Theorems · Theorem · functional analysis

ContinuousLinearMap.isStarNormal_iff_norm_eq_adjoint

∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
  {T : E →L[𝕜] E} [inst_3 : CompleteSpace E], IsStarNormal T ↔ ∀ (v : E), ‖T v‖ = ‖(ContinuousLinearMap.adjoint T) v‖

An operator T is normal iff ‖T v‖ = ‖(adjoint T) v‖ for all v.

Defined in

Mathlib.Analysis.InnerProductSpace.Adjoint

Cited by

2 results in Mathlib

Foundations

Depth 196 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceCompleteSpace