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An example of a quasinormable Fréchet function space which is not a Schwartz space

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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


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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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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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