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Marian Nowak, Aleksandra Rzepka ?=O IE@ JFCEAI  IF=?AI B LA?JHL=KA@ ?JEKKI BK?JEI

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Marian Nowak, Aleksandra Rzepka

Locally solid topologies on spaces of vector-valued continuous functions

Comment.Math.Univ.Carolinae 43,3 (2002) 473-483.

Abstract: 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 Cb(X, E) of allE-valued continuous and bounded functions from X into E. The mutual relationship between locally solid topologies on Cb(X, E) and Cb(X) (=

Cb(X,R)) is considered. In particular, the mutual relationship between strict topologies onCb(X) andCb(X, E) is established. It is shown that the strict topol- ogy βσ(X, E) (respectively βτ(X, E)) is the finest σ-Dini topology (respectively Dini topology) on Cb(X, E). A characterization ofσ-Dini and Dini topologies on Cb(X, E) in terms of their topological duals is given.

Keywords: vector-valued continuous functions, strict topologies, locally solid topologies, Dini topologies

AMS Subject Classification: 47A70, 46E05, 46E10

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