Theorems · Theorem · functional analysis

IsGreatest.nnnorm_cfc

∀ {𝕜 : Type u_1} {A : Type u_2} {p : outParam (A → Prop)} [inst : RCLike 𝕜] [inst_1 : NormedRing A]
  [inst_2 : StarRing A] [inst_3 : NormedAlgebra 𝕜 A] [inst_4 : IsometricContinuousFunctionalCalculus 𝕜 A p]
  [Nontrivial A] (f : 𝕜 → 𝕜) (a : A),
  autoParam (ContinuousOn f (spectrum 𝕜 a)) IsGreatest.nnnorm_cfc._auto_1 →
    autoParam (p a) IsGreatest.nnnorm_cfc._auto_3 → IsGreatest ((fun x => ‖f x‖₊) '' spectrum 𝕜 a) ‖cfc f a‖₊

Defined in

Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric

Cited by

2 results in Mathlib

Foundations

Depth 180 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedRingStarRingNormedAlgebraIsometricContinuousFunctionalCalculusNontrivial