Theorems · Theorem · functional analysis

OrthonormalBasis.orthogonalProjectionOnto_eq_sum_rankOne

∀ {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), U.orthogonalProjectionOnto = ∑ i, ((InnerProductSpace.rankOne 𝕜) (b i)) ↑(b i)

Defined in

Mathlib.Analysis.InnerProductSpace.PiL2

Cited by

2 results in Mathlib

Foundations

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

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceFintypeSubmodule.HasOrthogonalProjection