Theorems · Theorem · measure theory

ContinuousMap.toLp_def

∀ {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ENNReal} (μ : MeasureTheory.Measure α)
  [inst : TopologicalSpace α] [inst_1 : BorelSpace α] [inst_2 : NormedAddCommGroup E]
  [inst_3 : SecondCountableTopologyEither α E] [inst_4 : CompactSpace α] [inst_5 : MeasureTheory.IsFiniteMeasure μ]
  {𝕜 : Type u_3} [inst_6 : Fact (1 ≤ p)] [inst_7 : NormedRing 𝕜] [inst_8 : Module 𝕜 E] [inst_9 : IsBoundedSMul 𝕜 E]
  (f : C(α, E)),
  (ContinuousMap.toLp p μ 𝕜) f =
    (BoundedContinuousFunction.toLp p μ 𝕜) ((ContinuousMap.linearIsometryBoundedOfCompact α E 𝕜) f)

Defined in

Mathlib.MeasureTheory.Function.LpSpace.ContinuousFunctions

Cited by

0 results in Mathlib

Foundations

Depth 233 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceBorelSpaceNormedAddCommGroupSecondCountableTopologyEitherCompactSpaceMeasureTheory.IsFiniteMeasureFactNormedRingModuleIsBoundedSMul