This paper concerns functions on a balanced, open subset of a Banach space. It builds on the work of R. D. Bierstedt, J. Bonet and A. Galbis [Mich. Math. J. 40, No. 2, 271-297 (1993; Zbl 0803.46023)] who considered the Banach space to be finite dimensional. A countable family of continuous, non-negative weights defined on is used to construct a family of seminorms: if is a holomorphic function on then
The space consists of those holomorphic functions for which for all and is the subspace of those functions that vanish at infinity (this concept is defined suitably). The structure of these spaces under various conditions on the weights is investigated. Particular goals are conditions that (i) ensure that the -homogeneous polynomials form a Schauder decomposition; and (ii) ensure reflexivity. A useful catalogue of 6 Examples is given. The final section discusses the existence of a predual for these spaces.