Theorems · Theorem · several complex variables

AnalyticWithinAt.mul

∀ {𝕜 : Type u_2} [inst : NontriviallyNormedField 𝕜] {E : Type u_3} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {A : Type u_7} [inst_3 : NormedRing A] [inst_4 : NormedAlgebra 𝕜 A] {f g : E → A}
  {s : Set E} {z : E}, AnalyticWithinAt 𝕜 f s z → AnalyticWithinAt 𝕜 g s z → AnalyticWithinAt 𝕜 (fun x => f x * g x) s z

Multiplication of analytic functions (valued in a normed 𝕜-algebra) is analytic.

Defined in

Mathlib.Analysis.Analytic.Constructions

Cited by

5 results in Mathlib

Foundations

Depth 188 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedRingNormedAlgebra