Theorems · Definition · functional analysis

ContinuousLinearMap.holder

{α : Type u_1} →
  {𝕜 : Type u_2} →
    {E : Type u_3} →
      {F : Type u_4} →
        {G : Type u_5} →
          {m : MeasurableSpace α} →
            {μ : MeasureTheory.Measure α} →
              {p q : ENNReal} →
                (r : ENNReal) →
                  [hpqr : p.HolderTriple q r] →
                    [inst : NontriviallyNormedField 𝕜] →
                      [inst_1 : NormedAddCommGroup E] →
                        [inst_2 : NormedAddCommGroup F] →
                          [inst_3 : NormedAddCommGroup G] →
                            [inst_4 : NormedSpace 𝕜 E] →
                              [inst_5 : NormedSpace 𝕜 F] →
                                [inst_6 : NormedSpace 𝕜 G] →
                                  (E →L[𝕜] F →L[𝕜] G) →
                                    ↥(MeasureTheory.Lp E p μ) → ↥(MeasureTheory.Lp F q μ) → ↥(MeasureTheory.Lp G r μ)

The map between MeasureTheory.Lp spaces satisfying ENNReal.HolderTriple induced by a continuous bilinear map on the underlying spaces.

Defined in

Mathlib.MeasureTheory.Function.Holder

Cited by

11 results in Mathlib

Foundations

Depth 224 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

ENNReal.HolderTripleNontriviallyNormedFieldNormedAddCommGroupNormedAddCommGroupNormedAddCommGroupNormedSpaceNormedSpaceNormedSpace