`BookProof.FourierMultiplierEsa.essentiallySelfAdjointOn_of_real_symbol` (P : 𝓢(V, ℂ) →L[ℂ] 𝓢(V, ℂ)) (σ : V → ℝ) (hP : ∀ (f : 𝓢(V, ℂ)) (x : V), (𝓕 (P f) : 𝓢(V,...

Proved

BookProof.FourierMultiplierEsa.essentiallySelfAdjointOn_of_real_symbol

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterFourierMultiplierEsa.

BookProof.FourierMultiplierEsa.essentiallySelfAdjointOn_of_real_symbol (P : 𝓢(V, ℂ) →L[ℂ] 𝓢(V, ℂ)) (σ : V → ℝ) (hP : ∀ (f : 𝓢(V, ℂ)) (x : V), (𝓕 (P f) : 𝓢(V, ℂ)) x = ((σ x : ℝ) : ℂ) * (𝓕 f : 𝓢(V, ℂ)) x) (hσ : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) σ) : BookProof.FarisLavine.EssentiallySelfAdjointOn (schwartzDomain V) (opL2 P)

Formalization note: Lean 4 identifier BookProof.FourierMultiplierEsa.essentiallySelfAdjointOn_of_real_symbol.

Preamble

-- Generated from ChapterFourierMultiplierEsa.lean — theorem BookProof.FourierMultiplierEsa.essentiallySelfAdjointOn_of_real_symbol
import Mathlib
import Definitions.Def_ChapterFourierMultiplierEsa
import Definitions.Def_ChapterFarisLavineCore
import Definitions.Def_ChapterStrichartzWave
open BookProof.StrichartzWave
open BookProof.FourierMultiplierEsa



open MeasureTheory SchwartzMap FourierTransform ComplexInnerProductSpace LineDeriv
open BookProof.StrichartzWave

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

Formal statement

theorem BookProof.FourierMultiplierEsa.essentiallySelfAdjointOn_of_real_symbol (P : 𝓢(V, ℂ) →L[ℂ] 𝓢(V, ℂ)) (σ : V → ℝ)
    (hP : ∀ (f : 𝓢(V, ℂ)) (x : V),
      (𝓕 (P f) : 𝓢(V, ℂ)) x = ((σ x : ℝ) : ℂ) * (𝓕 f : 𝓢(V, ℂ)) x)
    (hσ : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) σ) :
    BookProof.FarisLavine.EssentiallySelfAdjointOn (schwartzDomain V) (opL2 P) := by sorry

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