Vector spaces of non-extendable holomorphic functions

Abstract

In this paper, the linear structure of the family He(G) of holomorphic functions in a domain G of the complex plane that are not analytically continuable beyond the boundary of G is analyzed. We prove that He(G) contains, except for zero, a dense algebra; and, under appropriate conditions, the subfamily of He(G) consisting of boundary-regular functions contains dense vector spaces with maximal dimension, as well as infinite dimensional closed vector spaces and large algebras. The case in which G is a domain of existence in a complex Banach space is also considered. The results obtained complete or extend a number of previous ones by several authors.