state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
case a.mk
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : IsROrC 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
K : Submodule 𝕜 E
v : E
hv : v ∈ K
w : E
hw : w ∈ Kᗮ
⊢ w ∈ LinearMap.ker ((innerSL 𝕜) ↑{ 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, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/ma... | simpa using hw _ hv | /-- `Kᗮ` 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 : Kᗮ = ⨅ v : K, LinearMap.ker (innerSL 𝕜 (v : E)) := by
apply le_antisymm
· rw [le_iInf_iff]
rintro ⟨v, hv⟩ w hw
| Mathlib.Analysis.InnerProductSpace.Orthogonal.116_0.QXx0GYqLoAbtfq1 | /-- `Kᗮ` 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 : Kᗮ = ⨅ v : K, LinearMap.ker (innerSL 𝕜 (v : E)) | Mathlib_Analysis_InnerProductSpace_Orthogonal |
case a
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : IsROrC 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
K : Submodule 𝕜 E
⊢ ⨅ v, LinearMap.ker ((innerSL 𝕜) ↑v) ≤ Kᗮ | /-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/ma... | intro v hv w hw | /-- `Kᗮ` 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 : Kᗮ = ⨅ v : K, LinearMap.ker (innerSL 𝕜 (v : E)) := by
apply le_antisymm
· rw [le_iInf_iff]
rintro ⟨v, hv⟩ w hw
simpa using hw _ hv
· | Mathlib.Analysis.InnerProductSpace.Orthogonal.116_0.QXx0GYqLoAbtfq1 | /-- `Kᗮ` 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 : Kᗮ = ⨅ v : K, LinearMap.ker (innerSL 𝕜 (v : E)) | Mathlib_Analysis_InnerProductSpace_Orthogonal |
case a
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : IsROrC 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
K : Submodule 𝕜 E
v : E
hv : v ∈ ⨅ v, LinearMap.ker ((innerSL 𝕜) ↑v)
w : E
hw : w ∈ K
⊢ 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, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/ma... | simp only [mem_iInf] at hv | /-- `Kᗮ` 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 : Kᗮ = ⨅ v : K, LinearMap.ker (innerSL 𝕜 (v : E)) := by
apply le_antisymm
· rw [le_iInf_iff]
rintro ⟨v, hv⟩ w hw
simpa using hw _ hv
· intro... | Mathlib.Analysis.InnerProductSpace.Orthogonal.116_0.QXx0GYqLoAbtfq1 | /-- `Kᗮ` 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 : Kᗮ = ⨅ v : K, LinearMap.ker (innerSL 𝕜 (v : E)) | Mathlib_Analysis_InnerProductSpace_Orthogonal |
case a
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : IsROrC 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
K : Submodule 𝕜 E
v w : E
hw : w ∈ K
hv : ∀ (i : ↥K), v ∈ LinearMap.ker ((innerSL 𝕜) ↑i)
⊢ 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, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/ma... | exact hv ⟨w, hw⟩ | /-- `Kᗮ` 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 : Kᗮ = ⨅ v : K, LinearMap.ker (innerSL 𝕜 (v : E)) := by
apply le_antisymm
· rw [le_iInf_iff]
rintro ⟨v, hv⟩ w hw
simpa using hw _ hv
· intro... | Mathlib.Analysis.InnerProductSpace.Orthogonal.116_0.QXx0GYqLoAbtfq1 | /-- `Kᗮ` 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 : Kᗮ = ⨅ v : K, LinearMap.ker (innerSL 𝕜 (v : E)) | Mathlib_Analysis_InnerProductSpace_Orthogonal |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : IsROrC 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
K : Submodule 𝕜 E
⊢ IsClosed ↑Kᗮ | /-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/ma... | rw [orthogonal_eq_inter K] | /-- The orthogonal complement of any submodule `K` is closed. -/
theorem isClosed_orthogonal : IsClosed (Kᗮ : Set E) := by
| Mathlib.Analysis.InnerProductSpace.Orthogonal.128_0.QXx0GYqLoAbtfq1 | /-- The orthogonal complement of any submodule `K` is closed. -/
theorem isClosed_orthogonal : IsClosed (Kᗮ : Set E) | Mathlib_Analysis_InnerProductSpace_Orthogonal |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : IsROrC 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
K : Submodule 𝕜 E
⊢ IsClosed ↑(⨅ v, LinearMap.ker ((innerSL 𝕜) ↑v)) | /-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/ma... | have := fun v : K => ContinuousLinearMap.isClosed_ker (innerSL 𝕜 (v : E)) | /-- The orthogonal complement of any submodule `K` is closed. -/
theorem isClosed_orthogonal : IsClosed (Kᗮ : 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 (Kᗮ : Set E) | Mathlib_Analysis_InnerProductSpace_Orthogonal |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : IsROrC 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
K : Submodule 𝕜 E
this : ∀ (v : ↥K), IsClosed ↑(LinearMap.ker ((innerSL 𝕜) ↑v))
⊢ IsClosed ↑(⨅ v, LinearMap.ker ((innerSL 𝕜) ↑v)) | /-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/ma... | convert isClosed_iInter this | /-- The orthogonal complement of any submodule `K` is closed. -/
theorem isClosed_orthogonal : IsClosed (Kᗮ : Set E) := by
rw [orthogonal_eq_inter K]
have := fun v : K => ContinuousLinearMap.isClosed_ker (innerSL 𝕜 (v : E))
| Mathlib.Analysis.InnerProductSpace.Orthogonal.128_0.QXx0GYqLoAbtfq1 | /-- The orthogonal complement of any submodule `K` is closed. -/
theorem isClosed_orthogonal : IsClosed (Kᗮ : Set E) | Mathlib_Analysis_InnerProductSpace_Orthogonal |
case h.e'_3
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : IsROrC 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
K : Submodule 𝕜 E
this : ∀ (v : ↥K), IsClosed ↑(LinearMap.ker ((innerSL 𝕜) ↑v))
⊢ ↑(⨅ v, LinearMap.ker ((innerSL 𝕜) ↑v)) = ... | /-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/ma... | simp only [iInf_coe] | /-- The orthogonal complement of any submodule `K` is closed. -/
theorem isClosed_orthogonal : IsClosed (Kᗮ : Set E) := by
rw [orthogonal_eq_inter K]
have := fun v : K => ContinuousLinearMap.isClosed_ker (innerSL 𝕜 (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 (Kᗮ : Set E) | Mathlib_Analysis_InnerProductSpace_Orthogonal |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : IsROrC 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
K : Submodule 𝕜 E
⊢ ⊤ᗮ = ⊥ | /-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/ma... | ext x | @[simp]
theorem top_orthogonal_eq_bot : (⊤ : Submodule 𝕜 E)ᗮ = ⊥ := by
| Mathlib.Analysis.InnerProductSpace.Orthogonal.186_0.QXx0GYqLoAbtfq1 | @[simp]
theorem top_orthogonal_eq_bot : (⊤ : Submodule 𝕜 E)ᗮ = ⊥ | Mathlib_Analysis_InnerProductSpace_Orthogonal |
case h
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : IsROrC 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
K : Submodule 𝕜 E
x : E
⊢ x ∈ ⊤ᗮ ↔ x ∈ ⊥ | /-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/ma... | rw [mem_bot, mem_orthogonal] | @[simp]
theorem top_orthogonal_eq_bot : (⊤ : Submodule 𝕜 E)ᗮ = ⊥ := by
ext x
| Mathlib.Analysis.InnerProductSpace.Orthogonal.186_0.QXx0GYqLoAbtfq1 | @[simp]
theorem top_orthogonal_eq_bot : (⊤ : Submodule 𝕜 E)ᗮ = ⊥ | Mathlib_Analysis_InnerProductSpace_Orthogonal |
case h
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : IsROrC 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
K : Submodule 𝕜 E
x : E
⊢ (∀ u ∈ ⊤, inner u x = 0) ↔ 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, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/ma... | exact
⟨fun h => inner_self_eq_zero.mp (h x mem_top), by
rintro rfl
simp⟩ | @[simp]
theorem top_orthogonal_eq_bot : (⊤ : Submodule 𝕜 E)ᗮ = ⊥ := by
ext x
rw [mem_bot, mem_orthogonal]
| Mathlib.Analysis.InnerProductSpace.Orthogonal.186_0.QXx0GYqLoAbtfq1 | @[simp]
theorem top_orthogonal_eq_bot : (⊤ : Submodule 𝕜 E)ᗮ = ⊥ | Mathlib_Analysis_InnerProductSpace_Orthogonal |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : IsROrC 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
K : Submodule 𝕜 E
x : E
⊢ x = 0 → ∀ u ∈ ⊤, 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, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/ma... | rintro rfl | @[simp]
theorem top_orthogonal_eq_bot : (⊤ : Submodule 𝕜 E)ᗮ = ⊥ := by
ext x
rw [mem_bot, mem_orthogonal]
exact
⟨fun h => inner_self_eq_zero.mp (h x mem_top), by
| Mathlib.Analysis.InnerProductSpace.Orthogonal.186_0.QXx0GYqLoAbtfq1 | @[simp]
theorem top_orthogonal_eq_bot : (⊤ : Submodule 𝕜 E)ᗮ = ⊥ | Mathlib_Analysis_InnerProductSpace_Orthogonal |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : IsROrC 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
K : Submodule 𝕜 E
⊢ ∀ u ∈ ⊤, 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, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/ma... | simp | @[simp]
theorem top_orthogonal_eq_bot : (⊤ : Submodule 𝕜 E)ᗮ = ⊥ := by
ext x
rw [mem_bot, mem_orthogonal]
exact
⟨fun 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 : (⊤ : Submodule 𝕜 E)ᗮ = ⊥ | Mathlib_Analysis_InnerProductSpace_Orthogonal |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : IsROrC 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
K : Submodule 𝕜 E
⊢ ⊥ᗮ = ⊤ | /-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/ma... | rw [← top_orthogonal_eq_bot, eq_top_iff] | @[simp]
