Theorems · Theorem · functional analysis

cfc_smul

∀ {R : Type u_1} {A : Type u_2} {p : A → Prop} [inst : CommSemiring R] [inst_1 : StarRing R] [inst_2 : MetricSpace R]
  [inst_3 : IsTopologicalSemiring R] [inst_4 : ContinuousStar R] [inst_5 : TopologicalSpace A] [inst_6 : Ring A]
  [inst_7 : StarRing A] [inst_8 : Algebra R A] [instCFC : ContinuousFunctionalCalculus R A p] {S : Type u_3}
  [inst_9 : SMul S R] [ContinuousConstSMul S R] [inst_11 : SMulZeroClass S A] [IsScalarTower S R A]
  [IsScalarTower S R (R → R)] (s : S) (f : R → R) (a : A),
  autoParam (ContinuousOn f (spectrum R a)) cfc_smul._auto_1 → cfc (fun x => s • f x) a = s • cfc f a

Defined in

Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Cited by

4 results in Mathlib

Foundations

Depth 101 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringStarRingMetricSpaceIsTopologicalSemiringContinuousStarTopologicalSpaceRingStarRingAlgebraContinuousFunctionalCalculusSMulContinuousConstSMulSMulZeroClassIsScalarTowerIsScalarTower