Theorems · Theorem · functional analysis

ContinuousLinearMap.holderL_apply_apply

∀ {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 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] (B : E →L[𝕜] F →L[𝕜] G)
  [inst_7 : Fact (1 ≤ p)] [inst_8 : Fact (1 ≤ q)] [inst_9 : Fact (1 ≤ r)] (x : ↥(MeasureTheory.Lp E p μ))
  (g : ↥(MeasureTheory.Lp F q μ)), ((ContinuousLinearMap.holderL μ p q r B) x) g = ContinuousLinearMap.holder r B x g

Defined in

Mathlib.MeasureTheory.Function.Holder

Cited by

1 results in Mathlib

Foundations

Depth 236 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

ENNReal.HolderTripleNontriviallyNormedFieldNormedAddCommGroupNormedAddCommGroupNormedAddCommGroupNormedSpaceNormedSpaceNormedSpaceFactFactFact