theorem bot_orthogonal_eq_top : (⊥ : Submodule 𝕜 E)ᗮ = ⊤ := by
| Mathlib.Analysis.InnerProductSpace.Orthogonal.196_0.QXx0GYqLoAbtfq1 | @[simp]
theorem bot_orthogonal_eq_top : (⊥ : Submodule 𝕜 E)ᗮ = ⊤ | Mathlib_Analysis_InnerProductSpace_Orthogonal |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : IsROrC 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
K : Submodule 𝕜 E
⊢ ⊤ ≤ ⊤ᗮᗮ | /-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/ma... | exact le_orthogonal_orthogonal ⊤ | @[simp]
theorem bot_orthogonal_eq_top : (⊥ : Submodule 𝕜 E)ᗮ = ⊤ := by
rw [← top_orthogonal_eq_bot, eq_top_iff]
| Mathlib.Analysis.InnerProductSpace.Orthogonal.196_0.QXx0GYqLoAbtfq1 | @[simp]
theorem bot_orthogonal_eq_top : (⊥ : Submodule 𝕜 E)ᗮ = ⊤ | Mathlib_Analysis_InnerProductSpace_Orthogonal |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : IsROrC 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
K : Submodule 𝕜 E
⊢ Kᗮ = ⊤ ↔ K = ⊥ | /-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/ma... | refine'
⟨_, by
rintro rfl
exact bot_orthogonal_eq_top⟩ | @[simp]
theorem orthogonal_eq_top_iff : Kᗮ = ⊤ ↔ K = ⊥ := by
| Mathlib.Analysis.InnerProductSpace.Orthogonal.202_0.QXx0GYqLoAbtfq1 | @[simp]
theorem orthogonal_eq_top_iff : Kᗮ = ⊤ ↔ K = ⊥ | Mathlib_Analysis_InnerProductSpace_Orthogonal |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : IsROrC 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
K : Submodule 𝕜 E
⊢ K = ⊥ → Kᗮ = ⊤ | /-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/ma... | rintro rfl | @[simp]
theorem orthogonal_eq_top_iff : Kᗮ = ⊤ ↔ K = ⊥ := by
refine'
⟨_, by
| Mathlib.Analysis.InnerProductSpace.Orthogonal.202_0.QXx0GYqLoAbtfq1 | @[simp]
theorem orthogonal_eq_top_iff : Kᗮ = ⊤ ↔ K = ⊥ | Mathlib_Analysis_InnerProductSpace_Orthogonal |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : IsROrC 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
⊢ ⊥ᗮ = ⊤ | /-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/ma... | exact bot_orthogonal_eq_top | @[simp]
theorem orthogonal_eq_top_iff : Kᗮ = ⊤ ↔ K = ⊥ := by
refine'
⟨_, by
rintro rfl
| Mathlib.Analysis.InnerProductSpace.Orthogonal.202_0.QXx0GYqLoAbtfq1 | @[simp]
theorem orthogonal_eq_top_iff : Kᗮ = ⊤ ↔ K = ⊥ | Mathlib_Analysis_InnerProductSpace_Orthogonal |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : IsROrC 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
K : Submodule 𝕜 E
⊢ Kᗮ = ⊤ → K = ⊥ | /-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/ma... | intro h | @[simp]
theorem orthogonal_eq_top_iff : Kᗮ = ⊤ ↔ K = ⊥ := by
refine'
⟨_, by
rintro rfl
exact bot_orthogonal_eq_top⟩
| Mathlib.Analysis.InnerProductSpace.Orthogonal.202_0.QXx0GYqLoAbtfq1 | @[simp]
theorem orthogonal_eq_top_iff : Kᗮ = ⊤ ↔ K = ⊥ | Mathlib_Analysis_InnerProductSpace_Orthogonal |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : IsROrC 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
K : Submodule 𝕜 E
h : Kᗮ = ⊤
⊢ K = ⊥ | /-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/ma... | have : K ⊓ Kᗮ = ⊥ := K.orthogonal_disjoint.eq_bot | @[simp]
theorem orthogonal_eq_top_iff : Kᗮ = ⊤ ↔ K = ⊥ := by
refine'
⟨_, by
rintro rfl
exact bot_orthogonal_eq_top⟩
intro h
| Mathlib.Analysis.InnerProductSpace.Orthogonal.202_0.QXx0GYqLoAbtfq1 | @[simp]
theorem orthogonal_eq_top_iff : Kᗮ = ⊤ ↔ K = ⊥ | Mathlib_Analysis_InnerProductSpace_Orthogonal |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : IsROrC 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
K : Submodule 𝕜 E
h : Kᗮ = ⊤
this : K ⊓ Kᗮ = ⊥
⊢ K = ⊥ | /-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/ma... | rwa [h, inf_comm, top_inf_eq] at this | @[simp]
theorem orthogonal_eq_top_iff : Kᗮ = ⊤ ↔ K = ⊥ := by
refine'
⟨_, by
rintro rfl
exact bot_orthogonal_eq_top⟩
intro h
have : K ⊓ Kᗮ = ⊥ := K.orthogonal_disjoint.eq_bot
| Mathlib.Analysis.InnerProductSpace.Orthogonal.202_0.QXx0GYqLoAbtfq1 | @[simp]
theorem orthogonal_eq_top_iff : Kᗮ = ⊤ ↔ K = ⊥ | Mathlib_Analysis_InnerProductSpace_Orthogonal |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : IsROrC 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
U : Submodule 𝕜 E
V : Set (Submodule 𝕜 E)
⊢ (∀ Uᵢ ∈ V, Uᵢ ⟂ U) ↔ ∀ Vᵢ ∈ V, U ⟂ Vᵢ | /-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/ma... | simp_rw [isOrtho_comm] | @[simp]
theorem isOrtho_sSup_right {U : Submodule 𝕜 E} {V : Set (Submodule 𝕜 E)} :
U ⟂ sSup V ↔ ∀ Vᵢ ∈ V, U ⟂ Vᵢ :=
isOrtho_comm.trans <| isOrtho_sSup_left.trans <| by | Mathlib.Analysis.InnerProductSpace.Orthogonal.355_0.QXx0GYqLoAbtfq1 | @[simp]
theorem isOrtho_sSup_right {U : Submodule 𝕜 E} {V : Set (Submodule 𝕜 E)} :
U ⟂ sSup V ↔ ∀ Vᵢ ∈ V, U ⟂ Vᵢ | Mathlib_Analysis_InnerProductSpace_Orthogonal |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : IsROrC 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
ι : Sort u_4
U : Submodule 𝕜 E
V : ι → Submodule 𝕜 E
⊢ (∀ (i : ι), V i ⟂ U) ↔ ∀ (i : ι), U ⟂ 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, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/ma... | simp_rw [isOrtho_comm] | @[simp]
theorem isOrtho_iSup_right {ι : Sort*} {U : Submodule 𝕜 E} {V : ι → Submodule 𝕜 E} :
U ⟂ iSup V ↔ ∀ i, U ⟂ V i :=
isOrtho_comm.trans <| isOrtho_iSup_left.trans <| by | Mathlib.Analysis.InnerProductSpace.Orthogonal.367_0.QXx0GYqLoAbtfq1 | @[simp]
theorem isOrtho_iSup_right {ι : Sort*} {U : Submodule 𝕜 E} {V : ι → Submodule 𝕜 E} :
U ⟂ iSup V ↔ ∀ i, U ⟂ V i | Mathlib_Analysis_InnerProductSpace_Orthogonal |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : IsROrC 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
s t : Set E
⊢ span 𝕜 s ⟂ span 𝕜 t ↔ ∀ ⦃u : E⦄, u ∈ s → ∀ ⦃v : E⦄, v ∈ t → 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, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/ma... | 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 𝕜 s ⟂ span 𝕜 t ↔ ∀ ⦃u⦄, u ∈ s → ∀ ⦃v⦄, v ∈ t → ⟪u, v⟫ = 0 := by
| Mathlib.Analysis.InnerProductSpace.Orthogonal.373_0.QXx0GYqLoAbtfq1 | @[simp]
theorem isOrtho_span {s t : Set E} :
span 𝕜 s ⟂ span 𝕜 t ↔ ∀ ⦃u⦄, u ∈ s → ∀ ⦃v⦄, v ∈ t → ⟪u, v⟫ = 0 | Mathlib_Analysis_InnerProductSpace_Orthogonal |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : IsROrC 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
f : E →ₗᵢ[𝕜] F
U V : Submodule 𝕜 E
h : U ⟂ V
⊢ Submodule.map f U ⟂ 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, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/ma... | rw [isOrtho_iff_inner_eq] at * | theorem IsOrtho.map (f : E →ₗᵢ[𝕜] F) {U V : Submodule 𝕜 E} (h : U ⟂ V) : U.map f ⟂ V.map f := by
| Mathlib.Analysis.InnerProductSpace.Orthogonal.381_0.QXx0GYqLoAbtfq1 | theorem IsOrtho.map (f : E →ₗᵢ[𝕜] F) {U V : Submodule 𝕜 E} (h : U ⟂ V) : U.map f ⟂ V.map f | Mathlib_Analysis_InnerProductSpace_Orthogonal |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : IsROrC 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
f : E →ₗᵢ[𝕜] F
U V : Submodule 𝕜 E
h : ∀ u ∈ U, ∀ v ∈ V, inner u v = 0
⊢ ∀ u ∈ Submodule.map f U, ∀ v ∈ 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, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/ma... | simp_rw [mem_map, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂,
LinearIsometry.inner_map_map] | theorem IsOrtho.map (f : E →ₗᵢ[𝕜] F) {U V : Submodule 𝕜 E} (h : U ⟂ V) : U.map f ⟂ V.map f := by
rw [isOrtho_iff_inner_eq] at *
| Mathlib.Analysis.InnerProductSpace.Orthogonal.381_0.QXx0GYqLoAbtfq1 | theorem IsOrtho.map (f : E →ₗᵢ[𝕜] F) {U V : Submodule 𝕜 E} (h : U ⟂ V) : U.map f ⟂ V.map f | Mathlib_Analysis_InnerProductSpace_Orthogonal |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : IsROrC 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
f : E →ₗᵢ[𝕜] F
U V : Submodule 𝕜 E
h : ∀ u ∈ U, ∀ v ∈ V, inner u v = 0
⊢ ∀ a ∈ U, ∀ a_2 ∈ 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, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/ma... | exact h | theorem IsOrtho.map (f : E →ₗᵢ[𝕜] F) {U V : Submodule 𝕜 E} (h : U ⟂ V) : U.map f ⟂ V.map f := by
rw [isOrtho_iff_inner_eq] at *
simp_rw [mem_map, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂,
LinearIsometry.inner_map_map]
| Mathlib.Analysis.InnerProductSpace.Orthogonal.381_0.QXx0GYqLoAbtfq1 | theorem IsOrtho.map (f : E →ₗᵢ[𝕜] F) {U V : Submodule 𝕜 E} (h : U ⟂ V) : U.map f ⟂ V.map f | Mathlib_Analysis_InnerProductSpace_Orthogonal |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : IsROrC 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
f : E →ₗᵢ[𝕜] F
U V : Submodule 𝕜 F
h : U ⟂ V
⊢ Submodule.comap f U ⟂ 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, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/ma... | rw [isOrtho_iff_inner_eq] at * | theorem IsOrtho.comap (f : E →ₗᵢ[𝕜] F) {U V : Submodule 𝕜 F} (h : U ⟂ V) :
U.comap f ⟂ V.comap f := by
