Martínez Ansemil, José MaríaPonte Miramontes, María Del SocorroMachado, Silvio2023-06-212023-06-211981Martínez Ansemil, J. M. & Ponte Miramontes, M. S. «An example of a quasi-normable Fréchet function space which is not a Schwartz space». Functional Analysis, Holomorphy, and Approximation Theory, editado por Silvio Machado, vol. 843, Springer Berlin Heidelberg, 1981, pp. 1-8. DOI.org (Crossref), https://doi.org/10.1007/BFb0089266.3-540-10560-310.1007/BFb0089266https://hdl.handle.net/20.500.14352/65471Proceedings of the Seminar held at the Universidade Federal do Rio de Janeiro, Rio de Janeiro, August 7–11, 1978If E and F are complex Banach spaces, and fixing a balanced open subset U of E, we let Hb=(Hb(U;F),τb) denote the space of all mappings f:U→F which are holomorphic of bounded type, endowed with its natural topology τb; 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 τ1 as well as the strong dual topology τb=β(S,Hb). Here, the authors prove that Hb is quasinormable (and hence distinguished) and τb=τ1 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;].book parthttps//doi.org/10.1007/BFb0089266http://link.springer.com/chapter/10.1007%2FBFb0089266metadata only access517.98Análisis funcional y teoría de operadores