Theorems · Theorem · functional analysis

TestFunction.integrable_bilin

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_3} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace ℝ E] {Ω : TopologicalSpace.Opens E} {n : ℕ∞} {m : MeasurableSpace E} [OpensMeasurableSpace E]
  {F₁ : Type u_6} {F₂ : Type u_7} {F₃ : Type u_8} [inst_4 : NormedAddCommGroup F₁] [inst_5 : NormedSpace 𝕜 F₁]
  [inst_6 : NormedSpace ℝ F₁] [inst_7 : NormedAddCommGroup F₂] [inst_8 : NormedSpace 𝕜 F₂]
  [inst_9 : NormedAddCommGroup F₃] [inst_10 : NormedSpace 𝕜 F₃] (B : F₁ →L[𝕜] F₂ →L[𝕜] F₃) {μ : MeasureTheory.Measure E}
  {φ : E → F₂},
  MeasureTheory.LocallyIntegrableOn φ (↑Ω) μ →
    ∀ (f : TestFunction Ω F₁ n), MeasureTheory.Integrable (fun x => (B (f x)) (φ x)) μ

Defined in

Mathlib.Analysis.Distribution.TestFunction

Cited by

0 results in Mathlib

Foundations

Depth 216 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceOpensMeasurableSpaceNormedAddCommGroupNormedSpaceNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace