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Resumen:
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Let H(U) be the space of holomorphic functions on an open subset U of a complex locally convex space E, and let H(K) be the space of holomorphic germs on a compact subset K of E. (For background on holomorphic functions on locally convex spaces and associated locally convex topologies, see a book by S. Dineen [Complex analysis in locally convex spaces, North-Holland, Amsterdam, 1981;].) This paper deals with the characterization of the barrelled topology associated to the compact open topology ?0 on the spaces H(U) and H(K). In an earlier paper [Proc. Royal Irish Acad. Sect. A 82 (1982), no. 1, 121–128;], the authors showed that ?? is not in general the barrelled topology associated with ?0 on H(U). Here, they show that in several natural situations, the barrelled topology associated with ?0 on H(U) [resp. H(K)] is ?? [resp. ??].
Following W. Ruess [in Functional analysis: surveys and recent results (Paderborn, 1976), 105–118, North-Holland, Amsterdam, 1977;], the authors define E to be gDF if it has a fundamental sequence of bounded sets and, for every locally convex space F, every sequence of continuous linear mappings from E to F that converges strongly to 0 is equicontinuous. The authors show that if E is a gDF space, then ?? is the barrelled topology associated with ?0 on H(U), for every balanced open subset U of E. Using a technique of Dineen, they show that if E is metrizable, then the barrelled topology associated with ?? is t0 on H(K), for an arbitrary compact subset K of E. It follows from a result of J. Mujica [J. Funct. Anal. 57 (1984), no. 1, 31–48;], that (H(K),??) is always complete in this situation, a result proved in a different way by Dineen. Several examples and counterexamples are given.
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