Theorems · Theorem · functional analysis

tendsto_cfc_fun

∀ {X : Type u_1} {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 : TopologicalSpace A] [inst_8 : Algebra R A]
  [inst_9 : ContinuousFunctionalCalculus R A p] {l : Filter X} {F : X → R → R} {f : R → R} {a : A},
  TendstoUniformlyOn F f l (spectrum R a) →
    (∀ᶠ (x : X) in l, ContinuousOn (F x) (spectrum R a)) → Filter.Tendsto (fun x => cfc (F x) a) l (nhds (cfc f a))

If F : X → R → R tends to f : R → R uniformly on the spectrum of a, and all these functions are continuous on the spectrum, then fun x ↦ cfc (F x) a tends to cfc f a.

Defined in

Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Cited by

3 results in Mathlib

Foundations

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

Assumes

CommSemiringStarRingMetricSpaceIsTopologicalSemiringContinuousStarRingStarRingTopologicalSpaceAlgebraContinuousFunctionalCalculus