Geodesic
Locally solid topologies on spaces of vector-valued continuous functions
Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002) no. 3, pp. 473-483.

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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.
Classification : 46A03, 46E05, 46E10, 46E40, 47A70
Keywords: vector-valued continuous functions; strict topologies; locally solid topologies; Dini topologies 
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     title = {Locally solid topologies on spaces of vector-valued continuous functions},
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Nowak, Marian; Rzepka, Aleksandra. Locally solid topologies on spaces of vector-valued continuous functions. Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002) no. 3, pp. 473-483. http://geodesic.mathdoc.fr/item/CMUC_2002__43_3_a6/