Theorems · Theorem · functional analysis

ContinuousMapZero.UniqueHom.eq_of_continuous_of_map_id

∀ {R : Type u_1} {A : Type u_2} {inst : CommSemiring R} {inst_1 : StarRing R} {inst_2 : MetricSpace R}
  {inst_3 : IsTopologicalSemiring R} {inst_4 : ContinuousStar R} {inst_5 : NonUnitalRing A} {inst_6 : StarRing A}
  {inst_7 : TopologicalSpace A} {inst_8 : Module R A} {inst_9 : IsScalarTower R A A} {inst_10 : SMulCommClass R A A}
  [self : ContinuousMapZero.UniqueHom R A] (s : Set R) [CompactSpace ↑s] [inst_12 : Fact (0 ∈ s)]
  (φ ψ : ContinuousMapZero (↑s) R →⋆ₙₐ[R] A),
  Continuous ⇑φ → Continuous ⇑ψ → φ (ContinuousMapZero.id s) = ψ (ContinuousMapZero.id s) → φ = ψ

Defined in

Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Cited by

4 results in Mathlib

Foundations

Depth 98 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

ContinuousMapZero.UniqueHomCompactSpaceFact