Theorems · Theorem · functional analysis

SchwartzMap.fourierMultiplierCLM_fourierMultiplierCLM_apply

∀ {𝕜 : Type u_2} {E : Type u_3} {F : Type u_4} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedAddCommGroup F] [inst_3 : InnerProductSpace ℝ E] [inst_4 : NormedSpace ℂ F] [inst_5 : NormedSpace 𝕜 F]
  [inst_6 : SMulCommClass ℂ 𝕜 F] [inst_7 : FiniteDimensional ℝ E] [inst_8 : MeasurableSpace E] [inst_9 : BorelSpace E]
  [CompleteSpace F] {g₁ g₂ : E → 𝕜},
  Function.HasTemperateGrowth g₁ →
    Function.HasTemperateGrowth g₂ →
      ∀ (f : SchwartzMap E F),
        (SchwartzMap.fourierMultiplierCLM F g₁) ((SchwartzMap.fourierMultiplierCLM F g₂) f) =
          (SchwartzMap.fourierMultiplierCLM F (g₁ * g₂)) f

Defined in

Mathlib.Analysis.Distribution.FourierMultiplier

Cited by

2 results in Mathlib

Foundations

Depth 304 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupNormedAddCommGroupInnerProductSpaceNormedSpaceNormedSpaceSMulCommClassFiniteDimensionalMeasurableSpaceBorelSpaceCompleteSpace