`BookProof.NsSpatialMultiplier.mulSymbolOp_apply` (σ : V → ℝ) (hσ : Function.HasTemperateGrowth (fun x : V => ((σ x : ℝ) : ℂ))) (f : 𝓢(V, ℂ)) (x : V) : (mulSymbolOp σ f :...

Proved

BookProof.NsSpatialMultiplier.mulSymbolOp_apply

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterNsSpatialMomentumMultiplier.

BookProof.NsSpatialMultiplier.mulSymbolOp_apply (σ : V → ℝ) (hσ : Function.HasTemperateGrowth (fun x : V => ((σ x : ℝ) : ℂ))) (f : 𝓢(V, ℂ)) (x : V) : (mulSymbolOp σ f : 𝓢(V, ℂ)) x = ((σ x : ℝ) : ℂ) * f x

Formalization note: Lean 4 identifier BookProof.NsSpatialMultiplier.mulSymbolOp_apply.

Preamble

-- Generated from ChapterNsSpatialMomentumMultiplier.lean — theorem BookProof.NsSpatialMultiplier.mulSymbolOp_apply
import Definitions.Def_ChapterStrichartzWave
import Definitions.Def_ChapterFourierMultiplierEsa
import Mathlib
import Definitions.Def_ChapterNsSpatialMomentumMultiplier
import Definitions.Def_ChapterFarisLavine
open BookProof.FarisLavine
open BookProof.NsSpatialMultiplier



open MeasureTheory SchwartzMap FourierTransform
open BookProof.StrichartzWave BookProof.FourierMultiplierEsa

noncomputable section

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

variable (V) in

omit [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] in

Formal statement

theorem BookProof.NsSpatialMultiplier.mulSymbolOp_apply (σ : V → ℝ)
    (hσ : Function.HasTemperateGrowth (fun x : V => ((σ x : ℝ) : ℂ)))
    (f : 𝓢(V, ℂ)) (x : V) :
    (mulSymbolOp σ f : 𝓢(V, ℂ)) x = ((σ x : ℝ) : ℂ) * f x := by sorry

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