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9eeb9c7 | 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 | import OAI.Analysis.IrrationalRotation.Model.ColumnOperator
import OAI.Analysis.IrrationalRotation.Analysis.StrongFieldMeasurability
import Mathlib.Topology.Instances.Rat
namespace OAI
namespace Invariant233.Fields
open MeasureTheory
variable {ι H F : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H]
[NormedAddCommGroup F] [NormedSpace ℂ F] [CompleteSpace F]
theorem denseRange_basis_linearCombination (b : HilbertBasis ι ℂ H) :
DenseRange (Finsupp.linearCombination ℂ b) := by
change Dense (LinearMap.range (Finsupp.linearCombination ℂ b) : Set H)
rw [Submodule.dense_iff_topologicalClosure_eq_top, Finsupp.range_linearCombination]
exact b.dense_span
theorem columnOperator_linearCombination (b : HilbertBasis ι ℂ H) (c : ι → F)
{C : ℝ}
(hbound : ∀ z : ι →₀ ℂ,
‖Finsupp.linearCombination ℂ c z‖ ≤ C * ‖Finsupp.linearCombination ℂ b z‖)
(z : ι →₀ ℂ) :
columnOperator b c (Finsupp.linearCombination ℂ b z) =
Finsupp.linearCombination ℂ c z := by
exact LinearMap.extendOfNorm_eq (denseRange_basis_linearCombination b) ⟨C, hbound⟩ z
theorem columnOperator_apply_basis (b : HilbertBasis ι ℂ H) (c : ι → F)
{C : ℝ}
(hbound : ∀ z : ι →₀ ℂ,
‖Finsupp.linearCombination ℂ c z‖ ≤ C * ‖Finsupp.linearCombination ℂ b z‖)
(i : ι) : columnOperator b c (b i) = c i := by
classical
simpa using columnOperator_linearCombination b c hbound (Finsupp.single i 1)
theorem norm_columnOperator_le (b : HilbertBasis ι ℂ H) (c : ι → F)
{C : ℝ} (hC : 0 ≤ C)
(hbound : ∀ z : ι →₀ ℂ,
‖Finsupp.linearCombination ℂ c z‖ ≤ C * ‖Finsupp.linearCombination ℂ b z‖) :
‖columnOperator b c‖ ≤ C :=
LinearMap.opNorm_extendOfNorm_le (denseRange_basis_linearCombination b) hC hbound
theorem columnOperator_unique (b : HilbertBasis ι ℂ H) (c : ι → F)
{C : ℝ}
(hbound : ∀ z : ι →₀ ℂ,
‖Finsupp.linearCombination ℂ c z‖ ≤ C * ‖Finsupp.linearCombination ℂ b z‖)
(T : H →L[ℂ] F) (hT : ∀ i, T (b i) = c i) : columnOperator b c = T := by
apply LinearMap.extendOfNorm_unique (denseRange_basis_linearCombination b) C hbound T
ext i
simp [hT]
theorem denseRange_rationalComplex : DenseRange rationalComplex := by
have hq := Rat.isDenseEmbedding_coe_real.dense
have hprod := hq.prodMap hq
have h := Complex.equivRealProdCLM.symm.surjective.denseRange.comp hprod
Complex.equivRealProdCLM.symm.continuous
change DenseRange (fun q : ℚ × ℚ =>
Complex.equivRealProdCLM.symm ((q.1 : ℝ), (q.2 : ℝ)))
simpa only [Function.comp_def, Prod.map] using h
omit [CompleteSpace F] in
theorem finite_combination_bound_of_rational
(b : HilbertBasis ι ℂ H) (c : ι → F) {C : ℝ}
(hbound : ∀ s : Finset ι, ∀ q : s → ℚ × ℚ,
‖∑ i : s, rationalComplex (q i) • c i‖ ≤
C * ‖∑ i : s, rationalComplex (q i) • b i‖) :
∀ z : ι →₀ ℂ,
‖Finsupp.linearCombination ℂ c z‖ ≤ C * ‖Finsupp.linearCombination ℂ b z‖ := by
classical
intro z
have hd : DenseRange (fun q : z.support → ℚ × ℚ =>
fun i : z.support => rationalComplex (q i)) :=
DenseRange.piMap (fun _ => denseRange_rationalComplex)
have h : ‖∑ i : z.support, z i • c i‖ ≤ C * ‖∑ i : z.support, z i • b i‖ := by
exact hd.induction_on
(p := fun a : z.support → ℂ =>
‖∑ i : z.support, a i • c i‖ ≤ C * ‖∑ i : z.support, a i • b i‖)
(fun i : z.support => z i)
(isClosed_le (by fun_prop) (by fun_prop)) (hbound z.support)
rw [z.support.sum_coe_sort (fun i => z i • c i),
z.support.sum_coe_sort (fun i => z i • b i)] at h
exact h
variable {X : Type*} [MeasurableSpace X] [Countable ι]
theorem stronglyMeasurable_columnOperator_apply
(b : HilbertBasis ι ℂ H) (c : X → ι → F) {C : ℝ}
(hbound : ∀ x, ∀ z : ι →₀ ℂ,
‖Finsupp.linearCombination ℂ (c x) z‖ ≤ C * ‖Finsupp.linearCombination ℂ b z‖)
(hc : ∀ i, StronglyMeasurable fun x => c x i)
{v : X → H} (hv : StronglyMeasurable v) :
StronglyMeasurable fun x => columnOperator b (c x) (v x) := by
apply StrongField.stronglyMeasurable_apply_of_columns b
· intro i
simpa only [columnOperator_apply_basis b _ (hbound _)] using hc i
· exact hv
theorem aestronglyMeasurable_columnOperator_apply
(b : HilbertBasis ι ℂ H) (c : X → ι → F) {C : ℝ} {μ : Measure X}
(hbound : ∀ᵐ x ∂μ, ∀ z : ι →₀ ℂ,
‖Finsupp.linearCombination ℂ (c x) z‖ ≤ C * ‖Finsupp.linearCombination ℂ b z‖)
(hc : ∀ i, AEStronglyMeasurable (fun x => c x i) μ)
{v : X → H} (hv : AEStronglyMeasurable v μ) :
AEStronglyMeasurable (fun x => columnOperator b (c x) (v x)) μ := by
apply StrongField.aestronglyMeasurable_apply_of_columns b
· intro i
apply (hc i).congr
filter_upwards [hbound] with x hx
exact (columnOperator_apply_basis b (c x) hx i).symm
· exact hv
omit [CompleteSpace F] in
theorem ae_finite_combination_bound_of_rational
(b : HilbertBasis ι ℂ H) (c : X → ι → F) {C : ℝ} {μ : Measure X}
(hbound : ∀ s : Finset ι, ∀ q : s → ℚ × ℚ, ∀ᵐ x ∂μ,
‖∑ i : s, rationalComplex (q i) • c x i‖ ≤
C * ‖∑ i : s, rationalComplex (q i) • b i‖) :
∀ᵐ x ∂μ, ∀ z : ι →₀ ℂ,
‖Finsupp.linearCombination ℂ (c x) z‖ ≤ C * ‖Finsupp.linearCombination ℂ b z‖ := by
have hs : ∀ s : Finset ι, ∀ᵐ x ∂μ, ∀ q : s → ℚ × ℚ,
‖∑ i : s, rationalComplex (q i) • c x i‖ ≤
C * ‖∑ i : s, rationalComplex (q i) • b i‖ := by
intro s
exact ae_all_iff.mpr (hbound s)
filter_upwards [ae_all_iff.mpr hs] with x hx
exact finite_combination_bound_of_rational b (c x) hx
theorem exists_bounded_field_of_rational_column_tests
(b : HilbertBasis ι ℂ H) (c : X → ι → F) {C : ℝ} {μ : Measure X}
(hC : 0 ≤ C)
(hbound : ∀ s : Finset ι, ∀ q : s → ℚ × ℚ, ∀ᵐ x ∂μ,
‖∑ i : s, rationalComplex (q i) • c x i‖ ≤
C * ‖∑ i : s, rationalComplex (q i) • b i‖)
(hc : ∀ i, AEStronglyMeasurable (fun x => c x i) μ) :
∃ A : X → H →L[ℂ] F,
(∀ᵐ x ∂μ, ‖A x‖ ≤ C ∧ ∀ i, A x (b i) = c x i) ∧
(∀ v : H, AEStronglyMeasurable (fun x => A x v) μ) := by
have hb := ae_finite_combination_bound_of_rational b c hbound
refine ⟨fun x => columnOperator b (c x), ?_, ?_⟩
· filter_upwards [hb] with x hx
exact ⟨norm_columnOperator_le b (c x) hC hx,
columnOperator_apply_basis b (c x) hx⟩
· intro v
exact aestronglyMeasurable_columnOperator_apply b c hb hc aestronglyMeasurable_const
end Invariant233.Fields
end OAI
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