Theorems · Theorem · functional analysis

SchwartzMap.integral_smul_lineDerivOp_right_eq_neg_left

∀ {𝕜 : Type u_2} {D : Type u_4} {F : Type u_8} [inst : NormedAddCommGroup F] [inst_1 : NormedSpace ℝ F]
  [inst_2 : NormedAddCommGroup D] [inst_3 : NormedSpace ℝ D] [inst_4 : MeasurableSpace D] {μ : MeasureTheory.Measure D}
  [BorelSpace D] [FiniteDimensional ℝ D] [μ.IsAddHaarMeasure] [inst_8 : RCLike 𝕜] [inst_9 : NormedSpace 𝕜 F]
  (f : SchwartzMap D 𝕜) (g : SchwartzMap D F) (v : D),
  ∫ (x : D), f x • (LineDeriv.lineDerivOp v g) x ∂μ = -∫ (x : D), (LineDeriv.lineDerivOp v f) x • g x ∂μ

Integration by parts of Schwartz functions for directional derivatives. Version for scalar multiplication.

Defined in

Mathlib.Analysis.Distribution.SchwartzSpace.Deriv

Cited by

1 results in Mathlib

Foundations

Depth 287 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceMeasurableSpaceBorelSpaceFiniteDimensionalMeasureTheory.Measure.IsAddHaarMeasureRCLikeNormedSpace