Theorems · Theorem · functional analysis

Submodule.orthogonalProjectionOnto_apply_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] {v : E}, v ∈ Kᗮ → K.orthogonalProjectionOnto v = 0

The orthogonal projection onto K of an element of Kᗮ is zero.

Defined in

Mathlib.Analysis.InnerProductSpace.Projection.Basic

Cited by

9 results in Mathlib

Foundations

Depth 181 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceSubmodule.HasOrthogonalProjection