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Title: Locally solid topologies on spaces of vector-valued continuous functions (English)
Author: Nowak, Marian
Author: Rzepka, Aleksandra
Language: English
Journal: Commentationes Mathematicae Universitatis Carolinae
ISSN: 0010-2628 (print)
ISSN: 1213-7243 (online)
Volume: 43
Issue: 3
Year: 2002
Pages: 473-483
Category: math
Summary: Let $X$ be a completely regular Hausdorff space and $E$ a real normed space. We examine the general properties of locally solid topologies on the space $C_b(X,E)$ of all $E$-valued continuous and bounded functions from $X$ into $E$. The mutual relationship between locally solid topologies on $C_b(X,E)$ and $C_b(X)$ $(=C_b(X,\Bbb R))$ is considered. In particular, the mutual relationship between strict topologies on $C_b(X)$ and $C_b(X,E)$ is established. It is shown that the strict topology $\beta _\sigma(X,E)$ (respectively $\beta _\tau(X,E)$) is the finest $\sigma $-Dini topology (respectively Dini topology) on $C_b(X,E)$. A characterization of $\sigma $-Dini and Dini topologies on $C_b(X,E)$ in terms of their topological duals is given. (English)
Keyword: vector-valued continuous functions
Keyword: strict topologies
Keyword: locally solid topologies
Keyword: Dini topologies
MSC: 46A03
MSC: 46E05
MSC: 46E10
MSC: 46E40
MSC: 47A70
idZBL: Zbl 1068.46023
idMR: MR1920522
Date available: 2009-01-08T19:24:00Z
Last updated: 2012-04-30
Stable URL:
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