Theorems · Theorem · functional analysis

ContinuousOn.cfc

∀ {X : Type u_1} {𝕜 : Type u_2} {A : Type u_3} {p : A → Prop} [inst : RCLike 𝕜] [inst_1 : NormedRing A]
  [inst_2 : StarRing A] [inst_3 : NormedAlgebra 𝕜 A] [inst_4 : IsometricContinuousFunctionalCalculus 𝕜 A p]
  [ContinuousStar A] [inst_6 : TopologicalSpace X] {s : X → Set 𝕜} (f : 𝕜 → 𝕜) {a : X → A} {t : Set X},
  (∀ x ∈ t, IsCompact (s x)) →
    ContinuousOn a t →
      (∀ x₀ ∈ t, ∀ᶠ (x : X) in nhdsWithin x₀ t, spectrum 𝕜 (a x) ⊆ s x₀) →
        (∀ x ∈ t, p (a x)) →
          autoParam (∀ x ∈ t, ContinuousOn f (s x)) ContinuousOn.cfc._auto_1 → ContinuousOn (fun x => cfc f (a x)) t

Suppose a : X → Set A is continuous on t : Set X and a x satisfies the predicate p for all x ∈ t. Suppose further that s : X → Set 𝕜 is a family of sets with s x compact when x ∈ t such that s x₀ contains the spectrum of a x for all sufficiently close x ∈ t. If f : 𝕜 → 𝕜 is continuous on s x, for each x ∈ t, then fun x ↦ cfc f (a x) is continuous on t.

Defined in

Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Cited by

3 results in Mathlib

Foundations

Depth 199 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedRingStarRingNormedAlgebraIsometricContinuousFunctionalCalculusContinuousStarTopologicalSpace