Theorems · Theorem · functional analysis

ContinuousLinearMap.lpPairing_eq_integral

∀ {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} [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 : p.HolderConjugate q] [inst_10 : NormedSpace ℝ G] [inst_11 : SMulCommClass ℝ 𝕜 G] [inst_12 : CompleteSpace G]
  (f : ↥(MeasureTheory.Lp E p μ)) (g : ↥(MeasureTheory.Lp F q μ)),
  ((ContinuousLinearMap.lpPairing μ p q B) f) g = ∫ (x : α), (B (↑↑f x)) (↑↑g x) ∂μ

Defined in

Mathlib.MeasureTheory.Function.Holder

Cited by

1 results in Mathlib

Foundations

Depth 253 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedAddCommGroupNormedAddCommGroupNormedSpaceNormedSpaceNormedSpaceFactFactENNReal.HolderConjugateNormedSpaceSMulCommClassCompleteSpace