| Mathlib.Analysis.InnerProductSpace.Orthogonal.388_0.QXx0GYqLoAbtfq1 | theorem IsOrtho.comap (f : E →ₗᵢ[𝕜] F) {U V : Submodule 𝕜 F} (h : U ⟂ V) :
U.comap f ⟂ V.comap f | Mathlib_Analysis_InnerProductSpace_Orthogonal |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : IsROrC 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
f : E →ₗᵢ[𝕜] F
U V : Submodule 𝕜 F
h : ∀ u ∈ U, ∀ v ∈ V, inner u v = 0
⊢ ∀ u ∈ Submodule.comap f U, ∀ v ∈ Submodule.comap f V, inner u v... | /-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/ma... | simp_rw [mem_comap, ← f.inner_map_map] | theorem IsOrtho.comap (f : E →ₗᵢ[𝕜] F) {U V : Submodule 𝕜 F} (h : U ⟂ V) :
U.comap f ⟂ V.comap f := by
rw [isOrtho_iff_inner_eq] at *
| Mathlib.Analysis.InnerProductSpace.Orthogonal.388_0.QXx0GYqLoAbtfq1 | theorem IsOrtho.comap (f : E →ₗᵢ[𝕜] F) {U V : Submodule 𝕜 F} (h : U ⟂ V) :
U.comap f ⟂ V.comap f | Mathlib_Analysis_InnerProductSpace_Orthogonal |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : IsROrC 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
f : E →ₗᵢ[𝕜] F
U V : Submodule 𝕜 F
h : ∀ u ∈ U, ∀ v ∈ V, inner u v = 0
⊢ ∀ (u : E), f u ∈ U → ∀ (v : E), f v ∈ V → 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, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/ma... | intro u hu v hv | theorem IsOrtho.comap (f : E →ₗᵢ[𝕜] F) {U V : Submodule 𝕜 F} (h : U ⟂ V) :
U.comap f ⟂ V.comap f := by
rw [isOrtho_iff_inner_eq] at *
simp_rw [mem_comap, ← f.inner_map_map]
| Mathlib.Analysis.InnerProductSpace.Orthogonal.388_0.QXx0GYqLoAbtfq1 | theorem IsOrtho.comap (f : E →ₗᵢ[𝕜] F) {U V : Submodule 𝕜 F} (h : U ⟂ V) :
U.comap f ⟂ V.comap f | Mathlib_Analysis_InnerProductSpace_Orthogonal |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : IsROrC 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
f : E →ₗᵢ[𝕜] F
U V : Submodule 𝕜 F
h : ∀ u ∈ U, ∀ v ∈ V, inner u v = 0
u : E
hu : f u ∈ U
v : E
hv : f v ∈ V
⊢ 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, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/ma... | exact h _ hu _ hv | theorem IsOrtho.comap (f : E →ₗᵢ[𝕜] F) {U V : Submodule 𝕜 F} (h : U ⟂ V) :
U.comap f ⟂ V.comap f := by
rw [isOrtho_iff_inner_eq] at *
simp_rw [mem_comap, ← f.inner_map_map]
intro u hu v hv
| Mathlib.Analysis.InnerProductSpace.Orthogonal.388_0.QXx0GYqLoAbtfq1 | theorem IsOrtho.comap (f : E →ₗᵢ[𝕜] F) {U V : Submodule 𝕜 F} (h : U ⟂ V) :
U.comap f ⟂ V.comap f | Mathlib_Analysis_InnerProductSpace_Orthogonal |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : IsROrC 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
f : E ≃ₗᵢ[𝕜] F
U V : Submodule 𝕜 E
h : Submodule.map f U ⟂ Submodule.map f V
⊢ U ⟂ V | /-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/ma... | have hf : ∀ p : Submodule 𝕜 E, (p.map f).comap f.toLinearIsometry = p :=
comap_map_eq_of_injective f.injective | @[simp]
theorem IsOrtho.map_iff (f : E ≃ₗᵢ[𝕜] F) {U V : Submodule 𝕜 E} : U.map f ⟂ V.map f ↔ U ⟂ V :=
⟨fun h => by
| Mathlib.Analysis.InnerProductSpace.Orthogonal.396_0.QXx0GYqLoAbtfq1 | @[simp]
theorem IsOrtho.map_iff (f : E ≃ₗᵢ[𝕜] F) {U V : Submodule 𝕜 E} : U.map f ⟂ V.map f ↔ U ⟂ V | Mathlib_Analysis_InnerProductSpace_Orthogonal |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : IsROrC 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
f : E ≃ₗᵢ[𝕜] F
U V : Submodule 𝕜 E
h : Submodule.map f U ⟂ Submodule.map f V
hf : ∀ (p : Submodule 𝕜 E), Submodule.comap (LinearIsometr... | /-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/ma... | simpa only [hf] using h.comap f.toLinearIsometry | @[simp]
theorem IsOrtho.map_iff (f : E ≃ₗᵢ[𝕜] F) {U V : Submodule 𝕜 E} : U.map f ⟂ V.map f ↔ U ⟂ V :=
⟨fun h => by
have hf : ∀ p : Submodule 𝕜 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 ≃ₗᵢ[𝕜] F) {U V : Submodule 𝕜 E} : U.map f ⟂ V.map f ↔ U ⟂ V | Mathlib_Analysis_InnerProductSpace_Orthogonal |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : IsROrC 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
f : E ≃ₗᵢ[𝕜] F
U V : Submodule 𝕜 F
h : Submodule.comap f U ⟂ Submodule.comap f V
⊢ U ⟂ V | /-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/ma... | have hf : ∀ p : Submodule 𝕜 F, (p.comap f).map f.toLinearIsometry = p :=
map_comap_eq_of_surjective f.surjective | @[simp]
theorem IsOrtho.comap_iff (f : E ≃ₗᵢ[𝕜] F) {U V : Submodule 𝕜 F} : U.comap f ⟂ V.comap f ↔ U ⟂ V :=
⟨fun h => by
| Mathlib.Analysis.InnerProductSpace.Orthogonal.404_0.QXx0GYqLoAbtfq1 | @[simp]
theorem IsOrtho.comap_iff (f : E ≃ₗᵢ[𝕜] F) {U V : Submodule 𝕜 F} : U.comap f ⟂ V.comap f ↔ U ⟂ V | Mathlib_Analysis_InnerProductSpace_Orthogonal |
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : IsROrC 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
f : E ≃ₗᵢ[𝕜] F
U V : Submodule 𝕜 F
h : Submodule.comap f U ⟂ Submodule.comap f V
hf : ∀ (p : Submodule 𝕜 F), Submodule.map (LinearIsome... | /-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/ma... | simpa only [hf] using h.map f.toLinearIsometry | @[simp]
theorem IsOrtho.comap_iff (f : E ≃ₗᵢ[𝕜] F) {U V : Submodule 𝕜 F} : U.comap f ⟂ V.comap f ↔ U ⟂ V :=
⟨fun h => by
have hf : ∀ p : Submodule 𝕜 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 ≃ₗᵢ[𝕜] F) {U V : Submodule 𝕜 F} : U.comap f ⟂ V.comap f ↔ U ⟂ V | Mathlib_Analysis_InnerProductSpace_Orthogonal |
α : Type u_1
inst✝ : HeytingAlgebra α
a b : α
⊢ IsRegular ⊥ | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.GaloisConnection
#align_import order.heyting.regular from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
/-!
# Heyting r... | rw [IsRegular, compl_bot, compl_top] | theorem isRegular_bot : IsRegular (⊥ : α) := by | Mathlib.Order.Heyting.Regular.60_0.HFWfWmvQ8w4gAZO | theorem isRegular_bot : IsRegular (⊥ : α) | Mathlib_Order_Heyting_Regular |
α : Type u_1
inst✝ : HeytingAlgebra α
a b : α
⊢ IsRegular ⊤ | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.GaloisConnection
#align_import order.heyting.regular from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
/-!
# Heyting r... | rw [IsRegular, compl_top, compl_bot] | theorem isRegular_top : IsRegular (⊤ : α) := by | Mathlib.Order.Heyting.Regular.63_0.HFWfWmvQ8w4gAZO | theorem isRegular_top : IsRegular (⊤ : α) | Mathlib_Order_Heyting_Regular |
α : Type u_1
inst✝ : HeytingAlgebra α
a b : α
ha : IsRegular a
hb : IsRegular b
⊢ IsRegular (a ⊓ b) | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.GaloisConnection
#align_import order.heyting.regular from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
/-!
# Heyting r... | rw [IsRegular, compl_compl_inf_distrib, ha.eq, hb.eq] | theorem IsRegular.inf (ha : IsRegular a) (hb : IsRegular b) : IsRegular (a ⊓ b) := by
| Mathlib.Order.Heyting.Regular.66_0.HFWfWmvQ8w4gAZO | theorem IsRegular.inf (ha : IsRegular a) (hb : IsRegular b) : IsRegular (a ⊓ b) | Mathlib_Order_Heyting_Regular |
α : Type u_1
inst✝ : HeytingAlgebra α
a b : α
ha : IsRegular a
hb : IsRegular b
⊢ IsRegular (a ⇨ b) | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.GaloisConnection
#align_import order.heyting.regular from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
/-!
# Heyting r... | rw [IsRegular, compl_compl_himp_distrib, ha.eq, hb.eq] | theorem IsRegular.himp (ha : IsRegular a) (hb : IsRegular b) : IsRegular (a ⇨ b) := by
| Mathlib.Order.Heyting.Regular.70_0.HFWfWmvQ8w4gAZO | theorem IsRegular.himp (ha : IsRegular a) (hb : IsRegular b) : IsRegular (a ⇨ b) | Mathlib_Order_Heyting_Regular |
α : Type u_1
inst✝ : HeytingAlgebra α
a b : α
ha : IsRegular a
⊢ Disjoint aᶜ b ↔ b ≤ a | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.GaloisConnection
#align_import order.heyting.regular from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
/-!
# Heyting r... | rw [← le_compl_iff_disjoint_left, ha.eq] | protected theorem IsRegular.disjoint_compl_left_iff (ha : IsRegular a) : Disjoint aᶜ b ↔ b ≤ a :=
by | Mathlib.Order.Heyting.Regular.78_0.HFWfWmvQ8w4gAZO | protected theorem IsRegular.disjoint_compl_left_iff (ha : IsRegular a) : Disjoint aᶜ b ↔ b ≤ a | Mathlib_Order_Heyting_Regular |
α : Type u_1
inst✝ : HeytingAlgebra α
a b : α
hb : IsRegular b
⊢ Disjoint a bᶜ ↔ a ≤ b | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.GaloisConnection
#align_import order.heyting.regular from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
/-!
# Heyting r... | rw [← le_compl_iff_disjoint_right, hb.eq] | protected theorem IsRegular.disjoint_compl_right_iff (hb : IsRegular b) : Disjoint a bᶜ ↔ a ≤ b :=
by | Mathlib.Order.Heyting.Regular.82_0.HFWfWmvQ8w4gAZO | protected theorem IsRegular.disjoint_compl_right_iff (hb : IsRegular b) : Disjoint a bᶜ ↔ a ≤ b | Mathlib_Order_Heyting_Regular |
α : Type u_1
inst✝ : HeytingAlgebra α
a✝ b : α
h : ∀ (a : α), IsRegular (a ⊔ aᶜ)
a : α
⊢ a ⊔ aᶜ = ⊤ | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.GaloisConnection
#align_import order.heyting.regular from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
/-!
