Theorems · Theorem · functional analysis
cfc_le_algebraMap_iff
∀ {R : Type u_1} {A : Type u_2} {p : A → Prop} [inst : CommRing R] [inst_1 : PartialOrder R] [inst_2 : StarRing R]
[inst_3 : MetricSpace R] [inst_4 : IsTopologicalRing 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) (r : R) (a : A),
autoParam (ContinuousOn f (spectrum R a)) cfc_le_algebraMap_iff._auto_1 →
autoParam (p a) cfc_le_algebraMap_iff._auto_3 → (cfc f a ≤ (algebraMap R A) r ↔ ∀ x ∈ spectrum R a, f x ≤ r)- Cited by
- 2 results in Mathlib
- Foundations
- Depth 102 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites23
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- Setstatement · cited by 53,352
- TopologicalSpacestatement and proof · cited by 24,529
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- RingHomstatement · cited by 10,189
- Ringstatement and proof · cited by 7,463
- PartialOrderstatement and proof · cited by 6,410
- Algebra.algebraMapstatement · cited by 4,706
- StarRingstatement and proof · cited by 1,686
- MetricSpacestatement and proof · cited by 1,684
- ContinuousOnstatement and proof · cited by 1,411
Cited by2
Results whose statement or proof uses this declaration.
- le_algebraMap_iff_spectrum_leproof · cited by 2
- cfc_le_one_iffproof · cited by 0