{F : Type*} [NormedAddCommGroup F] [InnerProductSpace ℂ F] [CompleteSpace F] (V : F →L[ℂ] E) (hV : (adjoint V).comp V = ContinuousLinearMap.id ℂ F) (x : E) {y : E} (hy : ∃ z, V z = y) : ‖x - V...

-- Generated from ChapterSirkGramCutoff.lean — theorem BookProof.ChapterSirkGramCutoff.norm_sub_proj_le_of_mem_range
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.norm_sub_proj_le_of_mem_range {F : Type*} [NormedAddCommGroup F]
    [InnerProductSpace ℂ F] [CompleteSpace F] (V : F →L[ℂ] E)
    (hV : (adjoint V).comp V = ContinuousLinearMap.id ℂ F)
    (x : E) {y : E} (hy : ∃ z, V z = y) :
    ‖x - V (adjoint V x)‖ ≤ ‖x - y‖ := by sorry