`BookProof.MixedLinearEsa.mixedLinearOp_apply` (b m : V) (f : 𝓢(V, ℂ)) (x : V) : (mixedLinearOp b m f) x = ((inner ℝ x b : ℝ) : ℂ) * f x + (-Complex.I) * (fderiv ℝ f x m)

Proved

BookProof.MixedLinearEsa.mixedLinearOp_apply

by leonardopedro · Oct 9, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterMixedLinearEsa.

BookProof.MixedLinearEsa.mixedLinearOp_apply (b m : V) (f : 𝓢(V, ℂ)) (x : V) : (mixedLinearOp b m f) x = ((inner ℝ x b : ℝ) : ℂ) * f x + (-Complex.I) * (fderiv ℝ f x m)

Formalization note: Lean 4 identifier BookProof.MixedLinearEsa.mixedLinearOp_apply.

Preamble

-- Generated from ChapterMixedLinearEsa.lean — theorem BookProof.MixedLinearEsa.mixedLinearOp_apply
import Definitions.Def_ChapterStrichartzWave
import Definitions.Def_ChapterFarisLavine
import Mathlib
import Definitions.Def_ChapterMixedLinearEsa
import Definitions.Def_ChapterFourierMultiplierEsa
open BookProof.FourierMultiplierEsa
open BookProof.MixedLinearEsa



open MeasureTheory SchwartzMap FourierTransform ComplexInnerProductSpace LineDeriv
open BookProof.StrichartzWave BookProof.FourierMultiplierEsa BookProof.FarisLavine

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

Formal statement

theorem BookProof.MixedLinearEsa.mixedLinearOp_apply (b m : V) (f : 𝓢(V, ℂ)) (x : V) :
    (mixedLinearOp b m f) x
      = ((inner ℝ x b : ℝ) : ℂ) * f x + (-Complex.I) * (fderiv ℝ f x m) := by sorry

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