The barrelled topology associated with the compact-open topology on H(U) and H(K)

Abstract

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 τ0 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 τδ is not in general the barrelled topology associated with τ0 on H(U). Here, they show that in several natural situations, the barrelled topology associated with τ0 on H(U) [resp. H(K)] is τδ [resp. τω]. 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 τδ is the barrelled topology associated with τ0 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 τω 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),τω) is always complete in this situation, a result proved in a different way by Dineen. Several examples and counterexamples are given.