| 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 |
|
|
