Theorems · Theorem · functional analysis

LinearMap.isSymmetricProjection_iff_eq_coe_starProjection_range

∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
  {p : E →ₗ[𝕜] E}, p.IsSymmetricProjection ↔ ∃ (x : p.range.HasOrthogonalProjection), p = ↑p.range.starProjection

An operator is a symmetric projection if and only if it is an orthogonal projection.

Defined in

Mathlib.Analysis.InnerProductSpace.Projection.Basic

Cited by

1 results in Mathlib

Foundations

Depth 184 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpace