| import Mathlib.Analysis.InnerProductSpace.l2Space |
| import Mathlib.Analysis.Normed.Operator.Basic |
| |
| namespace OAI |
|
|
| noncomputable section |
| open scoped ENNReal lp |
|
|
| namespace Invariant233 |
|
|
| abbrev SequenceHilbert (I : Type*) := lp (fun _ : I => ℂ) 2 |
|
|
| namespace SequenceHilbert |
| variable {I : Type*} |
|
|
| def diagonalLinear (a : I → ℂ) (C : ℝ) (ha : ∀ i, ‖a i‖ ≤ C) : |
| SequenceHilbert I →ₗ[ℂ] SequenceHilbert I where |
| toFun x := ⟨fun i => a i * x i, by |
| apply Memℓp.mono ((lp.memℓp x).norm.const_mul C) |
| intro i |
| rw [norm_mul] |
| exact mul_le_mul_of_nonneg_right (ha i) (norm_nonneg _)⟩ |
| map_add' x y := by |
| apply lp.ext |
| funext i |
| exact mul_add _ _ _ |
| map_smul' c x := by |
| apply lp.ext |
| funext i |
| change a i * (c * x i) = c * (a i * x i) |
| ring |
|
|
| def diagonal (a : I → ℂ) (C : ℝ) (hC : 0 ≤ C) (ha : ∀ i, ‖a i‖ ≤ C) : |
| SequenceHilbert I →L[ℂ] SequenceHilbert I := |
| (diagonalLinear a C ha).mkContinuous C (by |
| intro x |
| calc |
| ‖diagonalLinear a C ha x‖ ≤ ‖C • x‖ := by |
| apply lp.norm_mono (by norm_num : (2 : ℝ≥0∞) ≠ 0) |
| intro i |
| change ‖a i * x i‖ ≤ ‖C • x i‖ |
| rw [norm_mul, norm_smul, Real.norm_eq_abs, abs_of_nonneg hC] |
| exact mul_le_mul_of_nonneg_right (ha i) (norm_nonneg _) |
| _ = C * ‖x‖ := by rw [norm_smul, Real.norm_eq_abs, abs_of_nonneg hC]) |
|
|
| def reindex (e : Equiv.Perm I) : SequenceHilbert I ≃ₗᵢ[ℂ] SequenceHilbert I where |
| toFun x := ⟨fun i => x (e.symm i), by |
| apply memℓp_gen |
| exact e.symm.summable_iff.mpr ((lp.memℓp x).summable (by norm_num))⟩ |
| invFun x := ⟨fun i => x (e i), by |
| apply memℓp_gen |
| exact e.summable_iff.mpr ((lp.memℓp x).summable (by norm_num))⟩ |
| left_inv x := by |
| apply lp.ext |
| funext i |
| simp |
| right_inv x := by |
| apply lp.ext |
| funext i |
| simp |
| map_add' x y := by |
| apply lp.ext |
| funext i |
| rfl |
| map_smul' c x := by |
| apply lp.ext |
| funext i |
| rfl |
| norm_map' x := by |
| simp only [lp.norm_eq_tsum_rpow (by norm_num : 0 < (2 : ℝ≥0∞).toReal)] |
| exact congrArg (fun t : ℝ => t ^ (1 / (2 : ℝ≥0∞).toReal)) |
| (e.symm.tsum_eq (fun i => ‖x i‖ ^ (2 : ℝ≥0∞).toReal)) |
|
|
| end SequenceHilbert |
| end Invariant233 |
|
|
| end |
|
|
| end OAI |
|
|
