A Banach-Dieudonné theorem for germs of holomorphic functions
type: Item
instance of: scholarly article
title: A Banach-Dieudonné theorem for germs of holomorphic functions
zbMATH Open document ID: 0546.46021
DOI: 10.1016/0022-1236(84)90099-5
author: Jorge Mujica
published in: Journal of Functional Analysis
publication date: 1984-01-01T00:00:00Z
review text: 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)].
Mathematics Subject Classification ID: 46E10
Mathematics Subject Classification ID: 46G20
Mathematics Subject Classification ID: 46A04
Mathematics Subject Classification ID: 46A13
Mathematics Subject Classification ID: 46A11
zbMATH DE Number: 3869995
zbMATH Keywords: Banach-Dieudonné theorem
zbMATH Keywords: space of all holomorphic functions
zbMATH Keywords: Fréchet space
zbMATH Keywords: space of all holomorphic germs
zbMATH Keywords: inductive limit
zbMATH Keywords: compact-ported topology
zbMATH Keywords: compact-open topology
zbMATH Keywords: Fréchet-Schwartz space
MaRDI profile type: Publication
full work available at URL: 0022-1236(84)90099-5
OpenAlex ID: W2036675668
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description: scientific article; zbMATH DE number 3869995