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Ansemil, José María M. and Ponte, Socorro
(1981)
*An example of a quasinormable Fréchet function space which is not a Schwartz space.*
In
Functional analysis, holomorphy and approximation theory.
Lecture Notes in Mathematics
(843).
Springer, Berlín, pp. 1-8.
ISBN 3-540-10560-3

Official URL: http://link.springer.com/chapter/10.1007%2FBFb0089266

## Abstract

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;].

Item Type: | Book Section |
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Additional Information: | Proceedings of the Seminar held at the Universidade Federal do Rio de Janeiro, Rio de Janeiro, August 7–11, 1978 |

Subjects: | Sciences > Mathematics > Functional analysis and Operator theory |

ID Code: | 22113 |

Deposited On: | 26 Jun 2013 17:55 |

Last Modified: | 09 Dec 2013 17:32 |

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