Theorems · Theorem · functional analysis

CFC.abs_smul

∀ {𝕜 : Type u_1} {A : Type u_2} [inst : RCLike 𝕜] [inst_1 : NonUnitalRing A] [inst_2 : TopologicalSpace A]
  [inst_3 : Module 𝕜 A] [inst_4 : StarRing A] [inst_5 : PartialOrder A] [inst_6 : StarOrderedRing A]
  [IsScalarTower 𝕜 A A] [SMulCommClass 𝕜 A A] [inst_9 : Module ℝ A] [inst_10 : SMulCommClass ℝ A A]
  [inst_11 : IsScalarTower ℝ A A] [inst_12 : NonnegSpectrumClass ℝ A] [IsTopologicalRing A] [T2Space A]
  [inst_15 : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint] [StarModule 𝕜 A] [StarModule ℝ A]
  [IsScalarTower ℝ 𝕜 A] (r : 𝕜) (a : A), CFC.abs (r • a) = ‖r‖ • CFC.abs a

Defined in

Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs

Cited by

1 results in Mathlib

Foundations

Depth 216 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNonUnitalRingTopologicalSpaceModuleStarRingPartialOrderStarOrderedRingIsScalarTowerSMulCommClassModuleSMulCommClassIsScalarTowerNonnegSpectrumClassIsTopologicalRingT2SpaceNonUnitalContinuousFunctionalCalculusStarModuleStarModuleIsScalarTower