monotone {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎแฎ โค Kโแฎแฎ :=
orthogonal_le (orthogonal_le h)
#align submodule.orthogonal_orthogonal_monotone Submodule.orthogonal_orthogonal_monotone
/-- `K` is contained in `Kแฎแฎ`. -/
theorem le_orthogonal_orthogonal : K โค Kแฎแฎ :=
(orthogonal_gc ๐ E).le_u_l _
#align submodule.le_orthogonal_orthogonal Submodule.le_orthogonal_orthogonal
/-- The inf of two orthogonal subspaces equals the subspace orthogonal
to the sup. -/
theorem inf_orthogonal (Kโ Kโ : Submodule ๐ E) : Kโแฎ โ Kโแฎ = (Kโ โ Kโ)แฎ :=
(orthogonal_gc ๐ E).l_sup.symm
#align submodule.inf_orthogonal Submodule.inf_orthogonal
/-- The inf of an indexed family of orthogonal subspaces equals the
subspace orthogonal to the sup. -/
theorem iInf_orthogonal {ฮน : Type*} (K : ฮน โ Submodule ๐ E) : โจ
i, (K i)แฎ = (iSup K)แฎ :=
(orthogonal_gc ๐ E).l_iSup.symm
#align submodule.infi_orthogonal Submodule.iInf_orthogonal
/-- The inf of a set of orthogonal subspaces equals the subspace orthogonal to the sup. -/
theorem sInf_orthogonal (s : Set <| Submodule ๐ E) : โจ
K โ s, Kแฎ = (sSup s)แฎ :=
(orthogonal_gc ๐ E).l_sSup.symm
#align submodule.Inf_orthogonal Submodule.sInf_orthogonal
@[simp]
theorem top_orthogonal_eq_bot : (โค : Submodule ๐ E)แฎ = โฅ := by
ext x
rw [mem_bot, mem_orthogonal]
exact
โจfun h => inner_self_eq_zero.mp (h x mem_top), by
rintro rfl
simpโฉ
#align submodule.top_orthogonal_eq_bot Submodule.top_orthogonal_eq_bot
@[simp]
theorem bot_orthogonal_eq_top : (โฅ : Submodule ๐ E)แฎ = โค := by
rw [โ top_orthogonal_eq_bot, eq_top_iff]
exact le_orthogonal_orthogonal โค
#align submodule.bot_orthogonal_eq_top Submodule.bot_orthogonal_eq_top
@[simp]
theorem orthogonal_eq_top_iff : Kแฎ = โค โ K = โฅ := by
refine'
โจ_, by
rintro rfl
exact bot_orthogonal_eq_topโฉ
intro h
have : K โ Kแฎ = โฅ := K.orthogonal_disjoint.eq_bot
rwa [h, inf_comm, top_inf_eq] at this
#align submodule.orthogonal_eq_top_iff Submodule.orthogonal_eq_top_iff
theorem orthogonalFamily_self :
OrthogonalFamily ๐ (fun b => โฅ(cond b K Kแฎ)) fun b => (cond b K Kแฎ).subtypeโแตข
| true, true => absurd rfl
| true, false => fun _ x y => inner_right_of_mem_orthogonal x.prop y.prop
| false, true => fun _ x y => inner_left_of_mem_orthogonal y.prop x.prop
| false, false => absurd rfl
#align submodule.orthogonal_family_self Submodule.orthogonalFamily_self
end Submodule
@[simp]
theorem bilinFormOfRealInner_orthogonal {E} [NormedAddCommGroup E] [InnerProductSpace โ E]
(K : Submodule โ E) : bilinFormOfRealInner.orthogonal K = Kแฎ :=
rfl
#align bilin_form_of_real_inner_orthogonal bilinFormOfRealInner_orthogonal
/-!
### Orthogonality of submodules
In this section we define `Submodule.IsOrtho U V`, with notation `U โ V`.
The API roughly matches that of `Disjoint`.
-/
namespace Submodule
/-- The proposition that two submodules are orthogonal. Has notation `U โ V`. -/
def IsOrtho (U V : Submodule ๐ E) : Prop :=
U โค Vแฎ
#align submodule.is_ortho Submodule.IsOrtho
@[inherit_doc]
infixl:50 " โ " => Submodule.IsOrtho
theorem isOrtho_iff_le {U V : Submodule ๐ E} : U โ V โ U โค Vแฎ :=
Iff.rfl
#align submodule.is_ortho_iff_le Submodule.isOrtho_iff_le
@[symm]
theorem IsOrtho.symm {U V : Submodule ๐ E} (h : U โ V) : V โ U :=
(le_orthogonal_orthogonal _).trans (orthogonal_le h)
#align submodule.is_ortho.symm Submodule.IsOrtho.symm
theorem isOrtho_comm {U V : Submodule ๐ E} : U โ V โ V โ U :=
โจIsOrtho.symm, IsOrtho.symmโฉ
#align submodule.is_ortho_comm Submodule.isOrtho_comm
theorem symmetric_isOrtho : Symmetric (IsOrtho : Submodule ๐ E โ Submodule ๐ E โ Prop) := fun _ _ =>
IsOrtho.symm
#align submodule.symmetric_is_ortho Submodule.symmetric_isOrtho
theorem IsOrtho.inner_eq {U V : Submodule ๐ E} (h : U โ V) {u v : E} (hu : u โ U) (hv : v โ V) :
โชu, vโซ = 0 :=
h.symm hv _ hu
#align submodule.is_ortho.inner_eq Submodule.IsOrtho.inner_eq
theorem isOrtho_iff_inner_eq {U V : Submodule ๐ E} : U โ V โ โ u โ U, โ v โ V, โชu, vโซ = 0 :=
forallโ_congr fun _u _hu _v _hv => inner_eq_zero_symm
#align submodule.is_ortho_iff_inner_eq Submodule.isOrtho_iff_inner_eq
/- TODO: generalize `Submodule.mapโ` to semilinear maps, so that we can state
`U โ V โ Submodule.mapโ (innerโโ ๐) U V โค โฅ`. -/
@[simp]
theorem isOrtho_bot_left {V : Submodule ๐ E} : โฅ โ V :=
bot_le
#align submodule.is_ortho_bot_left Submodule.isOrtho_bot_left
@[simp]
theorem isOrtho_bot_right {U : Submodule ๐ E} : U โ โฅ :=
isOrtho_bot_left.symm
#align submodule.is_ortho_bot_right Submodule.isOrtho_bot_right
theorem IsOrtho.mono_left {Uโ Uโ V : Submodule ๐ E} (hU : Uโ โค Uโ) (h : Uโ โ V) : Uโ โ V :=
hU.trans h
#align submodule.is_ortho.mono_left Submodule.IsOrtho.mono_left
theorem IsOrtho.mono_right {U Vโ Vโ : Submodule ๐ E} (hV : Vโ โค Vโ) (h : U โ Vโ) : U โ Vโ :=
(h.symm.mono_left hV).symm
#align submodule.is_ortho.mono_right Submodule.IsOrtho.mono_right
theorem IsOrtho.mono {Uโ Vโ Uโ Vโ : Submodule ๐ E} (hU : Uโ โค Uโ) (hV : Vโ โค Vโ) (h : Uโ โ Vโ) :
Uโ โ Vโ :=
(h.mono_right hV).mono_left hU
#align submodule.is_ortho.mono Submodule.IsOrtho.mono
@[simp]
theorem isOrtho_self {U : Submodule ๐ E} : U โ U โ U = โฅ :=
โจfun h => eq_bot_iff.mpr fun x hx => inner_self_eq_zero.mp (h hx x hx), fun h =>
h.symm โธ isOrtho_bot_leftโฉ
#align submodule.is_ortho_self Submodule.isOrtho_self
@[simp]
theorem isOrtho_orthogonal_right (U : Submodule ๐ E) : U โ Uแฎ :=
le_orthogonal_orthogonal _
#align submodule.is_ortho_orthogonal_right Submodule.isOrtho_orthogonal_right
@[simp]
theorem isOrtho_orthogonal_left (U : Submodule ๐ E) : Uแฎ โ U :=
(isOrtho_orthogonal_right U).symm
#align submodule.is_ortho_orthogonal_left Submodule.isOrtho_orthogonal_left
theorem IsOrtho.le {U V : Submodule ๐ E} (h : U โ V) : U โค Vแฎ :=
h
#align submodule.is_ortho.le Submodule.IsOrtho.le
theorem IsOrtho.ge {U V : Submodule ๐ E} (h : U โ V) : V โค Uแฎ :=
h.symm
#align submodule.is_ortho.ge Submodule.IsOrtho.ge
@[simp]
theorem isOrtho_top_right {U : Submodule ๐ E} : U โ โค โ U = โฅ :=
โจfun h => eq_bot_iff.mpr fun _x hx => inner_self_eq_zero.mp (h hx _ mem_top), fun h =>
h.symm โธ isOrtho_bot_leftโฉ
#align submodule.is_ortho_top_right Submodule.isOrtho_top_right
@[simp]
theorem isOrtho_top_left {V : Submodule ๐ E} : โค โ V โ V = โฅ :=
isOrtho_comm.trans isOrtho_top_right
#align submodule.is_ortho_top_left Submodule.isOrtho_top_left
/-- Orthogonal submodules are disjoint. -/
theorem IsOrtho.disjoint {U V : Submodule ๐ E} (h : U โ V) : Disjoint U V :=
(Submodule.orthogonal_disjoint _).mono_right h.symm
#align submodule.is_ortho.disjoint Submodule.IsOrtho.disjoint
@[simp]
theorem isOrtho_sup_left {Uโ Uโ V : Submodule ๐ E} : Uโ โ Uโ โ V โ Uโ โ V โง Uโ โ V :=
sup_le_iff
#align submodule.is_ortho_sup_left Submodule.isOrtho_sup_left
@[simp]
theorem isOrtho_sup_right {U Vโ Vโ : Submodule ๐ E} : U โ Vโ โ Vโ โ U โ Vโ โง U โ Vโ :=
isOrtho_comm.trans <| isOrtho_sup_left.trans <| isOrtho_comm.and isOrtho_comm
#align submodule.is_ortho_sup_right Submodule.isOrtho_sup_right
@[simp]
theorem isOrtho_sSup_left {U : Set (Submodule ๐ E)} {V : Submodule ๐ E} :
sSup U โ V โ โ Uแตข โ U, Uแตข โ V :=
sSup_le_iff
#align submodule.is_ortho_Sup_left Submodule.isOrtho_sSup_left
@[simp]
theorem isOrtho_sSup_right {U : Submodule ๐ E} {V : Set (Submodule ๐ E)} :
U โ sSup V โ โ Vแตข โ V, U โ Vแตข :=
isOrtho_comm.trans <| isOrtho_sSup_left.trans <| by simp_rw [isOrtho_comm]
#align submodule.is_ortho_Sup_right Submodule.isOrtho_sSup_right
@[simp]
theorem isOrtho_iSup_left {ฮน : Sort*} {U : ฮน โ Submodule ๐ E} {V : Submodule ๐ E} :
iSup U โ V โ โ i, U i โ V :=
iSup_le_iff
#align submodule.is_ortho_supr_left Submodule.isOrtho_iSup_left
@[simp]
theorem isOrtho_iSup_right {ฮน : Sort*} {U : Submodule ๐ E} {V : ฮน โ Submodule ๐ E} :
U โ iSup V โ โ i, U โ V i :=
isOrtho_comm.trans <| isOrtho_iSup_left.trans <| by simp_rw [isOrtho_comm]
#align submodule.is_ortho_supr_right Submodule.isOrtho_iSup_right
@[simp]
theorem isOrtho_span {s t : Set E} :
span ๐ s โ span ๐ t โ โ โฆuโฆ, u โ s โ โ โฆvโฆ, v โ t โ โชu, vโซ = 0 := by
simp_rw [span_eq_iSup_of_singleton_spans s, span_eq_iSup_of_singleton_spans t, isOrtho_iSup_left,
isOrtho_iSup_right, isOrtho_iff_le, span_le, Set.subset_def, SetLike.mem_coe,
mem_orthogonal_singleton_iff_inner_left, Set.mem_singleton_iff, forall_eq]
#align submodule.is_ortho_span Submodule.isOrtho_span
theorem IsOrtho.map (f : E โโแตข[๐] F) {U V : Submodule ๐ E} (h : U โ V) : U.map f โ V.map f := by
rw [isOrtho_iff_inner_eq] at *
|
simp_rw [mem_map, forall_exists_index, and_imp, forall_apply_eq_imp_iffโ,
LinearIsometry.inner_map_map]
|
theorem IsOrtho.map (f : E โโแตข[๐] F) {U V : Submodule ๐ E} (h : U โ V) : U.map f โ V.map f := by
rw [isOrtho_iff_inner_eq] at *
|
Mathlib.Analysis.InnerProductSpace.Orthogonal.381_0.QXx0GYqLoAbtfq1
|
theorem IsOrtho.map (f : E โโแตข[๐] F) {U V : Submodule ๐ E} (h : U โ V) : U.map f โ V.map f
|
Mathlib_Analysis_InnerProductSpace_Orthogonal
|
๐ : Type u_1
E : Type u_2
F : Type u_3
instโโด : IsROrC ๐
instโยณ : NormedAddCommGroup E
instโยฒ : InnerProductSpace ๐ E
instโยน : NormedAddCommGroup F
instโ : InnerProductSpace ๐ F
f : E โโแตข[๐] F
U V : Submodule ๐ E
h : โ u โ U, โ v โ V, inner u v = 0
โข โ a โ U, โ a_2 โ V, inner a a_2 = 0
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sรฉbastien Gouรซzel, Frรฉdรฉric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
/-!
# Orthogonal complements of submodules
In this file, the `orthogonal` complement of a submodule `K` is defined, and basic API established.
Some of the more subtle results about the orthogonal complement are delayed to
`Analysis.InnerProductSpace.Projection`.
See also `BilinForm.orthogonal` for orthogonality with respect to a general bilinear form.
## Notation
The orthogonal complement of a submodule `K` is denoted by `Kแฎ`.
The proposition that two submodules are orthogonal, `Submodule.IsOrtho`, is denoted by `U โ V`.
Note this is not the same unicode symbol as `โฅ` (`Bot`).
-/
variable {๐ E F : Type*} [IsROrC ๐]
variable [NormedAddCommGroup E] [InnerProductSpace ๐ E]
variable [NormedAddCommGroup F] [InnerProductSpace ๐ F]
local notation "โช" x ", " y "โซ" => @inner ๐ _ _ x y
namespace Submodule
variable (K : Submodule ๐ E)
/-- The subspace of vectors orthogonal to a given subspace. -/
def orthogonal : Submodule ๐ E where
carrier := { v | โ u โ K, โชu, vโซ = 0 }
zero_mem' _ _ := inner_zero_right _
add_mem' hx hy u hu := by rw [inner_add_right, hx u hu, hy u hu, add_zero]
smul_mem' c x hx u hu := by rw [inner_smul_right, hx u hu, mul_zero]
#align submodule.orthogonal Submodule.orthogonal
@[inherit_doc]
notation:1200 K "แฎ" => orthogonal K
/-- When a vector is in `Kแฎ`. -/
theorem mem_orthogonal (v : E) : v โ Kแฎ โ โ u โ K, โชu, vโซ = 0 :=
Iff.rfl
#align submodule.mem_orthogonal Submodule.mem_orthogonal
/-- When a vector is in `Kแฎ`, with the inner product the
other way round. -/
theorem mem_orthogonal' (v : E) : v โ Kแฎ โ โ u โ K, โชv, uโซ = 0 := by
simp_rw [mem_orthogonal, inner_eq_zero_symm]
#align submodule.mem_orthogonal' Submodule.mem_orthogonal'
variable {K}
/-- A vector in `K` is orthogonal to one in `Kแฎ`. -/
theorem inner_right_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชu, vโซ = 0 :=
(K.mem_orthogonal v).1 hv u hu
#align submodule.inner_right_of_mem_orthogonal Submodule.inner_right_of_mem_orthogonal
/-- A vector in `Kแฎ` is orthogonal to one in `K`. -/
theorem inner_left_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชv, uโซ = 0 := by
rw [inner_eq_zero_symm]; exact inner_right_of_mem_orthogonal hu hv
#align submodule.inner_left_of_mem_orthogonal Submodule.inner_left_of_mem_orthogonal
/-- A vector is in `(๐ โ u)แฎ` iff it is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_right {u v : E} : v โ (๐ โ u)แฎ โ โชu, vโซ = 0 := by
refine' โจinner_right_of_mem_orthogonal (mem_span_singleton_self u), _โฉ
intro hv w hw
rw [mem_span_singleton] at hw
obtain โจc, rflโฉ := hw
simp [inner_smul_left, hv]
#align submodule.mem_orthogonal_singleton_iff_inner_right Submodule.mem_orthogonal_singleton_iff_inner_right
/-- A vector in `(๐ โ u)แฎ` is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_left {u v : E} : v โ (๐ โ u)แฎ โ โชv, uโซ = 0 := by
rw [mem_orthogonal_singleton_iff_inner_right, inner_eq_zero_symm]
#align submodule.mem_orthogonal_singleton_iff_inner_left Submodule.mem_orthogonal_singleton_iff_inner_left
theorem sub_mem_orthogonal_of_inner_left {x y : E} (h : โ v : K, โชx, vโซ = โชy, vโซ) : x - y โ Kแฎ := by
rw [mem_orthogonal']
intro u hu
rw [inner_sub_left, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_left Submodule.sub_mem_orthogonal_of_inner_left
theorem sub_mem_orthogonal_of_inner_right {x y : E} (h : โ v : K, โช(v : E), xโซ = โช(v : E), yโซ) :
x - y โ Kแฎ := by
intro u hu
rw [inner_sub_right, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_right Submodule.sub_mem_orthogonal_of_inner_right
variable (K)
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem inf_orthogonal_eq_bot : K โ Kแฎ = โฅ := by
rw [eq_bot_iff]
intro x
rw [mem_inf]
exact fun โจhx, hoโฉ => inner_self_eq_zero.1 (ho x hx)
#align submodule.inf_orthogonal_eq_bot Submodule.inf_orthogonal_eq_bot
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem orthogonal_disjoint : Disjoint K Kแฎ := by simp [disjoint_iff, K.inf_orthogonal_eq_bot]
#align submodule.orthogonal_disjoint Submodule.orthogonal_disjoint
/-- `Kแฎ` can be characterized as the intersection of the kernels of the operations of
inner product with each of the elements of `K`. -/
theorem orthogonal_eq_inter : Kแฎ = โจ
v : K, LinearMap.ker (innerSL ๐ (v : E)) := by
apply le_antisymm
ยท rw [le_iInf_iff]
rintro โจv, hvโฉ w hw
simpa using hw _ hv
ยท intro v hv w hw
simp only [mem_iInf] at hv
exact hv โจw, hwโฉ
#align submodule.orthogonal_eq_inter Submodule.orthogonal_eq_inter
/-- The orthogonal complement of any submodule `K` is closed. -/
theorem isClosed_orthogonal : IsClosed (Kแฎ : Set E) := by
rw [orthogonal_eq_inter K]
have := fun v : K => ContinuousLinearMap.isClosed_ker (innerSL ๐ (v : E))
convert isClosed_iInter this
simp only [iInf_coe]
#align submodule.is_closed_orthogonal Submodule.isClosed_orthogonal
/-- In a complete space, the orthogonal complement of any submodule `K` is complete. -/
instance instOrthogonalCompleteSpace [CompleteSpace E] : CompleteSpace Kแฎ :=
K.isClosed_orthogonal.completeSpace_coe
variable (๐ E)
/-- `orthogonal` gives a `GaloisConnection` between
`Submodule ๐ E` and its `OrderDual`. -/
theorem orthogonal_gc :
@GaloisConnection (Submodule ๐ E) (Submodule ๐ E)แตแต _ _ orthogonal orthogonal := fun _Kโ _Kโ =>
โจfun h _v hv _u hu => inner_left_of_mem_orthogonal hv (h hu), fun h _v hv _u hu =>
inner_left_of_mem_orthogonal hv (h hu)โฉ
#align submodule.orthogonal_gc Submodule.orthogonal_gc
variable {๐ E}
/-- `orthogonal` reverses the `โค` ordering of two
subspaces. -/
theorem orthogonal_le {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎ โค Kโแฎ :=
(orthogonal_gc ๐ E).monotone_l h
#align submodule.orthogonal_le Submodule.orthogonal_le
/-- `orthogonal.orthogonal` preserves the `โค` ordering of two
subspaces. -/
theorem orthogonal_orthogonal_monotone {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎแฎ โค Kโแฎแฎ :=
orthogonal_le (orthogonal_le h)
#align submodule.orthogonal_orthogonal_monotone Submodule.orthogonal_orthogonal_monotone
/-- `K` is contained in `Kแฎแฎ`. -/
theorem le_orthogonal_orthogonal : K โค Kแฎแฎ :=
(orthogonal_gc ๐ E).le_u_l _
#align submodule.le_orthogonal_orthogonal Submodule.le_orthogonal_orthogonal
/-- The inf of two orthogonal subspaces equals the subspace orthogonal
to the sup. -/
theorem inf_orthogonal (Kโ Kโ : Submodule ๐ E) : Kโแฎ โ Kโแฎ = (Kโ โ Kโ)แฎ :=
(orthogonal_gc ๐ E).l_sup.symm
#align submodule.inf_orthogonal Submodule.inf_orthogonal
/-- The inf of an indexed family of orthogonal subspaces equals the
subspace orthogonal to the sup. -/
theorem iInf_orthogonal {ฮน : Type*} (K : ฮน โ Submodule ๐ E) : โจ
i, (K i)แฎ = (iSup K)แฎ :=
(orthogonal_gc ๐ E).l_iSup.symm
#align submodule.infi_orthogonal Submodule.iInf_orthogonal
/-- The inf of a set of orthogonal subspaces equals the subspace orthogonal to the sup. -/
theorem sInf_orthogonal (s : Set <| Submodule ๐ E) : โจ
K โ s, Kแฎ = (sSup s)แฎ :=
(orthogonal_gc ๐ E).l_sSup.symm
#align submodule.Inf_orthogonal Submodule.sInf_orthogonal
@[simp]
theorem top_orthogonal_eq_bot : (โค : Submodule ๐ E)แฎ = โฅ := by
ext x
rw [mem_bot, mem_orthogonal]
exact
โจfun h => inner_self_eq_zero.mp (h x mem_top), by
rintro rfl
simpโฉ
#align submodule.top_orthogonal_eq_bot Submodule.top_orthogonal_eq_bot
@[simp]
theorem bot_orthogonal_eq_top : (โฅ : Submodule ๐ E)แฎ = โค := by
rw [โ top_orthogonal_eq_bot, eq_top_iff]
exact le_orthogonal_orthogonal โค
#align submodule.bot_orthogonal_eq_top Submodule.bot_orthogonal_eq_top
@[simp]
theorem orthogonal_eq_top_iff : Kแฎ = โค โ K = โฅ := by
refine'
โจ_, by
rintro rfl
exact bot_orthogonal_eq_topโฉ
intro h
have : K โ Kแฎ = โฅ := K.orthogonal_disjoint.eq_bot
rwa [h, inf_comm, top_inf_eq] at this
#align submodule.orthogonal_eq_top_iff Submodule.orthogonal_eq_top_iff
theorem orthogonalFamily_self :
OrthogonalFamily ๐ (fun b => โฅ(cond b K Kแฎ)) fun b => (cond b K Kแฎ).subtypeโแตข
| true, true => absurd rfl
| true, false => fun _ x y => inner_right_of_mem_orthogonal x.prop y.prop
| false, true => fun _ x y => inner_left_of_mem_orthogonal y.prop x.prop
| false, false => absurd rfl
#align submodule.orthogonal_family_self Submodule.orthogonalFamily_self
end Submodule
@[simp]
theorem bilinFormOfRealInner_orthogonal {E} [NormedAddCommGroup E] [InnerProductSpace โ E]
(K : Submodule โ E) : bilinFormOfRealInner.orthogonal K = Kแฎ :=
rfl
#align bilin_form_of_real_inner_orthogonal bilinFormOfRealInner_orthogonal
/-!
### Orthogonality of submodules
In this section we define `Submodule.IsOrtho U V`, with notation `U โ V`.
The API roughly matches that of `Disjoint`.
-/
namespace Submodule
/-- The proposition that two submodules are orthogonal. Has notation `U โ V`. -/
def IsOrtho (U V : Submodule ๐ E) : Prop :=
U โค Vแฎ
#align submodule.is_ortho Submodule.IsOrtho
@[inherit_doc]
infixl:50 " โ " => Submodule.IsOrtho
theorem isOrtho_iff_le {U V : Submodule ๐ E} : U โ V โ U โค Vแฎ :=
Iff.rfl
#align submodule.is_ortho_iff_le Submodule.isOrtho_iff_le
@[symm]
theorem IsOrtho.symm {U V : Submodule ๐ E} (h : U โ V) : V โ U :=
(le_orthogonal_orthogonal _).trans (orthogonal_le h)
#align submodule.is_ortho.symm Submodule.IsOrtho.symm
theorem isOrtho_comm {U V : Submodule ๐ E} : U โ V โ V โ U :=
โจIsOrtho.symm, IsOrtho.symmโฉ
#align submodule.is_ortho_comm Submodule.isOrtho_comm
theorem symmetric_isOrtho : Symmetric (IsOrtho : Submodule ๐ E โ Submodule ๐ E โ Prop) := fun _ _ =>
IsOrtho.symm
#align submodule.symmetric_is_ortho Submodule.symmetric_isOrtho
theorem IsOrtho.inner_eq {U V : Submodule ๐ E} (h : U โ V) {u v : E} (hu : u โ U) (hv : v โ V) :
โชu, vโซ = 0 :=
h.symm hv _ hu
#align submodule.is_ortho.inner_eq Submodule.IsOrtho.inner_eq
theorem isOrtho_iff_inner_eq {U V : Submodule ๐ E} : U โ V โ โ u โ U, โ v โ V, โชu, vโซ = 0 :=
forallโ_congr fun _u _hu _v _hv => inner_eq_zero_symm
#align submodule.is_ortho_iff_inner_eq Submodule.isOrtho_iff_inner_eq
/- TODO: generalize `Submodule.mapโ` to semilinear maps, so that we can state
`U โ V โ Submodule.mapโ (innerโโ ๐) U V โค โฅ`. -/
@[simp]
theorem isOrtho_bot_left {V : Submodule ๐ E} : โฅ โ V :=
bot_le
#align submodule.is_ortho_bot_left Submodule.isOrtho_bot_left
@[simp]
theorem isOrtho_bot_right {U : Submodule ๐ E} : U โ โฅ :=
isOrtho_bot_left.symm
#align submodule.is_ortho_bot_right Submodule.isOrtho_bot_right
theorem IsOrtho.mono_left {Uโ Uโ V : Submodule ๐ E} (hU : Uโ โค Uโ) (h : Uโ โ V) : Uโ โ V :=
hU.trans h
#align submodule.is_ortho.mono_left Submodule.IsOrtho.mono_left
theorem IsOrtho.mono_right {U Vโ Vโ : Submodule ๐ E} (hV : Vโ โค Vโ) (h : U โ Vโ) : U โ Vโ :=
(h.symm.mono_left hV).symm
#align submodule.is_ortho.mono_right Submodule.IsOrtho.mono_right
theorem IsOrtho.mono {Uโ Vโ Uโ Vโ : Submodule ๐ E} (hU : Uโ โค Uโ) (hV : Vโ โค Vโ) (h : Uโ โ Vโ) :
Uโ โ Vโ :=
(h.mono_right hV).mono_left hU
#align submodule.is_ortho.mono Submodule.IsOrtho.mono
@[simp]
theorem isOrtho_self {U : Submodule ๐ E} : U โ U โ U = โฅ :=
โจfun h => eq_bot_iff.mpr fun x hx => inner_self_eq_zero.mp (h hx x hx), fun h =>
h.symm โธ isOrtho_bot_leftโฉ
#align submodule.is_ortho_self Submodule.isOrtho_self
@[simp]
theorem isOrtho_orthogonal_right (U : Submodule ๐ E) : U โ Uแฎ :=
le_orthogonal_orthogonal _
#align submodule.is_ortho_orthogonal_right Submodule.isOrtho_orthogonal_right
@[simp]
theorem isOrtho_orthogonal_left (U : Submodule ๐ E) : Uแฎ โ U :=
(isOrtho_orthogonal_right U).symm
#align submodule.is_ortho_orthogonal_left Submodule.isOrtho_orthogonal_left
theorem IsOrtho.le {U V : Submodule ๐ E} (h : U โ V) : U โค Vแฎ :=
h
#align submodule.is_ortho.le Submodule.IsOrtho.le
theorem IsOrtho.ge {U V : Submodule ๐ E} (h : U โ V) : V โค Uแฎ :=
h.symm
#align submodule.is_ortho.ge Submodule.IsOrtho.ge
@[simp]
theorem isOrtho_top_right {U : Submodule ๐ E} : U โ โค โ U = โฅ :=
โจfun h => eq_bot_iff.mpr fun _x hx => inner_self_eq_zero.mp (h hx _ mem_top), fun h =>
h.symm โธ isOrtho_bot_leftโฉ
#align submodule.is_ortho_top_right Submodule.isOrtho_top_right
@[simp]
theorem isOrtho_top_left {V : Submodule ๐ E} : โค โ V โ V = โฅ :=
isOrtho_comm.trans isOrtho_top_right
#align submodule.is_ortho_top_left Submodule.isOrtho_top_left
/-- Orthogonal submodules are disjoint. -/
theorem IsOrtho.disjoint {U V : Submodule ๐ E} (h : U โ V) : Disjoint U V :=
(Submodule.orthogonal_disjoint _).mono_right h.symm
#align submodule.is_ortho.disjoint Submodule.IsOrtho.disjoint
@[simp]
theorem isOrtho_sup_left {Uโ Uโ V : Submodule ๐ E} : Uโ โ Uโ โ V โ Uโ โ V โง Uโ โ V :=
sup_le_iff
#align submodule.is_ortho_sup_left Submodule.isOrtho_sup_left
@[simp]
theorem isOrtho_sup_right {U Vโ Vโ : Submodule ๐ E} : U โ Vโ โ Vโ โ U โ Vโ โง U โ Vโ :=
isOrtho_comm.trans <| isOrtho_sup_left.trans <| isOrtho_comm.and isOrtho_comm
#align submodule.is_ortho_sup_right Submodule.isOrtho_sup_right
@[simp]
theorem isOrtho_sSup_left {U : Set (Submodule ๐ E)} {V : Submodule ๐ E} :
sSup U โ V โ โ Uแตข โ U, Uแตข โ V :=
sSup_le_iff
#align submodule.is_ortho_Sup_left Submodule.isOrtho_sSup_left
@[simp]
theorem isOrtho_sSup_right {U : Submodule ๐ E} {V : Set (Submodule ๐ E)} :
U โ sSup V โ โ Vแตข โ V, U โ Vแตข :=
isOrtho_comm.trans <| isOrtho_sSup_left.trans <| by simp_rw [isOrtho_comm]
#align submodule.is_ortho_Sup_right Submodule.isOrtho_sSup_right
@[simp]
theorem isOrtho_iSup_left {ฮน : Sort*} {U : ฮน โ Submodule ๐ E} {V : Submodule ๐ E} :
iSup U โ V โ โ i, U i โ V :=
iSup_le_iff
#align submodule.is_ortho_supr_left Submodule.isOrtho_iSup_left
@[simp]
theorem isOrtho_iSup_right {ฮน : Sort*} {U : Submodule ๐ E} {V : ฮน โ Submodule ๐ E} :
U โ iSup V โ โ i, U โ V i :=
isOrtho_comm.trans <| isOrtho_iSup_left.trans <| by simp_rw [isOrtho_comm]
#align submodule.is_ortho_supr_right Submodule.isOrtho_iSup_right
@[simp]
theorem isOrtho_span {s t : Set E} :
span ๐ s โ span ๐ t โ โ โฆuโฆ, u โ s โ โ โฆvโฆ, v โ t โ โชu, vโซ = 0 := by
simp_rw [span_eq_iSup_of_singleton_spans s, span_eq_iSup_of_singleton_spans t, isOrtho_iSup_left,
isOrtho_iSup_right, isOrtho_iff_le, span_le, Set.subset_def, SetLike.mem_coe,
mem_orthogonal_singleton_iff_inner_left, Set.mem_singleton_iff, forall_eq]
#align submodule.is_ortho_span Submodule.isOrtho_span
theorem IsOrtho.map (f : E โโแตข[๐] F) {U V : Submodule ๐ E} (h : U โ V) : U.map f โ V.map f := by
rw [isOrtho_iff_inner_eq] at *
simp_rw [mem_map, forall_exists_index, and_imp, forall_apply_eq_imp_iffโ,
LinearIsometry.inner_map_map]
|
exact h
|
theorem IsOrtho.map (f : E โโแตข[๐] F) {U V : Submodule ๐ E} (h : U โ V) : U.map f โ V.map f := by
rw [isOrtho_iff_inner_eq] at *
simp_rw [mem_map, forall_exists_index, and_imp, forall_apply_eq_imp_iffโ,
LinearIsometry.inner_map_map]
|
Mathlib.Analysis.InnerProductSpace.Orthogonal.381_0.QXx0GYqLoAbtfq1
|
theorem IsOrtho.map (f : E โโแตข[๐] F) {U V : Submodule ๐ E} (h : U โ V) : U.map f โ V.map f
|
Mathlib_Analysis_InnerProductSpace_Orthogonal
|
๐ : Type u_1
E : Type u_2
F : Type u_3
instโโด : IsROrC ๐
instโยณ : NormedAddCommGroup E
instโยฒ : InnerProductSpace ๐ E
instโยน : NormedAddCommGroup F
instโ : InnerProductSpace ๐ F
f : E โโแตข[๐] F
U V : Submodule ๐ F
h : U โ V
โข Submodule.comap f U โ Submodule.comap f V
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sรฉbastien Gouรซzel, Frรฉdรฉric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
/-!
# Orthogonal complements of submodules
In this file, the `orthogonal` complement of a submodule `K` is defined, and basic API established.
Some of the more subtle results about the orthogonal complement are delayed to
`Analysis.InnerProductSpace.Projection`.
See also `BilinForm.orthogonal` for orthogonality with respect to a general bilinear form.
## Notation
The orthogonal complement of a submodule `K` is denoted by `Kแฎ`.
The proposition that two submodules are orthogonal, `Submodule.IsOrtho`, is denoted by `U โ V`.
Note this is not the same unicode symbol as `โฅ` (`Bot`).
-/
variable {๐ E F : Type*} [IsROrC ๐]
variable [NormedAddCommGroup E] [InnerProductSpace ๐ E]
variable [NormedAddCommGroup F] [InnerProductSpace ๐ F]
local notation "โช" x ", " y "โซ" => @inner ๐ _ _ x y
namespace Submodule
variable (K : Submodule ๐ E)
/-- The subspace of vectors orthogonal to a given subspace. -/
def orthogonal : Submodule ๐ E where
carrier := { v | โ u โ K, โชu, vโซ = 0 }
zero_mem' _ _ := inner_zero_right _
add_mem' hx hy u hu := by rw [inner_add_right, hx u hu, hy u hu, add_zero]
smul_mem' c x hx u hu := by rw [inner_smul_right, hx u hu, mul_zero]
#align submodule.orthogonal Submodule.orthogonal
@[inherit_doc]
notation:1200 K "แฎ" => orthogonal K
/-- When a vector is in `Kแฎ`. -/
theorem mem_orthogonal (v : E) : v โ Kแฎ โ โ u โ K, โชu, vโซ = 0 :=
Iff.rfl
#align submodule.mem_orthogonal Submodule.mem_orthogonal
/-- When a vector is in `Kแฎ`, with the inner product the
other way round. -/
theorem mem_orthogonal' (v : E) : v โ Kแฎ โ โ u โ K, โชv, uโซ = 0 := by
simp_rw [mem_orthogonal, inner_eq_zero_symm]
#align submodule.mem_orthogonal' Submodule.mem_orthogonal'
variable {K}
/-- A vector in `K` is orthogonal to one in `Kแฎ`. -/
theorem inner_right_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชu, vโซ = 0 :=
(K.mem_orthogonal v).1 hv u hu
#align submodule.inner_right_of_mem_orthogonal Submodule.inner_right_of_mem_orthogonal
/-- A vector in `Kแฎ` is orthogonal to one in `K`. -/
theorem inner_left_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชv, uโซ = 0 := by
rw [inner_eq_zero_symm]; exact inner_right_of_mem_orthogonal hu hv
#align submodule.inner_left_of_mem_orthogonal Submodule.inner_left_of_mem_orthogonal
/-- A vector is in `(๐ โ u)แฎ` iff it is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_right {u v : E} : v โ (๐ โ u)แฎ โ โชu, vโซ = 0 := by
refine' โจinner_right_of_mem_orthogonal (mem_span_singleton_self u), _โฉ
intro hv w hw
rw [mem_span_singleton] at hw
obtain โจc, rflโฉ := hw
simp [inner_smul_left, hv]
#align submodule.mem_orthogonal_singleton_iff_inner_right Submodule.mem_orthogonal_singleton_iff_inner_right
/-- A vector in `(๐ โ u)แฎ` is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_left {u v : E} : v โ (๐ โ u)แฎ โ โชv, uโซ = 0 := by
rw [mem_orthogonal_singleton_iff_inner_right, inner_eq_zero_symm]
#align submodule.mem_orthogonal_singleton_iff_inner_left Submodule.mem_orthogonal_singleton_iff_inner_left
theorem sub_mem_orthogonal_of_inner_left {x y : E} (h : โ v : K, โชx, vโซ = โชy, vโซ) : x - y โ Kแฎ := by
rw [mem_orthogonal']
intro u hu
rw [inner_sub_left, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_left Submodule.sub_mem_orthogonal_of_inner_left
theorem sub_mem_orthogonal_of_inner_right {x y : E} (h : โ v : K, โช(v : E), xโซ = โช(v : E), yโซ) :
x - y โ Kแฎ := by
intro u hu
rw [inner_sub_right, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_right Submodule.sub_mem_orthogonal_of_inner_right
variable (K)
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem inf_orthogonal_eq_bot : K โ Kแฎ = โฅ := by
rw [eq_bot_iff]
intro x
rw [mem_inf]
exact fun โจhx, hoโฉ => inner_self_eq_zero.1 (ho x hx)
#align submodule.inf_orthogonal_eq_bot Submodule.inf_orthogonal_eq_bot
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem orthogonal_disjoint : Disjoint K Kแฎ := by simp [disjoint_iff, K.inf_orthogonal_eq_bot]
#align submodule.orthogonal_disjoint Submodule.orthogonal_disjoint
/-- `Kแฎ` can be characterized as the intersection of the kernels of the operations of
inner product with each of the elements of `K`. -/
theorem orthogonal_eq_inter : Kแฎ = โจ
v : K, LinearMap.ker (innerSL ๐ (v : E)) := by
apply le_antisymm
ยท rw [le_iInf_iff]
rintro โจv, hvโฉ w hw
simpa using hw _ hv
ยท intro v hv w hw
simp only [mem_iInf] at hv
exact hv โจw, hwโฉ
#align submodule.orthogonal_eq_inter Submodule.orthogonal_eq_inter
/-- The orthogonal complement of any submodule `K` is closed. -/
theorem isClosed_orthogonal : IsClosed (Kแฎ : Set E) := by
rw [orthogonal_eq_inter K]
have := fun v : K => ContinuousLinearMap.isClosed_ker (innerSL ๐ (v : E))
convert isClosed_iInter this
simp only [iInf_coe]
#align submodule.is_closed_orthogonal Submodule.isClosed_orthogonal
/-- In a complete space, the orthogonal complement of any submodule `K` is complete. -/
instance instOrthogonalCompleteSpace [CompleteSpace E] : CompleteSpace Kแฎ :=
K.isClosed_orthogonal.completeSpace_coe
variable (๐ E)
/-- `orthogonal` gives a `GaloisConnection` between
`Submodule ๐ E` and its `OrderDual`. -/
theorem orthogonal_gc :
@GaloisConnection (Submodule ๐ E) (Submodule ๐ E)แตแต _ _ orthogonal orthogonal := fun _Kโ _Kโ =>
โจfun h _v hv _u hu => inner_left_of_mem_orthogonal hv (h hu), fun h _v hv _u hu =>
inner_left_of_mem_orthogonal hv (h hu)โฉ
#align submodule.orthogonal_gc Submodule.orthogonal_gc
variable {๐ E}
/-- `orthogonal` reverses the `โค` ordering of two
subspaces. -/
theorem orthogonal_le {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎ โค Kโแฎ :=
(orthogonal_gc ๐ E).monotone_l h
#align submodule.orthogonal_le Submodule.orthogonal_le
/-- `orthogonal.orthogonal` preserves the `โค` ordering of two
subspaces. -/
theorem orthogonal_orthogonal_monotone {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎแฎ โค Kโแฎแฎ :=
orthogonal_le (orthogonal_le h)
#align submodule.orthogonal_orthogonal_monotone Submodule.orthogonal_orthogonal_monotone
/-- `K` is contained in `Kแฎแฎ`. -/
theorem le_orthogonal_orthogonal : K โค Kแฎแฎ :=
(orthogonal_gc ๐ E).le_u_l _
#align submodule.le_orthogonal_orthogonal Submodule.le_orthogonal_orthogonal
/-- The inf of two orthogonal subspaces equals the subspace orthogonal
to the sup. -/
theorem inf_orthogonal (Kโ Kโ : Submodule ๐ E) : Kโแฎ โ Kโแฎ = (Kโ โ Kโ)แฎ :=
(orthogonal_gc ๐ E).l_sup.symm
#align submodule.inf_orthogonal Submodule.inf_orthogonal
/-- The inf of an indexed family of orthogonal subspaces equals the
subspace orthogonal to the sup. -/
theorem iInf_orthogonal {ฮน : Type*} (K : ฮน โ Submodule ๐ E) : โจ
i, (K i)แฎ = (iSup K)แฎ :=
(orthogonal_gc ๐ E).l_iSup.symm
#align submodule.infi_orthogonal Submodule.iInf_orthogonal
/-- The inf of a set of orthogonal subspaces equals the subspace orthogonal to the sup. -/
theorem sInf_orthogonal (s : Set <| Submodule ๐ E) : โจ
K โ s, Kแฎ = (sSup s)แฎ :=
(orthogonal_gc ๐ E).l_sSup.symm
#align submodule.Inf_orthogonal Submodule.sInf_orthogonal
@[simp]
theorem top_orthogonal_eq_bot : (โค : Submodule ๐ E)แฎ = โฅ := by
ext x
rw [mem_bot, mem_orthogonal]
exact
โจfun h => inner_self_eq_zero.mp (h x mem_top), by
rintro rfl
simpโฉ
#align submodule.top_orthogonal_eq_bot Submodule.top_orthogonal_eq_bot
@[simp]
theorem bot_orthogonal_eq_top : (โฅ : Submodule ๐ E)แฎ = โค := by
rw [โ top_orthogonal_eq_bot, eq_top_iff]
exact le_orthogonal_orthogonal โค
#align submodule.bot_orthogonal_eq_top Submodule.bot_orthogonal_eq_top
@[simp]
theorem orthogonal_eq_top_iff : Kแฎ = โค โ K = โฅ := by
refine'
โจ_, by
rintro rfl
exact bot_orthogonal_eq_topโฉ
intro h
have : K โ Kแฎ = โฅ := K.orthogonal_disjoint.eq_bot
rwa [h, inf_comm, top_inf_eq] at this
#align submodule.orthogonal_eq_top_iff Submodule.orthogonal_eq_top_iff
theorem orthogonalFamily_self :
OrthogonalFamily ๐ (fun b => โฅ(cond b K Kแฎ)) fun b => (cond b K Kแฎ).subtypeโแตข
| true, true => absurd rfl
| true, false => fun _ x y => inner_right_of_mem_orthogonal x.prop y.prop
| false, true => fun _ x y => inner_left_of_mem_orthogonal y.prop x.prop
| false, false => absurd rfl
#align submodule.orthogonal_family_self Submodule.orthogonalFamily_self
end Submodule
@[simp]
theorem bilinFormOfRealInner_orthogonal {E} [NormedAddCommGroup E] [InnerProductSpace โ E]
(K : Submodule โ E) : bilinFormOfRealInner.orthogonal K = Kแฎ :=
rfl
#align bilin_form_of_real_inner_orthogonal bilinFormOfRealInner_orthogonal
/-!
### Orthogonality of submodules
In this section we define `Submodule.IsOrtho U V`, with notation `U โ V`.
The API roughly matches that of `Disjoint`.
-/
namespace Submodule
/-- The proposition that two submodules are orthogonal. Has notation `U โ V`. -/
def IsOrtho (U V : Submodule ๐ E) : Prop :=
U โค Vแฎ
#align submodule.is_ortho Submodule.IsOrtho
@[inherit_doc]
infixl:50 " โ " => Submodule.IsOrtho
theorem isOrtho_iff_le {U V : Submodule ๐ E} : U โ V โ U โค Vแฎ :=
Iff.rfl
#align submodule.is_ortho_iff_le Submodule.isOrtho_iff_le
@[symm]
theorem IsOrtho.symm {U V : Submodule ๐ E} (h : U โ V) : V โ U :=
(le_orthogonal_orthogonal _).trans (orthogonal_le h)
#align submodule.is_ortho.symm Submodule.IsOrtho.symm
theorem isOrtho_comm {U V : Submodule ๐ E} : U โ V โ V โ U :=
โจIsOrtho.symm, IsOrtho.symmโฉ
#align submodule.is_ortho_comm Submodule.isOrtho_comm
theorem symmetric_isOrtho : Symmetric (IsOrtho : Submodule ๐ E โ Submodule ๐ E โ Prop) := fun _ _ =>
IsOrtho.symm
#align submodule.symmetric_is_ortho Submodule.symmetric_isOrtho
theorem IsOrtho.inner_eq {U V : Submodule ๐ E} (h : U โ V) {u v : E} (hu : u โ U) (hv : v โ V) :
โชu, vโซ = 0 :=
h.symm hv _ hu
#align submodule.is_ortho.inner_eq Submodule.IsOrtho.inner_eq
theorem isOrtho_iff_inner_eq {U V : Submodule ๐ E} : U โ V โ โ u โ U, โ v โ V, โชu, vโซ = 0 :=
forallโ_congr fun _u _hu _v _hv => inner_eq_zero_symm
#align submodule.is_ortho_iff_inner_eq Submodule.isOrtho_iff_inner_eq
/- TODO: generalize `Submodule.mapโ` to semilinear maps, so that we can state
`U โ V โ Submodule.mapโ (innerโโ ๐) U V โค โฅ`. -/
@[simp]
theorem isOrtho_bot_left {V : Submodule ๐ E} : โฅ โ V :=
bot_le
#align submodule.is_ortho_bot_left Submodule.isOrtho_bot_left
@[simp]
theorem isOrtho_bot_right {U : Submodule ๐ E} : U โ โฅ :=
isOrtho_bot_left.symm
#align submodule.is_ortho_bot_right Submodule.isOrtho_bot_right
theorem IsOrtho.mono_left {Uโ Uโ V : Submodule ๐ E} (hU : Uโ โค Uโ) (h : Uโ โ V) : Uโ โ V :=
hU.trans h
#align submodule.is_ortho.mono_left Submodule.IsOrtho.mono_left
theorem IsOrtho.mono_right {U Vโ Vโ : Submodule ๐ E} (hV : Vโ โค Vโ) (h : U โ Vโ) : U โ Vโ :=
(h.symm.mono_left hV).symm
#align submodule.is_ortho.mono_right Submodule.IsOrtho.mono_right
theorem IsOrtho.mono {Uโ Vโ Uโ Vโ : Submodule ๐ E} (hU : Uโ โค Uโ) (hV : Vโ โค Vโ) (h : Uโ โ Vโ) :
Uโ โ Vโ :=
(h.mono_right hV).mono_left hU
#align submodule.is_ortho.mono Submodule.IsOrtho.mono
@[simp]
theorem isOrtho_self {U : Submodule ๐ E} : U โ U โ U = โฅ :=
โจfun h => eq_bot_iff.mpr fun x hx => inner_self_eq_zero.mp (h hx x hx), fun h =>
h.symm โธ isOrtho_bot_leftโฉ
#align submodule.is_ortho_self Submodule.isOrtho_self
@[simp]
theorem isOrtho_orthogonal_right (U : Submodule ๐ E) : U โ Uแฎ :=
le_orthogonal_orthogonal _
#align submodule.is_ortho_orthogonal_right Submodule.isOrtho_orthogonal_right
@[simp]
theorem isOrtho_orthogonal_left (U : Submodule ๐ E) : Uแฎ โ U :=
(isOrtho_orthogonal_right U).symm
#align submodule.is_ortho_orthogonal_left Submodule.isOrtho_orthogonal_left
theorem IsOrtho.le {U V : Submodule ๐ E} (h : U โ V) : U โค Vแฎ :=
h
#align submodule.is_ortho.le Submodule.IsOrtho.le
theorem IsOrtho.ge {U V : Submodule ๐ E} (h : U โ V) : V โค Uแฎ :=
h.symm
#align submodule.is_ortho.ge Submodule.IsOrtho.ge
@[simp]
theorem isOrtho_top_right {U : Submodule ๐ E} : U โ โค โ U = โฅ :=
โจfun h => eq_bot_iff.mpr fun _x hx => inner_self_eq_zero.mp (h hx _ mem_top), fun h =>
h.symm โธ isOrtho_bot_leftโฉ
#align submodule.is_ortho_top_right Submodule.isOrtho_top_right
@[simp]
theorem isOrtho_top_left {V : Submodule ๐ E} : โค โ V โ V = โฅ :=
isOrtho_comm.trans isOrtho_top_right
#align submodule.is_ortho_top_left Submodule.isOrtho_top_left
/-- Orthogonal submodules are disjoint. -/
theorem IsOrtho.disjoint {U V : Submodule ๐ E} (h : U โ V) : Disjoint U V :=
(Submodule.orthogonal_disjoint _).mono_right h.symm
#align submodule.is_ortho.disjoint Submodule.IsOrtho.disjoint
@[simp]
theorem isOrtho_sup_left {Uโ Uโ V : Submodule ๐ E} : Uโ โ Uโ โ V โ Uโ โ V โง Uโ โ V :=
sup_le_iff
#align submodule.is_ortho_sup_left Submodule.isOrtho_sup_left
@[simp]
theorem isOrtho_sup_right {U Vโ Vโ : Submodule ๐ E} : U โ Vโ โ Vโ โ U โ Vโ โง U โ Vโ :=
isOrtho_comm.trans <| isOrtho_sup_left.trans <| isOrtho_comm.and isOrtho_comm
#align submodule.is_ortho_sup_right Submodule.isOrtho_sup_right
@[simp]
theorem isOrtho_sSup_left {U : Set (Submodule ๐ E)} {V : Submodule ๐ E} :
sSup U โ V โ โ Uแตข โ U, Uแตข โ V :=
sSup_le_iff
#align submodule.is_ortho_Sup_left Submodule.isOrtho_sSup_left
@[simp]
theorem isOrtho_sSup_right {U : Submodule ๐ E} {V : Set (Submodule ๐ E)} :
U โ sSup V โ โ Vแตข โ V, U โ Vแตข :=
isOrtho_comm.trans <| isOrtho_sSup_left.trans <| by simp_rw [isOrtho_comm]
#align submodule.is_ortho_Sup_right Submodule.isOrtho_sSup_right
@[simp]
theorem isOrtho_iSup_left {ฮน : Sort*} {U : ฮน โ Submodule ๐ E} {V : Submodule ๐ E} :
iSup U โ V โ โ i, U i โ V :=
iSup_le_iff
#align submodule.is_ortho_supr_left Submodule.isOrtho_iSup_left
@[simp]
theorem isOrtho_iSup_right {ฮน : Sort*} {U : Submodule ๐ E} {V : ฮน โ Submodule ๐ E} :
U โ iSup V โ โ i, U โ V i :=
isOrtho_comm.trans <| isOrtho_iSup_left.trans <| by simp_rw [isOrtho_comm]
#align submodule.is_ortho_supr_right Submodule.isOrtho_iSup_right
@[simp]
theorem isOrtho_span {s t : Set E} :
span ๐ s โ span ๐ t โ โ โฆuโฆ, u โ s โ โ โฆvโฆ, v โ t โ โชu, vโซ = 0 := by
simp_rw [span_eq_iSup_of_singleton_spans s, span_eq_iSup_of_singleton_spans t, isOrtho_iSup_left,
isOrtho_iSup_right, isOrtho_iff_le, span_le, Set.subset_def, SetLike.mem_coe,
mem_orthogonal_singleton_iff_inner_left, Set.mem_singleton_iff, forall_eq]
#align submodule.is_ortho_span Submodule.isOrtho_span
theorem IsOrtho.map (f : E โโแตข[๐] F) {U V : Submodule ๐ E} (h : U โ V) : U.map f โ V.map f := by
rw [isOrtho_iff_inner_eq] at *
simp_rw [mem_map, forall_exists_index, and_imp, forall_apply_eq_imp_iffโ,
LinearIsometry.inner_map_map]
exact h
#align submodule.is_ortho.map Submodule.IsOrtho.map
theorem IsOrtho.comap (f : E โโแตข[๐] F) {U V : Submodule ๐ F} (h : U โ V) :
U.comap f โ V.comap f := by
|
rw [isOrtho_iff_inner_eq] at *
|
theorem IsOrtho.comap (f : E โโแตข[๐] F) {U V : Submodule ๐ F} (h : U โ V) :
U.comap f โ V.comap f := by
|
Mathlib.Analysis.InnerProductSpace.Orthogonal.388_0.QXx0GYqLoAbtfq1
|
theorem IsOrtho.comap (f : E โโแตข[๐] F) {U V : Submodule ๐ F} (h : U โ V) :
U.comap f โ V.comap f
|
Mathlib_Analysis_InnerProductSpace_Orthogonal
|
๐ : Type u_1
E : Type u_2
F : Type u_3
instโโด : IsROrC ๐
instโยณ : NormedAddCommGroup E
instโยฒ : InnerProductSpace ๐ E
instโยน : NormedAddCommGroup F
instโ : InnerProductSpace ๐ F
f : E โโแตข[๐] F
U V : Submodule ๐ F
h : โ u โ U, โ v โ V, inner u v = 0
โข โ u โ Submodule.comap f U, โ v โ Submodule.comap f V, inner u v = 0
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sรฉbastien Gouรซzel, Frรฉdรฉric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
/-!
# Orthogonal complements of submodules
In this file, the `orthogonal` complement of a submodule `K` is defined, and basic API established.
Some of the more subtle results about the orthogonal complement are delayed to
`Analysis.InnerProductSpace.Projection`.
See also `BilinForm.orthogonal` for orthogonality with respect to a general bilinear form.
## Notation
The orthogonal complement of a submodule `K` is denoted by `Kแฎ`.
The proposition that two submodules are orthogonal, `Submodule.IsOrtho`, is denoted by `U โ V`.
Note this is not the same unicode symbol as `โฅ` (`Bot`).
-/
variable {๐ E F : Type*} [IsROrC ๐]
variable [NormedAddCommGroup E] [InnerProductSpace ๐ E]
variable [NormedAddCommGroup F] [InnerProductSpace ๐ F]
local notation "โช" x ", " y "โซ" => @inner ๐ _ _ x y
namespace Submodule
variable (K : Submodule ๐ E)
/-- The subspace of vectors orthogonal to a given subspace. -/
def orthogonal : Submodule ๐ E where
carrier := { v | โ u โ K, โชu, vโซ = 0 }
zero_mem' _ _ := inner_zero_right _
add_mem' hx hy u hu := by rw [inner_add_right, hx u hu, hy u hu, add_zero]
smul_mem' c x hx u hu := by rw [inner_smul_right, hx u hu, mul_zero]
#align submodule.orthogonal Submodule.orthogonal
@[inherit_doc]
notation:1200 K "แฎ" => orthogonal K
/-- When a vector is in `Kแฎ`. -/
theorem mem_orthogonal (v : E) : v โ Kแฎ โ โ u โ K, โชu, vโซ = 0 :=
Iff.rfl
#align submodule.mem_orthogonal Submodule.mem_orthogonal
/-- When a vector is in `Kแฎ`, with the inner product the
other way round. -/
theorem mem_orthogonal' (v : E) : v โ Kแฎ โ โ u โ K, โชv, uโซ = 0 := by
simp_rw [mem_orthogonal, inner_eq_zero_symm]
#align submodule.mem_orthogonal' Submodule.mem_orthogonal'
variable {K}
/-- A vector in `K` is orthogonal to one in `Kแฎ`. -/
theorem inner_right_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชu, vโซ = 0 :=
(K.mem_orthogonal v).1 hv u hu
#align submodule.inner_right_of_mem_orthogonal Submodule.inner_right_of_mem_orthogonal
/-- A vector in `Kแฎ` is orthogonal to one in `K`. -/
theorem inner_left_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชv, uโซ = 0 := by
rw [inner_eq_zero_symm]; exact inner_right_of_mem_orthogonal hu hv
#align submodule.inner_left_of_mem_orthogonal Submodule.inner_left_of_mem_orthogonal
/-- A vector is in `(๐ โ u)แฎ` iff it is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_right {u v : E} : v โ (๐ โ u)แฎ โ โชu, vโซ = 0 := by
refine' โจinner_right_of_mem_orthogonal (mem_span_singleton_self u), _โฉ
intro hv w hw
rw [mem_span_singleton] at hw
obtain โจc, rflโฉ := hw
simp [inner_smul_left, hv]
#align submodule.mem_orthogonal_singleton_iff_inner_right Submodule.mem_orthogonal_singleton_iff_inner_right
/-- A vector in `(๐ โ u)แฎ` is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_left {u v : E} : v โ (๐ โ u)แฎ โ โชv, uโซ = 0 := by
rw [mem_orthogonal_singleton_iff_inner_right, inner_eq_zero_symm]
#align submodule.mem_orthogonal_singleton_iff_inner_left Submodule.mem_orthogonal_singleton_iff_inner_left
theorem sub_mem_orthogonal_of_inner_left {x y : E} (h : โ v : K, โชx, vโซ = โชy, vโซ) : x - y โ Kแฎ := by
rw [mem_orthogonal']
intro u hu
rw [inner_sub_left, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_left Submodule.sub_mem_orthogonal_of_inner_left
theorem sub_mem_orthogonal_of_inner_right {x y : E} (h : โ v : K, โช(v : E), xโซ = โช(v : E), yโซ) :
x - y โ Kแฎ := by
intro u hu
rw [inner_sub_right, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_right Submodule.sub_mem_orthogonal_of_inner_right
variable (K)
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem inf_orthogonal_eq_bot : K โ Kแฎ = โฅ := by
rw [eq_bot_iff]
intro x
rw [mem_inf]
exact fun โจhx, hoโฉ => inner_self_eq_zero.1 (ho x hx)
#align submodule.inf_orthogonal_eq_bot Submodule.inf_orthogonal_eq_bot
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem orthogonal_disjoint : Disjoint K Kแฎ := by simp [disjoint_iff, K.inf_orthogonal_eq_bot]
#align submodule.orthogonal_disjoint Submodule.orthogonal_disjoint
/-- `Kแฎ` can be characterized as the intersection of the kernels of the operations of
inner product with each of the elements of `K`. -/
theorem orthogonal_eq_inter : Kแฎ = โจ
v : K, LinearMap.ker (innerSL ๐ (v : E)) := by
apply le_antisymm
ยท rw [le_iInf_iff]
rintro โจv, hvโฉ w hw
simpa using hw _ hv
ยท intro v hv w hw
simp only [mem_iInf] at hv
exact hv โจw, hwโฉ
#align submodule.orthogonal_eq_inter Submodule.orthogonal_eq_inter
/-- The orthogonal complement of any submodule `K` is closed. -/
theorem isClosed_orthogonal : IsClosed (Kแฎ : Set E) := by
rw [orthogonal_eq_inter K]
have := fun v : K => ContinuousLinearMap.isClosed_ker (innerSL ๐ (v : E))
convert isClosed_iInter this
simp only [iInf_coe]
#align submodule.is_closed_orthogonal Submodule.isClosed_orthogonal
/-- In a complete space, the orthogonal complement of any submodule `K` is complete. -/
instance instOrthogonalCompleteSpace [CompleteSpace E] : CompleteSpace Kแฎ :=
K.isClosed_orthogonal.completeSpace_coe
variable (๐ E)
/-- `orthogonal` gives a `GaloisConnection` between
`Submodule ๐ E` and its `OrderDual`. -/
theorem orthogonal_gc :
@GaloisConnection (Submodule ๐ E) (Submodule ๐ E)แตแต _ _ orthogonal orthogonal := fun _Kโ _Kโ =>
โจfun h _v hv _u hu => inner_left_of_mem_orthogonal hv (h hu), fun h _v hv _u hu =>
inner_left_of_mem_orthogonal hv (h hu)โฉ
#align submodule.orthogonal_gc Submodule.orthogonal_gc
variable {๐ E}
/-- `orthogonal` reverses the `โค` ordering of two
subspaces. -/
theorem orthogonal_le {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎ โค Kโแฎ :=
(orthogonal_gc ๐ E).monotone_l h
#align submodule.orthogonal_le Submodule.orthogonal_le
/-- `orthogonal.orthogonal` preserves the `โค` ordering of two
subspaces. -/
theorem orthogonal_orthogonal_monotone {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎแฎ โค Kโแฎแฎ :=
orthogonal_le (orthogonal_le h)
#align submodule.orthogonal_orthogonal_monotone Submodule.orthogonal_orthogonal_monotone
/-- `K` is contained in `Kแฎแฎ`. -/
theorem le_orthogonal_orthogonal : K โค Kแฎแฎ :=
(orthogonal_gc ๐ E).le_u_l _
#align submodule.le_orthogonal_orthogonal Submodule.le_orthogonal_orthogonal
/-- The inf of two orthogonal subspaces equals the subspace orthogonal
to the sup. -/
theorem inf_orthogonal (Kโ Kโ : Submodule ๐ E) : Kโแฎ โ Kโแฎ = (Kโ โ Kโ)แฎ :=
(orthogonal_gc ๐ E).l_sup.symm
#align submodule.inf_orthogonal Submodule.inf_orthogonal
/-- The inf of an indexed family of orthogonal subspaces equals the
subspace orthogonal to the sup. -/
theorem iInf_orthogonal {ฮน : Type*} (K : ฮน โ Submodule ๐ E) : โจ
i, (K i)แฎ = (iSup K)แฎ :=
(orthogonal_gc ๐ E).l_iSup.symm
#align submodule.infi_orthogonal Submodule.iInf_orthogonal
/-- The inf of a set of orthogonal subspaces equals the subspace orthogonal to the sup. -/
theorem sInf_orthogonal (s : Set <| Submodule ๐ E) : โจ
K โ s, Kแฎ = (sSup s)แฎ :=
(orthogonal_gc ๐ E).l_sSup.symm
#align submodule.Inf_orthogonal Submodule.sInf_orthogonal
@[simp]
theorem top_orthogonal_eq_bot : (โค : Submodule ๐ E)แฎ = โฅ := by
ext x
rw [mem_bot, mem_orthogonal]
exact
โจfun h => inner_self_eq_zero.mp (h x mem_top), by
rintro rfl
simpโฉ
#align submodule.top_orthogonal_eq_bot Submodule.top_orthogonal_eq_bot
@[simp]
theorem bot_orthogonal_eq_top : (โฅ : Submodule ๐ E)แฎ = โค := by
rw [โ top_orthogonal_eq_bot, eq_top_iff]
exact le_orthogonal_orthogonal โค
#align submodule.bot_orthogonal_eq_top Submodule.bot_orthogonal_eq_top
@[simp]
theorem orthogonal_eq_top_iff : Kแฎ = โค โ K = โฅ := by
refine'
โจ_, by
rintro rfl
exact bot_orthogonal_eq_topโฉ
intro h
have : K โ Kแฎ = โฅ := K.orthogonal_disjoint.eq_bot
rwa [h, inf_comm, top_inf_eq] at this
#align submodule.orthogonal_eq_top_iff Submodule.orthogonal_eq_top_iff
theorem orthogonalFamily_self :
OrthogonalFamily ๐ (fun b => โฅ(cond b K Kแฎ)) fun b => (cond b K Kแฎ).subtypeโแตข
| true, true => absurd rfl
| true, false => fun _ x y => inner_right_of_mem_orthogonal x.prop y.prop
| false, true => fun _ x y => inner_left_of_mem_orthogonal y.prop x.prop
| false, false => absurd rfl
#align submodule.orthogonal_family_self Submodule.orthogonalFamily_self
end Submodule
@[simp]
theorem bilinFormOfRealInner_orthogonal {E} [NormedAddCommGroup E] [InnerProductSpace โ E]
(K : Submodule โ E) : bilinFormOfRealInner.orthogonal K = Kแฎ :=
rfl
#align bilin_form_of_real_inner_orthogonal bilinFormOfRealInner_orthogonal
/-!
### Orthogonality of submodules
In this section we define `Submodule.IsOrtho U V`, with notation `U โ V`.
The API roughly matches that of `Disjoint`.
-/
namespace Submodule
/-- The proposition that two submodules are orthogonal. Has notation `U โ V`. -/
def IsOrtho (U V : Submodule ๐ E) : Prop :=
U โค Vแฎ
#align submodule.is_ortho Submodule.IsOrtho
@[inherit_doc]
infixl:50 " โ " => Submodule.IsOrtho
theorem isOrtho_iff_le {U V : Submodule ๐ E} : U โ V โ U โค Vแฎ :=
Iff.rfl
#align submodule.is_ortho_iff_le Submodule.isOrtho_iff_le
@[symm]
theorem IsOrtho.symm {U V : Submodule ๐ E} (h : U โ V) : V โ U :=
(le_orthogonal_orthogonal _).trans (orthogonal_le h)
#align submodule.is_ortho.symm Submodule.IsOrtho.symm
theorem isOrtho_comm {U V : Submodule ๐ E} : U โ V โ V โ U :=
โจIsOrtho.symm, IsOrtho.symmโฉ
#align submodule.is_ortho_comm Submodule.isOrtho_comm
theorem symmetric_isOrtho : Symmetric (IsOrtho : Submodule ๐ E โ Submodule ๐ E โ Prop) := fun _ _ =>
IsOrtho.symm
#align submodule.symmetric_is_ortho Submodule.symmetric_isOrtho
theorem IsOrtho.inner_eq {U V : Submodule ๐ E} (h : U โ V) {u v : E} (hu : u โ U) (hv : v โ V) :
โชu, vโซ = 0 :=
h.symm hv _ hu
#align submodule.is_ortho.inner_eq Submodule.IsOrtho.inner_eq
theorem isOrtho_iff_inner_eq {U V : Submodule ๐ E} : U โ V โ โ u โ U, โ v โ V, โชu, vโซ = 0 :=
forallโ_congr fun _u _hu _v _hv => inner_eq_zero_symm
#align submodule.is_ortho_iff_inner_eq Submodule.isOrtho_iff_inner_eq
/- TODO: generalize `Submodule.mapโ` to semilinear maps, so that we can state
`U โ V โ Submodule.mapโ (innerโโ ๐) U V โค โฅ`. -/
@[simp]
theorem isOrtho_bot_left {V : Submodule ๐ E} : โฅ โ V :=
bot_le
#align submodule.is_ortho_bot_left Submodule.isOrtho_bot_left
@[simp]
theorem isOrtho_bot_right {U : Submodule ๐ E} : U โ โฅ :=
isOrtho_bot_left.symm
#align submodule.is_ortho_bot_right Submodule.isOrtho_bot_right
theorem IsOrtho.mono_left {Uโ Uโ V : Submodule ๐ E} (hU : Uโ โค Uโ) (h : Uโ โ V) : Uโ โ V :=
hU.trans h
#align submodule.is_ortho.mono_left Submodule.IsOrtho.mono_left
theorem IsOrtho.mono_right {U Vโ Vโ : Submodule ๐ E} (hV : Vโ โค Vโ) (h : U โ Vโ) : U โ Vโ :=
(h.symm.mono_left hV).symm
#align submodule.is_ortho.mono_right Submodule.IsOrtho.mono_right
theorem IsOrtho.mono {Uโ Vโ Uโ Vโ : Submodule ๐ E} (hU : Uโ โค Uโ) (hV : Vโ โค Vโ) (h : Uโ โ Vโ) :
Uโ โ Vโ :=
(h.mono_right hV).mono_left hU
#align submodule.is_ortho.mono Submodule.IsOrtho.mono
@[simp]
theorem isOrtho_self {U : Submodule ๐ E} : U โ U โ U = โฅ :=
โจfun h => eq_bot_iff.mpr fun x hx => inner_self_eq_zero.mp (h hx x hx), fun h =>
h.symm โธ isOrtho_bot_leftโฉ
#align submodule.is_ortho_self Submodule.isOrtho_self
@[simp]
theorem isOrtho_orthogonal_right (U : Submodule ๐ E) : U โ Uแฎ :=
le_orthogonal_orthogonal _
#align submodule.is_ortho_orthogonal_right Submodule.isOrtho_orthogonal_right
@[simp]
theorem isOrtho_orthogonal_left (U : Submodule ๐ E) : Uแฎ โ U :=
(isOrtho_orthogonal_right U).symm
#align submodule.is_ortho_orthogonal_left Submodule.isOrtho_orthogonal_left
theorem IsOrtho.le {U V : Submodule ๐ E} (h : U โ V) : U โค Vแฎ :=
h
#align submodule.is_ortho.le Submodule.IsOrtho.le
theorem IsOrtho.ge {U V : Submodule ๐ E} (h : U โ V) : V โค Uแฎ :=
h.symm
#align submodule.is_ortho.ge Submodule.IsOrtho.ge
@[simp]
theorem isOrtho_top_right {U : Submodule ๐ E} : U โ โค โ U = โฅ :=
โจfun h => eq_bot_iff.mpr fun _x hx => inner_self_eq_zero.mp (h hx _ mem_top), fun h =>
h.symm โธ isOrtho_bot_leftโฉ
#align submodule.is_ortho_top_right Submodule.isOrtho_top_right
@[simp]
theorem isOrtho_top_left {V : Submodule ๐ E} : โค โ V โ V = โฅ :=
isOrtho_comm.trans isOrtho_top_right
#align submodule.is_ortho_top_left Submodule.isOrtho_top_left
/-- Orthogonal submodules are disjoint. -/
theorem IsOrtho.disjoint {U V : Submodule ๐ E} (h : U โ V) : Disjoint U V :=
(Submodule.orthogonal_disjoint _).mono_right h.symm
#align submodule.is_ortho.disjoint Submodule.IsOrtho.disjoint
@[simp]
theorem isOrtho_sup_left {Uโ Uโ V : Submodule ๐ E} : Uโ โ Uโ โ V โ Uโ โ V โง Uโ โ V :=
sup_le_iff
#align submodule.is_ortho_sup_left Submodule.isOrtho_sup_left
@[simp]
theorem isOrtho_sup_right {U Vโ Vโ : Submodule ๐ E} : U โ Vโ โ Vโ โ U โ Vโ โง U โ Vโ :=
isOrtho_comm.trans <| isOrtho_sup_left.trans <| isOrtho_comm.and isOrtho_comm
#align submodule.is_ortho_sup_right Submodule.isOrtho_sup_right
@[simp]
theorem isOrtho_sSup_left {U : Set (Submodule ๐ E)} {V : Submodule ๐ E} :
sSup U โ V โ โ Uแตข โ U, Uแตข โ V :=
sSup_le_iff
#align submodule.is_ortho_Sup_left Submodule.isOrtho_sSup_left
@[simp]
theorem isOrtho_sSup_right {U : Submodule ๐ E} {V : Set (Submodule ๐ E)} :
U โ sSup V โ โ Vแตข โ V, U โ Vแตข :=
isOrtho_comm.trans <| isOrtho_sSup_left.trans <| by simp_rw [isOrtho_comm]
#align submodule.is_ortho_Sup_right Submodule.isOrtho_sSup_right
@[simp]
theorem isOrtho_iSup_left {ฮน : Sort*} {U : ฮน โ Submodule ๐ E} {V : Submodule ๐ E} :
iSup U โ V โ โ i, U i โ V :=
iSup_le_iff
#align submodule.is_ortho_supr_left Submodule.isOrtho_iSup_left
@[simp]
theorem isOrtho_iSup_right {ฮน : Sort*} {U : Submodule ๐ E} {V : ฮน โ Submodule ๐ E} :
U โ iSup V โ โ i, U โ V i :=
isOrtho_comm.trans <| isOrtho_iSup_left.trans <| by simp_rw [isOrtho_comm]
#align submodule.is_ortho_supr_right Submodule.isOrtho_iSup_right
@[simp]
theorem isOrtho_span {s t : Set E} :
span ๐ s โ span ๐ t โ โ โฆuโฆ, u โ s โ โ โฆvโฆ, v โ t โ โชu, vโซ = 0 := by
simp_rw [span_eq_iSup_of_singleton_spans s, span_eq_iSup_of_singleton_spans t, isOrtho_iSup_left,
isOrtho_iSup_right, isOrtho_iff_le, span_le, Set.subset_def, SetLike.mem_coe,
mem_orthogonal_singleton_iff_inner_left, Set.mem_singleton_iff, forall_eq]
#align submodule.is_ortho_span Submodule.isOrtho_span
theorem IsOrtho.map (f : E โโแตข[๐] F) {U V : Submodule ๐ E} (h : U โ V) : U.map f โ V.map f := by
rw [isOrtho_iff_inner_eq] at *
simp_rw [mem_map, forall_exists_index, and_imp, forall_apply_eq_imp_iffโ,
LinearIsometry.inner_map_map]
exact h
#align submodule.is_ortho.map Submodule.IsOrtho.map
theorem IsOrtho.comap (f : E โโแตข[๐] F) {U V : Submodule ๐ F} (h : U โ V) :
U.comap f โ V.comap f := by
rw [isOrtho_iff_inner_eq] at *
|
simp_rw [mem_comap, โ f.inner_map_map]
|
theorem IsOrtho.comap (f : E โโแตข[๐] F) {U V : Submodule ๐ F} (h : U โ V) :
U.comap f โ V.comap f := by
rw [isOrtho_iff_inner_eq] at *
|
Mathlib.Analysis.InnerProductSpace.Orthogonal.388_0.QXx0GYqLoAbtfq1
|
theorem IsOrtho.comap (f : E โโแตข[๐] F) {U V : Submodule ๐ F} (h : U โ V) :
U.comap f โ V.comap f
|
Mathlib_Analysis_InnerProductSpace_Orthogonal
|
๐ : Type u_1
E : Type u_2
F : Type u_3
instโโด : IsROrC ๐
instโยณ : NormedAddCommGroup E
instโยฒ : InnerProductSpace ๐ E
instโยน : NormedAddCommGroup F
instโ : InnerProductSpace ๐ F
f : E โโแตข[๐] F
U V : Submodule ๐ F
h : โ u โ U, โ v โ V, inner u v = 0
โข โ (u : E), f u โ U โ โ (v : E), f v โ V โ inner (f u) (f v) = 0
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sรฉbastien Gouรซzel, Frรฉdรฉric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
/-!
# Orthogonal complements of submodules
In this file, the `orthogonal` complement of a submodule `K` is defined, and basic API established.
Some of the more subtle results about the orthogonal complement are delayed to
`Analysis.InnerProductSpace.Projection`.
See also `BilinForm.orthogonal` for orthogonality with respect to a general bilinear form.
## Notation
The orthogonal complement of a submodule `K` is denoted by `Kแฎ`.
The proposition that two submodules are orthogonal, `Submodule.IsOrtho`, is denoted by `U โ V`.
Note this is not the same unicode symbol as `โฅ` (`Bot`).
-/
variable {๐ E F : Type*} [IsROrC ๐]
variable [NormedAddCommGroup E] [InnerProductSpace ๐ E]
variable [NormedAddCommGroup F] [InnerProductSpace ๐ F]
local notation "โช" x ", " y "โซ" => @inner ๐ _ _ x y
namespace Submodule
variable (K : Submodule ๐ E)
/-- The subspace of vectors orthogonal to a given subspace. -/
def orthogonal : Submodule ๐ E where
carrier := { v | โ u โ K, โชu, vโซ = 0 }
zero_mem' _ _ := inner_zero_right _
add_mem' hx hy u hu := by rw [inner_add_right, hx u hu, hy u hu, add_zero]
smul_mem' c x hx u hu := by rw [inner_smul_right, hx u hu, mul_zero]
#align submodule.orthogonal Submodule.orthogonal
@[inherit_doc]
notation:1200 K "แฎ" => orthogonal K
/-- When a vector is in `Kแฎ`. -/
theorem mem_orthogonal (v : E) : v โ Kแฎ โ โ u โ K, โชu, vโซ = 0 :=
Iff.rfl
#align submodule.mem_orthogonal Submodule.mem_orthogonal
/-- When a vector is in `Kแฎ`, with the inner product the
other way round. -/
theorem mem_orthogonal' (v : E) : v โ Kแฎ โ โ u โ K, โชv, uโซ = 0 := by
simp_rw [mem_orthogonal, inner_eq_zero_symm]
#align submodule.mem_orthogonal' Submodule.mem_orthogonal'
variable {K}
/-- A vector in `K` is orthogonal to one in `Kแฎ`. -/
theorem inner_right_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชu, vโซ = 0 :=
(K.mem_orthogonal v).1 hv u hu
#align submodule.inner_right_of_mem_orthogonal Submodule.inner_right_of_mem_orthogonal
/-- A vector in `Kแฎ` is orthogonal to one in `K`. -/
theorem inner_left_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชv, uโซ = 0 := by
rw [inner_eq_zero_symm]; exact inner_right_of_mem_orthogonal hu hv
#align submodule.inner_left_of_mem_orthogonal Submodule.inner_left_of_mem_orthogonal
/-- A vector is in `(๐ โ u)แฎ` iff it is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_right {u v : E} : v โ (๐ โ u)แฎ โ โชu, vโซ = 0 := by
refine' โจinner_right_of_mem_orthogonal (mem_span_singleton_self u), _โฉ
intro hv w hw
rw [mem_span_singleton] at hw
obtain โจc, rflโฉ := hw
simp [inner_smul_left, hv]
#align submodule.mem_orthogonal_singleton_iff_inner_right Submodule.mem_orthogonal_singleton_iff_inner_right
/-- A vector in `(๐ โ u)แฎ` is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_left {u v : E} : v โ (๐ โ u)แฎ โ โชv, uโซ = 0 := by
rw [mem_orthogonal_singleton_iff_inner_right, inner_eq_zero_symm]
#align submodule.mem_orthogonal_singleton_iff_inner_left Submodule.mem_orthogonal_singleton_iff_inner_left
theorem sub_mem_orthogonal_of_inner_left {x y : E} (h : โ v : K, โชx, vโซ = โชy, vโซ) : x - y โ Kแฎ := by
rw [mem_orthogonal']
intro u hu
rw [inner_sub_left, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_left Submodule.sub_mem_orthogonal_of_inner_left
theorem sub_mem_orthogonal_of_inner_right {x y : E} (h : โ v : K, โช(v : E), xโซ = โช(v : E), yโซ) :
x - y โ Kแฎ := by
intro u hu
rw [inner_sub_right, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_right Submodule.sub_mem_orthogonal_of_inner_right
variable (K)
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem inf_orthogonal_eq_bot : K โ Kแฎ = โฅ := by
rw [eq_bot_iff]
intro x
rw [mem_inf]
exact fun โจhx, hoโฉ => inner_self_eq_zero.1 (ho x hx)
#align submodule.inf_orthogonal_eq_bot Submodule.inf_orthogonal_eq_bot
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem orthogonal_disjoint : Disjoint K Kแฎ := by simp [disjoint_iff, K.inf_orthogonal_eq_bot]
#align submodule.orthogonal_disjoint Submodule.orthogonal_disjoint
/-- `Kแฎ` can be characterized as the intersection of the kernels of the operations of
inner product with each of the elements of `K`. -/
theorem orthogonal_eq_inter : Kแฎ = โจ
v : K, LinearMap.ker (innerSL ๐ (v : E)) := by
apply le_antisymm
ยท rw [le_iInf_iff]
rintro โจv, hvโฉ w hw
simpa using hw _ hv
ยท intro v hv w hw
simp only [mem_iInf] at hv
exact hv โจw, hwโฉ
#align submodule.orthogonal_eq_inter Submodule.orthogonal_eq_inter
/-- The orthogonal complement of any submodule `K` is closed. -/
theorem isClosed_orthogonal : IsClosed (Kแฎ : Set E) := by
rw [orthogonal_eq_inter K]
have := fun v : K => ContinuousLinearMap.isClosed_ker (innerSL ๐ (v : E))
convert isClosed_iInter this
simp only [iInf_coe]
#align submodule.is_closed_orthogonal Submodule.isClosed_orthogonal
/-- In a complete space, the orthogonal complement of any submodule `K` is complete. -/
instance instOrthogonalCompleteSpace [CompleteSpace E] : CompleteSpace Kแฎ :=
K.isClosed_orthogonal.completeSpace_coe
variable (๐ E)
/-- `orthogonal` gives a `GaloisConnection` between
`Submodule ๐ E` and its `OrderDual`. -/
theorem orthogonal_gc :
@GaloisConnection (Submodule ๐ E) (Submodule ๐ E)แตแต _ _ orthogonal orthogonal := fun _Kโ _Kโ =>
โจfun h _v hv _u hu => inner_left_of_mem_orthogonal hv (h hu), fun h _v hv _u hu =>
inner_left_of_mem_orthogonal hv (h hu)โฉ
#align submodule.orthogonal_gc Submodule.orthogonal_gc
variable {๐ E}
/-- `orthogonal` reverses the `โค` ordering of two
subspaces. -/
theorem orthogonal_le {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎ โค Kโแฎ :=
(orthogonal_gc ๐ E).monotone_l h
#align submodule.orthogonal_le Submodule.orthogonal_le
/-- `orthogonal.orthogonal` preserves the `โค` ordering of two
subspaces. -/
theorem orthogonal_orthogonal_monotone {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎแฎ โค Kโแฎแฎ :=
orthogonal_le (orthogonal_le h)
#align submodule.orthogonal_orthogonal_monotone Submodule.orthogonal_orthogonal_monotone
/-- `K` is contained in `Kแฎแฎ`. -/
theorem le_orthogonal_orthogonal : K โค Kแฎแฎ :=
(orthogonal_gc ๐ E).le_u_l _
#align submodule.le_orthogonal_orthogonal Submodule.le_orthogonal_orthogonal
/-- The inf of two orthogonal subspaces equals the subspace orthogonal
to the sup. -/
theorem inf_orthogonal (Kโ Kโ : Submodule ๐ E) : Kโแฎ โ Kโแฎ = (Kโ โ Kโ)แฎ :=
(orthogonal_gc ๐ E).l_sup.symm
#align submodule.inf_orthogonal Submodule.inf_orthogonal
/-- The inf of an indexed family of orthogonal subspaces equals the
subspace orthogonal to the sup. -/
theorem iInf_orthogonal {ฮน : Type*} (K : ฮน โ Submodule ๐ E) : โจ
i, (K i)แฎ = (iSup K)แฎ :=
(orthogonal_gc ๐ E).l_iSup.symm
#align submodule.infi_orthogonal Submodule.iInf_orthogonal
/-- The inf of a set of orthogonal subspaces equals the subspace orthogonal to the sup. -/
theorem sInf_orthogonal (s : Set <| Submodule ๐ E) : โจ
K โ s, Kแฎ = (sSup s)แฎ :=
(orthogonal_gc ๐ E).l_sSup.symm
#align submodule.Inf_orthogonal Submodule.sInf_orthogonal
@[simp]
theorem top_orthogonal_eq_bot : (โค : Submodule ๐ E)แฎ = โฅ := by
ext x
rw [mem_bot, mem_orthogonal]
exact
โจfun h => inner_self_eq_zero.mp (h x mem_top), by
rintro rfl
simpโฉ
#align submodule.top_orthogonal_eq_bot Submodule.top_orthogonal_eq_bot
@[simp]
theorem bot_orthogonal_eq_top : (โฅ : Submodule ๐ E)แฎ = โค := by
rw [โ top_orthogonal_eq_bot, eq_top_iff]
exact le_orthogonal_orthogonal โค
#align submodule.bot_orthogonal_eq_top Submodule.bot_orthogonal_eq_top
@[simp]
theorem orthogonal_eq_top_iff : Kแฎ = โค โ K = โฅ := by
refine'
โจ_, by
rintro rfl
exact bot_orthogonal_eq_topโฉ
intro h
have : K โ Kแฎ = โฅ := K.orthogonal_disjoint.eq_bot
rwa [h, inf_comm, top_inf_eq] at this
#align submodule.orthogonal_eq_top_iff Submodule.orthogonal_eq_top_iff
theorem orthogonalFamily_self :
OrthogonalFamily ๐ (fun b => โฅ(cond b K Kแฎ)) fun b => (cond b K Kแฎ).subtypeโแตข
| true, true => absurd rfl
| true, false => fun _ x y => inner_right_of_mem_orthogonal x.prop y.prop
| false, true => fun _ x y => inner_left_of_mem_orthogonal y.prop x.prop
| false, false => absurd rfl
#align submodule.orthogonal_family_self Submodule.orthogonalFamily_self
end Submodule
@[simp]
theorem bilinFormOfRealInner_orthogonal {E} [NormedAddCommGroup E] [InnerProductSpace โ E]
(K : Submodule โ E) : bilinFormOfRealInner.orthogonal K = Kแฎ :=
rfl
#align bilin_form_of_real_inner_orthogonal bilinFormOfRealInner_orthogonal
/-!
### Orthogonality of submodules
In this section we define `Submodule.IsOrtho U V`, with notation `U โ V`.
The API roughly matches that of `Disjoint`.
-/
namespace Submodule
/-- The proposition that two submodules are orthogonal. Has notation `U โ V`. -/
def IsOrtho (U V : Submodule ๐ E) : Prop :=
U โค Vแฎ
#align submodule.is_ortho Submodule.IsOrtho
@[inherit_doc]
infixl:50 " โ " => Submodule.IsOrtho
theorem isOrtho_iff_le {U V : Submodule ๐ E} : U โ V โ U โค Vแฎ :=
Iff.rfl
#align submodule.is_ortho_iff_le Submodule.isOrtho_iff_le
@[symm]
theorem IsOrtho.symm {U V : Submodule ๐ E} (h : U โ V) : V โ U :=
(le_orthogonal_orthogonal _).trans (orthogonal_le h)
#align submodule.is_ortho.symm Submodule.IsOrtho.symm
theorem isOrtho_comm {U V : Submodule ๐ E} : U โ V โ V โ U :=
โจIsOrtho.symm, IsOrtho.symmโฉ
#align submodule.is_ortho_comm Submodule.isOrtho_comm
theorem symmetric_isOrtho : Symmetric (IsOrtho : Submodule ๐ E โ Submodule ๐ E โ Prop) := fun _ _ =>
IsOrtho.symm
#align submodule.symmetric_is_ortho Submodule.symmetric_isOrtho
theorem IsOrtho.inner_eq {U V : Submodule ๐ E} (h : U โ V) {u v : E} (hu : u โ U) (hv : v โ V) :
โชu, vโซ = 0 :=
h.symm hv _ hu
#align submodule.is_ortho.inner_eq Submodule.IsOrtho.inner_eq
theorem isOrtho_iff_inner_eq {U V : Submodule ๐ E} : U โ V โ โ u โ U, โ v โ V, โชu, vโซ = 0 :=
forallโ_congr fun _u _hu _v _hv => inner_eq_zero_symm
#align submodule.is_ortho_iff_inner_eq Submodule.isOrtho_iff_inner_eq
/- TODO: generalize `Submodule.mapโ` to semilinear maps, so that we can state
`U โ V โ Submodule.mapโ (innerโโ ๐) U V โค โฅ`. -/
@[simp]
theorem isOrtho_bot_left {V : Submodule ๐ E} : โฅ โ V :=
bot_le
#align submodule.is_ortho_bot_left Submodule.isOrtho_bot_left
@[simp]
theorem isOrtho_bot_right {U : Submodule ๐ E} : U โ โฅ :=
isOrtho_bot_left.symm
#align submodule.is_ortho_bot_right Submodule.isOrtho_bot_right
theorem IsOrtho.mono_left {Uโ Uโ V : Submodule ๐ E} (hU : Uโ โค Uโ) (h : Uโ โ V) : Uโ โ V :=
hU.trans h
#align submodule.is_ortho.mono_left Submodule.IsOrtho.mono_left
theorem IsOrtho.mono_right {U Vโ Vโ : Submodule ๐ E} (hV : Vโ โค Vโ) (h : U โ Vโ) : U โ Vโ :=
(h.symm.mono_left hV).symm
#align submodule.is_ortho.mono_right Submodule.IsOrtho.mono_right
theorem IsOrtho.mono {Uโ Vโ Uโ Vโ : Submodule ๐ E} (hU : Uโ โค Uโ) (hV : Vโ โค Vโ) (h : Uโ โ Vโ) :
Uโ โ Vโ :=
(h.mono_right hV).mono_left hU
#align submodule.is_ortho.mono Submodule.IsOrtho.mono
@[simp]
theorem isOrtho_self {U : Submodule ๐ E} : U โ U โ U = โฅ :=
โจfun h => eq_bot_iff.mpr fun x hx => inner_self_eq_zero.mp (h hx x hx), fun h =>
h.symm โธ isOrtho_bot_leftโฉ
#align submodule.is_ortho_self Submodule.isOrtho_self
@[simp]
theorem isOrtho_orthogonal_right (U : Submodule ๐ E) : U โ Uแฎ :=
le_orthogonal_orthogonal _
#align submodule.is_ortho_orthogonal_right Submodule.isOrtho_orthogonal_right
@[simp]
theorem isOrtho_orthogonal_left (U : Submodule ๐ E) : Uแฎ โ U :=
(isOrtho_orthogonal_right U).symm
#align submodule.is_ortho_orthogonal_left Submodule.isOrtho_orthogonal_left
theorem IsOrtho.le {U V : Submodule ๐ E} (h : U โ V) : U โค Vแฎ :=
h
#align submodule.is_ortho.le Submodule.IsOrtho.le
theorem IsOrtho.ge {U V : Submodule ๐ E} (h : U โ V) : V โค Uแฎ :=
h.symm
#align submodule.is_ortho.ge Submodule.IsOrtho.ge
@[simp]
theorem isOrtho_top_right {U : Submodule ๐ E} : U โ โค โ U = โฅ :=
โจfun h => eq_bot_iff.mpr fun _x hx => inner_self_eq_zero.mp (h hx _ mem_top), fun h =>
h.symm โธ isOrtho_bot_leftโฉ
#align submodule.is_ortho_top_right Submodule.isOrtho_top_right
@[simp]
theorem isOrtho_top_left {V : Submodule ๐ E} : โค โ V โ V = โฅ :=
isOrtho_comm.trans isOrtho_top_right
#align submodule.is_ortho_top_left Submodule.isOrtho_top_left
/-- Orthogonal submodules are disjoint. -/
theorem IsOrtho.disjoint {U V : Submodule ๐ E} (h : U โ V) : Disjoint U V :=
(Submodule.orthogonal_disjoint _).mono_right h.symm
#align submodule.is_ortho.disjoint Submodule.IsOrtho.disjoint
@[simp]
theorem isOrtho_sup_left {Uโ Uโ V : Submodule ๐ E} : Uโ โ Uโ โ V โ Uโ โ V โง Uโ โ V :=
sup_le_iff
#align submodule.is_ortho_sup_left Submodule.isOrtho_sup_left
@[simp]
theorem isOrtho_sup_right {U Vโ Vโ : Submodule ๐ E} : U โ Vโ โ Vโ โ U โ Vโ โง U โ Vโ :=
isOrtho_comm.trans <| isOrtho_sup_left.trans <| isOrtho_comm.and isOrtho_comm
#align submodule.is_ortho_sup_right Submodule.isOrtho_sup_right
@[simp]
theorem isOrtho_sSup_left {U : Set (Submodule ๐ E)} {V : Submodule ๐ E} :
sSup U โ V โ โ Uแตข โ U, Uแตข โ V :=
sSup_le_iff
#align submodule.is_ortho_Sup_left Submodule.isOrtho_sSup_left
@[simp]
theorem isOrtho_sSup_right {U : Submodule ๐ E} {V : Set (Submodule ๐ E)} :
U โ sSup V โ โ Vแตข โ V, U โ Vแตข :=
isOrtho_comm.trans <| isOrtho_sSup_left.trans <| by simp_rw [isOrtho_comm]
#align submodule.is_ortho_Sup_right Submodule.isOrtho_sSup_right
@[simp]
theorem isOrtho_iSup_left {ฮน : Sort*} {U : ฮน โ Submodule ๐ E} {V : Submodule ๐ E} :
iSup U โ V โ โ i, U i โ V :=
iSup_le_iff
#align submodule.is_ortho_supr_left Submodule.isOrtho_iSup_left
@[simp]
theorem isOrtho_iSup_right {ฮน : Sort*} {U : Submodule ๐ E} {V : ฮน โ Submodule ๐ E} :
U โ iSup V โ โ i, U โ V i :=
isOrtho_comm.trans <| isOrtho_iSup_left.trans <| by simp_rw [isOrtho_comm]
#align submodule.is_ortho_supr_right Submodule.isOrtho_iSup_right
@[simp]
theorem isOrtho_span {s t : Set E} :
span ๐ s โ span ๐ t โ โ โฆuโฆ, u โ s โ โ โฆvโฆ, v โ t โ โชu, vโซ = 0 := by
simp_rw [span_eq_iSup_of_singleton_spans s, span_eq_iSup_of_singleton_spans t, isOrtho_iSup_left,
isOrtho_iSup_right, isOrtho_iff_le, span_le, Set.subset_def, SetLike.mem_coe,
mem_orthogonal_singleton_iff_inner_left, Set.mem_singleton_iff, forall_eq]
#align submodule.is_ortho_span Submodule.isOrtho_span
theorem IsOrtho.map (f : E โโแตข[๐] F) {U V : Submodule ๐ E} (h : U โ V) : U.map f โ V.map f := by
rw [isOrtho_iff_inner_eq] at *
simp_rw [mem_map, forall_exists_index, and_imp, forall_apply_eq_imp_iffโ,
LinearIsometry.inner_map_map]
exact h
#align submodule.is_ortho.map Submodule.IsOrtho.map
theorem IsOrtho.comap (f : E โโแตข[๐] F) {U V : Submodule ๐ F} (h : U โ V) :
U.comap f โ V.comap f := by
rw [isOrtho_iff_inner_eq] at *
simp_rw [mem_comap, โ f.inner_map_map]
|
intro u hu v hv
|
theorem IsOrtho.comap (f : E โโแตข[๐] F) {U V : Submodule ๐ F} (h : U โ V) :
U.comap f โ V.comap f := by
rw [isOrtho_iff_inner_eq] at *
simp_rw [mem_comap, โ f.inner_map_map]
|
Mathlib.Analysis.InnerProductSpace.Orthogonal.388_0.QXx0GYqLoAbtfq1
|
theorem IsOrtho.comap (f : E โโแตข[๐] F) {U V : Submodule ๐ F} (h : U โ V) :
U.comap f โ V.comap f
|
Mathlib_Analysis_InnerProductSpace_Orthogonal
|
๐ : Type u_1
E : Type u_2
F : Type u_3
instโโด : IsROrC ๐
instโยณ : NormedAddCommGroup E
instโยฒ : InnerProductSpace ๐ E
instโยน : NormedAddCommGroup F
instโ : InnerProductSpace ๐ F
f : E โโแตข[๐] F
U V : Submodule ๐ F
h : โ u โ U, โ v โ V, inner u v = 0
u : E
hu : f u โ U
v : E
hv : f v โ V
โข inner (f u) (f v) = 0
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sรฉbastien Gouรซzel, Frรฉdรฉric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
/-!
# Orthogonal complements of submodules
In this file, the `orthogonal` complement of a submodule `K` is defined, and basic API established.
Some of the more subtle results about the orthogonal complement are delayed to
`Analysis.InnerProductSpace.Projection`.
See also `BilinForm.orthogonal` for orthogonality with respect to a general bilinear form.
## Notation
The orthogonal complement of a submodule `K` is denoted by `Kแฎ`.
The proposition that two submodules are orthogonal, `Submodule.IsOrtho`, is denoted by `U โ V`.
Note this is not the same unicode symbol as `โฅ` (`Bot`).
-/
variable {๐ E F : Type*} [IsROrC ๐]
variable [NormedAddCommGroup E] [InnerProductSpace ๐ E]
variable [NormedAddCommGroup F] [InnerProductSpace ๐ F]
local notation "โช" x ", " y "โซ" => @inner ๐ _ _ x y
namespace Submodule
variable (K : Submodule ๐ E)
/-- The subspace of vectors orthogonal to a given subspace. -/
def orthogonal : Submodule ๐ E where
carrier := { v | โ u โ K, โชu, vโซ = 0 }
zero_mem' _ _ := inner_zero_right _
add_mem' hx hy u hu := by rw [inner_add_right, hx u hu, hy u hu, add_zero]
smul_mem' c x hx u hu := by rw [inner_smul_right, hx u hu, mul_zero]
#align submodule.orthogonal Submodule.orthogonal
@[inherit_doc]
notation:1200 K "แฎ" => orthogonal K
/-- When a vector is in `Kแฎ`. -/
theorem mem_orthogonal (v : E) : v โ Kแฎ โ โ u โ K, โชu, vโซ = 0 :=
Iff.rfl
#align submodule.mem_orthogonal Submodule.mem_orthogonal
/-- When a vector is in `Kแฎ`, with the inner product the
other way round. -/
theorem mem_orthogonal' (v : E) : v โ Kแฎ โ โ u โ K, โชv, uโซ = 0 := by
simp_rw [mem_orthogonal, inner_eq_zero_symm]
#align submodule.mem_orthogonal' Submodule.mem_orthogonal'
variable {K}
/-- A vector in `K` is orthogonal to one in `Kแฎ`. -/
theorem inner_right_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชu, vโซ = 0 :=
(K.mem_orthogonal v).1 hv u hu
#align submodule.inner_right_of_mem_orthogonal Submodule.inner_right_of_mem_orthogonal
/-- A vector in `Kแฎ` is orthogonal to one in `K`. -/
theorem inner_left_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชv, uโซ = 0 := by
rw [inner_eq_zero_symm]; exact inner_right_of_mem_orthogonal hu hv
#align submodule.inner_left_of_mem_orthogonal Submodule.inner_left_of_mem_orthogonal
/-- A vector is in `(๐ โ u)แฎ` iff it is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_right {u v : E} : v โ (๐ โ u)แฎ โ โชu, vโซ = 0 := by
refine' โจinner_right_of_mem_orthogonal (mem_span_singleton_self u), _โฉ
intro hv w hw
rw [mem_span_singleton] at hw
obtain โจc, rflโฉ := hw
simp [inner_smul_left, hv]
#align submodule.mem_orthogonal_singleton_iff_inner_right Submodule.mem_orthogonal_singleton_iff_inner_right
/-- A vector in `(๐ โ u)แฎ` is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_left {u v : E} : v โ (๐ โ u)แฎ โ โชv, uโซ = 0 := by
rw [mem_orthogonal_singleton_iff_inner_right, inner_eq_zero_symm]
#align submodule.mem_orthogonal_singleton_iff_inner_left Submodule.mem_orthogonal_singleton_iff_inner_left
theorem sub_mem_orthogonal_of_inner_left {x y : E} (h : โ v : K, โชx, vโซ = โชy, vโซ) : x - y โ Kแฎ := by
rw [mem_orthogonal']
intro u hu
rw [inner_sub_left, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_left Submodule.sub_mem_orthogonal_of_inner_left
theorem sub_mem_orthogonal_of_inner_right {x y : E} (h : โ v : K, โช(v : E), xโซ = โช(v : E), yโซ) :
x - y โ Kแฎ := by
intro u hu
rw [inner_sub_right, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_right Submodule.sub_mem_orthogonal_of_inner_right
variable (K)
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem inf_orthogonal_eq_bot : K โ Kแฎ = โฅ := by
rw [eq_bot_iff]
intro x
rw [mem_inf]
exact fun โจhx, hoโฉ => inner_self_eq_zero.1 (ho x hx)
#align submodule.inf_orthogonal_eq_bot Submodule.inf_orthogonal_eq_bot
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem orthogonal_disjoint : Disjoint K Kแฎ := by simp [disjoint_iff, K.inf_orthogonal_eq_bot]
#align submodule.orthogonal_disjoint Submodule.orthogonal_disjoint
/-- `Kแฎ` can be characterized as the intersection of the kernels of the operations of
inner product with each of the elements of `K`. -/
theorem orthogonal_eq_inter : Kแฎ = โจ
v : K, LinearMap.ker (innerSL ๐ (v : E)) := by
apply le_antisymm
ยท rw [le_iInf_iff]
rintro โจv, hvโฉ w hw
simpa using hw _ hv
ยท intro v hv w hw
simp only [mem_iInf] at hv
exact hv โจw, hwโฉ
#align submodule.orthogonal_eq_inter Submodule.orthogonal_eq_inter
/-- The orthogonal complement of any submodule `K` is closed. -/
theorem isClosed_orthogonal : IsClosed (Kแฎ : Set E) := by
rw [orthogonal_eq_inter K]
have := fun v : K => ContinuousLinearMap.isClosed_ker (innerSL ๐ (v : E))
convert isClosed_iInter this
simp only [iInf_coe]
#align submodule.is_closed_orthogonal Submodule.isClosed_orthogonal
/-- In a complete space, the orthogonal complement of any submodule `K` is complete. -/
instance instOrthogonalCompleteSpace [CompleteSpace E] : CompleteSpace Kแฎ :=
K.isClosed_orthogonal.completeSpace_coe
variable (๐ E)
/-- `orthogonal` gives a `GaloisConnection` between
`Submodule ๐ E` and its `OrderDual`. -/
theorem orthogonal_gc :
@GaloisConnection (Submodule ๐ E) (Submodule ๐ E)แตแต _ _ orthogonal orthogonal := fun _Kโ _Kโ =>
โจfun h _v hv _u hu => inner_left_of_mem_orthogonal hv (h hu), fun h _v hv _u hu =>
inner_left_of_mem_orthogonal hv (h hu)โฉ
#align submodule.orthogonal_gc Submodule.orthogonal_gc
variable {๐ E}
/-- `orthogonal` reverses the `โค` ordering of two
subspaces. -/
theorem orthogonal_le {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎ โค Kโแฎ :=
(orthogonal_gc ๐ E).monotone_l h
#align submodule.orthogonal_le Submodule.orthogonal_le
/-- `orthogonal.orthogonal` preserves the `โค` ordering of two
subspaces. -/
theorem orthogonal_orthogonal_monotone {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎแฎ โค Kโแฎแฎ :=
orthogonal_le (orthogonal_le h)
#align submodule.orthogonal_orthogonal_monotone Submodule.orthogonal_orthogonal_monotone
/-- `K` is contained in `Kแฎแฎ`. -/
theorem le_orthogonal_orthogonal : K โค Kแฎแฎ :=
(orthogonal_gc ๐ E).le_u_l _
#align submodule.le_orthogonal_orthogonal Submodule.le_orthogonal_orthogonal
/-- The inf of two orthogonal subspaces equals the subspace orthogonal
to the sup. -/
theorem inf_orthogonal (Kโ Kโ : Submodule ๐ E) : Kโแฎ โ Kโแฎ = (Kโ โ Kโ)แฎ :=
(orthogonal_gc ๐ E).l_sup.symm
#align submodule.inf_orthogonal Submodule.inf_orthogonal
/-- The inf of an indexed family of orthogonal subspaces equals the
subspace orthogonal to the sup. -/
theorem iInf_orthogonal {ฮน : Type*} (K : ฮน โ Submodule ๐ E) : โจ
i, (K i)แฎ = (iSup K)แฎ :=
(orthogonal_gc ๐ E).l_iSup.symm
#align submodule.infi_orthogonal Submodule.iInf_orthogonal
/-- The inf of a set of orthogonal subspaces equals the subspace orthogonal to the sup. -/
theorem sInf_orthogonal (s : Set <| Submodule ๐ E) : โจ
K โ s, Kแฎ = (sSup s)แฎ :=
(orthogonal_gc ๐ E).l_sSup.symm
#align submodule.Inf_orthogonal Submodule.sInf_orthogonal
@[simp]
theorem top_orthogonal_eq_bot : (โค : Submodule ๐ E)แฎ = โฅ := by
ext x
rw [mem_bot, mem_orthogonal]
exact
โจfun h => inner_self_eq_zero.mp (h x mem_top), by
rintro rfl
simpโฉ
#align submodule.top_orthogonal_eq_bot Submodule.top_orthogonal_eq_bot
@[simp]
theorem bot_orthogonal_eq_top : (โฅ : Submodule ๐ E)แฎ = โค := by
rw [โ top_orthogonal_eq_bot, eq_top_iff]
exact le_orthogonal_orthogonal โค
#align submodule.bot_orthogonal_eq_top Submodule.bot_orthogonal_eq_top
@[simp]
theorem orthogonal_eq_top_iff : Kแฎ = โค โ K = โฅ := by
refine'
โจ_, by
rintro rfl
exact bot_orthogonal_eq_topโฉ
intro h
have : K โ Kแฎ = โฅ := K.orthogonal_disjoint.eq_bot
rwa [h, inf_comm, top_inf_eq] at this
#align submodule.orthogonal_eq_top_iff Submodule.orthogonal_eq_top_iff
theorem orthogonalFamily_self :
OrthogonalFamily ๐ (fun b => โฅ(cond b K Kแฎ)) fun b => (cond b K Kแฎ).subtypeโแตข
| true, true => absurd rfl
| true, false => fun _ x y => inner_right_of_mem_orthogonal x.prop y.prop
| false, true => fun _ x y => inner_left_of_mem_orthogonal y.prop x.prop
| false, false => absurd rfl
#align submodule.orthogonal_family_self Submodule.orthogonalFamily_self
end Submodule
@[simp]
theorem bilinFormOfRealInner_orthogonal {E} [NormedAddCommGroup E] [InnerProductSpace โ E]
(K : Submodule โ E) : bilinFormOfRealInner.orthogonal K = Kแฎ :=
rfl
#align bilin_form_of_real_inner_orthogonal bilinFormOfRealInner_orthogonal
/-!
### Orthogonality of submodules
In this section we define `Submodule.IsOrtho U V`, with notation `U โ V`.
The API roughly matches that of `Disjoint`.
-/
namespace Submodule
/-- The proposition that two submodules are orthogonal. Has notation `U โ V`. -/
def IsOrtho (U V : Submodule ๐ E) : Prop :=
U โค Vแฎ
#align submodule.is_ortho Submodule.IsOrtho
@[inherit_doc]
infixl:50 " โ " => Submodule.IsOrtho
theorem isOrtho_iff_le {U V : Submodule ๐ E} : U โ V โ U โค Vแฎ :=
Iff.rfl
#align submodule.is_ortho_iff_le Submodule.isOrtho_iff_le
@[symm]
theorem IsOrtho.symm {U V : Submodule ๐ E} (h : U โ V) : V โ U :=
(le_orthogonal_orthogonal _).trans (orthogonal_le h)
#align submodule.is_ortho.symm Submodule.IsOrtho.symm
theorem isOrtho_comm {U V : Submodule ๐ E} : U โ V โ V โ U :=
โจIsOrtho.symm, IsOrtho.symmโฉ
#align submodule.is_ortho_comm Submodule.isOrtho_comm
theorem symmetric_isOrtho : Symmetric (IsOrtho : Submodule ๐ E โ Submodule ๐ E โ Prop) := fun _ _ =>
IsOrtho.symm
#align submodule.symmetric_is_ortho Submodule.symmetric_isOrtho
theorem IsOrtho.inner_eq {U V : Submodule ๐ E} (h : U โ V) {u v : E} (hu : u โ U) (hv : v โ V) :
โชu, vโซ = 0 :=
h.symm hv _ hu
#align submodule.is_ortho.inner_eq Submodule.IsOrtho.inner_eq
theorem isOrtho_iff_inner_eq {U V : Submodule ๐ E} : U โ V โ โ u โ U, โ v โ V, โชu, vโซ = 0 :=
forallโ_congr fun _u _hu _v _hv => inner_eq_zero_symm
#align submodule.is_ortho_iff_inner_eq Submodule.isOrtho_iff_inner_eq
/- TODO: generalize `Submodule.mapโ` to semilinear maps, so that we can state
`U โ V โ Submodule.mapโ (innerโโ ๐) U V โค โฅ`. -/
@[simp]
theorem isOrtho_bot_left {V : Submodule ๐ E} : โฅ โ V :=
bot_le
#align submodule.is_ortho_bot_left Submodule.isOrtho_bot_left
@[simp]
theorem isOrtho_bot_right {U : Submodule ๐ E} : U โ โฅ :=
isOrtho_bot_left.symm
#align submodule.is_ortho_bot_right Submodule.isOrtho_bot_right
theorem IsOrtho.mono_left {Uโ Uโ V : Submodule ๐ E} (hU : Uโ โค Uโ) (h : Uโ โ V) : Uโ โ V :=
hU.trans h
#align submodule.is_ortho.mono_left Submodule.IsOrtho.mono_left
theorem IsOrtho.mono_right {U Vโ Vโ : Submodule ๐ E} (hV : Vโ โค Vโ) (h : U โ Vโ) : U โ Vโ :=
(h.symm.mono_left hV).symm
#align submodule.is_ortho.mono_right Submodule.IsOrtho.mono_right
theorem IsOrtho.mono {Uโ Vโ Uโ Vโ : Submodule ๐ E} (hU : Uโ โค Uโ) (hV : Vโ โค Vโ) (h : Uโ โ Vโ) :
Uโ โ Vโ :=
(h.mono_right hV).mono_left hU
#align submodule.is_ortho.mono Submodule.IsOrtho.mono
@[simp]
theorem isOrtho_self {U : Submodule ๐ E} : U โ U โ U = โฅ :=
โจfun h => eq_bot_iff.mpr fun x hx => inner_self_eq_zero.mp (h hx x hx), fun h =>
h.symm โธ isOrtho_bot_leftโฉ
#align submodule.is_ortho_self Submodule.isOrtho_self
@[simp]
theorem isOrtho_orthogonal_right (U : Submodule ๐ E) : U โ Uแฎ :=
le_orthogonal_orthogonal _
#align submodule.is_ortho_orthogonal_right Submodule.isOrtho_orthogonal_right
@[simp]
theorem isOrtho_orthogonal_left (U : Submodule ๐ E) : Uแฎ โ U :=
(isOrtho_orthogonal_right U).symm
#align submodule.is_ortho_orthogonal_left Submodule.isOrtho_orthogonal_left
theorem IsOrtho.le {U V : Submodule ๐ E} (h : U โ V) : U โค Vแฎ :=
h
#align submodule.is_ortho.le Submodule.IsOrtho.le
theorem IsOrtho.ge {U V : Submodule ๐ E} (h : U โ V) : V โค Uแฎ :=
h.symm
#align submodule.is_ortho.ge Submodule.IsOrtho.ge
@[simp]
theorem isOrtho_top_right {U : Submodule ๐ E} : U โ โค โ U = โฅ :=
โจfun h => eq_bot_iff.mpr fun _x hx => inner_self_eq_zero.mp (h hx _ mem_top), fun h =>
h.symm โธ isOrtho_bot_leftโฉ
#align submodule.is_ortho_top_right Submodule.isOrtho_top_right
@[simp]
theorem isOrtho_top_left {V : Submodule ๐ E} : โค โ V โ V = โฅ :=
isOrtho_comm.trans isOrtho_top_right
#align submodule.is_ortho_top_left Submodule.isOrtho_top_left
/-- Orthogonal submodules are disjoint. -/
theorem IsOrtho.disjoint {U V : Submodule ๐ E} (h : U โ V) : Disjoint U V :=
(Submodule.orthogonal_disjoint _).mono_right h.symm
#align submodule.is_ortho.disjoint Submodule.IsOrtho.disjoint
@[simp]
theorem isOrtho_sup_left {Uโ Uโ V : Submodule ๐ E} : Uโ โ Uโ โ V โ Uโ โ V โง Uโ โ V :=
sup_le_iff
#align submodule.is_ortho_sup_left Submodule.isOrtho_sup_left
@[simp]
theorem isOrtho_sup_right {U Vโ Vโ : Submodule ๐ E} : U โ Vโ โ Vโ โ U โ Vโ โง U โ Vโ :=
isOrtho_comm.trans <| isOrtho_sup_left.trans <| isOrtho_comm.and isOrtho_comm
#align submodule.is_ortho_sup_right Submodule.isOrtho_sup_right
@[simp]
theorem isOrtho_sSup_left {U : Set (Submodule ๐ E)} {V : Submodule ๐ E} :
sSup U โ V โ โ Uแตข โ U, Uแตข โ V :=
sSup_le_iff
#align submodule.is_ortho_Sup_left Submodule.isOrtho_sSup_left
@[simp]
theorem isOrtho_sSup_right {U : Submodule ๐ E} {V : Set (Submodule ๐ E)} :
U โ sSup V โ โ Vแตข โ V, U โ Vแตข :=
isOrtho_comm.trans <| isOrtho_sSup_left.trans <| by simp_rw [isOrtho_comm]
#align submodule.is_ortho_Sup_right Submodule.isOrtho_sSup_right
@[simp]
theorem isOrtho_iSup_left {ฮน : Sort*} {U : ฮน โ Submodule ๐ E} {V : Submodule ๐ E} :
iSup U โ V โ โ i, U i โ V :=
iSup_le_iff
#align submodule.is_ortho_supr_left Submodule.isOrtho_iSup_left
@[simp]
theorem isOrtho_iSup_right {ฮน : Sort*} {U : Submodule ๐ E} {V : ฮน โ Submodule ๐ E} :
U โ iSup V โ โ i, U โ V i :=
isOrtho_comm.trans <| isOrtho_iSup_left.trans <| by simp_rw [isOrtho_comm]
#align submodule.is_ortho_supr_right Submodule.isOrtho_iSup_right
@[simp]
theorem isOrtho_span {s t : Set E} :
span ๐ s โ span ๐ t โ โ โฆuโฆ, u โ s โ โ โฆvโฆ, v โ t โ โชu, vโซ = 0 := by
simp_rw [span_eq_iSup_of_singleton_spans s, span_eq_iSup_of_singleton_spans t, isOrtho_iSup_left,
isOrtho_iSup_right, isOrtho_iff_le, span_le, Set.subset_def, SetLike.mem_coe,
mem_orthogonal_singleton_iff_inner_left, Set.mem_singleton_iff, forall_eq]
#align submodule.is_ortho_span Submodule.isOrtho_span
theorem IsOrtho.map (f : E โโแตข[๐] F) {U V : Submodule ๐ E} (h : U โ V) : U.map f โ V.map f := by
rw [isOrtho_iff_inner_eq] at *
simp_rw [mem_map, forall_exists_index, and_imp, forall_apply_eq_imp_iffโ,
LinearIsometry.inner_map_map]
exact h
#align submodule.is_ortho.map Submodule.IsOrtho.map
theorem IsOrtho.comap (f : E โโแตข[๐] F) {U V : Submodule ๐ F} (h : U โ V) :
U.comap f โ V.comap f := by
rw [isOrtho_iff_inner_eq] at *
simp_rw [mem_comap, โ f.inner_map_map]
intro u hu v hv
|
exact h _ hu _ hv
|
theorem IsOrtho.comap (f : E โโแตข[๐] F) {U V : Submodule ๐ F} (h : U โ V) :
U.comap f โ V.comap f := by
rw [isOrtho_iff_inner_eq] at *
simp_rw [mem_comap, โ f.inner_map_map]
intro u hu v hv
|
Mathlib.Analysis.InnerProductSpace.Orthogonal.388_0.QXx0GYqLoAbtfq1
|
theorem IsOrtho.comap (f : E โโแตข[๐] F) {U V : Submodule ๐ F} (h : U โ V) :
U.comap f โ V.comap f
|
Mathlib_Analysis_InnerProductSpace_Orthogonal
|
๐ : Type u_1
E : Type u_2
F : Type u_3
instโโด : IsROrC ๐
instโยณ : NormedAddCommGroup E
instโยฒ : InnerProductSpace ๐ E
instโยน : NormedAddCommGroup F
instโ : InnerProductSpace ๐ F
f : E โโแตข[๐] F
U V : Submodule ๐ E
h : Submodule.map f U โ Submodule.map f V
โข U โ V
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sรฉbastien Gouรซzel, Frรฉdรฉric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
/-!
# Orthogonal complements of submodules
In this file, the `orthogonal` complement of a submodule `K` is defined, and basic API established.
Some of the more subtle results about the orthogonal complement are delayed to
`Analysis.InnerProductSpace.Projection`.
See also `BilinForm.orthogonal` for orthogonality with respect to a general bilinear form.
## Notation
The orthogonal complement of a submodule `K` is denoted by `Kแฎ`.
The proposition that two submodules are orthogonal, `Submodule.IsOrtho`, is denoted by `U โ V`.
Note this is not the same unicode symbol as `โฅ` (`Bot`).
-/
variable {๐ E F : Type*} [IsROrC ๐]
variable [NormedAddCommGroup E] [InnerProductSpace ๐ E]
variable [NormedAddCommGroup F] [InnerProductSpace ๐ F]
local notation "โช" x ", " y "โซ" => @inner ๐ _ _ x y
namespace Submodule
variable (K : Submodule ๐ E)
/-- The subspace of vectors orthogonal to a given subspace. -/
def orthogonal : Submodule ๐ E where
carrier := { v | โ u โ K, โชu, vโซ = 0 }
zero_mem' _ _ := inner_zero_right _
add_mem' hx hy u hu := by rw [inner_add_right, hx u hu, hy u hu, add_zero]
smul_mem' c x hx u hu := by rw [inner_smul_right, hx u hu, mul_zero]
#align submodule.orthogonal Submodule.orthogonal
@[inherit_doc]
notation:1200 K "แฎ" => orthogonal K
/-- When a vector is in `Kแฎ`. -/
theorem mem_orthogonal (v : E) : v โ Kแฎ โ โ u โ K, โชu, vโซ = 0 :=
Iff.rfl
#align submodule.mem_orthogonal Submodule.mem_orthogonal
/-- When a vector is in `Kแฎ`, with the inner product the
other way round. -/
theorem mem_orthogonal' (v : E) : v โ Kแฎ โ โ u โ K, โชv, uโซ = 0 := by
simp_rw [mem_orthogonal, inner_eq_zero_symm]
#align submodule.mem_orthogonal' Submodule.mem_orthogonal'
variable {K}
/-- A vector in `K` is orthogonal to one in `Kแฎ`. -/
theorem inner_right_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชu, vโซ = 0 :=
(K.mem_orthogonal v).1 hv u hu
#align submodule.inner_right_of_mem_orthogonal Submodule.inner_right_of_mem_orthogonal
/-- A vector in `Kแฎ` is orthogonal to one in `K`. -/
theorem inner_left_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชv, uโซ = 0 := by
rw [inner_eq_zero_symm]; exact inner_right_of_mem_orthogonal hu hv
#align submodule.inner_left_of_mem_orthogonal Submodule.inner_left_of_mem_orthogonal
/-- A vector is in `(๐ โ u)แฎ` iff it is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_right {u v : E} : v โ (๐ โ u)แฎ โ โชu, vโซ = 0 := by
refine' โจinner_right_of_mem_orthogonal (mem_span_singleton_self u), _โฉ
intro hv w hw
rw [mem_span_singleton] at hw
obtain โจc, rflโฉ := hw
simp [inner_smul_left, hv]
#align submodule.mem_orthogonal_singleton_iff_inner_right Submodule.mem_orthogonal_singleton_iff_inner_right
/-- A vector in `(๐ โ u)แฎ` is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_left {u v : E} : v โ (๐ โ u)แฎ โ โชv, uโซ = 0 := by
rw [mem_orthogonal_singleton_iff_inner_right, inner_eq_zero_symm]
#align submodule.mem_orthogonal_singleton_iff_inner_left Submodule.mem_orthogonal_singleton_iff_inner_left
theorem sub_mem_orthogonal_of_inner_left {x y : E} (h : โ v : K, โชx, vโซ = โชy, vโซ) : x - y โ Kแฎ := by
rw [mem_orthogonal']
intro u hu
rw [inner_sub_left, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_left Submodule.sub_mem_orthogonal_of_inner_left
theorem sub_mem_orthogonal_of_inner_right {x y : E} (h : โ v : K, โช(v : E), xโซ = โช(v : E), yโซ) :
x - y โ Kแฎ := by
intro u hu
rw [inner_sub_right, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_right Submodule.sub_mem_orthogonal_of_inner_right
variable (K)
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem inf_orthogonal_eq_bot : K โ Kแฎ = โฅ := by
rw [eq_bot_iff]
intro x
rw [mem_inf]
exact fun โจhx, hoโฉ => inner_self_eq_zero.1 (ho x hx)
#align submodule.inf_orthogonal_eq_bot Submodule.inf_orthogonal_eq_bot
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem orthogonal_disjoint : Disjoint K Kแฎ := by simp [disjoint_iff, K.inf_orthogonal_eq_bot]
#align submodule.orthogonal_disjoint Submodule.orthogonal_disjoint
/-- `Kแฎ` can be characterized as the intersection of the kernels of the operations of
inner product with each of the elements of `K`. -/
theorem orthogonal_eq_inter : Kแฎ = โจ
v : K, LinearMap.ker (innerSL ๐ (v : E)) := by
apply le_antisymm
ยท rw [le_iInf_iff]
rintro โจv, hvโฉ w hw
simpa using hw _ hv
ยท intro v hv w hw
simp only [mem_iInf] at hv
exact hv โจw, hwโฉ
#align submodule.orthogonal_eq_inter Submodule.orthogonal_eq_inter
/-- The orthogonal complement of any submodule `K` is closed. -/
theorem isClosed_orthogonal : IsClosed (Kแฎ : Set E) := by
rw [orthogonal_eq_inter K]
have := fun v : K => ContinuousLinearMap.isClosed_ker (innerSL ๐ (v : E))
convert isClosed_iInter this
simp only [iInf_coe]
#align submodule.is_closed_orthogonal Submodule.isClosed_orthogonal
/-- In a complete space, the orthogonal complement of any submodule `K` is complete. -/
instance instOrthogonalCompleteSpace [CompleteSpace E] : CompleteSpace Kแฎ :=
K.isClosed_orthogonal.completeSpace_coe
variable (๐ E)
/-- `orthogonal` gives a `GaloisConnection` between
`Submodule ๐ E` and its `OrderDual`. -/
theorem orthogonal_gc :
@GaloisConnection (Submodule ๐ E) (Submodule ๐ E)แตแต _ _ orthogonal orthogonal := fun _Kโ _Kโ =>
โจfun h _v hv _u hu => inner_left_of_mem_orthogonal hv (h hu), fun h _v hv _u hu =>
inner_left_of_mem_orthogonal hv (h hu)โฉ
#align submodule.orthogonal_gc Submodule.orthogonal_gc
variable {๐ E}
/-- `orthogonal` reverses the `โค` ordering of two
subspaces. -/
theorem orthogonal_le {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎ โค Kโแฎ :=
(orthogonal_gc ๐ E).monotone_l h
#align submodule.orthogonal_le Submodule.orthogonal_le
/-- `orthogonal.orthogonal` preserves the `โค` ordering of two
subspaces. -/
theorem orthogonal_orthogonal_monotone {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎแฎ โค Kโแฎแฎ :=
orthogonal_le (orthogonal_le h)
#align submodule.orthogonal_orthogonal_monotone Submodule.orthogonal_orthogonal_monotone
/-- `K` is contained in `Kแฎแฎ`. -/
theorem le_orthogonal_orthogonal : K โค Kแฎแฎ :=
(orthogonal_gc ๐ E).le_u_l _
#align submodule.le_orthogonal_orthogonal Submodule.le_orthogonal_orthogonal
/-- The inf of two orthogonal subspaces equals the subspace orthogonal
to the sup. -/
theorem inf_orthogonal (Kโ Kโ : Submodule ๐ E) : Kโแฎ โ Kโแฎ = (Kโ โ Kโ)แฎ :=
(orthogonal_gc ๐ E).l_sup.symm
#align submodule.inf_orthogonal Submodule.inf_orthogonal
/-- The inf of an indexed family of orthogonal subspaces equals the
subspace orthogonal to the sup. -/
theorem iInf_orthogonal {ฮน : Type*} (K : ฮน โ Submodule ๐ E) : โจ
i, (K i)แฎ = (iSup K)แฎ :=
(orthogonal_gc ๐ E).l_iSup.symm
#align submodule.infi_orthogonal Submodule.iInf_orthogonal
/-- The inf of a set of orthogonal subspaces equals the subspace orthogonal to the sup. -/
theorem sInf_orthogonal (s : Set <| Submodule ๐ E) : โจ
K โ s, Kแฎ = (sSup s)แฎ :=
(orthogonal_gc ๐ E).l_sSup.symm
#align submodule.Inf_orthogonal Submodule.sInf_orthogonal
@[simp]
theorem top_orthogonal_eq_bot : (โค : Submodule ๐ E)แฎ = โฅ := by
ext x
rw [mem_bot, mem_orthogonal]
exact
โจfun h => inner_self_eq_zero.mp (h x mem_top), by
rintro rfl
simpโฉ
#align submodule.top_orthogonal_eq_bot Submodule.top_orthogonal_eq_bot
@[simp]
theorem bot_orthogonal_eq_top : (โฅ : Submodule ๐ E)แฎ = โค := by
rw [โ top_orthogonal_eq_bot, eq_top_iff]
exact le_orthogonal_orthogonal โค
#align submodule.bot_orthogonal_eq_top Submodule.bot_orthogonal_eq_top
@[simp]
theorem orthogonal_eq_top_iff : Kแฎ = โค โ K = โฅ := by
refine'
โจ_, by
rintro rfl
exact bot_orthogonal_eq_topโฉ
intro h
have : K โ Kแฎ = โฅ := K.orthogonal_disjoint.eq_bot
rwa [h, inf_comm, top_inf_eq] at this
#align submodule.orthogonal_eq_top_iff Submodule.orthogonal_eq_top_iff
theorem orthogonalFamily_self :
OrthogonalFamily ๐ (fun b => โฅ(cond b K Kแฎ)) fun b => (cond b K Kแฎ).subtypeโแตข
| true, true => absurd rfl
| true, false => fun _ x y => inner_right_of_mem_orthogonal x.prop y.prop
| false, true => fun _ x y => inner_left_of_mem_orthogonal y.prop x.prop
| false, false => absurd rfl
#align submodule.orthogonal_family_self Submodule.orthogonalFamily_self
end Submodule
@[simp]
theorem bilinFormOfRealInner_orthogonal {E} [NormedAddCommGroup E] [InnerProductSpace โ E]
(K : Submodule โ E) : bilinFormOfRealInner.orthogonal K = Kแฎ :=
rfl
#align bilin_form_of_real_inner_orthogonal bilinFormOfRealInner_orthogonal
/-!
### Orthogonality of submodules
In this section we define `Submodule.IsOrtho U V`, with notation `U โ V`.
The API roughly matches that of `Disjoint`.
-/
namespace Submodule
/-- The proposition that two submodules are orthogonal. Has notation `U โ V`. -/
def IsOrtho (U V : Submodule ๐ E) : Prop :=
U โค Vแฎ
#align submodule.is_ortho Submodule.IsOrtho
@[inherit_doc]
infixl:50 " โ " => Submodule.IsOrtho
theorem isOrtho_iff_le {U V : Submodule ๐ E} : U โ V โ U โค Vแฎ :=
Iff.rfl
#align submodule.is_ortho_iff_le Submodule.isOrtho_iff_le
@[symm]
theorem IsOrtho.symm {U V : Submodule ๐ E} (h : U โ V) : V โ U :=
(le_orthogonal_orthogonal _).trans (orthogonal_le h)
#align submodule.is_ortho.symm Submodule.IsOrtho.symm
theorem isOrtho_comm {U V : Submodule ๐ E} : U โ V โ V โ U :=
โจIsOrtho.symm, IsOrtho.symmโฉ
#align submodule.is_ortho_comm Submodule.isOrtho_comm
theorem symmetric_isOrtho : Symmetric (IsOrtho : Submodule ๐ E โ Submodule ๐ E โ Prop) := fun _ _ =>
IsOrtho.symm
#align submodule.symmetric_is_ortho Submodule.symmetric_isOrtho
theorem IsOrtho.inner_eq {U V : Submodule ๐ E} (h : U โ V) {u v : E} (hu : u โ U) (hv : v โ V) :
โชu, vโซ = 0 :=
h.symm hv _ hu
#align submodule.is_ortho.inner_eq Submodule.IsOrtho.inner_eq
theorem isOrtho_iff_inner_eq {U V : Submodule ๐ E} : U โ V โ โ u โ U, โ v โ V, โชu, vโซ = 0 :=
forallโ_congr fun _u _hu _v _hv => inner_eq_zero_symm
#align submodule.is_ortho_iff_inner_eq Submodule.isOrtho_iff_inner_eq
/- TODO: generalize `Submodule.mapโ` to semilinear maps, so that we can state
`U โ V โ Submodule.mapโ (innerโโ ๐) U V โค โฅ`. -/
@[simp]
theorem isOrtho_bot_left {V : Submodule ๐ E} : โฅ โ V :=
bot_le
#align submodule.is_ortho_bot_left Submodule.isOrtho_bot_left
@[simp]
theorem isOrtho_bot_right {U : Submodule ๐ E} : U โ โฅ :=
isOrtho_bot_left.symm
#align submodule.is_ortho_bot_right Submodule.isOrtho_bot_right
theorem IsOrtho.mono_left {Uโ Uโ V : Submodule ๐ E} (hU : Uโ โค Uโ) (h : Uโ โ V) : Uโ โ V :=
hU.trans h
#align submodule.is_ortho.mono_left Submodule.IsOrtho.mono_left
theorem IsOrtho.mono_right {U Vโ Vโ : Submodule ๐ E} (hV : Vโ โค Vโ) (h : U โ Vโ) : U โ Vโ :=
(h.symm.mono_left hV).symm
#align submodule.is_ortho.mono_right Submodule.IsOrtho.mono_right
theorem IsOrtho.mono {Uโ Vโ Uโ Vโ : Submodule ๐ E} (hU : Uโ โค Uโ) (hV : Vโ โค Vโ) (h : Uโ โ Vโ) :
Uโ โ Vโ :=
(h.mono_right hV).mono_left hU
#align submodule.is_ortho.mono Submodule.IsOrtho.mono
@[simp]
theorem isOrtho_self {U : Submodule ๐ E} : U โ U โ U = โฅ :=
โจfun h => eq_bot_iff.mpr fun x hx => inner_self_eq_zero.mp (h hx x hx), fun h =>
h.symm โธ isOrtho_bot_leftโฉ
#align submodule.is_ortho_self Submodule.isOrtho_self
@[simp]
theorem isOrtho_orthogonal_right (U : Submodule ๐ E) : U โ Uแฎ :=
le_orthogonal_orthogonal _
#align submodule.is_ortho_orthogonal_right Submodule.isOrtho_orthogonal_right
@[simp]
theorem isOrtho_orthogonal_left (U : Submodule ๐ E) : Uแฎ โ U :=
(isOrtho_orthogonal_right U).symm
#align submodule.is_ortho_orthogonal_left Submodule.isOrtho_orthogonal_left
theorem IsOrtho.le {U V : Submodule ๐ E} (h : U โ V) : U โค Vแฎ :=
h
#align submodule.is_ortho.le Submodule.IsOrtho.le
theorem IsOrtho.ge {U V : Submodule ๐ E} (h : U โ V) : V โค Uแฎ :=
h.symm
#align submodule.is_ortho.ge Submodule.IsOrtho.ge
@[simp]
theorem isOrtho_top_right {U : Submodule ๐ E} : U โ โค โ U = โฅ :=
โจfun h => eq_bot_iff.mpr fun _x hx => inner_self_eq_zero.mp (h hx _ mem_top), fun h =>
h.symm โธ isOrtho_bot_leftโฉ
#align submodule.is_ortho_top_right Submodule.isOrtho_top_right
@[simp]
theorem isOrtho_top_left {V : Submodule ๐ E} : โค โ V โ V = โฅ :=
isOrtho_comm.trans isOrtho_top_right
#align submodule.is_ortho_top_left Submodule.isOrtho_top_left
/-- Orthogonal submodules are disjoint. -/
theorem IsOrtho.disjoint {U V : Submodule ๐ E} (h : U โ V) : Disjoint U V :=
(Submodule.orthogonal_disjoint _).mono_right h.symm
#align submodule.is_ortho.disjoint Submodule.IsOrtho.disjoint
@[simp]
theorem isOrtho_sup_left {Uโ Uโ V : Submodule ๐ E} : Uโ โ Uโ โ V โ Uโ โ V โง Uโ โ V :=
sup_le_iff
#align submodule.is_ortho_sup_left Submodule.isOrtho_sup_left
@[simp]
theorem isOrtho_sup_right {U Vโ Vโ : Submodule ๐ E} : U โ Vโ โ Vโ โ U โ Vโ โง U โ Vโ :=
isOrtho_comm.trans <| isOrtho_sup_left.trans <| isOrtho_comm.and isOrtho_comm
#align submodule.is_ortho_sup_right Submodule.isOrtho_sup_right
@[simp]
theorem isOrtho_sSup_left {U : Set (Submodule ๐ E)} {V : Submodule ๐ E} :
sSup U โ V โ โ Uแตข โ U, Uแตข โ V :=
sSup_le_iff
#align submodule.is_ortho_Sup_left Submodule.isOrtho_sSup_left
@[simp]
theorem isOrtho_sSup_right {U : Submodule ๐ E} {V : Set (Submodule ๐ E)} :
U โ sSup V โ โ Vแตข โ V, U โ Vแตข :=
isOrtho_comm.trans <| isOrtho_sSup_left.trans <| by simp_rw [isOrtho_comm]
#align submodule.is_ortho_Sup_right Submodule.isOrtho_sSup_right
@[simp]
theorem isOrtho_iSup_left {ฮน : Sort*} {U : ฮน โ Submodule ๐ E} {V : Submodule ๐ E} :
iSup U โ V โ โ i, U i โ V :=
iSup_le_iff
#align submodule.is_ortho_supr_left Submodule.isOrtho_iSup_left
@[simp]
theorem isOrtho_iSup_right {ฮน : Sort*} {U : Submodule ๐ E} {V : ฮน โ Submodule ๐ E} :
U โ iSup V โ โ i, U โ V i :=
isOrtho_comm.trans <| isOrtho_iSup_left.trans <| by simp_rw [isOrtho_comm]
#align submodule.is_ortho_supr_right Submodule.isOrtho_iSup_right
@[simp]
theorem isOrtho_span {s t : Set E} :
span ๐ s โ span ๐ t โ โ โฆuโฆ, u โ s โ โ โฆvโฆ, v โ t โ โชu, vโซ = 0 := by
simp_rw [span_eq_iSup_of_singleton_spans s, span_eq_iSup_of_singleton_spans t, isOrtho_iSup_left,
isOrtho_iSup_right, isOrtho_iff_le, span_le, Set.subset_def, SetLike.mem_coe,
mem_orthogonal_singleton_iff_inner_left, Set.mem_singleton_iff, forall_eq]
#align submodule.is_ortho_span Submodule.isOrtho_span
theorem IsOrtho.map (f : E โโแตข[๐] F) {U V : Submodule ๐ E} (h : U โ V) : U.map f โ V.map f := by
rw [isOrtho_iff_inner_eq] at *
simp_rw [mem_map, forall_exists_index, and_imp, forall_apply_eq_imp_iffโ,
LinearIsometry.inner_map_map]
exact h
#align submodule.is_ortho.map Submodule.IsOrtho.map
theorem IsOrtho.comap (f : E โโแตข[๐] F) {U V : Submodule ๐ F} (h : U โ V) :
U.comap f โ V.comap f := by
rw [isOrtho_iff_inner_eq] at *
simp_rw [mem_comap, โ f.inner_map_map]
intro u hu v hv
exact h _ hu _ hv
#align submodule.is_ortho.comap Submodule.IsOrtho.comap
@[simp]
theorem IsOrtho.map_iff (f : E โโแตข[๐] F) {U V : Submodule ๐ E} : U.map f โ V.map f โ U โ V :=
โจfun h => by
|
have hf : โ p : Submodule ๐ E, (p.map f).comap f.toLinearIsometry = p :=
comap_map_eq_of_injective f.injective
|
@[simp]
theorem IsOrtho.map_iff (f : E โโแตข[๐] F) {U V : Submodule ๐ E} : U.map f โ V.map f โ U โ V :=
โจfun h => by
|
Mathlib.Analysis.InnerProductSpace.Orthogonal.396_0.QXx0GYqLoAbtfq1
|
@[simp]
theorem IsOrtho.map_iff (f : E โโแตข[๐] F) {U V : Submodule ๐ E} : U.map f โ V.map f โ U โ V
|
Mathlib_Analysis_InnerProductSpace_Orthogonal
|
๐ : Type u_1
E : Type u_2
F : Type u_3
instโโด : IsROrC ๐
instโยณ : NormedAddCommGroup E
instโยฒ : InnerProductSpace ๐ E
instโยน : NormedAddCommGroup F
instโ : InnerProductSpace ๐ F
f : E โโแตข[๐] F
U V : Submodule ๐ E
h : Submodule.map f U โ Submodule.map f V
hf : โ (p : Submodule ๐ E), Submodule.comap (LinearIsometryEquiv.toLinearIsometry f) (Submodule.map f p) = p
โข U โ V
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sรฉbastien Gouรซzel, Frรฉdรฉric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
/-!
# Orthogonal complements of submodules
In this file, the `orthogonal` complement of a submodule `K` is defined, and basic API established.
Some of the more subtle results about the orthogonal complement are delayed to
`Analysis.InnerProductSpace.Projection`.
See also `BilinForm.orthogonal` for orthogonality with respect to a general bilinear form.
## Notation
The orthogonal complement of a submodule `K` is denoted by `Kแฎ`.
The proposition that two submodules are orthogonal, `Submodule.IsOrtho`, is denoted by `U โ V`.
Note this is not the same unicode symbol as `โฅ` (`Bot`).
-/
variable {๐ E F : Type*} [IsROrC ๐]
variable [NormedAddCommGroup E] [InnerProductSpace ๐ E]
variable [NormedAddCommGroup F] [InnerProductSpace ๐ F]
local notation "โช" x ", " y "โซ" => @inner ๐ _ _ x y
namespace Submodule
variable (K : Submodule ๐ E)
/-- The subspace of vectors orthogonal to a given subspace. -/
def orthogonal : Submodule ๐ E where
carrier := { v | โ u โ K, โชu, vโซ = 0 }
zero_mem' _ _ := inner_zero_right _
add_mem' hx hy u hu := by rw [inner_add_right, hx u hu, hy u hu, add_zero]
smul_mem' c x hx u hu := by rw [inner_smul_right, hx u hu, mul_zero]
#align submodule.orthogonal Submodule.orthogonal
@[inherit_doc]
notation:1200 K "แฎ" => orthogonal K
/-- When a vector is in `Kแฎ`. -/
theorem mem_orthogonal (v : E) : v โ Kแฎ โ โ u โ K, โชu, vโซ = 0 :=
Iff.rfl
#align submodule.mem_orthogonal Submodule.mem_orthogonal
/-- When a vector is in `Kแฎ`, with the inner product the
other way round. -/
theorem mem_orthogonal' (v : E) : v โ Kแฎ โ โ u โ K, โชv, uโซ = 0 := by
simp_rw [mem_orthogonal, inner_eq_zero_symm]
#align submodule.mem_orthogonal' Submodule.mem_orthogonal'
variable {K}
/-- A vector in `K` is orthogonal to one in `Kแฎ`. -/
theorem inner_right_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชu, vโซ = 0 :=
(K.mem_orthogonal v).1 hv u hu
#align submodule.inner_right_of_mem_orthogonal Submodule.inner_right_of_mem_orthogonal
/-- A vector in `Kแฎ` is orthogonal to one in `K`. -/
theorem inner_left_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชv, uโซ = 0 := by
rw [inner_eq_zero_symm]; exact inner_right_of_mem_orthogonal hu hv
#align submodule.inner_left_of_mem_orthogonal Submodule.inner_left_of_mem_orthogonal
/-- A vector is in `(๐ โ u)แฎ` iff it is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_right {u v : E} : v โ (๐ โ u)แฎ โ โชu, vโซ = 0 := by
refine' โจinner_right_of_mem_orthogonal (mem_span_singleton_self u), _โฉ
intro hv w hw
rw [mem_span_singleton] at hw
obtain โจc, rflโฉ := hw
simp [inner_smul_left, hv]
#align submodule.mem_orthogonal_singleton_iff_inner_right Submodule.mem_orthogonal_singleton_iff_inner_right
/-- A vector in `(๐ โ u)แฎ` is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_left {u v : E} : v โ (๐ โ u)แฎ โ โชv, uโซ = 0 := by
rw [mem_orthogonal_singleton_iff_inner_right, inner_eq_zero_symm]
#align submodule.mem_orthogonal_singleton_iff_inner_left Submodule.mem_orthogonal_singleton_iff_inner_left
theorem sub_mem_orthogonal_of_inner_left {x y : E} (h : โ v : K, โชx, vโซ = โชy, vโซ) : x - y โ Kแฎ := by
rw [mem_orthogonal']
intro u hu
rw [inner_sub_left, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_left Submodule.sub_mem_orthogonal_of_inner_left
theorem sub_mem_orthogonal_of_inner_right {x y : E} (h : โ v : K, โช(v : E), xโซ = โช(v : E), yโซ) :
x - y โ Kแฎ := by
intro u hu
rw [inner_sub_right, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_right Submodule.sub_mem_orthogonal_of_inner_right
variable (K)
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem inf_orthogonal_eq_bot : K โ Kแฎ = โฅ := by
rw [eq_bot_iff]
intro x
rw [mem_inf]
exact fun โจhx, hoโฉ => inner_self_eq_zero.1 (ho x hx)
#align submodule.inf_orthogonal_eq_bot Submodule.inf_orthogonal_eq_bot
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem orthogonal_disjoint : Disjoint K Kแฎ := by simp [disjoint_iff, K.inf_orthogonal_eq_bot]
#align submodule.orthogonal_disjoint Submodule.orthogonal_disjoint
/-- `Kแฎ` can be characterized as the intersection of the kernels of the operations of
inner product with each of the elements of `K`. -/
theorem orthogonal_eq_inter : Kแฎ = โจ
v : K, LinearMap.ker (innerSL ๐ (v : E)) := by
apply le_antisymm
ยท rw [le_iInf_iff]
rintro โจv, hvโฉ w hw
simpa using hw _ hv
ยท intro v hv w hw
simp only [mem_iInf] at hv
exact hv โจw, hwโฉ
#align submodule.orthogonal_eq_inter Submodule.orthogonal_eq_inter
/-- The orthogonal complement of any submodule `K` is closed. -/
theorem isClosed_orthogonal : IsClosed (Kแฎ : Set E) := by
rw [orthogonal_eq_inter K]
have := fun v : K => ContinuousLinearMap.isClosed_ker (innerSL ๐ (v : E))
convert isClosed_iInter this
simp only [iInf_coe]
#align submodule.is_closed_orthogonal Submodule.isClosed_orthogonal
/-- In a complete space, the orthogonal complement of any submodule `K` is complete. -/
instance instOrthogonalCompleteSpace [CompleteSpace E] : CompleteSpace Kแฎ :=
K.isClosed_orthogonal.completeSpace_coe
variable (๐ E)
/-- `orthogonal` gives a `GaloisConnection` between
`Submodule ๐ E` and its `OrderDual`. -/
theorem orthogonal_gc :
@GaloisConnection (Submodule ๐ E) (Submodule ๐ E)แตแต _ _ orthogonal orthogonal := fun _Kโ _Kโ =>
โจfun h _v hv _u hu => inner_left_of_mem_orthogonal hv (h hu), fun h _v hv _u hu =>
inner_left_of_mem_orthogonal hv (h hu)โฉ
#align submodule.orthogonal_gc Submodule.orthogonal_gc
variable {๐ E}
/-- `orthogonal` reverses the `โค` ordering of two
subspaces. -/
theorem orthogonal_le {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎ โค Kโแฎ :=
(orthogonal_gc ๐ E).monotone_l h
#align submodule.orthogonal_le Submodule.orthogonal_le
/-- `orthogonal.orthogonal` preserves the `โค` ordering of two
subspaces. -/
theorem orthogonal_orthogonal_monotone {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎแฎ โค Kโแฎแฎ :=
orthogonal_le (orthogonal_le h)
#align submodule.orthogonal_orthogonal_monotone Submodule.orthogonal_orthogonal_monotone
/-- `K` is contained in `Kแฎแฎ`. -/
theorem le_orthogonal_orthogonal : K โค Kแฎแฎ :=
(orthogonal_gc ๐ E).le_u_l _
#align submodule.le_orthogonal_orthogonal Submodule.le_orthogonal_orthogonal
/-- The inf of two orthogonal subspaces equals the subspace orthogonal
to the sup. -/
theorem inf_orthogonal (Kโ Kโ : Submodule ๐ E) : Kโแฎ โ Kโแฎ = (Kโ โ Kโ)แฎ :=
(orthogonal_gc ๐ E).l_sup.symm
#align submodule.inf_orthogonal Submodule.inf_orthogonal
/-- The inf of an indexed family of orthogonal subspaces equals the
subspace orthogonal to the sup. -/
theorem iInf_orthogonal {ฮน : Type*} (K : ฮน โ Submodule ๐ E) : โจ
i, (K i)แฎ = (iSup K)แฎ :=
(orthogonal_gc ๐ E).l_iSup.symm
#align submodule.infi_orthogonal Submodule.iInf_orthogonal
/-- The inf of a set of orthogonal subspaces equals the subspace orthogonal to the sup. -/
theorem sInf_orthogonal (s : Set <| Submodule ๐ E) : โจ
K โ s, Kแฎ = (sSup s)แฎ :=
(orthogonal_gc ๐ E).l_sSup.symm
#align submodule.Inf_orthogonal Submodule.sInf_orthogonal
@[simp]
theorem top_orthogonal_eq_bot : (โค : Submodule ๐ E)แฎ = โฅ := by
ext x
rw [mem_bot, mem_orthogonal]
exact
โจfun h => inner_self_eq_zero.mp (h x mem_top), by
rintro rfl
simpโฉ
#align submodule.top_orthogonal_eq_bot Submodule.top_orthogonal_eq_bot
@[simp]
theorem bot_orthogonal_eq_top : (โฅ : Submodule ๐ E)แฎ = โค := by
rw [โ top_orthogonal_eq_bot, eq_top_iff]
exact le_orthogonal_orthogonal โค
#align submodule.bot_orthogonal_eq_top Submodule.bot_orthogonal_eq_top
@[simp]
theorem orthogonal_eq_top_iff : Kแฎ = โค โ K = โฅ := by
refine'
โจ_, by
rintro rfl
exact bot_orthogonal_eq_topโฉ
intro h
have : K โ Kแฎ = โฅ := K.orthogonal_disjoint.eq_bot
rwa [h, inf_comm, top_inf_eq] at this
#align submodule.orthogonal_eq_top_iff Submodule.orthogonal_eq_top_iff
theorem orthogonalFamily_self :
OrthogonalFamily ๐ (fun b => โฅ(cond b K Kแฎ)) fun b => (cond b K Kแฎ).subtypeโแตข
| true, true => absurd rfl
| true, false => fun _ x y => inner_right_of_mem_orthogonal x.prop y.prop
| false, true => fun _ x y => inner_left_of_mem_orthogonal y.prop x.prop
| false, false => absurd rfl
#align submodule.orthogonal_family_self Submodule.orthogonalFamily_self
end Submodule
@[simp]
theorem bilinFormOfRealInner_orthogonal {E} [NormedAddCommGroup E] [InnerProductSpace โ E]
(K : Submodule โ E) : bilinFormOfRealInner.orthogonal K = Kแฎ :=
rfl
#align bilin_form_of_real_inner_orthogonal bilinFormOfRealInner_orthogonal
/-!
### Orthogonality of submodules
In this section we define `Submodule.IsOrtho U V`, with notation `U โ V`.
The API roughly matches that of `Disjoint`.
-/
namespace Submodule
/-- The proposition that two submodules are orthogonal. Has notation `U โ V`. -/
def IsOrtho (U V : Submodule ๐ E) : Prop :=
U โค Vแฎ
#align submodule.is_ortho Submodule.IsOrtho
@[inherit_doc]
infixl:50 " โ " => Submodule.IsOrtho
theorem isOrtho_iff_le {U V : Submodule ๐ E} : U โ V โ U โค Vแฎ :=
Iff.rfl
#align submodule.is_ortho_iff_le Submodule.isOrtho_iff_le
@[symm]
theorem IsOrtho.symm {U V : Submodule ๐ E} (h : U โ V) : V โ U :=
(le_orthogonal_orthogonal _).trans (orthogonal_le h)
#align submodule.is_ortho.symm Submodule.IsOrtho.symm
theorem isOrtho_comm {U V : Submodule ๐ E} : U โ V โ V โ U :=
โจIsOrtho.symm, IsOrtho.symmโฉ
#align submodule.is_ortho_comm Submodule.isOrtho_comm
theorem symmetric_isOrtho : Symmetric (IsOrtho : Submodule ๐ E โ Submodule ๐ E โ Prop) := fun _ _ =>
IsOrtho.symm
#align submodule.symmetric_is_ortho Submodule.symmetric_isOrtho
theorem IsOrtho.inner_eq {U V : Submodule ๐ E} (h : U โ V) {u v : E} (hu : u โ U) (hv : v โ V) :
โชu, vโซ = 0 :=
h.symm hv _ hu
#align submodule.is_ortho.inner_eq Submodule.IsOrtho.inner_eq
theorem isOrtho_iff_inner_eq {U V : Submodule ๐ E} : U โ V โ โ u โ U, โ v โ V, โชu, vโซ = 0 :=
forallโ_congr fun _u _hu _v _hv => inner_eq_zero_symm
#align submodule.is_ortho_iff_inner_eq Submodule.isOrtho_iff_inner_eq
/- TODO: generalize `Submodule.mapโ` to semilinear maps, so that we can state
`U โ V โ Submodule.mapโ (innerโโ ๐) U V โค โฅ`. -/
@[simp]
theorem isOrtho_bot_left {V : Submodule ๐ E} : โฅ โ V :=
bot_le
#align submodule.is_ortho_bot_left Submodule.isOrtho_bot_left
@[simp]
theorem isOrtho_bot_right {U : Submodule ๐ E} : U โ โฅ :=
isOrtho_bot_left.symm
#align submodule.is_ortho_bot_right Submodule.isOrtho_bot_right
theorem IsOrtho.mono_left {Uโ Uโ V : Submodule ๐ E} (hU : Uโ โค Uโ) (h : Uโ โ V) : Uโ โ V :=
hU.trans h
#align submodule.is_ortho.mono_left Submodule.IsOrtho.mono_left
theorem IsOrtho.mono_right {U Vโ Vโ : Submodule ๐ E} (hV : Vโ โค Vโ) (h : U โ Vโ) : U โ Vโ :=
(h.symm.mono_left hV).symm
#align submodule.is_ortho.mono_right Submodule.IsOrtho.mono_right
theorem IsOrtho.mono {Uโ Vโ Uโ Vโ : Submodule ๐ E} (hU : Uโ โค Uโ) (hV : Vโ โค Vโ) (h : Uโ โ Vโ) :
Uโ โ Vโ :=
(h.mono_right hV).mono_left hU
#align submodule.is_ortho.mono Submodule.IsOrtho.mono
@[simp]
theorem isOrtho_self {U : Submodule ๐ E} : U โ U โ U = โฅ :=
โจfun h => eq_bot_iff.mpr fun x hx => inner_self_eq_zero.mp (h hx x hx), fun h =>
h.symm โธ isOrtho_bot_leftโฉ
#align submodule.is_ortho_self Submodule.isOrtho_self
@[simp]
theorem isOrtho_orthogonal_right (U : Submodule ๐ E) : U โ Uแฎ :=
le_orthogonal_orthogonal _
#align submodule.is_ortho_orthogonal_right Submodule.isOrtho_orthogonal_right
@[simp]
theorem isOrtho_orthogonal_left (U : Submodule ๐ E) : Uแฎ โ U :=
(isOrtho_orthogonal_right U).symm
#align submodule.is_ortho_orthogonal_left Submodule.isOrtho_orthogonal_left
theorem IsOrtho.le {U V : Submodule ๐ E} (h : U โ V) : U โค Vแฎ :=
h
#align submodule.is_ortho.le Submodule.IsOrtho.le
theorem IsOrtho.ge {U V : Submodule ๐ E} (h : U โ V) : V โค Uแฎ :=
h.symm
#align submodule.is_ortho.ge Submodule.IsOrtho.ge
@[simp]
theorem isOrtho_top_right {U : Submodule ๐ E} : U โ โค โ U = โฅ :=
โจfun h => eq_bot_iff.mpr fun _x hx => inner_self_eq_zero.mp (h hx _ mem_top), fun h =>
h.symm โธ isOrtho_bot_leftโฉ
#align submodule.is_ortho_top_right Submodule.isOrtho_top_right
@[simp]
theorem isOrtho_top_left {V : Submodule ๐ E} : โค โ V โ V = โฅ :=
isOrtho_comm.trans isOrtho_top_right
#align submodule.is_ortho_top_left Submodule.isOrtho_top_left
/-- Orthogonal submodules are disjoint. -/
theorem IsOrtho.disjoint {U V : Submodule ๐ E} (h : U โ V) : Disjoint U V :=
(Submodule.orthogonal_disjoint _).mono_right h.symm
#align submodule.is_ortho.disjoint Submodule.IsOrtho.disjoint
@[simp]
theorem isOrtho_sup_left {Uโ Uโ V : Submodule ๐ E} : Uโ โ Uโ โ V โ Uโ โ V โง Uโ โ V :=
sup_le_iff
#align submodule.is_ortho_sup_left Submodule.isOrtho_sup_left
@[simp]
theorem isOrtho_sup_right {U Vโ Vโ : Submodule ๐ E} : U โ Vโ โ Vโ โ U โ Vโ โง U โ Vโ :=
isOrtho_comm.trans <| isOrtho_sup_left.trans <| isOrtho_comm.and isOrtho_comm
#align submodule.is_ortho_sup_right Submodule.isOrtho_sup_right
@[simp]
theorem isOrtho_sSup_left {U : Set (Submodule ๐ E)} {V : Submodule ๐ E} :
sSup U โ V โ โ Uแตข โ U, Uแตข โ V :=
sSup_le_iff
#align submodule.is_ortho_Sup_left Submodule.isOrtho_sSup_left
@[simp]
theorem isOrtho_sSup_right {U : Submodule ๐ E} {V : Set (Submodule ๐ E)} :
U โ sSup V โ โ Vแตข โ V, U โ Vแตข :=
isOrtho_comm.trans <| isOrtho_sSup_left.trans <| by simp_rw [isOrtho_comm]
#align submodule.is_ortho_Sup_right Submodule.isOrtho_sSup_right
@[simp]
theorem isOrtho_iSup_left {ฮน : Sort*} {U : ฮน โ Submodule ๐ E} {V : Submodule ๐ E} :
iSup U โ V โ โ i, U i โ V :=
iSup_le_iff
#align submodule.is_ortho_supr_left Submodule.isOrtho_iSup_left
@[simp]
theorem isOrtho_iSup_right {ฮน : Sort*} {U : Submodule ๐ E} {V : ฮน โ Submodule ๐ E} :
U โ iSup V โ โ i, U โ V i :=
isOrtho_comm.trans <| isOrtho_iSup_left.trans <| by simp_rw [isOrtho_comm]
#align submodule.is_ortho_supr_right Submodule.isOrtho_iSup_right
@[simp]
theorem isOrtho_span {s t : Set E} :
span ๐ s โ span ๐ t โ โ โฆuโฆ, u โ s โ โ โฆvโฆ, v โ t โ โชu, vโซ = 0 := by
simp_rw [span_eq_iSup_of_singleton_spans s, span_eq_iSup_of_singleton_spans t, isOrtho_iSup_left,
isOrtho_iSup_right, isOrtho_iff_le, span_le, Set.subset_def, SetLike.mem_coe,
mem_orthogonal_singleton_iff_inner_left, Set.mem_singleton_iff, forall_eq]
#align submodule.is_ortho_span Submodule.isOrtho_span
theorem IsOrtho.map (f : E โโแตข[๐] F) {U V : Submodule ๐ E} (h : U โ V) : U.map f โ V.map f := by
rw [isOrtho_iff_inner_eq] at *
simp_rw [mem_map, forall_exists_index, and_imp, forall_apply_eq_imp_iffโ,
LinearIsometry.inner_map_map]
exact h
#align submodule.is_ortho.map Submodule.IsOrtho.map
theorem IsOrtho.comap (f : E โโแตข[๐] F) {U V : Submodule ๐ F} (h : U โ V) :
U.comap f โ V.comap f := by
rw [isOrtho_iff_inner_eq] at *
simp_rw [mem_comap, โ f.inner_map_map]
intro u hu v hv
exact h _ hu _ hv
#align submodule.is_ortho.comap Submodule.IsOrtho.comap
@[simp]
theorem IsOrtho.map_iff (f : E โโแตข[๐] F) {U V : Submodule ๐ E} : U.map f โ V.map f โ U โ V :=
โจfun h => by
have hf : โ p : Submodule ๐ E, (p.map f).comap f.toLinearIsometry = p :=
comap_map_eq_of_injective f.injective
|
simpa only [hf] using h.comap f.toLinearIsometry
|
@[simp]
theorem IsOrtho.map_iff (f : E โโแตข[๐] F) {U V : Submodule ๐ E} : U.map f โ V.map f โ U โ V :=
โจfun h => by
have hf : โ p : Submodule ๐ E, (p.map f).comap f.toLinearIsometry = p :=
comap_map_eq_of_injective f.injective
|
Mathlib.Analysis.InnerProductSpace.Orthogonal.396_0.QXx0GYqLoAbtfq1
|
@[simp]
theorem IsOrtho.map_iff (f : E โโแตข[๐] F) {U V : Submodule ๐ E} : U.map f โ V.map f โ U โ V
|
Mathlib_Analysis_InnerProductSpace_Orthogonal
|
๐ : Type u_1
E : Type u_2
F : Type u_3
instโโด : IsROrC ๐
instโยณ : NormedAddCommGroup E
instโยฒ : InnerProductSpace ๐ E
instโยน : NormedAddCommGroup F
instโ : InnerProductSpace ๐ F
f : E โโแตข[๐] F
U V : Submodule ๐ F
h : Submodule.comap f U โ Submodule.comap f V
โข U โ V
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sรฉbastien Gouรซzel, Frรฉdรฉric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
/-!
# Orthogonal complements of submodules
In this file, the `orthogonal` complement of a submodule `K` is defined, and basic API established.
Some of the more subtle results about the orthogonal complement are delayed to
`Analysis.InnerProductSpace.Projection`.
See also `BilinForm.orthogonal` for orthogonality with respect to a general bilinear form.
## Notation
The orthogonal complement of a submodule `K` is denoted by `Kแฎ`.
The proposition that two submodules are orthogonal, `Submodule.IsOrtho`, is denoted by `U โ V`.
Note this is not the same unicode symbol as `โฅ` (`Bot`).
-/
variable {๐ E F : Type*} [IsROrC ๐]
variable [NormedAddCommGroup E] [InnerProductSpace ๐ E]
variable [NormedAddCommGroup F] [InnerProductSpace ๐ F]
local notation "โช" x ", " y "โซ" => @inner ๐ _ _ x y
namespace Submodule
variable (K : Submodule ๐ E)
/-- The subspace of vectors orthogonal to a given subspace. -/
def orthogonal : Submodule ๐ E where
carrier := { v | โ u โ K, โชu, vโซ = 0 }
zero_mem' _ _ := inner_zero_right _
add_mem' hx hy u hu := by rw [inner_add_right, hx u hu, hy u hu, add_zero]
smul_mem' c x hx u hu := by rw [inner_smul_right, hx u hu, mul_zero]
#align submodule.orthogonal Submodule.orthogonal
@[inherit_doc]
notation:1200 K "แฎ" => orthogonal K
/-- When a vector is in `Kแฎ`. -/
theorem mem_orthogonal (v : E) : v โ Kแฎ โ โ u โ K, โชu, vโซ = 0 :=
Iff.rfl
#align submodule.mem_orthogonal Submodule.mem_orthogonal
/-- When a vector is in `Kแฎ`, with the inner product the
other way round. -/
theorem mem_orthogonal' (v : E) : v โ Kแฎ โ โ u โ K, โชv, uโซ = 0 := by
simp_rw [mem_orthogonal, inner_eq_zero_symm]
#align submodule.mem_orthogonal' Submodule.mem_orthogonal'
variable {K}
/-- A vector in `K` is orthogonal to one in `Kแฎ`. -/
theorem inner_right_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชu, vโซ = 0 :=
(K.mem_orthogonal v).1 hv u hu
#align submodule.inner_right_of_mem_orthogonal Submodule.inner_right_of_mem_orthogonal
/-- A vector in `Kแฎ` is orthogonal to one in `K`. -/
theorem inner_left_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชv, uโซ = 0 := by
rw [inner_eq_zero_symm]; exact inner_right_of_mem_orthogonal hu hv
#align submodule.inner_left_of_mem_orthogonal Submodule.inner_left_of_mem_orthogonal
/-- A vector is in `(๐ โ u)แฎ` iff it is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_right {u v : E} : v โ (๐ โ u)แฎ โ โชu, vโซ = 0 := by
refine' โจinner_right_of_mem_orthogonal (mem_span_singleton_self u), _โฉ
intro hv w hw
rw [mem_span_singleton] at hw
obtain โจc, rflโฉ := hw
simp [inner_smul_left, hv]
#align submodule.mem_orthogonal_singleton_iff_inner_right Submodule.mem_orthogonal_singleton_iff_inner_right
/-- A vector in `(๐ โ u)แฎ` is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_left {u v : E} : v โ (๐ โ u)แฎ โ โชv, uโซ = 0 := by
rw [mem_orthogonal_singleton_iff_inner_right, inner_eq_zero_symm]
#align submodule.mem_orthogonal_singleton_iff_inner_left Submodule.mem_orthogonal_singleton_iff_inner_left
theorem sub_mem_orthogonal_of_inner_left {x y : E} (h : โ v : K, โชx, vโซ = โชy, vโซ) : x - y โ Kแฎ := by
rw [mem_orthogonal']
intro u hu
rw [inner_sub_left, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_left Submodule.sub_mem_orthogonal_of_inner_left
theorem sub_mem_orthogonal_of_inner_right {x y : E} (h : โ v : K, โช(v : E), xโซ = โช(v : E), yโซ) :
x - y โ Kแฎ := by
intro u hu
rw [inner_sub_right, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_right Submodule.sub_mem_orthogonal_of_inner_right
variable (K)
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem inf_orthogonal_eq_bot : K โ Kแฎ = โฅ := by
rw [eq_bot_iff]
intro x
rw [mem_inf]
exact fun โจhx, hoโฉ => inner_self_eq_zero.1 (ho x hx)
#align submodule.inf_orthogonal_eq_bot Submodule.inf_orthogonal_eq_bot
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem orthogonal_disjoint : Disjoint K Kแฎ := by simp [disjoint_iff, K.inf_orthogonal_eq_bot]
#align submodule.orthogonal_disjoint Submodule.orthogonal_disjoint
/-- `Kแฎ` can be characterized as the intersection of the kernels of the operations of
inner product with each of the elements of `K`. -/
theorem orthogonal_eq_inter : Kแฎ = โจ
v : K, LinearMap.ker (innerSL ๐ (v : E)) := by
apply le_antisymm
ยท rw [le_iInf_iff]
rintro โจv, hvโฉ w hw
simpa using hw _ hv
ยท intro v hv w hw
simp only [mem_iInf] at hv
exact hv โจw, hwโฉ
#align submodule.orthogonal_eq_inter Submodule.orthogonal_eq_inter
/-- The orthogonal complement of any submodule `K` is closed. -/
theorem isClosed_orthogonal : IsClosed (Kแฎ : Set E) := by
rw [orthogonal_eq_inter K]
have := fun v : K => ContinuousLinearMap.isClosed_ker (innerSL ๐ (v : E))
convert isClosed_iInter this
simp only [iInf_coe]
#align submodule.is_closed_orthogonal Submodule.isClosed_orthogonal
/-- In a complete space, the orthogonal complement of any submodule `K` is complete. -/
instance instOrthogonalCompleteSpace [CompleteSpace E] : CompleteSpace Kแฎ :=
K.isClosed_orthogonal.completeSpace_coe
variable (๐ E)
/-- `orthogonal` gives a `GaloisConnection` between
`Submodule ๐ E` and its `OrderDual`. -/
theorem orthogonal_gc :
@GaloisConnection (Submodule ๐ E) (Submodule ๐ E)แตแต _ _ orthogonal orthogonal := fun _Kโ _Kโ =>
โจfun h _v hv _u hu => inner_left_of_mem_orthogonal hv (h hu), fun h _v hv _u hu =>
inner_left_of_mem_orthogonal hv (h hu)โฉ
#align submodule.orthogonal_gc Submodule.orthogonal_gc
variable {๐ E}
/-- `orthogonal` reverses the `โค` ordering of two
subspaces. -/
theorem orthogonal_le {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎ โค Kโแฎ :=
(orthogonal_gc ๐ E).monotone_l h
#align submodule.orthogonal_le Submodule.orthogonal_le
/-- `orthogonal.orthogonal` preserves the `โค` ordering of two
subspaces. -/
theorem orthogonal_orthogonal_monotone {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎแฎ โค Kโแฎแฎ :=
orthogonal_le (orthogonal_le h)
#align submodule.orthogonal_orthogonal_monotone Submodule.orthogonal_orthogonal_monotone
/-- `K` is contained in `Kแฎแฎ`. -/
theorem le_orthogonal_orthogonal : K โค Kแฎแฎ :=
(orthogonal_gc ๐ E).le_u_l _
#align submodule.le_orthogonal_orthogonal Submodule.le_orthogonal_orthogonal
/-- The inf of two orthogonal subspaces equals the subspace orthogonal
to the sup. -/
theorem inf_orthogonal (Kโ Kโ : Submodule ๐ E) : Kโแฎ โ Kโแฎ = (Kโ โ Kโ)แฎ :=
(orthogonal_gc ๐ E).l_sup.symm
#align submodule.inf_orthogonal Submodule.inf_orthogonal
/-- The inf of an indexed family of orthogonal subspaces equals the
subspace orthogonal to the sup. -/
theorem iInf_orthogonal {ฮน : Type*} (K : ฮน โ Submodule ๐ E) : โจ
i, (K i)แฎ = (iSup K)แฎ :=
(orthogonal_gc ๐ E).l_iSup.symm
#align submodule.infi_orthogonal Submodule.iInf_orthogonal
/-- The inf of a set of orthogonal subspaces equals the subspace orthogonal to the sup. -/
theorem sInf_orthogonal (s : Set <| Submodule ๐ E) : โจ
K โ s, Kแฎ = (sSup s)แฎ :=
(orthogonal_gc ๐ E).l_sSup.symm
#align submodule.Inf_orthogonal Submodule.sInf_orthogonal
@[simp]
theorem top_orthogonal_eq_bot : (โค : Submodule ๐ E)แฎ = โฅ := by
ext x
rw [mem_bot, mem_orthogonal]
exact
โจfun h => inner_self_eq_zero.mp (h x mem_top), by
rintro rfl
simpโฉ
#align submodule.top_orthogonal_eq_bot Submodule.top_orthogonal_eq_bot
@[simp]
theorem bot_orthogonal_eq_top : (โฅ : Submodule ๐ E)แฎ = โค := by
rw [โ top_orthogonal_eq_bot, eq_top_iff]
exact le_orthogonal_orthogonal โค
#align submodule.bot_orthogonal_eq_top Submodule.bot_orthogonal_eq_top
@[simp]
theorem orthogonal_eq_top_iff : Kแฎ = โค โ K = โฅ := by
refine'
โจ_, by
rintro rfl
exact bot_orthogonal_eq_topโฉ
intro h
have : K โ Kแฎ = โฅ := K.orthogonal_disjoint.eq_bot
rwa [h, inf_comm, top_inf_eq] at this
#align submodule.orthogonal_eq_top_iff Submodule.orthogonal_eq_top_iff
theorem orthogonalFamily_self :
OrthogonalFamily ๐ (fun b => โฅ(cond b K Kแฎ)) fun b => (cond b K Kแฎ).subtypeโแตข
| true, true => absurd rfl
| true, false => fun _ x y => inner_right_of_mem_orthogonal x.prop y.prop
| false, true => fun _ x y => inner_left_of_mem_orthogonal y.prop x.prop
| false, false => absurd rfl
#align submodule.orthogonal_family_self Submodule.orthogonalFamily_self
end Submodule
@[simp]
theorem bilinFormOfRealInner_orthogonal {E} [NormedAddCommGroup E] [InnerProductSpace โ E]
(K : Submodule โ E) : bilinFormOfRealInner.orthogonal K = Kแฎ :=
rfl
#align bilin_form_of_real_inner_orthogonal bilinFormOfRealInner_orthogonal
/-!
### Orthogonality of submodules
In this section we define `Submodule.IsOrtho U V`, with notation `U โ V`.
The API roughly matches that of `Disjoint`.
-/
namespace Submodule
/-- The proposition that two submodules are orthogonal. Has notation `U โ V`. -/
def IsOrtho (U V : Submodule ๐ E) : Prop :=
U โค Vแฎ
#align submodule.is_ortho Submodule.IsOrtho
@[inherit_doc]
infixl:50 " โ " => Submodule.IsOrtho
theorem isOrtho_iff_le {U V : Submodule ๐ E} : U โ V โ U โค Vแฎ :=
Iff.rfl
#align submodule.is_ortho_iff_le Submodule.isOrtho_iff_le
@[symm]
theorem IsOrtho.symm {U V : Submodule ๐ E} (h : U โ V) : V โ U :=
(le_orthogonal_orthogonal _).trans (orthogonal_le h)
#align submodule.is_ortho.symm Submodule.IsOrtho.symm
theorem isOrtho_comm {U V : Submodule ๐ E} : U โ V โ V โ U :=
โจIsOrtho.symm, IsOrtho.symmโฉ
#align submodule.is_ortho_comm Submodule.isOrtho_comm
theorem symmetric_isOrtho : Symmetric (IsOrtho : Submodule ๐ E โ Submodule ๐ E โ Prop) := fun _ _ =>
IsOrtho.symm
#align submodule.symmetric_is_ortho Submodule.symmetric_isOrtho
theorem IsOrtho.inner_eq {U V : Submodule ๐ E} (h : U โ V) {u v : E} (hu : u โ U) (hv : v โ V) :
โชu, vโซ = 0 :=
h.symm hv _ hu
#align submodule.is_ortho.inner_eq Submodule.IsOrtho.inner_eq
theorem isOrtho_iff_inner_eq {U V : Submodule ๐ E} : U โ V โ โ u โ U, โ v โ V, โชu, vโซ = 0 :=
forallโ_congr fun _u _hu _v _hv => inner_eq_zero_symm
#align submodule.is_ortho_iff_inner_eq Submodule.isOrtho_iff_inner_eq
/- TODO: generalize `Submodule.mapโ` to semilinear maps, so that we can state
`U โ V โ Submodule.mapโ (innerโโ ๐) U V โค โฅ`. -/
@[simp]
theorem isOrtho_bot_left {V : Submodule ๐ E} : โฅ โ V :=
bot_le
#align submodule.is_ortho_bot_left Submodule.isOrtho_bot_left
@[simp]
theorem isOrtho_bot_right {U : Submodule ๐ E} : U โ โฅ :=
isOrtho_bot_left.symm
#align submodule.is_ortho_bot_right Submodule.isOrtho_bot_right
theorem IsOrtho.mono_left {Uโ Uโ V : Submodule ๐ E} (hU : Uโ โค Uโ) (h : Uโ โ V) : Uโ โ V :=
hU.trans h
#align submodule.is_ortho.mono_left Submodule.IsOrtho.mono_left
theorem IsOrtho.mono_right {U Vโ Vโ : Submodule ๐ E} (hV : Vโ โค Vโ) (h : U โ Vโ) : U โ Vโ :=
(h.symm.mono_left hV).symm
#align submodule.is_ortho.mono_right Submodule.IsOrtho.mono_right
theorem IsOrtho.mono {Uโ Vโ Uโ Vโ : Submodule ๐ E} (hU : Uโ โค Uโ) (hV : Vโ โค Vโ) (h : Uโ โ Vโ) :
Uโ โ Vโ :=
(h.mono_right hV).mono_left hU
#align submodule.is_ortho.mono Submodule.IsOrtho.mono
@[simp]
theorem isOrtho_self {U : Submodule ๐ E} : U โ U โ U = โฅ :=
โจfun h => eq_bot_iff.mpr fun x hx => inner_self_eq_zero.mp (h hx x hx), fun h =>
h.symm โธ isOrtho_bot_leftโฉ
#align submodule.is_ortho_self Submodule.isOrtho_self
@[simp]
theorem isOrtho_orthogonal_right (U : Submodule ๐ E) : U โ Uแฎ :=
le_orthogonal_orthogonal _
#align submodule.is_ortho_orthogonal_right Submodule.isOrtho_orthogonal_right
@[simp]
theorem isOrtho_orthogonal_left (U : Submodule ๐ E) : Uแฎ โ U :=
(isOrtho_orthogonal_right U).symm
#align submodule.is_ortho_orthogonal_left Submodule.isOrtho_orthogonal_left
theorem IsOrtho.le {U V : Submodule ๐ E} (h : U โ V) : U โค Vแฎ :=
h
#align submodule.is_ortho.le Submodule.IsOrtho.le
theorem IsOrtho.ge {U V : Submodule ๐ E} (h : U โ V) : V โค Uแฎ :=
h.symm
#align submodule.is_ortho.ge Submodule.IsOrtho.ge
@[simp]
theorem isOrtho_top_right {U : Submodule ๐ E} : U โ โค โ U = โฅ :=
โจfun h => eq_bot_iff.mpr fun _x hx => inner_self_eq_zero.mp (h hx _ mem_top), fun h =>
h.symm โธ isOrtho_bot_leftโฉ
#align submodule.is_ortho_top_right Submodule.isOrtho_top_right
@[simp]
theorem isOrtho_top_left {V : Submodule ๐ E} : โค โ V โ V = โฅ :=
isOrtho_comm.trans isOrtho_top_right
#align submodule.is_ortho_top_left Submodule.isOrtho_top_left
/-- Orthogonal submodules are disjoint. -/
theorem IsOrtho.disjoint {U V : Submodule ๐ E} (h : U โ V) : Disjoint U V :=
(Submodule.orthogonal_disjoint _).mono_right h.symm
#align submodule.is_ortho.disjoint Submodule.IsOrtho.disjoint
@[simp]
theorem isOrtho_sup_left {Uโ Uโ V : Submodule ๐ E} : Uโ โ Uโ โ V โ Uโ โ V โง Uโ โ V :=
sup_le_iff
#align submodule.is_ortho_sup_left Submodule.isOrtho_sup_left
@[simp]
theorem isOrtho_sup_right {U Vโ Vโ : Submodule ๐ E} : U โ Vโ โ Vโ โ U โ Vโ โง U โ Vโ :=
isOrtho_comm.trans <| isOrtho_sup_left.trans <| isOrtho_comm.and isOrtho_comm
#align submodule.is_ortho_sup_right Submodule.isOrtho_sup_right
@[simp]
theorem isOrtho_sSup_left {U : Set (Submodule ๐ E)} {V : Submodule ๐ E} :
sSup U โ V โ โ Uแตข โ U, Uแตข โ V :=
sSup_le_iff
#align submodule.is_ortho_Sup_left Submodule.isOrtho_sSup_left
@[simp]
theorem isOrtho_sSup_right {U : Submodule ๐ E} {V : Set (Submodule ๐ E)} :
U โ sSup V โ โ Vแตข โ V, U โ Vแตข :=
isOrtho_comm.trans <| isOrtho_sSup_left.trans <| by simp_rw [isOrtho_comm]
#align submodule.is_ortho_Sup_right Submodule.isOrtho_sSup_right
@[simp]
theorem isOrtho_iSup_left {ฮน : Sort*} {U : ฮน โ Submodule ๐ E} {V : Submodule ๐ E} :
iSup U โ V โ โ i, U i โ V :=
iSup_le_iff
#align submodule.is_ortho_supr_left Submodule.isOrtho_iSup_left
@[simp]
theorem isOrtho_iSup_right {ฮน : Sort*} {U : Submodule ๐ E} {V : ฮน โ Submodule ๐ E} :
U โ iSup V โ โ i, U โ V i :=
isOrtho_comm.trans <| isOrtho_iSup_left.trans <| by simp_rw [isOrtho_comm]
#align submodule.is_ortho_supr_right Submodule.isOrtho_iSup_right
@[simp]
theorem isOrtho_span {s t : Set E} :
span ๐ s โ span ๐ t โ โ โฆuโฆ, u โ s โ โ โฆvโฆ, v โ t โ โชu, vโซ = 0 := by
simp_rw [span_eq_iSup_of_singleton_spans s, span_eq_iSup_of_singleton_spans t, isOrtho_iSup_left,
isOrtho_iSup_right, isOrtho_iff_le, span_le, Set.subset_def, SetLike.mem_coe,
mem_orthogonal_singleton_iff_inner_left, Set.mem_singleton_iff, forall_eq]
#align submodule.is_ortho_span Submodule.isOrtho_span
theorem IsOrtho.map (f : E โโแตข[๐] F) {U V : Submodule ๐ E} (h : U โ V) : U.map f โ V.map f := by
rw [isOrtho_iff_inner_eq] at *
simp_rw [mem_map, forall_exists_index, and_imp, forall_apply_eq_imp_iffโ,
LinearIsometry.inner_map_map]
exact h
#align submodule.is_ortho.map Submodule.IsOrtho.map
theorem IsOrtho.comap (f : E โโแตข[๐] F) {U V : Submodule ๐ F} (h : U โ V) :
U.comap f โ V.comap f := by
rw [isOrtho_iff_inner_eq] at *
simp_rw [mem_comap, โ f.inner_map_map]
intro u hu v hv
exact h _ hu _ hv
#align submodule.is_ortho.comap Submodule.IsOrtho.comap
@[simp]
theorem IsOrtho.map_iff (f : E โโแตข[๐] F) {U V : Submodule ๐ E} : U.map f โ V.map f โ U โ V :=
โจfun h => by
have hf : โ p : Submodule ๐ E, (p.map f).comap f.toLinearIsometry = p :=
comap_map_eq_of_injective f.injective
simpa only [hf] using h.comap f.toLinearIsometry, IsOrtho.map f.toLinearIsometryโฉ
#align submodule.is_ortho.map_iff Submodule.IsOrtho.map_iff
@[simp]
theorem IsOrtho.comap_iff (f : E โโแตข[๐] F) {U V : Submodule ๐ F} : U.comap f โ V.comap f โ U โ V :=
โจfun h => by
|
have hf : โ p : Submodule ๐ F, (p.comap f).map f.toLinearIsometry = p :=
map_comap_eq_of_surjective f.surjective
|
@[simp]
theorem IsOrtho.comap_iff (f : E โโแตข[๐] F) {U V : Submodule ๐ F} : U.comap f โ V.comap f โ U โ V :=
โจfun h => by
|
Mathlib.Analysis.InnerProductSpace.Orthogonal.404_0.QXx0GYqLoAbtfq1
|
@[simp]
theorem IsOrtho.comap_iff (f : E โโแตข[๐] F) {U V : Submodule ๐ F} : U.comap f โ V.comap f โ U โ V
|
Mathlib_Analysis_InnerProductSpace_Orthogonal
|
๐ : Type u_1
E : Type u_2
F : Type u_3
instโโด : IsROrC ๐
instโยณ : NormedAddCommGroup E
instโยฒ : InnerProductSpace ๐ E
instโยน : NormedAddCommGroup F
instโ : InnerProductSpace ๐ F
f : E โโแตข[๐] F
U V : Submodule ๐ F
h : Submodule.comap f U โ Submodule.comap f V
hf : โ (p : Submodule ๐ F), Submodule.map (LinearIsometryEquiv.toLinearIsometry f) (Submodule.comap f p) = p
โข U โ V
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sรฉbastien Gouรซzel, Frรฉdรฉric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
/-!
