Theorems · Theorem · operator theory

ContinuousLinearMap.orthogonalComplement_iSup_eigenspaces_eq_bot

∀ {𝕜 : Type u_1} [inst : RCLike 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
  [CompleteSpace E] {T : E →L[𝕜] E}, IsCompactOperator ⇑T → (↑T).IsSymmetric → (⨆ μ, Module.End.eigenspace (↑T) μ)ᗮ = ⊥

The Spectral Theorem for compact self-adjoint operators: the eigenspaces of a compact self-adjoint operator have trivial orthogonal complement.

Defined in

Mathlib.Analysis.InnerProductSpace.Spectrum

Cited by

0 results in Mathlib

Foundations

Depth 198 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceCompleteSpace