Theorems · Theorem · functional analysis

SchwartzMap.integral_smul_laplacian_right_eq_left

∀ {𝕜 : Type u_2} {E : Type u_5} {F : Type u_8} [inst : NormedAddCommGroup E] [inst_1 : NormedAddCommGroup F]
  [inst_2 : NormedSpace ℝ F] [inst_3 : InnerProductSpace ℝ E] [inst_4 : FiniteDimensional ℝ E]
  [inst_5 : MeasurableSpace E] {μ : MeasureTheory.Measure E} [BorelSpace E] [μ.IsAddHaarMeasure] [inst_8 : RCLike 𝕜]
  [inst_9 : NormedSpace 𝕜 F] (f : SchwartzMap E 𝕜) (g : SchwartzMap E F),
  ∫ (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 scalar multiplication.

Defined in

Mathlib.Analysis.Distribution.SchwartzSpace.Deriv

Cited by

1 results in Mathlib

Foundations

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

Assumes

NormedAddCommGroupNormedAddCommGroupNormedSpaceInnerProductSpaceFiniteDimensionalMeasurableSpaceBorelSpaceMeasureTheory.Measure.IsAddHaarMeasureRCLikeNormedSpace