# Orthogonal complements of submodules
In this file, the `orthogonal` complement of a submodule `K` is defined, and basic API established.
Some of the more subtle results about the orthogonal complement are delayed to
`Analysis.InnerProductSpace.Projection`.
See also `BilinForm.orthogonal` for orthogonality with respect to a general bilinear form.
## Notation
The orthogonal complement of a submodule `K` is denoted by `Kแฎ`.
The proposition that two submodules are orthogonal, `Submodule.IsOrtho`, is denoted by `U โ V`.
Note this is not the same unicode symbol as `โฅ` (`Bot`).
-/
variable {๐ E F : Type*} [IsROrC ๐]
variable [NormedAddCommGroup E] [InnerProductSpace ๐ E]
variable [NormedAddCommGroup F] [InnerProductSpace ๐ F]
local notation "โช" x ", " y "โซ" => @inner ๐ _ _ x y
namespace Submodule
variable (K : Submodule ๐ E)
/-- The subspace of vectors orthogonal to a given subspace. -/
def orthogonal : Submodule ๐ E where
carrier := { v | โ u โ K, โชu, vโซ = 0 }
zero_mem' _ _ := inner_zero_right _
add_mem' hx hy u hu := by rw [inner_add_right, hx u hu, hy u hu, add_zero]
smul_mem' c x hx u hu := by rw [inner_smul_right, hx u hu, mul_zero]
#align submodule.orthogonal Submodule.orthogonal
@[inherit_doc]
notation:1200 K "แฎ" => orthogonal K
/-- When a vector is in `Kแฎ`. -/
theorem mem_orthogonal (v : E) : v โ Kแฎ โ โ u โ K, โชu, vโซ = 0 :=
Iff.rfl
#align submodule.mem_orthogonal Submodule.mem_orthogonal
/-- When a vector is in `Kแฎ`, with the inner product the
other way round. -/
theorem mem_orthogonal' (v : E) : v โ Kแฎ โ โ u โ K, โชv, uโซ = 0 := by
simp_rw [mem_orthogonal, inner_eq_zero_symm]
#align submodule.mem_orthogonal' Submodule.mem_orthogonal'
variable {K}
/-- A vector in `K` is orthogonal to one in `Kแฎ`. -/
theorem inner_right_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชu, vโซ = 0 :=
(K.mem_orthogonal v).1 hv u hu
#align submodule.inner_right_of_mem_orthogonal Submodule.inner_right_of_mem_orthogonal
/-- A vector in `Kแฎ` is orthogonal to one in `K`. -/
theorem inner_left_of_mem_orthogonal {u v : E} (hu : u โ K) (hv : v โ Kแฎ) : โชv, uโซ = 0 := by
rw [inner_eq_zero_symm]; exact inner_right_of_mem_orthogonal hu hv
#align submodule.inner_left_of_mem_orthogonal Submodule.inner_left_of_mem_orthogonal
/-- A vector is in `(๐ โ u)แฎ` iff it is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_right {u v : E} : v โ (๐ โ u)แฎ โ โชu, vโซ = 0 := by
refine' โจinner_right_of_mem_orthogonal (mem_span_singleton_self u), _โฉ
intro hv w hw
rw [mem_span_singleton] at hw
obtain โจc, rflโฉ := hw
simp [inner_smul_left, hv]
#align submodule.mem_orthogonal_singleton_iff_inner_right Submodule.mem_orthogonal_singleton_iff_inner_right
/-- A vector in `(๐ โ u)แฎ` is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_left {u v : E} : v โ (๐ โ u)แฎ โ โชv, uโซ = 0 := by
rw [mem_orthogonal_singleton_iff_inner_right, inner_eq_zero_symm]
#align submodule.mem_orthogonal_singleton_iff_inner_left Submodule.mem_orthogonal_singleton_iff_inner_left
theorem sub_mem_orthogonal_of_inner_left {x y : E} (h : โ v : K, โชx, vโซ = โชy, vโซ) : x - y โ Kแฎ := by
rw [mem_orthogonal']
intro u hu
rw [inner_sub_left, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_left Submodule.sub_mem_orthogonal_of_inner_left
theorem sub_mem_orthogonal_of_inner_right {x y : E} (h : โ v : K, โช(v : E), xโซ = โช(v : E), yโซ) :
x - y โ Kแฎ := by
intro u hu
rw [inner_sub_right, sub_eq_zero]
exact h โจu, huโฉ
#align submodule.sub_mem_orthogonal_of_inner_right Submodule.sub_mem_orthogonal_of_inner_right
variable (K)
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem inf_orthogonal_eq_bot : K โ Kแฎ = โฅ := by
rw [eq_bot_iff]
intro x
rw [mem_inf]
exact fun โจhx, hoโฉ => inner_self_eq_zero.1 (ho x hx)
#align submodule.inf_orthogonal_eq_bot Submodule.inf_orthogonal_eq_bot
/-- `K` and `Kแฎ` have trivial intersection. -/
theorem orthogonal_disjoint : Disjoint K Kแฎ := by simp [disjoint_iff, K.inf_orthogonal_eq_bot]
#align submodule.orthogonal_disjoint Submodule.orthogonal_disjoint
/-- `Kแฎ` can be characterized as the intersection of the kernels of the operations of
inner product with each of the elements of `K`. -/
theorem orthogonal_eq_inter : Kแฎ = โจ
v : K, LinearMap.ker (innerSL ๐ (v : E)) := by
apply le_antisymm
ยท rw [le_iInf_iff]
rintro โจv, hvโฉ w hw
simpa using hw _ hv
ยท intro v hv w hw
simp only [mem_iInf] at hv
exact hv โจw, hwโฉ
#align submodule.orthogonal_eq_inter Submodule.orthogonal_eq_inter
/-- The orthogonal complement of any submodule `K` is closed. -/
theorem isClosed_orthogonal : IsClosed (Kแฎ : Set E) := by
rw [orthogonal_eq_inter K]
have := fun v : K => ContinuousLinearMap.isClosed_ker (innerSL ๐ (v : E))
convert isClosed_iInter this
simp only [iInf_coe]
#align submodule.is_closed_orthogonal Submodule.isClosed_orthogonal
/-- In a complete space, the orthogonal complement of any submodule `K` is complete. -/
instance instOrthogonalCompleteSpace [CompleteSpace E] : CompleteSpace Kแฎ :=
K.isClosed_orthogonal.completeSpace_coe
variable (๐ E)
/-- `orthogonal` gives a `GaloisConnection` between
`Submodule ๐ E` and its `OrderDual`. -/
theorem orthogonal_gc :
@GaloisConnection (Submodule ๐ E) (Submodule ๐ E)แตแต _ _ orthogonal orthogonal := fun _Kโ _Kโ =>
โจfun h _v hv _u hu => inner_left_of_mem_orthogonal hv (h hu), fun h _v hv _u hu =>
inner_left_of_mem_orthogonal hv (h hu)โฉ
#align submodule.orthogonal_gc Submodule.orthogonal_gc
variable {๐ E}
/-- `orthogonal` reverses the `โค` ordering of two
subspaces. -/
theorem orthogonal_le {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎ โค Kโแฎ :=
(orthogonal_gc ๐ E).monotone_l h
#align submodule.orthogonal_le Submodule.orthogonal_le
/-- `orthogonal.orthogonal` preserves the `โค` ordering of two
subspaces. -/
theorem orthogonal_orthogonal_monotone {Kโ Kโ : Submodule ๐ E} (h : Kโ โค Kโ) : Kโแฎแฎ โค Kโแฎแฎ :=
orthogonal_le (orthogonal_le h)
#align submodule.orthogonal_orthogonal_monotone Submodule.orthogonal_orthogonal_monotone
/-- `K` is contained in `Kแฎแฎ`. -/
theorem le_orthogonal_orthogonal : K โค Kแฎแฎ :=
(orthogonal_gc ๐ E).le_u_l _
#align submodule.le_orthogonal_orthogonal Submodule.le_orthogonal_orthogonal
/-- The inf of two orthogonal subspaces equals the subspace orthogonal
to the sup. -/
theorem inf_orthogonal (Kโ Kโ : Submodule ๐ E) : Kโแฎ โ Kโแฎ = (Kโ โ Kโ)แฎ :=
(orthogonal_gc ๐ E).l_sup.symm
#align submodule.inf_orthogonal Submodule.inf_orthogonal
/-- The inf of an indexed family of orthogonal subspaces equals the
subspace orthogonal to the sup. -/
theorem iInf_orthogonal {ฮน : Type*} (K : ฮน โ Submodule ๐ E) : โจ
i, (K i)แฎ = (iSup K)แฎ :=
(orthogonal_gc ๐ E).l_iSup.symm
#align submodule.infi_orthogonal Submodule.iInf_orthogonal
/-- The inf of a set of orthogonal subspaces equals the subspace orthogonal to the sup. -/
theorem sInf_orthogonal (s : Set <| Submodule ๐ E) : โจ
K โ s, Kแฎ = (sSup s)แฎ :=
(orthogonal_gc ๐ E).l_sSup.symm
#align submodule.Inf_orthogonal Submodule.sInf_orthogonal
@[simp]
theorem top_orthogonal_eq_bot : (โค : Submodule ๐ E)แฎ = โฅ := by
ext x
rw [mem_bot, mem_orthogonal]
exact
โจfun h => inner_self_eq_zero.mp (h x mem_top), by
rintro rfl
simpโฉ
#align submodule.top_orthogonal_eq_bot Submodule.top_orthogonal_eq_bot
@[simp]
theorem bot_orthogonal_eq_top : (โฅ : Submodule ๐ E)แฎ = โค := by
rw [โ top_orthogonal_eq_bot, eq_top_iff]
exact le_orthogonal_orthogonal โค
#align submodule.bot_orthogonal_eq_top Submodule.bot_orthogonal_eq_top
@[simp]
theorem orthogonal_eq_top_iff : Kแฎ = โค โ K = โฅ := by
refine'
โจ_, by
rintro rfl
exact bot_orthogonal_eq_topโฉ
intro h
have : K โ Kแฎ = โฅ := K.orthogonal_disjoint.eq_bot
rwa [h, inf_comm, top_inf_eq] at this
#align submodule.orthogonal_eq_top_iff Submodule.orthogonal_eq_top_iff
theorem orthogonalFamily_self :
OrthogonalFamily ๐ (fun b => โฅ(cond b K Kแฎ)) fun b => (cond b K Kแฎ).subtypeโแตข
| true, true => absurd rfl
| true, false => fun _ x y => inner_right_of_mem_orthogonal x.prop y.prop
| false, true => fun _ x y => inner_left_of_mem_orthogonal y.prop x.prop
| false, false => absurd rfl
#align submodule.orthogonal_family_self Submodule.orthogonalFamily_self
end Submodule
@[simp]
theorem bilinFormOfRealInner_orthogonal {E} [NormedAddCommGroup E] [InnerProductSpace โ E]
(K : Submodule โ E) : bilinFormOfRealInner.orthogonal K = Kแฎ :=
rfl
#align bilin_form_of_real_inner_orthogonal bilinFormOfRealInner_orthogonal
/-!
### Orthogonality of submodules
In this section we define `Submodule.IsOrtho U V`, with notation `U โ V`.
The API roughly matches that of `Disjoint`.
-/
namespace Submodule
/-- The proposition that two submodules are orthogonal. Has notation `U โ V`. -/
def IsOrtho (U V : Submodule ๐ E) : Prop :=
U โค Vแฎ
#align submodule.is_ortho Submodule.IsOrtho
@[inherit_doc]
infixl:50 " โ " => Submodule.IsOrtho
theorem isOrtho_iff_le {U V : Submodule ๐ E} : U โ V โ U โค Vแฎ :=
Iff.rfl
#align submodule.is_ortho_iff_le Submodule.isOrtho_iff_le
@[symm]
theorem IsOrtho.symm {U V : Submodule ๐ E} (h : U โ V) : V โ U :=
(le_orthogonal_orthogonal _).trans (orthogonal_le h)
#align submodule.is_ortho.symm Submodule.IsOrtho.symm
theorem isOrtho_comm {U V : Submodule ๐ E} : U โ V โ V โ U :=
โจIsOrtho.symm, IsOrtho.symmโฉ
#align submodule.is_ortho_comm Submodule.isOrtho_comm
theorem symmetric_isOrtho : Symmetric (IsOrtho : Submodule ๐ E โ Submodule ๐ E โ Prop) := fun _ _ =>
IsOrtho.symm
#align submodule.symmetric_is_ortho Submodule.symmetric_isOrtho
theorem IsOrtho.inner_eq {U V : Submodule ๐ E} (h : U โ V) {u v : E} (hu : u โ U) (hv : v โ V) :
โชu, vโซ = 0 :=
h.symm hv _ hu
#align submodule.is_ortho.inner_eq Submodule.IsOrtho.inner_eq
theorem isOrtho_iff_inner_eq {U V : Submodule ๐ E} : U โ V โ โ u โ U, โ v โ V, โชu, vโซ = 0 :=
forallโ_congr fun _u _hu _v _hv => inner_eq_zero_symm
#align submodule.is_ortho_iff_inner_eq Submodule.isOrtho_iff_inner_eq
/- TODO: generalize `Submodule.mapโ` to semilinear maps, so that we can state
`U โ V โ Submodule.mapโ (innerโโ ๐) U V โค โฅ`. -/
@[simp]
theorem isOrtho_bot_left {V : Submodule ๐ E} : โฅ โ V :=
bot_le
#align submodule.is_ortho_bot_left Submodule.isOrtho_bot_left
@[simp]
theorem isOrtho_bot_right {U : Submodule ๐ E} : U โ โฅ :=
isOrtho_bot_left.symm
#align submodule.is_ortho_bot_right Submodule.isOrtho_bot_right
theorem IsOrtho.mono_left {Uโ Uโ V : Submodule ๐ E} (hU : Uโ โค Uโ) (h : Uโ โ V) : Uโ โ V :=
hU.trans h
#align submodule.is_ortho.mono_left Submodule.IsOrtho.mono_left
theorem IsOrtho.mono_right {U Vโ Vโ : Submodule ๐ E} (hV : Vโ โค Vโ) (h : U โ Vโ) : U โ Vโ :=
(h.symm.mono_left hV).symm
#align submodule.is_ortho.mono_right Submodule.IsOrtho.mono_right
theorem IsOrtho.mono {Uโ Vโ Uโ Vโ : Submodule ๐ E} (hU : Uโ โค Uโ) (hV : Vโ โค Vโ) (h : Uโ โ Vโ) :
Uโ โ Vโ :=
(h.mono_right hV).mono_left hU
#align submodule.is_ortho.mono Submodule.IsOrtho.mono
@[simp]
theorem isOrtho_self {U : Submodule ๐ E} : U โ U โ U = โฅ :=
โจfun h => eq_bot_iff.mpr fun x hx => inner_self_eq_zero.mp (h hx x hx), fun h =>
h.symm โธ isOrtho_bot_leftโฉ
#align submodule.is_ortho_self Submodule.isOrtho_self
@[simp]
theorem isOrtho_orthogonal_right (U : Submodule ๐ E) : U โ Uแฎ :=
le_orthogonal_orthogonal _
#align submodule.is_ortho_orthogonal_right Submodule.isOrtho_orthogonal_right
@[simp]
theorem isOrtho_orthogonal_left (U : Submodule ๐ E) : Uแฎ โ U :=
(isOrtho_orthogonal_right U).symm
#align submodule.is_ortho_orthogonal_left Submodule.isOrtho_orthogonal_left
theorem IsOrtho.le {U V : Submodule ๐ E} (h : U โ V) : U โค Vแฎ :=
h
#align submodule.is_ortho.le Submodule.IsOrtho.le
theorem IsOrtho.ge {U V : Submodule ๐ E} (h : U โ V) : V โค Uแฎ :=
h.symm
#align submodule.is_ortho.ge Submodule.IsOrtho.ge
@[simp]
theorem isOrtho_top_right {U : Submodule ๐ E} : U โ โค โ U = โฅ :=
โจfun h => eq_bot_iff.mpr fun _x hx => inner_self_eq_zero.mp (h hx _ mem_top), fun h =>
h.symm โธ isOrtho_bot_leftโฉ
#align submodule.is_ortho_top_right Submodule.isOrtho_top_right
@[simp]
theorem isOrtho_top_left {V : Submodule ๐ E} : โค โ V โ V = โฅ :=
isOrtho_comm.trans isOrtho_top_right
#align submodule.is_ortho_top_left Submodule.isOrtho_top_left
/-- Orthogonal submodules are disjoint. -/
theorem IsOrtho.disjoint {U V : Submodule ๐ E} (h : U โ V) : Disjoint U V :=
(Submodule.orthogonal_disjoint _).mono_right h.symm
#align submodule.is_ortho.disjoint Submodule.IsOrtho.disjoint
@[simp]
theorem isOrtho_sup_left {Uโ Uโ V : Submodule ๐ E} : Uโ โ Uโ โ V โ Uโ โ V โง Uโ โ V :=
sup_le_iff
#align submodule.is_ortho_sup_left Submodule.isOrtho_sup_left
@[simp]
theorem isOrtho_sup_right {U Vโ Vโ : Submodule ๐ E} : U โ Vโ โ Vโ โ U โ Vโ โง U โ Vโ :=
isOrtho_comm.trans <| isOrtho_sup_left.trans <| isOrtho_comm.and isOrtho_comm
#align submodule.is_ortho_sup_right Submodule.isOrtho_sup_right
@[simp]
theorem isOrtho_sSup_left {U : Set (Submodule ๐ E)} {V : Submodule ๐ E} :
sSup U โ V โ โ Uแตข โ U, Uแตข โ V :=
sSup_le_iff
#align submodule.is_ortho_Sup_left Submodule.isOrtho_sSup_left
@[simp]
theorem isOrtho_sSup_right {U : Submodule ๐ E} {V : Set (Submodule ๐ E)} :
U โ sSup V โ โ Vแตข โ V, U โ Vแตข :=
isOrtho_comm.trans <| isOrtho_sSup_left.trans <| by simp_rw [isOrtho_comm]
#align submodule.is_ortho_Sup_right Submodule.isOrtho_sSup_right
@[simp]
theorem isOrtho_iSup_left {ฮน : Sort*} {U : ฮน โ Submodule ๐ E} {V : Submodule ๐ E} :
iSup U โ V โ โ i, U i โ V :=
iSup_le_iff
#align submodule.is_ortho_supr_left Submodule.isOrtho_iSup_left
@[simp]
theorem isOrtho_iSup_right {ฮน : Sort*} {U : Submodule ๐ E} {V : ฮน โ Submodule ๐ E} :
U โ iSup V โ โ i, U โ V i :=
isOrtho_comm.trans <| isOrtho_iSup_left.trans <| by simp_rw [isOrtho_comm]
#align submodule.is_ortho_supr_right Submodule.isOrtho_iSup_right
@[simp]
theorem isOrtho_span {s t : Set E} :
span ๐ s โ span ๐ t โ โ โฆuโฆ, u โ s โ โ โฆvโฆ, v โ t โ โชu, vโซ = 0 := by
simp_rw [span_eq_iSup_of_singleton_spans s, span_eq_iSup_of_singleton_spans t, isOrtho_iSup_left,
isOrtho_iSup_right, isOrtho_iff_le, span_le, Set.subset_def, SetLike.mem_coe,
mem_orthogonal_singleton_iff_inner_left, Set.mem_singleton_iff, forall_eq]
#align submodule.is_ortho_span Submodule.isOrtho_span
theorem IsOrtho.map (f : E โโแตข[๐] F) {U V : Submodule ๐ E} (h : U โ V) : U.map f โ V.map f := by
rw [isOrtho_iff_inner_eq] at *
simp_rw [mem_map, forall_exists_index, and_imp, forall_apply_eq_imp_iffโ,
LinearIsometry.inner_map_map]
exact h
#align submodule.is_ortho.map Submodule.IsOrtho.map
theorem IsOrtho.comap (f : E โโแตข[๐] F) {U V : Submodule ๐ F} (h : U โ V) :
U.comap f โ V.comap f := by
rw [isOrtho_iff_inner_eq] at *
simp_rw [mem_comap, โ f.inner_map_map]
intro u hu v hv
exact h _ hu _ hv
#align submodule.is_ortho.comap Submodule.IsOrtho.comap
@[simp]
theorem IsOrtho.map_iff (f : E โโแตข[๐] F) {U V : Submodule ๐ E} : U.map f โ V.map f โ U โ V :=
โจfun h => by
have hf : โ p : Submodule ๐ E, (p.map f).comap f.toLinearIsometry = p :=
comap_map_eq_of_injective f.injective
simpa only [hf] using h.comap f.toLinearIsometry, IsOrtho.map f.toLinearIsometryโฉ
#align submodule.is_ortho.map_iff Submodule.IsOrtho.map_iff
@[simp]
theorem IsOrtho.comap_iff (f : E โโแตข[๐] F) {U V : Submodule ๐ F} : U.comap f โ V.comap f โ U โ V :=
โจfun h => by
have hf : โ p : Submodule ๐ F, (p.comap f).map f.toLinearIsometry = p :=
map_comap_eq_of_surjective f.surjective
|
simpa only [hf] using h.map f.toLinearIsometry
|
@[simp]
theorem IsOrtho.comap_iff (f : E โโแตข[๐] F) {U V : Submodule ๐ F} : U.comap f โ V.comap f โ U โ V :=
โจfun h => by
have hf : โ p : Submodule ๐ F, (p.comap f).map f.toLinearIsometry = p :=
map_comap_eq_of_surjective f.surjective
|
Mathlib.Analysis.InnerProductSpace.Orthogonal.404_0.QXx0GYqLoAbtfq1
|
@[simp]
theorem IsOrtho.comap_iff (f : E โโแตข[๐] F) {U V : Submodule ๐ F} : U.comap f โ V.comap f โ U โ V
|
Mathlib_Analysis_InnerProductSpace_Orthogonal
|
ฮฑ : Type u_1
instโ : HeytingAlgebra ฮฑ
a b : ฮฑ
โข IsRegular โฅ
|
/-
Copyright (c) 2022 Yaรซl Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaรซl Dillies
-/
import Mathlib.Order.GaloisConnection
#align_import order.heyting.regular from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
/-!
# Heyting regular elements
This file defines Heyting regular elements, elements of a Heyting algebra that are their own double
complement, and proves that they form a boolean algebra.
From a logic standpoint, this means that we can perform classical logic within intuitionistic logic
by simply double-negating all propositions. This is practical for synthetic computability theory.
## Main declarations
* `IsRegular`: `a` is Heyting-regular if `aแถแถ = a`.
* `Regular`: The subtype of Heyting-regular elements.
* `Regular.BooleanAlgebra`: Heyting-regular elements form a boolean algebra.
## References
* [Francis Borceux, *Handbook of Categorical Algebra III*][borceux-vol3]
-/
open Function
variable {ฮฑ : Type*}
namespace Heyting
section HasCompl
variable [HasCompl ฮฑ] {a : ฮฑ}
/-- An element of a Heyting algebra is regular if its double complement is itself. -/
def IsRegular (a : ฮฑ) : Prop :=
aแถแถ = a
#align heyting.is_regular Heyting.IsRegular
protected theorem IsRegular.eq : IsRegular a โ aแถแถ = a :=
id
#align heyting.is_regular.eq Heyting.IsRegular.eq
instance IsRegular.decidablePred [DecidableEq ฮฑ] : @DecidablePred ฮฑ IsRegular := fun _ =>
โนDecidableEq ฮฑโบ _ _
#align heyting.is_regular.decidable_pred Heyting.IsRegular.decidablePred
end HasCompl
section HeytingAlgebra
variable [HeytingAlgebra ฮฑ] {a b : ฮฑ}
theorem isRegular_bot : IsRegular (โฅ : ฮฑ) := by
|
rw [IsRegular, compl_bot, compl_top]
|
theorem isRegular_bot : IsRegular (โฅ : ฮฑ) := by
|
Mathlib.Order.Heyting.Regular.60_0.HFWfWmvQ8w4gAZO
|
theorem isRegular_bot : IsRegular (โฅ : ฮฑ)
|
Mathlib_Order_Heyting_Regular
|
ฮฑ : Type u_1
instโ : HeytingAlgebra ฮฑ
a b : ฮฑ
โข IsRegular โค
|
/-
Copyright (c) 2022 Yaรซl Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaรซl Dillies
-/
import Mathlib.Order.GaloisConnection
#align_import order.heyting.regular from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
/-!
# Heyting regular elements
This file defines Heyting regular elements, elements of a Heyting algebra that are their own double
complement, and proves that they form a boolean algebra.
From a logic standpoint, this means that we can perform classical logic within intuitionistic logic
by simply double-negating all propositions. This is practical for synthetic computability theory.
## Main declarations
* `IsRegular`: `a` is Heyting-regular if `aแถแถ = a`.
* `Regular`: The subtype of Heyting-regular elements.
* `Regular.BooleanAlgebra`: Heyting-regular elements form a boolean algebra.
## References
* [Francis Borceux, *Handbook of Categorical Algebra III*][borceux-vol3]
-/
open Function
variable {ฮฑ : Type*}
namespace Heyting
section HasCompl
variable [HasCompl ฮฑ] {a : ฮฑ}
/-- An element of a Heyting algebra is regular if its double complement is itself. -/
def IsRegular (a : ฮฑ) : Prop :=
aแถแถ = a
#align heyting.is_regular Heyting.IsRegular
protected theorem IsRegular.eq : IsRegular a โ aแถแถ = a :=
id
#align heyting.is_regular.eq Heyting.IsRegular.eq
instance IsRegular.decidablePred [DecidableEq ฮฑ] : @DecidablePred ฮฑ IsRegular := fun _ =>
โนDecidableEq ฮฑโบ _ _
#align heyting.is_regular.decidable_pred Heyting.IsRegular.decidablePred
end HasCompl
section HeytingAlgebra
variable [HeytingAlgebra ฮฑ] {a b : ฮฑ}
theorem isRegular_bot : IsRegular (โฅ : ฮฑ) := by rw [IsRegular, compl_bot, compl_top]
#align heyting.is_regular_bot Heyting.isRegular_bot
theorem isRegular_top : IsRegular (โค : ฮฑ) := by
|
rw [IsRegular, compl_top, compl_bot]
|
theorem isRegular_top : IsRegular (โค : ฮฑ) := by
|
Mathlib.Order.Heyting.Regular.63_0.HFWfWmvQ8w4gAZO
|
theorem isRegular_top : IsRegular (โค : ฮฑ)
|
Mathlib_Order_Heyting_Regular
|
ฮฑ : Type u_1
instโ : HeytingAlgebra ฮฑ
a b : ฮฑ
ha : IsRegular a
hb : IsRegular b
โข IsRegular (a โ b)
|
/-
Copyright (c) 2022 Yaรซl Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaรซl Dillies
-/
import Mathlib.Order.GaloisConnection
#align_import order.heyting.regular from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
/-!
# Heyting regular elements
This file defines Heyting regular elements, elements of a Heyting algebra that are their own double
complement, and proves that they form a boolean algebra.
From a logic standpoint, this means that we can perform classical logic within intuitionistic logic
by simply double-negating all propositions. This is practical for synthetic computability theory.
## Main declarations
* `IsRegular`: `a` is Heyting-regular if `aแถแถ = a`.
* `Regular`: The subtype of Heyting-regular elements.
* `Regular.BooleanAlgebra`: Heyting-regular elements form a boolean algebra.
## References
* [Francis Borceux, *Handbook of Categorical Algebra III*][borceux-vol3]
-/
open Function
variable {ฮฑ : Type*}
namespace Heyting
section HasCompl
variable [HasCompl ฮฑ] {a : ฮฑ}
/-- An element of a Heyting algebra is regular if its double complement is itself. -/
def IsRegular (a : ฮฑ) : Prop :=
aแถแถ = a
#align heyting.is_regular Heyting.IsRegular
protected theorem IsRegular.eq : IsRegular a โ aแถแถ = a :=
id
#align heyting.is_regular.eq Heyting.IsRegular.eq
instance IsRegular.decidablePred [DecidableEq ฮฑ] : @DecidablePred ฮฑ IsRegular := fun _ =>
โนDecidableEq ฮฑโบ _ _
#align heyting.is_regular.decidable_pred Heyting.IsRegular.decidablePred
end HasCompl
section HeytingAlgebra
variable [HeytingAlgebra ฮฑ] {a b : ฮฑ}
theorem isRegular_bot : IsRegular (โฅ : ฮฑ) := by rw [IsRegular, compl_bot, compl_top]
#align heyting.is_regular_bot Heyting.isRegular_bot
theorem isRegular_top : IsRegular (โค : ฮฑ) := by rw [IsRegular, compl_top, compl_bot]
#align heyting.is_regular_top Heyting.isRegular_top
theorem IsRegular.inf (ha : IsRegular a) (hb : IsRegular b) : IsRegular (a โ b) := by
|
rw [IsRegular, compl_compl_inf_distrib, ha.eq, hb.eq]
|
theorem IsRegular.inf (ha : IsRegular a) (hb : IsRegular b) : IsRegular (a โ b) := by
|
Mathlib.Order.Heyting.Regular.66_0.HFWfWmvQ8w4gAZO
|
theorem IsRegular.inf (ha : IsRegular a) (hb : IsRegular b) : IsRegular (a โ b)
|
Mathlib_Order_Heyting_Regular
|
ฮฑ : Type u_1
instโ : HeytingAlgebra ฮฑ
a b : ฮฑ
ha : IsRegular a
hb : IsRegular b
โข IsRegular (a โจ b)
|
/-
Copyright (c) 2022 Yaรซl Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaรซl Dillies
-/
import Mathlib.Order.GaloisConnection
#align_import order.heyting.regular from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
/-!
# Heyting regular elements
This file defines Heyting regular elements, elements of a Heyting algebra that are their own double
complement, and proves that they form a boolean algebra.
From a logic standpoint, this means that we can perform classical logic within intuitionistic logic
by simply double-negating all propositions. This is practical for synthetic computability theory.
## Main declarations
* `IsRegular`: `a` is Heyting-regular if `aแถแถ = a`.
* `Regular`: The subtype of Heyting-regular elements.
* `Regular.BooleanAlgebra`: Heyting-regular elements form a boolean algebra.
## References
* [Francis Borceux, *Handbook of Categorical Algebra III*][borceux-vol3]
-/
open Function
variable {ฮฑ : Type*}
namespace Heyting
section HasCompl
variable [HasCompl ฮฑ] {a : ฮฑ}
/-- An element of a Heyting algebra is regular if its double complement is itself. -/
def IsRegular (a : ฮฑ) : Prop :=
aแถแถ = a
#align heyting.is_regular Heyting.IsRegular
protected theorem IsRegular.eq : IsRegular a โ aแถแถ = a :=
id
#align heyting.is_regular.eq Heyting.IsRegular.eq
instance IsRegular.decidablePred [DecidableEq ฮฑ] : @DecidablePred ฮฑ IsRegular := fun _ =>
โนDecidableEq ฮฑโบ _ _
#align heyting.is_regular.decidable_pred Heyting.IsRegular.decidablePred
end HasCompl
section HeytingAlgebra
variable [HeytingAlgebra ฮฑ] {a b : ฮฑ}
theorem isRegular_bot : IsRegular (โฅ : ฮฑ) := by rw [IsRegular, compl_bot, compl_top]
#align heyting.is_regular_bot Heyting.isRegular_bot
theorem isRegular_top : IsRegular (โค : ฮฑ) := by rw [IsRegular, compl_top, compl_bot]
#align heyting.is_regular_top Heyting.isRegular_top
theorem IsRegular.inf (ha : IsRegular a) (hb : IsRegular b) : IsRegular (a โ b) := by
rw [IsRegular, compl_compl_inf_distrib, ha.eq, hb.eq]
#align heyting.is_regular.inf Heyting.IsRegular.inf
theorem IsRegular.himp (ha : IsRegular a) (hb : IsRegular b) : IsRegular (a โจ b) := by
|
rw [IsRegular, compl_compl_himp_distrib, ha.eq, hb.eq]
|
theorem IsRegular.himp (ha : IsRegular a) (hb : IsRegular b) : IsRegular (a โจ b) := by
|
Mathlib.Order.Heyting.Regular.70_0.HFWfWmvQ8w4gAZO
|
theorem IsRegular.himp (ha : IsRegular a) (hb : IsRegular b) : IsRegular (a โจ b)
|
Mathlib_Order_Heyting_Regular
|
ฮฑ : Type u_1
instโ : HeytingAlgebra ฮฑ
a b : ฮฑ
ha : IsRegular a
โข Disjoint aแถ b โ b โค a
|
/-
Copyright (c) 2022 Yaรซl Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaรซl Dillies
-/
import Mathlib.Order.GaloisConnection
#align_import order.heyting.regular from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
/-!
# Heyting regular elements
This file defines Heyting regular elements, elements of a Heyting algebra that are their own double
complement, and proves that they form a boolean algebra.
From a logic standpoint, this means that we can perform classical logic within intuitionistic logic
by simply double-negating all propositions. This is practical for synthetic computability theory.
## Main declarations
* `IsRegular`: `a` is Heyting-regular if `aแถแถ = a`.
* `Regular`: The subtype of Heyting-regular elements.
* `Regular.BooleanAlgebra`: Heyting-regular elements form a boolean algebra.
## References
* [Francis Borceux, *Handbook of Categorical Algebra III*][borceux-vol3]
-/
open Function
variable {ฮฑ : Type*}
namespace Heyting
section HasCompl
variable [HasCompl ฮฑ] {a : ฮฑ}
/-- An element of a Heyting algebra is regular if its double complement is itself. -/
def IsRegular (a : ฮฑ) : Prop :=
aแถแถ = a
#align heyting.is_regular Heyting.IsRegular
protected theorem IsRegular.eq : IsRegular a โ aแถแถ = a :=
id
#align heyting.is_regular.eq Heyting.IsRegular.eq
instance IsRegular.decidablePred [DecidableEq ฮฑ] : @DecidablePred ฮฑ IsRegular := fun _ =>
โนDecidableEq ฮฑโบ _ _
#align heyting.is_regular.decidable_pred Heyting.IsRegular.decidablePred
end HasCompl
section HeytingAlgebra
variable [HeytingAlgebra ฮฑ] {a b : ฮฑ}
theorem isRegular_bot : IsRegular (โฅ : ฮฑ) := by rw [IsRegular, compl_bot, compl_top]
#align heyting.is_regular_bot Heyting.isRegular_bot
theorem isRegular_top : IsRegular (โค : ฮฑ) := by rw [IsRegular, compl_top, compl_bot]
#align heyting.is_regular_top Heyting.isRegular_top
theorem IsRegular.inf (ha : IsRegular a) (hb : IsRegular b) : IsRegular (a โ b) := by
rw [IsRegular, compl_compl_inf_distrib, ha.eq, hb.eq]
#align heyting.is_regular.inf Heyting.IsRegular.inf
theorem IsRegular.himp (ha : IsRegular a) (hb : IsRegular b) : IsRegular (a โจ b) := by
rw [IsRegular, compl_compl_himp_distrib, ha.eq, hb.eq]
#align heyting.is_regular.himp Heyting.IsRegular.himp
theorem isRegular_compl (a : ฮฑ) : IsRegular aแถ :=
compl_compl_compl _
#align heyting.is_regular_compl Heyting.isRegular_compl
protected theorem IsRegular.disjoint_compl_left_iff (ha : IsRegular a) : Disjoint aแถ b โ b โค a :=
by
|
rw [โ le_compl_iff_disjoint_left, ha.eq]
|
protected theorem IsRegular.disjoint_compl_left_iff (ha : IsRegular a) : Disjoint aแถ b โ b โค a :=
by
|
Mathlib.Order.Heyting.Regular.78_0.HFWfWmvQ8w4gAZO
|
protected theorem IsRegular.disjoint_compl_left_iff (ha : IsRegular a) : Disjoint aแถ b โ b โค a
|
Mathlib_Order_Heyting_Regular
|
ฮฑ : Type u_1
instโ : HeytingAlgebra ฮฑ
a b : ฮฑ
hb : IsRegular b
โข Disjoint a bแถ โ a โค b
|
/-
Copyright (c) 2022 Yaรซl Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaรซl Dillies
-/
import Mathlib.Order.GaloisConnection
#align_import order.heyting.regular from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
/-!
# Heyting regular elements
This file defines Heyting regular elements, elements of a Heyting algebra that are their own double
complement, and proves that they form a boolean algebra.
From a logic standpoint, this means that we can perform classical logic within intuitionistic logic
by simply double-negating all propositions. This is practical for synthetic computability theory.
## Main declarations
* `IsRegular`: `a` is Heyting-regular if `aแถแถ = a`.
* `Regular`: The subtype of Heyting-regular elements.
* `Regular.BooleanAlgebra`: Heyting-regular elements form a boolean algebra.
## References
* [Francis Borceux, *Handbook of Categorical Algebra III*][borceux-vol3]
-/
open Function
variable {ฮฑ : Type*}
namespace Heyting
section HasCompl
variable [HasCompl ฮฑ] {a : ฮฑ}
/-- An element of a Heyting algebra is regular if its double complement is itself. -/
def IsRegular (a : ฮฑ) : Prop :=
aแถแถ = a
#align heyting.is_regular Heyting.IsRegular
protected theorem IsRegular.eq : IsRegular a โ aแถแถ = a :=
id
#align heyting.is_regular.eq Heyting.IsRegular.eq
instance IsRegular.decidablePred [DecidableEq ฮฑ] : @DecidablePred ฮฑ IsRegular := fun _ =>
โนDecidableEq ฮฑโบ _ _
#align heyting.is_regular.decidable_pred Heyting.IsRegular.decidablePred
end HasCompl
section HeytingAlgebra
variable [HeytingAlgebra ฮฑ] {a b : ฮฑ}
theorem isRegular_bot : IsRegular (โฅ : ฮฑ) := by rw [IsRegular, compl_bot, compl_top]
#align heyting.is_regular_bot Heyting.isRegular_bot
theorem isRegular_top : IsRegular (โค : ฮฑ) := by rw [IsRegular, compl_top, compl_bot]
#align heyting.is_regular_top Heyting.isRegular_top
theorem IsRegular.inf (ha : IsRegular a) (hb : IsRegular b) : IsRegular (a โ b) := by
rw [IsRegular, compl_compl_inf_distrib, ha.eq, hb.eq]
#align heyting.is_regular.inf Heyting.IsRegular.inf
theorem IsRegular.himp (ha : IsRegular a) (hb : IsRegular b) : IsRegular (a โจ b) := by
rw [IsRegular, compl_compl_himp_distrib, ha.eq, hb.eq]
#align heyting.is_regular.himp Heyting.IsRegular.himp
theorem isRegular_compl (a : ฮฑ) : IsRegular aแถ :=
compl_compl_compl _
#align heyting.is_regular_compl Heyting.isRegular_compl
protected theorem IsRegular.disjoint_compl_left_iff (ha : IsRegular a) : Disjoint aแถ b โ b โค a :=
by rw [โ le_compl_iff_disjoint_left, ha.eq]
#align heyting.is_regular.disjoint_compl_left_iff Heyting.IsRegular.disjoint_compl_left_iff
protected theorem IsRegular.disjoint_compl_right_iff (hb : IsRegular b) : Disjoint a bแถ โ a โค b :=
by
|
rw [โ le_compl_iff_disjoint_right, hb.eq]
|
protected theorem IsRegular.disjoint_compl_right_iff (hb : IsRegular b) : Disjoint a bแถ โ a โค b :=
by
|
Mathlib.Order.Heyting.Regular.82_0.HFWfWmvQ8w4gAZO
|
protected theorem IsRegular.disjoint_compl_right_iff (hb : IsRegular b) : Disjoint a bแถ โ a โค b
|
Mathlib_Order_Heyting_Regular
|
ฮฑ : Type u_1
instโ : HeytingAlgebra ฮฑ
aโ b : ฮฑ
h : โ (a : ฮฑ), IsRegular (a โ aแถ)
a : ฮฑ
โข a โ aแถ = โค
|
/-
Copyright (c) 2022 Yaรซl Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaรซl Dillies
-/
import Mathlib.Order.GaloisConnection
#align_import order.heyting.regular from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
/-!
# Heyting regular elements
This file defines Heyting regular elements, elements of a Heyting algebra that are their own double
complement, and proves that they form a boolean algebra.
From a logic standpoint, this means that we can perform classical logic within intuitionistic logic
by simply double-negating all propositions. This is practical for synthetic computability theory.
## Main declarations
* `IsRegular`: `a` is Heyting-regular if `aแถแถ = a`.
* `Regular`: The subtype of Heyting-regular elements.
* `Regular.BooleanAlgebra`: Heyting-regular elements form a boolean algebra.
## References
* [Francis Borceux, *Handbook of Categorical Algebra III*][borceux-vol3]
-/
open Function
variable {ฮฑ : Type*}
namespace Heyting
section HasCompl
variable [HasCompl ฮฑ] {a : ฮฑ}
/-- An element of a Heyting algebra is regular if its double complement is itself. -/
def IsRegular (a : ฮฑ) : Prop :=
aแถแถ = a
#align heyting.is_regular Heyting.IsRegular
protected theorem IsRegular.eq : IsRegular a โ aแถแถ = a :=
id
#align heyting.is_regular.eq Heyting.IsRegular.eq
instance IsRegular.decidablePred [DecidableEq ฮฑ] : @DecidablePred ฮฑ IsRegular := fun _ =>
โนDecidableEq ฮฑโบ _ _
#align heyting.is_regular.decidable_pred Heyting.IsRegular.decidablePred
end HasCompl
section HeytingAlgebra
variable [HeytingAlgebra ฮฑ] {a b : ฮฑ}
theorem isRegular_bot : IsRegular (โฅ : ฮฑ) := by rw [IsRegular, compl_bot, compl_top]
#align heyting.is_regular_bot Heyting.isRegular_bot
theorem isRegular_top : IsRegular (โค : ฮฑ) := by rw [IsRegular, compl_top, compl_bot]
#align heyting.is_regular_top Heyting.isRegular_top
theorem IsRegular.inf (ha : IsRegular a) (hb : IsRegular b) : IsRegular (a โ b) := by
rw [IsRegular, compl_compl_inf_distrib, ha.eq, hb.eq]
#align heyting.is_regular.inf Heyting.IsRegular.inf
theorem IsRegular.himp (ha : IsRegular a) (hb : IsRegular b) : IsRegular (a โจ b) := by
rw [IsRegular, compl_compl_himp_distrib, ha.eq, hb.eq]
#align heyting.is_regular.himp Heyting.IsRegular.himp
theorem isRegular_compl (a : ฮฑ) : IsRegular aแถ :=
compl_compl_compl _
#align heyting.is_regular_compl Heyting.isRegular_compl
protected theorem IsRegular.disjoint_compl_left_iff (ha : IsRegular a) : Disjoint aแถ b โ b โค a :=
by rw [โ le_compl_iff_disjoint_left, ha.eq]
#align heyting.is_regular.disjoint_compl_left_iff Heyting.IsRegular.disjoint_compl_left_iff
protected theorem IsRegular.disjoint_compl_right_iff (hb : IsRegular b) : Disjoint a bแถ โ a โค b :=
by rw [โ le_compl_iff_disjoint_right, hb.eq]
#align heyting.is_regular.disjoint_compl_right_iff Heyting.IsRegular.disjoint_compl_right_iff
-- See note [reducible non-instances]
/-- A Heyting algebra with regular excluded middle is a boolean algebra. -/
@[reducible]
def _root_.BooleanAlgebra.ofRegular (h : โ a : ฮฑ, IsRegular (a โ aแถ)) : BooleanAlgebra ฮฑ :=
have : โ a : ฮฑ, IsCompl a aแถ := fun a =>
โจdisjoint_compl_right,
codisjoint_iff.2 <| by
|
erw [โ (h a), compl_sup, inf_compl_eq_bot, compl_bot]
|
/-- A Heyting algebra with regular excluded middle is a boolean algebra. -/
@[reducible]
def _root_.BooleanAlgebra.ofRegular (h : โ a : ฮฑ, IsRegular (a โ aแถ)) : BooleanAlgebra ฮฑ :=
have : โ a : ฮฑ, IsCompl a aแถ := fun a =>
โจdisjoint_compl_right,
codisjoint_iff.2 <| by
|
Mathlib.Order.Heyting.Regular.87_0.HFWfWmvQ8w4gAZO
|
/-- A Heyting algebra with regular excluded middle is a boolean algebra. -/
@[reducible]
def _root_.BooleanAlgebra.ofRegular (h : โ a : ฮฑ, IsRegular (a โ aแถ)) : BooleanAlgebra ฮฑ
|
Mathlib_Order_Heyting_Regular
|
ฮฑ : Type u_1
instโ : HeytingAlgebra ฮฑ
aโ bโ : ฮฑ
srcโยณ : Lattice (Regular ฮฑ) := lattice
srcโยฒ : BoundedOrder (Regular ฮฑ) := boundedOrder
srcโยน : HImp (Regular ฮฑ) := himp
srcโ : HasCompl (Regular ฮฑ) := hasCompl
a b c : Regular ฮฑ
โข โ((a โ b) โ (a โ c)) โค โ(a โ b โ c)
|
/-
Copyright (c) 2022 Yaรซl Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaรซl Dillies
-/
import Mathlib.Order.GaloisConnection
#align_import order.heyting.regular from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
/-!
# Heyting regular elements
This file defines Heyting regular elements, elements of a Heyting algebra that are their own double
complement, and proves that they form a boolean algebra.
From a logic standpoint, this means that we can perform classical logic within intuitionistic logic
by simply double-negating all propositions. This is practical for synthetic computability theory.
## Main declarations
* `IsRegular`: `a` is Heyting-regular if `aแถแถ = a`.
* `Regular`: The subtype of Heyting-regular elements.
* `Regular.BooleanAlgebra`: Heyting-regular elements form a boolean algebra.
## References
* [Francis Borceux, *Handbook of Categorical Algebra III*][borceux-vol3]
-/
open Function
variable {ฮฑ : Type*}
namespace Heyting
section HasCompl
variable [HasCompl ฮฑ] {a : ฮฑ}
/-- An element of a Heyting algebra is regular if its double complement is itself. -/
def IsRegular (a : ฮฑ) : Prop :=
aแถแถ = a
#align heyting.is_regular Heyting.IsRegular
protected theorem IsRegular.eq : IsRegular a โ aแถแถ = a :=
id
#align heyting.is_regular.eq Heyting.IsRegular.eq
instance IsRegular.decidablePred [DecidableEq ฮฑ] : @DecidablePred ฮฑ IsRegular := fun _ =>
โนDecidableEq ฮฑโบ _ _
#align heyting.is_regular.decidable_pred Heyting.IsRegular.decidablePred
end HasCompl
section HeytingAlgebra
variable [HeytingAlgebra ฮฑ] {a b : ฮฑ}
theorem isRegular_bot : IsRegular (โฅ : ฮฑ) := by rw [IsRegular, compl_bot, compl_top]
#align heyting.is_regular_bot Heyting.isRegular_bot
theorem isRegular_top : IsRegular (โค : ฮฑ) := by rw [IsRegular, compl_top, compl_bot]
#align heyting.is_regular_top Heyting.isRegular_top
theorem IsRegular.inf (ha : IsRegular a) (hb : IsRegular b) : IsRegular (a โ b) := by
rw [IsRegular, compl_compl_inf_distrib, ha.eq, hb.eq]
#align heyting.is_regular.inf Heyting.IsRegular.inf
theorem IsRegular.himp (ha : IsRegular a) (hb : IsRegular b) : IsRegular (a โจ b) := by
rw [IsRegular, compl_compl_himp_distrib, ha.eq, hb.eq]
#align heyting.is_regular.himp Heyting.IsRegular.himp
theorem isRegular_compl (a : ฮฑ) : IsRegular aแถ :=
compl_compl_compl _
#align heyting.is_regular_compl Heyting.isRegular_compl
protected theorem IsRegular.disjoint_compl_left_iff (ha : IsRegular a) : Disjoint aแถ b โ b โค a :=
by rw [โ le_compl_iff_disjoint_left, ha.eq]
#align heyting.is_regular.disjoint_compl_left_iff Heyting.IsRegular.disjoint_compl_left_iff
protected theorem IsRegular.disjoint_compl_right_iff (hb : IsRegular b) : Disjoint a bแถ โ a โค b :=
by rw [โ le_compl_iff_disjoint_right, hb.eq]
#align heyting.is_regular.disjoint_compl_right_iff Heyting.IsRegular.disjoint_compl_right_iff
-- See note [reducible non-instances]
/-- A Heyting algebra with regular excluded middle is a boolean algebra. -/
@[reducible]
def _root_.BooleanAlgebra.ofRegular (h : โ a : ฮฑ, IsRegular (a โ aแถ)) : BooleanAlgebra ฮฑ :=
have : โ a : ฮฑ, IsCompl a aแถ := fun a =>
โจdisjoint_compl_right,
codisjoint_iff.2 <| by erw [โ (h a), compl_sup, inf_compl_eq_bot, compl_bot]โฉ
{ โนHeytingAlgebra ฮฑโบ,
GeneralizedHeytingAlgebra.toDistribLattice with
himp_eq := fun a b =>
eq_of_forall_le_iff fun c => le_himp_iff.trans (this _).le_sup_right_iff_inf_left_le.symm
inf_compl_le_bot := fun a => (this _).1.le_bot
top_le_sup_compl := fun a => (this _).2.top_le }
#align boolean_algebra.of_regular BooleanAlgebra.ofRegular
variable (ฮฑ)
/-- The boolean algebra of Heyting regular elements. -/
def Regular : Type _ :=
{ a : ฮฑ // IsRegular a }
#align heyting.regular Heyting.Regular
variable {ฮฑ}
namespace Regular
--Porting note: `val` and `prop` are new
/-- The coercion `Regular ฮฑ โ ฮฑ` -/
@[coe] def val : Regular ฮฑ โ ฮฑ :=
Subtype.val
theorem prop : โ a : Regular ฮฑ, IsRegular a.val := Subtype.prop
instance : CoeOut (Regular ฮฑ) ฮฑ := โจRegular.valโฉ
theorem coe_injective : Injective ((โ) : Regular ฮฑ โ ฮฑ) :=
Subtype.coe_injective
#align heyting.regular.coe_injective Heyting.Regular.coe_injective
@[simp]
theorem coe_inj {a b : Regular ฮฑ} : (a : ฮฑ) = b โ a = b :=
Subtype.coe_inj
#align heyting.regular.coe_inj Heyting.Regular.coe_inj
instance top : Top (Regular ฮฑ) :=
โจโจโค, isRegular_topโฉโฉ
instance bot : Bot (Regular ฮฑ) :=
โจโจโฅ, isRegular_botโฉโฉ
instance inf : Inf (Regular ฮฑ) :=
โจfun a b => โจa โ b, a.2.inf b.2โฉโฉ
instance himp : HImp (Regular ฮฑ) :=
โจfun a b => โจa โจ b, a.2.himp b.2โฉโฉ
instance hasCompl : HasCompl (Regular ฮฑ) :=
โจfun a => โจaแถ, isRegular_compl _โฉโฉ
@[simp, norm_cast]
theorem coe_top : ((โค : Regular ฮฑ) : ฮฑ) = โค :=
rfl
#align heyting.regular.coe_top Heyting.Regular.coe_top
@[simp, norm_cast]
theorem coe_bot : ((โฅ : Regular ฮฑ) : ฮฑ) = โฅ :=
rfl
#align heyting.regular.coe_bot Heyting.Regular.coe_bot
@[simp, norm_cast]
theorem coe_inf (a b : Regular ฮฑ) : (โ(a โ b) : ฮฑ) = (a : ฮฑ) โ b :=
rfl
#align heyting.regular.coe_inf Heyting.Regular.coe_inf
@[simp, norm_cast]
theorem coe_himp (a b : Regular ฮฑ) : (โ(a โจ b) : ฮฑ) = (a : ฮฑ) โจ b :=
rfl
#align heyting.regular.coe_himp Heyting.Regular.coe_himp
@[simp, norm_cast]
theorem coe_compl (a : Regular ฮฑ) : (โaแถ : ฮฑ) = (a : ฮฑ)แถ :=
rfl
#align heyting.regular.coe_compl Heyting.Regular.coe_compl
instance : Inhabited (Regular ฮฑ) :=
โจโฅโฉ
instance : SemilatticeInf (Regular ฮฑ) :=
coe_injective.semilatticeInf _ coe_inf
instance boundedOrder : BoundedOrder (Regular ฮฑ) :=
BoundedOrder.lift ((โ) : Regular ฮฑ โ ฮฑ) (fun _ _ => id) coe_top coe_bot
@[simp, norm_cast]
theorem coe_le_coe {a b : Regular ฮฑ} : (a : ฮฑ) โค b โ a โค b :=
Iff.rfl
#align heyting.regular.coe_le_coe Heyting.Regular.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {a b : Regular ฮฑ} : (a : ฮฑ) < b โ a < b :=
Iff.rfl
#align heyting.regular.coe_lt_coe Heyting.Regular.coe_lt_coe
/-- **Regularization** of `a`. The smallest regular element greater than `a`. -/
def toRegular : ฮฑ โo Regular ฮฑ :=
โจfun a => โจaแถแถ, isRegular_compl _โฉ, fun _ _ h =>
coe_le_coe.1 <| compl_le_compl <| compl_le_compl hโฉ
#align heyting.regular.to_regular Heyting.Regular.toRegular
@[simp, norm_cast]
theorem coe_toRegular (a : ฮฑ) : (toRegular a : ฮฑ) = aแถแถ :=
rfl
#align heyting.regular.coe_to_regular Heyting.Regular.coe_toRegular
@[simp]
theorem toRegular_coe (a : Regular ฮฑ) : toRegular (a : ฮฑ) = a :=
coe_injective a.2
#align heyting.regular.to_regular_coe Heyting.Regular.toRegular_coe
/-- The Galois insertion between `Regular.toRegular` and `coe`. -/
def gi : GaloisInsertion toRegular ((โ) : Regular ฮฑ โ ฮฑ)
where
choice a ha := โจa, ha.antisymm le_compl_complโฉ
gc _ b :=
coe_le_coe.symm.trans <|
โจle_compl_compl.trans, fun h => (compl_anti <| compl_anti h).trans_eq b.2โฉ
le_l_u _ := le_compl_compl
choice_eq _ ha := coe_injective <| le_compl_compl.antisymm ha
#align heyting.regular.gi Heyting.Regular.gi
instance lattice : Lattice (Regular ฮฑ) :=
gi.liftLattice
@[simp, norm_cast]
theorem coe_sup (a b : Regular ฮฑ) : (โ(a โ b) : ฮฑ) = ((a : ฮฑ) โ b)แถแถ :=
rfl
#align heyting.regular.coe_sup Heyting.Regular.coe_sup
instance : BooleanAlgebra (Regular ฮฑ) :=
{ Regular.lattice, Regular.boundedOrder, Regular.himp,
Regular.hasCompl with
le_sup_inf := fun a b c =>
coe_le_coe.1 <| by
|
dsimp
|
instance : BooleanAlgebra (Regular ฮฑ) :=
{ Regular.lattice, Regular.boundedOrder, Regular.himp,
Regular.hasCompl with
le_sup_inf := fun a b c =>
coe_le_coe.1 <| by
|
Mathlib.Order.Heyting.Regular.224_0.HFWfWmvQ8w4gAZO
|
instance : BooleanAlgebra (Regular ฮฑ)
|
Mathlib_Order_Heyting_Regular
|
ฮฑ : Type u_1
instโ : HeytingAlgebra ฮฑ
aโ bโ : ฮฑ
srcโยณ : Lattice (Regular ฮฑ) := lattice
srcโยฒ : BoundedOrder (Regular ฮฑ) := boundedOrder
srcโยน : HImp (Regular ฮฑ) := himp
srcโ : HasCompl (Regular ฮฑ) := hasCompl
a b c : Regular ฮฑ
โข (โa โ โb)แถแถ โ (โa โ โc)แถแถ โค (โa โ โb โ โc)แถแถ
|
/-
Copyright (c) 2022 Yaรซl Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaรซl Dillies
-/
import Mathlib.Order.GaloisConnection
#align_import order.heyting.regular from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
/-!
# Heyting regular elements
This file defines Heyting regular elements, elements of a Heyting algebra that are their own double
complement, and proves that they form a boolean algebra.
From a logic standpoint, this means that we can perform classical logic within intuitionistic logic
by simply double-negating all propositions. This is practical for synthetic computability theory.
## Main declarations
* `IsRegular`: `a` is Heyting-regular if `aแถแถ = a`.
* `Regular`: The subtype of Heyting-regular elements.
* `Regular.BooleanAlgebra`: Heyting-regular elements form a boolean algebra.
## References
* [Francis Borceux, *Handbook of Categorical Algebra III*][borceux-vol3]
-/
open Function
variable {ฮฑ : Type*}
namespace Heyting
section HasCompl
variable [HasCompl ฮฑ] {a : ฮฑ}
/-- An element of a Heyting algebra is regular if its double complement is itself. -/
def IsRegular (a : ฮฑ) : Prop :=
aแถแถ = a
#align heyting.is_regular Heyting.IsRegular
protected theorem IsRegular.eq : IsRegular a โ aแถแถ = a :=
id
#align heyting.is_regular.eq Heyting.IsRegular.eq
instance IsRegular.decidablePred [DecidableEq ฮฑ] : @DecidablePred ฮฑ IsRegular := fun _ =>
โนDecidableEq ฮฑโบ _ _
#align heyting.is_regular.decidable_pred Heyting.IsRegular.decidablePred
end HasCompl
section HeytingAlgebra
variable [HeytingAlgebra ฮฑ] {a b : ฮฑ}
theorem isRegular_bot : IsRegular (โฅ : ฮฑ) := by rw [IsRegular, compl_bot, compl_top]
#align heyting.is_regular_bot Heyting.isRegular_bot
theorem isRegular_top : IsRegular (โค : ฮฑ) := by rw [IsRegular, compl_top, compl_bot]
#align heyting.is_regular_top Heyting.isRegular_top
theorem IsRegular.inf (ha : IsRegular a) (hb : IsRegular b) : IsRegular (a โ b) := by
rw [IsRegular, compl_compl_inf_distrib, ha.eq, hb.eq]
#align heyting.is_regular.inf Heyting.IsRegular.inf
theorem IsRegular.himp (ha : IsRegular a) (hb : IsRegular b) : IsRegular (a โจ b) := by
rw [IsRegular, compl_compl_himp_distrib, ha.eq, hb.eq]
#align heyting.is_regular.himp Heyting.IsRegular.himp
theorem isRegular_compl (a : ฮฑ) : IsRegular aแถ :=
compl_compl_compl _
#align heyting.is_regular_compl Heyting.isRegular_compl
protected theorem IsRegular.disjoint_compl_left_iff (ha : IsRegular a) : Disjoint aแถ b โ b โค a :=
by rw [โ le_compl_iff_disjoint_left, ha.eq]
#align heyting.is_regular.disjoint_compl_left_iff Heyting.IsRegular.disjoint_compl_left_iff
protected theorem IsRegular.disjoint_compl_right_iff (hb : IsRegular b) : Disjoint a bแถ โ a โค b :=
by rw [โ le_compl_iff_disjoint_right, hb.eq]
#align heyting.is_regular.disjoint_compl_right_iff Heyting.IsRegular.disjoint_compl_right_iff
-- See note [reducible non-instances]
/-- A Heyting algebra with regular excluded middle is a boolean algebra. -/
@[reducible]
def _root_.BooleanAlgebra.ofRegular (h : โ a : ฮฑ, IsRegular (a โ aแถ)) : BooleanAlgebra ฮฑ :=
have : โ a : ฮฑ, IsCompl a aแถ := fun a =>
โจdisjoint_compl_right,
codisjoint_iff.2 <| by erw [โ (h a), compl_sup, inf_compl_eq_bot, compl_bot]โฉ
{ โนHeytingAlgebra ฮฑโบ,
GeneralizedHeytingAlgebra.toDistribLattice with
himp_eq := fun a b =>
eq_of_forall_le_iff fun c => le_himp_iff.trans (this _).le_sup_right_iff_inf_left_le.symm
inf_compl_le_bot := fun a => (this _).1.le_bot
top_le_sup_compl := fun a => (this _).2.top_le }
#align boolean_algebra.of_regular BooleanAlgebra.ofRegular
variable (ฮฑ)
/-- The boolean algebra of Heyting regular elements. -/
def Regular : Type _ :=
{ a : ฮฑ // IsRegular a }
#align heyting.regular Heyting.Regular
variable {ฮฑ}
namespace Regular
--Porting note: `val` and `prop` are new
/-- The coercion `Regular ฮฑ โ ฮฑ` -/
@[coe] def val : Regular ฮฑ โ ฮฑ :=
Subtype.val
theorem prop : โ a : Regular ฮฑ, IsRegular a.val := Subtype.prop
instance : CoeOut (Regular ฮฑ) ฮฑ := โจRegular.valโฉ
theorem coe_injective : Injective ((โ) : Regular ฮฑ โ ฮฑ) :=
Subtype.coe_injective
#align heyting.regular.coe_injective Heyting.Regular.coe_injective
@[simp]
theorem coe_inj {a b : Regular ฮฑ} : (a : ฮฑ) = b โ a = b :=
Subtype.coe_inj
#align heyting.regular.coe_inj Heyting.Regular.coe_inj
instance top : Top (Regular ฮฑ) :=
โจโจโค, isRegular_topโฉโฉ
instance bot : Bot (Regular ฮฑ) :=
โจโจโฅ, isRegular_botโฉโฉ
instance inf : Inf (Regular ฮฑ) :=
โจfun a b => โจa โ b, a.2.inf b.2โฉโฉ
instance himp : HImp (Regular ฮฑ) :=
โจfun a b => โจa โจ b, a.2.himp b.2โฉโฉ
instance hasCompl : HasCompl (Regular ฮฑ) :=
โจfun a => โจaแถ, isRegular_compl _โฉโฉ
@[simp, norm_cast]
theorem coe_top : ((โค : Regular ฮฑ) : ฮฑ) = โค :=
rfl
#align heyting.regular.coe_top Heyting.Regular.coe_top
@[simp, norm_cast]
theorem coe_bot : ((โฅ : Regular ฮฑ) : ฮฑ) = โฅ :=
rfl
#align heyting.regular.coe_bot Heyting.Regular.coe_bot
@[simp, norm_cast]
theorem coe_inf (a b : Regular ฮฑ) : (โ(a โ b) : ฮฑ) = (a : ฮฑ) โ b :=
rfl
#align heyting.regular.coe_inf Heyting.Regular.coe_inf
@[simp, norm_cast]
theorem coe_himp (a b : Regular ฮฑ) : (โ(a โจ b) : ฮฑ) = (a : ฮฑ) โจ b :=
rfl
#align heyting.regular.coe_himp Heyting.Regular.coe_himp
@[simp, norm_cast]
theorem coe_compl (a : Regular ฮฑ) : (โaแถ : ฮฑ) = (a : ฮฑ)แถ :=
rfl
#align heyting.regular.coe_compl Heyting.Regular.coe_compl
instance : Inhabited (Regular ฮฑ) :=
โจโฅโฉ
instance : SemilatticeInf (Regular ฮฑ) :=
coe_injective.semilatticeInf _ coe_inf
instance boundedOrder : BoundedOrder (Regular ฮฑ) :=
BoundedOrder.lift ((โ) : Regular ฮฑ โ ฮฑ) (fun _ _ => id) coe_top coe_bot
@[simp, norm_cast]
theorem coe_le_coe {a b : Regular ฮฑ} : (a : ฮฑ) โค b โ a โค b :=
Iff.rfl
#align heyting.regular.coe_le_coe Heyting.Regular.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {a b : Regular ฮฑ} : (a : ฮฑ) < b โ a < b :=
Iff.rfl
#align heyting.regular.coe_lt_coe Heyting.Regular.coe_lt_coe
/-- **Regularization** of `a`. The smallest regular element greater than `a`. -/
def toRegular : ฮฑ โo Regular ฮฑ :=
โจfun a => โจaแถแถ, isRegular_compl _โฉ, fun _ _ h =>
coe_le_coe.1 <| compl_le_compl <| compl_le_compl hโฉ
#align heyting.regular.to_regular Heyting.Regular.toRegular
@[simp, norm_cast]
theorem coe_toRegular (a : ฮฑ) : (toRegular a : ฮฑ) = aแถแถ :=
rfl
#align heyting.regular.coe_to_regular Heyting.Regular.coe_toRegular
@[simp]
theorem toRegular_coe (a : Regular ฮฑ) : toRegular (a : ฮฑ) = a :=
coe_injective a.2
#align heyting.regular.to_regular_coe Heyting.Regular.toRegular_coe
/-- The Galois insertion between `Regular.toRegular` and `coe`. -/
def gi : GaloisInsertion toRegular ((โ) : Regular ฮฑ โ ฮฑ)
where
choice a ha := โจa, ha.antisymm le_compl_complโฉ
gc _ b :=
coe_le_coe.symm.trans <|
โจle_compl_compl.trans, fun h => (compl_anti <| compl_anti h).trans_eq b.2โฉ
le_l_u _ := le_compl_compl
choice_eq _ ha := coe_injective <| le_compl_compl.antisymm ha
#align heyting.regular.gi Heyting.Regular.gi
instance lattice : Lattice (Regular ฮฑ) :=
gi.liftLattice
@[simp, norm_cast]
theorem coe_sup (a b : Regular ฮฑ) : (โ(a โ b) : ฮฑ) = ((a : ฮฑ) โ b)แถแถ :=
rfl
#align heyting.regular.coe_sup Heyting.Regular.coe_sup
instance : BooleanAlgebra (Regular ฮฑ) :=
{ Regular.lattice, Regular.boundedOrder, Regular.himp,
Regular.hasCompl with
le_sup_inf := fun a b c =>
coe_le_coe.1 <| by
dsimp
|
rw [sup_inf_left, compl_compl_inf_distrib]
|
instance : BooleanAlgebra (Regular ฮฑ) :=
{ Regular.lattice, Regular.boundedOrder, Regular.himp,
Regular.hasCompl with
le_sup_inf := fun a b c =>
coe_le_coe.1 <| by
dsimp
|
Mathlib.Order.Heyting.Regular.224_0.HFWfWmvQ8w4gAZO
|
instance : BooleanAlgebra (Regular ฮฑ)
|
Mathlib_Order_Heyting_Regular
|
ฮฑ : Type u_1
instโ : HeytingAlgebra ฮฑ
aโ b : ฮฑ
srcโยณ : Lattice (Regular ฮฑ) := lattice
srcโยฒ : BoundedOrder (Regular ฮฑ) := boundedOrder
srcโยน : HImp (Regular ฮฑ) := himp
srcโ : HasCompl (Regular ฮฑ) := hasCompl
a : Regular ฮฑ
โข โโค โค โ(a โ aแถ)
|
/-
Copyright (c) 2022 Yaรซl Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaรซl Dillies
-/
import Mathlib.Order.GaloisConnection
#align_import order.heyting.regular from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
/-!
# Heyting regular elements
This file defines Heyting regular elements, elements of a Heyting algebra that are their own double
complement, and proves that they form a boolean algebra.
From a logic standpoint, this means that we can perform classical logic within intuitionistic logic
by simply double-negating all propositions. This is practical for synthetic computability theory.
## Main declarations
* `IsRegular`: `a` is Heyting-regular if `aแถแถ = a`.
* `Regular`: The subtype of Heyting-regular elements.
* `Regular.BooleanAlgebra`: Heyting-regular elements form a boolean algebra.
## References
* [Francis Borceux, *Handbook of Categorical Algebra III*][borceux-vol3]
-/
open Function
variable {ฮฑ : Type*}
namespace Heyting
section HasCompl
variable [HasCompl ฮฑ] {a : ฮฑ}
/-- An element of a Heyting algebra is regular if its double complement is itself. -/
def IsRegular (a : ฮฑ) : Prop :=
aแถแถ = a
#align heyting.is_regular Heyting.IsRegular
protected theorem IsRegular.eq : IsRegular a โ aแถแถ = a :=
id
#align heyting.is_regular.eq Heyting.IsRegular.eq
instance IsRegular.decidablePred [DecidableEq ฮฑ] : @DecidablePred ฮฑ IsRegular := fun _ =>
โนDecidableEq ฮฑโบ _ _
#align heyting.is_regular.decidable_pred Heyting.IsRegular.decidablePred
end HasCompl
section HeytingAlgebra
variable [HeytingAlgebra ฮฑ] {a b : ฮฑ}
theorem isRegular_bot : IsRegular (โฅ : ฮฑ) := by rw [IsRegular, compl_bot, compl_top]
#align heyting.is_regular_bot Heyting.isRegular_bot
theorem isRegular_top : IsRegular (โค : ฮฑ) := by rw [IsRegular, compl_top, compl_bot]
#align heyting.is_regular_top Heyting.isRegular_top
theorem IsRegular.inf (ha : IsRegular a) (hb : IsRegular b) : IsRegular (a โ b) := by
rw [IsRegular, compl_compl_inf_distrib, ha.eq, hb.eq]
#align heyting.is_regular.inf Heyting.IsRegular.inf
theorem IsRegular.himp (ha : IsRegular a) (hb : IsRegular b) : IsRegular (a โจ b) := by
rw [IsRegular, compl_compl_himp_distrib, ha.eq, hb.eq]
#align heyting.is_regular.himp Heyting.IsRegular.himp
theorem isRegular_compl (a : ฮฑ) : IsRegular aแถ :=
compl_compl_compl _
#align heyting.is_regular_compl Heyting.isRegular_compl
protected theorem IsRegular.disjoint_compl_left_iff (ha : IsRegular a) : Disjoint aแถ b โ b โค a :=
by rw [โ le_compl_iff_disjoint_left, ha.eq]
#align heyting.is_regular.disjoint_compl_left_iff Heyting.IsRegular.disjoint_compl_left_iff
protected theorem IsRegular.disjoint_compl_right_iff (hb : IsRegular b) : Disjoint a bแถ โ a โค b :=
by rw [โ le_compl_iff_disjoint_right, hb.eq]
#align heyting.is_regular.disjoint_compl_right_iff Heyting.IsRegular.disjoint_compl_right_iff
-- See note [reducible non-instances]
/-- A Heyting algebra with regular excluded middle is a boolean algebra. -/
@[reducible]
def _root_.BooleanAlgebra.ofRegular (h : โ a : ฮฑ, IsRegular (a โ aแถ)) : BooleanAlgebra ฮฑ :=
have : โ a : ฮฑ, IsCompl a aแถ := fun a =>
โจdisjoint_compl_right,
codisjoint_iff.2 <| by erw [โ (h a), compl_sup, inf_compl_eq_bot, compl_bot]โฉ
{ โนHeytingAlgebra ฮฑโบ,
GeneralizedHeytingAlgebra.toDistribLattice with
himp_eq := fun a b =>
eq_of_forall_le_iff fun c => le_himp_iff.trans (this _).le_sup_right_iff_inf_left_le.symm
inf_compl_le_bot := fun a => (this _).1.le_bot
top_le_sup_compl := fun a => (this _).2.top_le }
#align boolean_algebra.of_regular BooleanAlgebra.ofRegular
variable (ฮฑ)
/-- The boolean algebra of Heyting regular elements. -/
def Regular : Type _ :=
{ a : ฮฑ // IsRegular a }
#align heyting.regular Heyting.Regular
variable {ฮฑ}
namespace Regular
--Porting note: `val` and `prop` are new
/-- The coercion `Regular ฮฑ โ ฮฑ` -/
@[coe] def val : Regular ฮฑ โ ฮฑ :=
Subtype.val
theorem prop : โ a : Regular ฮฑ, IsRegular a.val := Subtype.prop
instance : CoeOut (Regular ฮฑ) ฮฑ := โจRegular.valโฉ
theorem coe_injective : Injective ((โ) : Regular ฮฑ โ ฮฑ) :=
Subtype.coe_injective
#align heyting.regular.coe_injective Heyting.Regular.coe_injective
@[simp]
theorem coe_inj {a b : Regular ฮฑ} : (a : ฮฑ) = b โ a = b :=
Subtype.coe_inj
#align heyting.regular.coe_inj Heyting.Regular.coe_inj
instance top : Top (Regular ฮฑ) :=
โจโจโค, isRegular_topโฉโฉ
instance bot : Bot (Regular ฮฑ) :=
โจโจโฅ, isRegular_botโฉโฉ
instance inf : Inf (Regular ฮฑ) :=
โจfun a b => โจa โ b, a.2.inf b.2โฉโฉ
instance himp : HImp (Regular ฮฑ) :=
โจfun a b => โจa โจ b, a.2.himp b.2โฉโฉ
instance hasCompl : HasCompl (Regular ฮฑ) :=
โจfun a => โจaแถ, isRegular_compl _โฉโฉ
@[simp, norm_cast]
theorem coe_top : ((โค : Regular ฮฑ) : ฮฑ) = โค :=
rfl
#align heyting.regular.coe_top Heyting.Regular.coe_top
@[simp, norm_cast]
theorem coe_bot : ((โฅ : Regular ฮฑ) : ฮฑ) = โฅ :=
rfl
#align heyting.regular.coe_bot Heyting.Regular.coe_bot
@[simp, norm_cast]
theorem coe_inf (a b : Regular ฮฑ) : (โ(a โ b) : ฮฑ) = (a : ฮฑ) โ b :=
rfl
#align heyting.regular.coe_inf Heyting.Regular.coe_inf
@[simp, norm_cast]
theorem coe_himp (a b : Regular ฮฑ) : (โ(a โจ b) : ฮฑ) = (a : ฮฑ) โจ b :=
rfl
#align heyting.regular.coe_himp Heyting.Regular.coe_himp
@[simp, norm_cast]
theorem coe_compl (a : Regular ฮฑ) : (โaแถ : ฮฑ) = (a : ฮฑ)แถ :=
rfl
#align heyting.regular.coe_compl Heyting.Regular.coe_compl
instance : Inhabited (Regular ฮฑ) :=
โจโฅโฉ
instance : SemilatticeInf (Regular ฮฑ) :=
coe_injective.semilatticeInf _ coe_inf
instance boundedOrder : BoundedOrder (Regular ฮฑ) :=
BoundedOrder.lift ((โ) : Regular ฮฑ โ ฮฑ) (fun _ _ => id) coe_top coe_bot
@[simp, norm_cast]
theorem coe_le_coe {a b : Regular ฮฑ} : (a : ฮฑ) โค b โ a โค b :=
Iff.rfl
#align heyting.regular.coe_le_coe Heyting.Regular.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {a b : Regular ฮฑ} : (a : ฮฑ) < b โ a < b :=
Iff.rfl
#align heyting.regular.coe_lt_coe Heyting.Regular.coe_lt_coe
/-- **Regularization** of `a`. The smallest regular element greater than `a`. -/
def toRegular : ฮฑ โo Regular ฮฑ :=
โจfun a => โจaแถแถ, isRegular_compl _โฉ, fun _ _ h =>
coe_le_coe.1 <| compl_le_compl <| compl_le_compl hโฉ
#align heyting.regular.to_regular Heyting.Regular.toRegular
@[simp, norm_cast]
theorem coe_toRegular (a : ฮฑ) : (toRegular a : ฮฑ) = aแถแถ :=
rfl
#align heyting.regular.coe_to_regular Heyting.Regular.coe_toRegular
@[simp]
theorem toRegular_coe (a : Regular ฮฑ) : toRegular (a : ฮฑ) = a :=
coe_injective a.2
#align heyting.regular.to_regular_coe Heyting.Regular.toRegular_coe
/-- The Galois insertion between `Regular.toRegular` and `coe`. -/
def gi : GaloisInsertion toRegular ((โ) : Regular ฮฑ โ ฮฑ)
where
choice a ha := โจa, ha.antisymm le_compl_complโฉ
gc _ b :=
coe_le_coe.symm.trans <|
โจle_compl_compl.trans, fun h => (compl_anti <| compl_anti h).trans_eq b.2โฉ
le_l_u _ := le_compl_compl
choice_eq _ ha := coe_injective <| le_compl_compl.antisymm ha
#align heyting.regular.gi Heyting.Regular.gi
instance lattice : Lattice (Regular ฮฑ) :=
gi.liftLattice
@[simp, norm_cast]
theorem coe_sup (a b : Regular ฮฑ) : (โ(a โ b) : ฮฑ) = ((a : ฮฑ) โ b)แถแถ :=
rfl
#align heyting.regular.coe_sup Heyting.Regular.coe_sup
instance : BooleanAlgebra (Regular ฮฑ) :=
{ Regular.lattice, Regular.boundedOrder, Regular.himp,
Regular.hasCompl with
le_sup_inf := fun a b c =>
coe_le_coe.1 <| by
dsimp
rw [sup_inf_left, compl_compl_inf_distrib]
inf_compl_le_bot := fun a => coe_le_coe.1 <| disjoint_iff_inf_le.1 disjoint_compl_right
top_le_sup_compl := fun a =>
coe_le_coe.1 <| by
|
dsimp
|
instance : BooleanAlgebra (Regular ฮฑ) :=
{ Regular.lattice, Regular.boundedOrder, Regular.himp,
Regular.hasCompl with
le_sup_inf := fun a b c =>
coe_le_coe.1 <| by
dsimp
rw [sup_inf_left, compl_compl_inf_distrib]
inf_compl_le_bot := fun a => coe_le_coe.1 <| disjoint_iff_inf_le.1 disjoint_compl_right
top_le_sup_compl := fun a =>
coe_le_coe.1 <| by
|
Mathlib.Order.Heyting.Regular.224_0.HFWfWmvQ8w4gAZO
|
instance : BooleanAlgebra (Regular ฮฑ)
|
Mathlib_Order_Heyting_Regular
|
ฮฑ : Type u_1
instโ : HeytingAlgebra ฮฑ
aโ b : ฮฑ
srcโยณ : Lattice (Regular ฮฑ) := lattice
srcโยฒ : BoundedOrder (Regular ฮฑ) := boundedOrder
srcโยน : HImp (Regular ฮฑ) := himp
srcโ : HasCompl (Regular ฮฑ) := hasCompl
a : Regular ฮฑ
โข โค โค (โa โ (โa)แถ)แถแถ
|
/-
Copyright (c) 2022 Yaรซl Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaรซl Dillies
-/
import Mathlib.Order.GaloisConnection
#align_import order.heyting.regular from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
/-!
# Heyting regular elements
This file defines Heyting regular elements, elements of a Heyting algebra that are their own double
complement, and proves that they form a boolean algebra.
From a logic standpoint, this means that we can perform classical logic within intuitionistic logic
by simply double-negating all propositions. This is practical for synthetic computability theory.
## Main declarations
* `IsRegular`: `a` is Heyting-regular if `aแถแถ = a`.
* `Regular`: The subtype of Heyting-regular elements.
* `Regular.BooleanAlgebra`: Heyting-regular elements form a boolean algebra.
## References
* [Francis Borceux, *Handbook of Categorical Algebra III*][borceux-vol3]
-/
open Function
variable {ฮฑ : Type*}
namespace Heyting
section HasCompl
variable [HasCompl ฮฑ] {a : ฮฑ}
/-- An element of a Heyting algebra is regular if its double complement is itself. -/
def IsRegular (a : ฮฑ) : Prop :=
aแถแถ = a
#align heyting.is_regular Heyting.IsRegular
protected theorem IsRegular.eq : IsRegular a โ aแถแถ = a :=
id
#align heyting.is_regular.eq Heyting.IsRegular.eq
instance IsRegular.decidablePred [DecidableEq ฮฑ] : @DecidablePred ฮฑ IsRegular := fun _ =>
โนDecidableEq ฮฑโบ _ _
#align heyting.is_regular.decidable_pred Heyting.IsRegular.decidablePred
end HasCompl
section HeytingAlgebra
variable [HeytingAlgebra ฮฑ] {a b : ฮฑ}
theorem isRegular_bot : IsRegular (โฅ : ฮฑ) := by rw [IsRegular, compl_bot, compl_top]
#align heyting.is_regular_bot Heyting.isRegular_bot
theorem isRegular_top : IsRegular (โค : ฮฑ) := by rw [IsRegular, compl_top, compl_bot]
#align heyting.is_regular_top Heyting.isRegular_top
theorem IsRegular.inf (ha : IsRegular a) (hb : IsRegular b) : IsRegular (a โ b) := by
rw [IsRegular, compl_compl_inf_distrib, ha.eq, hb.eq]
#align heyting.is_regular.inf Heyting.IsRegular.inf
theorem IsRegular.himp (ha : IsRegular a) (hb : IsRegular b) : IsRegular (a โจ b) := by
rw [IsRegular, compl_compl_himp_distrib, ha.eq, hb.eq]
#align heyting.is_regular.himp Heyting.IsRegular.himp
theorem isRegular_compl (a : ฮฑ) : IsRegular aแถ :=
compl_compl_compl _
#align heyting.is_regular_compl Heyting.isRegular_compl
protected theorem IsRegular.disjoint_compl_left_iff (ha : IsRegular a) : Disjoint aแถ b โ b โค a :=
by rw [โ le_compl_iff_disjoint_left, ha.eq]
#align heyting.is_regular.disjoint_compl_left_iff Heyting.IsRegular.disjoint_compl_left_iff
protected theorem IsRegular.disjoint_compl_right_iff (hb : IsRegular b) : Disjoint a bแถ โ a โค b :=
by rw [โ le_compl_iff_disjoint_right, hb.eq]
#align heyting.is_regular.disjoint_compl_right_iff Heyting.IsRegular.disjoint_compl_right_iff
-- See note [reducible non-instances]
/-- A Heyting algebra with regular excluded middle is a boolean algebra. -/
@[reducible]
def _root_.BooleanAlgebra.ofRegular (h : โ a : ฮฑ, IsRegular (a โ aแถ)) : BooleanAlgebra ฮฑ :=
have : โ a : ฮฑ, IsCompl a aแถ := fun a =>
โจdisjoint_compl_right,
codisjoint_iff.2 <| by erw [โ (h a), compl_sup, inf_compl_eq_bot, compl_bot]โฉ
{ โนHeytingAlgebra ฮฑโบ,
GeneralizedHeytingAlgebra.toDistribLattice with
himp_eq := fun a b =>
eq_of_forall_le_iff fun c => le_himp_iff.trans (this _).le_sup_right_iff_inf_left_le.symm
inf_compl_le_bot := fun a => (this _).1.le_bot
top_le_sup_compl := fun a => (this _).2.top_le }
#align boolean_algebra.of_regular BooleanAlgebra.ofRegular
variable (ฮฑ)
/-- The boolean algebra of Heyting regular elements. -/
def Regular : Type _ :=
{ a : ฮฑ // IsRegular a }
#align heyting.regular Heyting.Regular
variable {ฮฑ}
namespace Regular
--Porting note: `val` and `prop` are new
/-- The coercion `Regular ฮฑ โ ฮฑ` -/
@[coe] def val : Regular ฮฑ โ ฮฑ :=
Subtype.val
theorem prop : โ a : Regular ฮฑ, IsRegular a.val := Subtype.prop
instance : CoeOut (Regular ฮฑ) ฮฑ := โจRegular.valโฉ
theorem coe_injective : Injective ((โ) : Regular ฮฑ โ ฮฑ) :=
Subtype.coe_injective
#align heyting.regular.coe_injective Heyting.Regular.coe_injective
@[simp]
theorem coe_inj {a b : Regular ฮฑ} : (a : ฮฑ) = b โ a = b :=
Subtype.coe_inj
#align heyting.regular.coe_inj Heyting.Regular.coe_inj
instance top : Top (Regular ฮฑ) :=
โจโจโค, isRegular_topโฉโฉ
instance bot : Bot (Regular ฮฑ) :=
โจโจโฅ, isRegular_botโฉโฉ
instance inf : Inf (Regular ฮฑ) :=
โจfun a b => โจa โ b, a.2.inf b.2โฉโฉ
instance himp : HImp (Regular ฮฑ) :=
โจfun a b => โจa โจ b, a.2.himp b.2โฉโฉ
instance hasCompl : HasCompl (Regular ฮฑ) :=
โจfun a => โจaแถ, isRegular_compl _โฉโฉ
@[simp, norm_cast]
theorem coe_top : ((โค : Regular ฮฑ) : ฮฑ) = โค :=
rfl
#align heyting.regular.coe_top Heyting.Regular.coe_top
@[simp, norm_cast]
theorem coe_bot : ((โฅ : Regular ฮฑ) : ฮฑ) = โฅ :=
rfl
#align heyting.regular.coe_bot Heyting.Regular.coe_bot
@[simp, norm_cast]
theorem coe_inf (a b : Regular ฮฑ) : (โ(a โ b) : ฮฑ) = (a : ฮฑ) โ b :=
rfl
#align heyting.regular.coe_inf Heyting.Regular.coe_inf
@[simp, norm_cast]
theorem coe_himp (a b : Regular ฮฑ) : (โ(a โจ b) : ฮฑ) = (a : ฮฑ) โจ b :=
rfl
#align heyting.regular.coe_himp Heyting.Regular.coe_himp
@[simp, norm_cast]
theorem coe_compl (a : Regular ฮฑ) : (โaแถ : ฮฑ) = (a : ฮฑ)แถ :=
rfl
#align heyting.regular.coe_compl Heyting.Regular.coe_compl
instance : Inhabited (Regular ฮฑ) :=
โจโฅโฉ
instance : SemilatticeInf (Regular ฮฑ) :=
coe_injective.semilatticeInf _ coe_inf
instance boundedOrder : BoundedOrder (Regular ฮฑ) :=
BoundedOrder.lift ((โ) : Regular ฮฑ โ ฮฑ) (fun _ _ => id) coe_top coe_bot
@[simp, norm_cast]
theorem coe_le_coe {a b : Regular ฮฑ} : (a : ฮฑ) โค b โ a โค b :=
Iff.rfl
#align heyting.regular.coe_le_coe Heyting.Regular.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {a b : Regular ฮฑ} : (a : ฮฑ) < b โ a < b :=
Iff.rfl
#align heyting.regular.coe_lt_coe Heyting.Regular.coe_lt_coe
/-- **Regularization** of `a`. The smallest regular element greater than `a`. -/
def toRegular : ฮฑ โo Regular ฮฑ :=
โจfun a => โจaแถแถ, isRegular_compl _โฉ, fun _ _ h =>
coe_le_coe.1 <| compl_le_compl <| compl_le_compl hโฉ
#align heyting.regular.to_regular Heyting.Regular.toRegular
@[simp, norm_cast]
theorem coe_toRegular (a : ฮฑ) : (toRegular a : ฮฑ) = aแถแถ :=
rfl
#align heyting.regular.coe_to_regular Heyting.Regular.coe_toRegular
@[simp]
theorem toRegular_coe (a : Regular ฮฑ) : toRegular (a : ฮฑ) = a :=
coe_injective a.2
#align heyting.regular.to_regular_coe Heyting.Regular.toRegular_coe
/-- The Galois insertion between `Regular.toRegular` and `coe`. -/
def gi : GaloisInsertion toRegular ((โ) : Regular ฮฑ โ ฮฑ)
where
choice a ha := โจa, ha.antisymm le_compl_complโฉ
gc _ b :=
coe_le_coe.symm.trans <|
โจle_compl_compl.trans, fun h => (compl_anti <| compl_anti h).trans_eq b.2โฉ
le_l_u _ := le_compl_compl
choice_eq _ ha := coe_injective <| le_compl_compl.antisymm ha
#align heyting.regular.gi Heyting.Regular.gi
instance lattice : Lattice (Regular ฮฑ) :=
gi.liftLattice
@[simp, norm_cast]
theorem coe_sup (a b : Regular ฮฑ) : (โ(a โ b) : ฮฑ) = ((a : ฮฑ) โ b)แถแถ :=
rfl
#align heyting.regular.coe_sup Heyting.Regular.coe_sup
instance : BooleanAlgebra (Regular ฮฑ) :=
{ Regular.lattice, Regular.boundedOrder, Regular.himp,
Regular.hasCompl with
le_sup_inf := fun a b c =>
coe_le_coe.1 <| by
dsimp
rw [sup_inf_left, compl_compl_inf_distrib]
inf_compl_le_bot := fun a => coe_le_coe.1 <| disjoint_iff_inf_le.1 disjoint_compl_right
top_le_sup_compl := fun a =>
coe_le_coe.1 <| by
dsimp
|
rw [compl_sup, inf_compl_eq_bot, compl_bot]
|
instance : BooleanAlgebra (Regular ฮฑ) :=
{ Regular.lattice, Regular.boundedOrder, Regular.himp,
Regular.hasCompl with
le_sup_inf := fun a b c =>
coe_le_coe.1 <| by
dsimp
rw [sup_inf_left, compl_compl_inf_distrib]
inf_compl_le_bot := fun a => coe_le_coe.1 <| disjoint_iff_inf_le.1 disjoint_compl_right
top_le_sup_compl := fun a =>
coe_le_coe.1 <| by
dsimp
|
Mathlib.Order.Heyting.Regular.224_0.HFWfWmvQ8w4gAZO
|
instance : BooleanAlgebra (Regular ฮฑ)
|
Mathlib_Order_Heyting_Regular
|
ฮฑ : Type u_1
instโ : HeytingAlgebra ฮฑ
aโ bโ : ฮฑ
srcโยณ : Lattice (Regular ฮฑ) := lattice
srcโยฒ : BoundedOrder (Regular ฮฑ) := boundedOrder
srcโยน : HImp (Regular ฮฑ) := himp
srcโ : HasCompl (Regular ฮฑ) := hasCompl
a b : Regular ฮฑ
โข โ(a โจ b) = โ(b โ aแถ)
|
/-
Copyright (c) 2022 Yaรซl Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaรซl Dillies
-/
import Mathlib.Order.GaloisConnection
#align_import order.heyting.regular from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
/-!
