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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 | 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
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