Theorems · Theorem · functional analysis

Submodule.eq_starProjection_of_mem_orthogonal

∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
  {K : Submodule 𝕜 E} [inst_3 : K.HasOrthogonalProjection] {u v : E}, v ∈ K → u - v ∈ Kᗮ → K.starProjection u = v

A point in K with the orthogonality property (here characterized in terms of Kᗮ) must be the orthogonal projection.

Defined in

Mathlib.Analysis.InnerProductSpace.Projection.Basic

Cited by

3 results in Mathlib

Foundations

Depth 179 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceSubmodule.HasOrthogonalProjection