analyticOnNhd_fun_prod — Mathlib

English

Finite sums of analyticOnNhd functions are analyticOnNhd.

Русский

Конечная сумма аналитических на окрестности функций аналитична на окрестности.

LaTeX

$$$\forall N\,\text{Finset},\ (\forall n\in N,\ AnalyticOnNhd_{\mathfrak k}(f(n),s))\Rightarrow AnalyticOnNhd_{\mathfrak k}\bigl(\lambda z. \sum_{n\in N} f(n)(z)\bigr)s.$$$

Lean4

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

Похожие утверждения