| import Mathlib.Analysis.InnerProductSpace.l2Space |
| import OAI.Analysis.IrrationalRotation.Model.Analytic |
| import Mathlib.MeasureTheory.Measure.SeparableMeasure |
| import Mathlib.MeasureTheory.Function.LpSpace.Indicator |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace Invariant233.HilbertModelSize |
|
|
| open TopologicalSpace MeasureTheory |
| open scoped lp |
|
|
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℂ E] |
|
|
| theorem separable_of_countable_hilbertBasis {ι : Type*} [Countable ι] |
| (b : HilbertBasis ι ℂ E) : SeparableSpace E := by |
| have hs : IsSeparable (Submodule.span ℂ (Set.range b) : Set E) := |
| (Set.countable_range b).isSeparable.span |
| have hc := hs.closure |
| have heq : closure (Submodule.span ℂ (Set.range b) : Set E) = Set.univ := by |
| change ((Submodule.span ℂ (Set.range b)).topologicalClosure : Set E) = Set.univ |
| rw [b.dense_span] |
| rfl |
| rw [heq] at hc |
| exact isSeparable_univ_iff.mp hc |
|
|
| theorem separable_l2 (ι : Type*) [Countable ι] : SeparableSpace (ℓ²(ι, ℂ)) := |
| separable_of_countable_hilbertBasis (default : HilbertBasis ι ℂ (ℓ²(ι, ℂ))) |
|
|
| theorem not_finiteDimensional_of_infinite_orthonormal {ι : Type*} [Infinite ι] |
| {v : ι → E} (hv : Orthonormal ℂ v) : ¬ FiniteDimensional ℂ E := by |
| intro h |
| let := h |
| have : Finite ι := hv.linearIndependent.finite |
| exact not_finite ι |
|
|
| theorem not_finiteDimensional_l2 (ι : Type*) [Infinite ι] : |
| ¬ FiniteDimensional ℂ (ℓ²(ι, ℂ)) := |
| not_finiteDimensional_of_infinite_orthonormal |
| (default : HilbertBasis ι ℂ (ℓ²(ι, ℂ))).orthonormal |
|
|
| section BochnerLp |
|
|
| variable {X : Type*} [MeasurableSpace X] (μ : Measure X) [IsProbabilityMeasure μ] |
|
|
| def constL2Isometry : E →ₗᵢ[ℂ] Lp E 2 μ where |
| toLinearMap := Lp.constₗ 2 μ ℂ |
| norm_map' x := by |
| change ‖Lp.const 2 μ x‖ = ‖x‖ |
| rw [Lp.norm_const' 2 μ x (by norm_num) (by norm_num)] |
| simp |
|
|
| theorem not_finiteDimensional_L2 (hE : ¬ FiniteDimensional ℂ E) : |
| ¬ FiniteDimensional ℂ (Lp E 2 μ) := by |
| intro h |
| let := h |
| exact hE (FiniteDimensional.of_injective (constL2Isometry μ).toLinearMap |
| (constL2Isometry μ).injective) |
|
|
| theorem separable_L2_l2 [MeasurableSpace.CountablyGenerated X] |
| (ι : Type*) [Countable ι] : SeparableSpace (Lp (ℓ²(ι, ℂ)) 2 μ) := by |
| let := separable_l2 ι |
| let : Fact ((2 : ENNReal) ≠ ⊤) := ⟨by norm_num⟩ |
| infer_instance |
|
|
| theorem not_finiteDimensional_L2_l2 (ι : Type*) [Infinite ι] : |
| ¬ FiniteDimensional ℂ (Lp (ℓ²(ι, ℂ)) 2 μ) := |
| not_finiteDimensional_L2 μ (not_finiteDimensional_l2 ι) |
|
|
| end BochnerLp |
|
|
| theorem circleMeasure_isProbabilityMeasure : IsProbabilityMeasure circleMeasure := by |
| constructor |
| change (volume : Measure Circle) Set.univ = 1 |
| simp [AddCircle.measure_univ] |
|
|
| theorem circle_representation_separable : |
| SeparableSpace (Lp (ℓ²(ℤ, ℂ)) 2 circleMeasure) := by |
| let := circleMeasure_isProbabilityMeasure |
| exact separable_L2_l2 circleMeasure ℤ |
|
|
| theorem circle_representation_not_finiteDimensional : |
| ¬ FiniteDimensional ℂ (Lp (ℓ²(ℤ, ℂ)) 2 circleMeasure) := by |
| let := circleMeasure_isProbabilityMeasure |
| exact not_finiteDimensional_L2_l2 circleMeasure ℤ |
|
|
| end Invariant233.HilbertModelSize |
|
|
| end |
|
|
| end OAI |
|
|