# Heyting regular elements
This file defines Heyting regular elements, elements of a Heyting algebra that are their own double
complement, and proves that they form a boolean algebra.
From a logic standpoint, this means that we can perform classical logic within intuitionistic logic
by simply double-negating all propositions. This is practical for synthetic computability theory.
## Main declarations
* `IsRegular`: `a` is Heyting-regular if `aแถแถ = a`.
* `Regular`: The subtype of Heyting-regular elements.
* `Regular.BooleanAlgebra`: Heyting-regular elements form a boolean algebra.
## References
* [Francis Borceux, *Handbook of Categorical Algebra III*][borceux-vol3]
-/
open Function
variable {ฮฑ : Type*}
namespace Heyting
section HasCompl
variable [HasCompl ฮฑ] {a : ฮฑ}
/-- An element of a Heyting algebra is regular if its double complement is itself. -/
def IsRegular (a : ฮฑ) : Prop :=
aแถแถ = a
#align heyting.is_regular Heyting.IsRegular
protected theorem IsRegular.eq : IsRegular a โ aแถแถ = a :=
id
#align heyting.is_regular.eq Heyting.IsRegular.eq
instance IsRegular.decidablePred [DecidableEq ฮฑ] : @DecidablePred ฮฑ IsRegular := fun _ =>
โนDecidableEq ฮฑโบ _ _
#align heyting.is_regular.decidable_pred Heyting.IsRegular.decidablePred
end HasCompl
section HeytingAlgebra
variable [HeytingAlgebra ฮฑ] {a b : ฮฑ}
theorem isRegular_bot : IsRegular (โฅ : ฮฑ) := by rw [IsRegular, compl_bot, compl_top]
#align heyting.is_regular_bot Heyting.isRegular_bot
theorem isRegular_top : IsRegular (โค : ฮฑ) := by rw [IsRegular, compl_top, compl_bot]
#align heyting.is_regular_top Heyting.isRegular_top
theorem IsRegular.inf (ha : IsRegular a) (hb : IsRegular b) : IsRegular (a โ b) := by
rw [IsRegular, compl_compl_inf_distrib, ha.eq, hb.eq]
#align heyting.is_regular.inf Heyting.IsRegular.inf
theorem IsRegular.himp (ha : IsRegular a) (hb : IsRegular b) : IsRegular (a โจ b) := by
rw [IsRegular, compl_compl_himp_distrib, ha.eq, hb.eq]
#align heyting.is_regular.himp Heyting.IsRegular.himp
theorem isRegular_compl (a : ฮฑ) : IsRegular aแถ :=
compl_compl_compl _
#align heyting.is_regular_compl Heyting.isRegular_compl
protected theorem IsRegular.disjoint_compl_left_iff (ha : IsRegular a) : Disjoint aแถ b โ b โค a :=
by rw [โ le_compl_iff_disjoint_left, ha.eq]
#align heyting.is_regular.disjoint_compl_left_iff Heyting.IsRegular.disjoint_compl_left_iff
protected theorem IsRegular.disjoint_compl_right_iff (hb : IsRegular b) : Disjoint a bแถ โ a โค b :=
by rw [โ le_compl_iff_disjoint_right, hb.eq]
#align heyting.is_regular.disjoint_compl_right_iff Heyting.IsRegular.disjoint_compl_right_iff
-- See note [reducible non-instances]
/-- A Heyting algebra with regular excluded middle is a boolean algebra. -/
@[reducible]
def _root_.BooleanAlgebra.ofRegular (h : โ a : ฮฑ, IsRegular (a โ aแถ)) : BooleanAlgebra ฮฑ :=
have : โ a : ฮฑ, IsCompl a aแถ := fun a =>
โจdisjoint_compl_right,
codisjoint_iff.2 <| by erw [โ (h a), compl_sup, inf_compl_eq_bot, compl_bot]โฉ
{ โนHeytingAlgebra ฮฑโบ,
GeneralizedHeytingAlgebra.toDistribLattice with
himp_eq := fun a b =>
eq_of_forall_le_iff fun c => le_himp_iff.trans (this _).le_sup_right_iff_inf_left_le.symm
inf_compl_le_bot := fun a => (this _).1.le_bot
top_le_sup_compl := fun a => (this _).2.top_le }
#align boolean_algebra.of_regular BooleanAlgebra.ofRegular
variable (ฮฑ)
/-- The boolean algebra of Heyting regular elements. -/
def Regular : Type _ :=
{ a : ฮฑ // IsRegular a }
#align heyting.regular Heyting.Regular
variable {ฮฑ}
namespace Regular
--Porting note: `val` and `prop` are new
/-- The coercion `Regular ฮฑ โ ฮฑ` -/
@[coe] def val : Regular ฮฑ โ ฮฑ :=
Subtype.val
theorem prop : โ a : Regular ฮฑ, IsRegular a.val := Subtype.prop
instance : CoeOut (Regular ฮฑ) ฮฑ := โจRegular.valโฉ
theorem coe_injective : Injective ((โ) : Regular ฮฑ โ ฮฑ) :=
Subtype.coe_injective
#align heyting.regular.coe_injective Heyting.Regular.coe_injective
@[simp]
theorem coe_inj {a b : Regular ฮฑ} : (a : ฮฑ) = b โ a = b :=
Subtype.coe_inj
#align heyting.regular.coe_inj Heyting.Regular.coe_inj
instance top : Top (Regular ฮฑ) :=
โจโจโค, isRegular_topโฉโฉ
instance bot : Bot (Regular ฮฑ) :=
โจโจโฅ, isRegular_botโฉโฉ
instance inf : Inf (Regular ฮฑ) :=
โจfun a b => โจa โ b, a.2.inf b.2โฉโฉ
instance himp : HImp (Regular ฮฑ) :=
โจfun a b => โจa โจ b, a.2.himp b.2โฉโฉ
instance hasCompl : HasCompl (Regular ฮฑ) :=
โจfun a => โจaแถ, isRegular_compl _โฉโฉ
@[simp, norm_cast]
theorem coe_top : ((โค : Regular ฮฑ) : ฮฑ) = โค :=
rfl
#align heyting.regular.coe_top Heyting.Regular.coe_top
@[simp, norm_cast]
theorem coe_bot : ((โฅ : Regular ฮฑ) : ฮฑ) = โฅ :=
rfl
#align heyting.regular.coe_bot Heyting.Regular.coe_bot
@[simp, norm_cast]
theorem coe_inf (a b : Regular ฮฑ) : (โ(a โ b) : ฮฑ) = (a : ฮฑ) โ b :=
rfl
#align heyting.regular.coe_inf Heyting.Regular.coe_inf
@[simp, norm_cast]
theorem coe_himp (a b : Regular ฮฑ) : (โ(a โจ b) : ฮฑ) = (a : ฮฑ) โจ b :=
rfl
#align heyting.regular.coe_himp Heyting.Regular.coe_himp
@[simp, norm_cast]
theorem coe_compl (a : Regular ฮฑ) : (โaแถ : ฮฑ) = (a : ฮฑ)แถ :=
rfl
#align heyting.regular.coe_compl Heyting.Regular.coe_compl
instance : Inhabited (Regular ฮฑ) :=
โจโฅโฉ
instance : SemilatticeInf (Regular ฮฑ) :=
coe_injective.semilatticeInf _ coe_inf
instance boundedOrder : BoundedOrder (Regular ฮฑ) :=
BoundedOrder.lift ((โ) : Regular ฮฑ โ ฮฑ) (fun _ _ => id) coe_top coe_bot
@[simp, norm_cast]
theorem coe_le_coe {a b : Regular ฮฑ} : (a : ฮฑ) โค b โ a โค b :=
Iff.rfl
#align heyting.regular.coe_le_coe Heyting.Regular.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {a b : Regular ฮฑ} : (a : ฮฑ) < b โ a < b :=
Iff.rfl
#align heyting.regular.coe_lt_coe Heyting.Regular.coe_lt_coe
/-- **Regularization** of `a`. The smallest regular element greater than `a`. -/
def toRegular : ฮฑ โo Regular ฮฑ :=
โจfun a => โจaแถแถ, isRegular_compl _โฉ, fun _ _ h =>
coe_le_coe.1 <| compl_le_compl <| compl_le_compl hโฉ
#align heyting.regular.to_regular Heyting.Regular.toRegular
@[simp, norm_cast]
theorem coe_toRegular (a : ฮฑ) : (toRegular a : ฮฑ) = aแถแถ :=
rfl
#align heyting.regular.coe_to_regular Heyting.Regular.coe_toRegular
@[simp]
theorem toRegular_coe (a : Regular ฮฑ) : toRegular (a : ฮฑ) = a :=
coe_injective a.2
#align heyting.regular.to_regular_coe Heyting.Regular.toRegular_coe
/-- The Galois insertion between `Regular.toRegular` and `coe`. -/
def gi : GaloisInsertion toRegular ((โ) : Regular ฮฑ โ ฮฑ)
where
choice a ha := โจa, ha.antisymm le_compl_complโฉ
gc _ b :=
coe_le_coe.symm.trans <|
โจle_compl_compl.trans, fun h => (compl_anti <| compl_anti h).trans_eq b.2โฉ
le_l_u _ := le_compl_compl
choice_eq _ ha := coe_injective <| le_compl_compl.antisymm ha
#align heyting.regular.gi Heyting.Regular.gi
instance lattice : Lattice (Regular ฮฑ) :=
gi.liftLattice
@[simp, norm_cast]
theorem coe_sup (a b : Regular ฮฑ) : (โ(a โ b) : ฮฑ) = ((a : ฮฑ) โ b)แถแถ :=
rfl
#align heyting.regular.coe_sup Heyting.Regular.coe_sup
instance : BooleanAlgebra (Regular ฮฑ) :=
{ Regular.lattice, Regular.boundedOrder, Regular.himp,
Regular.hasCompl with
le_sup_inf := fun a b c =>
coe_le_coe.1 <| by
dsimp
rw [sup_inf_left, compl_compl_inf_distrib]
inf_compl_le_bot := fun a => coe_le_coe.1 <| disjoint_iff_inf_le.1 disjoint_compl_right
top_le_sup_compl := fun a =>
coe_le_coe.1 <| by
dsimp
rw [compl_sup, inf_compl_eq_bot, compl_bot]
himp_eq := fun a b =>
coe_injective
(by
|
dsimp
|
instance : BooleanAlgebra (Regular ฮฑ) :=
{ Regular.lattice, Regular.boundedOrder, Regular.himp,
Regular.hasCompl with
le_sup_inf := fun a b c =>
coe_le_coe.1 <| by
dsimp
rw [sup_inf_left, compl_compl_inf_distrib]
inf_compl_le_bot := fun a => coe_le_coe.1 <| disjoint_iff_inf_le.1 disjoint_compl_right
top_le_sup_compl := fun a =>
coe_le_coe.1 <| by
dsimp
rw [compl_sup, inf_compl_eq_bot, compl_bot]
himp_eq := fun a b =>
coe_injective
(by
|
Mathlib.Order.Heyting.Regular.224_0.HFWfWmvQ8w4gAZO
|
instance : BooleanAlgebra (Regular ฮฑ)
|
Mathlib_Order_Heyting_Regular
|
ฮฑ : Type u_1
instโ : HeytingAlgebra ฮฑ
aโ bโ : ฮฑ
srcโยณ : Lattice (Regular ฮฑ) := lattice
srcโยฒ : BoundedOrder (Regular ฮฑ) := boundedOrder
srcโยน : HImp (Regular ฮฑ) := himp
srcโ : HasCompl (Regular ฮฑ) := hasCompl
a b : Regular ฮฑ
โข โa โจ โb = (โb โ (โa)แถ)แถแถ
|
/-
Copyright (c) 2022 Yaรซl Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaรซl Dillies
-/
import Mathlib.Order.GaloisConnection
#align_import order.heyting.regular from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
/-!
# Heyting regular elements
This file defines Heyting regular elements, elements of a Heyting algebra that are their own double
complement, and proves that they form a boolean algebra.
From a logic standpoint, this means that we can perform classical logic within intuitionistic logic
by simply double-negating all propositions. This is practical for synthetic computability theory.
## Main declarations
* `IsRegular`: `a` is Heyting-regular if `aแถแถ = a`.
* `Regular`: The subtype of Heyting-regular elements.
* `Regular.BooleanAlgebra`: Heyting-regular elements form a boolean algebra.
## References
* [Francis Borceux, *Handbook of Categorical Algebra III*][borceux-vol3]
-/
open Function
variable {ฮฑ : Type*}
namespace Heyting
section HasCompl
variable [HasCompl ฮฑ] {a : ฮฑ}
/-- An element of a Heyting algebra is regular if its double complement is itself. -/
def IsRegular (a : ฮฑ) : Prop :=
aแถแถ = a
#align heyting.is_regular Heyting.IsRegular
protected theorem IsRegular.eq : IsRegular a โ aแถแถ = a :=
id
#align heyting.is_regular.eq Heyting.IsRegular.eq
instance IsRegular.decidablePred [DecidableEq ฮฑ] : @DecidablePred ฮฑ IsRegular := fun _ =>
โนDecidableEq ฮฑโบ _ _
#align heyting.is_regular.decidable_pred Heyting.IsRegular.decidablePred
end HasCompl
section HeytingAlgebra
variable [HeytingAlgebra ฮฑ] {a b : ฮฑ}
theorem isRegular_bot : IsRegular (โฅ : ฮฑ) := by rw [IsRegular, compl_bot, compl_top]
#align heyting.is_regular_bot Heyting.isRegular_bot
theorem isRegular_top : IsRegular (โค : ฮฑ) := by rw [IsRegular, compl_top, compl_bot]
#align heyting.is_regular_top Heyting.isRegular_top
theorem IsRegular.inf (ha : IsRegular a) (hb : IsRegular b) : IsRegular (a โ b) := by
rw [IsRegular, compl_compl_inf_distrib, ha.eq, hb.eq]
#align heyting.is_regular.inf Heyting.IsRegular.inf
theorem IsRegular.himp (ha : IsRegular a) (hb : IsRegular b) : IsRegular (a โจ b) := by
rw [IsRegular, compl_compl_himp_distrib, ha.eq, hb.eq]
#align heyting.is_regular.himp Heyting.IsRegular.himp
theorem isRegular_compl (a : ฮฑ) : IsRegular aแถ :=
compl_compl_compl _
#align heyting.is_regular_compl Heyting.isRegular_compl
protected theorem IsRegular.disjoint_compl_left_iff (ha : IsRegular a) : Disjoint aแถ b โ b โค a :=
by rw [โ le_compl_iff_disjoint_left, ha.eq]
#align heyting.is_regular.disjoint_compl_left_iff Heyting.IsRegular.disjoint_compl_left_iff
protected theorem IsRegular.disjoint_compl_right_iff (hb : IsRegular b) : Disjoint a bแถ โ a โค b :=
by rw [โ le_compl_iff_disjoint_right, hb.eq]
#align heyting.is_regular.disjoint_compl_right_iff Heyting.IsRegular.disjoint_compl_right_iff
-- See note [reducible non-instances]
/-- A Heyting algebra with regular excluded middle is a boolean algebra. -/
@[reducible]
def _root_.BooleanAlgebra.ofRegular (h : โ a : ฮฑ, IsRegular (a โ aแถ)) : BooleanAlgebra ฮฑ :=
have : โ a : ฮฑ, IsCompl a aแถ := fun a =>
โจdisjoint_compl_right,
codisjoint_iff.2 <| by erw [โ (h a), compl_sup, inf_compl_eq_bot, compl_bot]โฉ
{ โนHeytingAlgebra ฮฑโบ,
GeneralizedHeytingAlgebra.toDistribLattice with
himp_eq := fun a b =>
eq_of_forall_le_iff fun c => le_himp_iff.trans (this _).le_sup_right_iff_inf_left_le.symm
inf_compl_le_bot := fun a => (this _).1.le_bot
top_le_sup_compl := fun a => (this _).2.top_le }
#align boolean_algebra.of_regular BooleanAlgebra.ofRegular
variable (ฮฑ)
/-- The boolean algebra of Heyting regular elements. -/
def Regular : Type _ :=
{ a : ฮฑ // IsRegular a }
#align heyting.regular Heyting.Regular
variable {ฮฑ}
namespace Regular
--Porting note: `val` and `prop` are new
/-- The coercion `Regular ฮฑ โ ฮฑ` -/
@[coe] def val : Regular ฮฑ โ ฮฑ :=
Subtype.val
theorem prop : โ a : Regular ฮฑ, IsRegular a.val := Subtype.prop
instance : CoeOut (Regular ฮฑ) ฮฑ := โจRegular.valโฉ
theorem coe_injective : Injective ((โ) : Regular ฮฑ โ ฮฑ) :=
Subtype.coe_injective
#align heyting.regular.coe_injective Heyting.Regular.coe_injective
@[simp]
theorem coe_inj {a b : Regular ฮฑ} : (a : ฮฑ) = b โ a = b :=
Subtype.coe_inj
#align heyting.regular.coe_inj Heyting.Regular.coe_inj
instance top : Top (Regular ฮฑ) :=
โจโจโค, isRegular_topโฉโฉ
instance bot : Bot (Regular ฮฑ) :=
โจโจโฅ, isRegular_botโฉโฉ
instance inf : Inf (Regular ฮฑ) :=
โจfun a b => โจa โ b, a.2.inf b.2โฉโฉ
instance himp : HImp (Regular ฮฑ) :=
โจfun a b => โจa โจ b, a.2.himp b.2โฉโฉ
instance hasCompl : HasCompl (Regular ฮฑ) :=
โจfun a => โจaแถ, isRegular_compl _โฉโฉ
@[simp, norm_cast]
theorem coe_top : ((โค : Regular ฮฑ) : ฮฑ) = โค :=
rfl
#align heyting.regular.coe_top Heyting.Regular.coe_top
@[simp, norm_cast]
theorem coe_bot : ((โฅ : Regular ฮฑ) : ฮฑ) = โฅ :=
rfl
#align heyting.regular.coe_bot Heyting.Regular.coe_bot
@[simp, norm_cast]
theorem coe_inf (a b : Regular ฮฑ) : (โ(a โ b) : ฮฑ) = (a : ฮฑ) โ b :=
rfl
#align heyting.regular.coe_inf Heyting.Regular.coe_inf
@[simp, norm_cast]
theorem coe_himp (a b : Regular ฮฑ) : (โ(a โจ b) : ฮฑ) = (a : ฮฑ) โจ b :=
rfl
#align heyting.regular.coe_himp Heyting.Regular.coe_himp
@[simp, norm_cast]
theorem coe_compl (a : Regular ฮฑ) : (โaแถ : ฮฑ) = (a : ฮฑ)แถ :=
rfl
#align heyting.regular.coe_compl Heyting.Regular.coe_compl
instance : Inhabited (Regular ฮฑ) :=
โจโฅโฉ
instance : SemilatticeInf (Regular ฮฑ) :=
coe_injective.semilatticeInf _ coe_inf
instance boundedOrder : BoundedOrder (Regular ฮฑ) :=
BoundedOrder.lift ((โ) : Regular ฮฑ โ ฮฑ) (fun _ _ => id) coe_top coe_bot
@[simp, norm_cast]
theorem coe_le_coe {a b : Regular ฮฑ} : (a : ฮฑ) โค b โ a โค b :=
Iff.rfl
#align heyting.regular.coe_le_coe Heyting.Regular.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {a b : Regular ฮฑ} : (a : ฮฑ) < b โ a < b :=
Iff.rfl
#align heyting.regular.coe_lt_coe Heyting.Regular.coe_lt_coe
/-- **Regularization** of `a`. The smallest regular element greater than `a`. -/
def toRegular : ฮฑ โo Regular ฮฑ :=
โจfun a => โจaแถแถ, isRegular_compl _โฉ, fun _ _ h =>
coe_le_coe.1 <| compl_le_compl <| compl_le_compl hโฉ
#align heyting.regular.to_regular Heyting.Regular.toRegular
@[simp, norm_cast]
theorem coe_toRegular (a : ฮฑ) : (toRegular a : ฮฑ) = aแถแถ :=
rfl
#align heyting.regular.coe_to_regular Heyting.Regular.coe_toRegular
@[simp]
theorem toRegular_coe (a : Regular ฮฑ) : toRegular (a : ฮฑ) = a :=
coe_injective a.2
#align heyting.regular.to_regular_coe Heyting.Regular.toRegular_coe
/-- The Galois insertion between `Regular.toRegular` and `coe`. -/
def gi : GaloisInsertion toRegular ((โ) : Regular ฮฑ โ ฮฑ)
where
choice a ha := โจa, ha.antisymm le_compl_complโฉ
gc _ b :=
coe_le_coe.symm.trans <|
โจle_compl_compl.trans, fun h => (compl_anti <| compl_anti h).trans_eq b.2โฉ
le_l_u _ := le_compl_compl
choice_eq _ ha := coe_injective <| le_compl_compl.antisymm ha
#align heyting.regular.gi Heyting.Regular.gi
instance lattice : Lattice (Regular ฮฑ) :=
gi.liftLattice
@[simp, norm_cast]
theorem coe_sup (a b : Regular ฮฑ) : (โ(a โ b) : ฮฑ) = ((a : ฮฑ) โ b)แถแถ :=
rfl
#align heyting.regular.coe_sup Heyting.Regular.coe_sup
instance : BooleanAlgebra (Regular ฮฑ) :=
{ Regular.lattice, Regular.boundedOrder, Regular.himp,
Regular.hasCompl with
le_sup_inf := fun a b c =>
coe_le_coe.1 <| by
dsimp
rw [sup_inf_left, compl_compl_inf_distrib]
inf_compl_le_bot := fun a => coe_le_coe.1 <| disjoint_iff_inf_le.1 disjoint_compl_right
top_le_sup_compl := fun a =>
coe_le_coe.1 <| by
dsimp
rw [compl_sup, inf_compl_eq_bot, compl_bot]
himp_eq := fun a b =>
coe_injective
(by
dsimp
|
rw [compl_sup, a.prop.eq]
|
instance : BooleanAlgebra (Regular ฮฑ) :=
{ Regular.lattice, Regular.boundedOrder, Regular.himp,
Regular.hasCompl with
le_sup_inf := fun a b c =>
coe_le_coe.1 <| by
dsimp
rw [sup_inf_left, compl_compl_inf_distrib]
inf_compl_le_bot := fun a => coe_le_coe.1 <| disjoint_iff_inf_le.1 disjoint_compl_right
top_le_sup_compl := fun a =>
coe_le_coe.1 <| by
dsimp
rw [compl_sup, inf_compl_eq_bot, compl_bot]
himp_eq := fun a b =>
coe_injective
(by
dsimp
|
Mathlib.Order.Heyting.Regular.224_0.HFWfWmvQ8w4gAZO
|
instance : BooleanAlgebra (Regular ฮฑ)
|
Mathlib_Order_Heyting_Regular
|
ฮฑ : Type u_1
instโ : HeytingAlgebra ฮฑ
aโ bโ : ฮฑ
srcโยณ : Lattice (Regular ฮฑ) := lattice
srcโยฒ : BoundedOrder (Regular ฮฑ) := boundedOrder
srcโยน : HImp (Regular ฮฑ) := himp
srcโ : HasCompl (Regular ฮฑ) := hasCompl
a b : Regular ฮฑ
โข โa โจ โb = ((โb)แถ โ โa)แถ
|
/-
Copyright (c) 2022 Yaรซl Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaรซl Dillies
-/
import Mathlib.Order.GaloisConnection
#align_import order.heyting.regular from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
/-!
# Heyting regular elements
This file defines Heyting regular elements, elements of a Heyting algebra that are their own double
complement, and proves that they form a boolean algebra.
From a logic standpoint, this means that we can perform classical logic within intuitionistic logic
by simply double-negating all propositions. This is practical for synthetic computability theory.
## Main declarations
* `IsRegular`: `a` is Heyting-regular if `aแถแถ = a`.
* `Regular`: The subtype of Heyting-regular elements.
* `Regular.BooleanAlgebra`: Heyting-regular elements form a boolean algebra.
## References
* [Francis Borceux, *Handbook of Categorical Algebra III*][borceux-vol3]
-/
open Function
variable {ฮฑ : Type*}
namespace Heyting
section HasCompl
variable [HasCompl ฮฑ] {a : ฮฑ}
/-- An element of a Heyting algebra is regular if its double complement is itself. -/
def IsRegular (a : ฮฑ) : Prop :=
aแถแถ = a
#align heyting.is_regular Heyting.IsRegular
protected theorem IsRegular.eq : IsRegular a โ aแถแถ = a :=
id
#align heyting.is_regular.eq Heyting.IsRegular.eq
instance IsRegular.decidablePred [DecidableEq ฮฑ] : @DecidablePred ฮฑ IsRegular := fun _ =>
โนDecidableEq ฮฑโบ _ _
#align heyting.is_regular.decidable_pred Heyting.IsRegular.decidablePred
end HasCompl
section HeytingAlgebra
variable [HeytingAlgebra ฮฑ] {a b : ฮฑ}
theorem isRegular_bot : IsRegular (โฅ : ฮฑ) := by rw [IsRegular, compl_bot, compl_top]
#align heyting.is_regular_bot Heyting.isRegular_bot
theorem isRegular_top : IsRegular (โค : ฮฑ) := by rw [IsRegular, compl_top, compl_bot]
#align heyting.is_regular_top Heyting.isRegular_top
theorem IsRegular.inf (ha : IsRegular a) (hb : IsRegular b) : IsRegular (a โ b) := by
rw [IsRegular, compl_compl_inf_distrib, ha.eq, hb.eq]
#align heyting.is_regular.inf Heyting.IsRegular.inf
theorem IsRegular.himp (ha : IsRegular a) (hb : IsRegular b) : IsRegular (a โจ b) := by
rw [IsRegular, compl_compl_himp_distrib, ha.eq, hb.eq]
#align heyting.is_regular.himp Heyting.IsRegular.himp
theorem isRegular_compl (a : ฮฑ) : IsRegular aแถ :=
compl_compl_compl _
#align heyting.is_regular_compl Heyting.isRegular_compl
protected theorem IsRegular.disjoint_compl_left_iff (ha : IsRegular a) : Disjoint aแถ b โ b โค a :=
by rw [โ le_compl_iff_disjoint_left, ha.eq]
#align heyting.is_regular.disjoint_compl_left_iff Heyting.IsRegular.disjoint_compl_left_iff
protected theorem IsRegular.disjoint_compl_right_iff (hb : IsRegular b) : Disjoint a bแถ โ a โค b :=
by rw [โ le_compl_iff_disjoint_right, hb.eq]
#align heyting.is_regular.disjoint_compl_right_iff Heyting.IsRegular.disjoint_compl_right_iff
-- See note [reducible non-instances]
/-- A Heyting algebra with regular excluded middle is a boolean algebra. -/
@[reducible]
def _root_.BooleanAlgebra.ofRegular (h : โ a : ฮฑ, IsRegular (a โ aแถ)) : BooleanAlgebra ฮฑ :=
have : โ a : ฮฑ, IsCompl a aแถ := fun a =>
โจdisjoint_compl_right,
codisjoint_iff.2 <| by erw [โ (h a), compl_sup, inf_compl_eq_bot, compl_bot]โฉ
{ โนHeytingAlgebra ฮฑโบ,
GeneralizedHeytingAlgebra.toDistribLattice with
himp_eq := fun a b =>
eq_of_forall_le_iff fun c => le_himp_iff.trans (this _).le_sup_right_iff_inf_left_le.symm
inf_compl_le_bot := fun a => (this _).1.le_bot
top_le_sup_compl := fun a => (this _).2.top_le }
#align boolean_algebra.of_regular BooleanAlgebra.ofRegular
variable (ฮฑ)
/-- The boolean algebra of Heyting regular elements. -/
def Regular : Type _ :=
{ a : ฮฑ // IsRegular a }
#align heyting.regular Heyting.Regular
variable {ฮฑ}
namespace Regular
--Porting note: `val` and `prop` are new
/-- The coercion `Regular ฮฑ โ ฮฑ` -/
@[coe] def val : Regular ฮฑ โ ฮฑ :=
Subtype.val
theorem prop : โ a : Regular ฮฑ, IsRegular a.val := Subtype.prop
instance : CoeOut (Regular ฮฑ) ฮฑ := โจRegular.valโฉ
theorem coe_injective : Injective ((โ) : Regular ฮฑ โ ฮฑ) :=
Subtype.coe_injective
#align heyting.regular.coe_injective Heyting.Regular.coe_injective
@[simp]
theorem coe_inj {a b : Regular ฮฑ} : (a : ฮฑ) = b โ a = b :=
Subtype.coe_inj
#align heyting.regular.coe_inj Heyting.Regular.coe_inj
instance top : Top (Regular ฮฑ) :=
โจโจโค, isRegular_topโฉโฉ
instance bot : Bot (Regular ฮฑ) :=
โจโจโฅ, isRegular_botโฉโฉ
instance inf : Inf (Regular ฮฑ) :=
โจfun a b => โจa โ b, a.2.inf b.2โฉโฉ
instance himp : HImp (Regular ฮฑ) :=
โจfun a b => โจa โจ b, a.2.himp b.2โฉโฉ
instance hasCompl : HasCompl (Regular ฮฑ) :=
โจfun a => โจaแถ, isRegular_compl _โฉโฉ
@[simp, norm_cast]
theorem coe_top : ((โค : Regular ฮฑ) : ฮฑ) = โค :=
rfl
#align heyting.regular.coe_top Heyting.Regular.coe_top
@[simp, norm_cast]
theorem coe_bot : ((โฅ : Regular ฮฑ) : ฮฑ) = โฅ :=
rfl
#align heyting.regular.coe_bot Heyting.Regular.coe_bot
@[simp, norm_cast]
theorem coe_inf (a b : Regular ฮฑ) : (โ(a โ b) : ฮฑ) = (a : ฮฑ) โ b :=
rfl
#align heyting.regular.coe_inf Heyting.Regular.coe_inf
@[simp, norm_cast]
theorem coe_himp (a b : Regular ฮฑ) : (โ(a โจ b) : ฮฑ) = (a : ฮฑ) โจ b :=
rfl
#align heyting.regular.coe_himp Heyting.Regular.coe_himp
@[simp, norm_cast]
theorem coe_compl (a : Regular ฮฑ) : (โaแถ : ฮฑ) = (a : ฮฑ)แถ :=
rfl
#align heyting.regular.coe_compl Heyting.Regular.coe_compl
instance : Inhabited (Regular ฮฑ) :=
โจโฅโฉ
instance : SemilatticeInf (Regular ฮฑ) :=
coe_injective.semilatticeInf _ coe_inf
instance boundedOrder : BoundedOrder (Regular ฮฑ) :=
BoundedOrder.lift ((โ) : Regular ฮฑ โ ฮฑ) (fun _ _ => id) coe_top coe_bot
@[simp, norm_cast]
theorem coe_le_coe {a b : Regular ฮฑ} : (a : ฮฑ) โค b โ a โค b :=
Iff.rfl
#align heyting.regular.coe_le_coe Heyting.Regular.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {a b : Regular ฮฑ} : (a : ฮฑ) < b โ a < b :=
Iff.rfl
#align heyting.regular.coe_lt_coe Heyting.Regular.coe_lt_coe
/-- **Regularization** of `a`. The smallest regular element greater than `a`. -/
def toRegular : ฮฑ โo Regular ฮฑ :=
โจfun a => โจaแถแถ, isRegular_compl _โฉ, fun _ _ h =>
coe_le_coe.1 <| compl_le_compl <| compl_le_compl hโฉ
#align heyting.regular.to_regular Heyting.Regular.toRegular
@[simp, norm_cast]
theorem coe_toRegular (a : ฮฑ) : (toRegular a : ฮฑ) = aแถแถ :=
rfl
#align heyting.regular.coe_to_regular Heyting.Regular.coe_toRegular
@[simp]
theorem toRegular_coe (a : Regular ฮฑ) : toRegular (a : ฮฑ) = a :=
coe_injective a.2
#align heyting.regular.to_regular_coe Heyting.Regular.toRegular_coe
/-- The Galois insertion between `Regular.toRegular` and `coe`. -/
def gi : GaloisInsertion toRegular ((โ) : Regular ฮฑ โ ฮฑ)
where
choice a ha := โจa, ha.antisymm le_compl_complโฉ
gc _ b :=
coe_le_coe.symm.trans <|
โจle_compl_compl.trans, fun h => (compl_anti <| compl_anti h).trans_eq b.2โฉ
le_l_u _ := le_compl_compl
choice_eq _ ha := coe_injective <| le_compl_compl.antisymm ha
#align heyting.regular.gi Heyting.Regular.gi
instance lattice : Lattice (Regular ฮฑ) :=
gi.liftLattice
@[simp, norm_cast]
theorem coe_sup (a b : Regular ฮฑ) : (โ(a โ b) : ฮฑ) = ((a : ฮฑ) โ b)แถแถ :=
rfl
#align heyting.regular.coe_sup Heyting.Regular.coe_sup
instance : BooleanAlgebra (Regular ฮฑ) :=
{ Regular.lattice, Regular.boundedOrder, Regular.himp,
Regular.hasCompl with
le_sup_inf := fun a b c =>
coe_le_coe.1 <| by
dsimp
rw [sup_inf_left, compl_compl_inf_distrib]
inf_compl_le_bot := fun a => coe_le_coe.1 <| disjoint_iff_inf_le.1 disjoint_compl_right
top_le_sup_compl := fun a =>
coe_le_coe.1 <| by
dsimp
rw [compl_sup, inf_compl_eq_bot, compl_bot]
himp_eq := fun a b =>
coe_injective
(by
dsimp
rw [compl_sup, a.prop.eq]
|
refine' eq_of_forall_le_iff fun c => le_himp_iff.trans _
|
instance : BooleanAlgebra (Regular ฮฑ) :=
{ Regular.lattice, Regular.boundedOrder, Regular.himp,
Regular.hasCompl with
le_sup_inf := fun a b c =>
coe_le_coe.1 <| by
dsimp
rw [sup_inf_left, compl_compl_inf_distrib]
inf_compl_le_bot := fun a => coe_le_coe.1 <| disjoint_iff_inf_le.1 disjoint_compl_right
top_le_sup_compl := fun a =>
coe_le_coe.1 <| by
dsimp
rw [compl_sup, inf_compl_eq_bot, compl_bot]
himp_eq := fun a b =>
coe_injective
(by
dsimp
rw [compl_sup, a.prop.eq]
|
Mathlib.Order.Heyting.Regular.224_0.HFWfWmvQ8w4gAZO
|
instance : BooleanAlgebra (Regular ฮฑ)
|
Mathlib_Order_Heyting_Regular
|
ฮฑ : Type u_1
instโ : HeytingAlgebra ฮฑ
aโ bโ : ฮฑ
srcโยณ : Lattice (Regular ฮฑ) := lattice
srcโยฒ : BoundedOrder (Regular ฮฑ) := boundedOrder
srcโยน : HImp (Regular ฮฑ) := himp
srcโ : HasCompl (Regular ฮฑ) := hasCompl
a b : Regular ฮฑ
c : ฮฑ
โข c โ โa โค โb โ c โค ((โb)แถ โ โa)แถ
|
/-
Copyright (c) 2022 Yaรซl Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaรซl Dillies
-/
import Mathlib.Order.GaloisConnection
#align_import order.heyting.regular from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
/-!
# Heyting regular elements
This file defines Heyting regular elements, elements of a Heyting algebra that are their own double
complement, and proves that they form a boolean algebra.
From a logic standpoint, this means that we can perform classical logic within intuitionistic logic
by simply double-negating all propositions. This is practical for synthetic computability theory.
## Main declarations
* `IsRegular`: `a` is Heyting-regular if `aแถแถ = a`.
* `Regular`: The subtype of Heyting-regular elements.
* `Regular.BooleanAlgebra`: Heyting-regular elements form a boolean algebra.
## References
* [Francis Borceux, *Handbook of Categorical Algebra III*][borceux-vol3]
-/
open Function
variable {ฮฑ : Type*}
namespace Heyting
section HasCompl
variable [HasCompl ฮฑ] {a : ฮฑ}
/-- An element of a Heyting algebra is regular if its double complement is itself. -/
def IsRegular (a : ฮฑ) : Prop :=
aแถแถ = a
#align heyting.is_regular Heyting.IsRegular
protected theorem IsRegular.eq : IsRegular a โ aแถแถ = a :=
id
#align heyting.is_regular.eq Heyting.IsRegular.eq
instance IsRegular.decidablePred [DecidableEq ฮฑ] : @DecidablePred ฮฑ IsRegular := fun _ =>
โนDecidableEq ฮฑโบ _ _
#align heyting.is_regular.decidable_pred Heyting.IsRegular.decidablePred
end HasCompl
section HeytingAlgebra
variable [HeytingAlgebra ฮฑ] {a b : ฮฑ}
theorem isRegular_bot : IsRegular (โฅ : ฮฑ) := by rw [IsRegular, compl_bot, compl_top]
#align heyting.is_regular_bot Heyting.isRegular_bot
theorem isRegular_top : IsRegular (โค : ฮฑ) := by rw [IsRegular, compl_top, compl_bot]
#align heyting.is_regular_top Heyting.isRegular_top
theorem IsRegular.inf (ha : IsRegular a) (hb : IsRegular b) : IsRegular (a โ b) := by
rw [IsRegular, compl_compl_inf_distrib, ha.eq, hb.eq]
#align heyting.is_regular.inf Heyting.IsRegular.inf
theorem IsRegular.himp (ha : IsRegular a) (hb : IsRegular b) : IsRegular (a โจ b) := by
rw [IsRegular, compl_compl_himp_distrib, ha.eq, hb.eq]
#align heyting.is_regular.himp Heyting.IsRegular.himp
theorem isRegular_compl (a : ฮฑ) : IsRegular aแถ :=
compl_compl_compl _
#align heyting.is_regular_compl Heyting.isRegular_compl
protected theorem IsRegular.disjoint_compl_left_iff (ha : IsRegular a) : Disjoint aแถ b โ b โค a :=
by rw [โ le_compl_iff_disjoint_left, ha.eq]
#align heyting.is_regular.disjoint_compl_left_iff Heyting.IsRegular.disjoint_compl_left_iff
protected theorem IsRegular.disjoint_compl_right_iff (hb : IsRegular b) : Disjoint a bแถ โ a โค b :=
by rw [โ le_compl_iff_disjoint_right, hb.eq]
#align heyting.is_regular.disjoint_compl_right_iff Heyting.IsRegular.disjoint_compl_right_iff
-- See note [reducible non-instances]
/-- A Heyting algebra with regular excluded middle is a boolean algebra. -/
@[reducible]
def _root_.BooleanAlgebra.ofRegular (h : โ a : ฮฑ, IsRegular (a โ aแถ)) : BooleanAlgebra ฮฑ :=
have : โ a : ฮฑ, IsCompl a aแถ := fun a =>
โจdisjoint_compl_right,
codisjoint_iff.2 <| by erw [โ (h a), compl_sup, inf_compl_eq_bot, compl_bot]โฉ
{ โนHeytingAlgebra ฮฑโบ,
GeneralizedHeytingAlgebra.toDistribLattice with
himp_eq := fun a b =>
eq_of_forall_le_iff fun c => le_himp_iff.trans (this _).le_sup_right_iff_inf_left_le.symm
inf_compl_le_bot := fun a => (this _).1.le_bot
top_le_sup_compl := fun a => (this _).2.top_le }
#align boolean_algebra.of_regular BooleanAlgebra.ofRegular
variable (ฮฑ)
/-- The boolean algebra of Heyting regular elements. -/
def Regular : Type _ :=
{ a : ฮฑ // IsRegular a }
#align heyting.regular Heyting.Regular
variable {ฮฑ}
namespace Regular
--Porting note: `val` and `prop` are new
/-- The coercion `Regular ฮฑ โ ฮฑ` -/
@[coe] def val : Regular ฮฑ โ ฮฑ :=
Subtype.val
theorem prop : โ a : Regular ฮฑ, IsRegular a.val := Subtype.prop
instance : CoeOut (Regular ฮฑ) ฮฑ := โจRegular.valโฉ
theorem coe_injective : Injective ((โ) : Regular ฮฑ โ ฮฑ) :=
Subtype.coe_injective
#align heyting.regular.coe_injective Heyting.Regular.coe_injective
@[simp]
theorem coe_inj {a b : Regular ฮฑ} : (a : ฮฑ) = b โ a = b :=
Subtype.coe_inj
#align heyting.regular.coe_inj Heyting.Regular.coe_inj
instance top : Top (Regular ฮฑ) :=
โจโจโค, isRegular_topโฉโฉ
instance bot : Bot (Regular ฮฑ) :=
โจโจโฅ, isRegular_botโฉโฉ
instance inf : Inf (Regular ฮฑ) :=
โจfun a b => โจa โ b, a.2.inf b.2โฉโฉ
instance himp : HImp (Regular ฮฑ) :=
โจfun a b => โจa โจ b, a.2.himp b.2โฉโฉ
instance hasCompl : HasCompl (Regular ฮฑ) :=
โจfun a => โจaแถ, isRegular_compl _โฉโฉ
@[simp, norm_cast]
theorem coe_top : ((โค : Regular ฮฑ) : ฮฑ) = โค :=
rfl
#align heyting.regular.coe_top Heyting.Regular.coe_top
@[simp, norm_cast]
theorem coe_bot : ((โฅ : Regular ฮฑ) : ฮฑ) = โฅ :=
rfl
#align heyting.regular.coe_bot Heyting.Regular.coe_bot
@[simp, norm_cast]
theorem coe_inf (a b : Regular ฮฑ) : (โ(a โ b) : ฮฑ) = (a : ฮฑ) โ b :=
rfl
#align heyting.regular.coe_inf Heyting.Regular.coe_inf
@[simp, norm_cast]
theorem coe_himp (a b : Regular ฮฑ) : (โ(a โจ b) : ฮฑ) = (a : ฮฑ) โจ b :=
rfl
#align heyting.regular.coe_himp Heyting.Regular.coe_himp
@[simp, norm_cast]
theorem coe_compl (a : Regular ฮฑ) : (โaแถ : ฮฑ) = (a : ฮฑ)แถ :=
rfl
#align heyting.regular.coe_compl Heyting.Regular.coe_compl
instance : Inhabited (Regular ฮฑ) :=
โจโฅโฉ
instance : SemilatticeInf (Regular ฮฑ) :=
coe_injective.semilatticeInf _ coe_inf
instance boundedOrder : BoundedOrder (Regular ฮฑ) :=
BoundedOrder.lift ((โ) : Regular ฮฑ โ ฮฑ) (fun _ _ => id) coe_top coe_bot
@[simp, norm_cast]
theorem coe_le_coe {a b : Regular ฮฑ} : (a : ฮฑ) โค b โ a โค b :=
Iff.rfl
#align heyting.regular.coe_le_coe Heyting.Regular.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {a b : Regular ฮฑ} : (a : ฮฑ) < b โ a < b :=
Iff.rfl
#align heyting.regular.coe_lt_coe Heyting.Regular.coe_lt_coe
/-- **Regularization** of `a`. The smallest regular element greater than `a`. -/
def toRegular : ฮฑ โo Regular ฮฑ :=
โจfun a => โจaแถแถ, isRegular_compl _โฉ, fun _ _ h =>
coe_le_coe.1 <| compl_le_compl <| compl_le_compl hโฉ
#align heyting.regular.to_regular Heyting.Regular.toRegular
@[simp, norm_cast]
theorem coe_toRegular (a : ฮฑ) : (toRegular a : ฮฑ) = aแถแถ :=
rfl
#align heyting.regular.coe_to_regular Heyting.Regular.coe_toRegular
@[simp]
theorem toRegular_coe (a : Regular ฮฑ) : toRegular (a : ฮฑ) = a :=
coe_injective a.2
#align heyting.regular.to_regular_coe Heyting.Regular.toRegular_coe
/-- The Galois insertion between `Regular.toRegular` and `coe`. -/
def gi : GaloisInsertion toRegular ((โ) : Regular ฮฑ โ ฮฑ)
where
choice a ha := โจa, ha.antisymm le_compl_complโฉ
gc _ b :=
coe_le_coe.symm.trans <|
โจle_compl_compl.trans, fun h => (compl_anti <| compl_anti h).trans_eq b.2โฉ
le_l_u _ := le_compl_compl
choice_eq _ ha := coe_injective <| le_compl_compl.antisymm ha
#align heyting.regular.gi Heyting.Regular.gi
instance lattice : Lattice (Regular ฮฑ) :=
gi.liftLattice
@[simp, norm_cast]
theorem coe_sup (a b : Regular ฮฑ) : (โ(a โ b) : ฮฑ) = ((a : ฮฑ) โ b)แถแถ :=
rfl
#align heyting.regular.coe_sup Heyting.Regular.coe_sup
instance : BooleanAlgebra (Regular ฮฑ) :=
{ Regular.lattice, Regular.boundedOrder, Regular.himp,
Regular.hasCompl with
le_sup_inf := fun a b c =>
coe_le_coe.1 <| by
dsimp
rw [sup_inf_left, compl_compl_inf_distrib]
inf_compl_le_bot := fun a => coe_le_coe.1 <| disjoint_iff_inf_le.1 disjoint_compl_right
top_le_sup_compl := fun a =>
coe_le_coe.1 <| by
dsimp
rw [compl_sup, inf_compl_eq_bot, compl_bot]
himp_eq := fun a b =>
coe_injective
(by
dsimp
rw [compl_sup, a.prop.eq]
refine' eq_of_forall_le_iff fun c => le_himp_iff.trans _
|
rw [le_compl_iff_disjoint_right, disjoint_left_comm]
|
instance : BooleanAlgebra (Regular ฮฑ) :=
{ Regular.lattice, Regular.boundedOrder, Regular.himp,
Regular.hasCompl with
le_sup_inf := fun a b c =>
coe_le_coe.1 <| by
dsimp
rw [sup_inf_left, compl_compl_inf_distrib]
inf_compl_le_bot := fun a => coe_le_coe.1 <| disjoint_iff_inf_le.1 disjoint_compl_right
top_le_sup_compl := fun a =>
coe_le_coe.1 <| by
dsimp
rw [compl_sup, inf_compl_eq_bot, compl_bot]
himp_eq := fun a b =>
coe_injective
(by
dsimp
rw [compl_sup, a.prop.eq]
refine' eq_of_forall_le_iff fun c => le_himp_iff.trans _
|
Mathlib.Order.Heyting.Regular.224_0.HFWfWmvQ8w4gAZO
|
instance : BooleanAlgebra (Regular ฮฑ)
|
Mathlib_Order_Heyting_Regular
|
ฮฑ : Type u_1
instโ : HeytingAlgebra ฮฑ
aโ bโ : ฮฑ
srcโยณ : Lattice (Regular ฮฑ) := lattice
srcโยฒ : BoundedOrder (Regular ฮฑ) := boundedOrder
srcโยน : HImp (Regular ฮฑ) := himp
srcโ : HasCompl (Regular ฮฑ) := hasCompl
a b : Regular ฮฑ
c : ฮฑ
โข c โ โa โค โb โ Disjoint (โb)แถ (c โ โa)
|
/-
Copyright (c) 2022 Yaรซl Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaรซl Dillies
-/
import Mathlib.Order.GaloisConnection
#align_import order.heyting.regular from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
/-!
# Heyting regular elements
This file defines Heyting regular elements, elements of a Heyting algebra that are their own double
complement, and proves that they form a boolean algebra.
From a logic standpoint, this means that we can perform classical logic within intuitionistic logic
by simply double-negating all propositions. This is practical for synthetic computability theory.
## Main declarations
* `IsRegular`: `a` is Heyting-regular if `aแถแถ = a`.
* `Regular`: The subtype of Heyting-regular elements.
* `Regular.BooleanAlgebra`: Heyting-regular elements form a boolean algebra.
## References
* [Francis Borceux, *Handbook of Categorical Algebra III*][borceux-vol3]
-/
open Function
variable {ฮฑ : Type*}
namespace Heyting
section HasCompl
variable [HasCompl ฮฑ] {a : ฮฑ}
/-- An element of a Heyting algebra is regular if its double complement is itself. -/
def IsRegular (a : ฮฑ) : Prop :=
aแถแถ = a
#align heyting.is_regular Heyting.IsRegular
protected theorem IsRegular.eq : IsRegular a โ aแถแถ = a :=
id
#align heyting.is_regular.eq Heyting.IsRegular.eq
instance IsRegular.decidablePred [DecidableEq ฮฑ] : @DecidablePred ฮฑ IsRegular := fun _ =>
โนDecidableEq ฮฑโบ _ _
#align heyting.is_regular.decidable_pred Heyting.IsRegular.decidablePred
end HasCompl
section HeytingAlgebra
variable [HeytingAlgebra ฮฑ] {a b : ฮฑ}
theorem isRegular_bot : IsRegular (โฅ : ฮฑ) := by rw [IsRegular, compl_bot, compl_top]
#align heyting.is_regular_bot Heyting.isRegular_bot
theorem isRegular_top : IsRegular (โค : ฮฑ) := by rw [IsRegular, compl_top, compl_bot]
#align heyting.is_regular_top Heyting.isRegular_top
theorem IsRegular.inf (ha : IsRegular a) (hb : IsRegular b) : IsRegular (a โ b) := by
rw [IsRegular, compl_compl_inf_distrib, ha.eq, hb.eq]
#align heyting.is_regular.inf Heyting.IsRegular.inf
theorem IsRegular.himp (ha : IsRegular a) (hb : IsRegular b) : IsRegular (a โจ b) := by
rw [IsRegular, compl_compl_himp_distrib, ha.eq, hb.eq]
#align heyting.is_regular.himp Heyting.IsRegular.himp
theorem isRegular_compl (a : ฮฑ) : IsRegular aแถ :=
compl_compl_compl _
#align heyting.is_regular_compl Heyting.isRegular_compl
protected theorem IsRegular.disjoint_compl_left_iff (ha : IsRegular a) : Disjoint aแถ b โ b โค a :=
by rw [โ le_compl_iff_disjoint_left, ha.eq]
#align heyting.is_regular.disjoint_compl_left_iff Heyting.IsRegular.disjoint_compl_left_iff
protected theorem IsRegular.disjoint_compl_right_iff (hb : IsRegular b) : Disjoint a bแถ โ a โค b :=
by rw [โ le_compl_iff_disjoint_right, hb.eq]
#align heyting.is_regular.disjoint_compl_right_iff Heyting.IsRegular.disjoint_compl_right_iff
-- See note [reducible non-instances]
/-- A Heyting algebra with regular excluded middle is a boolean algebra. -/
@[reducible]
def _root_.BooleanAlgebra.ofRegular (h : โ a : ฮฑ, IsRegular (a โ aแถ)) : BooleanAlgebra ฮฑ :=
have : โ a : ฮฑ, IsCompl a aแถ := fun a =>
โจdisjoint_compl_right,
codisjoint_iff.2 <| by erw [โ (h a), compl_sup, inf_compl_eq_bot, compl_bot]โฉ
{ โนHeytingAlgebra ฮฑโบ,
GeneralizedHeytingAlgebra.toDistribLattice with
himp_eq := fun a b =>
eq_of_forall_le_iff fun c => le_himp_iff.trans (this _).le_sup_right_iff_inf_left_le.symm
inf_compl_le_bot := fun a => (this _).1.le_bot
top_le_sup_compl := fun a => (this _).2.top_le }
#align boolean_algebra.of_regular BooleanAlgebra.ofRegular
variable (ฮฑ)
/-- The boolean algebra of Heyting regular elements. -/
def Regular : Type _ :=
{ a : ฮฑ // IsRegular a }
#align heyting.regular Heyting.Regular
variable {ฮฑ}
namespace Regular
--Porting note: `val` and `prop` are new
/-- The coercion `Regular ฮฑ โ ฮฑ` -/
@[coe] def val : Regular ฮฑ โ ฮฑ :=
Subtype.val
theorem prop : โ a : Regular ฮฑ, IsRegular a.val := Subtype.prop
instance : CoeOut (Regular ฮฑ) ฮฑ := โจRegular.valโฉ
theorem coe_injective : Injective ((โ) : Regular ฮฑ โ ฮฑ) :=
Subtype.coe_injective
#align heyting.regular.coe_injective Heyting.Regular.coe_injective
@[simp]
theorem coe_inj {a b : Regular ฮฑ} : (a : ฮฑ) = b โ a = b :=
Subtype.coe_inj
#align heyting.regular.coe_inj Heyting.Regular.coe_inj
instance top : Top (Regular ฮฑ) :=
โจโจโค, isRegular_topโฉโฉ
instance bot : Bot (Regular ฮฑ) :=
โจโจโฅ, isRegular_botโฉโฉ
instance inf : Inf (Regular ฮฑ) :=
โจfun a b => โจa โ b, a.2.inf b.2โฉโฉ
instance himp : HImp (Regular ฮฑ) :=
โจfun a b => โจa โจ b, a.2.himp b.2โฉโฉ
instance hasCompl : HasCompl (Regular ฮฑ) :=
โจfun a => โจaแถ, isRegular_compl _โฉโฉ
@[simp, norm_cast]
theorem coe_top : ((โค : Regular ฮฑ) : ฮฑ) = โค :=
rfl
#align heyting.regular.coe_top Heyting.Regular.coe_top
@[simp, norm_cast]
theorem coe_bot : ((โฅ : Regular ฮฑ) : ฮฑ) = โฅ :=
rfl
#align heyting.regular.coe_bot Heyting.Regular.coe_bot
@[simp, norm_cast]
theorem coe_inf (a b : Regular ฮฑ) : (โ(a โ b) : ฮฑ) = (a : ฮฑ) โ b :=
rfl
#align heyting.regular.coe_inf Heyting.Regular.coe_inf
@[simp, norm_cast]
theorem coe_himp (a b : Regular ฮฑ) : (โ(a โจ b) : ฮฑ) = (a : ฮฑ) โจ b :=
rfl
#align heyting.regular.coe_himp Heyting.Regular.coe_himp
@[simp, norm_cast]
theorem coe_compl (a : Regular ฮฑ) : (โaแถ : ฮฑ) = (a : ฮฑ)แถ :=
rfl
#align heyting.regular.coe_compl Heyting.Regular.coe_compl
instance : Inhabited (Regular ฮฑ) :=
โจโฅโฉ
instance : SemilatticeInf (Regular ฮฑ) :=
coe_injective.semilatticeInf _ coe_inf
instance boundedOrder : BoundedOrder (Regular ฮฑ) :=
BoundedOrder.lift ((โ) : Regular ฮฑ โ ฮฑ) (fun _ _ => id) coe_top coe_bot
@[simp, norm_cast]
theorem coe_le_coe {a b : Regular ฮฑ} : (a : ฮฑ) โค b โ a โค b :=
Iff.rfl
#align heyting.regular.coe_le_coe Heyting.Regular.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {a b : Regular ฮฑ} : (a : ฮฑ) < b โ a < b :=
Iff.rfl
#align heyting.regular.coe_lt_coe Heyting.Regular.coe_lt_coe
/-- **Regularization** of `a`. The smallest regular element greater than `a`. -/
def toRegular : ฮฑ โo Regular ฮฑ :=
โจfun a => โจaแถแถ, isRegular_compl _โฉ, fun _ _ h =>
coe_le_coe.1 <| compl_le_compl <| compl_le_compl hโฉ
#align heyting.regular.to_regular Heyting.Regular.toRegular
@[simp, norm_cast]
theorem coe_toRegular (a : ฮฑ) : (toRegular a : ฮฑ) = aแถแถ :=
rfl
#align heyting.regular.coe_to_regular Heyting.Regular.coe_toRegular
@[simp]
theorem toRegular_coe (a : Regular ฮฑ) : toRegular (a : ฮฑ) = a :=
coe_injective a.2
#align heyting.regular.to_regular_coe Heyting.Regular.toRegular_coe
/-- The Galois insertion between `Regular.toRegular` and `coe`. -/
def gi : GaloisInsertion toRegular ((โ) : Regular ฮฑ โ ฮฑ)
where
choice a ha := โจa, ha.antisymm le_compl_complโฉ
gc _ b :=
coe_le_coe.symm.trans <|
โจle_compl_compl.trans, fun h => (compl_anti <| compl_anti h).trans_eq b.2โฉ
le_l_u _ := le_compl_compl
choice_eq _ ha := coe_injective <| le_compl_compl.antisymm ha
#align heyting.regular.gi Heyting.Regular.gi
instance lattice : Lattice (Regular ฮฑ) :=
gi.liftLattice
@[simp, norm_cast]
theorem coe_sup (a b : Regular ฮฑ) : (โ(a โ b) : ฮฑ) = ((a : ฮฑ) โ b)แถแถ :=
rfl
#align heyting.regular.coe_sup Heyting.Regular.coe_sup
instance : BooleanAlgebra (Regular ฮฑ) :=
{ Regular.lattice, Regular.boundedOrder, Regular.himp,
Regular.hasCompl with
le_sup_inf := fun a b c =>
coe_le_coe.1 <| by
dsimp
rw [sup_inf_left, compl_compl_inf_distrib]
inf_compl_le_bot := fun a => coe_le_coe.1 <| disjoint_iff_inf_le.1 disjoint_compl_right
top_le_sup_compl := fun a =>
coe_le_coe.1 <| by
dsimp
rw [compl_sup, inf_compl_eq_bot, compl_bot]
himp_eq := fun a b =>
coe_injective
(by
dsimp
rw [compl_sup, a.prop.eq]
refine' eq_of_forall_le_iff fun c => le_himp_iff.trans _
rw [le_compl_iff_disjoint_right, disjoint_left_comm]
|
rw [b.prop.disjoint_compl_left_iff]
|
instance : BooleanAlgebra (Regular ฮฑ) :=
{ Regular.lattice, Regular.boundedOrder, Regular.himp,
Regular.hasCompl with
le_sup_inf := fun a b c =>
coe_le_coe.1 <| by
dsimp
rw [sup_inf_left, compl_compl_inf_distrib]
inf_compl_le_bot := fun a => coe_le_coe.1 <| disjoint_iff_inf_le.1 disjoint_compl_right
top_le_sup_compl := fun a =>
coe_le_coe.1 <| by
dsimp
rw [compl_sup, inf_compl_eq_bot, compl_bot]
himp_eq := fun a b =>
coe_injective
(by
dsimp
rw [compl_sup, a.prop.eq]
refine' eq_of_forall_le_iff fun c => le_himp_iff.trans _
rw [le_compl_iff_disjoint_right, disjoint_left_comm]
|
Mathlib.Order.Heyting.Regular.224_0.HFWfWmvQ8w4gAZO
|
instance : BooleanAlgebra (Regular ฮฑ)
|
Mathlib_Order_Heyting_Regular
|
๐ : Type u_1
instโโด : NontriviallyNormedField ๐
E : Type u_2
instโยณ : SeminormedAddCommGroup E
instโยฒ : NormedSpace ๐ E
F : Type u_3
instโยน : NormedAddCommGroup F
instโ : NormedSpace ๐ F
โข โinclusionInDoubleDual ๐ Eโ โค 1
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# The topological dual of a normed space
In this file we define the topological dual `NormedSpace.Dual` of a normed space, and the
continuous linear map `NormedSpace.inclusionInDoubleDual` from a normed space into its double
dual.
For base field `๐ = โ` or `๐ = โ`, this map is actually an isometric embedding; we provide a
version `NormedSpace.inclusionInDoubleDualLi` of the map which is of type a bundled linear
isometric embedding, `E โโแตข[๐] (Dual ๐ (Dual ๐ E))`.
Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
theory for `SeminormedAddCommGroup` and we specialize to `NormedAddCommGroup` when needed.
## Main definitions
* `inclusionInDoubleDual` and `inclusionInDoubleDualLi` are the inclusion of a normed space
in its double dual, considered as a bounded linear map and as a linear isometry, respectively.
* `polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals `x'` for which
`โx' zโ โค 1` for every `z โ s`.
## Tags
dual
-/
noncomputable section
open Classical Topology Bornology
universe u v
namespace NormedSpace
section General
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable (E : Type*) [SeminormedAddCommGroup E] [NormedSpace ๐ E]
variable (F : Type*) [NormedAddCommGroup F] [NormedSpace ๐ F]
/-- The topological dual of a seminormed space `E`. -/
abbrev Dual : Type _ := E โL[๐] ๐
#align normed_space.dual NormedSpace.Dual
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_eq until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : NormedSpace ๐ (Dual ๐ E) := inferInstance
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_le until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : SeminormedAddCommGroup (Dual ๐ E) := inferInstance
/-- The inclusion of a normed space in its double (topological) dual, considered
as a bounded linear map. -/
def inclusionInDoubleDual : E โL[๐] Dual ๐ (Dual ๐ E) :=
ContinuousLinearMap.apply ๐ ๐
#align normed_space.inclusion_in_double_dual NormedSpace.inclusionInDoubleDual
@[simp]
theorem dual_def (x : E) (f : Dual ๐ E) : inclusionInDoubleDual ๐ E x f = f x :=
rfl
#align normed_space.dual_def NormedSpace.dual_def
theorem inclusionInDoubleDual_norm_eq :
โinclusionInDoubleDual ๐ Eโ = โContinuousLinearMap.id ๐ (Dual ๐ E)โ :=
ContinuousLinearMap.op_norm_flip _
#align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1 := by
|
rw [inclusionInDoubleDual_norm_eq]
|
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1 := by
|
Mathlib.Analysis.NormedSpace.Dual.83_0.WirVfj6f5oiZZ2w
|
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1
|
Mathlib_Analysis_NormedSpace_Dual
|
๐ : Type u_1
instโโด : NontriviallyNormedField ๐
E : Type u_2
instโยณ : SeminormedAddCommGroup E
instโยฒ : NormedSpace ๐ E
F : Type u_3
instโยน : NormedAddCommGroup F
instโ : NormedSpace ๐ F
โข โContinuousLinearMap.id ๐ (Dual ๐ E)โ โค 1
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# The topological dual of a normed space
In this file we define the topological dual `NormedSpace.Dual` of a normed space, and the
continuous linear map `NormedSpace.inclusionInDoubleDual` from a normed space into its double
dual.
For base field `๐ = โ` or `๐ = โ`, this map is actually an isometric embedding; we provide a
version `NormedSpace.inclusionInDoubleDualLi` of the map which is of type a bundled linear
isometric embedding, `E โโแตข[๐] (Dual ๐ (Dual ๐ E))`.
Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
theory for `SeminormedAddCommGroup` and we specialize to `NormedAddCommGroup` when needed.
## Main definitions
* `inclusionInDoubleDual` and `inclusionInDoubleDualLi` are the inclusion of a normed space
in its double dual, considered as a bounded linear map and as a linear isometry, respectively.
* `polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals `x'` for which
`โx' zโ โค 1` for every `z โ s`.
## Tags
dual
-/
noncomputable section
open Classical Topology Bornology
universe u v
namespace NormedSpace
section General
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable (E : Type*) [SeminormedAddCommGroup E] [NormedSpace ๐ E]
variable (F : Type*) [NormedAddCommGroup F] [NormedSpace ๐ F]
/-- The topological dual of a seminormed space `E`. -/
abbrev Dual : Type _ := E โL[๐] ๐
#align normed_space.dual NormedSpace.Dual
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_eq until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : NormedSpace ๐ (Dual ๐ E) := inferInstance
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_le until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : SeminormedAddCommGroup (Dual ๐ E) := inferInstance
/-- The inclusion of a normed space in its double (topological) dual, considered
as a bounded linear map. -/
def inclusionInDoubleDual : E โL[๐] Dual ๐ (Dual ๐ E) :=
ContinuousLinearMap.apply ๐ ๐
#align normed_space.inclusion_in_double_dual NormedSpace.inclusionInDoubleDual
@[simp]
theorem dual_def (x : E) (f : Dual ๐ E) : inclusionInDoubleDual ๐ E x f = f x :=
rfl
#align normed_space.dual_def NormedSpace.dual_def
theorem inclusionInDoubleDual_norm_eq :
โinclusionInDoubleDual ๐ Eโ = โContinuousLinearMap.id ๐ (Dual ๐ E)โ :=
ContinuousLinearMap.op_norm_flip _
#align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1 := by
rw [inclusionInDoubleDual_norm_eq]
|
exact ContinuousLinearMap.norm_id_le
|
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1 := by
rw [inclusionInDoubleDual_norm_eq]
|
Mathlib.Analysis.NormedSpace.Dual.83_0.WirVfj6f5oiZZ2w
|
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1
|
Mathlib_Analysis_NormedSpace_Dual
|
๐ : Type u_1
instโโด : NontriviallyNormedField ๐
E : Type u_2
instโยณ : SeminormedAddCommGroup E
instโยฒ : NormedSpace ๐ E
F : Type u_3
instโยน : NormedAddCommGroup F
instโ : NormedSpace ๐ F
x : E
โข โ(inclusionInDoubleDual ๐ E) xโ โค โxโ
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# The topological dual of a normed space
In this file we define the topological dual `NormedSpace.Dual` of a normed space, and the
continuous linear map `NormedSpace.inclusionInDoubleDual` from a normed space into its double
dual.
For base field `๐ = โ` or `๐ = โ`, this map is actually an isometric embedding; we provide a
version `NormedSpace.inclusionInDoubleDualLi` of the map which is of type a bundled linear
isometric embedding, `E โโแตข[๐] (Dual ๐ (Dual ๐ E))`.
Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
theory for `SeminormedAddCommGroup` and we specialize to `NormedAddCommGroup` when needed.
## Main definitions
* `inclusionInDoubleDual` and `inclusionInDoubleDualLi` are the inclusion of a normed space
in its double dual, considered as a bounded linear map and as a linear isometry, respectively.
* `polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals `x'` for which
`โx' zโ โค 1` for every `z โ s`.
## Tags
dual
-/
noncomputable section
open Classical Topology Bornology
universe u v
namespace NormedSpace
section General
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable (E : Type*) [SeminormedAddCommGroup E] [NormedSpace ๐ E]
variable (F : Type*) [NormedAddCommGroup F] [NormedSpace ๐ F]
/-- The topological dual of a seminormed space `E`. -/
abbrev Dual : Type _ := E โL[๐] ๐
#align normed_space.dual NormedSpace.Dual
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_eq until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : NormedSpace ๐ (Dual ๐ E) := inferInstance
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_le until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : SeminormedAddCommGroup (Dual ๐ E) := inferInstance
/-- The inclusion of a normed space in its double (topological) dual, considered
as a bounded linear map. -/
def inclusionInDoubleDual : E โL[๐] Dual ๐ (Dual ๐ E) :=
ContinuousLinearMap.apply ๐ ๐
#align normed_space.inclusion_in_double_dual NormedSpace.inclusionInDoubleDual
@[simp]
theorem dual_def (x : E) (f : Dual ๐ E) : inclusionInDoubleDual ๐ E x f = f x :=
rfl
#align normed_space.dual_def NormedSpace.dual_def
theorem inclusionInDoubleDual_norm_eq :
โinclusionInDoubleDual ๐ Eโ = โContinuousLinearMap.id ๐ (Dual ๐ E)โ :=
ContinuousLinearMap.op_norm_flip _
#align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1 := by
rw [inclusionInDoubleDual_norm_eq]
exact ContinuousLinearMap.norm_id_le
#align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
theorem double_dual_bound (x : E) : โ(inclusionInDoubleDual ๐ E) xโ โค โxโ := by
|
simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusionInDoubleDual_norm_le ๐ E) x
|
theorem double_dual_bound (x : E) : โ(inclusionInDoubleDual ๐ E) xโ โค โxโ := by
|
Mathlib.Analysis.NormedSpace.Dual.88_0.WirVfj6f5oiZZ2w
|
theorem double_dual_bound (x : E) : โ(inclusionInDoubleDual ๐ E) xโ โค โxโ
|
Mathlib_Analysis_NormedSpace_Dual
|
๐ : Type u_1
instโโด : NontriviallyNormedField ๐
E : Type u_2
instโยณ : SeminormedAddCommGroup E
instโยฒ : NormedSpace ๐ E
F : Type u_3
instโยน : NormedAddCommGroup F
instโ : NormedSpace ๐ F
โข LinearMap.SeparatingLeft (dualPairing ๐ E)
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# The topological dual of a normed space
In this file we define the topological dual `NormedSpace.Dual` of a normed space, and the
continuous linear map `NormedSpace.inclusionInDoubleDual` from a normed space into its double
dual.
For base field `๐ = โ` or `๐ = โ`, this map is actually an isometric embedding; we provide a
version `NormedSpace.inclusionInDoubleDualLi` of the map which is of type a bundled linear
isometric embedding, `E โโแตข[๐] (Dual ๐ (Dual ๐ E))`.
Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
theory for `SeminormedAddCommGroup` and we specialize to `NormedAddCommGroup` when needed.
## Main definitions
* `inclusionInDoubleDual` and `inclusionInDoubleDualLi` are the inclusion of a normed space
in its double dual, considered as a bounded linear map and as a linear isometry, respectively.
* `polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals `x'` for which
`โx' zโ โค 1` for every `z โ s`.
## Tags
dual
-/
noncomputable section
open Classical Topology Bornology
universe u v
namespace NormedSpace
section General
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable (E : Type*) [SeminormedAddCommGroup E] [NormedSpace ๐ E]
variable (F : Type*) [NormedAddCommGroup F] [NormedSpace ๐ F]
/-- The topological dual of a seminormed space `E`. -/
abbrev Dual : Type _ := E โL[๐] ๐
#align normed_space.dual NormedSpace.Dual
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_eq until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : NormedSpace ๐ (Dual ๐ E) := inferInstance
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_le until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : SeminormedAddCommGroup (Dual ๐ E) := inferInstance
/-- The inclusion of a normed space in its double (topological) dual, considered
as a bounded linear map. -/
def inclusionInDoubleDual : E โL[๐] Dual ๐ (Dual ๐ E) :=
ContinuousLinearMap.apply ๐ ๐
#align normed_space.inclusion_in_double_dual NormedSpace.inclusionInDoubleDual
@[simp]
theorem dual_def (x : E) (f : Dual ๐ E) : inclusionInDoubleDual ๐ E x f = f x :=
rfl
#align normed_space.dual_def NormedSpace.dual_def
theorem inclusionInDoubleDual_norm_eq :
โinclusionInDoubleDual ๐ Eโ = โContinuousLinearMap.id ๐ (Dual ๐ E)โ :=
ContinuousLinearMap.op_norm_flip _
#align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1 := by
rw [inclusionInDoubleDual_norm_eq]
exact ContinuousLinearMap.norm_id_le
#align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
theorem double_dual_bound (x : E) : โ(inclusionInDoubleDual ๐ E) xโ โค โxโ := by
simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusionInDoubleDual_norm_le ๐ E) x
#align normed_space.double_dual_bound NormedSpace.double_dual_bound
/-- The dual pairing as a bilinear form. -/
def dualPairing : Dual ๐ E โโ[๐] E โโ[๐] ๐ :=
ContinuousLinearMap.coeLM ๐
#align normed_space.dual_pairing NormedSpace.dualPairing
@[simp]
theorem dualPairing_apply {v : Dual ๐ E} {x : E} : dualPairing ๐ E v x = v x :=
rfl
#align normed_space.dual_pairing_apply NormedSpace.dualPairing_apply
theorem dualPairing_separatingLeft : (dualPairing ๐ E).SeparatingLeft := by
|
rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]
|
theorem dualPairing_separatingLeft : (dualPairing ๐ E).SeparatingLeft := by
|
Mathlib.Analysis.NormedSpace.Dual.102_0.WirVfj6f5oiZZ2w
|
theorem dualPairing_separatingLeft : (dualPairing ๐ E).SeparatingLeft
|
Mathlib_Analysis_NormedSpace_Dual
|
๐ : Type u_1
instโโด : NontriviallyNormedField ๐
E : Type u_2
instโยณ : SeminormedAddCommGroup E
instโยฒ : NormedSpace ๐ E
F : Type u_3
instโยน : NormedAddCommGroup F
instโ : NormedSpace ๐ F
โข Function.Injective โ(dualPairing ๐ E)
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# The topological dual of a normed space
In this file we define the topological dual `NormedSpace.Dual` of a normed space, and the
continuous linear map `NormedSpace.inclusionInDoubleDual` from a normed space into its double
dual.
For base field `๐ = โ` or `๐ = โ`, this map is actually an isometric embedding; we provide a
version `NormedSpace.inclusionInDoubleDualLi` of the map which is of type a bundled linear
isometric embedding, `E โโแตข[๐] (Dual ๐ (Dual ๐ E))`.
Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
theory for `SeminormedAddCommGroup` and we specialize to `NormedAddCommGroup` when needed.
## Main definitions
* `inclusionInDoubleDual` and `inclusionInDoubleDualLi` are the inclusion of a normed space
in its double dual, considered as a bounded linear map and as a linear isometry, respectively.
* `polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals `x'` for which
`โx' zโ โค 1` for every `z โ s`.
## Tags
dual
-/
noncomputable section
open Classical Topology Bornology
universe u v
namespace NormedSpace
section General
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable (E : Type*) [SeminormedAddCommGroup E] [NormedSpace ๐ E]
variable (F : Type*) [NormedAddCommGroup F] [NormedSpace ๐ F]
/-- The topological dual of a seminormed space `E`. -/
abbrev Dual : Type _ := E โL[๐] ๐
#align normed_space.dual NormedSpace.Dual
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_eq until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : NormedSpace ๐ (Dual ๐ E) := inferInstance
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_le until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : SeminormedAddCommGroup (Dual ๐ E) := inferInstance
/-- The inclusion of a normed space in its double (topological) dual, considered
as a bounded linear map. -/
def inclusionInDoubleDual : E โL[๐] Dual ๐ (Dual ๐ E) :=
ContinuousLinearMap.apply ๐ ๐
#align normed_space.inclusion_in_double_dual NormedSpace.inclusionInDoubleDual
@[simp]
theorem dual_def (x : E) (f : Dual ๐ E) : inclusionInDoubleDual ๐ E x f = f x :=
rfl
#align normed_space.dual_def NormedSpace.dual_def
theorem inclusionInDoubleDual_norm_eq :
โinclusionInDoubleDual ๐ Eโ = โContinuousLinearMap.id ๐ (Dual ๐ E)โ :=
ContinuousLinearMap.op_norm_flip _
#align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1 := by
rw [inclusionInDoubleDual_norm_eq]
exact ContinuousLinearMap.norm_id_le
#align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
theorem double_dual_bound (x : E) : โ(inclusionInDoubleDual ๐ E) xโ โค โxโ := by
simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusionInDoubleDual_norm_le ๐ E) x
#align normed_space.double_dual_bound NormedSpace.double_dual_bound
/-- The dual pairing as a bilinear form. -/
def dualPairing : Dual ๐ E โโ[๐] E โโ[๐] ๐ :=
ContinuousLinearMap.coeLM ๐
#align normed_space.dual_pairing NormedSpace.dualPairing
@[simp]
theorem dualPairing_apply {v : Dual ๐ E} {x : E} : dualPairing ๐ E v x = v x :=
rfl
#align normed_space.dual_pairing_apply NormedSpace.dualPairing_apply
theorem dualPairing_separatingLeft : (dualPairing ๐ E).SeparatingLeft := by
rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]
|
exact ContinuousLinearMap.coe_injective
|
theorem dualPairing_separatingLeft : (dualPairing ๐ E).SeparatingLeft := by
rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]
|
Mathlib.Analysis.NormedSpace.Dual.102_0.WirVfj6f5oiZZ2w
|
theorem dualPairing_separatingLeft : (dualPairing ๐ E).SeparatingLeft
|
Mathlib_Analysis_NormedSpace_Dual
|
๐ : Type v
instโยฒ : IsROrC ๐
E : Type u
instโยน : NormedAddCommGroup E
instโ : NormedSpace ๐ E
x : E
M : โ
hMp : 0 โค M
hM : โ (f : Dual ๐ E), โf xโ โค M * โfโ
โข โxโ โค M
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# The topological dual of a normed space
In this file we define the topological dual `NormedSpace.Dual` of a normed space, and the
continuous linear map `NormedSpace.inclusionInDoubleDual` from a normed space into its double
dual.
For base field `๐ = โ` or `๐ = โ`, this map is actually an isometric embedding; we provide a
version `NormedSpace.inclusionInDoubleDualLi` of the map which is of type a bundled linear
isometric embedding, `E โโแตข[๐] (Dual ๐ (Dual ๐ E))`.
Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
theory for `SeminormedAddCommGroup` and we specialize to `NormedAddCommGroup` when needed.
## Main definitions
* `inclusionInDoubleDual` and `inclusionInDoubleDualLi` are the inclusion of a normed space
in its double dual, considered as a bounded linear map and as a linear isometry, respectively.
* `polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals `x'` for which
`โx' zโ โค 1` for every `z โ s`.
## Tags
dual
-/
noncomputable section
open Classical Topology Bornology
universe u v
namespace NormedSpace
section General
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable (E : Type*) [SeminormedAddCommGroup E] [NormedSpace ๐ E]
variable (F : Type*) [NormedAddCommGroup F] [NormedSpace ๐ F]
/-- The topological dual of a seminormed space `E`. -/
abbrev Dual : Type _ := E โL[๐] ๐
#align normed_space.dual NormedSpace.Dual
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_eq until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : NormedSpace ๐ (Dual ๐ E) := inferInstance
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_le until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : SeminormedAddCommGroup (Dual ๐ E) := inferInstance
/-- The inclusion of a normed space in its double (topological) dual, considered
as a bounded linear map. -/
def inclusionInDoubleDual : E โL[๐] Dual ๐ (Dual ๐ E) :=
ContinuousLinearMap.apply ๐ ๐
#align normed_space.inclusion_in_double_dual NormedSpace.inclusionInDoubleDual
@[simp]
theorem dual_def (x : E) (f : Dual ๐ E) : inclusionInDoubleDual ๐ E x f = f x :=
rfl
#align normed_space.dual_def NormedSpace.dual_def
theorem inclusionInDoubleDual_norm_eq :
โinclusionInDoubleDual ๐ Eโ = โContinuousLinearMap.id ๐ (Dual ๐ E)โ :=
ContinuousLinearMap.op_norm_flip _
#align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1 := by
rw [inclusionInDoubleDual_norm_eq]
exact ContinuousLinearMap.norm_id_le
#align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
theorem double_dual_bound (x : E) : โ(inclusionInDoubleDual ๐ E) xโ โค โxโ := by
simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusionInDoubleDual_norm_le ๐ E) x
#align normed_space.double_dual_bound NormedSpace.double_dual_bound
/-- The dual pairing as a bilinear form. -/
def dualPairing : Dual ๐ E โโ[๐] E โโ[๐] ๐ :=
ContinuousLinearMap.coeLM ๐
#align normed_space.dual_pairing NormedSpace.dualPairing
@[simp]
theorem dualPairing_apply {v : Dual ๐ E} {x : E} : dualPairing ๐ E v x = v x :=
rfl
#align normed_space.dual_pairing_apply NormedSpace.dualPairing_apply
theorem dualPairing_separatingLeft : (dualPairing ๐ E).SeparatingLeft := by
rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]
exact ContinuousLinearMap.coe_injective
#align normed_space.dual_pairing_separating_left NormedSpace.dualPairing_separatingLeft
end General
section BidualIsometry
variable (๐ : Type v) [IsROrC ๐] {E : Type u} [NormedAddCommGroup E] [NormedSpace ๐ E]
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M := by
|
classical
by_cases h : x = 0
ยท simp only [h, hMp, norm_zero]
ยท obtain โจf, hfโ, hfxโฉ : โ f : E โL[๐] ๐, โfโ = 1 โง f x = โxโ := exists_dual_vector ๐ x h
calc
โxโ = โ(โxโ : ๐)โ := IsROrC.norm_coe_norm.symm
_ = โf xโ := by rw [hfx]
_ โค M * โfโ := (hM f)
_ = M := by rw [hfโ, mul_one]
|
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M := by
|
Mathlib.Analysis.NormedSpace.Dual.113_0.WirVfj6f5oiZZ2w
|
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M
|
Mathlib_Analysis_NormedSpace_Dual
|
๐ : Type v
instโยฒ : IsROrC ๐
E : Type u
instโยน : NormedAddCommGroup E
instโ : NormedSpace ๐ E
x : E
M : โ
hMp : 0 โค M
hM : โ (f : Dual ๐ E), โf xโ โค M * โfโ
โข โxโ โค M
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# The topological dual of a normed space
In this file we define the topological dual `NormedSpace.Dual` of a normed space, and the
continuous linear map `NormedSpace.inclusionInDoubleDual` from a normed space into its double
dual.
For base field `๐ = โ` or `๐ = โ`, this map is actually an isometric embedding; we provide a
version `NormedSpace.inclusionInDoubleDualLi` of the map which is of type a bundled linear
isometric embedding, `E โโแตข[๐] (Dual ๐ (Dual ๐ E))`.
Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
theory for `SeminormedAddCommGroup` and we specialize to `NormedAddCommGroup` when needed.
## Main definitions
* `inclusionInDoubleDual` and `inclusionInDoubleDualLi` are the inclusion of a normed space
in its double dual, considered as a bounded linear map and as a linear isometry, respectively.
* `polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals `x'` for which
`โx' zโ โค 1` for every `z โ s`.
## Tags
dual
-/
noncomputable section
open Classical Topology Bornology
universe u v
namespace NormedSpace
section General
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable (E : Type*) [SeminormedAddCommGroup E] [NormedSpace ๐ E]
variable (F : Type*) [NormedAddCommGroup F] [NormedSpace ๐ F]
/-- The topological dual of a seminormed space `E`. -/
abbrev Dual : Type _ := E โL[๐] ๐
#align normed_space.dual NormedSpace.Dual
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_eq until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : NormedSpace ๐ (Dual ๐ E) := inferInstance
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_le until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : SeminormedAddCommGroup (Dual ๐ E) := inferInstance
/-- The inclusion of a normed space in its double (topological) dual, considered
as a bounded linear map. -/
def inclusionInDoubleDual : E โL[๐] Dual ๐ (Dual ๐ E) :=
ContinuousLinearMap.apply ๐ ๐
#align normed_space.inclusion_in_double_dual NormedSpace.inclusionInDoubleDual
@[simp]
theorem dual_def (x : E) (f : Dual ๐ E) : inclusionInDoubleDual ๐ E x f = f x :=
rfl
#align normed_space.dual_def NormedSpace.dual_def
theorem inclusionInDoubleDual_norm_eq :
โinclusionInDoubleDual ๐ Eโ = โContinuousLinearMap.id ๐ (Dual ๐ E)โ :=
ContinuousLinearMap.op_norm_flip _
#align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1 := by
rw [inclusionInDoubleDual_norm_eq]
exact ContinuousLinearMap.norm_id_le
#align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
theorem double_dual_bound (x : E) : โ(inclusionInDoubleDual ๐ E) xโ โค โxโ := by
simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusionInDoubleDual_norm_le ๐ E) x
#align normed_space.double_dual_bound NormedSpace.double_dual_bound
/-- The dual pairing as a bilinear form. -/
def dualPairing : Dual ๐ E โโ[๐] E โโ[๐] ๐ :=
ContinuousLinearMap.coeLM ๐
#align normed_space.dual_pairing NormedSpace.dualPairing
@[simp]
theorem dualPairing_apply {v : Dual ๐ E} {x : E} : dualPairing ๐ E v x = v x :=
rfl
#align normed_space.dual_pairing_apply NormedSpace.dualPairing_apply
theorem dualPairing_separatingLeft : (dualPairing ๐ E).SeparatingLeft := by
rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]
exact ContinuousLinearMap.coe_injective
#align normed_space.dual_pairing_separating_left NormedSpace.dualPairing_separatingLeft
end General
section BidualIsometry
variable (๐ : Type v) [IsROrC ๐] {E : Type u} [NormedAddCommGroup E] [NormedSpace ๐ E]
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M := by
classical
|
by_cases h : x = 0
|
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M := by
classical
|
Mathlib.Analysis.NormedSpace.Dual.113_0.WirVfj6f5oiZZ2w
|
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M
|
Mathlib_Analysis_NormedSpace_Dual
|
case pos
๐ : Type v
instโยฒ : IsROrC ๐
E : Type u
instโยน : NormedAddCommGroup E
instโ : NormedSpace ๐ E
x : E
M : โ
hMp : 0 โค M
hM : โ (f : Dual ๐ E), โf xโ โค M * โfโ
h : x = 0
โข โxโ โค M
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# The topological dual of a normed space
In this file we define the topological dual `NormedSpace.Dual` of a normed space, and the
continuous linear map `NormedSpace.inclusionInDoubleDual` from a normed space into its double
dual.
For base field `๐ = โ` or `๐ = โ`, this map is actually an isometric embedding; we provide a
version `NormedSpace.inclusionInDoubleDualLi` of the map which is of type a bundled linear
isometric embedding, `E โโแตข[๐] (Dual ๐ (Dual ๐ E))`.
Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
theory for `SeminormedAddCommGroup` and we specialize to `NormedAddCommGroup` when needed.
## Main definitions
* `inclusionInDoubleDual` and `inclusionInDoubleDualLi` are the inclusion of a normed space
in its double dual, considered as a bounded linear map and as a linear isometry, respectively.
* `polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals `x'` for which
`โx' zโ โค 1` for every `z โ s`.
## Tags
dual
-/
noncomputable section
open Classical Topology Bornology
universe u v
namespace NormedSpace
section General
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable (E : Type*) [SeminormedAddCommGroup E] [NormedSpace ๐ E]
variable (F : Type*) [NormedAddCommGroup F] [NormedSpace ๐ F]
/-- The topological dual of a seminormed space `E`. -/
abbrev Dual : Type _ := E โL[๐] ๐
#align normed_space.dual NormedSpace.Dual
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_eq until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : NormedSpace ๐ (Dual ๐ E) := inferInstance
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_le until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : SeminormedAddCommGroup (Dual ๐ E) := inferInstance
/-- The inclusion of a normed space in its double (topological) dual, considered
as a bounded linear map. -/
def inclusionInDoubleDual : E โL[๐] Dual ๐ (Dual ๐ E) :=
ContinuousLinearMap.apply ๐ ๐
#align normed_space.inclusion_in_double_dual NormedSpace.inclusionInDoubleDual
@[simp]
theorem dual_def (x : E) (f : Dual ๐ E) : inclusionInDoubleDual ๐ E x f = f x :=
rfl
#align normed_space.dual_def NormedSpace.dual_def
theorem inclusionInDoubleDual_norm_eq :
โinclusionInDoubleDual ๐ Eโ = โContinuousLinearMap.id ๐ (Dual ๐ E)โ :=
ContinuousLinearMap.op_norm_flip _
#align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1 := by
rw [inclusionInDoubleDual_norm_eq]
exact ContinuousLinearMap.norm_id_le
#align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
theorem double_dual_bound (x : E) : โ(inclusionInDoubleDual ๐ E) xโ โค โxโ := by
simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusionInDoubleDual_norm_le ๐ E) x
#align normed_space.double_dual_bound NormedSpace.double_dual_bound
/-- The dual pairing as a bilinear form. -/
def dualPairing : Dual ๐ E โโ[๐] E โโ[๐] ๐ :=
ContinuousLinearMap.coeLM ๐
#align normed_space.dual_pairing NormedSpace.dualPairing
@[simp]
theorem dualPairing_apply {v : Dual ๐ E} {x : E} : dualPairing ๐ E v x = v x :=
rfl
#align normed_space.dual_pairing_apply NormedSpace.dualPairing_apply
theorem dualPairing_separatingLeft : (dualPairing ๐ E).SeparatingLeft := by
rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]
exact ContinuousLinearMap.coe_injective
#align normed_space.dual_pairing_separating_left NormedSpace.dualPairing_separatingLeft
end General
section BidualIsometry
variable (๐ : Type v) [IsROrC ๐] {E : Type u} [NormedAddCommGroup E] [NormedSpace ๐ E]
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M := by
classical
by_cases h : x = 0
ยท
|
simp only [h, hMp, norm_zero]
|
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M := by
classical
by_cases h : x = 0
ยท
|
Mathlib.Analysis.NormedSpace.Dual.113_0.WirVfj6f5oiZZ2w
|
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M
|
Mathlib_Analysis_NormedSpace_Dual
|
case neg
๐ : Type v
instโยฒ : IsROrC ๐
E : Type u
instโยน : NormedAddCommGroup E
instโ : NormedSpace ๐ E
x : E
M : โ
hMp : 0 โค M
hM : โ (f : Dual ๐ E), โf xโ โค M * โfโ
h : ยฌx = 0
โข โxโ โค M
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# The topological dual of a normed space
In this file we define the topological dual `NormedSpace.Dual` of a normed space, and the
continuous linear map `NormedSpace.inclusionInDoubleDual` from a normed space into its double
dual.
For base field `๐ = โ` or `๐ = โ`, this map is actually an isometric embedding; we provide a
version `NormedSpace.inclusionInDoubleDualLi` of the map which is of type a bundled linear
isometric embedding, `E โโแตข[๐] (Dual ๐ (Dual ๐ E))`.
Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
theory for `SeminormedAddCommGroup` and we specialize to `NormedAddCommGroup` when needed.
## Main definitions
* `inclusionInDoubleDual` and `inclusionInDoubleDualLi` are the inclusion of a normed space
in its double dual, considered as a bounded linear map and as a linear isometry, respectively.
* `polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals `x'` for which
`โx' zโ โค 1` for every `z โ s`.
## Tags
dual
-/
noncomputable section
open Classical Topology Bornology
universe u v
namespace NormedSpace
section General
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable (E : Type*) [SeminormedAddCommGroup E] [NormedSpace ๐ E]
variable (F : Type*) [NormedAddCommGroup F] [NormedSpace ๐ F]
/-- The topological dual of a seminormed space `E`. -/
abbrev Dual : Type _ := E โL[๐] ๐
#align normed_space.dual NormedSpace.Dual
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_eq until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : NormedSpace ๐ (Dual ๐ E) := inferInstance
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_le until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : SeminormedAddCommGroup (Dual ๐ E) := inferInstance
/-- The inclusion of a normed space in its double (topological) dual, considered
as a bounded linear map. -/
def inclusionInDoubleDual : E โL[๐] Dual ๐ (Dual ๐ E) :=
ContinuousLinearMap.apply ๐ ๐
#align normed_space.inclusion_in_double_dual NormedSpace.inclusionInDoubleDual
@[simp]
theorem dual_def (x : E) (f : Dual ๐ E) : inclusionInDoubleDual ๐ E x f = f x :=
rfl
#align normed_space.dual_def NormedSpace.dual_def
theorem inclusionInDoubleDual_norm_eq :
โinclusionInDoubleDual ๐ Eโ = โContinuousLinearMap.id ๐ (Dual ๐ E)โ :=
ContinuousLinearMap.op_norm_flip _
#align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1 := by
rw [inclusionInDoubleDual_norm_eq]
exact ContinuousLinearMap.norm_id_le
#align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
theorem double_dual_bound (x : E) : โ(inclusionInDoubleDual ๐ E) xโ โค โxโ := by
simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusionInDoubleDual_norm_le ๐ E) x
#align normed_space.double_dual_bound NormedSpace.double_dual_bound
/-- The dual pairing as a bilinear form. -/
def dualPairing : Dual ๐ E โโ[๐] E โโ[๐] ๐ :=
ContinuousLinearMap.coeLM ๐
#align normed_space.dual_pairing NormedSpace.dualPairing
@[simp]
theorem dualPairing_apply {v : Dual ๐ E} {x : E} : dualPairing ๐ E v x = v x :=
rfl
#align normed_space.dual_pairing_apply NormedSpace.dualPairing_apply
theorem dualPairing_separatingLeft : (dualPairing ๐ E).SeparatingLeft := by
rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]
exact ContinuousLinearMap.coe_injective
#align normed_space.dual_pairing_separating_left NormedSpace.dualPairing_separatingLeft
end General
section BidualIsometry
variable (๐ : Type v) [IsROrC ๐] {E : Type u} [NormedAddCommGroup E] [NormedSpace ๐ E]
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M := by
classical
by_cases h : x = 0
ยท simp only [h, hMp, norm_zero]
ยท
|
obtain โจf, hfโ, hfxโฉ : โ f : E โL[๐] ๐, โfโ = 1 โง f x = โxโ := exists_dual_vector ๐ x h
|
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M := by
classical
by_cases h : x = 0
ยท simp only [h, hMp, norm_zero]
ยท
|
Mathlib.Analysis.NormedSpace.Dual.113_0.WirVfj6f5oiZZ2w
|
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M
|
Mathlib_Analysis_NormedSpace_Dual
|
case neg.intro.intro
๐ : Type v
instโยฒ : IsROrC ๐
E : Type u
instโยน : NormedAddCommGroup E
instโ : NormedSpace ๐ E
x : E
M : โ
hMp : 0 โค M
hM : โ (f : Dual ๐ E), โf xโ โค M * โfโ
h : ยฌx = 0
f : E โL[๐] ๐
hfโ : โfโ = 1
hfx : f x = โโxโ
โข โxโ โค M
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# The topological dual of a normed space
In this file we define the topological dual `NormedSpace.Dual` of a normed space, and the
continuous linear map `NormedSpace.inclusionInDoubleDual` from a normed space into its double
dual.
For base field `๐ = โ` or `๐ = โ`, this map is actually an isometric embedding; we provide a
version `NormedSpace.inclusionInDoubleDualLi` of the map which is of type a bundled linear
isometric embedding, `E โโแตข[๐] (Dual ๐ (Dual ๐ E))`.
Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
theory for `SeminormedAddCommGroup` and we specialize to `NormedAddCommGroup` when needed.
## Main definitions
* `inclusionInDoubleDual` and `inclusionInDoubleDualLi` are the inclusion of a normed space
in its double dual, considered as a bounded linear map and as a linear isometry, respectively.
* `polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals `x'` for which
`โx' zโ โค 1` for every `z โ s`.
## Tags
dual
-/
noncomputable section
open Classical Topology Bornology
universe u v
namespace NormedSpace
section General
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable (E : Type*) [SeminormedAddCommGroup E] [NormedSpace ๐ E]
variable (F : Type*) [NormedAddCommGroup F] [NormedSpace ๐ F]
/-- The topological dual of a seminormed space `E`. -/
abbrev Dual : Type _ := E โL[๐] ๐
#align normed_space.dual NormedSpace.Dual
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_eq until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : NormedSpace ๐ (Dual ๐ E) := inferInstance
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_le until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : SeminormedAddCommGroup (Dual ๐ E) := inferInstance
/-- The inclusion of a normed space in its double (topological) dual, considered
as a bounded linear map. -/
def inclusionInDoubleDual : E โL[๐] Dual ๐ (Dual ๐ E) :=
ContinuousLinearMap.apply ๐ ๐
#align normed_space.inclusion_in_double_dual NormedSpace.inclusionInDoubleDual
@[simp]
theorem dual_def (x : E) (f : Dual ๐ E) : inclusionInDoubleDual ๐ E x f = f x :=
rfl
#align normed_space.dual_def NormedSpace.dual_def
theorem inclusionInDoubleDual_norm_eq :
โinclusionInDoubleDual ๐ Eโ = โContinuousLinearMap.id ๐ (Dual ๐ E)โ :=
ContinuousLinearMap.op_norm_flip _
#align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1 := by
rw [inclusionInDoubleDual_norm_eq]
exact ContinuousLinearMap.norm_id_le
#align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
theorem double_dual_bound (x : E) : โ(inclusionInDoubleDual ๐ E) xโ โค โxโ := by
simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusionInDoubleDual_norm_le ๐ E) x
#align normed_space.double_dual_bound NormedSpace.double_dual_bound
/-- The dual pairing as a bilinear form. -/
def dualPairing : Dual ๐ E โโ[๐] E โโ[๐] ๐ :=
ContinuousLinearMap.coeLM ๐
#align normed_space.dual_pairing NormedSpace.dualPairing
@[simp]
theorem dualPairing_apply {v : Dual ๐ E} {x : E} : dualPairing ๐ E v x = v x :=
rfl
#align normed_space.dual_pairing_apply NormedSpace.dualPairing_apply
theorem dualPairing_separatingLeft : (dualPairing ๐ E).SeparatingLeft := by
rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]
exact ContinuousLinearMap.coe_injective
#align normed_space.dual_pairing_separating_left NormedSpace.dualPairing_separatingLeft
end General
section BidualIsometry
variable (๐ : Type v) [IsROrC ๐] {E : Type u} [NormedAddCommGroup E] [NormedSpace ๐ E]
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M := by
classical
by_cases h : x = 0
ยท simp only [h, hMp, norm_zero]
ยท obtain โจf, hfโ, hfxโฉ : โ f : E โL[๐] ๐, โfโ = 1 โง f x = โxโ := exists_dual_vector ๐ x h
|
calc
โxโ = โ(โxโ : ๐)โ := IsROrC.norm_coe_norm.symm
_ = โf xโ := by rw [hfx]
_ โค M * โfโ := (hM f)
_ = M := by rw [hfโ, mul_one]
|
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M := by
classical
by_cases h : x = 0
ยท simp only [h, hMp, norm_zero]
ยท obtain โจf, hfโ, hfxโฉ : โ f : E โL[๐] ๐, โfโ = 1 โง f x = โxโ := exists_dual_vector ๐ x h
|
Mathlib.Analysis.NormedSpace.Dual.113_0.WirVfj6f5oiZZ2w
|
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M
|
Mathlib_Analysis_NormedSpace_Dual
|
๐ : Type v
instโยฒ : IsROrC ๐
E : Type u
instโยน : NormedAddCommGroup E
instโ : NormedSpace ๐ E
x : E
M : โ
hMp : 0 โค M
hM : โ (f : Dual ๐ E), โf xโ โค M * โfโ
h : ยฌx = 0
f : E โL[๐] ๐
hfโ : โfโ = 1
hfx : f x = โโxโ
โข โโโxโโ = โf xโ
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# The topological dual of a normed space
In this file we define the topological dual `NormedSpace.Dual` of a normed space, and the
continuous linear map `NormedSpace.inclusionInDoubleDual` from a normed space into its double
dual.
For base field `๐ = โ` or `๐ = โ`, this map is actually an isometric embedding; we provide a
version `NormedSpace.inclusionInDoubleDualLi` of the map which is of type a bundled linear
isometric embedding, `E โโแตข[๐] (Dual ๐ (Dual ๐ E))`.
Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
theory for `SeminormedAddCommGroup` and we specialize to `NormedAddCommGroup` when needed.
## Main definitions
* `inclusionInDoubleDual` and `inclusionInDoubleDualLi` are the inclusion of a normed space
in its double dual, considered as a bounded linear map and as a linear isometry, respectively.
* `polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals `x'` for which
`โx' zโ โค 1` for every `z โ s`.
## Tags
dual
-/
noncomputable section
open Classical Topology Bornology
universe u v
namespace NormedSpace
section General
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable (E : Type*) [SeminormedAddCommGroup E] [NormedSpace ๐ E]
variable (F : Type*) [NormedAddCommGroup F] [NormedSpace ๐ F]
/-- The topological dual of a seminormed space `E`. -/
abbrev Dual : Type _ := E โL[๐] ๐
#align normed_space.dual NormedSpace.Dual
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_eq until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : NormedSpace ๐ (Dual ๐ E) := inferInstance
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_le until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : SeminormedAddCommGroup (Dual ๐ E) := inferInstance
/-- The inclusion of a normed space in its double (topological) dual, considered
as a bounded linear map. -/
def inclusionInDoubleDual : E โL[๐] Dual ๐ (Dual ๐ E) :=
ContinuousLinearMap.apply ๐ ๐
#align normed_space.inclusion_in_double_dual NormedSpace.inclusionInDoubleDual
@[simp]
theorem dual_def (x : E) (f : Dual ๐ E) : inclusionInDoubleDual ๐ E x f = f x :=
rfl
#align normed_space.dual_def NormedSpace.dual_def
theorem inclusionInDoubleDual_norm_eq :
โinclusionInDoubleDual ๐ Eโ = โContinuousLinearMap.id ๐ (Dual ๐ E)โ :=
ContinuousLinearMap.op_norm_flip _
#align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1 := by
rw [inclusionInDoubleDual_norm_eq]
exact ContinuousLinearMap.norm_id_le
#align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
theorem double_dual_bound (x : E) : โ(inclusionInDoubleDual ๐ E) xโ โค โxโ := by
simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusionInDoubleDual_norm_le ๐ E) x
#align normed_space.double_dual_bound NormedSpace.double_dual_bound
/-- The dual pairing as a bilinear form. -/
def dualPairing : Dual ๐ E โโ[๐] E โโ[๐] ๐ :=
ContinuousLinearMap.coeLM ๐
#align normed_space.dual_pairing NormedSpace.dualPairing
@[simp]
theorem dualPairing_apply {v : Dual ๐ E} {x : E} : dualPairing ๐ E v x = v x :=
rfl
#align normed_space.dual_pairing_apply NormedSpace.dualPairing_apply
theorem dualPairing_separatingLeft : (dualPairing ๐ E).SeparatingLeft := by
rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]
exact ContinuousLinearMap.coe_injective
#align normed_space.dual_pairing_separating_left NormedSpace.dualPairing_separatingLeft
end General
section BidualIsometry
variable (๐ : Type v) [IsROrC ๐] {E : Type u} [NormedAddCommGroup E] [NormedSpace ๐ E]
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M := by
classical
by_cases h : x = 0
ยท simp only [h, hMp, norm_zero]
ยท obtain โจf, hfโ, hfxโฉ : โ f : E โL[๐] ๐, โfโ = 1 โง f x = โxโ := exists_dual_vector ๐ x h
calc
โxโ = โ(โxโ : ๐)โ := IsROrC.norm_coe_norm.symm
_ = โf xโ := by
|
rw [hfx]
|
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M := by
classical
by_cases h : x = 0
ยท simp only [h, hMp, norm_zero]
ยท obtain โจf, hfโ, hfxโฉ : โ f : E โL[๐] ๐, โfโ = 1 โง f x = โxโ := exists_dual_vector ๐ x h
calc
โxโ = โ(โxโ : ๐)โ := IsROrC.norm_coe_norm.symm
_ = โf xโ := by
|
Mathlib.Analysis.NormedSpace.Dual.113_0.WirVfj6f5oiZZ2w
|
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M
|
Mathlib_Analysis_NormedSpace_Dual
|
๐ : Type v
instโยฒ : IsROrC ๐
E : Type u
instโยน : NormedAddCommGroup E
instโ : NormedSpace ๐ E
x : E
M : โ
hMp : 0 โค M
hM : โ (f : Dual ๐ E), โf xโ โค M * โfโ
h : ยฌx = 0
f : E โL[๐] ๐
hfโ : โfโ = 1
hfx : f x = โโxโ
โข M * โfโ = M
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# The topological dual of a normed space
In this file we define the topological dual `NormedSpace.Dual` of a normed space, and the
continuous linear map `NormedSpace.inclusionInDoubleDual` from a normed space into its double
dual.
For base field `๐ = โ` or `๐ = โ`, this map is actually an isometric embedding; we provide a
version `NormedSpace.inclusionInDoubleDualLi` of the map which is of type a bundled linear
isometric embedding, `E โโแตข[๐] (Dual ๐ (Dual ๐ E))`.
Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
theory for `SeminormedAddCommGroup` and we specialize to `NormedAddCommGroup` when needed.
## Main definitions
* `inclusionInDoubleDual` and `inclusionInDoubleDualLi` are the inclusion of a normed space
in its double dual, considered as a bounded linear map and as a linear isometry, respectively.
* `polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals `x'` for which
`โx' zโ โค 1` for every `z โ s`.
## Tags
dual
-/
noncomputable section
open Classical Topology Bornology
universe u v
namespace NormedSpace
section General
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable (E : Type*) [SeminormedAddCommGroup E] [NormedSpace ๐ E]
variable (F : Type*) [NormedAddCommGroup F] [NormedSpace ๐ F]
/-- The topological dual of a seminormed space `E`. -/
abbrev Dual : Type _ := E โL[๐] ๐
#align normed_space.dual NormedSpace.Dual
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_eq until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : NormedSpace ๐ (Dual ๐ E) := inferInstance
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_le until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : SeminormedAddCommGroup (Dual ๐ E) := inferInstance
/-- The inclusion of a normed space in its double (topological) dual, considered
as a bounded linear map. -/
def inclusionInDoubleDual : E โL[๐] Dual ๐ (Dual ๐ E) :=
ContinuousLinearMap.apply ๐ ๐
#align normed_space.inclusion_in_double_dual NormedSpace.inclusionInDoubleDual
@[simp]
theorem dual_def (x : E) (f : Dual ๐ E) : inclusionInDoubleDual ๐ E x f = f x :=
rfl
#align normed_space.dual_def NormedSpace.dual_def
theorem inclusionInDoubleDual_norm_eq :
โinclusionInDoubleDual ๐ Eโ = โContinuousLinearMap.id ๐ (Dual ๐ E)โ :=
ContinuousLinearMap.op_norm_flip _
#align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1 := by
rw [inclusionInDoubleDual_norm_eq]
exact ContinuousLinearMap.norm_id_le
#align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
theorem double_dual_bound (x : E) : โ(inclusionInDoubleDual ๐ E) xโ โค โxโ := by
simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusionInDoubleDual_norm_le ๐ E) x
#align normed_space.double_dual_bound NormedSpace.double_dual_bound
/-- The dual pairing as a bilinear form. -/
def dualPairing : Dual ๐ E โโ[๐] E โโ[๐] ๐ :=
ContinuousLinearMap.coeLM ๐
#align normed_space.dual_pairing NormedSpace.dualPairing
@[simp]
theorem dualPairing_apply {v : Dual ๐ E} {x : E} : dualPairing ๐ E v x = v x :=
rfl
#align normed_space.dual_pairing_apply NormedSpace.dualPairing_apply
theorem dualPairing_separatingLeft : (dualPairing ๐ E).SeparatingLeft := by
rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]
exact ContinuousLinearMap.coe_injective
#align normed_space.dual_pairing_separating_left NormedSpace.dualPairing_separatingLeft
end General
section BidualIsometry
variable (๐ : Type v) [IsROrC ๐] {E : Type u} [NormedAddCommGroup E] [NormedSpace ๐ E]
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M := by
classical
by_cases h : x = 0
ยท simp only [h, hMp, norm_zero]
ยท obtain โจf, hfโ, hfxโฉ : โ f : E โL[๐] ๐, โfโ = 1 โง f x = โxโ := exists_dual_vector ๐ x h
calc
โxโ = โ(โxโ : ๐)โ := IsROrC.norm_coe_norm.symm
_ = โf xโ := by rw [hfx]
_ โค M * โfโ := (hM f)
_ = M := by
|
rw [hfโ, mul_one]
|
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M := by
classical
by_cases h : x = 0
ยท simp only [h, hMp, norm_zero]
ยท obtain โจf, hfโ, hfxโฉ : โ f : E โL[๐] ๐, โfโ = 1 โง f x = โxโ := exists_dual_vector ๐ x h
calc
โxโ = โ(โxโ : ๐)โ := IsROrC.norm_coe_norm.symm
_ = โf xโ := by rw [hfx]
_ โค M * โfโ := (hM f)
_ = M := by
|
Mathlib.Analysis.NormedSpace.Dual.113_0.WirVfj6f5oiZZ2w
|
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M
|
Mathlib_Analysis_NormedSpace_Dual
|
๐ : Type v
instโยฒ : IsROrC ๐
E : Type u
instโยน : NormedAddCommGroup E
instโ : NormedSpace ๐ E
x : E
h : โ (f : Dual ๐ E), f x = 0
f : Dual ๐ E
โข โf xโ โค 0 * โfโ
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# The topological dual of a normed space
In this file we define the topological dual `NormedSpace.Dual` of a normed space, and the
continuous linear map `NormedSpace.inclusionInDoubleDual` from a normed space into its double
dual.
For base field `๐ = โ` or `๐ = โ`, this map is actually an isometric embedding; we provide a
version `NormedSpace.inclusionInDoubleDualLi` of the map which is of type a bundled linear
isometric embedding, `E โโแตข[๐] (Dual ๐ (Dual ๐ E))`.
Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
theory for `SeminormedAddCommGroup` and we specialize to `NormedAddCommGroup` when needed.
## Main definitions
* `inclusionInDoubleDual` and `inclusionInDoubleDualLi` are the inclusion of a normed space
in its double dual, considered as a bounded linear map and as a linear isometry, respectively.
* `polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals `x'` for which
`โx' zโ โค 1` for every `z โ s`.
## Tags
dual
-/
noncomputable section
open Classical Topology Bornology
universe u v
namespace NormedSpace
section General
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable (E : Type*) [SeminormedAddCommGroup E] [NormedSpace ๐ E]
variable (F : Type*) [NormedAddCommGroup F] [NormedSpace ๐ F]
/-- The topological dual of a seminormed space `E`. -/
abbrev Dual : Type _ := E โL[๐] ๐
#align normed_space.dual NormedSpace.Dual
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_eq until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : NormedSpace ๐ (Dual ๐ E) := inferInstance
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_le until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : SeminormedAddCommGroup (Dual ๐ E) := inferInstance
/-- The inclusion of a normed space in its double (topological) dual, considered
as a bounded linear map. -/
def inclusionInDoubleDual : E โL[๐] Dual ๐ (Dual ๐ E) :=
ContinuousLinearMap.apply ๐ ๐
#align normed_space.inclusion_in_double_dual NormedSpace.inclusionInDoubleDual
@[simp]
theorem dual_def (x : E) (f : Dual ๐ E) : inclusionInDoubleDual ๐ E x f = f x :=
rfl
#align normed_space.dual_def NormedSpace.dual_def
theorem inclusionInDoubleDual_norm_eq :
โinclusionInDoubleDual ๐ Eโ = โContinuousLinearMap.id ๐ (Dual ๐ E)โ :=
ContinuousLinearMap.op_norm_flip _
#align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1 := by
rw [inclusionInDoubleDual_norm_eq]
exact ContinuousLinearMap.norm_id_le
#align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
theorem double_dual_bound (x : E) : โ(inclusionInDoubleDual ๐ E) xโ โค โxโ := by
simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusionInDoubleDual_norm_le ๐ E) x
#align normed_space.double_dual_bound NormedSpace.double_dual_bound
/-- The dual pairing as a bilinear form. -/
def dualPairing : Dual ๐ E โโ[๐] E โโ[๐] ๐ :=
ContinuousLinearMap.coeLM ๐
#align normed_space.dual_pairing NormedSpace.dualPairing
@[simp]
theorem dualPairing_apply {v : Dual ๐ E} {x : E} : dualPairing ๐ E v x = v x :=
rfl
#align normed_space.dual_pairing_apply NormedSpace.dualPairing_apply
theorem dualPairing_separatingLeft : (dualPairing ๐ E).SeparatingLeft := by
rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]
exact ContinuousLinearMap.coe_injective
#align normed_space.dual_pairing_separating_left NormedSpace.dualPairing_separatingLeft
end General
section BidualIsometry
variable (๐ : Type v) [IsROrC ๐] {E : Type u} [NormedAddCommGroup E] [NormedSpace ๐ E]
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M := by
classical
by_cases h : x = 0
ยท simp only [h, hMp, norm_zero]
ยท obtain โจf, hfโ, hfxโฉ : โ f : E โL[๐] ๐, โfโ = 1 โง f x = โxโ := exists_dual_vector ๐ x h
calc
โxโ = โ(โxโ : ๐)โ := IsROrC.norm_coe_norm.symm
_ = โf xโ := by rw [hfx]
_ โค M * โfโ := (hM f)
_ = M := by rw [hfโ, mul_one]
#align normed_space.norm_le_dual_bound NormedSpace.norm_le_dual_bound
theorem eq_zero_of_forall_dual_eq_zero {x : E} (h : โ f : Dual ๐ E, f x = (0 : ๐)) : x = 0 :=
norm_le_zero_iff.mp (norm_le_dual_bound ๐ x le_rfl fun f => by
|
simp [h f]
|
theorem eq_zero_of_forall_dual_eq_zero {x : E} (h : โ f : Dual ๐ E, f x = (0 : ๐)) : x = 0 :=
norm_le_zero_iff.mp (norm_le_dual_bound ๐ x le_rfl fun f => by
|
Mathlib.Analysis.NormedSpace.Dual.128_0.WirVfj6f5oiZZ2w
|
theorem eq_zero_of_forall_dual_eq_zero {x : E} (h : โ f : Dual ๐ E, f x = (0 : ๐)) : x = 0
|
Mathlib_Analysis_NormedSpace_Dual
|
๐ : Type v
instโยฒ : IsROrC ๐
E : Type u
instโยน : NormedAddCommGroup E
instโ : NormedSpace ๐ E
x : E
hx : x = 0
โข โ (g : Dual ๐ E), g x = 0
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# The topological dual of a normed space
In this file we define the topological dual `NormedSpace.Dual` of a normed space, and the
continuous linear map `NormedSpace.inclusionInDoubleDual` from a normed space into its double
dual.
For base field `๐ = โ` or `๐ = โ`, this map is actually an isometric embedding; we provide a
version `NormedSpace.inclusionInDoubleDualLi` of the map which is of type a bundled linear
isometric embedding, `E โโแตข[๐] (Dual ๐ (Dual ๐ E))`.
Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
theory for `SeminormedAddCommGroup` and we specialize to `NormedAddCommGroup` when needed.
## Main definitions
* `inclusionInDoubleDual` and `inclusionInDoubleDualLi` are the inclusion of a normed space
in its double dual, considered as a bounded linear map and as a linear isometry, respectively.
* `polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals `x'` for which
`โx' zโ โค 1` for every `z โ s`.
## Tags
dual
-/
noncomputable section
open Classical Topology Bornology
universe u v
namespace NormedSpace
section General
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable (E : Type*) [SeminormedAddCommGroup E] [NormedSpace ๐ E]
variable (F : Type*) [NormedAddCommGroup F] [NormedSpace ๐ F]
/-- The topological dual of a seminormed space `E`. -/
abbrev Dual : Type _ := E โL[๐] ๐
#align normed_space.dual NormedSpace.Dual
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_eq until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : NormedSpace ๐ (Dual ๐ E) := inferInstance
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_le until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : SeminormedAddCommGroup (Dual ๐ E) := inferInstance
/-- The inclusion of a normed space in its double (topological) dual, considered
as a bounded linear map. -/
def inclusionInDoubleDual : E โL[๐] Dual ๐ (Dual ๐ E) :=
ContinuousLinearMap.apply ๐ ๐
#align normed_space.inclusion_in_double_dual NormedSpace.inclusionInDoubleDual
@[simp]
theorem dual_def (x : E) (f : Dual ๐ E) : inclusionInDoubleDual ๐ E x f = f x :=
rfl
#align normed_space.dual_def NormedSpace.dual_def
theorem inclusionInDoubleDual_norm_eq :
โinclusionInDoubleDual ๐ Eโ = โContinuousLinearMap.id ๐ (Dual ๐ E)โ :=
ContinuousLinearMap.op_norm_flip _
#align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1 := by
rw [inclusionInDoubleDual_norm_eq]
exact ContinuousLinearMap.norm_id_le
#align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
theorem double_dual_bound (x : E) : โ(inclusionInDoubleDual ๐ E) xโ โค โxโ := by
simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusionInDoubleDual_norm_le ๐ E) x
#align normed_space.double_dual_bound NormedSpace.double_dual_bound
/-- The dual pairing as a bilinear form. -/
def dualPairing : Dual ๐ E โโ[๐] E โโ[๐] ๐ :=
ContinuousLinearMap.coeLM ๐
#align normed_space.dual_pairing NormedSpace.dualPairing
@[simp]
theorem dualPairing_apply {v : Dual ๐ E} {x : E} : dualPairing ๐ E v x = v x :=
rfl
#align normed_space.dual_pairing_apply NormedSpace.dualPairing_apply
theorem dualPairing_separatingLeft : (dualPairing ๐ E).SeparatingLeft := by
rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]
exact ContinuousLinearMap.coe_injective
#align normed_space.dual_pairing_separating_left NormedSpace.dualPairing_separatingLeft
end General
section BidualIsometry
variable (๐ : Type v) [IsROrC ๐] {E : Type u} [NormedAddCommGroup E] [NormedSpace ๐ E]
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M := by
classical
by_cases h : x = 0
ยท simp only [h, hMp, norm_zero]
ยท obtain โจf, hfโ, hfxโฉ : โ f : E โL[๐] ๐, โfโ = 1 โง f x = โxโ := exists_dual_vector ๐ x h
calc
โxโ = โ(โxโ : ๐)โ := IsROrC.norm_coe_norm.symm
_ = โf xโ := by rw [hfx]
_ โค M * โfโ := (hM f)
_ = M := by rw [hfโ, mul_one]
#align normed_space.norm_le_dual_bound NormedSpace.norm_le_dual_bound
theorem eq_zero_of_forall_dual_eq_zero {x : E} (h : โ f : Dual ๐ E, f x = (0 : ๐)) : x = 0 :=
norm_le_zero_iff.mp (norm_le_dual_bound ๐ x le_rfl fun f => by simp [h f])
#align normed_space.eq_zero_of_forall_dual_eq_zero NormedSpace.eq_zero_of_forall_dual_eq_zero
theorem eq_zero_iff_forall_dual_eq_zero (x : E) : x = 0 โ โ g : Dual ๐ E, g x = 0 :=
โจfun hx => by
|
simp [hx]
|
theorem eq_zero_iff_forall_dual_eq_zero (x : E) : x = 0 โ โ g : Dual ๐ E, g x = 0 :=
โจfun hx => by
|
Mathlib.Analysis.NormedSpace.Dual.132_0.WirVfj6f5oiZZ2w
|
theorem eq_zero_iff_forall_dual_eq_zero (x : E) : x = 0 โ โ g : Dual ๐ E, g x = 0
|
Mathlib_Analysis_NormedSpace_Dual
|
๐ : Type v
instโยฒ : IsROrC ๐
E : Type u
instโยน : NormedAddCommGroup E
instโ : NormedSpace ๐ E
x y : E
โข x = y โ โ (g : Dual ๐ E), g x = g y
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# The topological dual of a normed space
In this file we define the topological dual `NormedSpace.Dual` of a normed space, and the
continuous linear map `NormedSpace.inclusionInDoubleDual` from a normed space into its double
dual.
For base field `๐ = โ` or `๐ = โ`, this map is actually an isometric embedding; we provide a
version `NormedSpace.inclusionInDoubleDualLi` of the map which is of type a bundled linear
isometric embedding, `E โโแตข[๐] (Dual ๐ (Dual ๐ E))`.
Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
theory for `SeminormedAddCommGroup` and we specialize to `NormedAddCommGroup` when needed.
## Main definitions
* `inclusionInDoubleDual` and `inclusionInDoubleDualLi` are the inclusion of a normed space
in its double dual, considered as a bounded linear map and as a linear isometry, respectively.
* `polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals `x'` for which
`โx' zโ โค 1` for every `z โ s`.
## Tags
dual
-/
noncomputable section
open Classical Topology Bornology
universe u v
namespace NormedSpace
section General
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable (E : Type*) [SeminormedAddCommGroup E] [NormedSpace ๐ E]
variable (F : Type*) [NormedAddCommGroup F] [NormedSpace ๐ F]
/-- The topological dual of a seminormed space `E`. -/
abbrev Dual : Type _ := E โL[๐] ๐
#align normed_space.dual NormedSpace.Dual
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_eq until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : NormedSpace ๐ (Dual ๐ E) := inferInstance
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_le until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : SeminormedAddCommGroup (Dual ๐ E) := inferInstance
/-- The inclusion of a normed space in its double (topological) dual, considered
as a bounded linear map. -/
def inclusionInDoubleDual : E โL[๐] Dual ๐ (Dual ๐ E) :=
ContinuousLinearMap.apply ๐ ๐
#align normed_space.inclusion_in_double_dual NormedSpace.inclusionInDoubleDual
@[simp]
theorem dual_def (x : E) (f : Dual ๐ E) : inclusionInDoubleDual ๐ E x f = f x :=
rfl
#align normed_space.dual_def NormedSpace.dual_def
theorem inclusionInDoubleDual_norm_eq :
โinclusionInDoubleDual ๐ Eโ = โContinuousLinearMap.id ๐ (Dual ๐ E)โ :=
ContinuousLinearMap.op_norm_flip _
#align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1 := by
rw [inclusionInDoubleDual_norm_eq]
exact ContinuousLinearMap.norm_id_le
#align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
theorem double_dual_bound (x : E) : โ(inclusionInDoubleDual ๐ E) xโ โค โxโ := by
simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusionInDoubleDual_norm_le ๐ E) x
#align normed_space.double_dual_bound NormedSpace.double_dual_bound
/-- The dual pairing as a bilinear form. -/
def dualPairing : Dual ๐ E โโ[๐] E โโ[๐] ๐ :=
ContinuousLinearMap.coeLM ๐
#align normed_space.dual_pairing NormedSpace.dualPairing
@[simp]
theorem dualPairing_apply {v : Dual ๐ E} {x : E} : dualPairing ๐ E v x = v x :=
rfl
#align normed_space.dual_pairing_apply NormedSpace.dualPairing_apply
theorem dualPairing_separatingLeft : (dualPairing ๐ E).SeparatingLeft := by
rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]
exact ContinuousLinearMap.coe_injective
#align normed_space.dual_pairing_separating_left NormedSpace.dualPairing_separatingLeft
end General
section BidualIsometry
variable (๐ : Type v) [IsROrC ๐] {E : Type u} [NormedAddCommGroup E] [NormedSpace ๐ E]
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M := by
classical
by_cases h : x = 0
ยท simp only [h, hMp, norm_zero]
ยท obtain โจf, hfโ, hfxโฉ : โ f : E โL[๐] ๐, โfโ = 1 โง f x = โxโ := exists_dual_vector ๐ x h
calc
โxโ = โ(โxโ : ๐)โ := IsROrC.norm_coe_norm.symm
_ = โf xโ := by rw [hfx]
_ โค M * โfโ := (hM f)
_ = M := by rw [hfโ, mul_one]
#align normed_space.norm_le_dual_bound NormedSpace.norm_le_dual_bound
theorem eq_zero_of_forall_dual_eq_zero {x : E} (h : โ f : Dual ๐ E, f x = (0 : ๐)) : x = 0 :=
norm_le_zero_iff.mp (norm_le_dual_bound ๐ x le_rfl fun f => by simp [h f])
#align normed_space.eq_zero_of_forall_dual_eq_zero NormedSpace.eq_zero_of_forall_dual_eq_zero
theorem eq_zero_iff_forall_dual_eq_zero (x : E) : x = 0 โ โ g : Dual ๐ E, g x = 0 :=
โจfun hx => by simp [hx], fun h => eq_zero_of_forall_dual_eq_zero ๐ hโฉ
#align normed_space.eq_zero_iff_forall_dual_eq_zero NormedSpace.eq_zero_iff_forall_dual_eq_zero
/-- See also `geometric_hahn_banach_point_point`. -/
theorem eq_iff_forall_dual_eq {x y : E} : x = y โ โ g : Dual ๐ E, g x = g y := by
|
rw [โ sub_eq_zero, eq_zero_iff_forall_dual_eq_zero ๐ (x - y)]
|
/-- See also `geometric_hahn_banach_point_point`. -/
theorem eq_iff_forall_dual_eq {x y : E} : x = y โ โ g : Dual ๐ E, g x = g y := by
|
Mathlib.Analysis.NormedSpace.Dual.136_0.WirVfj6f5oiZZ2w
|
/-- See also `geometric_hahn_banach_point_point`. -/
theorem eq_iff_forall_dual_eq {x y : E} : x = y โ โ g : Dual ๐ E, g x = g y
|
Mathlib_Analysis_NormedSpace_Dual
|
๐ : Type v
instโยฒ : IsROrC ๐
E : Type u
instโยน : NormedAddCommGroup E
instโ : NormedSpace ๐ E
x y : E
โข (โ (g : Dual ๐ E), g (x - y) = 0) โ โ (g : Dual ๐ E), g x = g y
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# The topological dual of a normed space
In this file we define the topological dual `NormedSpace.Dual` of a normed space, and the
continuous linear map `NormedSpace.inclusionInDoubleDual` from a normed space into its double
dual.
For base field `๐ = โ` or `๐ = โ`, this map is actually an isometric embedding; we provide a
version `NormedSpace.inclusionInDoubleDualLi` of the map which is of type a bundled linear
isometric embedding, `E โโแตข[๐] (Dual ๐ (Dual ๐ E))`.
Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
theory for `SeminormedAddCommGroup` and we specialize to `NormedAddCommGroup` when needed.
## Main definitions
* `inclusionInDoubleDual` and `inclusionInDoubleDualLi` are the inclusion of a normed space
in its double dual, considered as a bounded linear map and as a linear isometry, respectively.
* `polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals `x'` for which
`โx' zโ โค 1` for every `z โ s`.
## Tags
dual
-/
noncomputable section
open Classical Topology Bornology
universe u v
namespace NormedSpace
section General
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable (E : Type*) [SeminormedAddCommGroup E] [NormedSpace ๐ E]
variable (F : Type*) [NormedAddCommGroup F] [NormedSpace ๐ F]
/-- The topological dual of a seminormed space `E`. -/
abbrev Dual : Type _ := E โL[๐] ๐
#align normed_space.dual NormedSpace.Dual
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_eq until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : NormedSpace ๐ (Dual ๐ E) := inferInstance
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_le until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : SeminormedAddCommGroup (Dual ๐ E) := inferInstance
/-- The inclusion of a normed space in its double (topological) dual, considered
as a bounded linear map. -/
def inclusionInDoubleDual : E โL[๐] Dual ๐ (Dual ๐ E) :=
ContinuousLinearMap.apply ๐ ๐
#align normed_space.inclusion_in_double_dual NormedSpace.inclusionInDoubleDual
@[simp]
theorem dual_def (x : E) (f : Dual ๐ E) : inclusionInDoubleDual ๐ E x f = f x :=
rfl
#align normed_space.dual_def NormedSpace.dual_def
theorem inclusionInDoubleDual_norm_eq :
โinclusionInDoubleDual ๐ Eโ = โContinuousLinearMap.id ๐ (Dual ๐ E)โ :=
ContinuousLinearMap.op_norm_flip _
#align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1 := by
rw [inclusionInDoubleDual_norm_eq]
exact ContinuousLinearMap.norm_id_le
#align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
theorem double_dual_bound (x : E) : โ(inclusionInDoubleDual ๐ E) xโ โค โxโ := by
simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusionInDoubleDual_norm_le ๐ E) x
#align normed_space.double_dual_bound NormedSpace.double_dual_bound
/-- The dual pairing as a bilinear form. -/
def dualPairing : Dual ๐ E โโ[๐] E โโ[๐] ๐ :=
ContinuousLinearMap.coeLM ๐
#align normed_space.dual_pairing NormedSpace.dualPairing
@[simp]
theorem dualPairing_apply {v : Dual ๐ E} {x : E} : dualPairing ๐ E v x = v x :=
rfl
#align normed_space.dual_pairing_apply NormedSpace.dualPairing_apply
theorem dualPairing_separatingLeft : (dualPairing ๐ E).SeparatingLeft := by
rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]
exact ContinuousLinearMap.coe_injective
#align normed_space.dual_pairing_separating_left NormedSpace.dualPairing_separatingLeft
end General
section BidualIsometry
variable (๐ : Type v) [IsROrC ๐] {E : Type u} [NormedAddCommGroup E] [NormedSpace ๐ E]
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M := by
classical
by_cases h : x = 0
ยท simp only [h, hMp, norm_zero]
ยท obtain โจf, hfโ, hfxโฉ : โ f : E โL[๐] ๐, โfโ = 1 โง f x = โxโ := exists_dual_vector ๐ x h
calc
โxโ = โ(โxโ : ๐)โ := IsROrC.norm_coe_norm.symm
_ = โf xโ := by rw [hfx]
_ โค M * โfโ := (hM f)
_ = M := by rw [hfโ, mul_one]
#align normed_space.norm_le_dual_bound NormedSpace.norm_le_dual_bound
theorem eq_zero_of_forall_dual_eq_zero {x : E} (h : โ f : Dual ๐ E, f x = (0 : ๐)) : x = 0 :=
norm_le_zero_iff.mp (norm_le_dual_bound ๐ x le_rfl fun f => by simp [h f])
#align normed_space.eq_zero_of_forall_dual_eq_zero NormedSpace.eq_zero_of_forall_dual_eq_zero
theorem eq_zero_iff_forall_dual_eq_zero (x : E) : x = 0 โ โ g : Dual ๐ E, g x = 0 :=
โจfun hx => by simp [hx], fun h => eq_zero_of_forall_dual_eq_zero ๐ hโฉ
#align normed_space.eq_zero_iff_forall_dual_eq_zero NormedSpace.eq_zero_iff_forall_dual_eq_zero
/-- See also `geometric_hahn_banach_point_point`. -/
theorem eq_iff_forall_dual_eq {x y : E} : x = y โ โ g : Dual ๐ E, g x = g y := by
rw [โ sub_eq_zero, eq_zero_iff_forall_dual_eq_zero ๐ (x - y)]
|
simp [sub_eq_zero]
|
/-- See also `geometric_hahn_banach_point_point`. -/
theorem eq_iff_forall_dual_eq {x y : E} : x = y โ โ g : Dual ๐ E, g x = g y := by
rw [โ sub_eq_zero, eq_zero_iff_forall_dual_eq_zero ๐ (x - y)]
|
Mathlib.Analysis.NormedSpace.Dual.136_0.WirVfj6f5oiZZ2w
|
/-- See also `geometric_hahn_banach_point_point`. -/
theorem eq_iff_forall_dual_eq {x y : E} : x = y โ โ g : Dual ๐ E, g x = g y
|
Mathlib_Analysis_NormedSpace_Dual
|
๐ : Type v
instโยฒ : IsROrC ๐
E : Type u
instโยน : NormedAddCommGroup E
instโ : NormedSpace ๐ E
srcโ : E โL[๐] Dual ๐ (Dual ๐ E) := inclusionInDoubleDual ๐ E
โข โ (x : E), โโsrcโ xโ = โxโ
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# The topological dual of a normed space
In this file we define the topological dual `NormedSpace.Dual` of a normed space, and the
continuous linear map `NormedSpace.inclusionInDoubleDual` from a normed space into its double
dual.
For base field `๐ = โ` or `๐ = โ`, this map is actually an isometric embedding; we provide a
version `NormedSpace.inclusionInDoubleDualLi` of the map which is of type a bundled linear
isometric embedding, `E โโแตข[๐] (Dual ๐ (Dual ๐ E))`.
Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
theory for `SeminormedAddCommGroup` and we specialize to `NormedAddCommGroup` when needed.
## Main definitions
* `inclusionInDoubleDual` and `inclusionInDoubleDualLi` are the inclusion of a normed space
in its double dual, considered as a bounded linear map and as a linear isometry, respectively.
* `polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals `x'` for which
`โx' zโ โค 1` for every `z โ s`.
## Tags
dual
-/
noncomputable section
open Classical Topology Bornology
universe u v
namespace NormedSpace
section General
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable (E : Type*) [SeminormedAddCommGroup E] [NormedSpace ๐ E]
variable (F : Type*) [NormedAddCommGroup F] [NormedSpace ๐ F]
/-- The topological dual of a seminormed space `E`. -/
abbrev Dual : Type _ := E โL[๐] ๐
#align normed_space.dual NormedSpace.Dual
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_eq until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : NormedSpace ๐ (Dual ๐ E) := inferInstance
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_le until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : SeminormedAddCommGroup (Dual ๐ E) := inferInstance
/-- The inclusion of a normed space in its double (topological) dual, considered
as a bounded linear map. -/
def inclusionInDoubleDual : E โL[๐] Dual ๐ (Dual ๐ E) :=
ContinuousLinearMap.apply ๐ ๐
#align normed_space.inclusion_in_double_dual NormedSpace.inclusionInDoubleDual
@[simp]
theorem dual_def (x : E) (f : Dual ๐ E) : inclusionInDoubleDual ๐ E x f = f x :=
rfl
#align normed_space.dual_def NormedSpace.dual_def
theorem inclusionInDoubleDual_norm_eq :
โinclusionInDoubleDual ๐ Eโ = โContinuousLinearMap.id ๐ (Dual ๐ E)โ :=
ContinuousLinearMap.op_norm_flip _
#align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1 := by
rw [inclusionInDoubleDual_norm_eq]
exact ContinuousLinearMap.norm_id_le
#align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
theorem double_dual_bound (x : E) : โ(inclusionInDoubleDual ๐ E) xโ โค โxโ := by
simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusionInDoubleDual_norm_le ๐ E) x
#align normed_space.double_dual_bound NormedSpace.double_dual_bound
/-- The dual pairing as a bilinear form. -/
def dualPairing : Dual ๐ E โโ[๐] E โโ[๐] ๐ :=
ContinuousLinearMap.coeLM ๐
#align normed_space.dual_pairing NormedSpace.dualPairing
@[simp]
theorem dualPairing_apply {v : Dual ๐ E} {x : E} : dualPairing ๐ E v x = v x :=
rfl
#align normed_space.dual_pairing_apply NormedSpace.dualPairing_apply
theorem dualPairing_separatingLeft : (dualPairing ๐ E).SeparatingLeft := by
rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]
exact ContinuousLinearMap.coe_injective
#align normed_space.dual_pairing_separating_left NormedSpace.dualPairing_separatingLeft
end General
section BidualIsometry
variable (๐ : Type v) [IsROrC ๐] {E : Type u} [NormedAddCommGroup E] [NormedSpace ๐ E]
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M := by
classical
by_cases h : x = 0
ยท simp only [h, hMp, norm_zero]
ยท obtain โจf, hfโ, hfxโฉ : โ f : E โL[๐] ๐, โfโ = 1 โง f x = โxโ := exists_dual_vector ๐ x h
calc
โxโ = โ(โxโ : ๐)โ := IsROrC.norm_coe_norm.symm
_ = โf xโ := by rw [hfx]
_ โค M * โfโ := (hM f)
_ = M := by rw [hfโ, mul_one]
#align normed_space.norm_le_dual_bound NormedSpace.norm_le_dual_bound
theorem eq_zero_of_forall_dual_eq_zero {x : E} (h : โ f : Dual ๐ E, f x = (0 : ๐)) : x = 0 :=
norm_le_zero_iff.mp (norm_le_dual_bound ๐ x le_rfl fun f => by simp [h f])
#align normed_space.eq_zero_of_forall_dual_eq_zero NormedSpace.eq_zero_of_forall_dual_eq_zero
theorem eq_zero_iff_forall_dual_eq_zero (x : E) : x = 0 โ โ g : Dual ๐ E, g x = 0 :=
โจfun hx => by simp [hx], fun h => eq_zero_of_forall_dual_eq_zero ๐ hโฉ
#align normed_space.eq_zero_iff_forall_dual_eq_zero NormedSpace.eq_zero_iff_forall_dual_eq_zero
/-- See also `geometric_hahn_banach_point_point`. -/
theorem eq_iff_forall_dual_eq {x y : E} : x = y โ โ g : Dual ๐ E, g x = g y := by
rw [โ sub_eq_zero, eq_zero_iff_forall_dual_eq_zero ๐ (x - y)]
simp [sub_eq_zero]
#align normed_space.eq_iff_forall_dual_eq NormedSpace.eq_iff_forall_dual_eq
/-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E โโแตข[๐] Dual ๐ (Dual ๐ E) :=
{ inclusionInDoubleDual ๐ E with
norm_map' := by
|
intro x
|
/-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E โโแตข[๐] Dual ๐ (Dual ๐ E) :=
{ inclusionInDoubleDual ๐ E with
norm_map' := by
|
Mathlib.Analysis.NormedSpace.Dual.142_0.WirVfj6f5oiZZ2w
|
/-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E โโแตข[๐] Dual ๐ (Dual ๐ E)
|
Mathlib_Analysis_NormedSpace_Dual
|
๐ : Type v
instโยฒ : IsROrC ๐
E : Type u
instโยน : NormedAddCommGroup E
instโ : NormedSpace ๐ E
srcโ : E โL[๐] Dual ๐ (Dual ๐ E) := inclusionInDoubleDual ๐ E
x : E
โข โโsrcโ xโ = โxโ
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# The topological dual of a normed space
In this file we define the topological dual `NormedSpace.Dual` of a normed space, and the
continuous linear map `NormedSpace.inclusionInDoubleDual` from a normed space into its double
dual.
For base field `๐ = โ` or `๐ = โ`, this map is actually an isometric embedding; we provide a
version `NormedSpace.inclusionInDoubleDualLi` of the map which is of type a bundled linear
isometric embedding, `E โโแตข[๐] (Dual ๐ (Dual ๐ E))`.
Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
theory for `SeminormedAddCommGroup` and we specialize to `NormedAddCommGroup` when needed.
## Main definitions
* `inclusionInDoubleDual` and `inclusionInDoubleDualLi` are the inclusion of a normed space
in its double dual, considered as a bounded linear map and as a linear isometry, respectively.
* `polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals `x'` for which
`โx' zโ โค 1` for every `z โ s`.
## Tags
dual
-/
noncomputable section
open Classical Topology Bornology
universe u v
namespace NormedSpace
section General
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable (E : Type*) [SeminormedAddCommGroup E] [NormedSpace ๐ E]
variable (F : Type*) [NormedAddCommGroup F] [NormedSpace ๐ F]
/-- The topological dual of a seminormed space `E`. -/
abbrev Dual : Type _ := E โL[๐] ๐
#align normed_space.dual NormedSpace.Dual
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_eq until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : NormedSpace ๐ (Dual ๐ E) := inferInstance
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_le until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : SeminormedAddCommGroup (Dual ๐ E) := inferInstance
/-- The inclusion of a normed space in its double (topological) dual, considered
as a bounded linear map. -/
def inclusionInDoubleDual : E โL[๐] Dual ๐ (Dual ๐ E) :=
ContinuousLinearMap.apply ๐ ๐
#align normed_space.inclusion_in_double_dual NormedSpace.inclusionInDoubleDual
@[simp]
theorem dual_def (x : E) (f : Dual ๐ E) : inclusionInDoubleDual ๐ E x f = f x :=
rfl
#align normed_space.dual_def NormedSpace.dual_def
theorem inclusionInDoubleDual_norm_eq :
โinclusionInDoubleDual ๐ Eโ = โContinuousLinearMap.id ๐ (Dual ๐ E)โ :=
ContinuousLinearMap.op_norm_flip _
#align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1 := by
rw [inclusionInDoubleDual_norm_eq]
exact ContinuousLinearMap.norm_id_le
#align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
theorem double_dual_bound (x : E) : โ(inclusionInDoubleDual ๐ E) xโ โค โxโ := by
simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusionInDoubleDual_norm_le ๐ E) x
#align normed_space.double_dual_bound NormedSpace.double_dual_bound
/-- The dual pairing as a bilinear form. -/
def dualPairing : Dual ๐ E โโ[๐] E โโ[๐] ๐ :=
ContinuousLinearMap.coeLM ๐
#align normed_space.dual_pairing NormedSpace.dualPairing
@[simp]
theorem dualPairing_apply {v : Dual ๐ E} {x : E} : dualPairing ๐ E v x = v x :=
rfl
#align normed_space.dual_pairing_apply NormedSpace.dualPairing_apply
theorem dualPairing_separatingLeft : (dualPairing ๐ E).SeparatingLeft := by
rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]
exact ContinuousLinearMap.coe_injective
#align normed_space.dual_pairing_separating_left NormedSpace.dualPairing_separatingLeft
end General
section BidualIsometry
variable (๐ : Type v) [IsROrC ๐] {E : Type u} [NormedAddCommGroup E] [NormedSpace ๐ E]
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M := by
classical
by_cases h : x = 0
ยท simp only [h, hMp, norm_zero]
ยท obtain โจf, hfโ, hfxโฉ : โ f : E โL[๐] ๐, โfโ = 1 โง f x = โxโ := exists_dual_vector ๐ x h
calc
โxโ = โ(โxโ : ๐)โ := IsROrC.norm_coe_norm.symm
_ = โf xโ := by rw [hfx]
_ โค M * โfโ := (hM f)
_ = M := by rw [hfโ, mul_one]
#align normed_space.norm_le_dual_bound NormedSpace.norm_le_dual_bound
theorem eq_zero_of_forall_dual_eq_zero {x : E} (h : โ f : Dual ๐ E, f x = (0 : ๐)) : x = 0 :=
norm_le_zero_iff.mp (norm_le_dual_bound ๐ x le_rfl fun f => by simp [h f])
#align normed_space.eq_zero_of_forall_dual_eq_zero NormedSpace.eq_zero_of_forall_dual_eq_zero
theorem eq_zero_iff_forall_dual_eq_zero (x : E) : x = 0 โ โ g : Dual ๐ E, g x = 0 :=
โจfun hx => by simp [hx], fun h => eq_zero_of_forall_dual_eq_zero ๐ hโฉ
#align normed_space.eq_zero_iff_forall_dual_eq_zero NormedSpace.eq_zero_iff_forall_dual_eq_zero
/-- See also `geometric_hahn_banach_point_point`. -/
theorem eq_iff_forall_dual_eq {x y : E} : x = y โ โ g : Dual ๐ E, g x = g y := by
rw [โ sub_eq_zero, eq_zero_iff_forall_dual_eq_zero ๐ (x - y)]
simp [sub_eq_zero]
#align normed_space.eq_iff_forall_dual_eq NormedSpace.eq_iff_forall_dual_eq
/-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E โโแตข[๐] Dual ๐ (Dual ๐ E) :=
{ inclusionInDoubleDual ๐ E with
norm_map' := by
intro x
|
apply le_antisymm
|
/-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E โโแตข[๐] Dual ๐ (Dual ๐ E) :=
{ inclusionInDoubleDual ๐ E with
norm_map' := by
intro x
|
Mathlib.Analysis.NormedSpace.Dual.142_0.WirVfj6f5oiZZ2w
|
/-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E โโแตข[๐] Dual ๐ (Dual ๐ E)
|
Mathlib_Analysis_NormedSpace_Dual
|
case a
๐ : Type v
instโยฒ : IsROrC ๐
E : Type u
instโยน : NormedAddCommGroup E
instโ : NormedSpace ๐ E
srcโ : E โL[๐] Dual ๐ (Dual ๐ E) := inclusionInDoubleDual ๐ E
x : E
โข โโsrcโ xโ โค โxโ
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# The topological dual of a normed space
In this file we define the topological dual `NormedSpace.Dual` of a normed space, and the
continuous linear map `NormedSpace.inclusionInDoubleDual` from a normed space into its double
dual.
For base field `๐ = โ` or `๐ = โ`, this map is actually an isometric embedding; we provide a
version `NormedSpace.inclusionInDoubleDualLi` of the map which is of type a bundled linear
isometric embedding, `E โโแตข[๐] (Dual ๐ (Dual ๐ E))`.
Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
theory for `SeminormedAddCommGroup` and we specialize to `NormedAddCommGroup` when needed.
## Main definitions
* `inclusionInDoubleDual` and `inclusionInDoubleDualLi` are the inclusion of a normed space
in its double dual, considered as a bounded linear map and as a linear isometry, respectively.
* `polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals `x'` for which
`โx' zโ โค 1` for every `z โ s`.
## Tags
dual
-/
noncomputable section
open Classical Topology Bornology
universe u v
namespace NormedSpace
section General
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable (E : Type*) [SeminormedAddCommGroup E] [NormedSpace ๐ E]
variable (F : Type*) [NormedAddCommGroup F] [NormedSpace ๐ F]
/-- The topological dual of a seminormed space `E`. -/
abbrev Dual : Type _ := E โL[๐] ๐
#align normed_space.dual NormedSpace.Dual
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_eq until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : NormedSpace ๐ (Dual ๐ E) := inferInstance
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_le until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : SeminormedAddCommGroup (Dual ๐ E) := inferInstance
/-- The inclusion of a normed space in its double (topological) dual, considered
as a bounded linear map. -/
def inclusionInDoubleDual : E โL[๐] Dual ๐ (Dual ๐ E) :=
ContinuousLinearMap.apply ๐ ๐
#align normed_space.inclusion_in_double_dual NormedSpace.inclusionInDoubleDual
@[simp]
theorem dual_def (x : E) (f : Dual ๐ E) : inclusionInDoubleDual ๐ E x f = f x :=
rfl
#align normed_space.dual_def NormedSpace.dual_def
theorem inclusionInDoubleDual_norm_eq :
โinclusionInDoubleDual ๐ Eโ = โContinuousLinearMap.id ๐ (Dual ๐ E)โ :=
ContinuousLinearMap.op_norm_flip _
#align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1 := by
rw [inclusionInDoubleDual_norm_eq]
exact ContinuousLinearMap.norm_id_le
#align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
theorem double_dual_bound (x : E) : โ(inclusionInDoubleDual ๐ E) xโ โค โxโ := by
simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusionInDoubleDual_norm_le ๐ E) x
#align normed_space.double_dual_bound NormedSpace.double_dual_bound
/-- The dual pairing as a bilinear form. -/
def dualPairing : Dual ๐ E โโ[๐] E โโ[๐] ๐ :=
ContinuousLinearMap.coeLM ๐
#align normed_space.dual_pairing NormedSpace.dualPairing
@[simp]
theorem dualPairing_apply {v : Dual ๐ E} {x : E} : dualPairing ๐ E v x = v x :=
rfl
#align normed_space.dual_pairing_apply NormedSpace.dualPairing_apply
theorem dualPairing_separatingLeft : (dualPairing ๐ E).SeparatingLeft := by
rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]
exact ContinuousLinearMap.coe_injective
#align normed_space.dual_pairing_separating_left NormedSpace.dualPairing_separatingLeft
end General
section BidualIsometry
variable (๐ : Type v) [IsROrC ๐] {E : Type u} [NormedAddCommGroup E] [NormedSpace ๐ E]
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M := by
classical
by_cases h : x = 0
ยท simp only [h, hMp, norm_zero]
ยท obtain โจf, hfโ, hfxโฉ : โ f : E โL[๐] ๐, โfโ = 1 โง f x = โxโ := exists_dual_vector ๐ x h
calc
โxโ = โ(โxโ : ๐)โ := IsROrC.norm_coe_norm.symm
_ = โf xโ := by rw [hfx]
_ โค M * โfโ := (hM f)
_ = M := by rw [hfโ, mul_one]
#align normed_space.norm_le_dual_bound NormedSpace.norm_le_dual_bound
theorem eq_zero_of_forall_dual_eq_zero {x : E} (h : โ f : Dual ๐ E, f x = (0 : ๐)) : x = 0 :=
norm_le_zero_iff.mp (norm_le_dual_bound ๐ x le_rfl fun f => by simp [h f])
#align normed_space.eq_zero_of_forall_dual_eq_zero NormedSpace.eq_zero_of_forall_dual_eq_zero
theorem eq_zero_iff_forall_dual_eq_zero (x : E) : x = 0 โ โ g : Dual ๐ E, g x = 0 :=
โจfun hx => by simp [hx], fun h => eq_zero_of_forall_dual_eq_zero ๐ hโฉ
#align normed_space.eq_zero_iff_forall_dual_eq_zero NormedSpace.eq_zero_iff_forall_dual_eq_zero
/-- See also `geometric_hahn_banach_point_point`. -/
theorem eq_iff_forall_dual_eq {x y : E} : x = y โ โ g : Dual ๐ E, g x = g y := by
rw [โ sub_eq_zero, eq_zero_iff_forall_dual_eq_zero ๐ (x - y)]
simp [sub_eq_zero]
#align normed_space.eq_iff_forall_dual_eq NormedSpace.eq_iff_forall_dual_eq
/-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E โโแตข[๐] Dual ๐ (Dual ๐ E) :=
{ inclusionInDoubleDual ๐ E with
norm_map' := by
intro x
apply le_antisymm
ยท
|
exact double_dual_bound ๐ E x
|
/-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E โโแตข[๐] Dual ๐ (Dual ๐ E) :=
{ inclusionInDoubleDual ๐ E with
norm_map' := by
intro x
apply le_antisymm
ยท
|
Mathlib.Analysis.NormedSpace.Dual.142_0.WirVfj6f5oiZZ2w
|
/-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E โโแตข[๐] Dual ๐ (Dual ๐ E)
|
Mathlib_Analysis_NormedSpace_Dual
|
case a
๐ : Type v
instโยฒ : IsROrC ๐
E : Type u
instโยน : NormedAddCommGroup E
instโ : NormedSpace ๐ E
srcโ : E โL[๐] Dual ๐ (Dual ๐ E) := inclusionInDoubleDual ๐ E
x : E
โข โxโ โค โโsrcโ xโ
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# The topological dual of a normed space
In this file we define the topological dual `NormedSpace.Dual` of a normed space, and the
continuous linear map `NormedSpace.inclusionInDoubleDual` from a normed space into its double
dual.
For base field `๐ = โ` or `๐ = โ`, this map is actually an isometric embedding; we provide a
version `NormedSpace.inclusionInDoubleDualLi` of the map which is of type a bundled linear
isometric embedding, `E โโแตข[๐] (Dual ๐ (Dual ๐ E))`.
Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
theory for `SeminormedAddCommGroup` and we specialize to `NormedAddCommGroup` when needed.
## Main definitions
* `inclusionInDoubleDual` and `inclusionInDoubleDualLi` are the inclusion of a normed space
in its double dual, considered as a bounded linear map and as a linear isometry, respectively.
* `polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals `x'` for which
`โx' zโ โค 1` for every `z โ s`.
## Tags
dual
-/
noncomputable section
open Classical Topology Bornology
universe u v
namespace NormedSpace
section General
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable (E : Type*) [SeminormedAddCommGroup E] [NormedSpace ๐ E]
variable (F : Type*) [NormedAddCommGroup F] [NormedSpace ๐ F]
/-- The topological dual of a seminormed space `E`. -/
abbrev Dual : Type _ := E โL[๐] ๐
#align normed_space.dual NormedSpace.Dual
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_eq until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : NormedSpace ๐ (Dual ๐ E) := inferInstance
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_le until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : SeminormedAddCommGroup (Dual ๐ E) := inferInstance
/-- The inclusion of a normed space in its double (topological) dual, considered
as a bounded linear map. -/
def inclusionInDoubleDual : E โL[๐] Dual ๐ (Dual ๐ E) :=
ContinuousLinearMap.apply ๐ ๐
#align normed_space.inclusion_in_double_dual NormedSpace.inclusionInDoubleDual
@[simp]
theorem dual_def (x : E) (f : Dual ๐ E) : inclusionInDoubleDual ๐ E x f = f x :=
rfl
#align normed_space.dual_def NormedSpace.dual_def
theorem inclusionInDoubleDual_norm_eq :
โinclusionInDoubleDual ๐ Eโ = โContinuousLinearMap.id ๐ (Dual ๐ E)โ :=
ContinuousLinearMap.op_norm_flip _
#align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1 := by
rw [inclusionInDoubleDual_norm_eq]
exact ContinuousLinearMap.norm_id_le
#align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
theorem double_dual_bound (x : E) : โ(inclusionInDoubleDual ๐ E) xโ โค โxโ := by
simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusionInDoubleDual_norm_le ๐ E) x
#align normed_space.double_dual_bound NormedSpace.double_dual_bound
/-- The dual pairing as a bilinear form. -/
def dualPairing : Dual ๐ E โโ[๐] E โโ[๐] ๐ :=
ContinuousLinearMap.coeLM ๐
#align normed_space.dual_pairing NormedSpace.dualPairing
@[simp]
theorem dualPairing_apply {v : Dual ๐ E} {x : E} : dualPairing ๐ E v x = v x :=
rfl
#align normed_space.dual_pairing_apply NormedSpace.dualPairing_apply
theorem dualPairing_separatingLeft : (dualPairing ๐ E).SeparatingLeft := by
rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]
exact ContinuousLinearMap.coe_injective
#align normed_space.dual_pairing_separating_left NormedSpace.dualPairing_separatingLeft
end General
section BidualIsometry
variable (๐ : Type v) [IsROrC ๐] {E : Type u} [NormedAddCommGroup E] [NormedSpace ๐ E]
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M := by
classical
by_cases h : x = 0
ยท simp only [h, hMp, norm_zero]
ยท obtain โจf, hfโ, hfxโฉ : โ f : E โL[๐] ๐, โfโ = 1 โง f x = โxโ := exists_dual_vector ๐ x h
calc
โxโ = โ(โxโ : ๐)โ := IsROrC.norm_coe_norm.symm
_ = โf xโ := by rw [hfx]
_ โค M * โfโ := (hM f)
_ = M := by rw [hfโ, mul_one]
#align normed_space.norm_le_dual_bound NormedSpace.norm_le_dual_bound
theorem eq_zero_of_forall_dual_eq_zero {x : E} (h : โ f : Dual ๐ E, f x = (0 : ๐)) : x = 0 :=
norm_le_zero_iff.mp (norm_le_dual_bound ๐ x le_rfl fun f => by simp [h f])
#align normed_space.eq_zero_of_forall_dual_eq_zero NormedSpace.eq_zero_of_forall_dual_eq_zero
theorem eq_zero_iff_forall_dual_eq_zero (x : E) : x = 0 โ โ g : Dual ๐ E, g x = 0 :=
โจfun hx => by simp [hx], fun h => eq_zero_of_forall_dual_eq_zero ๐ hโฉ
#align normed_space.eq_zero_iff_forall_dual_eq_zero NormedSpace.eq_zero_iff_forall_dual_eq_zero
/-- See also `geometric_hahn_banach_point_point`. -/
theorem eq_iff_forall_dual_eq {x y : E} : x = y โ โ g : Dual ๐ E, g x = g y := by
rw [โ sub_eq_zero, eq_zero_iff_forall_dual_eq_zero ๐ (x - y)]
simp [sub_eq_zero]
#align normed_space.eq_iff_forall_dual_eq NormedSpace.eq_iff_forall_dual_eq
/-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E โโแตข[๐] Dual ๐ (Dual ๐ E) :=
{ inclusionInDoubleDual ๐ E with
norm_map' := by
intro x
apply le_antisymm
ยท exact double_dual_bound ๐ E x
|
rw [ContinuousLinearMap.norm_def]
|
/-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E โโแตข[๐] Dual ๐ (Dual ๐ E) :=
{ inclusionInDoubleDual ๐ E with
norm_map' := by
intro x
apply le_antisymm
ยท exact double_dual_bound ๐ E x
|
Mathlib.Analysis.NormedSpace.Dual.142_0.WirVfj6f5oiZZ2w
|
/-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E โโแตข[๐] Dual ๐ (Dual ๐ E)
|
Mathlib_Analysis_NormedSpace_Dual
|
case a
๐ : Type v
instโยฒ : IsROrC ๐
E : Type u
instโยน : NormedAddCommGroup E
instโ : NormedSpace ๐ E
srcโ : E โL[๐] Dual ๐ (Dual ๐ E) := inclusionInDoubleDual ๐ E
x : E
โข โxโ โค sInf {c | 0 โค c โง โ (x_1 : Dual ๐ E), โ(โsrcโ x) x_1โ โค c * โx_1โ}
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# The topological dual of a normed space
In this file we define the topological dual `NormedSpace.Dual` of a normed space, and the
continuous linear map `NormedSpace.inclusionInDoubleDual` from a normed space into its double
dual.
For base field `๐ = โ` or `๐ = โ`, this map is actually an isometric embedding; we provide a
version `NormedSpace.inclusionInDoubleDualLi` of the map which is of type a bundled linear
isometric embedding, `E โโแตข[๐] (Dual ๐ (Dual ๐ E))`.
Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
theory for `SeminormedAddCommGroup` and we specialize to `NormedAddCommGroup` when needed.
## Main definitions
* `inclusionInDoubleDual` and `inclusionInDoubleDualLi` are the inclusion of a normed space
in its double dual, considered as a bounded linear map and as a linear isometry, respectively.
* `polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals `x'` for which
`โx' zโ โค 1` for every `z โ s`.
## Tags
dual
-/
noncomputable section
open Classical Topology Bornology
universe u v
namespace NormedSpace
section General
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable (E : Type*) [SeminormedAddCommGroup E] [NormedSpace ๐ E]
variable (F : Type*) [NormedAddCommGroup F] [NormedSpace ๐ F]
/-- The topological dual of a seminormed space `E`. -/
abbrev Dual : Type _ := E โL[๐] ๐
#align normed_space.dual NormedSpace.Dual
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_eq until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : NormedSpace ๐ (Dual ๐ E) := inferInstance
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_le until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : SeminormedAddCommGroup (Dual ๐ E) := inferInstance
/-- The inclusion of a normed space in its double (topological) dual, considered
as a bounded linear map. -/
def inclusionInDoubleDual : E โL[๐] Dual ๐ (Dual ๐ E) :=
ContinuousLinearMap.apply ๐ ๐
#align normed_space.inclusion_in_double_dual NormedSpace.inclusionInDoubleDual
@[simp]
theorem dual_def (x : E) (f : Dual ๐ E) : inclusionInDoubleDual ๐ E x f = f x :=
rfl
#align normed_space.dual_def NormedSpace.dual_def
theorem inclusionInDoubleDual_norm_eq :
โinclusionInDoubleDual ๐ Eโ = โContinuousLinearMap.id ๐ (Dual ๐ E)โ :=
ContinuousLinearMap.op_norm_flip _
#align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1 := by
rw [inclusionInDoubleDual_norm_eq]
exact ContinuousLinearMap.norm_id_le
#align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
theorem double_dual_bound (x : E) : โ(inclusionInDoubleDual ๐ E) xโ โค โxโ := by
simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusionInDoubleDual_norm_le ๐ E) x
#align normed_space.double_dual_bound NormedSpace.double_dual_bound
/-- The dual pairing as a bilinear form. -/
def dualPairing : Dual ๐ E โโ[๐] E โโ[๐] ๐ :=
ContinuousLinearMap.coeLM ๐
#align normed_space.dual_pairing NormedSpace.dualPairing
@[simp]
theorem dualPairing_apply {v : Dual ๐ E} {x : E} : dualPairing ๐ E v x = v x :=
rfl
#align normed_space.dual_pairing_apply NormedSpace.dualPairing_apply
theorem dualPairing_separatingLeft : (dualPairing ๐ E).SeparatingLeft := by
rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]
exact ContinuousLinearMap.coe_injective
#align normed_space.dual_pairing_separating_left NormedSpace.dualPairing_separatingLeft
end General
section BidualIsometry
variable (๐ : Type v) [IsROrC ๐] {E : Type u} [NormedAddCommGroup E] [NormedSpace ๐ E]
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M := by
classical
by_cases h : x = 0
ยท simp only [h, hMp, norm_zero]
ยท obtain โจf, hfโ, hfxโฉ : โ f : E โL[๐] ๐, โfโ = 1 โง f x = โxโ := exists_dual_vector ๐ x h
calc
โxโ = โ(โxโ : ๐)โ := IsROrC.norm_coe_norm.symm
_ = โf xโ := by rw [hfx]
_ โค M * โfโ := (hM f)
_ = M := by rw [hfโ, mul_one]
#align normed_space.norm_le_dual_bound NormedSpace.norm_le_dual_bound
theorem eq_zero_of_forall_dual_eq_zero {x : E} (h : โ f : Dual ๐ E, f x = (0 : ๐)) : x = 0 :=
norm_le_zero_iff.mp (norm_le_dual_bound ๐ x le_rfl fun f => by simp [h f])
#align normed_space.eq_zero_of_forall_dual_eq_zero NormedSpace.eq_zero_of_forall_dual_eq_zero
theorem eq_zero_iff_forall_dual_eq_zero (x : E) : x = 0 โ โ g : Dual ๐ E, g x = 0 :=
โจfun hx => by simp [hx], fun h => eq_zero_of_forall_dual_eq_zero ๐ hโฉ
#align normed_space.eq_zero_iff_forall_dual_eq_zero NormedSpace.eq_zero_iff_forall_dual_eq_zero
/-- See also `geometric_hahn_banach_point_point`. -/
theorem eq_iff_forall_dual_eq {x y : E} : x = y โ โ g : Dual ๐ E, g x = g y := by
rw [โ sub_eq_zero, eq_zero_iff_forall_dual_eq_zero ๐ (x - y)]
simp [sub_eq_zero]
#align normed_space.eq_iff_forall_dual_eq NormedSpace.eq_iff_forall_dual_eq
/-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E โโแตข[๐] Dual ๐ (Dual ๐ E) :=
{ inclusionInDoubleDual ๐ E with
norm_map' := by
intro x
apply le_antisymm
ยท exact double_dual_bound ๐ E x
rw [ContinuousLinearMap.norm_def]
|
refine' le_csInf ContinuousLinearMap.bounds_nonempty _
|
/-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E โโแตข[๐] Dual ๐ (Dual ๐ E) :=
{ inclusionInDoubleDual ๐ E with
norm_map' := by
intro x
apply le_antisymm
ยท exact double_dual_bound ๐ E x
rw [ContinuousLinearMap.norm_def]
|
Mathlib.Analysis.NormedSpace.Dual.142_0.WirVfj6f5oiZZ2w
|
/-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E โโแตข[๐] Dual ๐ (Dual ๐ E)
|
Mathlib_Analysis_NormedSpace_Dual
|
case a
๐ : Type v
instโยฒ : IsROrC ๐
E : Type u
instโยน : NormedAddCommGroup E
instโ : NormedSpace ๐ E
srcโ : E โL[๐] Dual ๐ (Dual ๐ E) := inclusionInDoubleDual ๐ E
x : E
โข โ b โ {c | 0 โค c โง โ (x_1 : Dual ๐ E), โ(โsrcโ x) x_1โ โค c * โx_1โ}, โxโ โค b
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# The topological dual of a normed space
In this file we define the topological dual `NormedSpace.Dual` of a normed space, and the
continuous linear map `NormedSpace.inclusionInDoubleDual` from a normed space into its double
dual.
For base field `๐ = โ` or `๐ = โ`, this map is actually an isometric embedding; we provide a
version `NormedSpace.inclusionInDoubleDualLi` of the map which is of type a bundled linear
isometric embedding, `E โโแตข[๐] (Dual ๐ (Dual ๐ E))`.
Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
theory for `SeminormedAddCommGroup` and we specialize to `NormedAddCommGroup` when needed.
## Main definitions
* `inclusionInDoubleDual` and `inclusionInDoubleDualLi` are the inclusion of a normed space
in its double dual, considered as a bounded linear map and as a linear isometry, respectively.
* `polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals `x'` for which
`โx' zโ โค 1` for every `z โ s`.
## Tags
dual
-/
noncomputable section
open Classical Topology Bornology
universe u v
namespace NormedSpace
section General
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable (E : Type*) [SeminormedAddCommGroup E] [NormedSpace ๐ E]
variable (F : Type*) [NormedAddCommGroup F] [NormedSpace ๐ F]
/-- The topological dual of a seminormed space `E`. -/
abbrev Dual : Type _ := E โL[๐] ๐
#align normed_space.dual NormedSpace.Dual
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_eq until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : NormedSpace ๐ (Dual ๐ E) := inferInstance
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_le until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : SeminormedAddCommGroup (Dual ๐ E) := inferInstance
/-- The inclusion of a normed space in its double (topological) dual, considered
as a bounded linear map. -/
def inclusionInDoubleDual : E โL[๐] Dual ๐ (Dual ๐ E) :=
ContinuousLinearMap.apply ๐ ๐
#align normed_space.inclusion_in_double_dual NormedSpace.inclusionInDoubleDual
@[simp]
theorem dual_def (x : E) (f : Dual ๐ E) : inclusionInDoubleDual ๐ E x f = f x :=
rfl
#align normed_space.dual_def NormedSpace.dual_def
theorem inclusionInDoubleDual_norm_eq :
โinclusionInDoubleDual ๐ Eโ = โContinuousLinearMap.id ๐ (Dual ๐ E)โ :=
ContinuousLinearMap.op_norm_flip _
#align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1 := by
rw [inclusionInDoubleDual_norm_eq]
exact ContinuousLinearMap.norm_id_le
#align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
theorem double_dual_bound (x : E) : โ(inclusionInDoubleDual ๐ E) xโ โค โxโ := by
simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusionInDoubleDual_norm_le ๐ E) x
#align normed_space.double_dual_bound NormedSpace.double_dual_bound
/-- The dual pairing as a bilinear form. -/
def dualPairing : Dual ๐ E โโ[๐] E โโ[๐] ๐ :=
ContinuousLinearMap.coeLM ๐
#align normed_space.dual_pairing NormedSpace.dualPairing
@[simp]
theorem dualPairing_apply {v : Dual ๐ E} {x : E} : dualPairing ๐ E v x = v x :=
rfl
#align normed_space.dual_pairing_apply NormedSpace.dualPairing_apply
theorem dualPairing_separatingLeft : (dualPairing ๐ E).SeparatingLeft := by
rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]
exact ContinuousLinearMap.coe_injective
#align normed_space.dual_pairing_separating_left NormedSpace.dualPairing_separatingLeft
end General
section BidualIsometry
variable (๐ : Type v) [IsROrC ๐] {E : Type u} [NormedAddCommGroup E] [NormedSpace ๐ E]
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M := by
classical
by_cases h : x = 0
ยท simp only [h, hMp, norm_zero]
ยท obtain โจf, hfโ, hfxโฉ : โ f : E โL[๐] ๐, โfโ = 1 โง f x = โxโ := exists_dual_vector ๐ x h
calc
โxโ = โ(โxโ : ๐)โ := IsROrC.norm_coe_norm.symm
_ = โf xโ := by rw [hfx]
_ โค M * โfโ := (hM f)
_ = M := by rw [hfโ, mul_one]
#align normed_space.norm_le_dual_bound NormedSpace.norm_le_dual_bound
theorem eq_zero_of_forall_dual_eq_zero {x : E} (h : โ f : Dual ๐ E, f x = (0 : ๐)) : x = 0 :=
norm_le_zero_iff.mp (norm_le_dual_bound ๐ x le_rfl fun f => by simp [h f])
#align normed_space.eq_zero_of_forall_dual_eq_zero NormedSpace.eq_zero_of_forall_dual_eq_zero
theorem eq_zero_iff_forall_dual_eq_zero (x : E) : x = 0 โ โ g : Dual ๐ E, g x = 0 :=
โจfun hx => by simp [hx], fun h => eq_zero_of_forall_dual_eq_zero ๐ hโฉ
#align normed_space.eq_zero_iff_forall_dual_eq_zero NormedSpace.eq_zero_iff_forall_dual_eq_zero
/-- See also `geometric_hahn_banach_point_point`. -/
theorem eq_iff_forall_dual_eq {x y : E} : x = y โ โ g : Dual ๐ E, g x = g y := by
rw [โ sub_eq_zero, eq_zero_iff_forall_dual_eq_zero ๐ (x - y)]
simp [sub_eq_zero]
#align normed_space.eq_iff_forall_dual_eq NormedSpace.eq_iff_forall_dual_eq
/-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E โโแตข[๐] Dual ๐ (Dual ๐ E) :=
{ inclusionInDoubleDual ๐ E with
norm_map' := by
intro x
apply le_antisymm
ยท exact double_dual_bound ๐ E x
rw [ContinuousLinearMap.norm_def]
refine' le_csInf ContinuousLinearMap.bounds_nonempty _
|
rintro c โจhc1, hc2โฉ
|
/-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E โโแตข[๐] Dual ๐ (Dual ๐ E) :=
{ inclusionInDoubleDual ๐ E with
norm_map' := by
intro x
apply le_antisymm
ยท exact double_dual_bound ๐ E x
rw [ContinuousLinearMap.norm_def]
refine' le_csInf ContinuousLinearMap.bounds_nonempty _
|
Mathlib.Analysis.NormedSpace.Dual.142_0.WirVfj6f5oiZZ2w
|
/-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E โโแตข[๐] Dual ๐ (Dual ๐ E)
|
Mathlib_Analysis_NormedSpace_Dual
|
case a.intro
๐ : Type v
instโยฒ : IsROrC ๐
E : Type u
instโยน : NormedAddCommGroup E
instโ : NormedSpace ๐ E
srcโ : E โL[๐] Dual ๐ (Dual ๐ E) := inclusionInDoubleDual ๐ E
x : E
c : โ
hc1 : 0 โค c
hc2 : โ (x_1 : Dual ๐ E), โ(โsrcโ x) x_1โ โค c * โx_1โ
โข โxโ โค c
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# The topological dual of a normed space
In this file we define the topological dual `NormedSpace.Dual` of a normed space, and the
continuous linear map `NormedSpace.inclusionInDoubleDual` from a normed space into its double
dual.
For base field `๐ = โ` or `๐ = โ`, this map is actually an isometric embedding; we provide a
version `NormedSpace.inclusionInDoubleDualLi` of the map which is of type a bundled linear
isometric embedding, `E โโแตข[๐] (Dual ๐ (Dual ๐ E))`.
Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
theory for `SeminormedAddCommGroup` and we specialize to `NormedAddCommGroup` when needed.
## Main definitions
* `inclusionInDoubleDual` and `inclusionInDoubleDualLi` are the inclusion of a normed space
in its double dual, considered as a bounded linear map and as a linear isometry, respectively.
* `polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals `x'` for which
`โx' zโ โค 1` for every `z โ s`.
## Tags
dual
-/
noncomputable section
open Classical Topology Bornology
universe u v
namespace NormedSpace
section General
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable (E : Type*) [SeminormedAddCommGroup E] [NormedSpace ๐ E]
variable (F : Type*) [NormedAddCommGroup F] [NormedSpace ๐ F]
/-- The topological dual of a seminormed space `E`. -/
abbrev Dual : Type _ := E โL[๐] ๐
#align normed_space.dual NormedSpace.Dual
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_eq until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : NormedSpace ๐ (Dual ๐ E) := inferInstance
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_le until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : SeminormedAddCommGroup (Dual ๐ E) := inferInstance
/-- The inclusion of a normed space in its double (topological) dual, considered
as a bounded linear map. -/
def inclusionInDoubleDual : E โL[๐] Dual ๐ (Dual ๐ E) :=
ContinuousLinearMap.apply ๐ ๐
#align normed_space.inclusion_in_double_dual NormedSpace.inclusionInDoubleDual
@[simp]
theorem dual_def (x : E) (f : Dual ๐ E) : inclusionInDoubleDual ๐ E x f = f x :=
rfl
#align normed_space.dual_def NormedSpace.dual_def
theorem inclusionInDoubleDual_norm_eq :
โinclusionInDoubleDual ๐ Eโ = โContinuousLinearMap.id ๐ (Dual ๐ E)โ :=
ContinuousLinearMap.op_norm_flip _
#align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1 := by
rw [inclusionInDoubleDual_norm_eq]
exact ContinuousLinearMap.norm_id_le
#align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
theorem double_dual_bound (x : E) : โ(inclusionInDoubleDual ๐ E) xโ โค โxโ := by
simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusionInDoubleDual_norm_le ๐ E) x
#align normed_space.double_dual_bound NormedSpace.double_dual_bound
/-- The dual pairing as a bilinear form. -/
def dualPairing : Dual ๐ E โโ[๐] E โโ[๐] ๐ :=
ContinuousLinearMap.coeLM ๐
#align normed_space.dual_pairing NormedSpace.dualPairing
@[simp]
theorem dualPairing_apply {v : Dual ๐ E} {x : E} : dualPairing ๐ E v x = v x :=
rfl
#align normed_space.dual_pairing_apply NormedSpace.dualPairing_apply
theorem dualPairing_separatingLeft : (dualPairing ๐ E).SeparatingLeft := by
rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]
exact ContinuousLinearMap.coe_injective
#align normed_space.dual_pairing_separating_left NormedSpace.dualPairing_separatingLeft
end General
section BidualIsometry
variable (๐ : Type v) [IsROrC ๐] {E : Type u} [NormedAddCommGroup E] [NormedSpace ๐ E]
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M := by
classical
by_cases h : x = 0
ยท simp only [h, hMp, norm_zero]
ยท obtain โจf, hfโ, hfxโฉ : โ f : E โL[๐] ๐, โfโ = 1 โง f x = โxโ := exists_dual_vector ๐ x h
calc
โxโ = โ(โxโ : ๐)โ := IsROrC.norm_coe_norm.symm
_ = โf xโ := by rw [hfx]
_ โค M * โfโ := (hM f)
_ = M := by rw [hfโ, mul_one]
#align normed_space.norm_le_dual_bound NormedSpace.norm_le_dual_bound
theorem eq_zero_of_forall_dual_eq_zero {x : E} (h : โ f : Dual ๐ E, f x = (0 : ๐)) : x = 0 :=
norm_le_zero_iff.mp (norm_le_dual_bound ๐ x le_rfl fun f => by simp [h f])
#align normed_space.eq_zero_of_forall_dual_eq_zero NormedSpace.eq_zero_of_forall_dual_eq_zero
theorem eq_zero_iff_forall_dual_eq_zero (x : E) : x = 0 โ โ g : Dual ๐ E, g x = 0 :=
โจfun hx => by simp [hx], fun h => eq_zero_of_forall_dual_eq_zero ๐ hโฉ
#align normed_space.eq_zero_iff_forall_dual_eq_zero NormedSpace.eq_zero_iff_forall_dual_eq_zero
/-- See also `geometric_hahn_banach_point_point`. -/
theorem eq_iff_forall_dual_eq {x y : E} : x = y โ โ g : Dual ๐ E, g x = g y := by
rw [โ sub_eq_zero, eq_zero_iff_forall_dual_eq_zero ๐ (x - y)]
simp [sub_eq_zero]
#align normed_space.eq_iff_forall_dual_eq NormedSpace.eq_iff_forall_dual_eq
/-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E โโแตข[๐] Dual ๐ (Dual ๐ E) :=
{ inclusionInDoubleDual ๐ E with
norm_map' := by
intro x
apply le_antisymm
ยท exact double_dual_bound ๐ E x
rw [ContinuousLinearMap.norm_def]
refine' le_csInf ContinuousLinearMap.bounds_nonempty _
rintro c โจhc1, hc2โฉ
|
exact norm_le_dual_bound ๐ x hc1 hc2
|
/-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E โโแตข[๐] Dual ๐ (Dual ๐ E) :=
{ inclusionInDoubleDual ๐ E with
norm_map' := by
intro x
apply le_antisymm
ยท exact double_dual_bound ๐ E x
rw [ContinuousLinearMap.norm_def]
refine' le_csInf ContinuousLinearMap.bounds_nonempty _
rintro c โจhc1, hc2โฉ
|
Mathlib.Analysis.NormedSpace.Dual.142_0.WirVfj6f5oiZZ2w
|
/-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E โโแตข[๐] Dual ๐ (Dual ๐ E)
|
Mathlib_Analysis_NormedSpace_Dual
|
๐ : Type u_1
instโยฒ : NontriviallyNormedField ๐
E : Type u_2
instโยน : SeminormedAddCommGroup E
instโ : NormedSpace ๐ E
s : Set E
โข IsClosed (polar ๐ s)
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# The topological dual of a normed space
In this file we define the topological dual `NormedSpace.Dual` of a normed space, and the
continuous linear map `NormedSpace.inclusionInDoubleDual` from a normed space into its double
dual.
For base field `๐ = โ` or `๐ = โ`, this map is actually an isometric embedding; we provide a
version `NormedSpace.inclusionInDoubleDualLi` of the map which is of type a bundled linear
isometric embedding, `E โโแตข[๐] (Dual ๐ (Dual ๐ E))`.
Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
theory for `SeminormedAddCommGroup` and we specialize to `NormedAddCommGroup` when needed.
## Main definitions
* `inclusionInDoubleDual` and `inclusionInDoubleDualLi` are the inclusion of a normed space
in its double dual, considered as a bounded linear map and as a linear isometry, respectively.
* `polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals `x'` for which
`โx' zโ โค 1` for every `z โ s`.
