Theorems · Theorem · functional analysis

SchwartzMap.integral_sesq_fourier_eq

∀ {E : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℂ E] {V : Type u_3} [inst_2 : NormedAddCommGroup V]
  [inst_3 : InnerProductSpace ℝ V] [inst_4 : FiniteDimensional ℝ V] [inst_5 : MeasurableSpace V] [inst_6 : BorelSpace V]
  {F : Type u_4} [inst_7 : NormedAddCommGroup F] [inst_8 : NormedSpace ℂ F] {G : Type u_5}
  [inst_9 : NormedAddCommGroup G] [inst_10 : NormedSpace ℂ G] [CompleteSpace E] [CompleteSpace F] (f : SchwartzMap V E)
  (g : SchwartzMap V F) (M : E →L⋆[ℂ] F →L[ℂ] G),
  ∫ (ξ : V), (M ((FourierTransform.fourier f) ξ)) (g ξ) = ∫ (x : V), (M (f x)) ((FourierTransformInv.fourierInv g) x)

Defined in

Mathlib.Analysis.Distribution.SchwartzSpace.Fourier

Cited by

1 results in Mathlib

Foundations

Depth 299 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceNormedAddCommGroupInnerProductSpaceFiniteDimensionalMeasurableSpaceBorelSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceCompleteSpaceCompleteSpace