Theorems · Theorem · functional analysis

OrthonormalBasis.orthogonalProjection_apply_eq_sum

Deprecated since 2026-05-05Use OrthonormalBasis.orthogonalProjectionOnto_apply_eq_sum instead.

∀ {ι : Type u_1} {𝕜 : Type u_3} [inst : RCLike 𝕜] {E : Type u_4} [inst_1 : NormedAddCommGroup E]
  [inst_2 : InnerProductSpace 𝕜 E] [inst_3 : Fintype ι] {U : Submodule 𝕜 E} [inst_4 : U.HasOrthogonalProjection]
  (b : OrthonormalBasis ι 𝕜 ↥U) (x : E), U.orthogonalProjectionOnto x = ∑ i, inner 𝕜 (↑(b i)) x • b i

Alias of OrthonormalBasis.orthogonalProjectionOnto_apply_eq_sum.

Defined in

Mathlib.Analysis.InnerProductSpace.PiL2

Cited by

0 results in Mathlib

Foundations

Depth 231 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceFintypeSubmodule.HasOrthogonalProjection