Theorems · Theorem · functional analysis

TestFunction.integralAgainstBilinCLM_ofSupportedIn

∀ {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}
  [inst_3 : 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] [inst_11 : Algebra ℝ 𝕜] [inst_12 : IsScalarTower ℝ 𝕜 F₁]
  [inst_13 : NormedSpace ℝ F₃] [inst_14 : IsScalarTower ℝ 𝕜 F₃] {B : F₁ →L[𝕜] F₂ →L[𝕜] F₃} {μ : MeasureTheory.Measure E}
  {φ : E → F₂},
  MeasureTheory.LocallyIntegrableOn φ (↑Ω) μ →
    ∀ {K : TopologicalSpace.Compacts E} (K_sub_Ω : ↑K ⊆ ↑Ω) {f : ContDiffMapSupportedIn E F₁ n K},
      (TestFunction.integralAgainstBilinCLM B μ φ) (TestFunction.ofSupportedIn K_sub_Ω f) =
        (ContDiffMapSupportedIn.integralAgainstBilinCLM B μ φ) f

Defined in

Mathlib.Analysis.Distribution.TestFunction

Cited by

0 results in Mathlib

Foundations

Depth 268 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceOpensMeasurableSpaceNormedAddCommGroupNormedSpaceNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceAlgebraIsScalarTowerNormedSpaceIsScalarTower