# Heyting r... | erw [← (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 : ∀ a : α, IsRegular (a ⊔ aᶜ)) : BooleanAlgebra α :=
have : ∀ a : α, IsCompl a aᶜ := fun a =>
⟨disjoint_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 : ∀ a : α, IsRegular (a ⊔ aᶜ)) : BooleanAlgebra α | Mathlib_Order_Heyting_Regular |
α : Type u_1
inst✝ : HeytingAlgebra α
a✝ b✝ : α
src✝³ : Lattice (Regular α) := lattice
src✝² : BoundedOrder (Regular α) := boundedOrder
src✝¹ : HImp (Regular α) := himp
src✝ : HasCompl (Regular α) := hasCompl
a b c : Regular α
⊢ ↑((a ⊔ b) ⊓ (a ⊔ c)) ≤ ↑(a ⊔ b ⊓ c) | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.GaloisConnection
#align_import order.heyting.regular from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
/-!
# Heyting r... | dsimp | instance : BooleanAlgebra (Regular α) :=
{ 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 α) | Mathlib_Order_Heyting_Regular |
α : Type u_1
inst✝ : HeytingAlgebra α
a✝ b✝ : α
src✝³ : Lattice (Regular α) := lattice
src✝² : BoundedOrder (Regular α) := boundedOrder
src✝¹ : HImp (Regular α) := himp
src✝ : HasCompl (Regular α) := hasCompl
a b c : Regular α
⊢ (↑a ⊔ ↑b)ᶜᶜ ⊓ (↑a ⊔ ↑c)ᶜᶜ ≤ (↑a ⊔ ↑b ⊓ ↑c)ᶜᶜ | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.GaloisConnection
#align_import order.heyting.regular from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
/-!
# Heyting r... | rw [sup_inf_left, compl_compl_inf_distrib] | instance : BooleanAlgebra (Regular α) :=
{ 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 α) | Mathlib_Order_Heyting_Regular |
α : Type u_1
inst✝ : HeytingAlgebra α
a✝ b : α
src✝³ : Lattice (Regular α) := lattice
src✝² : BoundedOrder (Regular α) := boundedOrder
src✝¹ : HImp (Regular α) := himp
src✝ : HasCompl (Regular α) := hasCompl
a : Regular α
⊢ ↑⊤ ≤ ↑(a ⊔ aᶜ) | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.GaloisConnection
#align_import order.heyting.regular from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
/-!
# Heyting r... | dsimp | instance : BooleanAlgebra (Regular α) :=
{ 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 disj... | Mathlib.Order.Heyting.Regular.224_0.HFWfWmvQ8w4gAZO | instance : BooleanAlgebra (Regular α) | Mathlib_Order_Heyting_Regular |
α : Type u_1
inst✝ : HeytingAlgebra α
a✝ b : α
src✝³ : Lattice (Regular α) := lattice
src✝² : BoundedOrder (Regular α) := boundedOrder
src✝¹ : HImp (Regular α) := himp
src✝ : HasCompl (Regular α) := hasCompl
a : Regular α
⊢ ⊤ ≤ (↑a ⊔ (↑a)ᶜ)ᶜᶜ | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.GaloisConnection
#align_import order.heyting.regular from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
/-!
# Heyting r... | rw [compl_sup, inf_compl_eq_bot, compl_bot] | instance : BooleanAlgebra (Regular α) :=
{ 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 disj... | Mathlib.Order.Heyting.Regular.224_0.HFWfWmvQ8w4gAZO | instance : BooleanAlgebra (Regular α) | Mathlib_Order_Heyting_Regular |
α : Type u_1
inst✝ : HeytingAlgebra α
a✝ b✝ : α
src✝³ : Lattice (Regular α) := lattice
src✝² : BoundedOrder (Regular α) := boundedOrder
src✝¹ : HImp (Regular α) := himp
src✝ : HasCompl (Regular α) := hasCompl
a b : Regular α
⊢ ↑(a ⇨ b) = ↑(b ⊔ aᶜ) | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.GaloisConnection
#align_import order.heyting.regular from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
/-!
# Heyting r... | dsimp | instance : BooleanAlgebra (Regular α) :=
{ 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 disj... | Mathlib.Order.Heyting.Regular.224_0.HFWfWmvQ8w4gAZO | instance : BooleanAlgebra (Regular α) | Mathlib_Order_Heyting_Regular |
α : Type u_1
inst✝ : HeytingAlgebra α
a✝ b✝ : α
src✝³ : Lattice (Regular α) := lattice
src✝² : BoundedOrder (Regular α) := boundedOrder
src✝¹ : HImp (Regular α) := himp
src✝ : HasCompl (Regular α) := hasCompl
a b : Regular α
⊢ ↑a ⇨ ↑b = (↑b ⊔ (↑a)ᶜ)ᶜᶜ | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.GaloisConnection
#align_import order.heyting.regular from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
/-!
# Heyting r... | rw [compl_sup, a.prop.eq] | instance : BooleanAlgebra (Regular α) :=
{ 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 disj... | Mathlib.Order.Heyting.Regular.224_0.HFWfWmvQ8w4gAZO | instance : BooleanAlgebra (Regular α) | Mathlib_Order_Heyting_Regular |
α : Type u_1
inst✝ : HeytingAlgebra α
a✝ b✝ : α
src✝³ : Lattice (Regular α) := lattice
src✝² : BoundedOrder (Regular α) := boundedOrder
src✝¹ : HImp (Regular α) := himp
src✝ : HasCompl (Regular α) := hasCompl
a b : Regular α
⊢ ↑a ⇨ ↑b = ((↑b)ᶜ ⊓ ↑a)ᶜ | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.GaloisConnection
#align_import order.heyting.regular from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
/-!
# Heyting r... | refine' eq_of_forall_le_iff fun c => le_himp_iff.trans _ | instance : BooleanAlgebra (Regular α) :=
{ 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 disj... | Mathlib.Order.Heyting.Regular.224_0.HFWfWmvQ8w4gAZO | instance : BooleanAlgebra (Regular α) | Mathlib_Order_Heyting_Regular |
α : Type u_1
inst✝ : HeytingAlgebra α
a✝ b✝ : α
src✝³ : Lattice (Regular α) := lattice
src✝² : BoundedOrder (Regular α) := boundedOrder
src✝¹ : HImp (Regular α) := himp
src✝ : HasCompl (Regular α) := hasCompl
a b : Regular α
c : α
⊢ c ⊓ ↑a ≤ ↑b ↔ c ≤ ((↑b)ᶜ ⊓ ↑a)ᶜ | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.GaloisConnection
#align_import order.heyting.regular from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
/-!
# Heyting r... | rw [le_compl_iff_disjoint_right, disjoint_left_comm] | instance : BooleanAlgebra (Regular α) :=
{ 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 disj... | Mathlib.Order.Heyting.Regular.224_0.HFWfWmvQ8w4gAZO | instance : BooleanAlgebra (Regular α) | Mathlib_Order_Heyting_Regular |
α : Type u_1
inst✝ : HeytingAlgebra α
a✝ b✝ : α
src✝³ : Lattice (Regular α) := lattice
src✝² : BoundedOrder (Regular α) := boundedOrder
src✝¹ : HImp (Regular α) := himp
src✝ : HasCompl (Regular α) := hasCompl
a b : Regular α
c : α
⊢ c ⊓ ↑a ≤ ↑b ↔ Disjoint (↑b)ᶜ (c ⊓ ↑a) | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.GaloisConnection
#align_import order.heyting.regular from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
/-!
# Heyting r... | rw [b.prop.disjoint_compl_left_iff] | instance : BooleanAlgebra (Regular α) :=
{ 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 disj... | Mathlib.Order.Heyting.Regular.224_0.HFWfWmvQ8w4gAZO | instance : BooleanAlgebra (Regular α) | Mathlib_Order_Heyting_Regular |
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : SeminormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
⊢ ‖inclusionInDoubleDual 𝕜 E‖ ≤ 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 analy... | rw [inclusionInDoubleDual_norm_eq] | theorem inclusionInDoubleDual_norm_le : ‖inclusionInDoubleDual 𝕜 E‖ ≤ 1 := by
| Mathlib.Analysis.NormedSpace.Dual.83_0.WirVfj6f5oiZZ2w | theorem inclusionInDoubleDual_norm_le : ‖inclusionInDoubleDual 𝕜 E‖ ≤ 1 | Mathlib_Analysis_NormedSpace_Dual |
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : SeminormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
⊢ ‖ContinuousLinearMap.id 𝕜 (Dual 𝕜 E)‖ ≤ 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 analy... | exact ContinuousLinearMap.norm_id_le | theorem inclusionInDoubleDual_norm_le : ‖inclusionInDoubleDual 𝕜 E‖ ≤ 1 := by
rw [inclusionInDoubleDual_norm_eq]
| Mathlib.Analysis.NormedSpace.Dual.83_0.WirVfj6f5oiZZ2w | theorem inclusionInDoubleDual_norm_le : ‖inclusionInDoubleDual 𝕜 E‖ ≤ 1 | Mathlib_Analysis_NormedSpace_Dual |
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : SeminormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
x : E
⊢ ‖(inclusionInDoubleDual 𝕜 E) x‖ ≤ ‖x‖ | /-
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 analy... | simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusionInDoubleDual_norm_le 𝕜 E) x | theorem double_dual_bound (x : E) : ‖(inclusionInDoubleDual 𝕜 E) x‖ ≤ ‖x‖ := by
| Mathlib.Analysis.NormedSpace.Dual.88_0.WirVfj6f5oiZZ2w | theorem double_dual_bound (x : E) : ‖(inclusionInDoubleDual 𝕜 E) x‖ ≤ ‖x‖ | Mathlib_Analysis_NormedSpace_Dual |
