Theorems · Theorem · functional analysis

Submodule.exists_add_mem_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} [K.HasOrthogonalProjection] (v : E), ∃ y ∈ K, ∃ z ∈ Kᗮ, v = y + z

If K is complete, any v in E can be expressed as a sum of elements of K and Kᗮ.

Defined in

Mathlib.Analysis.InnerProductSpace.Projection.Basic

Cited by

1 results in Mathlib

Foundations

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

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceSubmodule.HasOrthogonalProjection