Theorems · Theorem · harmonic analysis

SchwartzMap.convolution_continuous_left

∀ {𝕜 : Type u_1} {E : Type u_3} {F₁ : Type u_5} {F₂ : Type u_6} {F₃ : Type u_7} [inst : RCLike 𝕜]
  [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace ℝ E] [inst_3 : FiniteDimensional ℝ E]
  [inst_4 : MeasurableSpace E] [inst_5 : BorelSpace E] [inst_6 : NormedAddCommGroup F₁] [inst_7 : NormedSpace ℂ F₁]
  [inst_8 : NormedSpace 𝕜 F₁] [inst_9 : SMulCommClass ℂ 𝕜 F₁] [inst_10 : NormedAddCommGroup F₂]
  [inst_11 : NormedSpace ℂ F₂] [inst_12 : NormedSpace 𝕜 F₂] [inst_13 : SMulCommClass ℂ 𝕜 F₂]
  [inst_14 : NormedAddCommGroup F₃] [inst_15 : NormedSpace ℂ F₃] [inst_16 : NormedSpace 𝕜 F₃]
  [inst_17 : SMulCommClass ℂ 𝕜 F₃] (B : F₁ →L[𝕜] F₂ →L[𝕜] F₃) (g : SchwartzMap E F₂),
  Continuous fun x => ((SchwartzMap.convolution B) x) g

The convolution is continuous in the left argument. Note that since 𝓢(E, F) is not a normed space, uncurried and curried continuity do not coincide.

Defined in

Mathlib.Analysis.Fourier.Convolution

Cited by

0 results in Mathlib

Foundations

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

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceFiniteDimensionalMeasurableSpaceBorelSpaceNormedAddCommGroupNormedSpaceNormedSpaceSMulCommClassNormedAddCommGroupNormedSpaceNormedSpaceSMulCommClassNormedAddCommGroupNormedSpaceNormedSpaceSMulCommClass