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Resumen:
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If E and F are complex Banach spaces, and fixing a balanced open subset U of E, we let Hb=(Hb(U;F),?b) denote the space of all mappings f:U?F which are holomorphic of bounded type, endowed with its natural topology ?b; clearly, Hb is a Fréchet space. J. M. Isidro [Proc. Roy. Irish Acad. Sect. A 79 (1979), no. 12, 115–130;] characterized the topological dual of Hb as a certain space S=S(U;F) on which one has a natural inductive limit topology ?1 as well as the strong dual topology ?b=?(S,Hb). Here, the authors prove that Hb is quasinormable (and hence distinguished) and ?b=?1 on S whenever U is an open ball in E or U=E. But Hb is a (Montel or) Schwartz space if and only if both E and F are finite dimensional. The authors' main result remains true for arbitrary balanced open subsets U of E [see Isidro, J. Funct. Anal. 38 (1980), no. 2, 139–145;].
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