2026-06-28T04:56:19Zhttps://docta.ucm.es/rest/oai/requestoai:docta.ucm.es:20.500.14352/654712024-07-22T13:03:21Zcom_20.500.14352_14col_20.500.14352_21
Docta Complutense
author
Martínez Ansemil, José María
author
Ponte Miramontes, María Del Socorro
editor
Machado, Silvio
2023-06-21T02:43:01Z
2023-06-21T02:43:01Z
1981
Martínez 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.
3-540-10560-3
10.1007/BFb0089266
https://hdl.handle.net/20.500.14352/65471
https//doi.org/10.1007/BFb0089266
http://link.springer.com/chapter/10.1007%2FBFb0089266
If 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;].
An example of a quasinormable Fréchet function space which is not a Schwartz space
book part