Theorems · Theorem · functional analysis

SchwartzMap.integral_clm_comp_laplacian_right_eq_left

∀ {𝕜 : Type u_2} {E : Type u_5} {F₁ : Type u_9} {F₂ : Type u_10} [inst : NormedAddCommGroup E]
  [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E] [inst_3 : NormedAddCommGroup F₁]
  [inst_4 : NormedSpace ℝ F₁] [inst_5 : NormedAddCommGroup F₂] [inst_6 : NormedSpace ℝ F₂] [inst_7 : MeasurableSpace E]
  {μ : MeasureTheory.Measure E} [BorelSpace E] [μ.IsAddHaarMeasure] [inst_10 : RCLike 𝕜] [inst_11 : NormedSpace 𝕜 F₁]
  [inst_12 : NormedSpace 𝕜 F₂] (f : SchwartzMap E (F₁ →L[𝕜] F₂)) (g : SchwartzMap E F₁),
  ∫ (x : E), (f x) ((Laplacian.laplacian g) x) ∂μ = ∫ (x : E), ((Laplacian.laplacian f) x) (g x) ∂μ

Integration by parts of Schwartz functions for the Laplacian. Version for a Schwartz function with values in continuous linear maps.

Defined in

Mathlib.Analysis.Distribution.SchwartzSpace.Deriv

Cited by

0 results in Mathlib

Foundations

Depth 288 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupInnerProductSpaceFiniteDimensionalNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceMeasurableSpaceBorelSpaceMeasureTheory.Measure.IsAddHaarMeasureRCLikeNormedSpaceNormedSpace