ABSTRACT
This chapter deals with different spaces of holomorphic functions on a Banach space E. It explores whether there are natural maximal domains of existence for these function spaces and whether we can extend these function spaces to a larger space in some natural way without losing structure. The chapter considers these function spaces as spaces of holomorphic functions and follows the complex analytic approach of finding the envelope of holomorphy or as locally convex spaces. It presents a functional interpretation of the canonical mapping into the bidual. Thus, two different routes to consider in extending various spaces of holomorphic functions are presented, and the chapter considers the effect of each.
