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b317718 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 | import OAI.Probability.FreeIsing.Spectral.SpectralTransfer
namespace OAI
noncomputable section
namespace Problem367
namespace SpectralClosure
variable {𝕜 : Type*} [RCLike 𝕜]
variable {E F : Type*}
[NormedAddCommGroup E] [NormedSpace 𝕜 E]
[NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F]
lemma norm_apply_le_on_closure_range
(T : E →L[𝕜] E) (S : F →L[𝕜] F) (D : E →L[𝕜] F)
(hS : IsSelfAdjoint S) (hDS : ∀ x, S (D x) = D (T x))
{ρ : ℝ} (hρ : 0 ≤ ρ) (hT : ‖T‖ ≤ ρ)
{y : F} (hy : y ∈ closure (Set.range D)) : ‖S y‖ ≤ ρ * ‖y‖ :=
selfAdjoint_norm_apply_le_on_closure_range_of_intertwiner S hS T D hDS hρ hT hy
def rangeClosure (D : E →L[𝕜] F) : Submodule 𝕜 F :=
(LinearMap.range (D : E →ₗ[𝕜] F)).topologicalClosure
instance rangeClosure_completeSpace (D : E →L[𝕜] F) : CompleteSpace (rangeClosure D) :=
(Submodule.isClosed_topologicalClosure (LinearMap.range (D : E →ₗ[𝕜] F))).isComplete.completeSpace_coe
omit [CompleteSpace F] in
@[simp] lemma mem_rangeClosure (D : E →L[𝕜] F) (y : F) :
y ∈ rangeClosure D ↔ y ∈ closure (Set.range D) := Iff.rfl
omit [CompleteSpace F] in
lemma mapsTo_rangeClosure (T : E →L[𝕜] E) (S : F →L[𝕜] F) (D : E →L[𝕜] F)
(hDS : ∀ x, S (D x) = D (T x)) :
∀ y ∈ rangeClosure D, S y ∈ rangeClosure D := by
have hmap : Set.MapsTo S (Set.range D) (Set.range D) := by
rintro y ⟨x, rfl⟩
exact ⟨T x, (hDS x).symm⟩
exact hmap.closure S.continuous
def restrict (T : E →L[𝕜] E) (S : F →L[𝕜] F) (D : E →L[𝕜] F)
(hDS : ∀ x, S (D x) = D (T x)) : rangeClosure D →L[𝕜] rangeClosure D :=
S.restrict (mapsTo_rangeClosure T S D hDS)
lemma restrict_isSymmetric (T : E →L[𝕜] E) (S : F →L[𝕜] F) (D : E →L[𝕜] F)
(hS : IsSelfAdjoint S) (hDS : ∀ x, S (D x) = D (T x)) :
((restrict T S D hDS) : rangeClosure D →ₗ[𝕜] rangeClosure D).IsSymmetric := by
intro x y
exact hS.isSymmetric x y
lemma restrict_isSelfAdjoint (T : E →L[𝕜] E) (S : F →L[𝕜] F) (D : E →L[𝕜] F)
(hS : IsSelfAdjoint S) (hDS : ∀ x, S (D x) = D (T x)) :
IsSelfAdjoint (restrict T S D hDS) :=
(restrict_isSymmetric T S D hS hDS).isSelfAdjoint
lemma mapsTo_orthogonal_rangeClosure
(T : E →L[𝕜] E) (S : F →L[𝕜] F) (D : E →L[𝕜] F)
(hS : IsSelfAdjoint S) (hDS : ∀ x, S (D x) = D (T x)) :
∀ y ∈ (rangeClosure D)ᗮ, S y ∈ (rangeClosure D)ᗮ := by
intro y hy
rw [Submodule.mem_orthogonal]
intro x hx
calc
inner 𝕜 x (S y) = inner 𝕜 (S x) y := by
simpa only [ContinuousLinearMap.coe_coe] using (hS.isSymmetric x y).symm
_ = 0 := Submodule.inner_right_of_mem_orthogonal (mapsTo_rangeClosure T S D hDS x hx) hy
lemma norm_restrict_le (T : E →L[𝕜] E) (S : F →L[𝕜] F) (D : E →L[𝕜] F)
(hS : IsSelfAdjoint S) (hDS : ∀ x, S (D x) = D (T x))
{ρ : ℝ} (hρ : 0 ≤ ρ) (hT : ‖T‖ ≤ ρ) : ‖restrict T S D hDS‖ ≤ ρ := by
