The Spaces of Analytic Functions on Open Subsets of ${\mathbb{R}}^{\mathbb{N}}$ and ${\mathbb{C}}^{\mathbb{N}}$

Abstract

This paper is devoted to study the space $\mathcal{A}\left(U\right)$ of all analytic functions on an open subset $U$ of ${\mathbb{R}}^{\mathbb{N}}$ or ${\mathbb{C}}^{\mathbb{N}}$. It is proved that if $U$ satisfies a weak condition (that will be called 0-property), then every $f\in \mathcal{A}\left(U\right)$ depends only on a finite number of variables. Then several topologies on $\mathcal{A}\left(U\right)$ are studied: the compact open topology, the ${\tau }_{\delta }$ topology (already known in spaces of holomorphic functions) and a new one, defined by the inductive limit of the subspaces of analytic functions which only depend on the first variables.