Theorems · Theorem · functional analysis

SchwartzMap.integral_bilinear_lineDerivOp_right_eq_neg_left

∀ {D : Type u_4} {E : Type u_5} {V : Type u_7} {F : Type u_8} [inst : NormedAddCommGroup E]
  [inst_1 : NormedAddCommGroup F] [inst_2 : NormedSpace ℝ F] [inst_3 : NormedSpace ℝ E] [inst_4 : NormedAddCommGroup V]
  [inst_5 : NormedSpace ℝ V] [inst_6 : NormedAddCommGroup D] [inst_7 : NormedSpace ℝ D] [inst_8 : MeasurableSpace D]
  {μ : MeasureTheory.Measure D} [BorelSpace D] [FiniteDimensional ℝ D] [μ.IsAddHaarMeasure] (f : SchwartzMap D E)
  (g : SchwartzMap D F) (L : E →L[ℝ] F →L[ℝ] V) (v : D),
  ∫ (x : D), (L (f x)) ((LineDeriv.lineDerivOp v g) x) ∂μ = -∫ (x : D), (L ((LineDeriv.lineDerivOp v f) x)) (g x) ∂μ

Integration by parts of Schwartz functions for directional derivatives. Version for a general bilinear map.

Defined in

Mathlib.Analysis.Distribution.SchwartzSpace.Deriv

Cited by

4 results in Mathlib

Foundations

Depth 286 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedAddCommGroupNormedSpaceNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceMeasurableSpaceBorelSpaceFiniteDimensionalMeasureTheory.Measure.IsAddHaarMeasure