analyticOn_fun_prod — Mathlib

English

A finite product of analyticAt functions is analyticAt.

Русский

Конечный продукт аналитических функций в точке аналитичен в точке.

LaTeX

$$$\forall N\,\text{Finset},\ (\forall n\in N,\ AnalyticAt_{\mathfrak k}(f(n),c))\Rightarrow AnalyticAt_{\mathfrak k}\bigl(\prod_{n\in N} f(n)\bigr)(c).$$$

Lean4

/-- Finite products of analytic functions are analytic -/
theorem analyticOn_fun_prod {A : Type*} [NormedCommRing A] [NormedAlgebra 𝕜 A] {f : α → E → A} {s : Set E}
    (N : Finset α) (h : ∀ n ∈ N, AnalyticOn 𝕜 (f n) s) : AnalyticOn 𝕜 (fun z ↦ ∏ n ∈ N, f n z) s := fun z zs ↦
  N.analyticWithinAt_fun_prod (fun n m ↦ h n m z zs)

Похожие утверждения