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : SeminormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
⊢ LinearMap.SeparatingLeft (dualPairing 𝕜 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 analy... | rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot] | theorem dualPairing_separatingLeft : (dualPairing 𝕜 E).SeparatingLeft := by
| Mathlib.Analysis.NormedSpace.Dual.102_0.WirVfj6f5oiZZ2w | theorem dualPairing_separatingLeft : (dualPairing 𝕜 E).SeparatingLeft | Mathlib_Analysis_NormedSpace_Dual |
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : SeminormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
⊢ Function.Injective ⇑(dualPairing 𝕜 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 analy... | exact ContinuousLinearMap.coe_injective | theorem dualPairing_separatingLeft : (dualPairing 𝕜 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 𝕜 E).SeparatingLeft | Mathlib_Analysis_NormedSpace_Dual |
𝕜 : Type v
inst✝² : IsROrC 𝕜
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : E
M : ℝ
hMp : 0 ≤ M
hM : ∀ (f : Dual 𝕜 E), ‖f x‖ ≤ M * ‖f‖
⊢ ‖x‖ ≤ 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 analy... | classical
by_cases h : x = 0
· simp only [h, hMp, norm_zero]
· obtain ⟨f, hf₁, hfx⟩ : ∃ f : E →L[𝕜] 𝕜, ‖f‖ = 1 ∧ f x = ‖x‖ := exists_dual_vector 𝕜 x h
calc
‖x‖ = ‖(‖x‖ : 𝕜)‖ := IsROrC.norm_coe_norm.symm
_ = ‖f x‖ := by rw [hfx]
_ ≤ M * ‖f‖ := (hM f)
_ = M := by rw [... | /-- 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 : ℝ} (hMp : 0 ≤ M) (hM : ∀ f : Dual 𝕜 E, ‖f x‖ ≤ M * ‖f‖) :
‖x‖ ≤ 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 : ℝ} (hMp : 0 ≤ M) (hM : ∀ f : Dual 𝕜 E, ‖f x‖ ≤ M * ‖f‖) :
‖x‖ ≤ M | Mathlib_Analysis_NormedSpace_Dual |
𝕜 : Type v
inst✝² : IsROrC 𝕜
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : E
M : ℝ
hMp : 0 ≤ M
hM : ∀ (f : Dual 𝕜 E), ‖f x‖ ≤ M * ‖f‖
⊢ ‖x‖ ≤ 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 analy... | 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 : ℝ} (hMp : 0 ≤ M) (hM : ∀ f : Dual 𝕜 E, ‖f x‖ ≤ M * ‖f‖) :
‖x‖ ≤ 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 : ℝ} (hMp : 0 ≤ M) (hM : ∀ f : Dual 𝕜 E, ‖f x‖ ≤ M * ‖f‖) :
‖x‖ ≤ M | Mathlib_Analysis_NormedSpace_Dual |
case pos
𝕜 : Type v
inst✝² : IsROrC 𝕜
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : E
M : ℝ
hMp : 0 ≤ M
hM : ∀ (f : Dual 𝕜 E), ‖f x‖ ≤ M * ‖f‖
h : x = 0
⊢ ‖x‖ ≤ 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 analy... | 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 : ℝ} (hMp : 0 ≤ M) (hM : ∀ f : Dual 𝕜 E, ‖f x‖ ≤ M * ‖f‖) :
‖x‖ ≤ M := by
classical
by_cases h : x = 0
· | 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 : ℝ} (hMp : 0 ≤ M) (hM : ∀ f : Dual 𝕜 E, ‖f x‖ ≤ M * ‖f‖) :
‖x‖ ≤ M | Mathlib_Analysis_NormedSpace_Dual |
case neg
𝕜 : Type v
inst✝² : IsROrC 𝕜
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : E
M : ℝ
hMp : 0 ≤ M
hM : ∀ (f : Dual 𝕜 E), ‖f x‖ ≤ M * ‖f‖
h : ¬x = 0
⊢ ‖x‖ ≤ 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 analy... | obtain ⟨f, hf₁, hfx⟩ : ∃ f : E →L[𝕜] 𝕜, ‖f‖ = 1 ∧ f x = ‖x‖ := exists_dual_vector 𝕜 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 : ℝ} (hMp : 0 ≤ M) (hM : ∀ f : Dual 𝕜 E, ‖f x‖ ≤ M * ‖f‖) :
‖x‖ ≤ M := by
classical
by_cases h : x = 0
· simp only [h, hMp, norm_zer... | 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 : ℝ} (hMp : 0 ≤ M) (hM : ∀ f : Dual 𝕜 E, ‖f x‖ ≤ M * ‖f‖) :
‖x‖ ≤ M | Mathlib_Analysis_NormedSpace_Dual |
case neg.intro.intro
𝕜 : Type v
inst✝² : IsROrC 𝕜
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : E
M : ℝ
hMp : 0 ≤ M
hM : ∀ (f : Dual 𝕜 E), ‖f x‖ ≤ M * ‖f‖
h : ¬x = 0
f : E →L[𝕜] 𝕜
hf₁ : ‖f‖ = 1
hfx : f x = ↑‖x‖
⊢ ‖x‖ ≤ 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 analy... | calc
‖x‖ = ‖(‖x‖ : 𝕜)‖ := IsROrC.norm_coe_norm.symm
_ = ‖f x‖ := by rw [hfx]
_ ≤ M * ‖f‖ := (hM f)
_ = M := by rw [hf₁, 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 : ℝ} (hMp : 0 ≤ M) (hM : ∀ f : Dual 𝕜 E, ‖f x‖ ≤ M * ‖f‖) :
‖x‖ ≤ M := by
classical
by_cases h : x = 0
· simp only [h, hMp, norm_zer... | 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 : ℝ} (hMp : 0 ≤ M) (hM : ∀ f : Dual 𝕜 E, ‖f x‖ ≤ M * ‖f‖) :
‖x‖ ≤ M | Mathlib_Analysis_NormedSpace_Dual |
𝕜 : Type v
inst✝² : IsROrC 𝕜
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : E
M : ℝ
hMp : 0 ≤ M
hM : ∀ (f : Dual 𝕜 E), ‖f x‖ ≤ M * ‖f‖
h : ¬x = 0
f : E →L[𝕜] 𝕜
hf₁ : ‖f‖ = 1
hfx : f x = ↑‖x‖
⊢ ‖↑‖x‖‖ = ‖f x‖ | /-
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 analy... | 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 : ℝ} (hMp : 0 ≤ M) (hM : ∀ f : Dual 𝕜 E, ‖f x‖ ≤ M * ‖f‖) :
‖x‖ ≤ M := by
classical
by_cases h : x = 0
· simp only [h, hMp, norm_zer... | 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 : ℝ} (hMp : 0 ≤ M) (hM : ∀ f : Dual 𝕜 E, ‖f x‖ ≤ M * ‖f‖) :
‖x‖ ≤ M | Mathlib_Analysis_NormedSpace_Dual |
𝕜 : Type v
inst✝² : IsROrC 𝕜
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : E
M : ℝ
hMp : 0 ≤ M
hM : ∀ (f : Dual 𝕜 E), ‖f x‖ ≤ M * ‖f‖
h : ¬x = 0
f : E →L[𝕜] 𝕜
hf₁ : ‖f‖ = 1
hfx : f x = ↑‖x‖
⊢ M * ‖f‖ = 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 analy... | rw [hf₁, 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 : ℝ} (hMp : 0 ≤ M) (hM : ∀ f : Dual 𝕜 E, ‖f x‖ ≤ M * ‖f‖) :
‖x‖ ≤ M := by
classical
by_cases h : x = 0
· simp only [h, hMp, norm_zer... | 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 : ℝ} (hMp : 0 ≤ M) (hM : ∀ f : Dual 𝕜 E, ‖f x‖ ≤ M * ‖f‖) :
‖x‖ ≤ M | Mathlib_Analysis_NormedSpace_Dual |
𝕜 : Type v
inst✝² : IsROrC 𝕜
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : E
h : ∀ (f : Dual 𝕜 E), f x = 0
f : Dual 𝕜 E
⊢ ‖f x‖ ≤ 0 * ‖f‖ | /-
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 analy... | simp [h f] | theorem eq_zero_of_forall_dual_eq_zero {x : E} (h : ∀ f : Dual 𝕜 E, f x = (0 : 𝕜)) : x = 0 :=
norm_le_zero_iff.mp (norm_le_dual_bound 𝕜 x le_rfl fun f => by | Mathlib.Analysis.NormedSpace.Dual.128_0.WirVfj6f5oiZZ2w | theorem eq_zero_of_forall_dual_eq_zero {x : E} (h : ∀ f : Dual 𝕜 E, f x = (0 : 𝕜)) : x = 0 | Mathlib_Analysis_NormedSpace_Dual |
𝕜 : Type v
inst✝² : IsROrC 𝕜
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : E
hx : x = 0
⊢ ∀ (g : Dual 𝕜 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 analy... | simp [hx] | theorem eq_zero_iff_forall_dual_eq_zero (x : E) : x = 0 ↔ ∀ g : Dual 𝕜 E, g x = 0 :=
⟨fun hx => by | Mathlib.Analysis.NormedSpace.Dual.132_0.WirVfj6f5oiZZ2w | theorem eq_zero_iff_forall_dual_eq_zero (x : E) : x = 0 ↔ ∀ g : Dual 𝕜 E, g x = 0 | Mathlib_Analysis_NormedSpace_Dual |
𝕜 : Type v
inst✝² : IsROrC 𝕜
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x y : E
⊢ x = y ↔ ∀ (g : Dual 𝕜 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 analy... | rw [← sub_eq_zero, eq_zero_iff_forall_dual_eq_zero 𝕜 (x - y)] | /-- See also `geometric_hahn_banach_point_point`. -/
theorem eq_iff_forall_dual_eq {x y : E} : x = y ↔ ∀ g : Dual 𝕜 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 ↔ ∀ g : Dual 𝕜 E, g x = g y | Mathlib_Analysis_NormedSpace_Dual |
𝕜 : Type v
inst✝² : IsROrC 𝕜
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x y : E
⊢ (∀ (g : Dual 𝕜 E), g (x - y) = 0) ↔ ∀ (g : Dual 𝕜 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 analy... | simp [sub_eq_zero] | /-- See also `geometric_hahn_banach_point_point`. -/
theorem eq_iff_forall_dual_eq {x y : E} : x = y ↔ ∀ g : Dual 𝕜 E, g x = g y := by
rw [← sub_eq_zero, eq_zero_iff_forall_dual_eq_zero 𝕜 (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 ↔ ∀ g : Dual 𝕜 E, g x = g y | Mathlib_Analysis_NormedSpace_Dual |
𝕜 : Type v
inst✝² : IsROrC 𝕜
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
src✝ : E →L[𝕜] Dual 𝕜 (Dual 𝕜 E) := inclusionInDoubleDual 𝕜 E
⊢ ∀ (x : E), ‖↑src✝ x‖ = ‖x‖ | /-
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 analy... | intro x | /-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E →ₗᵢ[𝕜] Dual 𝕜 (Dual 𝕜 E) :=
{ inclusionInDoubleDual 𝕜 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 →ₗᵢ[𝕜] Dual 𝕜 (Dual 𝕜 E) | Mathlib_Analysis_NormedSpace_Dual |
𝕜 : Type v
inst✝² : IsROrC 𝕜
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
src✝ : E →L[𝕜] Dual 𝕜 (Dual 𝕜 E) := inclusionInDoubleDual 𝕜 E
x : E
⊢ ‖↑src✝ x‖ = ‖x‖ | /-
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 analy... | apply le_antisymm | /-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E →ₗᵢ[𝕜] Dual 𝕜 (Dual 𝕜 E) :=
{ inclusionInDoubleDual 𝕜 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 →ₗᵢ[𝕜] Dual 𝕜 (Dual 𝕜 E) | Mathlib_Analysis_NormedSpace_Dual |
case a
𝕜 : Type v
inst✝² : IsROrC 𝕜
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
src✝ : E →L[𝕜] Dual 𝕜 (Dual 𝕜 E) := inclusionInDoubleDual 𝕜 E
x : E
⊢ ‖↑src✝ x‖ ≤ ‖x‖ | /-
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 analy... | exact double_dual_bound 𝕜 E x | /-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E →ₗᵢ[𝕜] Dual 𝕜 (Dual 𝕜 E) :=
{ inclusionInDoubleDual 𝕜 E with
norm_map' := by
intro x
apply le_antisymm
· | 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 →ₗᵢ[𝕜] Dual 𝕜 (Dual 𝕜 E) | Mathlib_Analysis_NormedSpace_Dual |
case a
𝕜 : Type v
inst✝² : IsROrC 𝕜
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
src✝ : E →L[𝕜] Dual 𝕜 (Dual 𝕜 E) := inclusionInDoubleDual 𝕜 E
x : E
⊢ ‖x‖ ≤ ‖↑src✝ x‖ | /-
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 analy... | rw [ContinuousLinearMap.norm_def] | /-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E →ₗᵢ[𝕜] Dual 𝕜 (Dual 𝕜 E) :=
{ inclusionInDoubleDual 𝕜 E with
norm_map' := by
intro x
apply le_antisymm
· exact double_dual_bound 𝕜 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 →ₗᵢ[𝕜] Dual 𝕜 (Dual 𝕜 E) | Mathlib_Analysis_NormedSpace_Dual |
case a
𝕜 : Type v
inst✝² : IsROrC 𝕜
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
src✝ : E →L[𝕜] Dual 𝕜 (Dual 𝕜 E) := inclusionInDoubleDual 𝕜 E
x : E
⊢ ‖x‖ ≤ sInf {c | 0 ≤ c ∧ ∀ (x_1 : Dual 𝕜 E), ‖(↑src✝ x) x_1‖ ≤ c * ‖x_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 analy... | 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 →ₗᵢ[𝕜] Dual 𝕜 (Dual 𝕜 E) :=
{ inclusionInDoubleDual 𝕜 E with
norm_map' := by
intro x
apply le_antisymm
· exact double_dual_bound 𝕜 E x
rw [ContinuousLinearMap.norm_d... | 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 →ₗᵢ[𝕜] Dual 𝕜 (Dual 𝕜 E) | Mathlib_Analysis_NormedSpace_Dual |
case a
𝕜 : Type v
inst✝² : IsROrC 𝕜
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
src✝ : E →L[𝕜] Dual 𝕜 (Dual 𝕜 E) := inclusionInDoubleDual 𝕜 E
x : E
⊢ ∀ b ∈ {c | 0 ≤ c ∧ ∀ (x_1 : Dual 𝕜 E), ‖(↑src✝ x) x_1‖ ≤ c * ‖x_1‖}, ‖x‖ ≤ 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 analy... | rintro c ⟨hc1, hc2⟩ | /-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E →ₗᵢ[𝕜] Dual 𝕜 (Dual 𝕜 E) :=
{ inclusionInDoubleDual 𝕜 E with
norm_map' := by
intro x
apply le_antisymm
· exact double_dual_bound 𝕜 E x
rw [ContinuousLinearMap.norm_d... | 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 →ₗᵢ[𝕜] Dual 𝕜 (Dual 𝕜 E) | Mathlib_Analysis_NormedSpace_Dual |
case a.intro
𝕜 : Type v
inst✝² : IsROrC 𝕜
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
src✝ : E →L[𝕜] Dual 𝕜 (Dual 𝕜 E) := inclusionInDoubleDual 𝕜 E
x : E
c : ℝ
hc1 : 0 ≤ c
hc2 : ∀ (x_1 : Dual 𝕜 E), ‖(↑src✝ x) x_1‖ ≤ c * ‖x_1‖
⊢ ‖x‖ ≤ 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 analy... | exact norm_le_dual_bound 𝕜 x hc1 hc2 | /-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
def inclusionInDoubleDualLi : E →ₗᵢ[𝕜] Dual 𝕜 (Dual 𝕜 E) :=
{ inclusionInDoubleDual 𝕜 E with
norm_map' := by
intro x
apply le_antisymm
· exact double_dual_bound 𝕜 E x
rw [ContinuousLinearMap.norm_d... | 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 →ₗᵢ[𝕜] Dual 𝕜 (Dual 𝕜 E) | Mathlib_Analysis_NormedSpace_Dual |
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
⊢ IsClosed (polar 𝕜 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 analy... | dsimp only [NormedSpace.polar] | theorem isClosed_polar (s : Set E) : IsClosed (polar 𝕜 s) := by
| Mathlib.Analysis.NormedSpace.Dual.183_0.WirVfj6f5oiZZ2w | theorem isClosed_polar (s : Set E) : IsClosed (polar 𝕜 s) | Mathlib_Analysis_NormedSpace_Dual |
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
⊢ IsClosed (LinearMap.polar (LinearMap.flip (dualPairing 𝕜 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 analy... | simp only [LinearMap.polar_eq_iInter, LinearMap.flip_apply] | theorem isClosed_polar (s : Set E) : IsClosed (polar 𝕜 s) := by
dsimp only [NormedSpace.polar]
| Mathlib.Analysis.NormedSpace.Dual.183_0.WirVfj6f5oiZZ2w | theorem isClosed_polar (s : Set E) : IsClosed (polar 𝕜 s) | Mathlib_Analysis_NormedSpace_Dual |
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
⊢ IsClosed (⋂ x ∈ s, {y | ‖((dualPairing 𝕜 E) y) x‖ ≤ 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 analy... | refine' isClosed_biInter fun z _ => _ | theorem isClosed_polar (s : Set E) : IsClosed (polar 𝕜 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 𝕜 s) | Mathlib_Analysis_NormedSpace_Dual |
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
z : E
x✝ : z ∈ s
⊢ IsClosed {y | ‖((dualPairing 𝕜 E) y) z‖ ≤ 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 analy... | exact isClosed_Iic.preimage (ContinuousLinearMap.apply 𝕜 𝕜 z).continuous.norm | theorem isClosed_polar (s : Set E) : IsClosed (polar 𝕜 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 𝕜 s) | Mathlib_Analysis_NormedSpace_Dual |
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
⊢ IsClosed
((LinearMap.polar (LinearMap.flip (LinearMap.flip (dualPairing 𝕜 E))) ∘ ⇑OrderDual.ofDual)
(LinearMap.polar (LinearMap.flip (dualPairing 𝕜 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 analy... | simpa [LinearMap.flip_flip] using
(isClosed_polar _ _).preimage (inclusionInDoubleDual 𝕜 E).continuous | @[simp]
theorem polar_closure (s : Set E) : polar 𝕜 (closure s) = polar 𝕜 s :=
((dualPairing 𝕜 E).flip.polar_antitone subset_closure).antisymm <|
(dualPairing 𝕜 E).flip.polar_gc.l_le <|
closure_minimal ((dualPairing 𝕜 E).flip.polar_gc.le_u_l s) <| by
| Mathlib.Analysis.NormedSpace.Dual.190_0.WirVfj6f5oiZZ2w | @[simp]
theorem polar_closure (s : Set E) : polar 𝕜 (closure s) = polar 𝕜 s | Mathlib_Analysis_NormedSpace_Dual |
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
x' : Dual 𝕜 E
c : 𝕜
hc : ∀ z ∈ s, ‖x' z‖ ≤ ‖c‖
⊢ c⁻¹ • x' ∈ polar 𝕜 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 analy... | by_cases c_zero : c = 0 | /-- If `x'` is a dual element such that the norms `‖x' z‖` are bounded for `z ∈ s`, then a
small scalar multiple of `x'` is in `polar 𝕜 s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual 𝕜 E} {c : 𝕜} (hc : ∀ z, z ∈ s → ‖x' z‖ ≤ ‖c‖) :
c⁻¹ • x' ∈ polar 𝕜 s := by
| Mathlib.Analysis.NormedSpace.Dual.201_0.WirVfj6f5oiZZ2w | /-- If `x'` is a dual element such that the norms `‖x' z‖` are bounded for `z ∈ s`, then a
small scalar multiple of `x'` is in `polar 𝕜 s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual 𝕜 E} {c : 𝕜} (hc : ∀ z, z ∈ s → ‖x' z‖ ≤ ‖c‖) :
c⁻¹ • x' ∈ polar 𝕜 s | Mathlib_Analysis_NormedSpace_Dual |
case pos
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
x' : Dual 𝕜 E
c : 𝕜
hc : ∀ z ∈ s, ‖x' z‖ ≤ ‖c‖
c_zero : c = 0
⊢ c⁻¹ • x' ∈ polar 𝕜 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 analy... | simp only [c_zero, inv_zero, zero_smul] | /-- If `x'` is a dual element such that the norms `‖x' z‖` are bounded for `z ∈ s`, then a
small scalar multiple of `x'` is in `polar 𝕜 s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual 𝕜 E} {c : 𝕜} (hc : ∀ z, z ∈ s → ‖x' z‖ ≤ ‖c‖) :
c⁻¹ • x' ∈ polar 𝕜 s := by
by_cases c_zero : c = 0
· | Mathlib.Analysis.NormedSpace.Dual.201_0.WirVfj6f5oiZZ2w | /-- If `x'` is a dual element such that the norms `‖x' z‖` are bounded for `z ∈ s`, then a
small scalar multiple of `x'` is in `polar 𝕜 s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual 𝕜 E} {c : 𝕜} (hc : ∀ z, z ∈ s → ‖x' z‖ ≤ ‖c‖) :
c⁻¹ • x' ∈ polar 𝕜 s | Mathlib_Analysis_NormedSpace_Dual |
case pos
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
x' : Dual 𝕜 E
c : 𝕜
hc : ∀ z ∈ s, ‖x' z‖ ≤ ‖c‖
c_zero : c = 0
⊢ 0 ∈ polar 𝕜 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 analy... | exact (dualPairing 𝕜 E).flip.zero_mem_polar _ | /-- If `x'` is a dual element such that the norms `‖x' z‖` are bounded for `z ∈ s`, then a
