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Bulletin of the London Mathematical Society Advance Access originally published online on May 3, 2008
Bulletin of the London Mathematical Society 2008 40(3):505-515; doi:10.1112/blms/bdn033
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© 2008 London Mathematical Society

(LB)-spaces of vector-valued continuous functions

Leonhard Frerick

FB IV – Mathematik
Universität Trier
D–54286 Trier
Germany
frerick@uni-trier.de

Jochen Wengenroth

Département de Mathématique
Université de Liège,
Bât. B37 Analyse Mathéma-tique
B–4000 Liège,
Belgium

Received 24 October 2006. Revision received 26 February 2008.

We consider a question posed by Bierstedt and Schmets as to whether, for an (LB)-space E and a compact space K, the space C(K, E) of E-valued continuous functions endowed with the uniform topology is again an (LB)-space. We present proofs for the case of hilbertizable Montel (LB)-spaces as well as for the case where E is a weighted space of sequences or functions which even yield that Schwartz's {varepsilon}-product Y {varepsilon} E is an (LB)-space for every Formula -space Y. Although the problem for general (LB)-spaces remains open, we provide several relations and reductions. For instance, it is enough to consider curves, that is, the most natural case K=[0, 1] or, for the class of hilbertizable (LB)-spaces with metrizable bounded sets, the Stone–Cech compactification of N.

2000 Mathematics Subject Classification 46E40, 46A04, 46A13.


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