Theorems · Theorem · functional analysis

Submodule.orthogonalProjectionOnto_starProjection_of_le

∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
  {U V : Submodule 𝕜 E} [inst_3 : U.HasOrthogonalProjection] [inst_4 : V.HasOrthogonalProjection],
  U ≤ V → ∀ (x : E), U.orthogonalProjectionOnto (V.starProjection x) = U.orthogonalProjectionOnto x

If U ≤ V, then projecting on V and then on U is the same as projecting on U.

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.HasOrthogonalProjectionSubmodule.HasOrthogonalProjection