small scalar multiple of `x'` is in `polar 𝕜 s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual 𝕜 E} {c : 𝕜} (hc : ∀ z, z ∈ s → ‖x' z‖ ≤ ‖c‖) :
c⁻¹ • x' ∈ polar 𝕜 s := by
by_cases c_zero : c = 0
· simp only [c_zero,... | Mathlib.Analysis.NormedSpace.Dual.201_0.WirVfj6f5oiZZ2w | /-- If `x'` is a dual element such that the norms `‖x' z‖` are bounded for `z ∈ s`, then a
small scalar multiple of `x'` is in `polar 𝕜 s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual 𝕜 E} {c : 𝕜} (hc : ∀ z, z ∈ s → ‖x' z‖ ≤ ‖c‖) :
c⁻¹ • x' ∈ polar 𝕜 s | Mathlib_Analysis_NormedSpace_Dual |
case neg
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
x' : Dual 𝕜 E
c : 𝕜
hc : ∀ z ∈ s, ‖x' z‖ ≤ ‖c‖
c_zero : ¬c = 0
⊢ c⁻¹ • x' ∈ polar 𝕜 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 analy... | have eq : ∀ z, ‖c⁻¹ • x' z‖ = ‖c⁻¹‖ * ‖x' z‖ := fun z => norm_smul c⁻¹ _ | /-- If `x'` is a dual element such that the norms `‖x' z‖` are bounded for `z ∈ s`, then a
small scalar multiple of `x'` is in `polar 𝕜 s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual 𝕜 E} {c : 𝕜} (hc : ∀ z, z ∈ s → ‖x' z‖ ≤ ‖c‖) :
c⁻¹ • x' ∈ polar 𝕜 s := by
by_cases c_zero : c = 0
· simp only [c_zero,... | Mathlib.Analysis.NormedSpace.Dual.201_0.WirVfj6f5oiZZ2w | /-- If `x'` is a dual element such that the norms `‖x' z‖` are bounded for `z ∈ s`, then a
small scalar multiple of `x'` is in `polar 𝕜 s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual 𝕜 E} {c : 𝕜} (hc : ∀ z, z ∈ s → ‖x' z‖ ≤ ‖c‖) :
c⁻¹ • x' ∈ polar 𝕜 s | Mathlib_Analysis_NormedSpace_Dual |
case neg
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
x' : Dual 𝕜 E
c : 𝕜
hc : ∀ z ∈ s, ‖x' z‖ ≤ ‖c‖
c_zero : ¬c = 0
eq : ∀ (z : E), ‖c⁻¹ • x' z‖ = ‖c⁻¹‖ * ‖x' z‖
⊢ c⁻¹ • x' ∈ polar 𝕜 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 analy... | have le : ∀ z, z ∈ s → ‖c⁻¹ • x' z‖ ≤ ‖c⁻¹‖ * ‖c‖ := 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 `‖x' z‖` are bounded for `z ∈ s`, then a
small scalar multiple of `x'` is in `polar 𝕜 s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual 𝕜 E} {c : 𝕜} (hc : ∀ z, z ∈ s → ‖x' z‖ ≤ ‖c‖) :
c⁻¹ • x' ∈ polar 𝕜 s := by
by_cases c_zero : c = 0
· simp only [c_zero,... | Mathlib.Analysis.NormedSpace.Dual.201_0.WirVfj6f5oiZZ2w | /-- If `x'` is a dual element such that the norms `‖x' z‖` are bounded for `z ∈ s`, then a
small scalar multiple of `x'` is in `polar 𝕜 s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual 𝕜 E} {c : 𝕜} (hc : ∀ z, z ∈ s → ‖x' z‖ ≤ ‖c‖) :
c⁻¹ • x' ∈ polar 𝕜 s | Mathlib_Analysis_NormedSpace_Dual |
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
x' : Dual 𝕜 E
c : 𝕜
hc : ∀ z ∈ s, ‖x' z‖ ≤ ‖c‖
c_zero : ¬c = 0
eq : ∀ (z : E), ‖c⁻¹ • x' z‖ = ‖c⁻¹‖ * ‖x' z‖
⊢ ∀ z ∈ s, ‖c⁻¹ • x' z‖ ≤ ‖c⁻¹‖ * ‖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 analy... | intro z hzs | /-- If `x'` is a dual element such that the norms `‖x' z‖` are bounded for `z ∈ s`, then a
small scalar multiple of `x'` is in `polar 𝕜 s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual 𝕜 E} {c : 𝕜} (hc : ∀ z, z ∈ s → ‖x' z‖ ≤ ‖c‖) :
c⁻¹ • x' ∈ polar 𝕜 s := by
by_cases c_zero : c = 0
· simp only [c_zero,... | Mathlib.Analysis.NormedSpace.Dual.201_0.WirVfj6f5oiZZ2w | /-- If `x'` is a dual element such that the norms `‖x' z‖` are bounded for `z ∈ s`, then a
small scalar multiple of `x'` is in `polar 𝕜 s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual 𝕜 E} {c : 𝕜} (hc : ∀ z, z ∈ s → ‖x' z‖ ≤ ‖c‖) :
c⁻¹ • x' ∈ polar 𝕜 s | Mathlib_Analysis_NormedSpace_Dual |
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
x' : Dual 𝕜 E
c : 𝕜
hc : ∀ z ∈ s, ‖x' z‖ ≤ ‖c‖
c_zero : ¬c = 0
eq : ∀ (z : E), ‖c⁻¹ • x' z‖ = ‖c⁻¹‖ * ‖x' z‖
z : E
hzs : z ∈ s
⊢ ‖c⁻¹ • x' z‖ ≤ ‖c⁻¹‖ * ‖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 analy... | rw [eq z] | /-- If `x'` is a dual element such that the norms `‖x' z‖` are bounded for `z ∈ s`, then a
small scalar multiple of `x'` is in `polar 𝕜 s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual 𝕜 E} {c : 𝕜} (hc : ∀ z, z ∈ s → ‖x' z‖ ≤ ‖c‖) :
c⁻¹ • x' ∈ polar 𝕜 s := by
by_cases c_zero : c = 0
· simp only [c_zero,... | Mathlib.Analysis.NormedSpace.Dual.201_0.WirVfj6f5oiZZ2w | /-- If `x'` is a dual element such that the norms `‖x' z‖` are bounded for `z ∈ s`, then a
small scalar multiple of `x'` is in `polar 𝕜 s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual 𝕜 E} {c : 𝕜} (hc : ∀ z, z ∈ s → ‖x' z‖ ≤ ‖c‖) :
c⁻¹ • x' ∈ polar 𝕜 s | Mathlib_Analysis_NormedSpace_Dual |
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
x' : Dual 𝕜 E
c : 𝕜
hc : ∀ z ∈ s, ‖x' z‖ ≤ ‖c‖
c_zero : ¬c = 0
eq : ∀ (z : E), ‖c⁻¹ • x' z‖ = ‖c⁻¹‖ * ‖x' z‖
z : E
hzs : z ∈ s
⊢ ‖c⁻¹‖ * ‖x' z‖ ≤ ‖c⁻¹‖ * ‖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 analy... | 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 `‖x' z‖` are bounded for `z ∈ s`, then a
small scalar multiple of `x'` is in `polar 𝕜 s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual 𝕜 E} {c : 𝕜} (hc : ∀ z, z ∈ s → ‖x' z‖ ≤ ‖c‖) :
c⁻¹ • x' ∈ polar 𝕜 s := by
by_cases c_zero : c = 0
· simp only [c_zero,... | Mathlib.Analysis.NormedSpace.Dual.201_0.WirVfj6f5oiZZ2w | /-- If `x'` is a dual element such that the norms `‖x' z‖` are bounded for `z ∈ s`, then a
small scalar multiple of `x'` is in `polar 𝕜 s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual 𝕜 E} {c : 𝕜} (hc : ∀ z, z ∈ s → ‖x' z‖ ≤ ‖c‖) :
c⁻¹ • x' ∈ polar 𝕜 s | Mathlib_Analysis_NormedSpace_Dual |
case neg
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
x' : Dual 𝕜 E
c : 𝕜
hc : ∀ z ∈ s, ‖x' z‖ ≤ ‖c‖
c_zero : ¬c = 0
eq : ∀ (z : E), ‖c⁻¹ • x' z‖ = ‖c⁻¹‖ * ‖x' z‖
le : ∀ z ∈ s, ‖c⁻¹ • x' z‖ ≤ ‖c⁻¹‖ * ‖c‖
⊢ c⁻¹ • x' ∈ polar 𝕜 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 analy... | have cancel : ‖c⁻¹‖ * ‖c‖ = 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 `‖x' z‖` are bounded for `z ∈ s`, then a
small scalar multiple of `x'` is in `polar 𝕜 s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual 𝕜 E} {c : 𝕜} (hc : ∀ z, z ∈ s → ‖x' z‖ ≤ ‖c‖) :
c⁻¹ • x' ∈ polar 𝕜 s := by
by_cases c_zero : c = 0
· simp only [c_zero,... | Mathlib.Analysis.NormedSpace.Dual.201_0.WirVfj6f5oiZZ2w | /-- If `x'` is a dual element such that the norms `‖x' z‖` are bounded for `z ∈ s`, then a
small scalar multiple of `x'` is in `polar 𝕜 s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual 𝕜 E} {c : 𝕜} (hc : ∀ z, z ∈ s → ‖x' z‖ ≤ ‖c‖) :
c⁻¹ • x' ∈ polar 𝕜 s | Mathlib_Analysis_NormedSpace_Dual |
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
x' : Dual 𝕜 E
c : 𝕜
hc : ∀ z ∈ s, ‖x' z‖ ≤ ‖c‖
c_zero : ¬c = 0
eq : ∀ (z : E), ‖c⁻¹ • x' z‖ = ‖c⁻¹‖ * ‖x' z‖
le : ∀ z ∈ s, ‖c⁻¹ • x' z‖ ≤ ‖c⁻¹‖ * ‖c‖
⊢ ‖c⁻¹‖ * ‖c‖ = 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 analy... | 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 `‖x' z‖` are bounded for `z ∈ s`, then a
small scalar multiple of `x'` is in `polar 𝕜 s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual 𝕜 E} {c : 𝕜} (hc : ∀ z, z ∈ s → ‖x' z‖ ≤ ‖c‖) :
c⁻¹ • x' ∈ polar 𝕜 s := by
by_cases c_zero : c = 0
· simp only [c_zero,... | Mathlib.Analysis.NormedSpace.Dual.201_0.WirVfj6f5oiZZ2w | /-- If `x'` is a dual element such that the norms `‖x' z‖` are bounded for `z ∈ s`, then a
small scalar multiple of `x'` is in `polar 𝕜 s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual 𝕜 E} {c : 𝕜} (hc : ∀ z, z ∈ s → ‖x' z‖ ≤ ‖c‖) :
c⁻¹ • x' ∈ polar 𝕜 s | Mathlib_Analysis_NormedSpace_Dual |
case neg
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
x' : Dual 𝕜 E
c : 𝕜
hc : ∀ z ∈ s, ‖x' z‖ ≤ ‖c‖
c_zero : ¬c = 0
eq : ∀ (z : E), ‖c⁻¹ • x' z‖ = ‖c⁻¹‖ * ‖x' z‖
le : ∀ z ∈ s, ‖c⁻¹ • x' z‖ ≤ ‖c⁻¹‖ * ‖c‖
cancel : ‖c⁻¹‖ * ‖c‖ = 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 analy... | rwa [cancel] at le | /-- If `x'` is a dual element such that the norms `‖x' z‖` are bounded for `z ∈ s`, then a
small scalar multiple of `x'` is in `polar 𝕜 s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual 𝕜 E} {c : 𝕜} (hc : ∀ z, z ∈ s → ‖x' z‖ ≤ ‖c‖) :
c⁻¹ • x' ∈ polar 𝕜 s := by
by_cases c_zero : c = 0
