Theorems · Theorem · functional analysis

cfcHomSuperset_apply

∀ {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 : Ring A] [inst_6 : StarRing A]
  [inst_7 : TopologicalSpace A] [inst_8 : Algebra R A] [instCFC : ContinuousFunctionalCalculus R A p] {a : A} (ha : p a)
  {s : Set R} (hs : spectrum R a ⊆ s) (x : C(↑s, R)),
  (cfcHomSuperset ha hs) x = (cfcHom ha) (x.comp { toFun := Subtype.map id hs, continuous_toFun := ⋯ })

Defined in

Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Cited by

2 results in Mathlib

Foundations

Depth 96 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringStarRingMetricSpaceIsTopologicalSemiringContinuousStarRingStarRingTopologicalSpaceAlgebraContinuousFunctionalCalculus