*(English)*Zbl 0960.46025

This paper concerns functions on a balanced, open subset $\mathcal{U}$ 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 $V$ of continuous, non-negative *weights* $v$ defined on $\mathcal{U}$ is used to construct a family of seminorms: if $f$ is a holomorphic function on $\mathcal{U}$ then

The space $HV\left(\mathcal{U}\right)$ consists of those holomorphic functions for which ${p}_{v}\left(f\right)<\infty $ for all $v\in V$ and $H{V}_{0}\left(\mathcal{U}\right)$ 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 $m$-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.

##### MSC:

46E50 | Spaces of differentiable or holomorphic functions on infinite-dimensional spaces |

46E10 | Topological linear spaces of continuous, differentiable or analytic functions |

46B15 | Summability and bases in normed spaces |