`BookProof.MixedLinearEsa.integral_conj_mul_pos_sub_eq_zero` (b : V) (z : ℂ) (u : Lp ℂ 2 (volume : Measure V)) (hu : ∀ v : schwartzDomain V, (inner ℂ (opL2 (posOp b) v) u : ℂ) = z

Proved

BookProof.MixedLinearEsa.integral_conj_mul_pos_sub_eq_zero

by leonardopedro · Oct 9, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterMixedLinearEsa.

BookProof.MixedLinearEsa.integral_conj_mul_pos_sub_eq_zero (b : V) (z : ℂ) (u : Lp ℂ 2 (volume : Measure V)) (hu : ∀ v : schwartzDomain V, (inner ℂ (opL2 (posOp b) v) u : ℂ) = z * inner ℂ (v : Lp ℂ 2 _) u) (f : 𝓢(V, ℂ)) : ∫ x, (starRingEnd ℂ) (f x) * (((inner ℝ x b : ℝ) : ℂ) - z) * (u x) = 0

Formalization note: Lean 4 identifier BookProof.MixedLinearEsa.integral_conj_mul_pos_sub_eq_zero.

Preamble

-- Generated from ChapterMixedLinearEsa.lean — theorem BookProof.MixedLinearEsa.integral_conj_mul_pos_sub_eq_zero
import Definitions.Def_ChapterFourierMultiplierEsa
import Definitions.Def_ChapterFarisLavine
import Mathlib
import Definitions.Def_ChapterMixedLinearEsa
import Definitions.Def_ChapterStrichartzWave
open BookProof.StrichartzWave
open BookProof.MixedLinearEsa



open MeasureTheory SchwartzMap FourierTransform ComplexInnerProductSpace LineDeriv
open BookProof.StrichartzWave BookProof.FourierMultiplierEsa BookProof.FarisLavine

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

Formal statement

theorem BookProof.MixedLinearEsa.integral_conj_mul_pos_sub_eq_zero (b : V) (z : ℂ) (u : Lp ℂ 2 (volume : Measure V))
    (hu : ∀ v : schwartzDomain V,
      (inner ℂ (opL2 (posOp b) v) u : ℂ) = z * inner ℂ (v : Lp ℂ 2 _) u)
    (f : 𝓢(V, ℂ)) :
    ∫ x, (starRingEnd ℂ) (f x) * (((inner ℝ x b : ℝ) : ℂ) - z) * (u x) = 0 := by sorry

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