· simp only [c_zero,... | Mathlib.Analysis.NormedSpace.Dual.201_0.WirVfj6f5oiZZ2w | /-- If `x'` is a dual element such that the norms `‖x' z‖` are bounded for `z ∈ s`, then a
small scalar multiple of `x'` is in `polar 𝕜 s`. -/
theorem smul_mem_polar {s : Set E} {x' : Dual 𝕜 E} {c : 𝕜} (hc : ∀ z, z ∈ s → ‖x' z‖ ≤ ‖c‖) :
c⁻¹ • x' ∈ polar 𝕜 s | Mathlib_Analysis_NormedSpace_Dual |
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
hc : 1 < ‖c‖
r : ℝ
hr : 0 < r
⊢ polar 𝕜 (ball 0 r) ⊆ closedBall 0 (‖c‖ / 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 analy... | intro x' hx' | theorem polar_ball_subset_closedBall_div {c : 𝕜} (hc : 1 < ‖c‖) {r : ℝ} (hr : 0 < r) :
polar 𝕜 (ball (0 : E) r) ⊆ closedBall (0 : Dual 𝕜 E) (‖c‖ / r) := by
| Mathlib.Analysis.NormedSpace.Dual.218_0.WirVfj6f5oiZZ2w | theorem polar_ball_subset_closedBall_div {c : 𝕜} (hc : 1 < ‖c‖) {r : ℝ} (hr : 0 < r) :
polar 𝕜 (ball (0 : E) r) ⊆ closedBall (0 : Dual 𝕜 E) (‖c‖ / r) | Mathlib_Analysis_NormedSpace_Dual |
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
hc : 1 < ‖c‖
r : ℝ
hr : 0 < r
x' : Dual 𝕜 E
hx' : x' ∈ polar 𝕜 (ball 0 r)
⊢ x' ∈ closedBall 0 (‖c‖ / 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 analy... | rw [mem_polar_iff] at hx' | theorem polar_ball_subset_closedBall_div {c : 𝕜} (hc : 1 < ‖c‖) {r : ℝ} (hr : 0 < r) :
polar 𝕜 (ball (0 : E) r) ⊆ closedBall (0 : Dual 𝕜 E) (‖c‖ / r) := by
intro x' hx'
| Mathlib.Analysis.NormedSpace.Dual.218_0.WirVfj6f5oiZZ2w | theorem polar_ball_subset_closedBall_div {c : 𝕜} (hc : 1 < ‖c‖) {r : ℝ} (hr : 0 < r) :
polar 𝕜 (ball (0 : E) r) ⊆ closedBall (0 : Dual 𝕜 E) (‖c‖ / r) | Mathlib_Analysis_NormedSpace_Dual |
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
hc : 1 < ‖c‖
r : ℝ
hr : 0 < r
x' : Dual 𝕜 E
hx' : ∀ z ∈ ball 0 r, ‖x' z‖ ≤ 1
⊢ x' ∈ closedBall 0 (‖c‖ / 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 analy... | simp only [polar, mem_setOf, mem_closedBall_zero_iff, mem_ball_zero_iff] at * | theorem polar_ball_subset_closedBall_div {c : 𝕜} (hc : 1 < ‖c‖) {r : ℝ} (hr : 0 < r) :
polar 𝕜 (ball (0 : E) r) ⊆ closedBall (0 : Dual 𝕜 E) (‖c‖ / 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 : 𝕜} (hc : 1 < ‖c‖) {r : ℝ} (hr : 0 < r) :
polar 𝕜 (ball (0 : E) r) ⊆ closedBall (0 : Dual 𝕜 E) (‖c‖ / r) | Mathlib_Analysis_NormedSpace_Dual |
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
hc : 1 < ‖c‖
r : ℝ
hr : 0 < r
x' : Dual 𝕜 E
hx' : ∀ (z : E), ‖z‖ < r → ‖x' z‖ ≤ 1
⊢ ‖x'‖ ≤ ‖c‖ / 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 analy... | have hcr : 0 < ‖c‖ / r := div_pos (zero_lt_one.trans hc) hr | theorem polar_ball_subset_closedBall_div {c : 𝕜} (hc : 1 < ‖c‖) {r : ℝ} (hr : 0 < r) :
polar 𝕜 (ball (0 : E) r) ⊆ closedBall (0 : Dual 𝕜 E) (‖c‖ / 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 : 𝕜} (hc : 1 < ‖c‖) {r : ℝ} (hr : 0 < r) :
polar 𝕜 (ball (0 : E) r) ⊆ closedBall (0 : Dual 𝕜 E) (‖c‖ / r) | Mathlib_Analysis_NormedSpace_Dual |
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
hc : 1 < ‖c‖
r : ℝ
hr : 0 < r
x' : Dual 𝕜 E
hx' : ∀ (z : E), ‖z‖ < r → ‖x' z‖ ≤ 1
hcr : 0 < ‖c‖ / r
⊢ ‖x'‖ ≤ ‖c‖ / 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 analy... | refine' ContinuousLinearMap.op_norm_le_of_shell hr hcr.le hc fun x h₁ h₂ => _ | theorem polar_ball_subset_closedBall_div {c : 𝕜} (hc : 1 < ‖c‖) {r : ℝ} (hr : 0 < r) :
polar 𝕜 (ball (0 : E) r) ⊆ closedBall (0 : Dual 𝕜 E) (‖c‖ / 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 < ‖c‖ / r := div_po... | Mathlib.Analysis.NormedSpace.Dual.218_0.WirVfj6f5oiZZ2w | theorem polar_ball_subset_closedBall_div {c : 𝕜} (hc : 1 < ‖c‖) {r : ℝ} (hr : 0 < r) :
polar 𝕜 (ball (0 : E) r) ⊆ closedBall (0 : Dual 𝕜 E) (‖c‖ / r) | Mathlib_Analysis_NormedSpace_Dual |
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
hc : 1 < ‖c‖
r : ℝ
hr : 0 < r
x' : Dual 𝕜 E
hx' : ∀ (z : E), ‖z‖ < r → ‖x' z‖ ≤ 1
hcr : 0 < ‖c‖ / r
x : E
h₁ : r / ‖c‖ ≤ ‖x‖
h₂ : ‖x‖ < r
⊢ ‖x' x‖ ≤ ‖c‖ / r * ‖x‖ | /-
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 analy... | calc
‖x' x‖ ≤ 1 := hx' _ h₂
_ ≤ ‖c‖ / r * ‖x‖ := (inv_pos_le_iff_one_le_mul' hcr).1 (by rwa [inv_div]) | theorem polar_ball_subset_closedBall_div {c : 𝕜} (hc : 1 < ‖c‖) {r : ℝ} (hr : 0 < r) :
polar 𝕜 (ball (0 : E) r) ⊆ closedBall (0 : Dual 𝕜 E) (‖c‖ / 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 < ‖c‖ / r := div_po... | Mathlib.Analysis.NormedSpace.Dual.218_0.WirVfj6f5oiZZ2w | theorem polar_ball_subset_closedBall_div {c : 𝕜} (hc : 1 < ‖c‖) {r : ℝ} (hr : 0 < r) :
polar 𝕜 (ball (0 : E) r) ⊆ closedBall (0 : Dual 𝕜 E) (‖c‖ / r) | Mathlib_Analysis_NormedSpace_Dual |
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
hc : 1 < ‖c‖
r : ℝ
hr : 0 < r
x' : Dual 𝕜 E
hx' : ∀ (z : E), ‖z‖ < r → ‖x' z‖ ≤ 1
hcr : 0 < ‖c‖ / r
x : E
h₁ : r / ‖c‖ ≤ ‖x‖
h₂ : ‖x‖ < r
⊢ (‖c‖ / r)⁻¹ ≤ ‖x‖ | /-
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 analy... | rwa [inv_div] | theorem polar_ball_subset_closedBall_div {c : 𝕜} (hc : 1 < ‖c‖) {r : ℝ} (hr : 0 < r) :
polar 𝕜 (ball (0 : E) r) ⊆ closedBall (0 : Dual 𝕜 E) (‖c‖ / 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 < ‖c‖ / r := div_po... | Mathlib.Analysis.NormedSpace.Dual.218_0.WirVfj6f5oiZZ2w | theorem polar_ball_subset_closedBall_div {c : 𝕜} (hc : 1 < ‖c‖) {r : ℝ} (hr : 0 < r) :
polar 𝕜 (ball (0 : E) r) ⊆ closedBall (0 : Dual 𝕜 E) (‖c‖ / r) | Mathlib_Analysis_NormedSpace_Dual |
𝕜✝ : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜✝
E✝ : Type u_2
inst✝⁴ : SeminormedAddCommGroup E✝
inst✝³ : NormedSpace 𝕜✝ E✝
𝕜 : Type u_3
E : Type u_4
inst✝² : IsROrC 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
r : ℝ
hr : 0 < r
⊢ polar 𝕜 (closedBall 0 r) = closedBall 0 r⁻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 analy... | refine' Subset.antisymm _ (closedBall_inv_subset_polar_closedBall 𝕜) | /-- The `polar` of closed ball in a normed space `E` is the closed ball of the dual with
inverse radius. -/
theorem polar_closedBall {𝕜 E : Type*} [IsROrC 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {r : ℝ}
(hr : 0 < r) : polar 𝕜 (closedBall (0 : E) r) = closedBall (0 : Dual 𝕜 E) r⁻¹ := 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 {𝕜 E : Type*} [IsROrC 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {r : ℝ}
(hr : 0 < r) : polar 𝕜 (closedBall (0 : E) r) = closedBall (0 : Dual 𝕜 E) r⁻¹ | Mathlib_Analysis_NormedSpace_Dual |
𝕜✝ : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜✝
E✝ : Type u_2
inst✝⁴ : SeminormedAddCommGroup E✝
inst✝³ : NormedSpace 𝕜✝ E✝
𝕜 : Type u_3
E : Type u_4
inst✝² : IsROrC 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
r : ℝ
hr : 0 < r
⊢ polar 𝕜 (closedBall 0 r) ⊆ closedBall 0 r⁻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 analy... | 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 {𝕜 E : Type*} [IsROrC 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {r : ℝ}
(hr : 0 < r) : polar 𝕜 (closedBall (0 : E) r) = closedBall (0 : Dual 𝕜 E) r⁻¹ := by
refine' Subset.ant... | 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 {𝕜 E : Type*} [IsROrC 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {r : ℝ}
(hr : 0 < r) : polar 𝕜 (closedBall (0 : E) r) = closedBall (0 : Dual 𝕜 E) r⁻¹ | Mathlib_Analysis_NormedSpace_Dual |
𝕜✝ : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜✝
E✝ : Type u_2
inst✝⁴ : SeminormedAddCommGroup E✝
inst✝³ : NormedSpace 𝕜✝ E✝
𝕜 : Type u_3
E : Type u_4
inst✝² : IsROrC 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
r : ℝ
hr : 0 < r
x' : Dual 𝕜 E
h : x' ∈ polar 𝕜 (closedBall 0 r)
⊢ x' ∈ closedBall 0 r⁻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 analy... | 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 {𝕜 E : Type*} [IsROrC 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {r : ℝ}
(hr : 0 < r) : polar 𝕜 (closedBall (0 : E) r) = closedBall (0 : Dual 𝕜 E) r⁻¹ := by
refine' Subset.ant... | 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 {𝕜 E : Type*} [IsROrC 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {r : ℝ}
(hr : 0 < r) : polar 𝕜 (closedBall (0 : E) r) = closedBall (0 : Dual 𝕜 E) r⁻¹ | Mathlib_Analysis_NormedSpace_Dual |
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