Theorems · Theorem · functional analysis

cfc_isStrictlyPositive_iff

∀ {R : Type u_1} {A : Type u_2} {p : A → Prop} [inst : CommSemiring R] [inst_1 : PartialOrder R] [inst_2 : StarRing R]
  [inst_3 : MetricSpace R] [inst_4 : IsTopologicalSemiring R] [inst_5 : ContinuousStar R] [ContinuousSqrt R]
  [StarOrderedRing R] [inst_8 : TopologicalSpace A] [inst_9 : Ring A] [inst_10 : StarRing A] [inst_11 : PartialOrder A]
  [StarOrderedRing A] [inst_13 : Algebra R A] [instCFC : ContinuousFunctionalCalculus R A p] [NonnegSpectrumClass R A]
  (f : R → R) (a : A),
  autoParam (ContinuousOn f (spectrum R a)) cfc_isStrictlyPositive_iff._auto_1 →
    autoParam (p a) cfc_isStrictlyPositive_iff._auto_3 → (IsStrictlyPositive (cfc f a) ↔ ∀ x ∈ spectrum R a, 0 < f x)

Defined in

Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Cited by

1 results in Mathlib

Foundations

Depth 101 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringPartialOrderStarRingMetricSpaceIsTopologicalSemiringContinuousStarContinuousSqrtStarOrderedRingTopologicalSpaceRingStarRingPartialOrderStarOrderedRingAlgebraContinuousFunctionalCalculusNonnegSpectrumClass