`BookProof.StrichartzWave.mulL2_symmetric` (W : Lp ℂ (⊤ : ℝ≥0∞) (volume : Measure V)) (hW : ∀ᵐ x ∂(volume : Measure V), (starRingEnd ℂ) (W x) = W x) (u w : Lp ℂ 2 (volume : Measure

Proved

BookProof.StrichartzWave.mulL2_symmetric

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterWaveBoundedPotential.

BookProof.StrichartzWave.mulL2_symmetric (W : Lp ℂ (⊤ : ℝ≥0∞) (volume : Measure V)) (hW : ∀ᵐ x ∂(volume : Measure V), (starRingEnd ℂ) (W x) = W x) (u w : Lp ℂ 2 (volume : Measure V)) : (inner ℂ (mulL2 W u) w : ℂ) = inner ℂ u (mulL2 W w)

Formalization note: Lean 4 identifier BookProof.StrichartzWave.mulL2_symmetric.

Preamble

-- Generated from ChapterWaveBoundedPotential.lean — theorem BookProof.StrichartzWave.mulL2_symmetric
import Mathlib
import Definitions.Def_ChapterWaveBoundedPotential
import Definitions.Def_ChapterStrichartzWave
open BookProof.StrichartzWave



open MeasureTheory SchwartzMap FourierTransform ComplexInnerProductSpace ENNReal

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

Formal statement

theorem BookProof.StrichartzWave.mulL2_symmetric (W : Lp ℂ (⊤ : ℝ≥0∞) (volume : Measure V))
    (hW : ∀ᵐ x ∂(volume : Measure V), (starRingEnd ℂ) (W x) = W x)
    (u w : Lp ℂ 2 (volume : Measure V)) :
    (inner ℂ (mulL2 W u) w : ℂ) = inner ℂ u (mulL2 W w) := by sorry

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