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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 | 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
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