File size: 2,692 Bytes
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 | import Mathlib.Analysis.InnerProductSpace.l2Space
import Mathlib.Topology.Bases
import Mathlib.Basic.Denumerable
namespace OAI
namespace Hyperinvariant
open Submodule TopologicalSpace
universe u v
variable {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H]
/-- An orthonormal subset of a separable Hilbert space is countable. -/
theorem orthonormal_subtype_countable [CompleteSpace H] [SeparableSpace H] (w : Set H)
(hw : Orthonormal ℂ (Subtype.val : w → H)) : Countable w := by
classical
let : DiscreteTopology w := discreteTopology_iff_isOpen_singleton.mpr (by
intro i
have ho : IsOpen {j : w | (1 / 2 : ℝ) < ‖inner ℂ (i : H) (j : H)‖} :=
isOpen_lt continuous_const (continuous_const.inner continuous_subtype_val).norm
have he : {j : w | (1 / 2 : ℝ) < ‖inner ℂ (i : H) (j : H)‖} = {i} := by
ext j
rw [Set.mem_ofPred_eq, orthonormal_iff_ite.mp hw i j]
by_cases hij : i = j <;> simp [hij, eq_comm]; norm_num
rwa [he] at ho)
exact separableSpace_iff_countable.mp inferInstance
variable [CompleteSpace H] [SeparableSpace H]
/-- Every infinite-dimensional separable complex Hilbert space has a
Hilbert basis indexed by the natural numbers. -/
theorem exists_nat_hilbertBasis (hInf : ¬FiniteDimensional ℂ H) :
Nonempty (HilbertBasis ℕ ℂ H) := by
classical
obtain ⟨w, b, hb⟩ := exists_hilbertBasis ℂ H
have hw : Orthonormal ℂ (Subtype.val : w → H) := by
simpa only [hb] using b.orthonormal
let : Countable w := orthonormal_subtype_countable w hw
let : Infinite w := by
refine not_finite_iff_infinite.mp ?_
intro hfin
let : Finite w := hfin
let : Fintype w := Fintype.ofFinite w
exact hInf (Module.Finite.of_basis b.toOrthonormalBasis.toBasis)
let e : ℕ ≃ w := Classical.choice (inferInstance : Nonempty (ℕ ≃ w))
have ho : Orthonormal ℂ (b ∘ e) := b.orthonormal.comp e e.injective
have hr : Set.range (b ∘ e) = Set.range b := by
ext x
constructor
· rintro ⟨n, rfl⟩
exact ⟨e n, rfl⟩
· rintro ⟨i, rfl⟩
exact ⟨e.symm i, by simp⟩
exact ⟨HilbertBasis.mk ho (by rw [hr, b.dense_span])⟩
/-- This handles arbitrary ambient Hilbert spaces, not just a canonical model. -/
theorem nonempty_linearIsometryEquiv
{K : Type v} [NormedAddCommGroup K] [InnerProductSpace ℂ K]
[CompleteSpace K] [SeparableSpace K]
(hH : ¬FiniteDimensional ℂ H) (hK : ¬FiniteDimensional ℂ K) :
Nonempty (H ≃ₗᵢ[ℂ] K) := by
obtain ⟨bH⟩ := exists_nat_hilbertBasis hH
obtain ⟨bK⟩ := exists_nat_hilbertBasis hK
exact ⟨bH.repr.trans bK.repr.symm⟩
end Hyperinvariant
end OAI
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