A Banach-Dieudonné theorem for germs of holomorphic functions
Let H(U) denote the vector space of all holomorphic functions on an open subset U of a Fréchet space E. Let H(K) denote the vector space of all holomorphic germs on a compact subset K of E. In this paper it is shown that H(K), endowed with a natural locally convex topology, is the inductive limit of a suitable sequence of compact subsets within the category of all topological spaces. As an application of this result it is shown that the compact-ported topology introduced by L. Nachbin coincides with the compact-open topology on H(U) whenever U is a balanced open subset of a Fréchet-Schwartz space. This last result improves earlier results of \textit{P. J. Boland} and \textit{S. Dineen} [Bull. Soc. Math. France 106, 311-336 (1978; Zbl 0402.46017)], \textit{R. Meise} [Proc. R. Ir. Acad. 81, 217-223 (1981; Zbl 0483.46019)] and other authors, in a series of efforts to solve completely one of the problems listed by \textit{K. D. Bierstedt} and \textit{R. Meise} in [Advances in Holomorphy, Rio de Janeiro 1977, North-Holland Math. Studies, 34, 111-178 (1979; Zbl 0409.46051)].
scientific article; zbMATH DE number 3869995
scientific article