(heig : IsGramEigen w u lam) {d : ℕ} {e : Fin d → Fin m} (he : Function.Injective e) (hpos : ∀ j, 0 < lam (e j)) : (adjoint (retainedEmbedding w u lam e)).comp (retainedEmbedding w u lam e)...

Proved

BookProof.ChapterSirkGramCutoff.retainedEmbedding_isometry

by leonardopedro · Sep 12, 2026 · Mathlib 0df444a (Lean v4.33.1)

sirkspectral-theorytimepiece

Lean 4 theorem BookProof.ChapterSirkGramCutoff.retainedEmbedding_isometry (module BookProof.ChapterSirkGramCutoff), source chapter BookProof/ChapterChapterSirkGramCutoff.lean.

Preamble

-- Generated from ChapterSirkGramCutoff.lean — theorem BookProof.ChapterSirkGramCutoff.retainedEmbedding_isometry
import Mathlib
import Definitions.Def_ChapterSirkGramCutoff
open BookProof.ChapterSirkGramCutoff









noncomputable section


open scoped InnerProductSpace
open BookProof.ChapterSirkGramWhitening
open ContinuousLinearMap

variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℂ E] [CompleteSpace E]






variable {m : ℕ} {w : Fin m → E}
variable {u : OrthonormalBasis (Fin m) ℂ (EuclideanSpace ℂ (Fin m))} {lam : Fin m → ℝ}

Formal statement

theorem BookProof.ChapterSirkGramCutoff.retainedEmbedding_isometry (heig : IsGramEigen w u lam) {d : ℕ} {e : Fin d → Fin m}
    (he : Function.Injective e) (hpos : ∀ j, 0 < lam (e j)) :
    (adjoint (retainedEmbedding w u lam e)).comp (retainedEmbedding w u lam e)
      = ContinuousLinearMap.id ℂ (EuclideanSpace ℂ (Fin d)) := by sorry

Source

https://github.com/leonardopedrio/timepiece/blob/61595bc/BookProof/ChapterChapterSirkGramCutoff.lean

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