Theorems · Theorem · operator theory

LinearMap.IsSymmetric.iSup_iInf_eq_top_of_commute

∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
  [FiniteDimensional 𝕜 E] {ι : Type u_5} {T : ι → E →ₗ[𝕜] E},
  (∀ (i : ι), (T i).IsSymmetric) → Pairwise (Function.onFun Commute T) → ⨆ χ, ⨅ i, Module.End.eigenspace (T i) (χ i) = ⊤

A commuting family of symmetric linear maps on a finite-dimensional inner product space is simultaneously diagonalizable.

Defined in

Mathlib.Analysis.InnerProductSpace.JointEigenspace

Cited by

1 results in Mathlib

Foundations

Depth 204 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceFiniteDimensional