openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Operators /HilbertConjugationExistence.lean
| import OAI.Analysis.IrrationalRotation.Operators.Conjugation | |
| import Mathlib.Analysis.InnerProductSpace.l2Space | |
| namespace OAI | |
| noncomputable section | |
| namespace Invariant233 | |
| open scoped lp | |
| variable {H ι : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] | |
| theorem exists_conjugation_fixing_hilbertBasis (b : HilbertBasis ι ℂ H) : | |
| ∃ C : HilbertConjugation H, ∀ i, C.val (b i) = b i := by | |
| classical | |
| let C : H → H := fun x => b.repr.symm (star (b.repr x)) | |
| have hC : IsConjugation C := by | |
| refine ⟨fun x => ?_, fun x y => ?_, fun c x => ?_, fun x => ?_⟩ | |
| · simp [C] | |
| · simp [C] | |
| · simp [C] | |
| · simp [C] | |
| refine ⟨hC.toHilbertConjugation, fun i => ?_⟩ | |
| change b.repr.symm (star (b.repr (b i))) = b i | |
| rw [b.repr_self] | |
| have hs : star (lp.single 2 i (1 : ℂ) : ℓ²(ι, ℂ)) = lp.single 2 i (1 : ℂ) := by | |
| ext j | |
| by_cases hij : j = i | |
| · subst j; simp [lp.single_apply] | |
| · simp [lp.single_apply, Pi.single_eq_of_ne hij] | |
| rw [hs, b.repr_symm_single] | |
| theorem nonempty_hilbertConjugation [CompleteSpace H] : Nonempty (HilbertConjugation H) := by | |
| obtain ⟨w, b, _⟩ := exists_hilbertBasis ℂ H | |
| obtain ⟨C, _⟩ := exists_conjugation_fixing_hilbertBasis b | |
| exact ⟨C⟩ | |
| theorem exists_isConjugation [CompleteSpace H] : ∃ C : H → H, IsConjugation C := by | |
| obtain ⟨C⟩ := nonempty_hilbertConjugation (H := H) | |
| exact ⟨C.val, C.isConjugation⟩ | |
| end Invariant233 | |
| end | |
| end OAI | |