## Tags
dual
-/
noncomputable section
open Classical Topology Bornology
universe u v
namespace NormedSpace
section General
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable (E : Type*) [SeminormedAddCommGroup E] [NormedSpace ๐ E]
variable (F : Type*) [NormedAddCommGroup F] [NormedSpace ๐ F]
/-- The topological dual of a seminormed space `E`. -/
abbrev Dual : Type _ := E โL[๐] ๐
#align normed_space.dual NormedSpace.Dual
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_eq until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : NormedSpace ๐ (Dual ๐ E) := inferInstance
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_le until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : SeminormedAddCommGroup (Dual ๐ E) := inferInstance
/-- The inclusion of a normed space in its double (topological) dual, considered
as a bounded linear map. -/
def inclusionInDoubleDual : E โL[๐] Dual ๐ (Dual ๐ E) :=
ContinuousLinearMap.apply ๐ ๐
#align normed_space.inclusion_in_double_dual NormedSpace.inclusionInDoubleDual
@[simp]
theorem dual_def (x : E) (f : Dual ๐ E) : inclusionInDoubleDual ๐ E x f = f x :=
rfl
#align normed_space.dual_def NormedSpace.dual_def
theorem inclusionInDoubleDual_norm_eq :
โinclusionInDoubleDual ๐ Eโ = โContinuousLinearMap.id ๐ (Dual ๐ E)โ :=
ContinuousLinearMap.op_norm_flip _
#align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1 := by
rw [inclusionInDoubleDual_norm_eq]
exact ContinuousLinearMap.norm_id_le
#align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
theorem double_dual_bound (x : E) : โ(inclusionInDoubleDual ๐ E) xโ โค โxโ := by
simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusionInDoubleDual_norm_le ๐ E) x
#align normed_space.double_dual_bound NormedSpace.double_dual_bound
/-- The dual pairing as a bilinear form. -/
def dualPairing : Dual ๐ E โโ[๐] E โโ[๐] ๐ :=
ContinuousLinearMap.coeLM ๐
#align normed_space.dual_pairing NormedSpace.dualPairing
@[simp]
theorem dualPairing_apply {v : Dual ๐ E} {x : E} : dualPairing ๐ E v x = v x :=
rfl
#align normed_space.dual_pairing_apply NormedSpace.dualPairing_apply
theorem dualPairing_separatingLeft : (dualPairing ๐ E).SeparatingLeft := by
rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]
exact ContinuousLinearMap.coe_injective
#align normed_space.dual_pairing_separating_left NormedSpace.dualPairing_separatingLeft
end General
section BidualIsometry
variable (๐ : Type v) [IsROrC ๐] {E : Type u} [NormedAddCommGroup E] [NormedSpace ๐ E]
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M := by
classical
by_cases h : x = 0
ยท simp only [h, hMp, norm_zero]
ยท obtain โจf, hfโ, hfxโฉ : โ f : E โL[๐] ๐, โfโ = 1 โง f x = โxโ := exists_dual_vector ๐ x h
calc
โxโ = โ(โxโ : ๐)โ := IsROrC.norm_coe_norm.symm
_ = โf xโ := by rw [hfx]
_ โค M * โfโ := (hM f)
_ = M := by rw [hfโ, mul_one]
#align normed_space.norm_le_dual_bound NormedSpace.norm_le_dual_bound
theorem eq_zero_of_forall_dual_eq_zero {x : E} (h : โ f : Dual ๐ E, f x = (0 : ๐)) : x = 0 :=
norm_le_zero_iff.mp (norm_le_dual_bound ๐ x le_rfl fun f => by simp [h f])
#align normed_space.eq_zero_of_forall_dual_eq_zero NormedSpace.eq_zero_of_forall_dual_eq_zero
theorem eq_zero_iff_forall_dual_eq_zero (x : E) : x = 0 โ โ g : Dual ๐ E, g x = 0 :=
โจfun hx => by simp [hx], fun h => eq_zero_of_forall_dual_eq_zero ๐ hโฉ
#align normed_space.eq_zero_iff_forall_dual_eq_zero NormedSpace.eq_zero_iff_forall_dual_eq_zero
/-- See also `geometric_hahn_banach_point_point`. -/
theorem eq_iff_forall_dual_eq {x y : E} : x = y โ โ g : Dual ๐ E, g x = g y := by
rw [โ sub_eq_zero, eq_zero_iff_forall_dual_eq_zero ๐ (x - y)]
simp [sub_eq_zero]
#align normed_space.eq_iff_forall_dual_eq NormedSpace.eq_iff_forall_dual_eq
/-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E โโแตข[๐] Dual ๐ (Dual ๐ E) :=
{ inclusionInDoubleDual ๐ E with
norm_map' := by
intro x
apply le_antisymm
ยท exact double_dual_bound ๐ E x
rw [ContinuousLinearMap.norm_def]
refine' le_csInf ContinuousLinearMap.bounds_nonempty _
rintro c โจhc1, hc2โฉ
exact norm_le_dual_bound ๐ x hc1 hc2 }
#align normed_space.inclusion_in_double_dual_li NormedSpace.inclusionInDoubleDualLi
end BidualIsometry
section PolarSets
open Metric Set NormedSpace
/-- Given a subset `s` in a normed space `E` (over a field `๐`), the polar
`polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals which
evaluate to something of norm at most one at all points `z โ s`. -/
def polar (๐ : Type*) [NontriviallyNormedField ๐] {E : Type*} [SeminormedAddCommGroup E]
[NormedSpace ๐ E] : Set E โ Set (Dual ๐ E) :=
(dualPairing ๐ E).flip.polar
#align normed_space.polar NormedSpace.polar
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable {E : Type*} [SeminormedAddCommGroup E] [NormedSpace ๐ E]
theorem mem_polar_iff {x' : Dual ๐ E} (s : Set E) : x' โ polar ๐ s โ โ z โ s, โx' zโ โค 1 :=
Iff.rfl
#align normed_space.mem_polar_iff NormedSpace.mem_polar_iff
@[simp]
theorem polar_univ : polar ๐ (univ : Set E) = {(0 : Dual ๐ E)} :=
(dualPairing ๐ E).flip.polar_univ
(LinearMap.flip_separatingRight.mpr (dualPairing_separatingLeft ๐ E))
#align normed_space.polar_univ NormedSpace.polar_univ
theorem isClosed_polar (s : Set E) : IsClosed (polar ๐ s) := by
|
dsimp only [NormedSpace.polar]
|
theorem isClosed_polar (s : Set E) : IsClosed (polar ๐ s) := by
|
Mathlib.Analysis.NormedSpace.Dual.183_0.WirVfj6f5oiZZ2w
|
theorem isClosed_polar (s : Set E) : IsClosed (polar ๐ s)
|
Mathlib_Analysis_NormedSpace_Dual
|
๐ : Type u_1
instโยฒ : NontriviallyNormedField ๐
E : Type u_2
instโยน : SeminormedAddCommGroup E
instโ : NormedSpace ๐ E
s : Set E
โข IsClosed (LinearMap.polar (LinearMap.flip (dualPairing ๐ E)) s)
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# The topological dual of a normed space
In this file we define the topological dual `NormedSpace.Dual` of a normed space, and the
continuous linear map `NormedSpace.inclusionInDoubleDual` from a normed space into its double
dual.
For base field `๐ = โ` or `๐ = โ`, this map is actually an isometric embedding; we provide a
version `NormedSpace.inclusionInDoubleDualLi` of the map which is of type a bundled linear
isometric embedding, `E โโแตข[๐] (Dual ๐ (Dual ๐ E))`.
Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
theory for `SeminormedAddCommGroup` and we specialize to `NormedAddCommGroup` when needed.
## Main definitions
* `inclusionInDoubleDual` and `inclusionInDoubleDualLi` are the inclusion of a normed space
in its double dual, considered as a bounded linear map and as a linear isometry, respectively.
* `polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals `x'` for which
`โx' zโ โค 1` for every `z โ s`.
## Tags
dual
-/
noncomputable section
open Classical Topology Bornology
universe u v
namespace NormedSpace
section General
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable (E : Type*) [SeminormedAddCommGroup E] [NormedSpace ๐ E]
variable (F : Type*) [NormedAddCommGroup F] [NormedSpace ๐ F]
/-- The topological dual of a seminormed space `E`. -/
abbrev Dual : Type _ := E โL[๐] ๐
#align normed_space.dual NormedSpace.Dual
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_eq until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : NormedSpace ๐ (Dual ๐ E) := inferInstance
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_le until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : SeminormedAddCommGroup (Dual ๐ E) := inferInstance
/-- The inclusion of a normed space in its double (topological) dual, considered
as a bounded linear map. -/
def inclusionInDoubleDual : E โL[๐] Dual ๐ (Dual ๐ E) :=
ContinuousLinearMap.apply ๐ ๐
#align normed_space.inclusion_in_double_dual NormedSpace.inclusionInDoubleDual
@[simp]
theorem dual_def (x : E) (f : Dual ๐ E) : inclusionInDoubleDual ๐ E x f = f x :=
rfl
#align normed_space.dual_def NormedSpace.dual_def
theorem inclusionInDoubleDual_norm_eq :
โinclusionInDoubleDual ๐ Eโ = โContinuousLinearMap.id ๐ (Dual ๐ E)โ :=
ContinuousLinearMap.op_norm_flip _
#align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1 := by
rw [inclusionInDoubleDual_norm_eq]
exact ContinuousLinearMap.norm_id_le
#align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
theorem double_dual_bound (x : E) : โ(inclusionInDoubleDual ๐ E) xโ โค โxโ := by
simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusionInDoubleDual_norm_le ๐ E) x
#align normed_space.double_dual_bound NormedSpace.double_dual_bound
/-- The dual pairing as a bilinear form. -/
def dualPairing : Dual ๐ E โโ[๐] E โโ[๐] ๐ :=
ContinuousLinearMap.coeLM ๐
#align normed_space.dual_pairing NormedSpace.dualPairing
@[simp]
theorem dualPairing_apply {v : Dual ๐ E} {x : E} : dualPairing ๐ E v x = v x :=
rfl
#align normed_space.dual_pairing_apply NormedSpace.dualPairing_apply
theorem dualPairing_separatingLeft : (dualPairing ๐ E).SeparatingLeft := by
rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]
exact ContinuousLinearMap.coe_injective
#align normed_space.dual_pairing_separating_left NormedSpace.dualPairing_separatingLeft
end General
section BidualIsometry
variable (๐ : Type v) [IsROrC ๐] {E : Type u} [NormedAddCommGroup E] [NormedSpace ๐ E]
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M := by
classical
by_cases h : x = 0
ยท simp only [h, hMp, norm_zero]
ยท obtain โจf, hfโ, hfxโฉ : โ f : E โL[๐] ๐, โfโ = 1 โง f x = โxโ := exists_dual_vector ๐ x h
calc
โxโ = โ(โxโ : ๐)โ := IsROrC.norm_coe_norm.symm
_ = โf xโ := by rw [hfx]
_ โค M * โfโ := (hM f)
_ = M := by rw [hfโ, mul_one]
#align normed_space.norm_le_dual_bound NormedSpace.norm_le_dual_bound
theorem eq_zero_of_forall_dual_eq_zero {x : E} (h : โ f : Dual ๐ E, f x = (0 : ๐)) : x = 0 :=
norm_le_zero_iff.mp (norm_le_dual_bound ๐ x le_rfl fun f => by simp [h f])
#align normed_space.eq_zero_of_forall_dual_eq_zero NormedSpace.eq_zero_of_forall_dual_eq_zero
theorem eq_zero_iff_forall_dual_eq_zero (x : E) : x = 0 โ โ g : Dual ๐ E, g x = 0 :=
โจfun hx => by simp [hx], fun h => eq_zero_of_forall_dual_eq_zero ๐ hโฉ
#align normed_space.eq_zero_iff_forall_dual_eq_zero NormedSpace.eq_zero_iff_forall_dual_eq_zero
/-- See also `geometric_hahn_banach_point_point`. -/
theorem eq_iff_forall_dual_eq {x y : E} : x = y โ โ g : Dual ๐ E, g x = g y := by
rw [โ sub_eq_zero, eq_zero_iff_forall_dual_eq_zero ๐ (x - y)]
simp [sub_eq_zero]
#align normed_space.eq_iff_forall_dual_eq NormedSpace.eq_iff_forall_dual_eq
/-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E โโแตข[๐] Dual ๐ (Dual ๐ E) :=
{ inclusionInDoubleDual ๐ E with
norm_map' := by
intro x
apply le_antisymm
ยท exact double_dual_bound ๐ E x
rw [ContinuousLinearMap.norm_def]
refine' le_csInf ContinuousLinearMap.bounds_nonempty _
rintro c โจhc1, hc2โฉ
exact norm_le_dual_bound ๐ x hc1 hc2 }
#align normed_space.inclusion_in_double_dual_li NormedSpace.inclusionInDoubleDualLi
end BidualIsometry
section PolarSets
open Metric Set NormedSpace
/-- Given a subset `s` in a normed space `E` (over a field `๐`), the polar
`polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals which
evaluate to something of norm at most one at all points `z โ s`. -/
def polar (๐ : Type*) [NontriviallyNormedField ๐] {E : Type*} [SeminormedAddCommGroup E]
[NormedSpace ๐ E] : Set E โ Set (Dual ๐ E) :=
(dualPairing ๐ E).flip.polar
#align normed_space.polar NormedSpace.polar
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable {E : Type*} [SeminormedAddCommGroup E] [NormedSpace ๐ E]
theorem mem_polar_iff {x' : Dual ๐ E} (s : Set E) : x' โ polar ๐ s โ โ z โ s, โx' zโ โค 1 :=
Iff.rfl
#align normed_space.mem_polar_iff NormedSpace.mem_polar_iff
@[simp]
theorem polar_univ : polar ๐ (univ : Set E) = {(0 : Dual ๐ E)} :=
(dualPairing ๐ E).flip.polar_univ
(LinearMap.flip_separatingRight.mpr (dualPairing_separatingLeft ๐ E))
#align normed_space.polar_univ NormedSpace.polar_univ
theorem isClosed_polar (s : Set E) : IsClosed (polar ๐ s) := by
dsimp only [NormedSpace.polar]
|
simp only [LinearMap.polar_eq_iInter, LinearMap.flip_apply]
|
theorem isClosed_polar (s : Set E) : IsClosed (polar ๐ s) := by
dsimp only [NormedSpace.polar]
|
Mathlib.Analysis.NormedSpace.Dual.183_0.WirVfj6f5oiZZ2w
|
theorem isClosed_polar (s : Set E) : IsClosed (polar ๐ s)
|
Mathlib_Analysis_NormedSpace_Dual
|
๐ : Type u_1
instโยฒ : NontriviallyNormedField ๐
E : Type u_2
instโยน : SeminormedAddCommGroup E
instโ : NormedSpace ๐ E
s : Set E
โข IsClosed (โ x โ s, {y | โ((dualPairing ๐ E) y) xโ โค 1})
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# The topological dual of a normed space
In this file we define the topological dual `NormedSpace.Dual` of a normed space, and the
continuous linear map `NormedSpace.inclusionInDoubleDual` from a normed space into its double
dual.
For base field `๐ = โ` or `๐ = โ`, this map is actually an isometric embedding; we provide a
version `NormedSpace.inclusionInDoubleDualLi` of the map which is of type a bundled linear
isometric embedding, `E โโแตข[๐] (Dual ๐ (Dual ๐ E))`.
Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
theory for `SeminormedAddCommGroup` and we specialize to `NormedAddCommGroup` when needed.
## Main definitions
* `inclusionInDoubleDual` and `inclusionInDoubleDualLi` are the inclusion of a normed space
in its double dual, considered as a bounded linear map and as a linear isometry, respectively.
* `polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals `x'` for which
`โx' zโ โค 1` for every `z โ s`.
## Tags
dual
-/
noncomputable section
open Classical Topology Bornology
universe u v
namespace NormedSpace
section General
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable (E : Type*) [SeminormedAddCommGroup E] [NormedSpace ๐ E]
variable (F : Type*) [NormedAddCommGroup F] [NormedSpace ๐ F]
/-- The topological dual of a seminormed space `E`. -/
abbrev Dual : Type _ := E โL[๐] ๐
#align normed_space.dual NormedSpace.Dual
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_eq until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : NormedSpace ๐ (Dual ๐ E) := inferInstance
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_le until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : SeminormedAddCommGroup (Dual ๐ E) := inferInstance
/-- The inclusion of a normed space in its double (topological) dual, considered
as a bounded linear map. -/
def inclusionInDoubleDual : E โL[๐] Dual ๐ (Dual ๐ E) :=
ContinuousLinearMap.apply ๐ ๐
#align normed_space.inclusion_in_double_dual NormedSpace.inclusionInDoubleDual
@[simp]
theorem dual_def (x : E) (f : Dual ๐ E) : inclusionInDoubleDual ๐ E x f = f x :=
rfl
#align normed_space.dual_def NormedSpace.dual_def
theorem inclusionInDoubleDual_norm_eq :
โinclusionInDoubleDual ๐ Eโ = โContinuousLinearMap.id ๐ (Dual ๐ E)โ :=
ContinuousLinearMap.op_norm_flip _
#align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1 := by
rw [inclusionInDoubleDual_norm_eq]
exact ContinuousLinearMap.norm_id_le
#align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
theorem double_dual_bound (x : E) : โ(inclusionInDoubleDual ๐ E) xโ โค โxโ := by
simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusionInDoubleDual_norm_le ๐ E) x
#align normed_space.double_dual_bound NormedSpace.double_dual_bound
/-- The dual pairing as a bilinear form. -/
def dualPairing : Dual ๐ E โโ[๐] E โโ[๐] ๐ :=
ContinuousLinearMap.coeLM ๐
#align normed_space.dual_pairing NormedSpace.dualPairing
@[simp]
theorem dualPairing_apply {v : Dual ๐ E} {x : E} : dualPairing ๐ E v x = v x :=
rfl
#align normed_space.dual_pairing_apply NormedSpace.dualPairing_apply
theorem dualPairing_separatingLeft : (dualPairing ๐ E).SeparatingLeft := by
rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]
exact ContinuousLinearMap.coe_injective
#align normed_space.dual_pairing_separating_left NormedSpace.dualPairing_separatingLeft
end General
section BidualIsometry
variable (๐ : Type v) [IsROrC ๐] {E : Type u} [NormedAddCommGroup E] [NormedSpace ๐ E]
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M := by
classical
by_cases h : x = 0
ยท simp only [h, hMp, norm_zero]
ยท obtain โจf, hfโ, hfxโฉ : โ f : E โL[๐] ๐, โfโ = 1 โง f x = โxโ := exists_dual_vector ๐ x h
calc
โxโ = โ(โxโ : ๐)โ := IsROrC.norm_coe_norm.symm
_ = โf xโ := by rw [hfx]
_ โค M * โfโ := (hM f)
_ = M := by rw [hfโ, mul_one]
#align normed_space.norm_le_dual_bound NormedSpace.norm_le_dual_bound
theorem eq_zero_of_forall_dual_eq_zero {x : E} (h : โ f : Dual ๐ E, f x = (0 : ๐)) : x = 0 :=
norm_le_zero_iff.mp (norm_le_dual_bound ๐ x le_rfl fun f => by simp [h f])
#align normed_space.eq_zero_of_forall_dual_eq_zero NormedSpace.eq_zero_of_forall_dual_eq_zero
theorem eq_zero_iff_forall_dual_eq_zero (x : E) : x = 0 โ โ g : Dual ๐ E, g x = 0 :=
โจfun hx => by simp [hx], fun h => eq_zero_of_forall_dual_eq_zero ๐ hโฉ
#align normed_space.eq_zero_iff_forall_dual_eq_zero NormedSpace.eq_zero_iff_forall_dual_eq_zero
/-- See also `geometric_hahn_banach_point_point`. -/
theorem eq_iff_forall_dual_eq {x y : E} : x = y โ โ g : Dual ๐ E, g x = g y := by
rw [โ sub_eq_zero, eq_zero_iff_forall_dual_eq_zero ๐ (x - y)]
simp [sub_eq_zero]
#align normed_space.eq_iff_forall_dual_eq NormedSpace.eq_iff_forall_dual_eq
/-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E โโแตข[๐] Dual ๐ (Dual ๐ E) :=
{ inclusionInDoubleDual ๐ E with
norm_map' := by
intro x
apply le_antisymm
ยท exact double_dual_bound ๐ E x
rw [ContinuousLinearMap.norm_def]
refine' le_csInf ContinuousLinearMap.bounds_nonempty _
rintro c โจhc1, hc2โฉ
exact norm_le_dual_bound ๐ x hc1 hc2 }
#align normed_space.inclusion_in_double_dual_li NormedSpace.inclusionInDoubleDualLi
end BidualIsometry
section PolarSets
open Metric Set NormedSpace
/-- Given a subset `s` in a normed space `E` (over a field `๐`), the polar
`polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals which
evaluate to something of norm at most one at all points `z โ s`. -/
def polar (๐ : Type*) [NontriviallyNormedField ๐] {E : Type*} [SeminormedAddCommGroup E]
[NormedSpace ๐ E] : Set E โ Set (Dual ๐ E) :=
(dualPairing ๐ E).flip.polar
#align normed_space.polar NormedSpace.polar
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable {E : Type*} [SeminormedAddCommGroup E] [NormedSpace ๐ E]
theorem mem_polar_iff {x' : Dual ๐ E} (s : Set E) : x' โ polar ๐ s โ โ z โ s, โx' zโ โค 1 :=
Iff.rfl
#align normed_space.mem_polar_iff NormedSpace.mem_polar_iff
@[simp]
theorem polar_univ : polar ๐ (univ : Set E) = {(0 : Dual ๐ E)} :=
(dualPairing ๐ E).flip.polar_univ
(LinearMap.flip_separatingRight.mpr (dualPairing_separatingLeft ๐ E))
#align normed_space.polar_univ NormedSpace.polar_univ
theorem isClosed_polar (s : Set E) : IsClosed (polar ๐ s) := by
dsimp only [NormedSpace.polar]
simp only [LinearMap.polar_eq_iInter, LinearMap.flip_apply]
|
refine' isClosed_biInter fun z _ => _
|
theorem isClosed_polar (s : Set E) : IsClosed (polar ๐ s) := by
dsimp only [NormedSpace.polar]
simp only [LinearMap.polar_eq_iInter, LinearMap.flip_apply]
|
Mathlib.Analysis.NormedSpace.Dual.183_0.WirVfj6f5oiZZ2w
|
theorem isClosed_polar (s : Set E) : IsClosed (polar ๐ s)
|
Mathlib_Analysis_NormedSpace_Dual
|
๐ : Type u_1
instโยฒ : NontriviallyNormedField ๐
E : Type u_2
instโยน : SeminormedAddCommGroup E
instโ : NormedSpace ๐ E
s : Set E
z : E
xโ : z โ s
โข IsClosed {y | โ((dualPairing ๐ E) y) zโ โค 1}
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# The topological dual of a normed space
In this file we define the topological dual `NormedSpace.Dual` of a normed space, and the
continuous linear map `NormedSpace.inclusionInDoubleDual` from a normed space into its double
dual.
For base field `๐ = โ` or `๐ = โ`, this map is actually an isometric embedding; we provide a
version `NormedSpace.inclusionInDoubleDualLi` of the map which is of type a bundled linear
isometric embedding, `E โโแตข[๐] (Dual ๐ (Dual ๐ E))`.
Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
theory for `SeminormedAddCommGroup` and we specialize to `NormedAddCommGroup` when needed.
## Main definitions
* `inclusionInDoubleDual` and `inclusionInDoubleDualLi` are the inclusion of a normed space
in its double dual, considered as a bounded linear map and as a linear isometry, respectively.
* `polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals `x'` for which
`โx' zโ โค 1` for every `z โ s`.
## Tags
dual
-/
noncomputable section
open Classical Topology Bornology
universe u v
namespace NormedSpace
section General
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable (E : Type*) [SeminormedAddCommGroup E] [NormedSpace ๐ E]
variable (F : Type*) [NormedAddCommGroup F] [NormedSpace ๐ F]
/-- The topological dual of a seminormed space `E`. -/
abbrev Dual : Type _ := E โL[๐] ๐
#align normed_space.dual NormedSpace.Dual
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_eq until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : NormedSpace ๐ (Dual ๐ E) := inferInstance
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_le until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : SeminormedAddCommGroup (Dual ๐ E) := inferInstance
/-- The inclusion of a normed space in its double (topological) dual, considered
as a bounded linear map. -/
def inclusionInDoubleDual : E โL[๐] Dual ๐ (Dual ๐ E) :=
ContinuousLinearMap.apply ๐ ๐
#align normed_space.inclusion_in_double_dual NormedSpace.inclusionInDoubleDual
@[simp]
theorem dual_def (x : E) (f : Dual ๐ E) : inclusionInDoubleDual ๐ E x f = f x :=
rfl
#align normed_space.dual_def NormedSpace.dual_def
theorem inclusionInDoubleDual_norm_eq :
โinclusionInDoubleDual ๐ Eโ = โContinuousLinearMap.id ๐ (Dual ๐ E)โ :=
ContinuousLinearMap.op_norm_flip _
#align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1 := by
rw [inclusionInDoubleDual_norm_eq]
exact ContinuousLinearMap.norm_id_le
#align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
theorem double_dual_bound (x : E) : โ(inclusionInDoubleDual ๐ E) xโ โค โxโ := by
simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusionInDoubleDual_norm_le ๐ E) x
#align normed_space.double_dual_bound NormedSpace.double_dual_bound
/-- The dual pairing as a bilinear form. -/
def dualPairing : Dual ๐ E โโ[๐] E โโ[๐] ๐ :=
ContinuousLinearMap.coeLM ๐
#align normed_space.dual_pairing NormedSpace.dualPairing
@[simp]
theorem dualPairing_apply {v : Dual ๐ E} {x : E} : dualPairing ๐ E v x = v x :=
rfl
#align normed_space.dual_pairing_apply NormedSpace.dualPairing_apply
theorem dualPairing_separatingLeft : (dualPairing ๐ E).SeparatingLeft := by
rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]
exact ContinuousLinearMap.coe_injective
#align normed_space.dual_pairing_separating_left NormedSpace.dualPairing_separatingLeft
end General
section BidualIsometry
variable (๐ : Type v) [IsROrC ๐] {E : Type u} [NormedAddCommGroup E] [NormedSpace ๐ E]
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M := by
classical
by_cases h : x = 0
ยท simp only [h, hMp, norm_zero]
ยท obtain โจf, hfโ, hfxโฉ : โ f : E โL[๐] ๐, โfโ = 1 โง f x = โxโ := exists_dual_vector ๐ x h
calc
โxโ = โ(โxโ : ๐)โ := IsROrC.norm_coe_norm.symm
_ = โf xโ := by rw [hfx]
_ โค M * โfโ := (hM f)
_ = M := by rw [hfโ, mul_one]
#align normed_space.norm_le_dual_bound NormedSpace.norm_le_dual_bound
theorem eq_zero_of_forall_dual_eq_zero {x : E} (h : โ f : Dual ๐ E, f x = (0 : ๐)) : x = 0 :=
norm_le_zero_iff.mp (norm_le_dual_bound ๐ x le_rfl fun f => by simp [h f])
#align normed_space.eq_zero_of_forall_dual_eq_zero NormedSpace.eq_zero_of_forall_dual_eq_zero
theorem eq_zero_iff_forall_dual_eq_zero (x : E) : x = 0 โ โ g : Dual ๐ E, g x = 0 :=
โจfun hx => by simp [hx], fun h => eq_zero_of_forall_dual_eq_zero ๐ hโฉ
#align normed_space.eq_zero_iff_forall_dual_eq_zero NormedSpace.eq_zero_iff_forall_dual_eq_zero
/-- See also `geometric_hahn_banach_point_point`. -/
theorem eq_iff_forall_dual_eq {x y : E} : x = y โ โ g : Dual ๐ E, g x = g y := by
rw [โ sub_eq_zero, eq_zero_iff_forall_dual_eq_zero ๐ (x - y)]
simp [sub_eq_zero]
#align normed_space.eq_iff_forall_dual_eq NormedSpace.eq_iff_forall_dual_eq
/-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E โโแตข[๐] Dual ๐ (Dual ๐ E) :=
{ inclusionInDoubleDual ๐ E with
norm_map' := by
intro x
apply le_antisymm
ยท exact double_dual_bound ๐ E x
rw [ContinuousLinearMap.norm_def]
refine' le_csInf ContinuousLinearMap.bounds_nonempty _
rintro c โจhc1, hc2โฉ
exact norm_le_dual_bound ๐ x hc1 hc2 }
#align normed_space.inclusion_in_double_dual_li NormedSpace.inclusionInDoubleDualLi
end BidualIsometry
section PolarSets
open Metric Set NormedSpace
/-- Given a subset `s` in a normed space `E` (over a field `๐`), the polar
`polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals which
evaluate to something of norm at most one at all points `z โ s`. -/
def polar (๐ : Type*) [NontriviallyNormedField ๐] {E : Type*} [SeminormedAddCommGroup E]
[NormedSpace ๐ E] : Set E โ Set (Dual ๐ E) :=
(dualPairing ๐ E).flip.polar
#align normed_space.polar NormedSpace.polar
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable {E : Type*} [SeminormedAddCommGroup E] [NormedSpace ๐ E]
theorem mem_polar_iff {x' : Dual ๐ E} (s : Set E) : x' โ polar ๐ s โ โ z โ s, โx' zโ โค 1 :=
Iff.rfl
#align normed_space.mem_polar_iff NormedSpace.mem_polar_iff
@[simp]
theorem polar_univ : polar ๐ (univ : Set E) = {(0 : Dual ๐ E)} :=
(dualPairing ๐ E).flip.polar_univ
(LinearMap.flip_separatingRight.mpr (dualPairing_separatingLeft ๐ E))
#align normed_space.polar_univ NormedSpace.polar_univ
theorem isClosed_polar (s : Set E) : IsClosed (polar ๐ s) := by
dsimp only [NormedSpace.polar]
simp only [LinearMap.polar_eq_iInter, LinearMap.flip_apply]
refine' isClosed_biInter fun z _ => _
|
exact isClosed_Iic.preimage (ContinuousLinearMap.apply ๐ ๐ z).continuous.norm
|
theorem isClosed_polar (s : Set E) : IsClosed (polar ๐ s) := by
dsimp only [NormedSpace.polar]
simp only [LinearMap.polar_eq_iInter, LinearMap.flip_apply]
refine' isClosed_biInter fun z _ => _
|
Mathlib.Analysis.NormedSpace.Dual.183_0.WirVfj6f5oiZZ2w
|
theorem isClosed_polar (s : Set E) : IsClosed (polar ๐ s)
|
Mathlib_Analysis_NormedSpace_Dual
|
๐ : Type u_1
instโยฒ : NontriviallyNormedField ๐
E : Type u_2
instโยน : SeminormedAddCommGroup E
instโ : NormedSpace ๐ E
s : Set E
โข IsClosed
((LinearMap.polar (LinearMap.flip (LinearMap.flip (dualPairing ๐ E))) โ โOrderDual.ofDual)
(LinearMap.polar (LinearMap.flip (dualPairing ๐ E)) s))
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# The topological dual of a normed space
In this file we define the topological dual `NormedSpace.Dual` of a normed space, and the
continuous linear map `NormedSpace.inclusionInDoubleDual` from a normed space into its double
dual.
For base field `๐ = โ` or `๐ = โ`, this map is actually an isometric embedding; we provide a
version `NormedSpace.inclusionInDoubleDualLi` of the map which is of type a bundled linear
isometric embedding, `E โโแตข[๐] (Dual ๐ (Dual ๐ E))`.
Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
theory for `SeminormedAddCommGroup` and we specialize to `NormedAddCommGroup` when needed.
## Main definitions
* `inclusionInDoubleDual` and `inclusionInDoubleDualLi` are the inclusion of a normed space
in its double dual, considered as a bounded linear map and as a linear isometry, respectively.
* `polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals `x'` for which
`โx' zโ โค 1` for every `z โ s`.
## Tags
dual
-/
noncomputable section
open Classical Topology Bornology
universe u v
namespace NormedSpace
section General
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable (E : Type*) [SeminormedAddCommGroup E] [NormedSpace ๐ E]
variable (F : Type*) [NormedAddCommGroup F] [NormedSpace ๐ F]
/-- The topological dual of a seminormed space `E`. -/
abbrev Dual : Type _ := E โL[๐] ๐
#align normed_space.dual NormedSpace.Dual
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_eq until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : NormedSpace ๐ (Dual ๐ E) := inferInstance
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_le until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : SeminormedAddCommGroup (Dual ๐ E) := inferInstance
/-- The inclusion of a normed space in its double (topological) dual, considered
as a bounded linear map. -/
def inclusionInDoubleDual : E โL[๐] Dual ๐ (Dual ๐ E) :=
ContinuousLinearMap.apply ๐ ๐
#align normed_space.inclusion_in_double_dual NormedSpace.inclusionInDoubleDual
@[simp]
theorem dual_def (x : E) (f : Dual ๐ E) : inclusionInDoubleDual ๐ E x f = f x :=
rfl
#align normed_space.dual_def NormedSpace.dual_def
theorem inclusionInDoubleDual_norm_eq :
โinclusionInDoubleDual ๐ Eโ = โContinuousLinearMap.id ๐ (Dual ๐ E)โ :=
ContinuousLinearMap.op_norm_flip _
#align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1 := by
rw [inclusionInDoubleDual_norm_eq]
exact ContinuousLinearMap.norm_id_le
#align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
theorem double_dual_bound (x : E) : โ(inclusionInDoubleDual ๐ E) xโ โค โxโ := by
simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusionInDoubleDual_norm_le ๐ E) x
#align normed_space.double_dual_bound NormedSpace.double_dual_bound
/-- The dual pairing as a bilinear form. -/
def dualPairing : Dual ๐ E โโ[๐] E โโ[๐] ๐ :=
ContinuousLinearMap.coeLM ๐
#align normed_space.dual_pairing NormedSpace.dualPairing
@[simp]
theorem dualPairing_apply {v : Dual ๐ E} {x : E} : dualPairing ๐ E v x = v x :=
rfl
#align normed_space.dual_pairing_apply NormedSpace.dualPairing_apply
theorem dualPairing_separatingLeft : (dualPairing ๐ E).SeparatingLeft := by
rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]
exact ContinuousLinearMap.coe_injective
#align normed_space.dual_pairing_separating_left NormedSpace.dualPairing_separatingLeft
end General
section BidualIsometry
variable (๐ : Type v) [IsROrC ๐] {E : Type u} [NormedAddCommGroup E] [NormedSpace ๐ E]
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M := by
classical
by_cases h : x = 0
ยท simp only [h, hMp, norm_zero]
ยท obtain โจf, hfโ, hfxโฉ : โ f : E โL[๐] ๐, โfโ = 1 โง f x = โxโ := exists_dual_vector ๐ x h
calc
โxโ = โ(โxโ : ๐)โ := IsROrC.norm_coe_norm.symm
_ = โf xโ := by rw [hfx]
_ โค M * โfโ := (hM f)
_ = M := by rw [hfโ, mul_one]
#align normed_space.norm_le_dual_bound NormedSpace.norm_le_dual_bound
theorem eq_zero_of_forall_dual_eq_zero {x : E} (h : โ f : Dual ๐ E, f x = (0 : ๐)) : x = 0 :=
norm_le_zero_iff.mp (norm_le_dual_bound ๐ x le_rfl fun f => by simp [h f])
#align normed_space.eq_zero_of_forall_dual_eq_zero NormedSpace.eq_zero_of_forall_dual_eq_zero
theorem eq_zero_iff_forall_dual_eq_zero (x : E) : x = 0 โ โ g : Dual ๐ E, g x = 0 :=
โจfun hx => by simp [hx], fun h => eq_zero_of_forall_dual_eq_zero ๐ hโฉ
#align normed_space.eq_zero_iff_forall_dual_eq_zero NormedSpace.eq_zero_iff_forall_dual_eq_zero
/-- See also `geometric_hahn_banach_point_point`. -/
theorem eq_iff_forall_dual_eq {x y : E} : x = y โ โ g : Dual ๐ E, g x = g y := by
rw [โ sub_eq_zero, eq_zero_iff_forall_dual_eq_zero ๐ (x - y)]
simp [sub_eq_zero]
#align normed_space.eq_iff_forall_dual_eq NormedSpace.eq_iff_forall_dual_eq
/-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E โโแตข[๐] Dual ๐ (Dual ๐ E) :=
{ inclusionInDoubleDual ๐ E with
norm_map' := by
intro x
apply le_antisymm
ยท exact double_dual_bound ๐ E x
rw [ContinuousLinearMap.norm_def]
refine' le_csInf ContinuousLinearMap.bounds_nonempty _
rintro c โจhc1, hc2โฉ
exact norm_le_dual_bound ๐ x hc1 hc2 }
#align normed_space.inclusion_in_double_dual_li NormedSpace.inclusionInDoubleDualLi
end BidualIsometry
section PolarSets
open Metric Set NormedSpace
/-- Given a subset `s` in a normed space `E` (over a field `๐`), the polar
`polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals which
evaluate to something of norm at most one at all points `z โ s`. -/
def polar (๐ : Type*) [NontriviallyNormedField ๐] {E : Type*} [SeminormedAddCommGroup E]
[NormedSpace ๐ E] : Set E โ Set (Dual ๐ E) :=
(dualPairing ๐ E).flip.polar
#align normed_space.polar NormedSpace.polar
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable {E : Type*} [SeminormedAddCommGroup E] [NormedSpace ๐ E]
theorem mem_polar_iff {x' : Dual ๐ E} (s : Set E) : x' โ polar ๐ s โ โ z โ s, โx' zโ โค 1 :=
Iff.rfl
#align normed_space.mem_polar_iff NormedSpace.mem_polar_iff
@[simp]
theorem polar_univ : polar ๐ (univ : Set E) = {(0 : Dual ๐ E)} :=
(dualPairing ๐ E).flip.polar_univ
(LinearMap.flip_separatingRight.mpr (dualPairing_separatingLeft ๐ E))
#align normed_space.polar_univ NormedSpace.polar_univ
theorem isClosed_polar (s : Set E) : IsClosed (polar ๐ s) := by
dsimp only [NormedSpace.polar]
simp only [LinearMap.polar_eq_iInter, LinearMap.flip_apply]
refine' isClosed_biInter fun z _ => _
exact isClosed_Iic.preimage (ContinuousLinearMap.apply ๐ ๐ z).continuous.norm
#align normed_space.is_closed_polar NormedSpace.isClosed_polar
@[simp]
theorem polar_closure (s : Set E) : polar ๐ (closure s) = polar ๐ s :=
((dualPairing ๐ E).flip.polar_antitone subset_closure).antisymm <|
(dualPairing ๐ E).flip.polar_gc.l_le <|
closure_minimal ((dualPairing ๐ E).flip.polar_gc.le_u_l s) <| by
|
simpa [LinearMap.flip_flip] using
(isClosed_polar _ _).preimage (inclusionInDoubleDual ๐ E).continuous
|
@[simp]
theorem polar_closure (s : Set E) : polar ๐ (closure s) = polar ๐ s :=
((dualPairing ๐ E).flip.polar_antitone subset_closure).antisymm <|
(dualPairing ๐ E).flip.polar_gc.l_le <|
closure_minimal ((dualPairing ๐ E).flip.polar_gc.le_u_l s) <| by
|
Mathlib.Analysis.NormedSpace.Dual.190_0.WirVfj6f5oiZZ2w
|
@[simp]
theorem polar_closure (s : Set E) : polar ๐ (closure s) = polar ๐ s
|
Mathlib_Analysis_NormedSpace_Dual
|
๐ : Type u_1
instโยฒ : NontriviallyNormedField ๐
E : Type u_2
instโยน : SeminormedAddCommGroup E
instโ : NormedSpace ๐ E
s : Set E
x' : Dual ๐ E
c : ๐
hc : โ z โ s, โx' zโ โค โcโ
โข cโปยน โข x' โ polar ๐ s
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# The topological dual of a normed space
In this file we define the topological dual `NormedSpace.Dual` of a normed space, and the
continuous linear map `NormedSpace.inclusionInDoubleDual` from a normed space into its double
dual.
For base field `๐ = โ` or `๐ = โ`, this map is actually an isometric embedding; we provide a
version `NormedSpace.inclusionInDoubleDualLi` of the map which is of type a bundled linear
isometric embedding, `E โโแตข[๐] (Dual ๐ (Dual ๐ E))`.
Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
theory for `SeminormedAddCommGroup` and we specialize to `NormedAddCommGroup` when needed.
## Main definitions
* `inclusionInDoubleDual` and `inclusionInDoubleDualLi` are the inclusion of a normed space
in its double dual, considered as a bounded linear map and as a linear isometry, respectively.
* `polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals `x'` for which
`โx' zโ โค 1` for every `z โ s`.
## Tags
dual
-/
noncomputable section
open Classical Topology Bornology
universe u v
namespace NormedSpace
section General
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable (E : Type*) [SeminormedAddCommGroup E] [NormedSpace ๐ E]
variable (F : Type*) [NormedAddCommGroup F] [NormedSpace ๐ F]
/-- The topological dual of a seminormed space `E`. -/
abbrev Dual : Type _ := E โL[๐] ๐
#align normed_space.dual NormedSpace.Dual
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_eq until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : NormedSpace ๐ (Dual ๐ E) := inferInstance
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_le until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : SeminormedAddCommGroup (Dual ๐ E) := inferInstance
/-- The inclusion of a normed space in its double (topological) dual, considered
as a bounded linear map. -/
def inclusionInDoubleDual : E โL[๐] Dual ๐ (Dual ๐ E) :=
ContinuousLinearMap.apply ๐ ๐
#align normed_space.inclusion_in_double_dual NormedSpace.inclusionInDoubleDual
@[simp]
theorem dual_def (x : E) (f : Dual ๐ E) : inclusionInDoubleDual ๐ E x f = f x :=
rfl
#align normed_space.dual_def NormedSpace.dual_def
theorem inclusionInDoubleDual_norm_eq :
โinclusionInDoubleDual ๐ Eโ = โContinuousLinearMap.id ๐ (Dual ๐ E)โ :=
ContinuousLinearMap.op_norm_flip _
#align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1 := by
rw [inclusionInDoubleDual_norm_eq]
exact ContinuousLinearMap.norm_id_le
#align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
theorem double_dual_bound (x : E) : โ(inclusionInDoubleDual ๐ E) xโ โค โxโ := by
simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusionInDoubleDual_norm_le ๐ E) x
#align normed_space.double_dual_bound NormedSpace.double_dual_bound
/-- The dual pairing as a bilinear form. -/
def dualPairing : Dual ๐ E โโ[๐] E โโ[๐] ๐ :=
ContinuousLinearMap.coeLM ๐
#align normed_space.dual_pairing NormedSpace.dualPairing
@[simp]
theorem dualPairing_apply {v : Dual ๐ E} {x : E} : dualPairing ๐ E v x = v x :=
rfl
#align normed_space.dual_pairing_apply NormedSpace.dualPairing_apply
theorem dualPairing_separatingLeft : (dualPairing ๐ E).SeparatingLeft := by
rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]
exact ContinuousLinearMap.coe_injective
#align normed_space.dual_pairing_separating_left NormedSpace.dualPairing_separatingLeft
end General
section BidualIsometry
variable (๐ : Type v) [IsROrC ๐] {E : Type u} [NormedAddCommGroup E] [NormedSpace ๐ E]
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M := by
classical
by_cases h : x = 0
ยท simp only [h, hMp, norm_zero]
ยท obtain โจf, hfโ, hfxโฉ : โ f : E โL[๐] ๐, โfโ = 1 โง f x = โxโ := exists_dual_vector ๐ x h
calc
โxโ = โ(โxโ : ๐)โ := IsROrC.norm_coe_norm.symm
_ = โf xโ := by rw [hfx]
_ โค M * โfโ := (hM f)
_ = M := by rw [hfโ, mul_one]
#align normed_space.norm_le_dual_bound NormedSpace.norm_le_dual_bound
theorem eq_zero_of_forall_dual_eq_zero {x : E} (h : โ f : Dual ๐ E, f x = (0 : ๐)) : x = 0 :=
norm_le_zero_iff.mp (norm_le_dual_bound ๐ x le_rfl fun f => by simp [h f])
#align normed_space.eq_zero_of_forall_dual_eq_zero NormedSpace.eq_zero_of_forall_dual_eq_zero
theorem eq_zero_iff_forall_dual_eq_zero (x : E) : x = 0 โ โ g : Dual ๐ E, g x = 0 :=
โจfun hx => by simp [hx], fun h => eq_zero_of_forall_dual_eq_zero ๐ hโฉ
#align normed_space.eq_zero_iff_forall_dual_eq_zero NormedSpace.eq_zero_iff_forall_dual_eq_zero
/-- See also `geometric_hahn_banach_point_point`. -/
theorem eq_iff_forall_dual_eq {x y : E} : x = y โ โ g : Dual ๐ E, g x = g y := by
rw [โ sub_eq_zero, eq_zero_iff_forall_dual_eq_zero ๐ (x - y)]
simp [sub_eq_zero]
#align normed_space.eq_iff_forall_dual_eq NormedSpace.eq_iff_forall_dual_eq
/-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E โโแตข[๐] Dual ๐ (Dual ๐ E) :=
{ inclusionInDoubleDual ๐ E with
norm_map' := by
intro x
apply le_antisymm
ยท exact double_dual_bound ๐ E x
rw [ContinuousLinearMap.norm_def]
refine' le_csInf ContinuousLinearMap.bounds_nonempty _
rintro c โจhc1, hc2โฉ
exact norm_le_dual_bound ๐ x hc1 hc2 }
#align normed_space.inclusion_in_double_dual_li NormedSpace.inclusionInDoubleDualLi
end BidualIsometry
section PolarSets
open Metric Set NormedSpace
/-- Given a subset `s` in a normed space `E` (over a field `๐`), the polar
`polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals which
evaluate to something of norm at most one at all points `z โ s`. -/
def polar (๐ : Type*) [NontriviallyNormedField ๐] {E : Type*} [SeminormedAddCommGroup E]
[NormedSpace ๐ E] : Set E โ Set (Dual ๐ E) :=
(dualPairing ๐ E).flip.polar
#align normed_space.polar NormedSpace.polar
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable {E : Type*} [SeminormedAddCommGroup E] [NormedSpace ๐ E]
theorem mem_polar_iff {x' : Dual ๐ E} (s : Set E) : x' โ polar ๐ s โ โ z โ s, โx' zโ โค 1 :=
Iff.rfl
#align normed_space.mem_polar_iff NormedSpace.mem_polar_iff
@[simp]
theorem polar_univ : polar ๐ (univ : Set E) = {(0 : Dual ๐ E)} :=
(dualPairing ๐ E).flip.polar_univ
(LinearMap.flip_separatingRight.mpr (dualPairing_separatingLeft ๐ E))
#align normed_space.polar_univ NormedSpace.polar_univ
theorem isClosed_polar (s : Set E) : IsClosed (polar ๐ s) := by
dsimp only [NormedSpace.polar]
simp only [LinearMap.polar_eq_iInter, LinearMap.flip_apply]
refine' isClosed_biInter fun z _ => _
exact isClosed_Iic.preimage (ContinuousLinearMap.apply ๐ ๐ z).continuous.norm
#align normed_space.is_closed_polar NormedSpace.isClosed_polar
@[simp]
theorem polar_closure (s : Set E) : polar ๐ (closure s) = polar ๐ s :=
((dualPairing ๐ E).flip.polar_antitone subset_closure).antisymm <|
(dualPairing ๐ E).flip.polar_gc.l_le <|
closure_minimal ((dualPairing ๐ E).flip.polar_gc.le_u_l s) <| by
simpa [LinearMap.flip_flip] using
(isClosed_polar _ _).preimage (inclusionInDoubleDual ๐ E).continuous
#align normed_space.polar_closure NormedSpace.polar_closure
variable {๐}
/-- If `x'` is a dual element such that the norms `โx' zโ` are bounded for `z โ s`, then a
small scalar multiple of `x'` is in `polar ๐ s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual ๐ E} {c : ๐} (hc : โ z, z โ s โ โx' zโ โค โcโ) :
cโปยน โข x' โ polar ๐ s := by
|
by_cases c_zero : c = 0
|
/-- If `x'` is a dual element such that the norms `โx' zโ` are bounded for `z โ s`, then a
small scalar multiple of `x'` is in `polar ๐ s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual ๐ E} {c : ๐} (hc : โ z, z โ s โ โx' zโ โค โcโ) :
cโปยน โข x' โ polar ๐ s := by
|
Mathlib.Analysis.NormedSpace.Dual.201_0.WirVfj6f5oiZZ2w
|
/-- If `x'` is a dual element such that the norms `โx' zโ` are bounded for `z โ s`, then a
small scalar multiple of `x'` is in `polar ๐ s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual ๐ E} {c : ๐} (hc : โ z, z โ s โ โx' zโ โค โcโ) :
cโปยน โข x' โ polar ๐ s
|
Mathlib_Analysis_NormedSpace_Dual
|
case pos
๐ : Type u_1
instโยฒ : NontriviallyNormedField ๐
E : Type u_2
instโยน : SeminormedAddCommGroup E
instโ : NormedSpace ๐ E
s : Set E
x' : Dual ๐ E
c : ๐
hc : โ z โ s, โx' zโ โค โcโ
c_zero : c = 0
โข cโปยน โข x' โ polar ๐ s
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# The topological dual of a normed space
In this file we define the topological dual `NormedSpace.Dual` of a normed space, and the
continuous linear map `NormedSpace.inclusionInDoubleDual` from a normed space into its double
dual.
For base field `๐ = โ` or `๐ = โ`, this map is actually an isometric embedding; we provide a
version `NormedSpace.inclusionInDoubleDualLi` of the map which is of type a bundled linear
isometric embedding, `E โโแตข[๐] (Dual ๐ (Dual ๐ E))`.
Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
theory for `SeminormedAddCommGroup` and we specialize to `NormedAddCommGroup` when needed.
## Main definitions
* `inclusionInDoubleDual` and `inclusionInDoubleDualLi` are the inclusion of a normed space
in its double dual, considered as a bounded linear map and as a linear isometry, respectively.
* `polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals `x'` for which
`โx' zโ โค 1` for every `z โ s`.
## Tags
dual
-/
noncomputable section
open Classical Topology Bornology
universe u v
namespace NormedSpace
section General
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable (E : Type*) [SeminormedAddCommGroup E] [NormedSpace ๐ E]
variable (F : Type*) [NormedAddCommGroup F] [NormedSpace ๐ F]
/-- The topological dual of a seminormed space `E`. -/
abbrev Dual : Type _ := E โL[๐] ๐
#align normed_space.dual NormedSpace.Dual
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_eq until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : NormedSpace ๐ (Dual ๐ E) := inferInstance
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_le until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : SeminormedAddCommGroup (Dual ๐ E) := inferInstance
/-- The inclusion of a normed space in its double (topological) dual, considered
as a bounded linear map. -/
def inclusionInDoubleDual : E โL[๐] Dual ๐ (Dual ๐ E) :=
ContinuousLinearMap.apply ๐ ๐
#align normed_space.inclusion_in_double_dual NormedSpace.inclusionInDoubleDual
@[simp]
theorem dual_def (x : E) (f : Dual ๐ E) : inclusionInDoubleDual ๐ E x f = f x :=
rfl
#align normed_space.dual_def NormedSpace.dual_def
theorem inclusionInDoubleDual_norm_eq :
โinclusionInDoubleDual ๐ Eโ = โContinuousLinearMap.id ๐ (Dual ๐ E)โ :=
ContinuousLinearMap.op_norm_flip _
#align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1 := by
rw [inclusionInDoubleDual_norm_eq]
exact ContinuousLinearMap.norm_id_le
#align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
theorem double_dual_bound (x : E) : โ(inclusionInDoubleDual ๐ E) xโ โค โxโ := by
simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusionInDoubleDual_norm_le ๐ E) x
#align normed_space.double_dual_bound NormedSpace.double_dual_bound
/-- The dual pairing as a bilinear form. -/
def dualPairing : Dual ๐ E โโ[๐] E โโ[๐] ๐ :=
ContinuousLinearMap.coeLM ๐
#align normed_space.dual_pairing NormedSpace.dualPairing
@[simp]
theorem dualPairing_apply {v : Dual ๐ E} {x : E} : dualPairing ๐ E v x = v x :=
rfl
#align normed_space.dual_pairing_apply NormedSpace.dualPairing_apply
theorem dualPairing_separatingLeft : (dualPairing ๐ E).SeparatingLeft := by
rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]
exact ContinuousLinearMap.coe_injective
#align normed_space.dual_pairing_separating_left NormedSpace.dualPairing_separatingLeft
end General
section BidualIsometry
variable (๐ : Type v) [IsROrC ๐] {E : Type u} [NormedAddCommGroup E] [NormedSpace ๐ E]
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M := by
classical
by_cases h : x = 0
ยท simp only [h, hMp, norm_zero]
ยท obtain โจf, hfโ, hfxโฉ : โ f : E โL[๐] ๐, โfโ = 1 โง f x = โxโ := exists_dual_vector ๐ x h
calc
โxโ = โ(โxโ : ๐)โ := IsROrC.norm_coe_norm.symm
_ = โf xโ := by rw [hfx]
_ โค M * โfโ := (hM f)
_ = M := by rw [hfโ, mul_one]
#align normed_space.norm_le_dual_bound NormedSpace.norm_le_dual_bound
theorem eq_zero_of_forall_dual_eq_zero {x : E} (h : โ f : Dual ๐ E, f x = (0 : ๐)) : x = 0 :=
norm_le_zero_iff.mp (norm_le_dual_bound ๐ x le_rfl fun f => by simp [h f])
#align normed_space.eq_zero_of_forall_dual_eq_zero NormedSpace.eq_zero_of_forall_dual_eq_zero
theorem eq_zero_iff_forall_dual_eq_zero (x : E) : x = 0 โ โ g : Dual ๐ E, g x = 0 :=
โจfun hx => by simp [hx], fun h => eq_zero_of_forall_dual_eq_zero ๐ hโฉ
#align normed_space.eq_zero_iff_forall_dual_eq_zero NormedSpace.eq_zero_iff_forall_dual_eq_zero
/-- See also `geometric_hahn_banach_point_point`. -/
theorem eq_iff_forall_dual_eq {x y : E} : x = y โ โ g : Dual ๐ E, g x = g y := by
rw [โ sub_eq_zero, eq_zero_iff_forall_dual_eq_zero ๐ (x - y)]
simp [sub_eq_zero]
#align normed_space.eq_iff_forall_dual_eq NormedSpace.eq_iff_forall_dual_eq
/-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E โโแตข[๐] Dual ๐ (Dual ๐ E) :=
{ inclusionInDoubleDual ๐ E with
norm_map' := by
intro x
apply le_antisymm
ยท exact double_dual_bound ๐ E x
rw [ContinuousLinearMap.norm_def]
refine' le_csInf ContinuousLinearMap.bounds_nonempty _
rintro c โจhc1, hc2โฉ
exact norm_le_dual_bound ๐ x hc1 hc2 }
#align normed_space.inclusion_in_double_dual_li NormedSpace.inclusionInDoubleDualLi
end BidualIsometry
section PolarSets
open Metric Set NormedSpace
/-- Given a subset `s` in a normed space `E` (over a field `๐`), the polar
`polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals which
evaluate to something of norm at most one at all points `z โ s`. -/
def polar (๐ : Type*) [NontriviallyNormedField ๐] {E : Type*} [SeminormedAddCommGroup E]
[NormedSpace ๐ E] : Set E โ Set (Dual ๐ E) :=
(dualPairing ๐ E).flip.polar
#align normed_space.polar NormedSpace.polar
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable {E : Type*} [SeminormedAddCommGroup E] [NormedSpace ๐ E]
theorem mem_polar_iff {x' : Dual ๐ E} (s : Set E) : x' โ polar ๐ s โ โ z โ s, โx' zโ โค 1 :=
Iff.rfl
#align normed_space.mem_polar_iff NormedSpace.mem_polar_iff
@[simp]
theorem polar_univ : polar ๐ (univ : Set E) = {(0 : Dual ๐ E)} :=
(dualPairing ๐ E).flip.polar_univ
(LinearMap.flip_separatingRight.mpr (dualPairing_separatingLeft ๐ E))
#align normed_space.polar_univ NormedSpace.polar_univ
theorem isClosed_polar (s : Set E) : IsClosed (polar ๐ s) := by
dsimp only [NormedSpace.polar]
simp only [LinearMap.polar_eq_iInter, LinearMap.flip_apply]
refine' isClosed_biInter fun z _ => _
exact isClosed_Iic.preimage (ContinuousLinearMap.apply ๐ ๐ z).continuous.norm
#align normed_space.is_closed_polar NormedSpace.isClosed_polar
@[simp]
theorem polar_closure (s : Set E) : polar ๐ (closure s) = polar ๐ s :=
((dualPairing ๐ E).flip.polar_antitone subset_closure).antisymm <|
(dualPairing ๐ E).flip.polar_gc.l_le <|
closure_minimal ((dualPairing ๐ E).flip.polar_gc.le_u_l s) <| by
simpa [LinearMap.flip_flip] using
(isClosed_polar _ _).preimage (inclusionInDoubleDual ๐ E).continuous
#align normed_space.polar_closure NormedSpace.polar_closure
variable {๐}
/-- If `x'` is a dual element such that the norms `โx' zโ` are bounded for `z โ s`, then a
small scalar multiple of `x'` is in `polar ๐ s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual ๐ E} {c : ๐} (hc : โ z, z โ s โ โx' zโ โค โcโ) :
cโปยน โข x' โ polar ๐ s := by
by_cases c_zero : c = 0
ยท
|
simp only [c_zero, inv_zero, zero_smul]
|
/-- If `x'` is a dual element such that the norms `โx' zโ` are bounded for `z โ s`, then a
small scalar multiple of `x'` is in `polar ๐ s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual ๐ E} {c : ๐} (hc : โ z, z โ s โ โx' zโ โค โcโ) :
cโปยน โข x' โ polar ๐ s := by
by_cases c_zero : c = 0
ยท
|
Mathlib.Analysis.NormedSpace.Dual.201_0.WirVfj6f5oiZZ2w
|
/-- If `x'` is a dual element such that the norms `โx' zโ` are bounded for `z โ s`, then a
small scalar multiple of `x'` is in `polar ๐ s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual ๐ E} {c : ๐} (hc : โ z, z โ s โ โx' zโ โค โcโ) :
cโปยน โข x' โ polar ๐ s
|
Mathlib_Analysis_NormedSpace_Dual
|
case pos
๐ : Type u_1
instโยฒ : NontriviallyNormedField ๐
E : Type u_2
instโยน : SeminormedAddCommGroup E
instโ : NormedSpace ๐ E
s : Set E
x' : Dual ๐ E
c : ๐
hc : โ z โ s, โx' zโ โค โcโ
c_zero : c = 0
โข 0 โ polar ๐ s
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# The topological dual of a normed space
In this file we define the topological dual `NormedSpace.Dual` of a normed space, and the
continuous linear map `NormedSpace.inclusionInDoubleDual` from a normed space into its double
dual.
For base field `๐ = โ` or `๐ = โ`, this map is actually an isometric embedding; we provide a
version `NormedSpace.inclusionInDoubleDualLi` of the map which is of type a bundled linear
isometric embedding, `E โโแตข[๐] (Dual ๐ (Dual ๐ E))`.
Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
theory for `SeminormedAddCommGroup` and we specialize to `NormedAddCommGroup` when needed.
## Main definitions
* `inclusionInDoubleDual` and `inclusionInDoubleDualLi` are the inclusion of a normed space
in its double dual, considered as a bounded linear map and as a linear isometry, respectively.
* `polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals `x'` for which
`โx' zโ โค 1` for every `z โ s`.
## Tags
dual
-/
noncomputable section
open Classical Topology Bornology
universe u v
namespace NormedSpace
section General
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable (E : Type*) [SeminormedAddCommGroup E] [NormedSpace ๐ E]
variable (F : Type*) [NormedAddCommGroup F] [NormedSpace ๐ F]
/-- The topological dual of a seminormed space `E`. -/
abbrev Dual : Type _ := E โL[๐] ๐
#align normed_space.dual NormedSpace.Dual
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_eq until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : NormedSpace ๐ (Dual ๐ E) := inferInstance
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_le until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : SeminormedAddCommGroup (Dual ๐ E) := inferInstance
/-- The inclusion of a normed space in its double (topological) dual, considered
as a bounded linear map. -/
def inclusionInDoubleDual : E โL[๐] Dual ๐ (Dual ๐ E) :=
ContinuousLinearMap.apply ๐ ๐
#align normed_space.inclusion_in_double_dual NormedSpace.inclusionInDoubleDual
@[simp]
theorem dual_def (x : E) (f : Dual ๐ E) : inclusionInDoubleDual ๐ E x f = f x :=
rfl
#align normed_space.dual_def NormedSpace.dual_def
theorem inclusionInDoubleDual_norm_eq :
โinclusionInDoubleDual ๐ Eโ = โContinuousLinearMap.id ๐ (Dual ๐ E)โ :=
ContinuousLinearMap.op_norm_flip _
#align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1 := by
rw [inclusionInDoubleDual_norm_eq]
exact ContinuousLinearMap.norm_id_le
#align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
theorem double_dual_bound (x : E) : โ(inclusionInDoubleDual ๐ E) xโ โค โxโ := by
simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusionInDoubleDual_norm_le ๐ E) x
#align normed_space.double_dual_bound NormedSpace.double_dual_bound
/-- The dual pairing as a bilinear form. -/
def dualPairing : Dual ๐ E โโ[๐] E โโ[๐] ๐ :=
ContinuousLinearMap.coeLM ๐
#align normed_space.dual_pairing NormedSpace.dualPairing
@[simp]
theorem dualPairing_apply {v : Dual ๐ E} {x : E} : dualPairing ๐ E v x = v x :=
rfl
#align normed_space.dual_pairing_apply NormedSpace.dualPairing_apply
theorem dualPairing_separatingLeft : (dualPairing ๐ E).SeparatingLeft := by
rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]
exact ContinuousLinearMap.coe_injective
#align normed_space.dual_pairing_separating_left NormedSpace.dualPairing_separatingLeft
end General
section BidualIsometry
variable (๐ : Type v) [IsROrC ๐] {E : Type u} [NormedAddCommGroup E] [NormedSpace ๐ E]
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M := by
classical
by_cases h : x = 0
ยท simp only [h, hMp, norm_zero]
ยท obtain โจf, hfโ, hfxโฉ : โ f : E โL[๐] ๐, โfโ = 1 โง f x = โxโ := exists_dual_vector ๐ x h
calc
โxโ = โ(โxโ : ๐)โ := IsROrC.norm_coe_norm.symm
_ = โf xโ := by rw [hfx]
_ โค M * โfโ := (hM f)
_ = M := by rw [hfโ, mul_one]
#align normed_space.norm_le_dual_bound NormedSpace.norm_le_dual_bound
theorem eq_zero_of_forall_dual_eq_zero {x : E} (h : โ f : Dual ๐ E, f x = (0 : ๐)) : x = 0 :=
norm_le_zero_iff.mp (norm_le_dual_bound ๐ x le_rfl fun f => by simp [h f])
#align normed_space.eq_zero_of_forall_dual_eq_zero NormedSpace.eq_zero_of_forall_dual_eq_zero
theorem eq_zero_iff_forall_dual_eq_zero (x : E) : x = 0 โ โ g : Dual ๐ E, g x = 0 :=
โจfun hx => by simp [hx], fun h => eq_zero_of_forall_dual_eq_zero ๐ hโฉ
#align normed_space.eq_zero_iff_forall_dual_eq_zero NormedSpace.eq_zero_iff_forall_dual_eq_zero
/-- See also `geometric_hahn_banach_point_point`. -/
theorem eq_iff_forall_dual_eq {x y : E} : x = y โ โ g : Dual ๐ E, g x = g y := by
rw [โ sub_eq_zero, eq_zero_iff_forall_dual_eq_zero ๐ (x - y)]
simp [sub_eq_zero]
#align normed_space.eq_iff_forall_dual_eq NormedSpace.eq_iff_forall_dual_eq
/-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E โโแตข[๐] Dual ๐ (Dual ๐ E) :=
{ inclusionInDoubleDual ๐ E with
norm_map' := by
intro x
apply le_antisymm
ยท exact double_dual_bound ๐ E x
rw [ContinuousLinearMap.norm_def]
refine' le_csInf ContinuousLinearMap.bounds_nonempty _
rintro c โจhc1, hc2โฉ
exact norm_le_dual_bound ๐ x hc1 hc2 }
#align normed_space.inclusion_in_double_dual_li NormedSpace.inclusionInDoubleDualLi
end BidualIsometry
section PolarSets
open Metric Set NormedSpace
/-- Given a subset `s` in a normed space `E` (over a field `๐`), the polar
`polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals which
evaluate to something of norm at most one at all points `z โ s`. -/
def polar (๐ : Type*) [NontriviallyNormedField ๐] {E : Type*} [SeminormedAddCommGroup E]
[NormedSpace ๐ E] : Set E โ Set (Dual ๐ E) :=
(dualPairing ๐ E).flip.polar
#align normed_space.polar NormedSpace.polar
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable {E : Type*} [SeminormedAddCommGroup E] [NormedSpace ๐ E]
theorem mem_polar_iff {x' : Dual ๐ E} (s : Set E) : x' โ polar ๐ s โ โ z โ s, โx' zโ โค 1 :=
Iff.rfl
#align normed_space.mem_polar_iff NormedSpace.mem_polar_iff
@[simp]
theorem polar_univ : polar ๐ (univ : Set E) = {(0 : Dual ๐ E)} :=
(dualPairing ๐ E).flip.polar_univ
(LinearMap.flip_separatingRight.mpr (dualPairing_separatingLeft ๐ E))
#align normed_space.polar_univ NormedSpace.polar_univ
theorem isClosed_polar (s : Set E) : IsClosed (polar ๐ s) := by
dsimp only [NormedSpace.polar]
simp only [LinearMap.polar_eq_iInter, LinearMap.flip_apply]
refine' isClosed_biInter fun z _ => _
exact isClosed_Iic.preimage (ContinuousLinearMap.apply ๐ ๐ z).continuous.norm
#align normed_space.is_closed_polar NormedSpace.isClosed_polar
@[simp]
theorem polar_closure (s : Set E) : polar ๐ (closure s) = polar ๐ s :=
((dualPairing ๐ E).flip.polar_antitone subset_closure).antisymm <|
(dualPairing ๐ E).flip.polar_gc.l_le <|
closure_minimal ((dualPairing ๐ E).flip.polar_gc.le_u_l s) <| by
simpa [LinearMap.flip_flip] using
(isClosed_polar _ _).preimage (inclusionInDoubleDual ๐ E).continuous
#align normed_space.polar_closure NormedSpace.polar_closure
variable {๐}
/-- If `x'` is a dual element such that the norms `โx' zโ` are bounded for `z โ s`, then a
small scalar multiple of `x'` is in `polar ๐ s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual ๐ E} {c : ๐} (hc : โ z, z โ s โ โx' zโ โค โcโ) :
cโปยน โข x' โ polar ๐ s := by
by_cases c_zero : c = 0
ยท simp only [c_zero, inv_zero, zero_smul]
|
exact (dualPairing ๐ E).flip.zero_mem_polar _
|
/-- If `x'` is a dual element such that the norms `โx' zโ` are bounded for `z โ s`, then a
small scalar multiple of `x'` is in `polar ๐ s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual ๐ E} {c : ๐} (hc : โ z, z โ s โ โx' zโ โค โcโ) :
cโปยน โข x' โ polar ๐ s := by
by_cases c_zero : c = 0
ยท simp only [c_zero, inv_zero, zero_smul]
|
Mathlib.Analysis.NormedSpace.Dual.201_0.WirVfj6f5oiZZ2w
|
/-- If `x'` is a dual element such that the norms `โx' zโ` are bounded for `z โ s`, then a
small scalar multiple of `x'` is in `polar ๐ s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual ๐ E} {c : ๐} (hc : โ z, z โ s โ โx' zโ โค โcโ) :
cโปยน โข x' โ polar ๐ s
|
Mathlib_Analysis_NormedSpace_Dual
|
case neg
๐ : Type u_1
instโยฒ : NontriviallyNormedField ๐
E : Type u_2
instโยน : SeminormedAddCommGroup E
instโ : NormedSpace ๐ E
s : Set E
x' : Dual ๐ E
c : ๐
hc : โ z โ s, โx' zโ โค โcโ
c_zero : ยฌc = 0
โข cโปยน โข x' โ polar ๐ s
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# The topological dual of a normed space
In this file we define the topological dual `NormedSpace.Dual` of a normed space, and the
continuous linear map `NormedSpace.inclusionInDoubleDual` from a normed space into its double
dual.
For base field `๐ = โ` or `๐ = โ`, this map is actually an isometric embedding; we provide a
version `NormedSpace.inclusionInDoubleDualLi` of the map which is of type a bundled linear
isometric embedding, `E โโแตข[๐] (Dual ๐ (Dual ๐ E))`.
Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
theory for `SeminormedAddCommGroup` and we specialize to `NormedAddCommGroup` when needed.
## Main definitions
* `inclusionInDoubleDual` and `inclusionInDoubleDualLi` are the inclusion of a normed space
in its double dual, considered as a bounded linear map and as a linear isometry, respectively.
* `polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals `x'` for which
`โx' zโ โค 1` for every `z โ s`.
## Tags
dual
-/
noncomputable section
open Classical Topology Bornology
universe u v
namespace NormedSpace
section General
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable (E : Type*) [SeminormedAddCommGroup E] [NormedSpace ๐ E]
variable (F : Type*) [NormedAddCommGroup F] [NormedSpace ๐ F]
/-- The topological dual of a seminormed space `E`. -/
abbrev Dual : Type _ := E โL[๐] ๐
#align normed_space.dual NormedSpace.Dual
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_eq until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : NormedSpace ๐ (Dual ๐ E) := inferInstance
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_le until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : SeminormedAddCommGroup (Dual ๐ E) := inferInstance
/-- The inclusion of a normed space in its double (topological) dual, considered
as a bounded linear map. -/
def inclusionInDoubleDual : E โL[๐] Dual ๐ (Dual ๐ E) :=
ContinuousLinearMap.apply ๐ ๐
#align normed_space.inclusion_in_double_dual NormedSpace.inclusionInDoubleDual
@[simp]
theorem dual_def (x : E) (f : Dual ๐ E) : inclusionInDoubleDual ๐ E x f = f x :=
rfl
#align normed_space.dual_def NormedSpace.dual_def
theorem inclusionInDoubleDual_norm_eq :
โinclusionInDoubleDual ๐ Eโ = โContinuousLinearMap.id ๐ (Dual ๐ E)โ :=
ContinuousLinearMap.op_norm_flip _
#align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1 := by
rw [inclusionInDoubleDual_norm_eq]
exact ContinuousLinearMap.norm_id_le
#align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
theorem double_dual_bound (x : E) : โ(inclusionInDoubleDual ๐ E) xโ โค โxโ := by
simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusionInDoubleDual_norm_le ๐ E) x
#align normed_space.double_dual_bound NormedSpace.double_dual_bound
/-- The dual pairing as a bilinear form. -/
def dualPairing : Dual ๐ E โโ[๐] E โโ[๐] ๐ :=
ContinuousLinearMap.coeLM ๐
#align normed_space.dual_pairing NormedSpace.dualPairing
@[simp]
theorem dualPairing_apply {v : Dual ๐ E} {x : E} : dualPairing ๐ E v x = v x :=
rfl
#align normed_space.dual_pairing_apply NormedSpace.dualPairing_apply
theorem dualPairing_separatingLeft : (dualPairing ๐ E).SeparatingLeft := by
rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]
exact ContinuousLinearMap.coe_injective
#align normed_space.dual_pairing_separating_left NormedSpace.dualPairing_separatingLeft
end General
section BidualIsometry
variable (๐ : Type v) [IsROrC ๐] {E : Type u} [NormedAddCommGroup E] [NormedSpace ๐ E]
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M := by
classical
by_cases h : x = 0
ยท simp only [h, hMp, norm_zero]
ยท obtain โจf, hfโ, hfxโฉ : โ f : E โL[๐] ๐, โfโ = 1 โง f x = โxโ := exists_dual_vector ๐ x h
calc
โxโ = โ(โxโ : ๐)โ := IsROrC.norm_coe_norm.symm
_ = โf xโ := by rw [hfx]
_ โค M * โfโ := (hM f)
_ = M := by rw [hfโ, mul_one]
#align normed_space.norm_le_dual_bound NormedSpace.norm_le_dual_bound
theorem eq_zero_of_forall_dual_eq_zero {x : E} (h : โ f : Dual ๐ E, f x = (0 : ๐)) : x = 0 :=
norm_le_zero_iff.mp (norm_le_dual_bound ๐ x le_rfl fun f => by simp [h f])
#align normed_space.eq_zero_of_forall_dual_eq_zero NormedSpace.eq_zero_of_forall_dual_eq_zero
theorem eq_zero_iff_forall_dual_eq_zero (x : E) : x = 0 โ โ g : Dual ๐ E, g x = 0 :=
โจfun hx => by simp [hx], fun h => eq_zero_of_forall_dual_eq_zero ๐ hโฉ
#align normed_space.eq_zero_iff_forall_dual_eq_zero NormedSpace.eq_zero_iff_forall_dual_eq_zero
/-- See also `geometric_hahn_banach_point_point`. -/
theorem eq_iff_forall_dual_eq {x y : E} : x = y โ โ g : Dual ๐ E, g x = g y := by
rw [โ sub_eq_zero, eq_zero_iff_forall_dual_eq_zero ๐ (x - y)]
simp [sub_eq_zero]
#align normed_space.eq_iff_forall_dual_eq NormedSpace.eq_iff_forall_dual_eq
/-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E โโแตข[๐] Dual ๐ (Dual ๐ E) :=
{ inclusionInDoubleDual ๐ E with
norm_map' := by
intro x
apply le_antisymm
ยท exact double_dual_bound ๐ E x
rw [ContinuousLinearMap.norm_def]
refine' le_csInf ContinuousLinearMap.bounds_nonempty _
rintro c โจhc1, hc2โฉ
exact norm_le_dual_bound ๐ x hc1 hc2 }
#align normed_space.inclusion_in_double_dual_li NormedSpace.inclusionInDoubleDualLi
end BidualIsometry
section PolarSets
open Metric Set NormedSpace
/-- Given a subset `s` in a normed space `E` (over a field `๐`), the polar
`polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals which
evaluate to something of norm at most one at all points `z โ s`. -/
def polar (๐ : Type*) [NontriviallyNormedField ๐] {E : Type*} [SeminormedAddCommGroup E]
[NormedSpace ๐ E] : Set E โ Set (Dual ๐ E) :=
(dualPairing ๐ E).flip.polar
#align normed_space.polar NormedSpace.polar
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable {E : Type*} [SeminormedAddCommGroup E] [NormedSpace ๐ E]
theorem mem_polar_iff {x' : Dual ๐ E} (s : Set E) : x' โ polar ๐ s โ โ z โ s, โx' zโ โค 1 :=
Iff.rfl
#align normed_space.mem_polar_iff NormedSpace.mem_polar_iff
@[simp]
theorem polar_univ : polar ๐ (univ : Set E) = {(0 : Dual ๐ E)} :=
(dualPairing ๐ E).flip.polar_univ
(LinearMap.flip_separatingRight.mpr (dualPairing_separatingLeft ๐ E))
#align normed_space.polar_univ NormedSpace.polar_univ
theorem isClosed_polar (s : Set E) : IsClosed (polar ๐ s) := by
dsimp only [NormedSpace.polar]
simp only [LinearMap.polar_eq_iInter, LinearMap.flip_apply]
refine' isClosed_biInter fun z _ => _
exact isClosed_Iic.preimage (ContinuousLinearMap.apply ๐ ๐ z).continuous.norm
#align normed_space.is_closed_polar NormedSpace.isClosed_polar
@[simp]
theorem polar_closure (s : Set E) : polar ๐ (closure s) = polar ๐ s :=
((dualPairing ๐ E).flip.polar_antitone subset_closure).antisymm <|
(dualPairing ๐ E).flip.polar_gc.l_le <|
closure_minimal ((dualPairing ๐ E).flip.polar_gc.le_u_l s) <| by
simpa [LinearMap.flip_flip] using
(isClosed_polar _ _).preimage (inclusionInDoubleDual ๐ E).continuous
#align normed_space.polar_closure NormedSpace.polar_closure
variable {๐}
/-- If `x'` is a dual element such that the norms `โx' zโ` are bounded for `z โ s`, then a
small scalar multiple of `x'` is in `polar ๐ s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual ๐ E} {c : ๐} (hc : โ z, z โ s โ โx' zโ โค โcโ) :
cโปยน โข x' โ polar ๐ s := by
by_cases c_zero : c = 0
ยท simp only [c_zero, inv_zero, zero_smul]
exact (dualPairing ๐ E).flip.zero_mem_polar _
|
have eq : โ z, โcโปยน โข x' zโ = โcโปยนโ * โx' zโ := fun z => norm_smul cโปยน _
|
/-- If `x'` is a dual element such that the norms `โx' zโ` are bounded for `z โ s`, then a
small scalar multiple of `x'` is in `polar ๐ s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual ๐ E} {c : ๐} (hc : โ z, z โ s โ โx' zโ โค โcโ) :
cโปยน โข x' โ polar ๐ s := by
by_cases c_zero : c = 0
ยท simp only [c_zero, inv_zero, zero_smul]
exact (dualPairing ๐ E).flip.zero_mem_polar _
|
Mathlib.Analysis.NormedSpace.Dual.201_0.WirVfj6f5oiZZ2w
|
/-- If `x'` is a dual element such that the norms `โx' zโ` are bounded for `z โ s`, then a
small scalar multiple of `x'` is in `polar ๐ s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual ๐ E} {c : ๐} (hc : โ z, z โ s โ โx' zโ โค โcโ) :
cโปยน โข x' โ polar ๐ s
|
Mathlib_Analysis_NormedSpace_Dual
|
case neg
๐ : Type u_1
instโยฒ : NontriviallyNormedField ๐
E : Type u_2
instโยน : SeminormedAddCommGroup E
instโ : NormedSpace ๐ E
s : Set E
x' : Dual ๐ E
c : ๐
hc : โ z โ s, โx' zโ โค โcโ
c_zero : ยฌc = 0
eq : โ (z : E), โcโปยน โข x' zโ = โcโปยนโ * โx' zโ
โข cโปยน โข x' โ polar ๐ s
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# The topological dual of a normed space
In this file we define the topological dual `NormedSpace.Dual` of a normed space, and the
continuous linear map `NormedSpace.inclusionInDoubleDual` from a normed space into its double
dual.
For base field `๐ = โ` or `๐ = โ`, this map is actually an isometric embedding; we provide a
version `NormedSpace.inclusionInDoubleDualLi` of the map which is of type a bundled linear
isometric embedding, `E โโแตข[๐] (Dual ๐ (Dual ๐ E))`.
Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
theory for `SeminormedAddCommGroup` and we specialize to `NormedAddCommGroup` when needed.
## Main definitions
* `inclusionInDoubleDual` and `inclusionInDoubleDualLi` are the inclusion of a normed space
in its double dual, considered as a bounded linear map and as a linear isometry, respectively.
* `polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals `x'` for which
`โx' zโ โค 1` for every `z โ s`.
## Tags
dual
-/
noncomputable section
open Classical Topology Bornology
universe u v
namespace NormedSpace
section General
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable (E : Type*) [SeminormedAddCommGroup E] [NormedSpace ๐ E]
variable (F : Type*) [NormedAddCommGroup F] [NormedSpace ๐ F]
/-- The topological dual of a seminormed space `E`. -/
abbrev Dual : Type _ := E โL[๐] ๐
#align normed_space.dual NormedSpace.Dual
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_eq until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : NormedSpace ๐ (Dual ๐ E) := inferInstance
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_le until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : SeminormedAddCommGroup (Dual ๐ E) := inferInstance
/-- The inclusion of a normed space in its double (topological) dual, considered
as a bounded linear map. -/
def inclusionInDoubleDual : E โL[๐] Dual ๐ (Dual ๐ E) :=
ContinuousLinearMap.apply ๐ ๐
#align normed_space.inclusion_in_double_dual NormedSpace.inclusionInDoubleDual
@[simp]
theorem dual_def (x : E) (f : Dual ๐ E) : inclusionInDoubleDual ๐ E x f = f x :=
rfl
#align normed_space.dual_def NormedSpace.dual_def
theorem inclusionInDoubleDual_norm_eq :
โinclusionInDoubleDual ๐ Eโ = โContinuousLinearMap.id ๐ (Dual ๐ E)โ :=
ContinuousLinearMap.op_norm_flip _
#align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1 := by
rw [inclusionInDoubleDual_norm_eq]
exact ContinuousLinearMap.norm_id_le
#align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
theorem double_dual_bound (x : E) : โ(inclusionInDoubleDual ๐ E) xโ โค โxโ := by
simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusionInDoubleDual_norm_le ๐ E) x
#align normed_space.double_dual_bound NormedSpace.double_dual_bound
/-- The dual pairing as a bilinear form. -/
def dualPairing : Dual ๐ E โโ[๐] E โโ[๐] ๐ :=
ContinuousLinearMap.coeLM ๐
#align normed_space.dual_pairing NormedSpace.dualPairing
@[simp]
theorem dualPairing_apply {v : Dual ๐ E} {x : E} : dualPairing ๐ E v x = v x :=
rfl
#align normed_space.dual_pairing_apply NormedSpace.dualPairing_apply
theorem dualPairing_separatingLeft : (dualPairing ๐ E).SeparatingLeft := by
rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]
exact ContinuousLinearMap.coe_injective
#align normed_space.dual_pairing_separating_left NormedSpace.dualPairing_separatingLeft
end General
section BidualIsometry
variable (๐ : Type v) [IsROrC ๐] {E : Type u} [NormedAddCommGroup E] [NormedSpace ๐ E]
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M := by
classical
by_cases h : x = 0
ยท simp only [h, hMp, norm_zero]
ยท obtain โจf, hfโ, hfxโฉ : โ f : E โL[๐] ๐, โfโ = 1 โง f x = โxโ := exists_dual_vector ๐ x h
calc
โxโ = โ(โxโ : ๐)โ := IsROrC.norm_coe_norm.symm
_ = โf xโ := by rw [hfx]
_ โค M * โfโ := (hM f)
_ = M := by rw [hfโ, mul_one]
#align normed_space.norm_le_dual_bound NormedSpace.norm_le_dual_bound
theorem eq_zero_of_forall_dual_eq_zero {x : E} (h : โ f : Dual ๐ E, f x = (0 : ๐)) : x = 0 :=
norm_le_zero_iff.mp (norm_le_dual_bound ๐ x le_rfl fun f => by simp [h f])
#align normed_space.eq_zero_of_forall_dual_eq_zero NormedSpace.eq_zero_of_forall_dual_eq_zero
theorem eq_zero_iff_forall_dual_eq_zero (x : E) : x = 0 โ โ g : Dual ๐ E, g x = 0 :=
โจfun hx => by simp [hx], fun h => eq_zero_of_forall_dual_eq_zero ๐ hโฉ
#align normed_space.eq_zero_iff_forall_dual_eq_zero NormedSpace.eq_zero_iff_forall_dual_eq_zero
/-- See also `geometric_hahn_banach_point_point`. -/
theorem eq_iff_forall_dual_eq {x y : E} : x = y โ โ g : Dual ๐ E, g x = g y := by
rw [โ sub_eq_zero, eq_zero_iff_forall_dual_eq_zero ๐ (x - y)]
simp [sub_eq_zero]
#align normed_space.eq_iff_forall_dual_eq NormedSpace.eq_iff_forall_dual_eq
/-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E โโแตข[๐] Dual ๐ (Dual ๐ E) :=
{ inclusionInDoubleDual ๐ E with
norm_map' := by
intro x
apply le_antisymm
ยท exact double_dual_bound ๐ E x
rw [ContinuousLinearMap.norm_def]
refine' le_csInf ContinuousLinearMap.bounds_nonempty _
rintro c โจhc1, hc2โฉ
exact norm_le_dual_bound ๐ x hc1 hc2 }
#align normed_space.inclusion_in_double_dual_li NormedSpace.inclusionInDoubleDualLi
end BidualIsometry
section PolarSets
open Metric Set NormedSpace
/-- Given a subset `s` in a normed space `E` (over a field `๐`), the polar
`polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals which
evaluate to something of norm at most one at all points `z โ s`. -/
def polar (๐ : Type*) [NontriviallyNormedField ๐] {E : Type*} [SeminormedAddCommGroup E]
[NormedSpace ๐ E] : Set E โ Set (Dual ๐ E) :=
(dualPairing ๐ E).flip.polar
#align normed_space.polar NormedSpace.polar
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable {E : Type*} [SeminormedAddCommGroup E] [NormedSpace ๐ E]
theorem mem_polar_iff {x' : Dual ๐ E} (s : Set E) : x' โ polar ๐ s โ โ z โ s, โx' zโ โค 1 :=
Iff.rfl
#align normed_space.mem_polar_iff NormedSpace.mem_polar_iff
@[simp]
theorem polar_univ : polar ๐ (univ : Set E) = {(0 : Dual ๐ E)} :=
(dualPairing ๐ E).flip.polar_univ
(LinearMap.flip_separatingRight.mpr (dualPairing_separatingLeft ๐ E))
#align normed_space.polar_univ NormedSpace.polar_univ
theorem isClosed_polar (s : Set E) : IsClosed (polar ๐ s) := by
dsimp only [NormedSpace.polar]
simp only [LinearMap.polar_eq_iInter, LinearMap.flip_apply]
refine' isClosed_biInter fun z _ => _
exact isClosed_Iic.preimage (ContinuousLinearMap.apply ๐ ๐ z).continuous.norm
#align normed_space.is_closed_polar NormedSpace.isClosed_polar
@[simp]
theorem polar_closure (s : Set E) : polar ๐ (closure s) = polar ๐ s :=
((dualPairing ๐ E).flip.polar_antitone subset_closure).antisymm <|
(dualPairing ๐ E).flip.polar_gc.l_le <|
closure_minimal ((dualPairing ๐ E).flip.polar_gc.le_u_l s) <| by
simpa [LinearMap.flip_flip] using
(isClosed_polar _ _).preimage (inclusionInDoubleDual ๐ E).continuous
#align normed_space.polar_closure NormedSpace.polar_closure
variable {๐}
/-- If `x'` is a dual element such that the norms `โx' zโ` are bounded for `z โ s`, then a
small scalar multiple of `x'` is in `polar ๐ s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual ๐ E} {c : ๐} (hc : โ z, z โ s โ โx' zโ โค โcโ) :
cโปยน โข x' โ polar ๐ s := by
by_cases c_zero : c = 0
ยท simp only [c_zero, inv_zero, zero_smul]
exact (dualPairing ๐ E).flip.zero_mem_polar _
have eq : โ z, โcโปยน โข x' zโ = โcโปยนโ * โx' zโ := fun z => norm_smul cโปยน _
|
have le : โ z, z โ s โ โcโปยน โข x' zโ โค โcโปยนโ * โcโ := by
intro z hzs
rw [eq z]
apply mul_le_mul (le_of_eq rfl) (hc z hzs) (norm_nonneg _) (norm_nonneg _)
|
/-- If `x'` is a dual element such that the norms `โx' zโ` are bounded for `z โ s`, then a
small scalar multiple of `x'` is in `polar ๐ s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual ๐ E} {c : ๐} (hc : โ z, z โ s โ โx' zโ โค โcโ) :
cโปยน โข x' โ polar ๐ s := by
by_cases c_zero : c = 0
ยท simp only [c_zero, inv_zero, zero_smul]
exact (dualPairing ๐ E).flip.zero_mem_polar _
have eq : โ z, โcโปยน โข x' zโ = โcโปยนโ * โx' zโ := fun z => norm_smul cโปยน _
|
Mathlib.Analysis.NormedSpace.Dual.201_0.WirVfj6f5oiZZ2w
|
/-- If `x'` is a dual element such that the norms `โx' zโ` are bounded for `z โ s`, then a
small scalar multiple of `x'` is in `polar ๐ s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual ๐ E} {c : ๐} (hc : โ z, z โ s โ โx' zโ โค โcโ) :
cโปยน โข x' โ polar ๐ s
|
Mathlib_Analysis_NormedSpace_Dual
|
๐ : Type u_1
instโยฒ : NontriviallyNormedField ๐
E : Type u_2
instโยน : SeminormedAddCommGroup E
instโ : NormedSpace ๐ E
s : Set E
x' : Dual ๐ E
c : ๐
hc : โ z โ s, โx' zโ โค โcโ
c_zero : ยฌc = 0
eq : โ (z : E), โcโปยน โข x' zโ = โcโปยนโ * โx' zโ
โข โ z โ s, โcโปยน โข x' zโ โค โcโปยนโ * โcโ
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# The topological dual of a normed space
In this file we define the topological dual `NormedSpace.Dual` of a normed space, and the
continuous linear map `NormedSpace.inclusionInDoubleDual` from a normed space into its double
dual.
For base field `๐ = โ` or `๐ = โ`, this map is actually an isometric embedding; we provide a
version `NormedSpace.inclusionInDoubleDualLi` of the map which is of type a bundled linear
isometric embedding, `E โโแตข[๐] (Dual ๐ (Dual ๐ E))`.
Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
theory for `SeminormedAddCommGroup` and we specialize to `NormedAddCommGroup` when needed.
## Main definitions
* `inclusionInDoubleDual` and `inclusionInDoubleDualLi` are the inclusion of a normed space
in its double dual, considered as a bounded linear map and as a linear isometry, respectively.
