(LB)-spaces of vector-valued continuous functions
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 f...
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| Veröffentlicht in: | The Bulletin of the London Mathematical Society Jg. 40; H. 3; S. 505 - 515 |
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| Hauptverfasser: | , |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
Oxford University Press
01.06.2008
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| ISSN: | 0024-6093, 1469-2120 |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | 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 ε-product Y ε E is an (LB)-space for every -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–Čech compactification of ℕ. |
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| Bibliographie: | istex:4F5BC09DBF686BB262C353FF4426F30B25251BC1 ArticleID:bdn033 2000 Mathematics Subject Classification 46E40, 46A04, 46A13. ark:/67375/HXZ-3Z1Q6QXD-0 46E40, 46A04, 46A13. 2000 Mathematics Subject Classification |
| ISSN: | 0024-6093 1469-2120 |
| DOI: | 10.1112/blms/bdn033 |