Theorems · Theorem · functional analysis
cfcHom_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]
{a : A} (ha : p a) {f : C(↑(spectrum R a), R)}, IsStrictlyPositive ((cfcHom ha) f) ↔ ∀ (x : ↑(spectrum R a)), 0 < f x- Cited by
- 1 results in Mathlib
- Foundations
- Depth 97 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites27
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- Setstatement · cited by 53,352
- TopologicalSpacestatement and proof · cited by 24,529
- Algebrastatement and proof · cited by 11,388
- CommSemiringstatement and proof · cited by 10,911
- Ringstatement and proof · cited by 7,463
- Set.Elemstatement and proof · cited by 7,166
- PartialOrderstatement and proof · cited by 6,410
- ContinuousMapstatement and proof · cited by 2,491
- StarRingstatement and proof · cited by 1,686
- MetricSpacestatement and proof · cited by 1,684
- le_of_ltproof · cited by 1,175
Cited by1
Results whose statement or proof uses this declaration.
- cfc_isStrictlyPositive_iffproof · cited by 1