Preduals of spaces of holomorphic functions and the approximation property

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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.