Preduals of spaces of holomorphic functions and the approximation property

Let U be a balanced open subset of a Hausdorff complex locally convex space E. In this paper we represent the predual G(U) of the space of holomorphic functions as the projective limit of the preduals of spaces of holomorphic functions that are bounded on certain subsets of U. As an application we prove that if E is a Banach space, then it has the approximation property if and only if G(U) has the approximation property. We also give a new proof of a result due to Aron and Schottenloher stating that if E is a complex Banach space with the approximation property then the space of holomorphic functions H(U), with the compact-open topology, has the approximation property.