Theorems · Theorem · functional analysis

nnnorm_apply_le_nnnorm_cfc

∀ {𝕜 : Type u_1} {A : Type u_2} {p : outParam (A → Prop)} [inst : RCLike 𝕜] [inst_1 : NormedRing A]
  [inst_2 : StarRing A] [inst_3 : NormedAlgebra 𝕜 A] [inst_4 : IsometricContinuousFunctionalCalculus 𝕜 A p] (f : 𝕜 → 𝕜)
  (a : A) ⦃x : 𝕜⦄,
  x ∈ spectrum 𝕜 a →
    autoParam (ContinuousOn f (spectrum 𝕜 a)) nnnorm_apply_le_nnnorm_cfc._auto_1 →
      autoParam (p a) nnnorm_apply_le_nnnorm_cfc._auto_3 → ‖f x‖₊ ≤ ‖cfc f a‖₊

Defined in: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric
Cited by: 1 results in Mathlib
Foundations: Depth 181 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: RCLike NormedRing StarRing NormedAlgebra IsometricContinuousFunctionalCalculus

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Cites12

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Setstatement · cited by 53,352
NNRealstatement · cited by 4,310
RCLikestatement and proof · cited by 2,829
StarRingstatement and proof · cited by 1,686
ContinuousOnstatement and proof · cited by 1,411
NormedAlgebrastatement and proof · cited by 1,165
NNNorm.nnnormstatement · cited by 952
NormedRingstatement and proof · cited by 924
spectrumstatement and proof · cited by 510
cfcstatement · cited by 228
IsometricContinuousFunctionalCalculusstatement and proof · cited by 62
norm_apply_le_norm_cfcproof · cited by 5

Cited by1

Results whose statement or proof uses this declaration.

IsometricContinuousFunctionalCalculus.nnnorm_spectrum_leproof · cited by 0