apply ContinuousLinearMap.opNorm_le_bound _ hρ
intro y
exact norm_apply_le_on_closure_range T S D hS hDS hρ hT y.property
omit [CompleteSpace F] in
lemma range_domRestrict_eq (D : E →L[𝕜] F) (K : Submodule 𝕜 E)
(hcover : ∀ x : E, ∃ z ∈ K, D z = D x) :
Set.range (D.domRestrict K) = Set.range D := by
ext y
constructor
· rintro ⟨x, rfl⟩
exact ⟨x, rfl⟩
· rintro ⟨x, rfl⟩
obtain ⟨z, hz, heq⟩ := hcover x
exact ⟨⟨z, hz⟩, heq⟩
omit [CompleteSpace F] in
lemma rangeClosure_domRestrict_eq (D : E →L[𝕜] F) (K : Submodule 𝕜 E)
(hcover : ∀ x : E, ∃ z ∈ K, D z = D x) :
rangeClosure (D.domRestrict K) = rangeClosure D := by
ext y
simp only [mem_rangeClosure, range_domRestrict_eq D K hcover]
omit [CompleteSpace F] in
lemma source_cover_of_kills_direction (D : E →L[𝕜] F) (ℓ : E →L[𝕜] 𝕜)
(e : E) (he : ℓ e = 1) (hDe : D e = 0) :
∀ x : E, ∃ z ∈ LinearMap.ker (ℓ : E →ₗ[𝕜] 𝕜), D z = D x := by
intro x
refine ⟨x - ℓ x • e, ?_, ?_⟩
· change ℓ (x - ℓ x • e) = 0
simp [he]
· simp [hDe]
omit [CompleteSpace F] in
lemma rangeClosure_ker_eq_of_kills_direction (D : E →L[𝕜] F) (ℓ : E →L[𝕜] 𝕜)
(e : E) (he : ℓ e = 1) (hDe : D e = 0) :
rangeClosure (D.domRestrict (LinearMap.ker (ℓ : E →ₗ[𝕜] 𝕜))) = rangeClosure D :=
rangeClosure_domRestrict_eq D _ (source_cover_of_kills_direction D ℓ e he hDe)
lemma norm_apply_le_of_source_subspace
(T : E →L[𝕜] E) (S : F →L[𝕜] F) (D : E →L[𝕜] F)
(K : Submodule 𝕜 E) (hK : ∀ x ∈ K, T x ∈ K)
(hS : IsSelfAdjoint S) (hDS : ∀ x, S (D x) = D (T x))
(hcover : ∀ x : E, ∃ z ∈ K, D z = D x)
{ρ : ℝ} (hρ : 0 ≤ ρ) (hT : ‖T.restrict hK‖ ≤ ρ)
{y : F} (hy : y ∈ closure (Set.range D)) : ‖S y‖ ≤ ρ * ‖y‖ := by
apply norm_apply_le_on_closure_range (T.restrict hK) S (D.domRestrict K) hS
(fun x => hDS x) hρ hT
rwa [range_domRestrict_eq D K hcover]
lemma norm_restrict_le_of_source_subspace
(T : E →L[𝕜] E) (S : F →L[𝕜] F) (D : E →L[𝕜] F)
(K : Submodule 𝕜 E) (hK : ∀ x ∈ K, T x ∈ K)
(hS : IsSelfAdjoint S) (hDS : ∀ x, S (D x) = D (T x))
(hcover : ∀ x : E, ∃ z ∈ K, D z = D x)
{ρ : ℝ} (hρ : 0 ≤ ρ) (hT : ‖T.restrict hK‖ ≤ ρ) :
‖restrict T S D hDS‖ ≤ ρ := by
apply ContinuousLinearMap.opNorm_le_bound _ hρ
intro y
exact norm_apply_le_of_source_subspace T S D K hK hS hDS hcover hρ hT y.property
lemma norm_restrict_le_of_kills_direction
(T : E →L[𝕜] E) (S : F →L[𝕜] F) (D : E →L[𝕜] F)
(ℓ : E →L[𝕜] 𝕜) (e : E) (he : ℓ e = 1) (hDe : D e = 0)
(hK : ∀ x ∈ LinearMap.ker (ℓ : E →ₗ[𝕜] 𝕜),
T x ∈ LinearMap.ker (ℓ : E →ₗ[𝕜] 𝕜))
(hS : IsSelfAdjoint S) (hDS : ∀ x, S (D x) = D (T x))
{ρ : ℝ} (hρ : 0 ≤ ρ) (hT : ‖T.restrict hK‖ ≤ ρ) :
‖restrict T S D hDS‖ ≤ ρ :=
norm_restrict_le_of_source_subspace T S D _ hK hS hDS
(source_cover_of_kills_direction D ℓ e he hDe) hρ hT
end SpectralClosure
end Problem367
end
end OAI
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