Theorems · Theorem · functional analysis

lipschitzOnWith_cfc_fun_of_subset

∀ {R : Type u_2} {A : Type u_3} {p : A → Prop} [inst : CommSemiring R] [inst_1 : StarRing R] [inst_2 : MetricSpace R]
  [inst_3 : IsTopologicalSemiring R] [inst_4 : ContinuousStar R] [inst_5 : Ring A] [inst_6 : StarRing A]
  [inst_7 : MetricSpace A] [inst_8 : Algebra R A] [inst_9 : IsometricContinuousFunctionalCalculus R A p] (a : A)
  {s : Set R},
  spectrum R a ⊆ s →
    LipschitzOnWith 1 (fun f => cfc ((UniformOnFun.toFun {s}) f) a) {f | ContinuousOn ((UniformOnFun.toFun {s}) f) s}

The function f ↦ cfc f a is Lipschitz with constant 1 with respect to supremum metric (on R →ᵤ[{s}] R) on those functions which are continuous on a set s containing the spectrum.

Defined in

Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Cited by

2 results in Mathlib

Foundations

Depth 160 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringStarRingMetricSpaceIsTopologicalSemiringContinuousStarRingStarRingMetricSpaceAlgebraIsometricContinuousFunctionalCalculus