Theorems · Theorem · functional analysis

SchwartzMap.inner_fourier_toL2_eq

∀ {V : Type u_3} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] [inst_2 : FiniteDimensional ℝ V]
  [inst_3 : MeasurableSpace V] [inst_4 : BorelSpace V] {H : Type u_4} [inst_5 : NormedAddCommGroup H]
  [inst_6 : InnerProductSpace ℂ H] [CompleteSpace H] (f g : SchwartzMap V H),
  inner ℂ ((FourierTransform.fourier f).toLp 2 MeasureTheory.volume)
      ((FourierTransform.fourier g).toLp 2 MeasureTheory.volume) =
    inner ℂ (f.toLp 2 MeasureTheory.volume) (g.toLp 2 MeasureTheory.volume)

Defined in

Mathlib.Analysis.Distribution.SchwartzSpace.Fourier

Cited by

1 results in Mathlib

Foundations

Depth 305 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupInnerProductSpaceFiniteDimensionalMeasurableSpaceBorelSpaceNormedAddCommGroupInnerProductSpaceCompleteSpace