Theorems · Definition · measure theory

ContinuousMap.toLp

{α : Type u_1} →
  {E : Type u_2} →
    {m0 : MeasurableSpace α} →
      (p : ENNReal) →
        (μ : MeasureTheory.Measure α) →
          [inst : TopologicalSpace α] →
            [BorelSpace α] →
              [inst_2 : NormedAddCommGroup E] →
                [SecondCountableTopologyEither α E] →
                  [CompactSpace α] →
                    [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] → C(α, E) →L[𝕜] ↥(MeasureTheory.Lp E p μ)

The bounded linear map of considering a continuous function on a compact finite-measure space α as an element of Lp. By definition, the norm on C(α, E) is the sup-norm, transferred from the space α →ᵇ E of bounded continuous functions, so this construction is just a matter of transferring the structure from BoundedContinuousFunction.toLp along the isometry.

Defined in

Mathlib.MeasureTheory.Function.LpSpace.ContinuousFunctions

Cited by

21 results in Mathlib

Foundations

Depth 232 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceBorelSpaceNormedAddCommGroupSecondCountableTopologyEitherCompactSpaceMeasureTheory.IsFiniteMeasureFactNormedRingModuleIsBoundedSMul