A Montel-Type Theorem for Hardy Spaces of Holomorphic Functions

Abstract

We give a version of the Montel theorem for Hardy spaces of holomorphic functions on an infinite dimensional space. Precisely, we show that any bounded sequence of holomorphic functions in some Hardy space, has a subsequence that converges uniformly over compact subsets to a function that also belongs to the same Hardy space. As a by-product of our results for spaces of functions on infinitely many variables, we also provide an elementary proof of a Montel-type theorem for the Hardy space of Dirichlet series.