Theorems · Theorem · functional analysis

ContinuousLinearMap.bilinear_hasTemperateGrowth

∀ {𝕜 : Type u_2} {D : Type u_4} {E : Type u_5} {F : Type u_6} {G : Type u_7} [inst : NormedAddCommGroup E]
  [inst_1 : NormedSpace ℝ E] [inst_2 : NormedAddCommGroup F] [inst_3 : NormedSpace ℝ F]
  [inst_4 : NontriviallyNormedField 𝕜] [NormedAlgebra ℝ 𝕜] [inst_6 : NormedAddCommGroup D] [inst_7 : NormedSpace ℝ D]
  [inst_8 : NormedAddCommGroup G] [inst_9 : NormedSpace ℝ G] [inst_10 : NormedSpace 𝕜 F] [inst_11 : NormedSpace 𝕜 G]
  [inst_12 : NormedSpace 𝕜 E] (B : E →L[𝕜] F →L[𝕜] G) {f : D → E} {g : D → F},
  Function.HasTemperateGrowth f → Function.HasTemperateGrowth g → Function.HasTemperateGrowth fun x => (B (f x)) (g x)

The product of two functions of temperate growth is again of temperate growth. Version for bilinear maps.

Defined in

Mathlib.Analysis.Distribution.TemperateGrowth

Cited by

2 results in Mathlib

Foundations

Depth 203 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNontriviallyNormedFieldNormedAlgebraNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedSpaceNormedSpaceNormedSpace