Theorems · Theorem · functional analysis
apply_le_nnnorm_cfc_nnreal
∀ {A : Type u_1} [inst : NormedRing A] [inst_1 : StarRing A] [inst_2 : NormedAlgebra ℝ A] [inst_3 : PartialOrder A]
[inst_4 : StarOrderedRing A] [inst_5 : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint]
[inst_6 : NonnegSpectrumClass ℝ A] (f : NNReal → NNReal) (a : A) ⦃x : NNReal⦄,
x ∈ spectrum NNReal a →
autoParam (ContinuousOn f (spectrum NNReal a)) apply_le_nnnorm_cfc_nnreal._auto_1 →
autoParam (0 ≤ a) apply_le_nnnorm_cfc_nnreal._auto_3 → f x ≤ ‖cfc f a‖₊- Cited by
- 3 results in Mathlib
- Foundations
- Depth 200 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites20
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- Realstatement and proof · cited by 25,697
- PartialOrderstatement and proof · cited by 6,410
- NNRealstatement and proof · cited by 4,310
- Nontrivialproof · cited by 2,416
- StarRingstatement and proof · cited by 1,686
- ContinuousOnstatement and proof · cited by 1,411
- NormedAlgebrastatement and proof · cited by 1,165
- NNNorm.nnnormstatement and proof · cited by 952
- NormedRingstatement and proof · cited by 924
- StarOrderedRingstatement and proof · cited by 587
- IsSelfAdjointstatement and proof · cited by 545
Cited by3
Results whose statement or proof uses this declaration.
- nnnorm_cfc_nnreal_lt_iffproof · cited by 0
- IsometricContinuousFunctionalCalculus.spectrum_leproof · cited by 0
- nnnorm_cfc_nnreal_le_iffproof · cited by 0