Prove2Me

`BookProof.NsPartialFourier.partialFourier_inner` (v u : Lp F 2 (volume : Measure V)) : (inner ℂ (partialFourier V F v) (partialFourier V F u) : ℂ) = inner ℂ v u

Proved
BookProof.NsPartialFourier.partialFourier_inner

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterNsPartialFourier.

BookProof.NsPartialFourier.partialFourier_inner (v u : Lp F 2 (volume : Measure V)) : (inner ℂ (partialFourier V F v) (partialFourier V F u) : ℂ) = inner ℂ v u

Formalization note: Lean 4 identifier BookProof.NsPartialFourier.partialFourier_inner.

Preamble
-- Generated from ChapterNsPartialFourier.lean — theorem BookProof.NsPartialFourier.partialFourier_inner
import Mathlib
import Definitions.Def_ChapterNsPartialFourier
open BookProof.NsPartialFourier



open MeasureTheory SchwartzMap FourierTransform LineDeriv

noncomputable section

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

variable {F G : Type*} [NormedAddCommGroup F] [NormedSpace ℂ F]
  [NormedAddCommGroup G] [NormedSpace ℂ G]
variable {F G : Type*} [NormedAddCommGroup F] [InnerProductSpace ℂ F] [CompleteSpace F]
  [NormedAddCommGroup G] [InnerProductSpace ℂ G] [CompleteSpace G]
variable (V F) in
Formal statement
theorem BookProof.NsPartialFourier.partialFourier_inner (v u : Lp F 2 (volume : Measure V)) :
    (inner ℂ (partialFourier V F v) (partialFourier V F u) : ℂ) = inner ℂ v u := by sorry

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