Preduals of spaces of holomorphic functions and the approximation property (Q1754629)

Let \(U\) be a balanced open subset of a Hausdorff complex locally convex space \(E\) and let \(H(U)\) denote the space of holomorphic functions from \(U\) into \(\mathbb{C}\). \textit{P. Mazet} [Analytic sets in locally convex spaces. Transl. from the French by Paul Barry. Amsterdam-New York-Oxford: North-Holland (1984; Zbl 0588.46032)] proved the existence of a complete locally convex space \(G(U)\) which is a predual of \(H(U)\). \textit{J. Mujica} [in: Progress in functional analysis. Proceedings of the international functional meeting on the occasion of the 60th birthday of Professor M. Valdivia in Peñíscola. Amsterdam etc.: North-Holland. 149--162 (1992; Zbl 0795.46036)] gave a new proof of this result and a useful description of the space \(G(U)\). The authors give a representation of \(G(U)\) as the projective limit of the preduals of spaces of holomorphic functions that are bounded on certain subsets of \(U\). As a consequence, they prove that, if \(E\) is a Banach space, then it has the approximation property if and only if \(G(U)\) has the approximation property. A new proof of a result due to \textit{R. M. Aron} and \textit{M. Schottenloher} [J. Funct. Anal. 21, 7--30 (1976; Zbl 0328.46046)], 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, is also presented.