Theorems · Theorem · functional analysis

SchwartzMap.integral_clm_comp_lineDerivOp_right_eq_neg_left

∀ {𝕜 : Type u_2} {D : Type u_4} {V : Type u_7} {F : Type u_8} [inst : NormedAddCommGroup F] [inst_1 : NormedSpace ℝ F]
  [inst_2 : NormedAddCommGroup V] [inst_3 : NormedSpace ℝ V] [inst_4 : NormedAddCommGroup D] [inst_5 : NormedSpace ℝ D]
  [inst_6 : MeasurableSpace D] {μ : MeasureTheory.Measure D} [BorelSpace D] [FiniteDimensional ℝ D] [μ.IsAddHaarMeasure]
  [inst_10 : RCLike 𝕜] [inst_11 : NormedSpace 𝕜 F] [inst_12 : NormedSpace 𝕜 V] (f : SchwartzMap D (F →L[𝕜] V))
  (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 a Schwartz function with values in continuous linear maps.

Defined in

Mathlib.Analysis.Distribution.SchwartzSpace.Deriv

Cited by

0 results in Mathlib

Foundations

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

Assumes

NormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceMeasurableSpaceBorelSpaceFiniteDimensionalMeasureTheory.Measure.IsAddHaarMeasureRCLikeNormedSpaceNormedSpace