* `polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals `x'` for which
`โx' zโ โค 1` for every `z โ s`.
## Tags
dual
-/
noncomputable section
open Classical Topology Bornology
universe u v
namespace NormedSpace
section General
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable (E : Type*) [SeminormedAddCommGroup E] [NormedSpace ๐ E]
variable (F : Type*) [NormedAddCommGroup F] [NormedSpace ๐ F]
/-- The topological dual of a seminormed space `E`. -/
abbrev Dual : Type _ := E โL[๐] ๐
#align normed_space.dual NormedSpace.Dual
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_eq until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : NormedSpace ๐ (Dual ๐ E) := inferInstance
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_le until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : SeminormedAddCommGroup (Dual ๐ E) := inferInstance
/-- The inclusion of a normed space in its double (topological) dual, considered
as a bounded linear map. -/
def inclusionInDoubleDual : E โL[๐] Dual ๐ (Dual ๐ E) :=
ContinuousLinearMap.apply ๐ ๐
#align normed_space.inclusion_in_double_dual NormedSpace.inclusionInDoubleDual
@[simp]
theorem dual_def (x : E) (f : Dual ๐ E) : inclusionInDoubleDual ๐ E x f = f x :=
rfl
#align normed_space.dual_def NormedSpace.dual_def
theorem inclusionInDoubleDual_norm_eq :
โinclusionInDoubleDual ๐ Eโ = โContinuousLinearMap.id ๐ (Dual ๐ E)โ :=
ContinuousLinearMap.op_norm_flip _
#align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1 := by
rw [inclusionInDoubleDual_norm_eq]
exact ContinuousLinearMap.norm_id_le
#align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
theorem double_dual_bound (x : E) : โ(inclusionInDoubleDual ๐ E) xโ โค โxโ := by
simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusionInDoubleDual_norm_le ๐ E) x
#align normed_space.double_dual_bound NormedSpace.double_dual_bound
/-- The dual pairing as a bilinear form. -/
def dualPairing : Dual ๐ E โโ[๐] E โโ[๐] ๐ :=
ContinuousLinearMap.coeLM ๐
#align normed_space.dual_pairing NormedSpace.dualPairing
@[simp]
theorem dualPairing_apply {v : Dual ๐ E} {x : E} : dualPairing ๐ E v x = v x :=
rfl
#align normed_space.dual_pairing_apply NormedSpace.dualPairing_apply
theorem dualPairing_separatingLeft : (dualPairing ๐ E).SeparatingLeft := by
rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]
exact ContinuousLinearMap.coe_injective
#align normed_space.dual_pairing_separating_left NormedSpace.dualPairing_separatingLeft
end General
section BidualIsometry
variable (๐ : Type v) [IsROrC ๐] {E : Type u} [NormedAddCommGroup E] [NormedSpace ๐ E]
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M := by
classical
by_cases h : x = 0
ยท simp only [h, hMp, norm_zero]
ยท obtain โจf, hfโ, hfxโฉ : โ f : E โL[๐] ๐, โfโ = 1 โง f x = โxโ := exists_dual_vector ๐ x h
calc
โxโ = โ(โxโ : ๐)โ := IsROrC.norm_coe_norm.symm
_ = โf xโ := by rw [hfx]
_ โค M * โfโ := (hM f)
_ = M := by rw [hfโ, mul_one]
#align normed_space.norm_le_dual_bound NormedSpace.norm_le_dual_bound
theorem eq_zero_of_forall_dual_eq_zero {x : E} (h : โ f : Dual ๐ E, f x = (0 : ๐)) : x = 0 :=
norm_le_zero_iff.mp (norm_le_dual_bound ๐ x le_rfl fun f => by simp [h f])
#align normed_space.eq_zero_of_forall_dual_eq_zero NormedSpace.eq_zero_of_forall_dual_eq_zero
theorem eq_zero_iff_forall_dual_eq_zero (x : E) : x = 0 โ โ g : Dual ๐ E, g x = 0 :=
โจfun hx => by simp [hx], fun h => eq_zero_of_forall_dual_eq_zero ๐ hโฉ
#align normed_space.eq_zero_iff_forall_dual_eq_zero NormedSpace.eq_zero_iff_forall_dual_eq_zero
/-- See also `geometric_hahn_banach_point_point`. -/
theorem eq_iff_forall_dual_eq {x y : E} : x = y โ โ g : Dual ๐ E, g x = g y := by
rw [โ sub_eq_zero, eq_zero_iff_forall_dual_eq_zero ๐ (x - y)]
simp [sub_eq_zero]
#align normed_space.eq_iff_forall_dual_eq NormedSpace.eq_iff_forall_dual_eq
/-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E โโแตข[๐] Dual ๐ (Dual ๐ E) :=
{ inclusionInDoubleDual ๐ E with
norm_map' := by
intro x
apply le_antisymm
ยท exact double_dual_bound ๐ E x
rw [ContinuousLinearMap.norm_def]
refine' le_csInf ContinuousLinearMap.bounds_nonempty _
rintro c โจhc1, hc2โฉ
exact norm_le_dual_bound ๐ x hc1 hc2 }
#align normed_space.inclusion_in_double_dual_li NormedSpace.inclusionInDoubleDualLi
end BidualIsometry
section PolarSets
open Metric Set NormedSpace
/-- Given a subset `s` in a normed space `E` (over a field `๐`), the polar
`polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals which
evaluate to something of norm at most one at all points `z โ s`. -/
def polar (๐ : Type*) [NontriviallyNormedField ๐] {E : Type*} [SeminormedAddCommGroup E]
[NormedSpace ๐ E] : Set E โ Set (Dual ๐ E) :=
(dualPairing ๐ E).flip.polar
#align normed_space.polar NormedSpace.polar
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable {E : Type*} [SeminormedAddCommGroup E] [NormedSpace ๐ E]
theorem mem_polar_iff {x' : Dual ๐ E} (s : Set E) : x' โ polar ๐ s โ โ z โ s, โx' zโ โค 1 :=
Iff.rfl
#align normed_space.mem_polar_iff NormedSpace.mem_polar_iff
@[simp]
theorem polar_univ : polar ๐ (univ : Set E) = {(0 : Dual ๐ E)} :=
(dualPairing ๐ E).flip.polar_univ
(LinearMap.flip_separatingRight.mpr (dualPairing_separatingLeft ๐ E))
#align normed_space.polar_univ NormedSpace.polar_univ
theorem isClosed_polar (s : Set E) : IsClosed (polar ๐ s) := by
dsimp only [NormedSpace.polar]
simp only [LinearMap.polar_eq_iInter, LinearMap.flip_apply]
refine' isClosed_biInter fun z _ => _
exact isClosed_Iic.preimage (ContinuousLinearMap.apply ๐ ๐ z).continuous.norm
#align normed_space.is_closed_polar NormedSpace.isClosed_polar
@[simp]
theorem polar_closure (s : Set E) : polar ๐ (closure s) = polar ๐ s :=
((dualPairing ๐ E).flip.polar_antitone subset_closure).antisymm <|
(dualPairing ๐ E).flip.polar_gc.l_le <|
closure_minimal ((dualPairing ๐ E).flip.polar_gc.le_u_l s) <| by
simpa [LinearMap.flip_flip] using
(isClosed_polar _ _).preimage (inclusionInDoubleDual ๐ E).continuous
#align normed_space.polar_closure NormedSpace.polar_closure
variable {๐}
/-- If `x'` is a dual element such that the norms `โx' zโ` are bounded for `z โ s`, then a
small scalar multiple of `x'` is in `polar ๐ s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual ๐ E} {c : ๐} (hc : โ z, z โ s โ โx' zโ โค โcโ) :
cโปยน โข x' โ polar ๐ s := by
by_cases c_zero : c = 0
ยท simp only [c_zero, inv_zero, zero_smul]
exact (dualPairing ๐ E).flip.zero_mem_polar _
have eq : โ z, โcโปยน โข x' zโ = โcโปยนโ * โx' zโ := fun z => norm_smul cโปยน _
have le : โ z, z โ s โ โcโปยน โข x' zโ โค โcโปยนโ * โcโ := by
|
intro z hzs
|
/-- If `x'` is a dual element such that the norms `โx' zโ` are bounded for `z โ s`, then a
small scalar multiple of `x'` is in `polar ๐ s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual ๐ E} {c : ๐} (hc : โ z, z โ s โ โx' zโ โค โcโ) :
cโปยน โข x' โ polar ๐ s := by
by_cases c_zero : c = 0
ยท simp only [c_zero, inv_zero, zero_smul]
exact (dualPairing ๐ E).flip.zero_mem_polar _
have eq : โ z, โcโปยน โข x' zโ = โcโปยนโ * โx' zโ := fun z => norm_smul cโปยน _
have le : โ z, z โ s โ โcโปยน โข x' zโ โค โcโปยนโ * โcโ := by
|
Mathlib.Analysis.NormedSpace.Dual.201_0.WirVfj6f5oiZZ2w
|
/-- If `x'` is a dual element such that the norms `โx' zโ` are bounded for `z โ s`, then a
small scalar multiple of `x'` is in `polar ๐ s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual ๐ E} {c : ๐} (hc : โ z, z โ s โ โx' zโ โค โcโ) :
cโปยน โข x' โ polar ๐ s
|
Mathlib_Analysis_NormedSpace_Dual
|
๐ : Type u_1
instโยฒ : NontriviallyNormedField ๐
E : Type u_2
instโยน : SeminormedAddCommGroup E
instโ : NormedSpace ๐ E
s : Set E
x' : Dual ๐ E
c : ๐
hc : โ z โ s, โx' zโ โค โcโ
c_zero : ยฌc = 0
eq : โ (z : E), โcโปยน โข x' zโ = โcโปยนโ * โx' zโ
z : E
hzs : z โ s
โข โcโปยน โข x' zโ โค โcโปยนโ * โcโ
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# The topological dual of a normed space
In this file we define the topological dual `NormedSpace.Dual` of a normed space, and the
continuous linear map `NormedSpace.inclusionInDoubleDual` from a normed space into its double
dual.
For base field `๐ = โ` or `๐ = โ`, this map is actually an isometric embedding; we provide a
version `NormedSpace.inclusionInDoubleDualLi` of the map which is of type a bundled linear
isometric embedding, `E โโแตข[๐] (Dual ๐ (Dual ๐ E))`.
Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
theory for `SeminormedAddCommGroup` and we specialize to `NormedAddCommGroup` when needed.
## Main definitions
* `inclusionInDoubleDual` and `inclusionInDoubleDualLi` are the inclusion of a normed space
in its double dual, considered as a bounded linear map and as a linear isometry, respectively.
* `polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals `x'` for which
`โx' zโ โค 1` for every `z โ s`.
## Tags
dual
-/
noncomputable section
open Classical Topology Bornology
universe u v
namespace NormedSpace
section General
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable (E : Type*) [SeminormedAddCommGroup E] [NormedSpace ๐ E]
variable (F : Type*) [NormedAddCommGroup F] [NormedSpace ๐ F]
/-- The topological dual of a seminormed space `E`. -/
abbrev Dual : Type _ := E โL[๐] ๐
#align normed_space.dual NormedSpace.Dual
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_eq until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : NormedSpace ๐ (Dual ๐ E) := inferInstance
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_le until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : SeminormedAddCommGroup (Dual ๐ E) := inferInstance
/-- The inclusion of a normed space in its double (topological) dual, considered
as a bounded linear map. -/
def inclusionInDoubleDual : E โL[๐] Dual ๐ (Dual ๐ E) :=
ContinuousLinearMap.apply ๐ ๐
#align normed_space.inclusion_in_double_dual NormedSpace.inclusionInDoubleDual
@[simp]
theorem dual_def (x : E) (f : Dual ๐ E) : inclusionInDoubleDual ๐ E x f = f x :=
rfl
#align normed_space.dual_def NormedSpace.dual_def
theorem inclusionInDoubleDual_norm_eq :
โinclusionInDoubleDual ๐ Eโ = โContinuousLinearMap.id ๐ (Dual ๐ E)โ :=
ContinuousLinearMap.op_norm_flip _
#align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1 := by
rw [inclusionInDoubleDual_norm_eq]
exact ContinuousLinearMap.norm_id_le
#align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
theorem double_dual_bound (x : E) : โ(inclusionInDoubleDual ๐ E) xโ โค โxโ := by
simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusionInDoubleDual_norm_le ๐ E) x
#align normed_space.double_dual_bound NormedSpace.double_dual_bound
/-- The dual pairing as a bilinear form. -/
def dualPairing : Dual ๐ E โโ[๐] E โโ[๐] ๐ :=
ContinuousLinearMap.coeLM ๐
#align normed_space.dual_pairing NormedSpace.dualPairing
@[simp]
theorem dualPairing_apply {v : Dual ๐ E} {x : E} : dualPairing ๐ E v x = v x :=
rfl
#align normed_space.dual_pairing_apply NormedSpace.dualPairing_apply
theorem dualPairing_separatingLeft : (dualPairing ๐ E).SeparatingLeft := by
rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]
exact ContinuousLinearMap.coe_injective
#align normed_space.dual_pairing_separating_left NormedSpace.dualPairing_separatingLeft
end General
section BidualIsometry
variable (๐ : Type v) [IsROrC ๐] {E : Type u} [NormedAddCommGroup E] [NormedSpace ๐ E]
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M := by
classical
by_cases h : x = 0
ยท simp only [h, hMp, norm_zero]
ยท obtain โจf, hfโ, hfxโฉ : โ f : E โL[๐] ๐, โfโ = 1 โง f x = โxโ := exists_dual_vector ๐ x h
calc
โxโ = โ(โxโ : ๐)โ := IsROrC.norm_coe_norm.symm
_ = โf xโ := by rw [hfx]
_ โค M * โfโ := (hM f)
_ = M := by rw [hfโ, mul_one]
#align normed_space.norm_le_dual_bound NormedSpace.norm_le_dual_bound
theorem eq_zero_of_forall_dual_eq_zero {x : E} (h : โ f : Dual ๐ E, f x = (0 : ๐)) : x = 0 :=
norm_le_zero_iff.mp (norm_le_dual_bound ๐ x le_rfl fun f => by simp [h f])
#align normed_space.eq_zero_of_forall_dual_eq_zero NormedSpace.eq_zero_of_forall_dual_eq_zero
theorem eq_zero_iff_forall_dual_eq_zero (x : E) : x = 0 โ โ g : Dual ๐ E, g x = 0 :=
โจfun hx => by simp [hx], fun h => eq_zero_of_forall_dual_eq_zero ๐ hโฉ
#align normed_space.eq_zero_iff_forall_dual_eq_zero NormedSpace.eq_zero_iff_forall_dual_eq_zero
/-- See also `geometric_hahn_banach_point_point`. -/
theorem eq_iff_forall_dual_eq {x y : E} : x = y โ โ g : Dual ๐ E, g x = g y := by
rw [โ sub_eq_zero, eq_zero_iff_forall_dual_eq_zero ๐ (x - y)]
simp [sub_eq_zero]
#align normed_space.eq_iff_forall_dual_eq NormedSpace.eq_iff_forall_dual_eq
/-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E โโแตข[๐] Dual ๐ (Dual ๐ E) :=
{ inclusionInDoubleDual ๐ E with
norm_map' := by
intro x
apply le_antisymm
ยท exact double_dual_bound ๐ E x
rw [ContinuousLinearMap.norm_def]
refine' le_csInf ContinuousLinearMap.bounds_nonempty _
rintro c โจhc1, hc2โฉ
exact norm_le_dual_bound ๐ x hc1 hc2 }
#align normed_space.inclusion_in_double_dual_li NormedSpace.inclusionInDoubleDualLi
end BidualIsometry
section PolarSets
open Metric Set NormedSpace
/-- Given a subset `s` in a normed space `E` (over a field `๐`), the polar
`polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals which
evaluate to something of norm at most one at all points `z โ s`. -/
def polar (๐ : Type*) [NontriviallyNormedField ๐] {E : Type*} [SeminormedAddCommGroup E]
[NormedSpace ๐ E] : Set E โ Set (Dual ๐ E) :=
(dualPairing ๐ E).flip.polar
#align normed_space.polar NormedSpace.polar
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable {E : Type*} [SeminormedAddCommGroup E] [NormedSpace ๐ E]
theorem mem_polar_iff {x' : Dual ๐ E} (s : Set E) : x' โ polar ๐ s โ โ z โ s, โx' zโ โค 1 :=
Iff.rfl
#align normed_space.mem_polar_iff NormedSpace.mem_polar_iff
@[simp]
theorem polar_univ : polar ๐ (univ : Set E) = {(0 : Dual ๐ E)} :=
(dualPairing ๐ E).flip.polar_univ
(LinearMap.flip_separatingRight.mpr (dualPairing_separatingLeft ๐ E))
#align normed_space.polar_univ NormedSpace.polar_univ
theorem isClosed_polar (s : Set E) : IsClosed (polar ๐ s) := by
dsimp only [NormedSpace.polar]
simp only [LinearMap.polar_eq_iInter, LinearMap.flip_apply]
refine' isClosed_biInter fun z _ => _
exact isClosed_Iic.preimage (ContinuousLinearMap.apply ๐ ๐ z).continuous.norm
#align normed_space.is_closed_polar NormedSpace.isClosed_polar
@[simp]
theorem polar_closure (s : Set E) : polar ๐ (closure s) = polar ๐ s :=
((dualPairing ๐ E).flip.polar_antitone subset_closure).antisymm <|
(dualPairing ๐ E).flip.polar_gc.l_le <|
closure_minimal ((dualPairing ๐ E).flip.polar_gc.le_u_l s) <| by
simpa [LinearMap.flip_flip] using
(isClosed_polar _ _).preimage (inclusionInDoubleDual ๐ E).continuous
#align normed_space.polar_closure NormedSpace.polar_closure
variable {๐}
/-- If `x'` is a dual element such that the norms `โx' zโ` are bounded for `z โ s`, then a
small scalar multiple of `x'` is in `polar ๐ s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual ๐ E} {c : ๐} (hc : โ z, z โ s โ โx' zโ โค โcโ) :
cโปยน โข x' โ polar ๐ s := by
by_cases c_zero : c = 0
ยท simp only [c_zero, inv_zero, zero_smul]
exact (dualPairing ๐ E).flip.zero_mem_polar _
have eq : โ z, โcโปยน โข x' zโ = โcโปยนโ * โx' zโ := fun z => norm_smul cโปยน _
have le : โ z, z โ s โ โcโปยน โข x' zโ โค โcโปยนโ * โcโ := by
intro z hzs
|
rw [eq z]
|
/-- If `x'` is a dual element such that the norms `โx' zโ` are bounded for `z โ s`, then a
small scalar multiple of `x'` is in `polar ๐ s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual ๐ E} {c : ๐} (hc : โ z, z โ s โ โx' zโ โค โcโ) :
cโปยน โข x' โ polar ๐ s := by
by_cases c_zero : c = 0
ยท simp only [c_zero, inv_zero, zero_smul]
exact (dualPairing ๐ E).flip.zero_mem_polar _
have eq : โ z, โcโปยน โข x' zโ = โcโปยนโ * โx' zโ := fun z => norm_smul cโปยน _
have le : โ z, z โ s โ โcโปยน โข x' zโ โค โcโปยนโ * โcโ := by
intro z hzs
|
Mathlib.Analysis.NormedSpace.Dual.201_0.WirVfj6f5oiZZ2w
|
/-- If `x'` is a dual element such that the norms `โx' zโ` are bounded for `z โ s`, then a
small scalar multiple of `x'` is in `polar ๐ s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual ๐ E} {c : ๐} (hc : โ z, z โ s โ โx' zโ โค โcโ) :
cโปยน โข x' โ polar ๐ s
|
Mathlib_Analysis_NormedSpace_Dual
|
๐ : Type u_1
instโยฒ : NontriviallyNormedField ๐
E : Type u_2
instโยน : SeminormedAddCommGroup E
instโ : NormedSpace ๐ E
s : Set E
x' : Dual ๐ E
c : ๐
hc : โ z โ s, โx' zโ โค โcโ
c_zero : ยฌc = 0
eq : โ (z : E), โcโปยน โข x' zโ = โcโปยนโ * โx' zโ
z : E
hzs : z โ s
โข โcโปยนโ * โx' zโ โค โcโปยนโ * โcโ
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# The topological dual of a normed space
In this file we define the topological dual `NormedSpace.Dual` of a normed space, and the
continuous linear map `NormedSpace.inclusionInDoubleDual` from a normed space into its double
dual.
For base field `๐ = โ` or `๐ = โ`, this map is actually an isometric embedding; we provide a
version `NormedSpace.inclusionInDoubleDualLi` of the map which is of type a bundled linear
isometric embedding, `E โโแตข[๐] (Dual ๐ (Dual ๐ E))`.
Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
theory for `SeminormedAddCommGroup` and we specialize to `NormedAddCommGroup` when needed.
## Main definitions
* `inclusionInDoubleDual` and `inclusionInDoubleDualLi` are the inclusion of a normed space
in its double dual, considered as a bounded linear map and as a linear isometry, respectively.
* `polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals `x'` for which
`โx' zโ โค 1` for every `z โ s`.
## Tags
dual
-/
noncomputable section
open Classical Topology Bornology
universe u v
namespace NormedSpace
section General
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable (E : Type*) [SeminormedAddCommGroup E] [NormedSpace ๐ E]
variable (F : Type*) [NormedAddCommGroup F] [NormedSpace ๐ F]
/-- The topological dual of a seminormed space `E`. -/
abbrev Dual : Type _ := E โL[๐] ๐
#align normed_space.dual NormedSpace.Dual
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_eq until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : NormedSpace ๐ (Dual ๐ E) := inferInstance
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_le until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : SeminormedAddCommGroup (Dual ๐ E) := inferInstance
/-- The inclusion of a normed space in its double (topological) dual, considered
as a bounded linear map. -/
def inclusionInDoubleDual : E โL[๐] Dual ๐ (Dual ๐ E) :=
ContinuousLinearMap.apply ๐ ๐
#align normed_space.inclusion_in_double_dual NormedSpace.inclusionInDoubleDual
@[simp]
theorem dual_def (x : E) (f : Dual ๐ E) : inclusionInDoubleDual ๐ E x f = f x :=
rfl
#align normed_space.dual_def NormedSpace.dual_def
theorem inclusionInDoubleDual_norm_eq :
โinclusionInDoubleDual ๐ Eโ = โContinuousLinearMap.id ๐ (Dual ๐ E)โ :=
ContinuousLinearMap.op_norm_flip _
#align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1 := by
rw [inclusionInDoubleDual_norm_eq]
exact ContinuousLinearMap.norm_id_le
#align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
theorem double_dual_bound (x : E) : โ(inclusionInDoubleDual ๐ E) xโ โค โxโ := by
simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusionInDoubleDual_norm_le ๐ E) x
#align normed_space.double_dual_bound NormedSpace.double_dual_bound
/-- The dual pairing as a bilinear form. -/
def dualPairing : Dual ๐ E โโ[๐] E โโ[๐] ๐ :=
ContinuousLinearMap.coeLM ๐
#align normed_space.dual_pairing NormedSpace.dualPairing
@[simp]
theorem dualPairing_apply {v : Dual ๐ E} {x : E} : dualPairing ๐ E v x = v x :=
rfl
#align normed_space.dual_pairing_apply NormedSpace.dualPairing_apply
theorem dualPairing_separatingLeft : (dualPairing ๐ E).SeparatingLeft := by
rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]
exact ContinuousLinearMap.coe_injective
#align normed_space.dual_pairing_separating_left NormedSpace.dualPairing_separatingLeft
end General
section BidualIsometry
variable (๐ : Type v) [IsROrC ๐] {E : Type u} [NormedAddCommGroup E] [NormedSpace ๐ E]
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M := by
classical
by_cases h : x = 0
ยท simp only [h, hMp, norm_zero]
ยท obtain โจf, hfโ, hfxโฉ : โ f : E โL[๐] ๐, โfโ = 1 โง f x = โxโ := exists_dual_vector ๐ x h
calc
โxโ = โ(โxโ : ๐)โ := IsROrC.norm_coe_norm.symm
_ = โf xโ := by rw [hfx]
_ โค M * โfโ := (hM f)
_ = M := by rw [hfโ, mul_one]
#align normed_space.norm_le_dual_bound NormedSpace.norm_le_dual_bound
theorem eq_zero_of_forall_dual_eq_zero {x : E} (h : โ f : Dual ๐ E, f x = (0 : ๐)) : x = 0 :=
norm_le_zero_iff.mp (norm_le_dual_bound ๐ x le_rfl fun f => by simp [h f])
#align normed_space.eq_zero_of_forall_dual_eq_zero NormedSpace.eq_zero_of_forall_dual_eq_zero
theorem eq_zero_iff_forall_dual_eq_zero (x : E) : x = 0 โ โ g : Dual ๐ E, g x = 0 :=
โจfun hx => by simp [hx], fun h => eq_zero_of_forall_dual_eq_zero ๐ hโฉ
#align normed_space.eq_zero_iff_forall_dual_eq_zero NormedSpace.eq_zero_iff_forall_dual_eq_zero
/-- See also `geometric_hahn_banach_point_point`. -/
theorem eq_iff_forall_dual_eq {x y : E} : x = y โ โ g : Dual ๐ E, g x = g y := by
rw [โ sub_eq_zero, eq_zero_iff_forall_dual_eq_zero ๐ (x - y)]
simp [sub_eq_zero]
#align normed_space.eq_iff_forall_dual_eq NormedSpace.eq_iff_forall_dual_eq
/-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E โโแตข[๐] Dual ๐ (Dual ๐ E) :=
{ inclusionInDoubleDual ๐ E with
norm_map' := by
intro x
apply le_antisymm
ยท exact double_dual_bound ๐ E x
rw [ContinuousLinearMap.norm_def]
refine' le_csInf ContinuousLinearMap.bounds_nonempty _
rintro c โจhc1, hc2โฉ
exact norm_le_dual_bound ๐ x hc1 hc2 }
#align normed_space.inclusion_in_double_dual_li NormedSpace.inclusionInDoubleDualLi
end BidualIsometry
section PolarSets
open Metric Set NormedSpace
/-- Given a subset `s` in a normed space `E` (over a field `๐`), the polar
`polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals which
evaluate to something of norm at most one at all points `z โ s`. -/
def polar (๐ : Type*) [NontriviallyNormedField ๐] {E : Type*} [SeminormedAddCommGroup E]
[NormedSpace ๐ E] : Set E โ Set (Dual ๐ E) :=
(dualPairing ๐ E).flip.polar
#align normed_space.polar NormedSpace.polar
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable {E : Type*} [SeminormedAddCommGroup E] [NormedSpace ๐ E]
theorem mem_polar_iff {x' : Dual ๐ E} (s : Set E) : x' โ polar ๐ s โ โ z โ s, โx' zโ โค 1 :=
Iff.rfl
#align normed_space.mem_polar_iff NormedSpace.mem_polar_iff
@[simp]
theorem polar_univ : polar ๐ (univ : Set E) = {(0 : Dual ๐ E)} :=
(dualPairing ๐ E).flip.polar_univ
(LinearMap.flip_separatingRight.mpr (dualPairing_separatingLeft ๐ E))
#align normed_space.polar_univ NormedSpace.polar_univ
theorem isClosed_polar (s : Set E) : IsClosed (polar ๐ s) := by
dsimp only [NormedSpace.polar]
simp only [LinearMap.polar_eq_iInter, LinearMap.flip_apply]
refine' isClosed_biInter fun z _ => _
exact isClosed_Iic.preimage (ContinuousLinearMap.apply ๐ ๐ z).continuous.norm
#align normed_space.is_closed_polar NormedSpace.isClosed_polar
@[simp]
theorem polar_closure (s : Set E) : polar ๐ (closure s) = polar ๐ s :=
((dualPairing ๐ E).flip.polar_antitone subset_closure).antisymm <|
(dualPairing ๐ E).flip.polar_gc.l_le <|
closure_minimal ((dualPairing ๐ E).flip.polar_gc.le_u_l s) <| by
simpa [LinearMap.flip_flip] using
(isClosed_polar _ _).preimage (inclusionInDoubleDual ๐ E).continuous
#align normed_space.polar_closure NormedSpace.polar_closure
variable {๐}
/-- If `x'` is a dual element such that the norms `โx' zโ` are bounded for `z โ s`, then a
small scalar multiple of `x'` is in `polar ๐ s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual ๐ E} {c : ๐} (hc : โ z, z โ s โ โx' zโ โค โcโ) :
cโปยน โข x' โ polar ๐ s := by
by_cases c_zero : c = 0
ยท simp only [c_zero, inv_zero, zero_smul]
exact (dualPairing ๐ E).flip.zero_mem_polar _
have eq : โ z, โcโปยน โข x' zโ = โcโปยนโ * โx' zโ := fun z => norm_smul cโปยน _
have le : โ z, z โ s โ โcโปยน โข x' zโ โค โcโปยนโ * โcโ := by
intro z hzs
rw [eq z]
|
apply mul_le_mul (le_of_eq rfl) (hc z hzs) (norm_nonneg _) (norm_nonneg _)
|
/-- If `x'` is a dual element such that the norms `โx' zโ` are bounded for `z โ s`, then a
small scalar multiple of `x'` is in `polar ๐ s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual ๐ E} {c : ๐} (hc : โ z, z โ s โ โx' zโ โค โcโ) :
cโปยน โข x' โ polar ๐ s := by
by_cases c_zero : c = 0
ยท simp only [c_zero, inv_zero, zero_smul]
exact (dualPairing ๐ E).flip.zero_mem_polar _
have eq : โ z, โcโปยน โข x' zโ = โcโปยนโ * โx' zโ := fun z => norm_smul cโปยน _
have le : โ z, z โ s โ โcโปยน โข x' zโ โค โcโปยนโ * โcโ := by
intro z hzs
rw [eq z]
|
Mathlib.Analysis.NormedSpace.Dual.201_0.WirVfj6f5oiZZ2w
|
/-- If `x'` is a dual element such that the norms `โx' zโ` are bounded for `z โ s`, then a
small scalar multiple of `x'` is in `polar ๐ s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual ๐ E} {c : ๐} (hc : โ z, z โ s โ โx' zโ โค โcโ) :
cโปยน โข x' โ polar ๐ s
|
Mathlib_Analysis_NormedSpace_Dual
|
case neg
๐ : Type u_1
instโยฒ : NontriviallyNormedField ๐
E : Type u_2
instโยน : SeminormedAddCommGroup E
instโ : NormedSpace ๐ E
s : Set E
x' : Dual ๐ E
c : ๐
hc : โ z โ s, โx' zโ โค โcโ
c_zero : ยฌc = 0
eq : โ (z : E), โcโปยน โข x' zโ = โcโปยนโ * โx' zโ
le : โ z โ s, โcโปยน โข x' zโ โค โcโปยนโ * โcโ
โข cโปยน โข x' โ polar ๐ s
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# The topological dual of a normed space
In this file we define the topological dual `NormedSpace.Dual` of a normed space, and the
continuous linear map `NormedSpace.inclusionInDoubleDual` from a normed space into its double
dual.
For base field `๐ = โ` or `๐ = โ`, this map is actually an isometric embedding; we provide a
version `NormedSpace.inclusionInDoubleDualLi` of the map which is of type a bundled linear
isometric embedding, `E โโแตข[๐] (Dual ๐ (Dual ๐ E))`.
Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
theory for `SeminormedAddCommGroup` and we specialize to `NormedAddCommGroup` when needed.
## Main definitions
* `inclusionInDoubleDual` and `inclusionInDoubleDualLi` are the inclusion of a normed space
in its double dual, considered as a bounded linear map and as a linear isometry, respectively.
* `polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals `x'` for which
`โx' zโ โค 1` for every `z โ s`.
## Tags
dual
-/
noncomputable section
open Classical Topology Bornology
universe u v
namespace NormedSpace
section General
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable (E : Type*) [SeminormedAddCommGroup E] [NormedSpace ๐ E]
variable (F : Type*) [NormedAddCommGroup F] [NormedSpace ๐ F]
/-- The topological dual of a seminormed space `E`. -/
abbrev Dual : Type _ := E โL[๐] ๐
#align normed_space.dual NormedSpace.Dual
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_eq until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : NormedSpace ๐ (Dual ๐ E) := inferInstance
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_le until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : SeminormedAddCommGroup (Dual ๐ E) := inferInstance
/-- The inclusion of a normed space in its double (topological) dual, considered
as a bounded linear map. -/
def inclusionInDoubleDual : E โL[๐] Dual ๐ (Dual ๐ E) :=
ContinuousLinearMap.apply ๐ ๐
#align normed_space.inclusion_in_double_dual NormedSpace.inclusionInDoubleDual
@[simp]
theorem dual_def (x : E) (f : Dual ๐ E) : inclusionInDoubleDual ๐ E x f = f x :=
rfl
#align normed_space.dual_def NormedSpace.dual_def
theorem inclusionInDoubleDual_norm_eq :
โinclusionInDoubleDual ๐ Eโ = โContinuousLinearMap.id ๐ (Dual ๐ E)โ :=
ContinuousLinearMap.op_norm_flip _
#align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1 := by
rw [inclusionInDoubleDual_norm_eq]
exact ContinuousLinearMap.norm_id_le
#align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
theorem double_dual_bound (x : E) : โ(inclusionInDoubleDual ๐ E) xโ โค โxโ := by
simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusionInDoubleDual_norm_le ๐ E) x
#align normed_space.double_dual_bound NormedSpace.double_dual_bound
/-- The dual pairing as a bilinear form. -/
def dualPairing : Dual ๐ E โโ[๐] E โโ[๐] ๐ :=
ContinuousLinearMap.coeLM ๐
#align normed_space.dual_pairing NormedSpace.dualPairing
@[simp]
theorem dualPairing_apply {v : Dual ๐ E} {x : E} : dualPairing ๐ E v x = v x :=
rfl
#align normed_space.dual_pairing_apply NormedSpace.dualPairing_apply
theorem dualPairing_separatingLeft : (dualPairing ๐ E).SeparatingLeft := by
rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]
exact ContinuousLinearMap.coe_injective
#align normed_space.dual_pairing_separating_left NormedSpace.dualPairing_separatingLeft
end General
section BidualIsometry
variable (๐ : Type v) [IsROrC ๐] {E : Type u} [NormedAddCommGroup E] [NormedSpace ๐ E]
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M := by
classical
by_cases h : x = 0
ยท simp only [h, hMp, norm_zero]
ยท obtain โจf, hfโ, hfxโฉ : โ f : E โL[๐] ๐, โfโ = 1 โง f x = โxโ := exists_dual_vector ๐ x h
calc
โxโ = โ(โxโ : ๐)โ := IsROrC.norm_coe_norm.symm
_ = โf xโ := by rw [hfx]
_ โค M * โfโ := (hM f)
_ = M := by rw [hfโ, mul_one]
#align normed_space.norm_le_dual_bound NormedSpace.norm_le_dual_bound
theorem eq_zero_of_forall_dual_eq_zero {x : E} (h : โ f : Dual ๐ E, f x = (0 : ๐)) : x = 0 :=
norm_le_zero_iff.mp (norm_le_dual_bound ๐ x le_rfl fun f => by simp [h f])
#align normed_space.eq_zero_of_forall_dual_eq_zero NormedSpace.eq_zero_of_forall_dual_eq_zero
theorem eq_zero_iff_forall_dual_eq_zero (x : E) : x = 0 โ โ g : Dual ๐ E, g x = 0 :=
โจfun hx => by simp [hx], fun h => eq_zero_of_forall_dual_eq_zero ๐ hโฉ
#align normed_space.eq_zero_iff_forall_dual_eq_zero NormedSpace.eq_zero_iff_forall_dual_eq_zero
/-- See also `geometric_hahn_banach_point_point`. -/
theorem eq_iff_forall_dual_eq {x y : E} : x = y โ โ g : Dual ๐ E, g x = g y := by
rw [โ sub_eq_zero, eq_zero_iff_forall_dual_eq_zero ๐ (x - y)]
simp [sub_eq_zero]
#align normed_space.eq_iff_forall_dual_eq NormedSpace.eq_iff_forall_dual_eq
/-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E โโแตข[๐] Dual ๐ (Dual ๐ E) :=
{ inclusionInDoubleDual ๐ E with
norm_map' := by
intro x
apply le_antisymm
ยท exact double_dual_bound ๐ E x
rw [ContinuousLinearMap.norm_def]
refine' le_csInf ContinuousLinearMap.bounds_nonempty _
rintro c โจhc1, hc2โฉ
exact norm_le_dual_bound ๐ x hc1 hc2 }
#align normed_space.inclusion_in_double_dual_li NormedSpace.inclusionInDoubleDualLi
end BidualIsometry
section PolarSets
open Metric Set NormedSpace
/-- Given a subset `s` in a normed space `E` (over a field `๐`), the polar
`polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals which
evaluate to something of norm at most one at all points `z โ s`. -/
def polar (๐ : Type*) [NontriviallyNormedField ๐] {E : Type*} [SeminormedAddCommGroup E]
[NormedSpace ๐ E] : Set E โ Set (Dual ๐ E) :=
(dualPairing ๐ E).flip.polar
#align normed_space.polar NormedSpace.polar
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable {E : Type*} [SeminormedAddCommGroup E] [NormedSpace ๐ E]
theorem mem_polar_iff {x' : Dual ๐ E} (s : Set E) : x' โ polar ๐ s โ โ z โ s, โx' zโ โค 1 :=
Iff.rfl
#align normed_space.mem_polar_iff NormedSpace.mem_polar_iff
@[simp]
theorem polar_univ : polar ๐ (univ : Set E) = {(0 : Dual ๐ E)} :=
(dualPairing ๐ E).flip.polar_univ
(LinearMap.flip_separatingRight.mpr (dualPairing_separatingLeft ๐ E))
#align normed_space.polar_univ NormedSpace.polar_univ
theorem isClosed_polar (s : Set E) : IsClosed (polar ๐ s) := by
dsimp only [NormedSpace.polar]
simp only [LinearMap.polar_eq_iInter, LinearMap.flip_apply]
refine' isClosed_biInter fun z _ => _
exact isClosed_Iic.preimage (ContinuousLinearMap.apply ๐ ๐ z).continuous.norm
#align normed_space.is_closed_polar NormedSpace.isClosed_polar
@[simp]
theorem polar_closure (s : Set E) : polar ๐ (closure s) = polar ๐ s :=
((dualPairing ๐ E).flip.polar_antitone subset_closure).antisymm <|
(dualPairing ๐ E).flip.polar_gc.l_le <|
closure_minimal ((dualPairing ๐ E).flip.polar_gc.le_u_l s) <| by
simpa [LinearMap.flip_flip] using
(isClosed_polar _ _).preimage (inclusionInDoubleDual ๐ E).continuous
#align normed_space.polar_closure NormedSpace.polar_closure
variable {๐}
/-- If `x'` is a dual element such that the norms `โx' zโ` are bounded for `z โ s`, then a
small scalar multiple of `x'` is in `polar ๐ s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual ๐ E} {c : ๐} (hc : โ z, z โ s โ โx' zโ โค โcโ) :
cโปยน โข x' โ polar ๐ s := by
by_cases c_zero : c = 0
ยท simp only [c_zero, inv_zero, zero_smul]
exact (dualPairing ๐ E).flip.zero_mem_polar _
have eq : โ z, โcโปยน โข x' zโ = โcโปยนโ * โx' zโ := fun z => norm_smul cโปยน _
have le : โ z, z โ s โ โcโปยน โข x' zโ โค โcโปยนโ * โcโ := by
intro z hzs
rw [eq z]
apply mul_le_mul (le_of_eq rfl) (hc z hzs) (norm_nonneg _) (norm_nonneg _)
|
have cancel : โcโปยนโ * โcโ = 1 := by
simp only [c_zero, norm_eq_zero, Ne.def, not_false_iff, inv_mul_cancel, norm_inv]
|
/-- If `x'` is a dual element such that the norms `โx' zโ` are bounded for `z โ s`, then a
small scalar multiple of `x'` is in `polar ๐ s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual ๐ E} {c : ๐} (hc : โ z, z โ s โ โx' zโ โค โcโ) :
cโปยน โข x' โ polar ๐ s := by
by_cases c_zero : c = 0
ยท simp only [c_zero, inv_zero, zero_smul]
exact (dualPairing ๐ E).flip.zero_mem_polar _
have eq : โ z, โcโปยน โข x' zโ = โcโปยนโ * โx' zโ := fun z => norm_smul cโปยน _
have le : โ z, z โ s โ โcโปยน โข x' zโ โค โcโปยนโ * โcโ := by
intro z hzs
rw [eq z]
apply mul_le_mul (le_of_eq rfl) (hc z hzs) (norm_nonneg _) (norm_nonneg _)
|
Mathlib.Analysis.NormedSpace.Dual.201_0.WirVfj6f5oiZZ2w
|
/-- If `x'` is a dual element such that the norms `โx' zโ` are bounded for `z โ s`, then a
small scalar multiple of `x'` is in `polar ๐ s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual ๐ E} {c : ๐} (hc : โ z, z โ s โ โx' zโ โค โcโ) :
cโปยน โข x' โ polar ๐ s
|
Mathlib_Analysis_NormedSpace_Dual
|
๐ : Type u_1
instโยฒ : NontriviallyNormedField ๐
E : Type u_2
instโยน : SeminormedAddCommGroup E
instโ : NormedSpace ๐ E
s : Set E
x' : Dual ๐ E
c : ๐
hc : โ z โ s, โx' zโ โค โcโ
c_zero : ยฌc = 0
eq : โ (z : E), โcโปยน โข x' zโ = โcโปยนโ * โx' zโ
le : โ z โ s, โcโปยน โข x' zโ โค โcโปยนโ * โcโ
โข โcโปยนโ * โcโ = 1
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# The topological dual of a normed space
In this file we define the topological dual `NormedSpace.Dual` of a normed space, and the
continuous linear map `NormedSpace.inclusionInDoubleDual` from a normed space into its double
dual.
For base field `๐ = โ` or `๐ = โ`, this map is actually an isometric embedding; we provide a
version `NormedSpace.inclusionInDoubleDualLi` of the map which is of type a bundled linear
isometric embedding, `E โโแตข[๐] (Dual ๐ (Dual ๐ E))`.
Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
theory for `SeminormedAddCommGroup` and we specialize to `NormedAddCommGroup` when needed.
## Main definitions
* `inclusionInDoubleDual` and `inclusionInDoubleDualLi` are the inclusion of a normed space
in its double dual, considered as a bounded linear map and as a linear isometry, respectively.
* `polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals `x'` for which
`โx' zโ โค 1` for every `z โ s`.
## Tags
dual
-/
noncomputable section
open Classical Topology Bornology
universe u v
namespace NormedSpace
section General
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable (E : Type*) [SeminormedAddCommGroup E] [NormedSpace ๐ E]
variable (F : Type*) [NormedAddCommGroup F] [NormedSpace ๐ F]
/-- The topological dual of a seminormed space `E`. -/
abbrev Dual : Type _ := E โL[๐] ๐
#align normed_space.dual NormedSpace.Dual
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_eq until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : NormedSpace ๐ (Dual ๐ E) := inferInstance
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_le until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : SeminormedAddCommGroup (Dual ๐ E) := inferInstance
/-- The inclusion of a normed space in its double (topological) dual, considered
as a bounded linear map. -/
def inclusionInDoubleDual : E โL[๐] Dual ๐ (Dual ๐ E) :=
ContinuousLinearMap.apply ๐ ๐
#align normed_space.inclusion_in_double_dual NormedSpace.inclusionInDoubleDual
@[simp]
theorem dual_def (x : E) (f : Dual ๐ E) : inclusionInDoubleDual ๐ E x f = f x :=
rfl
#align normed_space.dual_def NormedSpace.dual_def
theorem inclusionInDoubleDual_norm_eq :
โinclusionInDoubleDual ๐ Eโ = โContinuousLinearMap.id ๐ (Dual ๐ E)โ :=
ContinuousLinearMap.op_norm_flip _
#align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1 := by
rw [inclusionInDoubleDual_norm_eq]
exact ContinuousLinearMap.norm_id_le
#align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
theorem double_dual_bound (x : E) : โ(inclusionInDoubleDual ๐ E) xโ โค โxโ := by
simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusionInDoubleDual_norm_le ๐ E) x
#align normed_space.double_dual_bound NormedSpace.double_dual_bound
/-- The dual pairing as a bilinear form. -/
def dualPairing : Dual ๐ E โโ[๐] E โโ[๐] ๐ :=
ContinuousLinearMap.coeLM ๐
#align normed_space.dual_pairing NormedSpace.dualPairing
@[simp]
theorem dualPairing_apply {v : Dual ๐ E} {x : E} : dualPairing ๐ E v x = v x :=
rfl
#align normed_space.dual_pairing_apply NormedSpace.dualPairing_apply
theorem dualPairing_separatingLeft : (dualPairing ๐ E).SeparatingLeft := by
rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]
exact ContinuousLinearMap.coe_injective
#align normed_space.dual_pairing_separating_left NormedSpace.dualPairing_separatingLeft
end General
section BidualIsometry
variable (๐ : Type v) [IsROrC ๐] {E : Type u} [NormedAddCommGroup E] [NormedSpace ๐ E]
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M := by
classical
by_cases h : x = 0
ยท simp only [h, hMp, norm_zero]
ยท obtain โจf, hfโ, hfxโฉ : โ f : E โL[๐] ๐, โfโ = 1 โง f x = โxโ := exists_dual_vector ๐ x h
calc
โxโ = โ(โxโ : ๐)โ := IsROrC.norm_coe_norm.symm
_ = โf xโ := by rw [hfx]
_ โค M * โfโ := (hM f)
_ = M := by rw [hfโ, mul_one]
#align normed_space.norm_le_dual_bound NormedSpace.norm_le_dual_bound
theorem eq_zero_of_forall_dual_eq_zero {x : E} (h : โ f : Dual ๐ E, f x = (0 : ๐)) : x = 0 :=
norm_le_zero_iff.mp (norm_le_dual_bound ๐ x le_rfl fun f => by simp [h f])
#align normed_space.eq_zero_of_forall_dual_eq_zero NormedSpace.eq_zero_of_forall_dual_eq_zero
theorem eq_zero_iff_forall_dual_eq_zero (x : E) : x = 0 โ โ g : Dual ๐ E, g x = 0 :=
โจfun hx => by simp [hx], fun h => eq_zero_of_forall_dual_eq_zero ๐ hโฉ
#align normed_space.eq_zero_iff_forall_dual_eq_zero NormedSpace.eq_zero_iff_forall_dual_eq_zero
/-- See also `geometric_hahn_banach_point_point`. -/
theorem eq_iff_forall_dual_eq {x y : E} : x = y โ โ g : Dual ๐ E, g x = g y := by
rw [โ sub_eq_zero, eq_zero_iff_forall_dual_eq_zero ๐ (x - y)]
simp [sub_eq_zero]
#align normed_space.eq_iff_forall_dual_eq NormedSpace.eq_iff_forall_dual_eq
/-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E โโแตข[๐] Dual ๐ (Dual ๐ E) :=
{ inclusionInDoubleDual ๐ E with
norm_map' := by
intro x
apply le_antisymm
ยท exact double_dual_bound ๐ E x
rw [ContinuousLinearMap.norm_def]
refine' le_csInf ContinuousLinearMap.bounds_nonempty _
rintro c โจhc1, hc2โฉ
exact norm_le_dual_bound ๐ x hc1 hc2 }
#align normed_space.inclusion_in_double_dual_li NormedSpace.inclusionInDoubleDualLi
end BidualIsometry
section PolarSets
open Metric Set NormedSpace
/-- Given a subset `s` in a normed space `E` (over a field `๐`), the polar
`polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals which
evaluate to something of norm at most one at all points `z โ s`. -/
def polar (๐ : Type*) [NontriviallyNormedField ๐] {E : Type*} [SeminormedAddCommGroup E]
[NormedSpace ๐ E] : Set E โ Set (Dual ๐ E) :=
(dualPairing ๐ E).flip.polar
#align normed_space.polar NormedSpace.polar
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable {E : Type*} [SeminormedAddCommGroup E] [NormedSpace ๐ E]
theorem mem_polar_iff {x' : Dual ๐ E} (s : Set E) : x' โ polar ๐ s โ โ z โ s, โx' zโ โค 1 :=
Iff.rfl
#align normed_space.mem_polar_iff NormedSpace.mem_polar_iff
@[simp]
theorem polar_univ : polar ๐ (univ : Set E) = {(0 : Dual ๐ E)} :=
(dualPairing ๐ E).flip.polar_univ
(LinearMap.flip_separatingRight.mpr (dualPairing_separatingLeft ๐ E))
#align normed_space.polar_univ NormedSpace.polar_univ
theorem isClosed_polar (s : Set E) : IsClosed (polar ๐ s) := by
dsimp only [NormedSpace.polar]
simp only [LinearMap.polar_eq_iInter, LinearMap.flip_apply]
refine' isClosed_biInter fun z _ => _
exact isClosed_Iic.preimage (ContinuousLinearMap.apply ๐ ๐ z).continuous.norm
#align normed_space.is_closed_polar NormedSpace.isClosed_polar
@[simp]
theorem polar_closure (s : Set E) : polar ๐ (closure s) = polar ๐ s :=
((dualPairing ๐ E).flip.polar_antitone subset_closure).antisymm <|
(dualPairing ๐ E).flip.polar_gc.l_le <|
closure_minimal ((dualPairing ๐ E).flip.polar_gc.le_u_l s) <| by
simpa [LinearMap.flip_flip] using
(isClosed_polar _ _).preimage (inclusionInDoubleDual ๐ E).continuous
#align normed_space.polar_closure NormedSpace.polar_closure
variable {๐}
/-- If `x'` is a dual element such that the norms `โx' zโ` are bounded for `z โ s`, then a
small scalar multiple of `x'` is in `polar ๐ s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual ๐ E} {c : ๐} (hc : โ z, z โ s โ โx' zโ โค โcโ) :
cโปยน โข x' โ polar ๐ s := by
by_cases c_zero : c = 0
ยท simp only [c_zero, inv_zero, zero_smul]
exact (dualPairing ๐ E).flip.zero_mem_polar _
have eq : โ z, โcโปยน โข x' zโ = โcโปยนโ * โx' zโ := fun z => norm_smul cโปยน _
have le : โ z, z โ s โ โcโปยน โข x' zโ โค โcโปยนโ * โcโ := by
intro z hzs
rw [eq z]
apply mul_le_mul (le_of_eq rfl) (hc z hzs) (norm_nonneg _) (norm_nonneg _)
have cancel : โcโปยนโ * โcโ = 1 := by
|
simp only [c_zero, norm_eq_zero, Ne.def, not_false_iff, inv_mul_cancel, norm_inv]
|
/-- If `x'` is a dual element such that the norms `โx' zโ` are bounded for `z โ s`, then a
small scalar multiple of `x'` is in `polar ๐ s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual ๐ E} {c : ๐} (hc : โ z, z โ s โ โx' zโ โค โcโ) :
cโปยน โข x' โ polar ๐ s := by
by_cases c_zero : c = 0
ยท simp only [c_zero, inv_zero, zero_smul]
exact (dualPairing ๐ E).flip.zero_mem_polar _
have eq : โ z, โcโปยน โข x' zโ = โcโปยนโ * โx' zโ := fun z => norm_smul cโปยน _
have le : โ z, z โ s โ โcโปยน โข x' zโ โค โcโปยนโ * โcโ := by
intro z hzs
rw [eq z]
apply mul_le_mul (le_of_eq rfl) (hc z hzs) (norm_nonneg _) (norm_nonneg _)
have cancel : โcโปยนโ * โcโ = 1 := by
|
Mathlib.Analysis.NormedSpace.Dual.201_0.WirVfj6f5oiZZ2w
|
/-- If `x'` is a dual element such that the norms `โx' zโ` are bounded for `z โ s`, then a
small scalar multiple of `x'` is in `polar ๐ s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual ๐ E} {c : ๐} (hc : โ z, z โ s โ โx' zโ โค โcโ) :
cโปยน โข x' โ polar ๐ s
|
Mathlib_Analysis_NormedSpace_Dual
|
case neg
๐ : Type u_1
instโยฒ : NontriviallyNormedField ๐
E : Type u_2
instโยน : SeminormedAddCommGroup E
instโ : NormedSpace ๐ E
s : Set E
x' : Dual ๐ E
c : ๐
hc : โ z โ s, โx' zโ โค โcโ
c_zero : ยฌc = 0
eq : โ (z : E), โcโปยน โข x' zโ = โcโปยนโ * โx' zโ
le : โ z โ s, โcโปยน โข x' zโ โค โcโปยนโ * โcโ
cancel : โcโปยนโ * โcโ = 1
โข cโปยน โข x' โ polar ๐ s
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# The topological dual of a normed space
In this file we define the topological dual `NormedSpace.Dual` of a normed space, and the
continuous linear map `NormedSpace.inclusionInDoubleDual` from a normed space into its double
dual.
For base field `๐ = โ` or `๐ = โ`, this map is actually an isometric embedding; we provide a
version `NormedSpace.inclusionInDoubleDualLi` of the map which is of type a bundled linear
isometric embedding, `E โโแตข[๐] (Dual ๐ (Dual ๐ E))`.
Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
theory for `SeminormedAddCommGroup` and we specialize to `NormedAddCommGroup` when needed.
## Main definitions
* `inclusionInDoubleDual` and `inclusionInDoubleDualLi` are the inclusion of a normed space
in its double dual, considered as a bounded linear map and as a linear isometry, respectively.
* `polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals `x'` for which
`โx' zโ โค 1` for every `z โ s`.
## Tags
dual
-/
noncomputable section
open Classical Topology Bornology
universe u v
namespace NormedSpace
section General
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable (E : Type*) [SeminormedAddCommGroup E] [NormedSpace ๐ E]
variable (F : Type*) [NormedAddCommGroup F] [NormedSpace ๐ F]
/-- The topological dual of a seminormed space `E`. -/
abbrev Dual : Type _ := E โL[๐] ๐
#align normed_space.dual NormedSpace.Dual
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_eq until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : NormedSpace ๐ (Dual ๐ E) := inferInstance
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_le until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : SeminormedAddCommGroup (Dual ๐ E) := inferInstance
/-- The inclusion of a normed space in its double (topological) dual, considered
as a bounded linear map. -/
def inclusionInDoubleDual : E โL[๐] Dual ๐ (Dual ๐ E) :=
ContinuousLinearMap.apply ๐ ๐
#align normed_space.inclusion_in_double_dual NormedSpace.inclusionInDoubleDual
@[simp]
theorem dual_def (x : E) (f : Dual ๐ E) : inclusionInDoubleDual ๐ E x f = f x :=
rfl
#align normed_space.dual_def NormedSpace.dual_def
theorem inclusionInDoubleDual_norm_eq :
โinclusionInDoubleDual ๐ Eโ = โContinuousLinearMap.id ๐ (Dual ๐ E)โ :=
ContinuousLinearMap.op_norm_flip _
#align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1 := by
rw [inclusionInDoubleDual_norm_eq]
exact ContinuousLinearMap.norm_id_le
#align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
theorem double_dual_bound (x : E) : โ(inclusionInDoubleDual ๐ E) xโ โค โxโ := by
simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusionInDoubleDual_norm_le ๐ E) x
#align normed_space.double_dual_bound NormedSpace.double_dual_bound
/-- The dual pairing as a bilinear form. -/
def dualPairing : Dual ๐ E โโ[๐] E โโ[๐] ๐ :=
ContinuousLinearMap.coeLM ๐
#align normed_space.dual_pairing NormedSpace.dualPairing
@[simp]
theorem dualPairing_apply {v : Dual ๐ E} {x : E} : dualPairing ๐ E v x = v x :=
rfl
#align normed_space.dual_pairing_apply NormedSpace.dualPairing_apply
theorem dualPairing_separatingLeft : (dualPairing ๐ E).SeparatingLeft := by
rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]
exact ContinuousLinearMap.coe_injective
#align normed_space.dual_pairing_separating_left NormedSpace.dualPairing_separatingLeft
end General
section BidualIsometry
variable (๐ : Type v) [IsROrC ๐] {E : Type u} [NormedAddCommGroup E] [NormedSpace ๐ E]
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M := by
classical
by_cases h : x = 0
ยท simp only [h, hMp, norm_zero]
ยท obtain โจf, hfโ, hfxโฉ : โ f : E โL[๐] ๐, โfโ = 1 โง f x = โxโ := exists_dual_vector ๐ x h
calc
โxโ = โ(โxโ : ๐)โ := IsROrC.norm_coe_norm.symm
_ = โf xโ := by rw [hfx]
_ โค M * โfโ := (hM f)
_ = M := by rw [hfโ, mul_one]
#align normed_space.norm_le_dual_bound NormedSpace.norm_le_dual_bound
theorem eq_zero_of_forall_dual_eq_zero {x : E} (h : โ f : Dual ๐ E, f x = (0 : ๐)) : x = 0 :=
norm_le_zero_iff.mp (norm_le_dual_bound ๐ x le_rfl fun f => by simp [h f])
#align normed_space.eq_zero_of_forall_dual_eq_zero NormedSpace.eq_zero_of_forall_dual_eq_zero
theorem eq_zero_iff_forall_dual_eq_zero (x : E) : x = 0 โ โ g : Dual ๐ E, g x = 0 :=
โจfun hx => by simp [hx], fun h => eq_zero_of_forall_dual_eq_zero ๐ hโฉ
#align normed_space.eq_zero_iff_forall_dual_eq_zero NormedSpace.eq_zero_iff_forall_dual_eq_zero
/-- See also `geometric_hahn_banach_point_point`. -/
theorem eq_iff_forall_dual_eq {x y : E} : x = y โ โ g : Dual ๐ E, g x = g y := by
rw [โ sub_eq_zero, eq_zero_iff_forall_dual_eq_zero ๐ (x - y)]
simp [sub_eq_zero]
#align normed_space.eq_iff_forall_dual_eq NormedSpace.eq_iff_forall_dual_eq
/-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E โโแตข[๐] Dual ๐ (Dual ๐ E) :=
{ inclusionInDoubleDual ๐ E with
norm_map' := by
intro x
apply le_antisymm
ยท exact double_dual_bound ๐ E x
rw [ContinuousLinearMap.norm_def]
refine' le_csInf ContinuousLinearMap.bounds_nonempty _
rintro c โจhc1, hc2โฉ
exact norm_le_dual_bound ๐ x hc1 hc2 }
#align normed_space.inclusion_in_double_dual_li NormedSpace.inclusionInDoubleDualLi
end BidualIsometry
section PolarSets
open Metric Set NormedSpace
/-- Given a subset `s` in a normed space `E` (over a field `๐`), the polar
`polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals which
evaluate to something of norm at most one at all points `z โ s`. -/
def polar (๐ : Type*) [NontriviallyNormedField ๐] {E : Type*} [SeminormedAddCommGroup E]
[NormedSpace ๐ E] : Set E โ Set (Dual ๐ E) :=
(dualPairing ๐ E).flip.polar
#align normed_space.polar NormedSpace.polar
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable {E : Type*} [SeminormedAddCommGroup E] [NormedSpace ๐ E]
theorem mem_polar_iff {x' : Dual ๐ E} (s : Set E) : x' โ polar ๐ s โ โ z โ s, โx' zโ โค 1 :=
Iff.rfl
#align normed_space.mem_polar_iff NormedSpace.mem_polar_iff
@[simp]
theorem polar_univ : polar ๐ (univ : Set E) = {(0 : Dual ๐ E)} :=
(dualPairing ๐ E).flip.polar_univ
(LinearMap.flip_separatingRight.mpr (dualPairing_separatingLeft ๐ E))
#align normed_space.polar_univ NormedSpace.polar_univ
theorem isClosed_polar (s : Set E) : IsClosed (polar ๐ s) := by
dsimp only [NormedSpace.polar]
simp only [LinearMap.polar_eq_iInter, LinearMap.flip_apply]
refine' isClosed_biInter fun z _ => _
exact isClosed_Iic.preimage (ContinuousLinearMap.apply ๐ ๐ z).continuous.norm
#align normed_space.is_closed_polar NormedSpace.isClosed_polar
@[simp]
theorem polar_closure (s : Set E) : polar ๐ (closure s) = polar ๐ s :=
((dualPairing ๐ E).flip.polar_antitone subset_closure).antisymm <|
(dualPairing ๐ E).flip.polar_gc.l_le <|
closure_minimal ((dualPairing ๐ E).flip.polar_gc.le_u_l s) <| by
simpa [LinearMap.flip_flip] using
(isClosed_polar _ _).preimage (inclusionInDoubleDual ๐ E).continuous
#align normed_space.polar_closure NormedSpace.polar_closure
variable {๐}
/-- If `x'` is a dual element such that the norms `โx' zโ` are bounded for `z โ s`, then a
small scalar multiple of `x'` is in `polar ๐ s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual ๐ E} {c : ๐} (hc : โ z, z โ s โ โx' zโ โค โcโ) :
cโปยน โข x' โ polar ๐ s := by
by_cases c_zero : c = 0
ยท simp only [c_zero, inv_zero, zero_smul]
exact (dualPairing ๐ E).flip.zero_mem_polar _
have eq : โ z, โcโปยน โข x' zโ = โcโปยนโ * โx' zโ := fun z => norm_smul cโปยน _
have le : โ z, z โ s โ โcโปยน โข x' zโ โค โcโปยนโ * โcโ := by
intro z hzs
rw [eq z]
apply mul_le_mul (le_of_eq rfl) (hc z hzs) (norm_nonneg _) (norm_nonneg _)
have cancel : โcโปยนโ * โcโ = 1 := by
simp only [c_zero, norm_eq_zero, Ne.def, not_false_iff, inv_mul_cancel, norm_inv]
|
rwa [cancel] at le
|
/-- If `x'` is a dual element such that the norms `โx' zโ` are bounded for `z โ s`, then a
small scalar multiple of `x'` is in `polar ๐ s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual ๐ E} {c : ๐} (hc : โ z, z โ s โ โx' zโ โค โcโ) :
cโปยน โข x' โ polar ๐ s := by
by_cases c_zero : c = 0
ยท simp only [c_zero, inv_zero, zero_smul]
exact (dualPairing ๐ E).flip.zero_mem_polar _
have eq : โ z, โcโปยน โข x' zโ = โcโปยนโ * โx' zโ := fun z => norm_smul cโปยน _
have le : โ z, z โ s โ โcโปยน โข x' zโ โค โcโปยนโ * โcโ := by
intro z hzs
rw [eq z]
apply mul_le_mul (le_of_eq rfl) (hc z hzs) (norm_nonneg _) (norm_nonneg _)
have cancel : โcโปยนโ * โcโ = 1 := by
simp only [c_zero, norm_eq_zero, Ne.def, not_false_iff, inv_mul_cancel, norm_inv]
|
Mathlib.Analysis.NormedSpace.Dual.201_0.WirVfj6f5oiZZ2w
|
/-- If `x'` is a dual element such that the norms `โx' zโ` are bounded for `z โ s`, then a
small scalar multiple of `x'` is in `polar ๐ s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual ๐ E} {c : ๐} (hc : โ z, z โ s โ โx' zโ โค โcโ) :
cโปยน โข x' โ polar ๐ s
|
Mathlib_Analysis_NormedSpace_Dual
|
๐ : Type u_1
instโยฒ : NontriviallyNormedField ๐
E : Type u_2
instโยน : SeminormedAddCommGroup E
instโ : NormedSpace ๐ E
c : ๐
hc : 1 < โcโ
r : โ
hr : 0 < r
โข polar ๐ (ball 0 r) โ closedBall 0 (โcโ / r)
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# The topological dual of a normed space
In this file we define the topological dual `NormedSpace.Dual` of a normed space, and the
continuous linear map `NormedSpace.inclusionInDoubleDual` from a normed space into its double
dual.
For base field `๐ = โ` or `๐ = โ`, this map is actually an isometric embedding; we provide a
version `NormedSpace.inclusionInDoubleDualLi` of the map which is of type a bundled linear
isometric embedding, `E โโแตข[๐] (Dual ๐ (Dual ๐ E))`.
Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
theory for `SeminormedAddCommGroup` and we specialize to `NormedAddCommGroup` when needed.
## Main definitions
* `inclusionInDoubleDual` and `inclusionInDoubleDualLi` are the inclusion of a normed space
in its double dual, considered as a bounded linear map and as a linear isometry, respectively.
* `polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals `x'` for which
`โx' zโ โค 1` for every `z โ s`.
## Tags
dual
-/
noncomputable section
open Classical Topology Bornology
universe u v
namespace NormedSpace
section General
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable (E : Type*) [SeminormedAddCommGroup E] [NormedSpace ๐ E]
variable (F : Type*) [NormedAddCommGroup F] [NormedSpace ๐ F]
/-- The topological dual of a seminormed space `E`. -/
abbrev Dual : Type _ := E โL[๐] ๐
#align normed_space.dual NormedSpace.Dual
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_eq until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : NormedSpace ๐ (Dual ๐ E) := inferInstance
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_le until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : SeminormedAddCommGroup (Dual ๐ E) := inferInstance
/-- The inclusion of a normed space in its double (topological) dual, considered
as a bounded linear map. -/
def inclusionInDoubleDual : E โL[๐] Dual ๐ (Dual ๐ E) :=
ContinuousLinearMap.apply ๐ ๐
#align normed_space.inclusion_in_double_dual NormedSpace.inclusionInDoubleDual
@[simp]
theorem dual_def (x : E) (f : Dual ๐ E) : inclusionInDoubleDual ๐ E x f = f x :=
rfl
#align normed_space.dual_def NormedSpace.dual_def
theorem inclusionInDoubleDual_norm_eq :
โinclusionInDoubleDual ๐ Eโ = โContinuousLinearMap.id ๐ (Dual ๐ E)โ :=
ContinuousLinearMap.op_norm_flip _
#align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1 := by
rw [inclusionInDoubleDual_norm_eq]
exact ContinuousLinearMap.norm_id_le
#align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
theorem double_dual_bound (x : E) : โ(inclusionInDoubleDual ๐ E) xโ โค โxโ := by
simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusionInDoubleDual_norm_le ๐ E) x
#align normed_space.double_dual_bound NormedSpace.double_dual_bound
/-- The dual pairing as a bilinear form. -/
def dualPairing : Dual ๐ E โโ[๐] E โโ[๐] ๐ :=
ContinuousLinearMap.coeLM ๐
#align normed_space.dual_pairing NormedSpace.dualPairing
@[simp]
theorem dualPairing_apply {v : Dual ๐ E} {x : E} : dualPairing ๐ E v x = v x :=
rfl
#align normed_space.dual_pairing_apply NormedSpace.dualPairing_apply
theorem dualPairing_separatingLeft : (dualPairing ๐ E).SeparatingLeft := by
rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]
exact ContinuousLinearMap.coe_injective
#align normed_space.dual_pairing_separating_left NormedSpace.dualPairing_separatingLeft
end General
section BidualIsometry
variable (๐ : Type v) [IsROrC ๐] {E : Type u} [NormedAddCommGroup E] [NormedSpace ๐ E]
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M := by
classical
by_cases h : x = 0
ยท simp only [h, hMp, norm_zero]
ยท obtain โจf, hfโ, hfxโฉ : โ f : E โL[๐] ๐, โfโ = 1 โง f x = โxโ := exists_dual_vector ๐ x h
calc
โxโ = โ(โxโ : ๐)โ := IsROrC.norm_coe_norm.symm
_ = โf xโ := by rw [hfx]
_ โค M * โfโ := (hM f)
_ = M := by rw [hfโ, mul_one]
#align normed_space.norm_le_dual_bound NormedSpace.norm_le_dual_bound
theorem eq_zero_of_forall_dual_eq_zero {x : E} (h : โ f : Dual ๐ E, f x = (0 : ๐)) : x = 0 :=
norm_le_zero_iff.mp (norm_le_dual_bound ๐ x le_rfl fun f => by simp [h f])
#align normed_space.eq_zero_of_forall_dual_eq_zero NormedSpace.eq_zero_of_forall_dual_eq_zero
theorem eq_zero_iff_forall_dual_eq_zero (x : E) : x = 0 โ โ g : Dual ๐ E, g x = 0 :=
โจfun hx => by simp [hx], fun h => eq_zero_of_forall_dual_eq_zero ๐ hโฉ
#align normed_space.eq_zero_iff_forall_dual_eq_zero NormedSpace.eq_zero_iff_forall_dual_eq_zero
/-- See also `geometric_hahn_banach_point_point`. -/
theorem eq_iff_forall_dual_eq {x y : E} : x = y โ โ g : Dual ๐ E, g x = g y := by
rw [โ sub_eq_zero, eq_zero_iff_forall_dual_eq_zero ๐ (x - y)]
simp [sub_eq_zero]
#align normed_space.eq_iff_forall_dual_eq NormedSpace.eq_iff_forall_dual_eq
/-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E โโแตข[๐] Dual ๐ (Dual ๐ E) :=
{ inclusionInDoubleDual ๐ E with
norm_map' := by
intro x
apply le_antisymm
ยท exact double_dual_bound ๐ E x
rw [ContinuousLinearMap.norm_def]
refine' le_csInf ContinuousLinearMap.bounds_nonempty _
rintro c โจhc1, hc2โฉ
exact norm_le_dual_bound ๐ x hc1 hc2 }
#align normed_space.inclusion_in_double_dual_li NormedSpace.inclusionInDoubleDualLi
end BidualIsometry
section PolarSets
open Metric Set NormedSpace
/-- Given a subset `s` in a normed space `E` (over a field `๐`), the polar
`polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals which
evaluate to something of norm at most one at all points `z โ s`. -/
def polar (๐ : Type*) [NontriviallyNormedField ๐] {E : Type*} [SeminormedAddCommGroup E]
[NormedSpace ๐ E] : Set E โ Set (Dual ๐ E) :=
(dualPairing ๐ E).flip.polar
#align normed_space.polar NormedSpace.polar
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable {E : Type*} [SeminormedAddCommGroup E] [NormedSpace ๐ E]
theorem mem_polar_iff {x' : Dual ๐ E} (s : Set E) : x' โ polar ๐ s โ โ z โ s, โx' zโ โค 1 :=
Iff.rfl
#align normed_space.mem_polar_iff NormedSpace.mem_polar_iff
@[simp]
theorem polar_univ : polar ๐ (univ : Set E) = {(0 : Dual ๐ E)} :=
(dualPairing ๐ E).flip.polar_univ
(LinearMap.flip_separatingRight.mpr (dualPairing_separatingLeft ๐ E))
#align normed_space.polar_univ NormedSpace.polar_univ
theorem isClosed_polar (s : Set E) : IsClosed (polar ๐ s) := by
dsimp only [NormedSpace.polar]
simp only [LinearMap.polar_eq_iInter, LinearMap.flip_apply]
refine' isClosed_biInter fun z _ => _
exact isClosed_Iic.preimage (ContinuousLinearMap.apply ๐ ๐ z).continuous.norm
#align normed_space.is_closed_polar NormedSpace.isClosed_polar
@[simp]
theorem polar_closure (s : Set E) : polar ๐ (closure s) = polar ๐ s :=
((dualPairing ๐ E).flip.polar_antitone subset_closure).antisymm <|
(dualPairing ๐ E).flip.polar_gc.l_le <|
closure_minimal ((dualPairing ๐ E).flip.polar_gc.le_u_l s) <| by
simpa [LinearMap.flip_flip] using
(isClosed_polar _ _).preimage (inclusionInDoubleDual ๐ E).continuous
#align normed_space.polar_closure NormedSpace.polar_closure
variable {๐}
/-- If `x'` is a dual element such that the norms `โx' zโ` are bounded for `z โ s`, then a
small scalar multiple of `x'` is in `polar ๐ s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual ๐ E} {c : ๐} (hc : โ z, z โ s โ โx' zโ โค โcโ) :
cโปยน โข x' โ polar ๐ s := by
by_cases c_zero : c = 0
ยท simp only [c_zero, inv_zero, zero_smul]
exact (dualPairing ๐ E).flip.zero_mem_polar _
have eq : โ z, โcโปยน โข x' zโ = โcโปยนโ * โx' zโ := fun z => norm_smul cโปยน _
have le : โ z, z โ s โ โcโปยน โข x' zโ โค โcโปยนโ * โcโ := by
intro z hzs
rw [eq z]
apply mul_le_mul (le_of_eq rfl) (hc z hzs) (norm_nonneg _) (norm_nonneg _)
have cancel : โcโปยนโ * โcโ = 1 := by
simp only [c_zero, norm_eq_zero, Ne.def, not_false_iff, inv_mul_cancel, norm_inv]
rwa [cancel] at le
#align normed_space.smul_mem_polar NormedSpace.smul_mem_polar
theorem polar_ball_subset_closedBall_div {c : ๐} (hc : 1 < โcโ) {r : โ} (hr : 0 < r) :
polar ๐ (ball (0 : E) r) โ closedBall (0 : Dual ๐ E) (โcโ / r) := by
|
intro x' hx'
|
theorem polar_ball_subset_closedBall_div {c : ๐} (hc : 1 < โcโ) {r : โ} (hr : 0 < r) :
polar ๐ (ball (0 : E) r) โ closedBall (0 : Dual ๐ E) (โcโ / r) := by
|
Mathlib.Analysis.NormedSpace.Dual.218_0.WirVfj6f5oiZZ2w
|
theorem polar_ball_subset_closedBall_div {c : ๐} (hc : 1 < โcโ) {r : โ} (hr : 0 < r) :
polar ๐ (ball (0 : E) r) โ closedBall (0 : Dual ๐ E) (โcโ / r)
|
Mathlib_Analysis_NormedSpace_Dual
|
๐ : Type u_1
instโยฒ : NontriviallyNormedField ๐
E : Type u_2
instโยน : SeminormedAddCommGroup E
instโ : NormedSpace ๐ E
c : ๐
hc : 1 < โcโ
r : โ
hr : 0 < r
x' : Dual ๐ E
hx' : x' โ polar ๐ (ball 0 r)
โข x' โ closedBall 0 (โcโ / r)
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# The topological dual of a normed space
In this file we define the topological dual `NormedSpace.Dual` of a normed space, and the
continuous linear map `NormedSpace.inclusionInDoubleDual` from a normed space into its double
dual.
For base field `๐ = โ` or `๐ = โ`, this map is actually an isometric embedding; we provide a
version `NormedSpace.inclusionInDoubleDualLi` of the map which is of type a bundled linear
isometric embedding, `E โโแตข[๐] (Dual ๐ (Dual ๐ E))`.
Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
theory for `SeminormedAddCommGroup` and we specialize to `NormedAddCommGroup` when needed.
## Main definitions
* `inclusionInDoubleDual` and `inclusionInDoubleDualLi` are the inclusion of a normed space
in its double dual, considered as a bounded linear map and as a linear isometry, respectively.
* `polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals `x'` for which
`โx' zโ โค 1` for every `z โ s`.
## Tags
dual
-/
noncomputable section
open Classical Topology Bornology
universe u v
namespace NormedSpace
section General
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable (E : Type*) [SeminormedAddCommGroup E] [NormedSpace ๐ E]
variable (F : Type*) [NormedAddCommGroup F] [NormedSpace ๐ F]
/-- The topological dual of a seminormed space `E`. -/
abbrev Dual : Type _ := E โL[๐] ๐
#align normed_space.dual NormedSpace.Dual
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_eq until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : NormedSpace ๐ (Dual ๐ E) := inferInstance
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_le until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : SeminormedAddCommGroup (Dual ๐ E) := inferInstance
/-- The inclusion of a normed space in its double (topological) dual, considered
as a bounded linear map. -/
def inclusionInDoubleDual : E โL[๐] Dual ๐ (Dual ๐ E) :=
ContinuousLinearMap.apply ๐ ๐
#align normed_space.inclusion_in_double_dual NormedSpace.inclusionInDoubleDual
@[simp]
theorem dual_def (x : E) (f : Dual ๐ E) : inclusionInDoubleDual ๐ E x f = f x :=
rfl
#align normed_space.dual_def NormedSpace.dual_def
theorem inclusionInDoubleDual_norm_eq :
โinclusionInDoubleDual ๐ Eโ = โContinuousLinearMap.id ๐ (Dual ๐ E)โ :=
ContinuousLinearMap.op_norm_flip _
#align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1 := by
rw [inclusionInDoubleDual_norm_eq]
exact ContinuousLinearMap.norm_id_le
#align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
theorem double_dual_bound (x : E) : โ(inclusionInDoubleDual ๐ E) xโ โค โxโ := by
simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusionInDoubleDual_norm_le ๐ E) x
#align normed_space.double_dual_bound NormedSpace.double_dual_bound
/-- The dual pairing as a bilinear form. -/
def dualPairing : Dual ๐ E โโ[๐] E โโ[๐] ๐ :=
ContinuousLinearMap.coeLM ๐
#align normed_space.dual_pairing NormedSpace.dualPairing
@[simp]
theorem dualPairing_apply {v : Dual ๐ E} {x : E} : dualPairing ๐ E v x = v x :=
rfl
#align normed_space.dual_pairing_apply NormedSpace.dualPairing_apply
theorem dualPairing_separatingLeft : (dualPairing ๐ E).SeparatingLeft := by
rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]
exact ContinuousLinearMap.coe_injective
#align normed_space.dual_pairing_separating_left NormedSpace.dualPairing_separatingLeft
end General
section BidualIsometry
variable (๐ : Type v) [IsROrC ๐] {E : Type u} [NormedAddCommGroup E] [NormedSpace ๐ E]
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M := by
classical
by_cases h : x = 0
ยท simp only [h, hMp, norm_zero]
ยท obtain โจf, hfโ, hfxโฉ : โ f : E โL[๐] ๐, โfโ = 1 โง f x = โxโ := exists_dual_vector ๐ x h
calc
โxโ = โ(โxโ : ๐)โ := IsROrC.norm_coe_norm.symm
_ = โf xโ := by rw [hfx]
_ โค M * โfโ := (hM f)
_ = M := by rw [hfโ, mul_one]
#align normed_space.norm_le_dual_bound NormedSpace.norm_le_dual_bound
theorem eq_zero_of_forall_dual_eq_zero {x : E} (h : โ f : Dual ๐ E, f x = (0 : ๐)) : x = 0 :=
norm_le_zero_iff.mp (norm_le_dual_bound ๐ x le_rfl fun f => by simp [h f])
#align normed_space.eq_zero_of_forall_dual_eq_zero NormedSpace.eq_zero_of_forall_dual_eq_zero
theorem eq_zero_iff_forall_dual_eq_zero (x : E) : x = 0 โ โ g : Dual ๐ E, g x = 0 :=
โจfun hx => by simp [hx], fun h => eq_zero_of_forall_dual_eq_zero ๐ hโฉ
#align normed_space.eq_zero_iff_forall_dual_eq_zero NormedSpace.eq_zero_iff_forall_dual_eq_zero
/-- See also `geometric_hahn_banach_point_point`. -/
theorem eq_iff_forall_dual_eq {x y : E} : x = y โ โ g : Dual ๐ E, g x = g y := by
rw [โ sub_eq_zero, eq_zero_iff_forall_dual_eq_zero ๐ (x - y)]
simp [sub_eq_zero]
#align normed_space.eq_iff_forall_dual_eq NormedSpace.eq_iff_forall_dual_eq
/-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E โโแตข[๐] Dual ๐ (Dual ๐ E) :=
{ inclusionInDoubleDual ๐ E with
norm_map' := by
intro x
apply le_antisymm
ยท exact double_dual_bound ๐ E x
rw [ContinuousLinearMap.norm_def]
refine' le_csInf ContinuousLinearMap.bounds_nonempty _
rintro c โจhc1, hc2โฉ
exact norm_le_dual_bound ๐ x hc1 hc2 }
#align normed_space.inclusion_in_double_dual_li NormedSpace.inclusionInDoubleDualLi
end BidualIsometry
section PolarSets
open Metric Set NormedSpace
/-- Given a subset `s` in a normed space `E` (over a field `๐`), the polar
`polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals which
evaluate to something of norm at most one at all points `z โ s`. -/
def polar (๐ : Type*) [NontriviallyNormedField ๐] {E : Type*} [SeminormedAddCommGroup E]
[NormedSpace ๐ E] : Set E โ Set (Dual ๐ E) :=
(dualPairing ๐ E).flip.polar
#align normed_space.polar NormedSpace.polar
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable {E : Type*} [SeminormedAddCommGroup E] [NormedSpace ๐ E]
theorem mem_polar_iff {x' : Dual ๐ E} (s : Set E) : x' โ polar ๐ s โ โ z โ s, โx' zโ โค 1 :=
Iff.rfl
#align normed_space.mem_polar_iff NormedSpace.mem_polar_iff
@[simp]
theorem polar_univ : polar ๐ (univ : Set E) = {(0 : Dual ๐ E)} :=
(dualPairing ๐ E).flip.polar_univ
(LinearMap.flip_separatingRight.mpr (dualPairing_separatingLeft ๐ E))
#align normed_space.polar_univ NormedSpace.polar_univ
theorem isClosed_polar (s : Set E) : IsClosed (polar ๐ s) := by
dsimp only [NormedSpace.polar]
simp only [LinearMap.polar_eq_iInter, LinearMap.flip_apply]
refine' isClosed_biInter fun z _ => _
exact isClosed_Iic.preimage (ContinuousLinearMap.apply ๐ ๐ z).continuous.norm
#align normed_space.is_closed_polar NormedSpace.isClosed_polar
@[simp]
theorem polar_closure (s : Set E) : polar ๐ (closure s) = polar ๐ s :=
((dualPairing ๐ E).flip.polar_antitone subset_closure).antisymm <|
(dualPairing ๐ E).flip.polar_gc.l_le <|
closure_minimal ((dualPairing ๐ E).flip.polar_gc.le_u_l s) <| by
simpa [LinearMap.flip_flip] using
(isClosed_polar _ _).preimage (inclusionInDoubleDual ๐ E).continuous
#align normed_space.polar_closure NormedSpace.polar_closure
variable {๐}
/-- If `x'` is a dual element such that the norms `โx' zโ` are bounded for `z โ s`, then a
small scalar multiple of `x'` is in `polar ๐ s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual ๐ E} {c : ๐} (hc : โ z, z โ s โ โx' zโ โค โcโ) :
cโปยน โข x' โ polar ๐ s := by
by_cases c_zero : c = 0
ยท simp only [c_zero, inv_zero, zero_smul]
exact (dualPairing ๐ E).flip.zero_mem_polar _
have eq : โ z, โcโปยน โข x' zโ = โcโปยนโ * โx' zโ := fun z => norm_smul cโปยน _
have le : โ z, z โ s โ โcโปยน โข x' zโ โค โcโปยนโ * โcโ := by
intro z hzs
rw [eq z]
apply mul_le_mul (le_of_eq rfl) (hc z hzs) (norm_nonneg _) (norm_nonneg _)
have cancel : โcโปยนโ * โcโ = 1 := by
simp only [c_zero, norm_eq_zero, Ne.def, not_false_iff, inv_mul_cancel, norm_inv]
rwa [cancel] at le
#align normed_space.smul_mem_polar NormedSpace.smul_mem_polar
theorem polar_ball_subset_closedBall_div {c : ๐} (hc : 1 < โcโ) {r : โ} (hr : 0 < r) :
polar ๐ (ball (0 : E) r) โ closedBall (0 : Dual ๐ E) (โcโ / r) := by
intro x' hx'
|
rw [mem_polar_iff] at hx'
|
theorem polar_ball_subset_closedBall_div {c : ๐} (hc : 1 < โcโ) {r : โ} (hr : 0 < r) :
polar ๐ (ball (0 : E) r) โ closedBall (0 : Dual ๐ E) (โcโ / r) := by
intro x' hx'
|
Mathlib.Analysis.NormedSpace.Dual.218_0.WirVfj6f5oiZZ2w
|
theorem polar_ball_subset_closedBall_div {c : ๐} (hc : 1 < โcโ) {r : โ} (hr : 0 < r) :
polar ๐ (ball (0 : E) r) โ closedBall (0 : Dual ๐ E) (โcโ / r)
|
Mathlib_Analysis_NormedSpace_Dual
|
๐ : Type u_1
instโยฒ : NontriviallyNormedField ๐
E : Type u_2
instโยน : SeminormedAddCommGroup E
instโ : NormedSpace ๐ E
c : ๐
hc : 1 < โcโ
r : โ
hr : 0 < r
x' : Dual ๐ E
hx' : โ z โ ball 0 r, โx' zโ โค 1
โข x' โ closedBall 0 (โcโ / r)
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# The topological dual of a normed space
In this file we define the topological dual `NormedSpace.Dual` of a normed space, and the
continuous linear map `NormedSpace.inclusionInDoubleDual` from a normed space into its double
dual.
For base field `๐ = โ` or `๐ = โ`, this map is actually an isometric embedding; we provide a
version `NormedSpace.inclusionInDoubleDualLi` of the map which is of type a bundled linear
isometric embedding, `E โโแตข[๐] (Dual ๐ (Dual ๐ E))`.
Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
theory for `SeminormedAddCommGroup` and we specialize to `NormedAddCommGroup` when needed.
## Main definitions
* `inclusionInDoubleDual` and `inclusionInDoubleDualLi` are the inclusion of a normed space
in its double dual, considered as a bounded linear map and as a linear isometry, respectively.
* `polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals `x'` for which
`โx' zโ โค 1` for every `z โ s`.
