Theorems · Theorem · functional analysis

Submodule.starProjection_add_starProjection_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] (w : E), K.starProjection w + Kᗮ.starProjection w = w

If the orthogonal projection to K is well-defined, then a vector splits as the sum of its orthogonal projections onto a complete submodule K and onto the orthogonal complement of K.

Defined in

Mathlib.Analysis.InnerProductSpace.Projection.Basic

Cited by

3 results in Mathlib

Foundations

Depth 182 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceSubmodule.HasOrthogonalProjection