(heig : IsGramEigen w u lam) (k l : Fin m) : ⟪synthesis w (u k), synthesis w (u l)⟫_ℂ = if k = l then (lam l : ℂ) else 0

-- Generated from ChapterSirkGramCutoff.lean — theorem BookProof.ChapterSirkGramCutoff.inner_synthesis_gramEigen
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}

theorem BookProof.ChapterSirkGramCutoff.inner_synthesis_gramEigen (heig : IsGramEigen w u lam) (k l : Fin m) :
    ⟪synthesis w (u k), synthesis w (u l)⟫_ℂ = if k = l then (lam l : ℂ) else 0 := by sorry