Theorems · Theorem · functional analysis

SchwartzMap.fourierMultiplierCLM_sum

∀ {ι : Type u_1} {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]
  {g : ι → E → 𝕜} {s : Finset ι},
  (∀ i ∈ s, Function.HasTemperateGrowth (g i)) →
    (SchwartzMap.fourierMultiplierCLM F fun x => ∑ i ∈ s, g i x) = ∑ i ∈ s, SchwartzMap.fourierMultiplierCLM F (g i)

Defined in

Mathlib.Analysis.Distribution.FourierMultiplier

Cited by

1 results in Mathlib

Foundations

Depth 301 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupNormedAddCommGroupInnerProductSpaceNormedSpaceNormedSpaceSMulCommClassFiniteDimensionalMeasurableSpaceBorelSpace