Theorems · Theorem · functional analysis

ContinuousWithinAt.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 : Set 𝕜},
  IsCompact s →
    ∀ (f : 𝕜 → 𝕜) {a : X → A} {x₀ : X} {t : Set X},
      x₀ ∈ t →
        ContinuousWithinAt a t x₀ →
          (∀ᶠ (x : X) in nhdsWithin x₀ t, spectrum 𝕜 (a x) ⊆ s) →
            (∀ᶠ (x : X) in nhdsWithin x₀ t, p (a x)) →
              autoParam (ContinuousOn f s) ContinuousWithinAt.cfc._auto_1 →
                ContinuousWithinAt (fun x => cfc f (a x)) t x₀

If f : 𝕜 → 𝕜 is continuous on a compact set s and a : X → A is continuous at x₀ within a set t : Set X, and eventually a x satisfies the predicate p associated to 𝕜 and has spectrum contained in s, then fun x ↦ cfc f (a x) is continuous at x₀ within t.

Defined in

Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Cited by

1 results in Mathlib

Foundations

Depth 198 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedRingStarRingNormedAlgebraIsometricContinuousFunctionalCalculusContinuousStarTopologicalSpace