Prove2Me

`BookProof.energyTransform_symm` (Θ : (Lp F 2 (volume : Measure E)) ≃ₗᵢ[ℝ] P) (fourierTime : (Lp F 2 (volume : Measure E)) ≃ₗᵢ[ℝ] (Lp F 2 (volume : Measure E))) : (energyTransform

Proved
BookProof.energyTransform_symm

by leonardopedro · Oct 7, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterA4.

BookProof.energyTransform_symm (Θ : (Lp F 2 (volume : Measure E)) ≃ₗᵢ[ℝ] P) (fourierTime : (Lp F 2 (volume : Measure E)) ≃ₗᵢ[ℝ] (Lp F 2 (volume : Measure E))) : (energyTransform E F Θ fourierTime).symm = conjugateₗᵢ Θ fourierTime.symm

Formalization note: Lean 4 identifier BookProof.energyTransform_symm.

Preamble
-- Generated from ChapterA4.lean — theorem BookProof.energyTransform_symm
import Mathlib
import Definitions.Def_ChapterA4
open BookProof



open MeasureTheory




variable {R E E' : Type*} [Semiring R]
    [SeminormedAddCommGroup E] [SeminormedAddCommGroup E'] [Module R E] [Module R E']

variable (E F : Type*) [NormedAddCommGroup E] [MeasurableSpace E] [BorelSpace E]
    [NormedAddCommGroup F] [InnerProductSpace ℂ F] [CompleteSpace F]
    [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
variable {P : Type*} [NormedAddCommGroup P] [InnerProductSpace ℝ P]
Formal statement
theorem BookProof.energyTransform_symm
    (Θ : (Lp F 2 (volume : Measure E)) ≃ₗᵢ[ℝ] P)
    (fourierTime : (Lp F 2 (volume : Measure E)) ≃ₗᵢ[ℝ] (Lp F 2 (volume : Measure E))) :
    (energyTransform E F Θ fourierTime).symm = conjugateₗᵢ Θ fourierTime.symm := by sorry

View graph