Prove2Me

`BookProof.NsSpatialMultiplier.l2Fourier_inner` (v u : Lp ℂ 2 (volume : Measure V)) : (inner ℂ (l2Fourier V v) (l2Fourier V u) : ℂ) = inner ℂ v u

Proved
BookProof.NsSpatialMultiplier.l2Fourier_inner

by leonardopedro · Oct 10, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterNsSpatialMomentumMultiplier.

BookProof.NsSpatialMultiplier.l2Fourier_inner (v u : Lp ℂ 2 (volume : Measure V)) : (inner ℂ (l2Fourier V v) (l2Fourier V u) : ℂ) = inner ℂ v u

Formalization note: Lean 4 identifier BookProof.NsSpatialMultiplier.l2Fourier_inner.

Preamble
-- Generated from ChapterNsSpatialMomentumMultiplier.lean — theorem BookProof.NsSpatialMultiplier.l2Fourier_inner
import Definitions.Def_ChapterStrichartzWave
import Definitions.Def_ChapterFourierMultiplierEsa
import Mathlib
import Definitions.Def_ChapterNsSpatialMomentumMultiplier
open BookProof.NsSpatialMultiplier



open MeasureTheory SchwartzMap FourierTransform
open BookProof.StrichartzWave BookProof.FourierMultiplierEsa

noncomputable section

variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V]
  [MeasurableSpace V] [BorelSpace V]
variable {ι : Type*} [Fintype ι]

variable (V) in
Formal statement
theorem BookProof.NsSpatialMultiplier.l2Fourier_inner (v u : Lp ℂ 2 (volume : Measure V)) :
    (inner ℂ (l2Fourier V v) (l2Fourier V u) : ℂ) = inner ℂ v u := by sorry

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