## Tags
dual
-/
noncomputable section
open Classical Topology Bornology
universe u v
namespace NormedSpace
section General
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable (E : Type*) [SeminormedAddCommGroup E] [NormedSpace ๐ E]
variable (F : Type*) [NormedAddCommGroup F] [NormedSpace ๐ F]
/-- The topological dual of a seminormed space `E`. -/
abbrev Dual : Type _ := E โL[๐] ๐
#align normed_space.dual NormedSpace.Dual
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_eq until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : NormedSpace ๐ (Dual ๐ E) := inferInstance
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_le until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : SeminormedAddCommGroup (Dual ๐ E) := inferInstance
/-- The inclusion of a normed space in its double (topological) dual, considered
as a bounded linear map. -/
def inclusionInDoubleDual : E โL[๐] Dual ๐ (Dual ๐ E) :=
ContinuousLinearMap.apply ๐ ๐
#align normed_space.inclusion_in_double_dual NormedSpace.inclusionInDoubleDual
@[simp]
theorem dual_def (x : E) (f : Dual ๐ E) : inclusionInDoubleDual ๐ E x f = f x :=
rfl
#align normed_space.dual_def NormedSpace.dual_def
theorem inclusionInDoubleDual_norm_eq :
โinclusionInDoubleDual ๐ Eโ = โContinuousLinearMap.id ๐ (Dual ๐ E)โ :=
ContinuousLinearMap.op_norm_flip _
#align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1 := by
rw [inclusionInDoubleDual_norm_eq]
exact ContinuousLinearMap.norm_id_le
#align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
theorem double_dual_bound (x : E) : โ(inclusionInDoubleDual ๐ E) xโ โค โxโ := by
simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusionInDoubleDual_norm_le ๐ E) x
#align normed_space.double_dual_bound NormedSpace.double_dual_bound
/-- The dual pairing as a bilinear form. -/
def dualPairing : Dual ๐ E โโ[๐] E โโ[๐] ๐ :=
ContinuousLinearMap.coeLM ๐
#align normed_space.dual_pairing NormedSpace.dualPairing
@[simp]
theorem dualPairing_apply {v : Dual ๐ E} {x : E} : dualPairing ๐ E v x = v x :=
rfl
#align normed_space.dual_pairing_apply NormedSpace.dualPairing_apply
theorem dualPairing_separatingLeft : (dualPairing ๐ E).SeparatingLeft := by
rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]
exact ContinuousLinearMap.coe_injective
#align normed_space.dual_pairing_separating_left NormedSpace.dualPairing_separatingLeft
end General
section BidualIsometry
variable (๐ : Type v) [IsROrC ๐] {E : Type u} [NormedAddCommGroup E] [NormedSpace ๐ E]
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M := by
classical
by_cases h : x = 0
ยท simp only [h, hMp, norm_zero]
ยท obtain โจf, hfโ, hfxโฉ : โ f : E โL[๐] ๐, โfโ = 1 โง f x = โxโ := exists_dual_vector ๐ x h
calc
โxโ = โ(โxโ : ๐)โ := IsROrC.norm_coe_norm.symm
_ = โf xโ := by rw [hfx]
_ โค M * โfโ := (hM f)
_ = M := by rw [hfโ, mul_one]
#align normed_space.norm_le_dual_bound NormedSpace.norm_le_dual_bound
theorem eq_zero_of_forall_dual_eq_zero {x : E} (h : โ f : Dual ๐ E, f x = (0 : ๐)) : x = 0 :=
norm_le_zero_iff.mp (norm_le_dual_bound ๐ x le_rfl fun f => by simp [h f])
#align normed_space.eq_zero_of_forall_dual_eq_zero NormedSpace.eq_zero_of_forall_dual_eq_zero
theorem eq_zero_iff_forall_dual_eq_zero (x : E) : x = 0 โ โ g : Dual ๐ E, g x = 0 :=
โจfun hx => by simp [hx], fun h => eq_zero_of_forall_dual_eq_zero ๐ hโฉ
#align normed_space.eq_zero_iff_forall_dual_eq_zero NormedSpace.eq_zero_iff_forall_dual_eq_zero
/-- See also `geometric_hahn_banach_point_point`. -/
theorem eq_iff_forall_dual_eq {x y : E} : x = y โ โ g : Dual ๐ E, g x = g y := by
rw [โ sub_eq_zero, eq_zero_iff_forall_dual_eq_zero ๐ (x - y)]
simp [sub_eq_zero]
#align normed_space.eq_iff_forall_dual_eq NormedSpace.eq_iff_forall_dual_eq
/-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E โโแตข[๐] Dual ๐ (Dual ๐ E) :=
{ inclusionInDoubleDual ๐ E with
norm_map' := by
intro x
apply le_antisymm
ยท exact double_dual_bound ๐ E x
rw [ContinuousLinearMap.norm_def]
refine' le_csInf ContinuousLinearMap.bounds_nonempty _
rintro c โจhc1, hc2โฉ
exact norm_le_dual_bound ๐ x hc1 hc2 }
#align normed_space.inclusion_in_double_dual_li NormedSpace.inclusionInDoubleDualLi
end BidualIsometry
section PolarSets
open Metric Set NormedSpace
/-- Given a subset `s` in a normed space `E` (over a field `๐`), the polar
`polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals which
evaluate to something of norm at most one at all points `z โ s`. -/
def polar (๐ : Type*) [NontriviallyNormedField ๐] {E : Type*} [SeminormedAddCommGroup E]
[NormedSpace ๐ E] : Set E โ Set (Dual ๐ E) :=
(dualPairing ๐ E).flip.polar
#align normed_space.polar NormedSpace.polar
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable {E : Type*} [SeminormedAddCommGroup E] [NormedSpace ๐ E]
theorem mem_polar_iff {x' : Dual ๐ E} (s : Set E) : x' โ polar ๐ s โ โ z โ s, โx' zโ โค 1 :=
Iff.rfl
#align normed_space.mem_polar_iff NormedSpace.mem_polar_iff
@[simp]
theorem polar_univ : polar ๐ (univ : Set E) = {(0 : Dual ๐ E)} :=
(dualPairing ๐ E).flip.polar_univ
(LinearMap.flip_separatingRight.mpr (dualPairing_separatingLeft ๐ E))
#align normed_space.polar_univ NormedSpace.polar_univ
theorem isClosed_polar (s : Set E) : IsClosed (polar ๐ s) := by
dsimp only [NormedSpace.polar]
simp only [LinearMap.polar_eq_iInter, LinearMap.flip_apply]
refine' isClosed_biInter fun z _ => _
exact isClosed_Iic.preimage (ContinuousLinearMap.apply ๐ ๐ z).continuous.norm
#align normed_space.is_closed_polar NormedSpace.isClosed_polar
@[simp]
theorem polar_closure (s : Set E) : polar ๐ (closure s) = polar ๐ s :=
((dualPairing ๐ E).flip.polar_antitone subset_closure).antisymm <|
(dualPairing ๐ E).flip.polar_gc.l_le <|
closure_minimal ((dualPairing ๐ E).flip.polar_gc.le_u_l s) <| by
simpa [LinearMap.flip_flip] using
(isClosed_polar _ _).preimage (inclusionInDoubleDual ๐ E).continuous
#align normed_space.polar_closure NormedSpace.polar_closure
variable {๐}
/-- If `x'` is a dual element such that the norms `โx' zโ` are bounded for `z โ s`, then a
small scalar multiple of `x'` is in `polar ๐ s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual ๐ E} {c : ๐} (hc : โ z, z โ s โ โx' zโ โค โcโ) :
cโปยน โข x' โ polar ๐ s := by
by_cases c_zero : c = 0
ยท simp only [c_zero, inv_zero, zero_smul]
exact (dualPairing ๐ E).flip.zero_mem_polar _
have eq : โ z, โcโปยน โข x' zโ = โcโปยนโ * โx' zโ := fun z => norm_smul cโปยน _
have le : โ z, z โ s โ โcโปยน โข x' zโ โค โcโปยนโ * โcโ := by
intro z hzs
rw [eq z]
apply mul_le_mul (le_of_eq rfl) (hc z hzs) (norm_nonneg _) (norm_nonneg _)
have cancel : โcโปยนโ * โcโ = 1 := by
simp only [c_zero, norm_eq_zero, Ne.def, not_false_iff, inv_mul_cancel, norm_inv]
rwa [cancel] at le
#align normed_space.smul_mem_polar NormedSpace.smul_mem_polar
theorem polar_ball_subset_closedBall_div {c : ๐} (hc : 1 < โcโ) {r : โ} (hr : 0 < r) :
polar ๐ (ball (0 : E) r) โ closedBall (0 : Dual ๐ E) (โcโ / r) := by
intro x' hx'
rw [mem_polar_iff] at hx'
|
simp only [polar, mem_setOf, mem_closedBall_zero_iff, mem_ball_zero_iff] at *
|
theorem polar_ball_subset_closedBall_div {c : ๐} (hc : 1 < โcโ) {r : โ} (hr : 0 < r) :
polar ๐ (ball (0 : E) r) โ closedBall (0 : Dual ๐ E) (โcโ / r) := by
intro x' hx'
rw [mem_polar_iff] at hx'
|
Mathlib.Analysis.NormedSpace.Dual.218_0.WirVfj6f5oiZZ2w
|
theorem polar_ball_subset_closedBall_div {c : ๐} (hc : 1 < โcโ) {r : โ} (hr : 0 < r) :
polar ๐ (ball (0 : E) r) โ closedBall (0 : Dual ๐ E) (โcโ / r)
|
Mathlib_Analysis_NormedSpace_Dual
|
๐ : Type u_1
instโยฒ : NontriviallyNormedField ๐
E : Type u_2
instโยน : SeminormedAddCommGroup E
instโ : NormedSpace ๐ E
c : ๐
hc : 1 < โcโ
r : โ
hr : 0 < r
x' : Dual ๐ E
hx' : โ (z : E), โzโ < r โ โx' zโ โค 1
โข โx'โ โค โcโ / r
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# The topological dual of a normed space
In this file we define the topological dual `NormedSpace.Dual` of a normed space, and the
continuous linear map `NormedSpace.inclusionInDoubleDual` from a normed space into its double
dual.
For base field `๐ = โ` or `๐ = โ`, this map is actually an isometric embedding; we provide a
version `NormedSpace.inclusionInDoubleDualLi` of the map which is of type a bundled linear
isometric embedding, `E โโแตข[๐] (Dual ๐ (Dual ๐ E))`.
Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
theory for `SeminormedAddCommGroup` and we specialize to `NormedAddCommGroup` when needed.
## Main definitions
* `inclusionInDoubleDual` and `inclusionInDoubleDualLi` are the inclusion of a normed space
in its double dual, considered as a bounded linear map and as a linear isometry, respectively.
* `polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals `x'` for which
`โx' zโ โค 1` for every `z โ s`.
## Tags
dual
-/
noncomputable section
open Classical Topology Bornology
universe u v
namespace NormedSpace
section General
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable (E : Type*) [SeminormedAddCommGroup E] [NormedSpace ๐ E]
variable (F : Type*) [NormedAddCommGroup F] [NormedSpace ๐ F]
/-- The topological dual of a seminormed space `E`. -/
abbrev Dual : Type _ := E โL[๐] ๐
#align normed_space.dual NormedSpace.Dual
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_eq until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : NormedSpace ๐ (Dual ๐ E) := inferInstance
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_le until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : SeminormedAddCommGroup (Dual ๐ E) := inferInstance
/-- The inclusion of a normed space in its double (topological) dual, considered
as a bounded linear map. -/
def inclusionInDoubleDual : E โL[๐] Dual ๐ (Dual ๐ E) :=
ContinuousLinearMap.apply ๐ ๐
#align normed_space.inclusion_in_double_dual NormedSpace.inclusionInDoubleDual
@[simp]
theorem dual_def (x : E) (f : Dual ๐ E) : inclusionInDoubleDual ๐ E x f = f x :=
rfl
#align normed_space.dual_def NormedSpace.dual_def
theorem inclusionInDoubleDual_norm_eq :
โinclusionInDoubleDual ๐ Eโ = โContinuousLinearMap.id ๐ (Dual ๐ E)โ :=
ContinuousLinearMap.op_norm_flip _
#align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1 := by
rw [inclusionInDoubleDual_norm_eq]
exact ContinuousLinearMap.norm_id_le
#align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
theorem double_dual_bound (x : E) : โ(inclusionInDoubleDual ๐ E) xโ โค โxโ := by
simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusionInDoubleDual_norm_le ๐ E) x
#align normed_space.double_dual_bound NormedSpace.double_dual_bound
/-- The dual pairing as a bilinear form. -/
def dualPairing : Dual ๐ E โโ[๐] E โโ[๐] ๐ :=
ContinuousLinearMap.coeLM ๐
#align normed_space.dual_pairing NormedSpace.dualPairing
@[simp]
theorem dualPairing_apply {v : Dual ๐ E} {x : E} : dualPairing ๐ E v x = v x :=
rfl
#align normed_space.dual_pairing_apply NormedSpace.dualPairing_apply
theorem dualPairing_separatingLeft : (dualPairing ๐ E).SeparatingLeft := by
rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]
exact ContinuousLinearMap.coe_injective
#align normed_space.dual_pairing_separating_left NormedSpace.dualPairing_separatingLeft
end General
section BidualIsometry
variable (๐ : Type v) [IsROrC ๐] {E : Type u} [NormedAddCommGroup E] [NormedSpace ๐ E]
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M := by
classical
by_cases h : x = 0
ยท simp only [h, hMp, norm_zero]
ยท obtain โจf, hfโ, hfxโฉ : โ f : E โL[๐] ๐, โfโ = 1 โง f x = โxโ := exists_dual_vector ๐ x h
calc
โxโ = โ(โxโ : ๐)โ := IsROrC.norm_coe_norm.symm
_ = โf xโ := by rw [hfx]
_ โค M * โfโ := (hM f)
_ = M := by rw [hfโ, mul_one]
#align normed_space.norm_le_dual_bound NormedSpace.norm_le_dual_bound
theorem eq_zero_of_forall_dual_eq_zero {x : E} (h : โ f : Dual ๐ E, f x = (0 : ๐)) : x = 0 :=
norm_le_zero_iff.mp (norm_le_dual_bound ๐ x le_rfl fun f => by simp [h f])
#align normed_space.eq_zero_of_forall_dual_eq_zero NormedSpace.eq_zero_of_forall_dual_eq_zero
theorem eq_zero_iff_forall_dual_eq_zero (x : E) : x = 0 โ โ g : Dual ๐ E, g x = 0 :=
โจfun hx => by simp [hx], fun h => eq_zero_of_forall_dual_eq_zero ๐ hโฉ
#align normed_space.eq_zero_iff_forall_dual_eq_zero NormedSpace.eq_zero_iff_forall_dual_eq_zero
/-- See also `geometric_hahn_banach_point_point`. -/
theorem eq_iff_forall_dual_eq {x y : E} : x = y โ โ g : Dual ๐ E, g x = g y := by
rw [โ sub_eq_zero, eq_zero_iff_forall_dual_eq_zero ๐ (x - y)]
simp [sub_eq_zero]
#align normed_space.eq_iff_forall_dual_eq NormedSpace.eq_iff_forall_dual_eq
/-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E โโแตข[๐] Dual ๐ (Dual ๐ E) :=
{ inclusionInDoubleDual ๐ E with
norm_map' := by
intro x
apply le_antisymm
ยท exact double_dual_bound ๐ E x
rw [ContinuousLinearMap.norm_def]
refine' le_csInf ContinuousLinearMap.bounds_nonempty _
rintro c โจhc1, hc2โฉ
exact norm_le_dual_bound ๐ x hc1 hc2 }
#align normed_space.inclusion_in_double_dual_li NormedSpace.inclusionInDoubleDualLi
end BidualIsometry
section PolarSets
open Metric Set NormedSpace
/-- Given a subset `s` in a normed space `E` (over a field `๐`), the polar
`polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals which
evaluate to something of norm at most one at all points `z โ s`. -/
def polar (๐ : Type*) [NontriviallyNormedField ๐] {E : Type*} [SeminormedAddCommGroup E]
[NormedSpace ๐ E] : Set E โ Set (Dual ๐ E) :=
(dualPairing ๐ E).flip.polar
#align normed_space.polar NormedSpace.polar
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable {E : Type*} [SeminormedAddCommGroup E] [NormedSpace ๐ E]
theorem mem_polar_iff {x' : Dual ๐ E} (s : Set E) : x' โ polar ๐ s โ โ z โ s, โx' zโ โค 1 :=
Iff.rfl
#align normed_space.mem_polar_iff NormedSpace.mem_polar_iff
@[simp]
theorem polar_univ : polar ๐ (univ : Set E) = {(0 : Dual ๐ E)} :=
(dualPairing ๐ E).flip.polar_univ
(LinearMap.flip_separatingRight.mpr (dualPairing_separatingLeft ๐ E))
#align normed_space.polar_univ NormedSpace.polar_univ
theorem isClosed_polar (s : Set E) : IsClosed (polar ๐ s) := by
dsimp only [NormedSpace.polar]
simp only [LinearMap.polar_eq_iInter, LinearMap.flip_apply]
refine' isClosed_biInter fun z _ => _
exact isClosed_Iic.preimage (ContinuousLinearMap.apply ๐ ๐ z).continuous.norm
#align normed_space.is_closed_polar NormedSpace.isClosed_polar
@[simp]
theorem polar_closure (s : Set E) : polar ๐ (closure s) = polar ๐ s :=
((dualPairing ๐ E).flip.polar_antitone subset_closure).antisymm <|
(dualPairing ๐ E).flip.polar_gc.l_le <|
closure_minimal ((dualPairing ๐ E).flip.polar_gc.le_u_l s) <| by
simpa [LinearMap.flip_flip] using
(isClosed_polar _ _).preimage (inclusionInDoubleDual ๐ E).continuous
#align normed_space.polar_closure NormedSpace.polar_closure
variable {๐}
/-- If `x'` is a dual element such that the norms `โx' zโ` are bounded for `z โ s`, then a
small scalar multiple of `x'` is in `polar ๐ s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual ๐ E} {c : ๐} (hc : โ z, z โ s โ โx' zโ โค โcโ) :
cโปยน โข x' โ polar ๐ s := by
by_cases c_zero : c = 0
ยท simp only [c_zero, inv_zero, zero_smul]
exact (dualPairing ๐ E).flip.zero_mem_polar _
have eq : โ z, โcโปยน โข x' zโ = โcโปยนโ * โx' zโ := fun z => norm_smul cโปยน _
have le : โ z, z โ s โ โcโปยน โข x' zโ โค โcโปยนโ * โcโ := by
intro z hzs
rw [eq z]
apply mul_le_mul (le_of_eq rfl) (hc z hzs) (norm_nonneg _) (norm_nonneg _)
have cancel : โcโปยนโ * โcโ = 1 := by
simp only [c_zero, norm_eq_zero, Ne.def, not_false_iff, inv_mul_cancel, norm_inv]
rwa [cancel] at le
#align normed_space.smul_mem_polar NormedSpace.smul_mem_polar
theorem polar_ball_subset_closedBall_div {c : ๐} (hc : 1 < โcโ) {r : โ} (hr : 0 < r) :
polar ๐ (ball (0 : E) r) โ closedBall (0 : Dual ๐ E) (โcโ / r) := by
intro x' hx'
rw [mem_polar_iff] at hx'
simp only [polar, mem_setOf, mem_closedBall_zero_iff, mem_ball_zero_iff] at *
|
have hcr : 0 < โcโ / r := div_pos (zero_lt_one.trans hc) hr
|
theorem polar_ball_subset_closedBall_div {c : ๐} (hc : 1 < โcโ) {r : โ} (hr : 0 < r) :
polar ๐ (ball (0 : E) r) โ closedBall (0 : Dual ๐ E) (โcโ / r) := by
intro x' hx'
rw [mem_polar_iff] at hx'
simp only [polar, mem_setOf, mem_closedBall_zero_iff, mem_ball_zero_iff] at *
|
Mathlib.Analysis.NormedSpace.Dual.218_0.WirVfj6f5oiZZ2w
|
theorem polar_ball_subset_closedBall_div {c : ๐} (hc : 1 < โcโ) {r : โ} (hr : 0 < r) :
polar ๐ (ball (0 : E) r) โ closedBall (0 : Dual ๐ E) (โcโ / r)
|
Mathlib_Analysis_NormedSpace_Dual
|
๐ : Type u_1
instโยฒ : NontriviallyNormedField ๐
E : Type u_2
instโยน : SeminormedAddCommGroup E
instโ : NormedSpace ๐ E
c : ๐
hc : 1 < โcโ
r : โ
hr : 0 < r
x' : Dual ๐ E
hx' : โ (z : E), โzโ < r โ โx' zโ โค 1
hcr : 0 < โcโ / r
โข โx'โ โค โcโ / r
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# The topological dual of a normed space
In this file we define the topological dual `NormedSpace.Dual` of a normed space, and the
continuous linear map `NormedSpace.inclusionInDoubleDual` from a normed space into its double
dual.
For base field `๐ = โ` or `๐ = โ`, this map is actually an isometric embedding; we provide a
version `NormedSpace.inclusionInDoubleDualLi` of the map which is of type a bundled linear
isometric embedding, `E โโแตข[๐] (Dual ๐ (Dual ๐ E))`.
Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
theory for `SeminormedAddCommGroup` and we specialize to `NormedAddCommGroup` when needed.
## Main definitions
* `inclusionInDoubleDual` and `inclusionInDoubleDualLi` are the inclusion of a normed space
in its double dual, considered as a bounded linear map and as a linear isometry, respectively.
* `polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals `x'` for which
`โx' zโ โค 1` for every `z โ s`.
## Tags
dual
-/
noncomputable section
open Classical Topology Bornology
universe u v
namespace NormedSpace
section General
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable (E : Type*) [SeminormedAddCommGroup E] [NormedSpace ๐ E]
variable (F : Type*) [NormedAddCommGroup F] [NormedSpace ๐ F]
/-- The topological dual of a seminormed space `E`. -/
abbrev Dual : Type _ := E โL[๐] ๐
#align normed_space.dual NormedSpace.Dual
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_eq until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : NormedSpace ๐ (Dual ๐ E) := inferInstance
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_le until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : SeminormedAddCommGroup (Dual ๐ E) := inferInstance
/-- The inclusion of a normed space in its double (topological) dual, considered
as a bounded linear map. -/
def inclusionInDoubleDual : E โL[๐] Dual ๐ (Dual ๐ E) :=
ContinuousLinearMap.apply ๐ ๐
#align normed_space.inclusion_in_double_dual NormedSpace.inclusionInDoubleDual
@[simp]
theorem dual_def (x : E) (f : Dual ๐ E) : inclusionInDoubleDual ๐ E x f = f x :=
rfl
#align normed_space.dual_def NormedSpace.dual_def
theorem inclusionInDoubleDual_norm_eq :
โinclusionInDoubleDual ๐ Eโ = โContinuousLinearMap.id ๐ (Dual ๐ E)โ :=
ContinuousLinearMap.op_norm_flip _
#align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1 := by
rw [inclusionInDoubleDual_norm_eq]
exact ContinuousLinearMap.norm_id_le
#align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
theorem double_dual_bound (x : E) : โ(inclusionInDoubleDual ๐ E) xโ โค โxโ := by
simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusionInDoubleDual_norm_le ๐ E) x
#align normed_space.double_dual_bound NormedSpace.double_dual_bound
/-- The dual pairing as a bilinear form. -/
def dualPairing : Dual ๐ E โโ[๐] E โโ[๐] ๐ :=
ContinuousLinearMap.coeLM ๐
#align normed_space.dual_pairing NormedSpace.dualPairing
@[simp]
theorem dualPairing_apply {v : Dual ๐ E} {x : E} : dualPairing ๐ E v x = v x :=
rfl
#align normed_space.dual_pairing_apply NormedSpace.dualPairing_apply
theorem dualPairing_separatingLeft : (dualPairing ๐ E).SeparatingLeft := by
rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]
exact ContinuousLinearMap.coe_injective
#align normed_space.dual_pairing_separating_left NormedSpace.dualPairing_separatingLeft
end General
section BidualIsometry
variable (๐ : Type v) [IsROrC ๐] {E : Type u} [NormedAddCommGroup E] [NormedSpace ๐ E]
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M := by
classical
by_cases h : x = 0
ยท simp only [h, hMp, norm_zero]
ยท obtain โจf, hfโ, hfxโฉ : โ f : E โL[๐] ๐, โfโ = 1 โง f x = โxโ := exists_dual_vector ๐ x h
calc
โxโ = โ(โxโ : ๐)โ := IsROrC.norm_coe_norm.symm
_ = โf xโ := by rw [hfx]
_ โค M * โfโ := (hM f)
_ = M := by rw [hfโ, mul_one]
#align normed_space.norm_le_dual_bound NormedSpace.norm_le_dual_bound
theorem eq_zero_of_forall_dual_eq_zero {x : E} (h : โ f : Dual ๐ E, f x = (0 : ๐)) : x = 0 :=
norm_le_zero_iff.mp (norm_le_dual_bound ๐ x le_rfl fun f => by simp [h f])
#align normed_space.eq_zero_of_forall_dual_eq_zero NormedSpace.eq_zero_of_forall_dual_eq_zero
theorem eq_zero_iff_forall_dual_eq_zero (x : E) : x = 0 โ โ g : Dual ๐ E, g x = 0 :=
โจfun hx => by simp [hx], fun h => eq_zero_of_forall_dual_eq_zero ๐ hโฉ
#align normed_space.eq_zero_iff_forall_dual_eq_zero NormedSpace.eq_zero_iff_forall_dual_eq_zero
/-- See also `geometric_hahn_banach_point_point`. -/
theorem eq_iff_forall_dual_eq {x y : E} : x = y โ โ g : Dual ๐ E, g x = g y := by
rw [โ sub_eq_zero, eq_zero_iff_forall_dual_eq_zero ๐ (x - y)]
simp [sub_eq_zero]
#align normed_space.eq_iff_forall_dual_eq NormedSpace.eq_iff_forall_dual_eq
/-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E โโแตข[๐] Dual ๐ (Dual ๐ E) :=
{ inclusionInDoubleDual ๐ E with
norm_map' := by
intro x
apply le_antisymm
ยท exact double_dual_bound ๐ E x
rw [ContinuousLinearMap.norm_def]
refine' le_csInf ContinuousLinearMap.bounds_nonempty _
rintro c โจhc1, hc2โฉ
exact norm_le_dual_bound ๐ x hc1 hc2 }
#align normed_space.inclusion_in_double_dual_li NormedSpace.inclusionInDoubleDualLi
end BidualIsometry
section PolarSets
open Metric Set NormedSpace
/-- Given a subset `s` in a normed space `E` (over a field `๐`), the polar
`polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals which
evaluate to something of norm at most one at all points `z โ s`. -/
def polar (๐ : Type*) [NontriviallyNormedField ๐] {E : Type*} [SeminormedAddCommGroup E]
[NormedSpace ๐ E] : Set E โ Set (Dual ๐ E) :=
(dualPairing ๐ E).flip.polar
#align normed_space.polar NormedSpace.polar
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable {E : Type*} [SeminormedAddCommGroup E] [NormedSpace ๐ E]
theorem mem_polar_iff {x' : Dual ๐ E} (s : Set E) : x' โ polar ๐ s โ โ z โ s, โx' zโ โค 1 :=
Iff.rfl
#align normed_space.mem_polar_iff NormedSpace.mem_polar_iff
@[simp]
theorem polar_univ : polar ๐ (univ : Set E) = {(0 : Dual ๐ E)} :=
(dualPairing ๐ E).flip.polar_univ
(LinearMap.flip_separatingRight.mpr (dualPairing_separatingLeft ๐ E))
#align normed_space.polar_univ NormedSpace.polar_univ
theorem isClosed_polar (s : Set E) : IsClosed (polar ๐ s) := by
dsimp only [NormedSpace.polar]
simp only [LinearMap.polar_eq_iInter, LinearMap.flip_apply]
refine' isClosed_biInter fun z _ => _
exact isClosed_Iic.preimage (ContinuousLinearMap.apply ๐ ๐ z).continuous.norm
#align normed_space.is_closed_polar NormedSpace.isClosed_polar
@[simp]
theorem polar_closure (s : Set E) : polar ๐ (closure s) = polar ๐ s :=
((dualPairing ๐ E).flip.polar_antitone subset_closure).antisymm <|
(dualPairing ๐ E).flip.polar_gc.l_le <|
closure_minimal ((dualPairing ๐ E).flip.polar_gc.le_u_l s) <| by
simpa [LinearMap.flip_flip] using
(isClosed_polar _ _).preimage (inclusionInDoubleDual ๐ E).continuous
#align normed_space.polar_closure NormedSpace.polar_closure
variable {๐}
/-- If `x'` is a dual element such that the norms `โx' zโ` are bounded for `z โ s`, then a
small scalar multiple of `x'` is in `polar ๐ s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual ๐ E} {c : ๐} (hc : โ z, z โ s โ โx' zโ โค โcโ) :
cโปยน โข x' โ polar ๐ s := by
by_cases c_zero : c = 0
ยท simp only [c_zero, inv_zero, zero_smul]
exact (dualPairing ๐ E).flip.zero_mem_polar _
have eq : โ z, โcโปยน โข x' zโ = โcโปยนโ * โx' zโ := fun z => norm_smul cโปยน _
have le : โ z, z โ s โ โcโปยน โข x' zโ โค โcโปยนโ * โcโ := by
intro z hzs
rw [eq z]
apply mul_le_mul (le_of_eq rfl) (hc z hzs) (norm_nonneg _) (norm_nonneg _)
have cancel : โcโปยนโ * โcโ = 1 := by
simp only [c_zero, norm_eq_zero, Ne.def, not_false_iff, inv_mul_cancel, norm_inv]
rwa [cancel] at le
#align normed_space.smul_mem_polar NormedSpace.smul_mem_polar
theorem polar_ball_subset_closedBall_div {c : ๐} (hc : 1 < โcโ) {r : โ} (hr : 0 < r) :
polar ๐ (ball (0 : E) r) โ closedBall (0 : Dual ๐ E) (โcโ / r) := by
intro x' hx'
rw [mem_polar_iff] at hx'
simp only [polar, mem_setOf, mem_closedBall_zero_iff, mem_ball_zero_iff] at *
have hcr : 0 < โcโ / r := div_pos (zero_lt_one.trans hc) hr
|
refine' ContinuousLinearMap.op_norm_le_of_shell hr hcr.le hc fun x hโ hโ => _
|
theorem polar_ball_subset_closedBall_div {c : ๐} (hc : 1 < โcโ) {r : โ} (hr : 0 < r) :
polar ๐ (ball (0 : E) r) โ closedBall (0 : Dual ๐ E) (โcโ / r) := by
intro x' hx'
rw [mem_polar_iff] at hx'
simp only [polar, mem_setOf, mem_closedBall_zero_iff, mem_ball_zero_iff] at *
have hcr : 0 < โcโ / r := div_pos (zero_lt_one.trans hc) hr
|
Mathlib.Analysis.NormedSpace.Dual.218_0.WirVfj6f5oiZZ2w
|
theorem polar_ball_subset_closedBall_div {c : ๐} (hc : 1 < โcโ) {r : โ} (hr : 0 < r) :
polar ๐ (ball (0 : E) r) โ closedBall (0 : Dual ๐ E) (โcโ / r)
|
Mathlib_Analysis_NormedSpace_Dual
|
๐ : Type u_1
instโยฒ : NontriviallyNormedField ๐
E : Type u_2
instโยน : SeminormedAddCommGroup E
instโ : NormedSpace ๐ E
c : ๐
hc : 1 < โcโ
r : โ
hr : 0 < r
x' : Dual ๐ E
hx' : โ (z : E), โzโ < r โ โx' zโ โค 1
hcr : 0 < โcโ / r
x : E
hโ : r / โcโ โค โxโ
hโ : โxโ < r
โข โx' xโ โค โcโ / r * โxโ
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# The topological dual of a normed space
In this file we define the topological dual `NormedSpace.Dual` of a normed space, and the
continuous linear map `NormedSpace.inclusionInDoubleDual` from a normed space into its double
dual.
For base field `๐ = โ` or `๐ = โ`, this map is actually an isometric embedding; we provide a
version `NormedSpace.inclusionInDoubleDualLi` of the map which is of type a bundled linear
isometric embedding, `E โโแตข[๐] (Dual ๐ (Dual ๐ E))`.
Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
theory for `SeminormedAddCommGroup` and we specialize to `NormedAddCommGroup` when needed.
## Main definitions
* `inclusionInDoubleDual` and `inclusionInDoubleDualLi` are the inclusion of a normed space
in its double dual, considered as a bounded linear map and as a linear isometry, respectively.
* `polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals `x'` for which
`โx' zโ โค 1` for every `z โ s`.
## Tags
dual
-/
noncomputable section
open Classical Topology Bornology
universe u v
namespace NormedSpace
section General
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable (E : Type*) [SeminormedAddCommGroup E] [NormedSpace ๐ E]
variable (F : Type*) [NormedAddCommGroup F] [NormedSpace ๐ F]
/-- The topological dual of a seminormed space `E`. -/
abbrev Dual : Type _ := E โL[๐] ๐
#align normed_space.dual NormedSpace.Dual
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_eq until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : NormedSpace ๐ (Dual ๐ E) := inferInstance
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_le until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : SeminormedAddCommGroup (Dual ๐ E) := inferInstance
/-- The inclusion of a normed space in its double (topological) dual, considered
as a bounded linear map. -/
def inclusionInDoubleDual : E โL[๐] Dual ๐ (Dual ๐ E) :=
ContinuousLinearMap.apply ๐ ๐
#align normed_space.inclusion_in_double_dual NormedSpace.inclusionInDoubleDual
@[simp]
theorem dual_def (x : E) (f : Dual ๐ E) : inclusionInDoubleDual ๐ E x f = f x :=
rfl
#align normed_space.dual_def NormedSpace.dual_def
theorem inclusionInDoubleDual_norm_eq :
โinclusionInDoubleDual ๐ Eโ = โContinuousLinearMap.id ๐ (Dual ๐ E)โ :=
ContinuousLinearMap.op_norm_flip _
#align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1 := by
rw [inclusionInDoubleDual_norm_eq]
exact ContinuousLinearMap.norm_id_le
#align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
theorem double_dual_bound (x : E) : โ(inclusionInDoubleDual ๐ E) xโ โค โxโ := by
simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusionInDoubleDual_norm_le ๐ E) x
#align normed_space.double_dual_bound NormedSpace.double_dual_bound
/-- The dual pairing as a bilinear form. -/
def dualPairing : Dual ๐ E โโ[๐] E โโ[๐] ๐ :=
ContinuousLinearMap.coeLM ๐
#align normed_space.dual_pairing NormedSpace.dualPairing
@[simp]
theorem dualPairing_apply {v : Dual ๐ E} {x : E} : dualPairing ๐ E v x = v x :=
rfl
#align normed_space.dual_pairing_apply NormedSpace.dualPairing_apply
theorem dualPairing_separatingLeft : (dualPairing ๐ E).SeparatingLeft := by
rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]
exact ContinuousLinearMap.coe_injective
#align normed_space.dual_pairing_separating_left NormedSpace.dualPairing_separatingLeft
end General
section BidualIsometry
variable (๐ : Type v) [IsROrC ๐] {E : Type u} [NormedAddCommGroup E] [NormedSpace ๐ E]
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M := by
classical
by_cases h : x = 0
ยท simp only [h, hMp, norm_zero]
ยท obtain โจf, hfโ, hfxโฉ : โ f : E โL[๐] ๐, โfโ = 1 โง f x = โxโ := exists_dual_vector ๐ x h
calc
โxโ = โ(โxโ : ๐)โ := IsROrC.norm_coe_norm.symm
_ = โf xโ := by rw [hfx]
_ โค M * โfโ := (hM f)
_ = M := by rw [hfโ, mul_one]
#align normed_space.norm_le_dual_bound NormedSpace.norm_le_dual_bound
theorem eq_zero_of_forall_dual_eq_zero {x : E} (h : โ f : Dual ๐ E, f x = (0 : ๐)) : x = 0 :=
norm_le_zero_iff.mp (norm_le_dual_bound ๐ x le_rfl fun f => by simp [h f])
#align normed_space.eq_zero_of_forall_dual_eq_zero NormedSpace.eq_zero_of_forall_dual_eq_zero
theorem eq_zero_iff_forall_dual_eq_zero (x : E) : x = 0 โ โ g : Dual ๐ E, g x = 0 :=
โจfun hx => by simp [hx], fun h => eq_zero_of_forall_dual_eq_zero ๐ hโฉ
#align normed_space.eq_zero_iff_forall_dual_eq_zero NormedSpace.eq_zero_iff_forall_dual_eq_zero
/-- See also `geometric_hahn_banach_point_point`. -/
theorem eq_iff_forall_dual_eq {x y : E} : x = y โ โ g : Dual ๐ E, g x = g y := by
rw [โ sub_eq_zero, eq_zero_iff_forall_dual_eq_zero ๐ (x - y)]
simp [sub_eq_zero]
#align normed_space.eq_iff_forall_dual_eq NormedSpace.eq_iff_forall_dual_eq
/-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E โโแตข[๐] Dual ๐ (Dual ๐ E) :=
{ inclusionInDoubleDual ๐ E with
norm_map' := by
intro x
apply le_antisymm
ยท exact double_dual_bound ๐ E x
rw [ContinuousLinearMap.norm_def]
refine' le_csInf ContinuousLinearMap.bounds_nonempty _
rintro c โจhc1, hc2โฉ
exact norm_le_dual_bound ๐ x hc1 hc2 }
#align normed_space.inclusion_in_double_dual_li NormedSpace.inclusionInDoubleDualLi
end BidualIsometry
section PolarSets
open Metric Set NormedSpace
/-- Given a subset `s` in a normed space `E` (over a field `๐`), the polar
`polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals which
evaluate to something of norm at most one at all points `z โ s`. -/
def polar (๐ : Type*) [NontriviallyNormedField ๐] {E : Type*} [SeminormedAddCommGroup E]
[NormedSpace ๐ E] : Set E โ Set (Dual ๐ E) :=
(dualPairing ๐ E).flip.polar
#align normed_space.polar NormedSpace.polar
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable {E : Type*} [SeminormedAddCommGroup E] [NormedSpace ๐ E]
theorem mem_polar_iff {x' : Dual ๐ E} (s : Set E) : x' โ polar ๐ s โ โ z โ s, โx' zโ โค 1 :=
Iff.rfl
#align normed_space.mem_polar_iff NormedSpace.mem_polar_iff
@[simp]
theorem polar_univ : polar ๐ (univ : Set E) = {(0 : Dual ๐ E)} :=
(dualPairing ๐ E).flip.polar_univ
(LinearMap.flip_separatingRight.mpr (dualPairing_separatingLeft ๐ E))
#align normed_space.polar_univ NormedSpace.polar_univ
theorem isClosed_polar (s : Set E) : IsClosed (polar ๐ s) := by
dsimp only [NormedSpace.polar]
simp only [LinearMap.polar_eq_iInter, LinearMap.flip_apply]
refine' isClosed_biInter fun z _ => _
exact isClosed_Iic.preimage (ContinuousLinearMap.apply ๐ ๐ z).continuous.norm
#align normed_space.is_closed_polar NormedSpace.isClosed_polar
@[simp]
theorem polar_closure (s : Set E) : polar ๐ (closure s) = polar ๐ s :=
((dualPairing ๐ E).flip.polar_antitone subset_closure).antisymm <|
(dualPairing ๐ E).flip.polar_gc.l_le <|
closure_minimal ((dualPairing ๐ E).flip.polar_gc.le_u_l s) <| by
simpa [LinearMap.flip_flip] using
(isClosed_polar _ _).preimage (inclusionInDoubleDual ๐ E).continuous
#align normed_space.polar_closure NormedSpace.polar_closure
variable {๐}
/-- If `x'` is a dual element such that the norms `โx' zโ` are bounded for `z โ s`, then a
small scalar multiple of `x'` is in `polar ๐ s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual ๐ E} {c : ๐} (hc : โ z, z โ s โ โx' zโ โค โcโ) :
cโปยน โข x' โ polar ๐ s := by
by_cases c_zero : c = 0
ยท simp only [c_zero, inv_zero, zero_smul]
exact (dualPairing ๐ E).flip.zero_mem_polar _
have eq : โ z, โcโปยน โข x' zโ = โcโปยนโ * โx' zโ := fun z => norm_smul cโปยน _
have le : โ z, z โ s โ โcโปยน โข x' zโ โค โcโปยนโ * โcโ := by
intro z hzs
rw [eq z]
apply mul_le_mul (le_of_eq rfl) (hc z hzs) (norm_nonneg _) (norm_nonneg _)
have cancel : โcโปยนโ * โcโ = 1 := by
simp only [c_zero, norm_eq_zero, Ne.def, not_false_iff, inv_mul_cancel, norm_inv]
rwa [cancel] at le
#align normed_space.smul_mem_polar NormedSpace.smul_mem_polar
theorem polar_ball_subset_closedBall_div {c : ๐} (hc : 1 < โcโ) {r : โ} (hr : 0 < r) :
polar ๐ (ball (0 : E) r) โ closedBall (0 : Dual ๐ E) (โcโ / r) := by
intro x' hx'
rw [mem_polar_iff] at hx'
simp only [polar, mem_setOf, mem_closedBall_zero_iff, mem_ball_zero_iff] at *
have hcr : 0 < โcโ / r := div_pos (zero_lt_one.trans hc) hr
refine' ContinuousLinearMap.op_norm_le_of_shell hr hcr.le hc fun x hโ hโ => _
|
calc
โx' xโ โค 1 := hx' _ hโ
_ โค โcโ / r * โxโ := (inv_pos_le_iff_one_le_mul' hcr).1 (by rwa [inv_div])
|
theorem polar_ball_subset_closedBall_div {c : ๐} (hc : 1 < โcโ) {r : โ} (hr : 0 < r) :
polar ๐ (ball (0 : E) r) โ closedBall (0 : Dual ๐ E) (โcโ / r) := by
intro x' hx'
rw [mem_polar_iff] at hx'
simp only [polar, mem_setOf, mem_closedBall_zero_iff, mem_ball_zero_iff] at *
have hcr : 0 < โcโ / r := div_pos (zero_lt_one.trans hc) hr
refine' ContinuousLinearMap.op_norm_le_of_shell hr hcr.le hc fun x hโ hโ => _
|
Mathlib.Analysis.NormedSpace.Dual.218_0.WirVfj6f5oiZZ2w
|
theorem polar_ball_subset_closedBall_div {c : ๐} (hc : 1 < โcโ) {r : โ} (hr : 0 < r) :
polar ๐ (ball (0 : E) r) โ closedBall (0 : Dual ๐ E) (โcโ / r)
|
Mathlib_Analysis_NormedSpace_Dual
|
๐ : Type u_1
instโยฒ : NontriviallyNormedField ๐
E : Type u_2
instโยน : SeminormedAddCommGroup E
instโ : NormedSpace ๐ E
c : ๐
hc : 1 < โcโ
r : โ
hr : 0 < r
x' : Dual ๐ E
hx' : โ (z : E), โzโ < r โ โx' zโ โค 1
hcr : 0 < โcโ / r
x : E
hโ : r / โcโ โค โxโ
hโ : โxโ < r
โข (โcโ / r)โปยน โค โxโ
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# The topological dual of a normed space
In this file we define the topological dual `NormedSpace.Dual` of a normed space, and the
continuous linear map `NormedSpace.inclusionInDoubleDual` from a normed space into its double
dual.
For base field `๐ = โ` or `๐ = โ`, this map is actually an isometric embedding; we provide a
version `NormedSpace.inclusionInDoubleDualLi` of the map which is of type a bundled linear
isometric embedding, `E โโแตข[๐] (Dual ๐ (Dual ๐ E))`.
Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
theory for `SeminormedAddCommGroup` and we specialize to `NormedAddCommGroup` when needed.
## Main definitions
* `inclusionInDoubleDual` and `inclusionInDoubleDualLi` are the inclusion of a normed space
in its double dual, considered as a bounded linear map and as a linear isometry, respectively.
* `polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals `x'` for which
`โx' zโ โค 1` for every `z โ s`.
## Tags
dual
-/
noncomputable section
open Classical Topology Bornology
universe u v
namespace NormedSpace
section General
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable (E : Type*) [SeminormedAddCommGroup E] [NormedSpace ๐ E]
variable (F : Type*) [NormedAddCommGroup F] [NormedSpace ๐ F]
/-- The topological dual of a seminormed space `E`. -/
abbrev Dual : Type _ := E โL[๐] ๐
#align normed_space.dual NormedSpace.Dual
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_eq until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : NormedSpace ๐ (Dual ๐ E) := inferInstance
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_le until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : SeminormedAddCommGroup (Dual ๐ E) := inferInstance
/-- The inclusion of a normed space in its double (topological) dual, considered
as a bounded linear map. -/
def inclusionInDoubleDual : E โL[๐] Dual ๐ (Dual ๐ E) :=
ContinuousLinearMap.apply ๐ ๐
#align normed_space.inclusion_in_double_dual NormedSpace.inclusionInDoubleDual
@[simp]
theorem dual_def (x : E) (f : Dual ๐ E) : inclusionInDoubleDual ๐ E x f = f x :=
rfl
#align normed_space.dual_def NormedSpace.dual_def
theorem inclusionInDoubleDual_norm_eq :
โinclusionInDoubleDual ๐ Eโ = โContinuousLinearMap.id ๐ (Dual ๐ E)โ :=
ContinuousLinearMap.op_norm_flip _
#align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1 := by
rw [inclusionInDoubleDual_norm_eq]
exact ContinuousLinearMap.norm_id_le
#align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
theorem double_dual_bound (x : E) : โ(inclusionInDoubleDual ๐ E) xโ โค โxโ := by
simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusionInDoubleDual_norm_le ๐ E) x
#align normed_space.double_dual_bound NormedSpace.double_dual_bound
/-- The dual pairing as a bilinear form. -/
def dualPairing : Dual ๐ E โโ[๐] E โโ[๐] ๐ :=
ContinuousLinearMap.coeLM ๐
#align normed_space.dual_pairing NormedSpace.dualPairing
@[simp]
theorem dualPairing_apply {v : Dual ๐ E} {x : E} : dualPairing ๐ E v x = v x :=
rfl
#align normed_space.dual_pairing_apply NormedSpace.dualPairing_apply
theorem dualPairing_separatingLeft : (dualPairing ๐ E).SeparatingLeft := by
rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]
exact ContinuousLinearMap.coe_injective
#align normed_space.dual_pairing_separating_left NormedSpace.dualPairing_separatingLeft
end General
section BidualIsometry
variable (๐ : Type v) [IsROrC ๐] {E : Type u} [NormedAddCommGroup E] [NormedSpace ๐ E]
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M := by
classical
by_cases h : x = 0
ยท simp only [h, hMp, norm_zero]
ยท obtain โจf, hfโ, hfxโฉ : โ f : E โL[๐] ๐, โfโ = 1 โง f x = โxโ := exists_dual_vector ๐ x h
calc
โxโ = โ(โxโ : ๐)โ := IsROrC.norm_coe_norm.symm
_ = โf xโ := by rw [hfx]
_ โค M * โfโ := (hM f)
_ = M := by rw [hfโ, mul_one]
#align normed_space.norm_le_dual_bound NormedSpace.norm_le_dual_bound
theorem eq_zero_of_forall_dual_eq_zero {x : E} (h : โ f : Dual ๐ E, f x = (0 : ๐)) : x = 0 :=
norm_le_zero_iff.mp (norm_le_dual_bound ๐ x le_rfl fun f => by simp [h f])
#align normed_space.eq_zero_of_forall_dual_eq_zero NormedSpace.eq_zero_of_forall_dual_eq_zero
theorem eq_zero_iff_forall_dual_eq_zero (x : E) : x = 0 โ โ g : Dual ๐ E, g x = 0 :=
โจfun hx => by simp [hx], fun h => eq_zero_of_forall_dual_eq_zero ๐ hโฉ
#align normed_space.eq_zero_iff_forall_dual_eq_zero NormedSpace.eq_zero_iff_forall_dual_eq_zero
/-- See also `geometric_hahn_banach_point_point`. -/
theorem eq_iff_forall_dual_eq {x y : E} : x = y โ โ g : Dual ๐ E, g x = g y := by
rw [โ sub_eq_zero, eq_zero_iff_forall_dual_eq_zero ๐ (x - y)]
simp [sub_eq_zero]
#align normed_space.eq_iff_forall_dual_eq NormedSpace.eq_iff_forall_dual_eq
/-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E โโแตข[๐] Dual ๐ (Dual ๐ E) :=
{ inclusionInDoubleDual ๐ E with
norm_map' := by
intro x
apply le_antisymm
ยท exact double_dual_bound ๐ E x
rw [ContinuousLinearMap.norm_def]
refine' le_csInf ContinuousLinearMap.bounds_nonempty _
rintro c โจhc1, hc2โฉ
exact norm_le_dual_bound ๐ x hc1 hc2 }
#align normed_space.inclusion_in_double_dual_li NormedSpace.inclusionInDoubleDualLi
end BidualIsometry
section PolarSets
open Metric Set NormedSpace
/-- Given a subset `s` in a normed space `E` (over a field `๐`), the polar
`polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals which
evaluate to something of norm at most one at all points `z โ s`. -/
def polar (๐ : Type*) [NontriviallyNormedField ๐] {E : Type*} [SeminormedAddCommGroup E]
[NormedSpace ๐ E] : Set E โ Set (Dual ๐ E) :=
(dualPairing ๐ E).flip.polar
#align normed_space.polar NormedSpace.polar
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable {E : Type*} [SeminormedAddCommGroup E] [NormedSpace ๐ E]
theorem mem_polar_iff {x' : Dual ๐ E} (s : Set E) : x' โ polar ๐ s โ โ z โ s, โx' zโ โค 1 :=
Iff.rfl
#align normed_space.mem_polar_iff NormedSpace.mem_polar_iff
@[simp]
theorem polar_univ : polar ๐ (univ : Set E) = {(0 : Dual ๐ E)} :=
(dualPairing ๐ E).flip.polar_univ
(LinearMap.flip_separatingRight.mpr (dualPairing_separatingLeft ๐ E))
#align normed_space.polar_univ NormedSpace.polar_univ
theorem isClosed_polar (s : Set E) : IsClosed (polar ๐ s) := by
dsimp only [NormedSpace.polar]
simp only [LinearMap.polar_eq_iInter, LinearMap.flip_apply]
refine' isClosed_biInter fun z _ => _
exact isClosed_Iic.preimage (ContinuousLinearMap.apply ๐ ๐ z).continuous.norm
#align normed_space.is_closed_polar NormedSpace.isClosed_polar
@[simp]
theorem polar_closure (s : Set E) : polar ๐ (closure s) = polar ๐ s :=
((dualPairing ๐ E).flip.polar_antitone subset_closure).antisymm <|
(dualPairing ๐ E).flip.polar_gc.l_le <|
closure_minimal ((dualPairing ๐ E).flip.polar_gc.le_u_l s) <| by
simpa [LinearMap.flip_flip] using
(isClosed_polar _ _).preimage (inclusionInDoubleDual ๐ E).continuous
#align normed_space.polar_closure NormedSpace.polar_closure
variable {๐}
/-- If `x'` is a dual element such that the norms `โx' zโ` are bounded for `z โ s`, then a
small scalar multiple of `x'` is in `polar ๐ s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual ๐ E} {c : ๐} (hc : โ z, z โ s โ โx' zโ โค โcโ) :
cโปยน โข x' โ polar ๐ s := by
by_cases c_zero : c = 0
ยท simp only [c_zero, inv_zero, zero_smul]
exact (dualPairing ๐ E).flip.zero_mem_polar _
have eq : โ z, โcโปยน โข x' zโ = โcโปยนโ * โx' zโ := fun z => norm_smul cโปยน _
have le : โ z, z โ s โ โcโปยน โข x' zโ โค โcโปยนโ * โcโ := by
intro z hzs
rw [eq z]
apply mul_le_mul (le_of_eq rfl) (hc z hzs) (norm_nonneg _) (norm_nonneg _)
have cancel : โcโปยนโ * โcโ = 1 := by
simp only [c_zero, norm_eq_zero, Ne.def, not_false_iff, inv_mul_cancel, norm_inv]
rwa [cancel] at le
#align normed_space.smul_mem_polar NormedSpace.smul_mem_polar
theorem polar_ball_subset_closedBall_div {c : ๐} (hc : 1 < โcโ) {r : โ} (hr : 0 < r) :
polar ๐ (ball (0 : E) r) โ closedBall (0 : Dual ๐ E) (โcโ / r) := by
intro x' hx'
rw [mem_polar_iff] at hx'
simp only [polar, mem_setOf, mem_closedBall_zero_iff, mem_ball_zero_iff] at *
have hcr : 0 < โcโ / r := div_pos (zero_lt_one.trans hc) hr
refine' ContinuousLinearMap.op_norm_le_of_shell hr hcr.le hc fun x hโ hโ => _
calc
โx' xโ โค 1 := hx' _ hโ
_ โค โcโ / r * โxโ := (inv_pos_le_iff_one_le_mul' hcr).1 (by
|
rwa [inv_div]
|
theorem polar_ball_subset_closedBall_div {c : ๐} (hc : 1 < โcโ) {r : โ} (hr : 0 < r) :
polar ๐ (ball (0 : E) r) โ closedBall (0 : Dual ๐ E) (โcโ / r) := by
intro x' hx'
rw [mem_polar_iff] at hx'
simp only [polar, mem_setOf, mem_closedBall_zero_iff, mem_ball_zero_iff] at *
have hcr : 0 < โcโ / r := div_pos (zero_lt_one.trans hc) hr
refine' ContinuousLinearMap.op_norm_le_of_shell hr hcr.le hc fun x hโ hโ => _
calc
โx' xโ โค 1 := hx' _ hโ
_ โค โcโ / r * โxโ := (inv_pos_le_iff_one_le_mul' hcr).1 (by
|
Mathlib.Analysis.NormedSpace.Dual.218_0.WirVfj6f5oiZZ2w
|
theorem polar_ball_subset_closedBall_div {c : ๐} (hc : 1 < โcโ) {r : โ} (hr : 0 < r) :
polar ๐ (ball (0 : E) r) โ closedBall (0 : Dual ๐ E) (โcโ / r)
|
Mathlib_Analysis_NormedSpace_Dual
|
๐โ : Type u_1
instโโต : NontriviallyNormedField ๐โ
Eโ : Type u_2
instโโด : SeminormedAddCommGroup Eโ
instโยณ : NormedSpace ๐โ Eโ
๐ : Type u_3
E : Type u_4
instโยฒ : IsROrC ๐
instโยน : NormedAddCommGroup E
instโ : NormedSpace ๐ E
r : โ
hr : 0 < r
โข polar ๐ (closedBall 0 r) = closedBall 0 rโปยน
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# The topological dual of a normed space
In this file we define the topological dual `NormedSpace.Dual` of a normed space, and the
continuous linear map `NormedSpace.inclusionInDoubleDual` from a normed space into its double
dual.
For base field `๐ = โ` or `๐ = โ`, this map is actually an isometric embedding; we provide a
version `NormedSpace.inclusionInDoubleDualLi` of the map which is of type a bundled linear
isometric embedding, `E โโแตข[๐] (Dual ๐ (Dual ๐ E))`.
Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
theory for `SeminormedAddCommGroup` and we specialize to `NormedAddCommGroup` when needed.
## Main definitions
* `inclusionInDoubleDual` and `inclusionInDoubleDualLi` are the inclusion of a normed space
in its double dual, considered as a bounded linear map and as a linear isometry, respectively.
* `polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals `x'` for which
`โx' zโ โค 1` for every `z โ s`.
## Tags
dual
-/
noncomputable section
open Classical Topology Bornology
universe u v
namespace NormedSpace
section General
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable (E : Type*) [SeminormedAddCommGroup E] [NormedSpace ๐ E]
variable (F : Type*) [NormedAddCommGroup F] [NormedSpace ๐ F]
/-- The topological dual of a seminormed space `E`. -/
abbrev Dual : Type _ := E โL[๐] ๐
#align normed_space.dual NormedSpace.Dual
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_eq until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : NormedSpace ๐ (Dual ๐ E) := inferInstance
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_le until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : SeminormedAddCommGroup (Dual ๐ E) := inferInstance
/-- The inclusion of a normed space in its double (topological) dual, considered
as a bounded linear map. -/
def inclusionInDoubleDual : E โL[๐] Dual ๐ (Dual ๐ E) :=
ContinuousLinearMap.apply ๐ ๐
#align normed_space.inclusion_in_double_dual NormedSpace.inclusionInDoubleDual
@[simp]
theorem dual_def (x : E) (f : Dual ๐ E) : inclusionInDoubleDual ๐ E x f = f x :=
rfl
#align normed_space.dual_def NormedSpace.dual_def
theorem inclusionInDoubleDual_norm_eq :
โinclusionInDoubleDual ๐ Eโ = โContinuousLinearMap.id ๐ (Dual ๐ E)โ :=
ContinuousLinearMap.op_norm_flip _
#align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1 := by
rw [inclusionInDoubleDual_norm_eq]
exact ContinuousLinearMap.norm_id_le
#align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
theorem double_dual_bound (x : E) : โ(inclusionInDoubleDual ๐ E) xโ โค โxโ := by
simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusionInDoubleDual_norm_le ๐ E) x
#align normed_space.double_dual_bound NormedSpace.double_dual_bound
/-- The dual pairing as a bilinear form. -/
def dualPairing : Dual ๐ E โโ[๐] E โโ[๐] ๐ :=
ContinuousLinearMap.coeLM ๐
#align normed_space.dual_pairing NormedSpace.dualPairing
@[simp]
theorem dualPairing_apply {v : Dual ๐ E} {x : E} : dualPairing ๐ E v x = v x :=
rfl
#align normed_space.dual_pairing_apply NormedSpace.dualPairing_apply
theorem dualPairing_separatingLeft : (dualPairing ๐ E).SeparatingLeft := by
rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]
exact ContinuousLinearMap.coe_injective
#align normed_space.dual_pairing_separating_left NormedSpace.dualPairing_separatingLeft
end General
section BidualIsometry
variable (๐ : Type v) [IsROrC ๐] {E : Type u} [NormedAddCommGroup E] [NormedSpace ๐ E]
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M := by
classical
by_cases h : x = 0
ยท simp only [h, hMp, norm_zero]
ยท obtain โจf, hfโ, hfxโฉ : โ f : E โL[๐] ๐, โfโ = 1 โง f x = โxโ := exists_dual_vector ๐ x h
calc
โxโ = โ(โxโ : ๐)โ := IsROrC.norm_coe_norm.symm
_ = โf xโ := by rw [hfx]
_ โค M * โfโ := (hM f)
_ = M := by rw [hfโ, mul_one]
#align normed_space.norm_le_dual_bound NormedSpace.norm_le_dual_bound
theorem eq_zero_of_forall_dual_eq_zero {x : E} (h : โ f : Dual ๐ E, f x = (0 : ๐)) : x = 0 :=
norm_le_zero_iff.mp (norm_le_dual_bound ๐ x le_rfl fun f => by simp [h f])
#align normed_space.eq_zero_of_forall_dual_eq_zero NormedSpace.eq_zero_of_forall_dual_eq_zero
theorem eq_zero_iff_forall_dual_eq_zero (x : E) : x = 0 โ โ g : Dual ๐ E, g x = 0 :=
โจfun hx => by simp [hx], fun h => eq_zero_of_forall_dual_eq_zero ๐ hโฉ
#align normed_space.eq_zero_iff_forall_dual_eq_zero NormedSpace.eq_zero_iff_forall_dual_eq_zero
/-- See also `geometric_hahn_banach_point_point`. -/
theorem eq_iff_forall_dual_eq {x y : E} : x = y โ โ g : Dual ๐ E, g x = g y := by
rw [โ sub_eq_zero, eq_zero_iff_forall_dual_eq_zero ๐ (x - y)]
simp [sub_eq_zero]
#align normed_space.eq_iff_forall_dual_eq NormedSpace.eq_iff_forall_dual_eq
/-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E โโแตข[๐] Dual ๐ (Dual ๐ E) :=
{ inclusionInDoubleDual ๐ E with
norm_map' := by
intro x
apply le_antisymm
ยท exact double_dual_bound ๐ E x
rw [ContinuousLinearMap.norm_def]
refine' le_csInf ContinuousLinearMap.bounds_nonempty _
rintro c โจhc1, hc2โฉ
exact norm_le_dual_bound ๐ x hc1 hc2 }
#align normed_space.inclusion_in_double_dual_li NormedSpace.inclusionInDoubleDualLi
end BidualIsometry
section PolarSets
open Metric Set NormedSpace
/-- Given a subset `s` in a normed space `E` (over a field `๐`), the polar
`polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals which
evaluate to something of norm at most one at all points `z โ s`. -/
def polar (๐ : Type*) [NontriviallyNormedField ๐] {E : Type*} [SeminormedAddCommGroup E]
[NormedSpace ๐ E] : Set E โ Set (Dual ๐ E) :=
(dualPairing ๐ E).flip.polar
#align normed_space.polar NormedSpace.polar
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable {E : Type*} [SeminormedAddCommGroup E] [NormedSpace ๐ E]
theorem mem_polar_iff {x' : Dual ๐ E} (s : Set E) : x' โ polar ๐ s โ โ z โ s, โx' zโ โค 1 :=
Iff.rfl
#align normed_space.mem_polar_iff NormedSpace.mem_polar_iff
@[simp]
theorem polar_univ : polar ๐ (univ : Set E) = {(0 : Dual ๐ E)} :=
(dualPairing ๐ E).flip.polar_univ
(LinearMap.flip_separatingRight.mpr (dualPairing_separatingLeft ๐ E))
#align normed_space.polar_univ NormedSpace.polar_univ
theorem isClosed_polar (s : Set E) : IsClosed (polar ๐ s) := by
dsimp only [NormedSpace.polar]
simp only [LinearMap.polar_eq_iInter, LinearMap.flip_apply]
refine' isClosed_biInter fun z _ => _
exact isClosed_Iic.preimage (ContinuousLinearMap.apply ๐ ๐ z).continuous.norm
#align normed_space.is_closed_polar NormedSpace.isClosed_polar
@[simp]
theorem polar_closure (s : Set E) : polar ๐ (closure s) = polar ๐ s :=
((dualPairing ๐ E).flip.polar_antitone subset_closure).antisymm <|
(dualPairing ๐ E).flip.polar_gc.l_le <|
closure_minimal ((dualPairing ๐ E).flip.polar_gc.le_u_l s) <| by
simpa [LinearMap.flip_flip] using
(isClosed_polar _ _).preimage (inclusionInDoubleDual ๐ E).continuous
#align normed_space.polar_closure NormedSpace.polar_closure
variable {๐}
/-- If `x'` is a dual element such that the norms `โx' zโ` are bounded for `z โ s`, then a
small scalar multiple of `x'` is in `polar ๐ s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual ๐ E} {c : ๐} (hc : โ z, z โ s โ โx' zโ โค โcโ) :
cโปยน โข x' โ polar ๐ s := by
by_cases c_zero : c = 0
ยท simp only [c_zero, inv_zero, zero_smul]
exact (dualPairing ๐ E).flip.zero_mem_polar _
have eq : โ z, โcโปยน โข x' zโ = โcโปยนโ * โx' zโ := fun z => norm_smul cโปยน _
have le : โ z, z โ s โ โcโปยน โข x' zโ โค โcโปยนโ * โcโ := by
intro z hzs
rw [eq z]
apply mul_le_mul (le_of_eq rfl) (hc z hzs) (norm_nonneg _) (norm_nonneg _)
have cancel : โcโปยนโ * โcโ = 1 := by
simp only [c_zero, norm_eq_zero, Ne.def, not_false_iff, inv_mul_cancel, norm_inv]
rwa [cancel] at le
#align normed_space.smul_mem_polar NormedSpace.smul_mem_polar
theorem polar_ball_subset_closedBall_div {c : ๐} (hc : 1 < โcโ) {r : โ} (hr : 0 < r) :
polar ๐ (ball (0 : E) r) โ closedBall (0 : Dual ๐ E) (โcโ / r) := by
intro x' hx'
rw [mem_polar_iff] at hx'
simp only [polar, mem_setOf, mem_closedBall_zero_iff, mem_ball_zero_iff] at *
have hcr : 0 < โcโ / r := div_pos (zero_lt_one.trans hc) hr
refine' ContinuousLinearMap.op_norm_le_of_shell hr hcr.le hc fun x hโ hโ => _
calc
โx' xโ โค 1 := hx' _ hโ
_ โค โcโ / r * โxโ := (inv_pos_le_iff_one_le_mul' hcr).1 (by rwa [inv_div])
#align normed_space.polar_ball_subset_closed_ball_div NormedSpace.polar_ball_subset_closedBall_div
variable (๐)
theorem closedBall_inv_subset_polar_closedBall {r : โ} :
closedBall (0 : Dual ๐ E) rโปยน โ polar ๐ (closedBall (0 : E) r) := fun x' hx' x hx =>
calc
โx' xโ โค โx'โ * โxโ := x'.le_op_norm x
_ โค rโปยน * r :=
(mul_le_mul (mem_closedBall_zero_iff.1 hx') (mem_closedBall_zero_iff.1 hx) (norm_nonneg _)
(dist_nonneg.trans hx'))
_ = r / r := (inv_mul_eq_div _ _)
_ โค 1 := div_self_le_one r
#align normed_space.closed_ball_inv_subset_polar_closed_ball NormedSpace.closedBall_inv_subset_polar_closedBall
/-- The `polar` of closed ball in a normed space `E` is the closed ball of the dual with
inverse radius. -/
theorem polar_closedBall {๐ E : Type*} [IsROrC ๐] [NormedAddCommGroup E] [NormedSpace ๐ E] {r : โ}
(hr : 0 < r) : polar ๐ (closedBall (0 : E) r) = closedBall (0 : Dual ๐ E) rโปยน := by
|
refine' Subset.antisymm _ (closedBall_inv_subset_polar_closedBall ๐)
|
/-- The `polar` of closed ball in a normed space `E` is the closed ball of the dual with
inverse radius. -/
theorem polar_closedBall {๐ E : Type*} [IsROrC ๐] [NormedAddCommGroup E] [NormedSpace ๐ E] {r : โ}
(hr : 0 < r) : polar ๐ (closedBall (0 : E) r) = closedBall (0 : Dual ๐ E) rโปยน := by
|
Mathlib.Analysis.NormedSpace.Dual.243_0.WirVfj6f5oiZZ2w
|
/-- The `polar` of closed ball in a normed space `E` is the closed ball of the dual with
inverse radius. -/
theorem polar_closedBall {๐ E : Type*} [IsROrC ๐] [NormedAddCommGroup E] [NormedSpace ๐ E] {r : โ}
(hr : 0 < r) : polar ๐ (closedBall (0 : E) r) = closedBall (0 : Dual ๐ E) rโปยน
|
Mathlib_Analysis_NormedSpace_Dual
|
๐โ : Type u_1
instโโต : NontriviallyNormedField ๐โ
Eโ : Type u_2
instโโด : SeminormedAddCommGroup Eโ
instโยณ : NormedSpace ๐โ Eโ
๐ : Type u_3
E : Type u_4
instโยฒ : IsROrC ๐
instโยน : NormedAddCommGroup E
instโ : NormedSpace ๐ E
r : โ
hr : 0 < r
โข polar ๐ (closedBall 0 r) โ closedBall 0 rโปยน
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# The topological dual of a normed space
In this file we define the topological dual `NormedSpace.Dual` of a normed space, and the
continuous linear map `NormedSpace.inclusionInDoubleDual` from a normed space into its double
dual.
For base field `๐ = โ` or `๐ = โ`, this map is actually an isometric embedding; we provide a
version `NormedSpace.inclusionInDoubleDualLi` of the map which is of type a bundled linear
isometric embedding, `E โโแตข[๐] (Dual ๐ (Dual ๐ E))`.
Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
theory for `SeminormedAddCommGroup` and we specialize to `NormedAddCommGroup` when needed.
## Main definitions
* `inclusionInDoubleDual` and `inclusionInDoubleDualLi` are the inclusion of a normed space
in its double dual, considered as a bounded linear map and as a linear isometry, respectively.
* `polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals `x'` for which
`โx' zโ โค 1` for every `z โ s`.
## Tags
dual
-/
noncomputable section
open Classical Topology Bornology
universe u v
namespace NormedSpace
section General
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable (E : Type*) [SeminormedAddCommGroup E] [NormedSpace ๐ E]
variable (F : Type*) [NormedAddCommGroup F] [NormedSpace ๐ F]
/-- The topological dual of a seminormed space `E`. -/
abbrev Dual : Type _ := E โL[๐] ๐
#align normed_space.dual NormedSpace.Dual
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_eq until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : NormedSpace ๐ (Dual ๐ E) := inferInstance
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_le until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : SeminormedAddCommGroup (Dual ๐ E) := inferInstance
/-- The inclusion of a normed space in its double (topological) dual, considered
as a bounded linear map. -/
def inclusionInDoubleDual : E โL[๐] Dual ๐ (Dual ๐ E) :=
ContinuousLinearMap.apply ๐ ๐
#align normed_space.inclusion_in_double_dual NormedSpace.inclusionInDoubleDual
@[simp]
theorem dual_def (x : E) (f : Dual ๐ E) : inclusionInDoubleDual ๐ E x f = f x :=
rfl
#align normed_space.dual_def NormedSpace.dual_def
theorem inclusionInDoubleDual_norm_eq :
โinclusionInDoubleDual ๐ Eโ = โContinuousLinearMap.id ๐ (Dual ๐ E)โ :=
ContinuousLinearMap.op_norm_flip _
#align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1 := by
rw [inclusionInDoubleDual_norm_eq]
exact ContinuousLinearMap.norm_id_le
#align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
theorem double_dual_bound (x : E) : โ(inclusionInDoubleDual ๐ E) xโ โค โxโ := by
simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusionInDoubleDual_norm_le ๐ E) x
#align normed_space.double_dual_bound NormedSpace.double_dual_bound
/-- The dual pairing as a bilinear form. -/
def dualPairing : Dual ๐ E โโ[๐] E โโ[๐] ๐ :=
ContinuousLinearMap.coeLM ๐
#align normed_space.dual_pairing NormedSpace.dualPairing
@[simp]
theorem dualPairing_apply {v : Dual ๐ E} {x : E} : dualPairing ๐ E v x = v x :=
rfl
#align normed_space.dual_pairing_apply NormedSpace.dualPairing_apply
theorem dualPairing_separatingLeft : (dualPairing ๐ E).SeparatingLeft := by
rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]
exact ContinuousLinearMap.coe_injective
#align normed_space.dual_pairing_separating_left NormedSpace.dualPairing_separatingLeft
end General
section BidualIsometry
variable (๐ : Type v) [IsROrC ๐] {E : Type u} [NormedAddCommGroup E] [NormedSpace ๐ E]
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M := by
classical
by_cases h : x = 0
ยท simp only [h, hMp, norm_zero]
ยท obtain โจf, hfโ, hfxโฉ : โ f : E โL[๐] ๐, โfโ = 1 โง f x = โxโ := exists_dual_vector ๐ x h
calc
โxโ = โ(โxโ : ๐)โ := IsROrC.norm_coe_norm.symm
_ = โf xโ := by rw [hfx]
_ โค M * โfโ := (hM f)
_ = M := by rw [hfโ, mul_one]
#align normed_space.norm_le_dual_bound NormedSpace.norm_le_dual_bound
theorem eq_zero_of_forall_dual_eq_zero {x : E} (h : โ f : Dual ๐ E, f x = (0 : ๐)) : x = 0 :=
norm_le_zero_iff.mp (norm_le_dual_bound ๐ x le_rfl fun f => by simp [h f])
#align normed_space.eq_zero_of_forall_dual_eq_zero NormedSpace.eq_zero_of_forall_dual_eq_zero
theorem eq_zero_iff_forall_dual_eq_zero (x : E) : x = 0 โ โ g : Dual ๐ E, g x = 0 :=
โจfun hx => by simp [hx], fun h => eq_zero_of_forall_dual_eq_zero ๐ hโฉ
#align normed_space.eq_zero_iff_forall_dual_eq_zero NormedSpace.eq_zero_iff_forall_dual_eq_zero
/-- See also `geometric_hahn_banach_point_point`. -/
theorem eq_iff_forall_dual_eq {x y : E} : x = y โ โ g : Dual ๐ E, g x = g y := by
rw [โ sub_eq_zero, eq_zero_iff_forall_dual_eq_zero ๐ (x - y)]
simp [sub_eq_zero]
#align normed_space.eq_iff_forall_dual_eq NormedSpace.eq_iff_forall_dual_eq
/-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E โโแตข[๐] Dual ๐ (Dual ๐ E) :=
{ inclusionInDoubleDual ๐ E with
norm_map' := by
intro x
apply le_antisymm
ยท exact double_dual_bound ๐ E x
rw [ContinuousLinearMap.norm_def]
refine' le_csInf ContinuousLinearMap.bounds_nonempty _
rintro c โจhc1, hc2โฉ
exact norm_le_dual_bound ๐ x hc1 hc2 }
#align normed_space.inclusion_in_double_dual_li NormedSpace.inclusionInDoubleDualLi
end BidualIsometry
section PolarSets
open Metric Set NormedSpace
/-- Given a subset `s` in a normed space `E` (over a field `๐`), the polar
`polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals which
evaluate to something of norm at most one at all points `z โ s`. -/
def polar (๐ : Type*) [NontriviallyNormedField ๐] {E : Type*} [SeminormedAddCommGroup E]
[NormedSpace ๐ E] : Set E โ Set (Dual ๐ E) :=
(dualPairing ๐ E).flip.polar
#align normed_space.polar NormedSpace.polar
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable {E : Type*} [SeminormedAddCommGroup E] [NormedSpace ๐ E]
theorem mem_polar_iff {x' : Dual ๐ E} (s : Set E) : x' โ polar ๐ s โ โ z โ s, โx' zโ โค 1 :=
Iff.rfl
#align normed_space.mem_polar_iff NormedSpace.mem_polar_iff
@[simp]
theorem polar_univ : polar ๐ (univ : Set E) = {(0 : Dual ๐ E)} :=
(dualPairing ๐ E).flip.polar_univ
(LinearMap.flip_separatingRight.mpr (dualPairing_separatingLeft ๐ E))
#align normed_space.polar_univ NormedSpace.polar_univ
theorem isClosed_polar (s : Set E) : IsClosed (polar ๐ s) := by
dsimp only [NormedSpace.polar]
simp only [LinearMap.polar_eq_iInter, LinearMap.flip_apply]
refine' isClosed_biInter fun z _ => _
exact isClosed_Iic.preimage (ContinuousLinearMap.apply ๐ ๐ z).continuous.norm
#align normed_space.is_closed_polar NormedSpace.isClosed_polar
@[simp]
theorem polar_closure (s : Set E) : polar ๐ (closure s) = polar ๐ s :=
((dualPairing ๐ E).flip.polar_antitone subset_closure).antisymm <|
(dualPairing ๐ E).flip.polar_gc.l_le <|
closure_minimal ((dualPairing ๐ E).flip.polar_gc.le_u_l s) <| by
simpa [LinearMap.flip_flip] using
(isClosed_polar _ _).preimage (inclusionInDoubleDual ๐ E).continuous
#align normed_space.polar_closure NormedSpace.polar_closure
variable {๐}
/-- If `x'` is a dual element such that the norms `โx' zโ` are bounded for `z โ s`, then a
small scalar multiple of `x'` is in `polar ๐ s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual ๐ E} {c : ๐} (hc : โ z, z โ s โ โx' zโ โค โcโ) :
cโปยน โข x' โ polar ๐ s := by
by_cases c_zero : c = 0
ยท simp only [c_zero, inv_zero, zero_smul]
exact (dualPairing ๐ E).flip.zero_mem_polar _
have eq : โ z, โcโปยน โข x' zโ = โcโปยนโ * โx' zโ := fun z => norm_smul cโปยน _
have le : โ z, z โ s โ โcโปยน โข x' zโ โค โcโปยนโ * โcโ := by
intro z hzs
rw [eq z]
apply mul_le_mul (le_of_eq rfl) (hc z hzs) (norm_nonneg _) (norm_nonneg _)
have cancel : โcโปยนโ * โcโ = 1 := by
simp only [c_zero, norm_eq_zero, Ne.def, not_false_iff, inv_mul_cancel, norm_inv]
rwa [cancel] at le
#align normed_space.smul_mem_polar NormedSpace.smul_mem_polar
theorem polar_ball_subset_closedBall_div {c : ๐} (hc : 1 < โcโ) {r : โ} (hr : 0 < r) :
polar ๐ (ball (0 : E) r) โ closedBall (0 : Dual ๐ E) (โcโ / r) := by
intro x' hx'
rw [mem_polar_iff] at hx'
simp only [polar, mem_setOf, mem_closedBall_zero_iff, mem_ball_zero_iff] at *
have hcr : 0 < โcโ / r := div_pos (zero_lt_one.trans hc) hr
refine' ContinuousLinearMap.op_norm_le_of_shell hr hcr.le hc fun x hโ hโ => _
calc
โx' xโ โค 1 := hx' _ hโ
_ โค โcโ / r * โxโ := (inv_pos_le_iff_one_le_mul' hcr).1 (by rwa [inv_div])
#align normed_space.polar_ball_subset_closed_ball_div NormedSpace.polar_ball_subset_closedBall_div
variable (๐)
theorem closedBall_inv_subset_polar_closedBall {r : โ} :
closedBall (0 : Dual ๐ E) rโปยน โ polar ๐ (closedBall (0 : E) r) := fun x' hx' x hx =>
calc
โx' xโ โค โx'โ * โxโ := x'.le_op_norm x
_ โค rโปยน * r :=
(mul_le_mul (mem_closedBall_zero_iff.1 hx') (mem_closedBall_zero_iff.1 hx) (norm_nonneg _)
(dist_nonneg.trans hx'))
_ = r / r := (inv_mul_eq_div _ _)
_ โค 1 := div_self_le_one r
#align normed_space.closed_ball_inv_subset_polar_closed_ball NormedSpace.closedBall_inv_subset_polar_closedBall
/-- The `polar` of closed ball in a normed space `E` is the closed ball of the dual with
inverse radius. -/
theorem polar_closedBall {๐ E : Type*} [IsROrC ๐] [NormedAddCommGroup E] [NormedSpace ๐ E] {r : โ}
(hr : 0 < r) : polar ๐ (closedBall (0 : E) r) = closedBall (0 : Dual ๐ E) rโปยน := by
refine' Subset.antisymm _ (closedBall_inv_subset_polar_closedBall ๐)
|
intro x' h
|
/-- The `polar` of closed ball in a normed space `E` is the closed ball of the dual with
inverse radius. -/
theorem polar_closedBall {๐ E : Type*} [IsROrC ๐] [NormedAddCommGroup E] [NormedSpace ๐ E] {r : โ}
(hr : 0 < r) : polar ๐ (closedBall (0 : E) r) = closedBall (0 : Dual ๐ E) rโปยน := by
refine' Subset.antisymm _ (closedBall_inv_subset_polar_closedBall ๐)
|
Mathlib.Analysis.NormedSpace.Dual.243_0.WirVfj6f5oiZZ2w
|
/-- The `polar` of closed ball in a normed space `E` is the closed ball of the dual with
inverse radius. -/
theorem polar_closedBall {๐ E : Type*} [IsROrC ๐] [NormedAddCommGroup E] [NormedSpace ๐ E] {r : โ}
(hr : 0 < r) : polar ๐ (closedBall (0 : E) r) = closedBall (0 : Dual ๐ E) rโปยน
|
Mathlib_Analysis_NormedSpace_Dual
|
๐โ : Type u_1
instโโต : NontriviallyNormedField ๐โ
Eโ : Type u_2
instโโด : SeminormedAddCommGroup Eโ
instโยณ : NormedSpace ๐โ Eโ
๐ : Type u_3
E : Type u_4
instโยฒ : IsROrC ๐
instโยน : NormedAddCommGroup E
instโ : NormedSpace ๐ E
r : โ
hr : 0 < r
x' : Dual ๐ E
h : x' โ polar ๐ (closedBall 0 r)
โข x' โ closedBall 0 rโปยน
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.IsROrC
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# The topological dual of a normed space
In this file we define the topological dual `NormedSpace.Dual` of a normed space, and the
continuous linear map `NormedSpace.inclusionInDoubleDual` from a normed space into its double
dual.
For base field `๐ = โ` or `๐ = โ`, this map is actually an isometric embedding; we provide a
version `NormedSpace.inclusionInDoubleDualLi` of the map which is of type a bundled linear
isometric embedding, `E โโแตข[๐] (Dual ๐ (Dual ๐ E))`.
Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
theory for `SeminormedAddCommGroup` and we specialize to `NormedAddCommGroup` when needed.
## Main definitions
* `inclusionInDoubleDual` and `inclusionInDoubleDualLi` are the inclusion of a normed space
in its double dual, considered as a bounded linear map and as a linear isometry, respectively.
* `polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals `x'` for which
`โx' zโ โค 1` for every `z โ s`.
## Tags
dual
-/
noncomputable section
open Classical Topology Bornology
universe u v
namespace NormedSpace
section General
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable (E : Type*) [SeminormedAddCommGroup E] [NormedSpace ๐ E]
variable (F : Type*) [NormedAddCommGroup F] [NormedSpace ๐ F]
/-- The topological dual of a seminormed space `E`. -/
abbrev Dual : Type _ := E โL[๐] ๐
#align normed_space.dual NormedSpace.Dual
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_eq until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : NormedSpace ๐ (Dual ๐ E) := inferInstance
-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_le until
-- leanprover/lean4#2522 is resolved; remove once fixed
instance : SeminormedAddCommGroup (Dual ๐ E) := inferInstance
/-- The inclusion of a normed space in its double (topological) dual, considered
as a bounded linear map. -/
def inclusionInDoubleDual : E โL[๐] Dual ๐ (Dual ๐ E) :=
ContinuousLinearMap.apply ๐ ๐
#align normed_space.inclusion_in_double_dual NormedSpace.inclusionInDoubleDual
@[simp]
theorem dual_def (x : E) (f : Dual ๐ E) : inclusionInDoubleDual ๐ E x f = f x :=
rfl
#align normed_space.dual_def NormedSpace.dual_def
theorem inclusionInDoubleDual_norm_eq :
โinclusionInDoubleDual ๐ Eโ = โContinuousLinearMap.id ๐ (Dual ๐ E)โ :=
ContinuousLinearMap.op_norm_flip _
#align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
theorem inclusionInDoubleDual_norm_le : โinclusionInDoubleDual ๐ Eโ โค 1 := by
rw [inclusionInDoubleDual_norm_eq]
exact ContinuousLinearMap.norm_id_le
#align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
theorem double_dual_bound (x : E) : โ(inclusionInDoubleDual ๐ E) xโ โค โxโ := by
simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusionInDoubleDual_norm_le ๐ E) x
#align normed_space.double_dual_bound NormedSpace.double_dual_bound
/-- The dual pairing as a bilinear form. -/
def dualPairing : Dual ๐ E โโ[๐] E โโ[๐] ๐ :=
ContinuousLinearMap.coeLM ๐
#align normed_space.dual_pairing NormedSpace.dualPairing
@[simp]
theorem dualPairing_apply {v : Dual ๐ E} {x : E} : dualPairing ๐ E v x = v x :=
rfl
#align normed_space.dual_pairing_apply NormedSpace.dualPairing_apply
theorem dualPairing_separatingLeft : (dualPairing ๐ E).SeparatingLeft := by
rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]
exact ContinuousLinearMap.coe_injective
#align normed_space.dual_pairing_separating_left NormedSpace.dualPairing_separatingLeft
end General
section BidualIsometry
variable (๐ : Type v) [IsROrC ๐] {E : Type u} [NormedAddCommGroup E] [NormedSpace ๐ E]
/-- If one controls the norm of every `f x`, then one controls the norm of `x`.
Compare `ContinuousLinearMap.op_norm_le_bound`. -/
theorem norm_le_dual_bound (x : E) {M : โ} (hMp : 0 โค M) (hM : โ f : Dual ๐ E, โf xโ โค M * โfโ) :
โxโ โค M := by
classical
by_cases h : x = 0
ยท simp only [h, hMp, norm_zero]
ยท obtain โจf, hfโ, hfxโฉ : โ f : E โL[๐] ๐, โfโ = 1 โง f x = โxโ := exists_dual_vector ๐ x h
calc
โxโ = โ(โxโ : ๐)โ := IsROrC.norm_coe_norm.symm
_ = โf xโ := by rw [hfx]
_ โค M * โfโ := (hM f)
_ = M := by rw [hfโ, mul_one]
#align normed_space.norm_le_dual_bound NormedSpace.norm_le_dual_bound
theorem eq_zero_of_forall_dual_eq_zero {x : E} (h : โ f : Dual ๐ E, f x = (0 : ๐)) : x = 0 :=
norm_le_zero_iff.mp (norm_le_dual_bound ๐ x le_rfl fun f => by simp [h f])
#align normed_space.eq_zero_of_forall_dual_eq_zero NormedSpace.eq_zero_of_forall_dual_eq_zero
theorem eq_zero_iff_forall_dual_eq_zero (x : E) : x = 0 โ โ g : Dual ๐ E, g x = 0 :=
โจfun hx => by simp [hx], fun h => eq_zero_of_forall_dual_eq_zero ๐ hโฉ
#align normed_space.eq_zero_iff_forall_dual_eq_zero NormedSpace.eq_zero_iff_forall_dual_eq_zero
/-- See also `geometric_hahn_banach_point_point`. -/
theorem eq_iff_forall_dual_eq {x y : E} : x = y โ โ g : Dual ๐ E, g x = g y := by
rw [โ sub_eq_zero, eq_zero_iff_forall_dual_eq_zero ๐ (x - y)]
simp [sub_eq_zero]
#align normed_space.eq_iff_forall_dual_eq NormedSpace.eq_iff_forall_dual_eq
/-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E โโแตข[๐] Dual ๐ (Dual ๐ E) :=
{ inclusionInDoubleDual ๐ E with
norm_map' := by
intro x
apply le_antisymm
ยท exact double_dual_bound ๐ E x
rw [ContinuousLinearMap.norm_def]
refine' le_csInf ContinuousLinearMap.bounds_nonempty _
rintro c โจhc1, hc2โฉ
exact norm_le_dual_bound ๐ x hc1 hc2 }
#align normed_space.inclusion_in_double_dual_li NormedSpace.inclusionInDoubleDualLi
end BidualIsometry
section PolarSets
open Metric Set NormedSpace
/-- Given a subset `s` in a normed space `E` (over a field `๐`), the polar
`polar ๐ s` is the subset of `Dual ๐ E` consisting of those functionals which
evaluate to something of norm at most one at all points `z โ s`. -/
def polar (๐ : Type*) [NontriviallyNormedField ๐] {E : Type*} [SeminormedAddCommGroup E]
[NormedSpace ๐ E] : Set E โ Set (Dual ๐ E) :=
(dualPairing ๐ E).flip.polar
#align normed_space.polar NormedSpace.polar
variable (๐ : Type*) [NontriviallyNormedField ๐]
variable {E : Type*} [SeminormedAddCommGroup E] [NormedSpace ๐ E]
theorem mem_polar_iff {x' : Dual ๐ E} (s : Set E) : x' โ polar ๐ s โ โ z โ s, โx' zโ โค 1 :=
Iff.rfl
#align normed_space.mem_polar_iff NormedSpace.mem_polar_iff
@[simp]
theorem polar_univ : polar ๐ (univ : Set E) = {(0 : Dual ๐ E)} :=
(dualPairing ๐ E).flip.polar_univ
(LinearMap.flip_separatingRight.mpr (dualPairing_separatingLeft ๐ E))
#align normed_space.polar_univ NormedSpace.polar_univ
theorem isClosed_polar (s : Set E) : IsClosed (polar ๐ s) := by
dsimp only [NormedSpace.polar]
simp only [LinearMap.polar_eq_iInter, LinearMap.flip_apply]
refine' isClosed_biInter fun z _ => _
exact isClosed_Iic.preimage (ContinuousLinearMap.apply ๐ ๐ z).continuous.norm
#align normed_space.is_closed_polar NormedSpace.isClosed_polar
@[simp]
theorem polar_closure (s : Set E) : polar ๐ (closure s) = polar ๐ s :=
((dualPairing ๐ E).flip.polar_antitone subset_closure).antisymm <|
(dualPairing ๐ E).flip.polar_gc.l_le <|
closure_minimal ((dualPairing ๐ E).flip.polar_gc.le_u_l s) <| by
simpa [LinearMap.flip_flip] using
(isClosed_polar _ _).preimage (inclusionInDoubleDual ๐ E).continuous
#align normed_space.polar_closure NormedSpace.polar_closure
variable {๐}
/-- If `x'` is a dual element such that the norms `โx' zโ` are bounded for `z โ s`, then a
small scalar multiple of `x'` is in `polar ๐ s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual ๐ E} {c : ๐} (hc : โ z, z โ s โ โx' zโ โค โcโ) :
cโปยน โข x' โ polar ๐ s := by
by_cases c_zero : c = 0
ยท simp only [c_zero, inv_zero, zero_smul]
exact (dualPairing ๐ E).flip.zero_mem_polar _
have eq : โ z, โcโปยน โข x' zโ = โcโปยนโ * โx' zโ := fun z => norm_smul cโปยน _
have le : โ z, z โ s โ โcโปยน โข x' zโ โค โcโปยนโ * โcโ := by
intro z hzs
rw [eq z]
apply mul_le_mul (le_of_eq rfl) (hc z hzs) (norm_nonneg _) (norm_nonneg _)
have cancel : โcโปยนโ * โcโ = 1 := by
simp only [c_zero, norm_eq_zero, Ne.def, not_false_iff, inv_mul_cancel, norm_inv]
rwa [cancel] at le
#align normed_space.smul_mem_polar NormedSpace.smul_mem_polar
theorem polar_ball_subset_closedBall_div {c : ๐} (hc : 1 < โcโ) {r : โ} (hr : 0 < r) :
polar ๐ (ball (0 : E) r) โ closedBall (0 : Dual ๐ E) (โcโ / r) := by
intro x' hx'
rw [mem_polar_iff] at hx'
simp only [polar, mem_setOf, mem_closedBall_zero_iff, mem_ball_zero_iff] at *
have hcr : 0 < โcโ / r := div_pos (zero_lt_one.trans hc) hr
refine' ContinuousLinearMap.op_norm_le_of_shell hr hcr.le hc fun x hโ hโ => _
calc
โx' xโ โค 1 := hx' _ hโ
_ โค โcโ / r * โxโ := (inv_pos_le_iff_one_le_mul' hcr).1 (by rwa [inv_div])
#align normed_space.polar_ball_subset_closed_ball_div NormedSpace.polar_ball_subset_closedBall_div
variable (๐)
theorem closedBall_inv_subset_polar_closedBall {r : โ} :
closedBall (0 : Dual ๐ E) rโปยน โ polar ๐ (closedBall (0 : E) r) := fun x' hx' x hx =>
calc
โx' xโ โค โx'โ * โxโ := x'.le_op_norm x
_ โค rโปยน * r :=
(mul_le_mul (mem_closedBall_zero_iff.1 hx') (mem_closedBall_zero_iff.1 hx) (norm_nonneg _)
(dist_nonneg.trans hx'))
_ = r / r := (inv_mul_eq_div _ _)
_ โค 1 := div_self_le_one r
#align normed_space.closed_ball_inv_subset_polar_closed_ball NormedSpace.closedBall_inv_subset_polar_closedBall
/-- The `polar` of closed ball in a normed space `E` is the closed ball of the dual with
inverse radius. -/
theorem polar_closedBall {๐ E : Type*} [IsROrC ๐] [NormedAddCommGroup E] [NormedSpace ๐ E] {r : โ}
(hr : 0 < r) : polar ๐ (closedBall (0 : E) r) = closedBall (0 : Dual ๐ E) rโปยน := by
refine' Subset.antisymm _ (closedBall_inv_subset_polar_closedBall ๐)
intro x' h
|
simp only [mem_closedBall_zero_iff]
|
/-- The `polar` of closed ball in a normed space `E` is the closed ball of the dual with
inverse radius. -/
theorem polar_closedBall {๐ E : Type*} [IsROrC ๐] [NormedAddCommGroup E] [NormedSpace ๐ E] {r : โ}
(hr : 0 < r) : polar ๐ (closedBall (0 : E) r) = closedBall (0 : Dual ๐ E) rโปยน := by
refine' Subset.antisymm _ (closedBall_inv_subset_polar_closedBall ๐)
intro x' h
|
Mathlib.Analysis.NormedSpace.Dual.243_0.WirVfj6f5oiZZ2w
|
/-- The `polar` of closed ball in a normed space `E` is the closed ball of the dual with
inverse radius. -/
theorem polar_closedBall {๐ E : Type*} [IsROrC ๐] [NormedAddCommGroup E] [NormedSpace ๐ E] {r : โ}
(hr : 0 < r) : polar ๐ (closedBall (0 : E) r) = closedBall (0 : Dual ๐ E) rโปยน
|
Mathlib_Analysis_NormedSpace_Dual
|
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