Theorems · Theorem · functional analysis

Filter.Tendsto.cfc_nnreal

∀ {X : Type u_1} {A : Type u_2} [inst : NormedRing A] [inst_1 : StarRing A] [inst_2 : NormedAlgebra ℝ A]
  [inst_3 : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint] [ContinuousStar A] [inst_5 : PartialOrder A]
  [inst_6 : StarOrderedRing A] [inst_7 : NonnegSpectrumClass ℝ A] [T2Space A] [IsSemitopologicalRing A]
  {s : Set NNReal},
  IsCompact s →
    ∀ (f : NNReal → NNReal) {a : X → A} {a₀ : A} {l : Filter X},
      Filter.Tendsto a l (nhds a₀) →
        (∀ᶠ (x : X) in l, spectrum NNReal (a x) ⊆ s) →
          (∀ᶠ (x : X) in l, 0 ≤ a x) →
            spectrum NNReal a₀ ⊆ s →
              0 ≤ a₀ →
                autoParam (ContinuousOn f s) Filter.Tendsto.cfc_nnreal._auto_1 →
                  Filter.Tendsto (fun x => cfc f (a x)) l (nhds (cfc f a₀))

If f : ℝ≥0 → ℝ≥0 is continuous on a compact set s and a : X → A tends to a₀ : A along a filter l (such that eventually 0 ≤ a x and has spectrum contained in s, as does a₀), then fun x ↦ cfc f (a x) tends to cfc f a₀.

Defined in

Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Cited by

2 results in Mathlib

Foundations

Depth 200 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedRingStarRingNormedAlgebraIsometricContinuousFunctionalCalculusContinuousStarPartialOrderStarOrderedRingNonnegSpectrumClassT2SpaceIsSemitopologicalRing