Theorems · Theorem · functional analysis

Submodule.starProjection_minimal

∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
  {U : Submodule 𝕜 E} [inst_3 : U.HasOrthogonalProjection] (y : E), ‖y - U.starProjection y‖ = ⨅ x, ‖y - ↑x‖

The orthogonal projection of y on U minimizes the distance ‖y - x‖ for x ∈ U.

Defined in

Mathlib.Analysis.InnerProductSpace.Projection.Basic

Cited by

1 results in Mathlib

Foundations

Depth 178 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceSubmodule.HasOrthogonalProjection