`BookProof.ChapterF7.schwartz_integration_by_parts` (f g : 𝓢(ℝ, ℂ)) : (∫ x, (starRingEnd ℂ) (deriv (f : ℝ → ℂ) x) * g x) = - ∫ x, (starRingEnd ℂ) (f x) * deriv (g : ℝ → ℂ) x

Proved

BookProof.ChapterF7.schwartz_integration_by_parts

by leonardopedro · Oct 8, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterF7.

BookProof.ChapterF7.schwartz_integration_by_parts (f g : 𝓢(ℝ, ℂ)) : (∫ x, (starRingEnd ℂ) (deriv (f : ℝ → ℂ) x) * g x) = - ∫ x, (starRingEnd ℂ) (f x) * deriv (g : ℝ → ℂ) x

Formalization note: Lean 4 identifier BookProof.ChapterF7.schwartz_integration_by_parts.

Preamble

-- Generated from ChapterF7.lean — theorem BookProof.ChapterF7.schwartz_integration_by_parts
import Mathlib
import Definitions.Def_ChapterF7
open BookProof.ChapterF7


open SchwartzMap MeasureTheory Complex
open scoped BigOperators


noncomputable section

Formal statement

theorem BookProof.ChapterF7.schwartz_integration_by_parts (f g : 𝓢(ℝ, ℂ)) :
    (∫ x, (starRingEnd ℂ) (deriv (f : ℝ → ℂ) x) * g x)
      = - ∫ x, (starRingEnd ℂ) (f x) * deriv (g : ℝ → ℂ) x := by sorry

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