Theorems · Theorem · operator theory

Submodule.IsCompl.projection_isSymmetric_iff

Deprecated since 2026-05-05Use Submodule.isSymmetric_projection_iff instead.

∀ {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} (hUV : IsCompl U V), (U.projection V hUV).IsSymmetric ↔ U ⟂ V

Alias of Submodule.isSymmetric_projection_iff. A linear projection onto U along its complement V is symmetric if and only if U and V are pairwise orthogonal.

Defined in

Mathlib.Analysis.InnerProductSpace.Symmetric

Cited by

0 results in Mathlib

Foundations

Depth 165 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpace