Theorems · Theorem · functional analysis

Submodule.norm_orthogonalProjectionOnto

∀ {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], K ≠ ⊥ → ‖K.orthogonalProjectionOnto‖ = 1

The operator norm of the orthogonal projection onto a nontrivial subspace is 1.

Defined in

Mathlib.Analysis.InnerProductSpace.Projection.Basic

Cited by

2 results in Mathlib

Foundations

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

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceSubmodule.HasOrthogonalProjection