Duality theory of spaces of vector-valued continuous functions

Nowak, Marian ; Rzepka, Aleksandra

Commentationes Mathematicae Universitatis Carolinae, Tome 46 (2005) no. 1, pp. 55-73.

Résumé

Let $X$ be a completely regular Hausdorff space, $E$ a real normed space, and let $C_b(X,E)$ be the space of all bounded continuous $E$-valued functions on $X$. We develop the general duality theory of the space $C_b(X,E)$ endowed with locally solid topologies; in particular with the strict topologies $\beta_z(X,E)$ for $z=\sigma, \tau, t$. As an application, we consider criteria for relative weak-star compactness in the spaces of vector measures $M_z(X,E')$ for $z=\sigma, \tau, t$. It is shown that if a subset $H$ of $M_z(X,E')$ is relatively $\sigma(M_z(X,E'), C_b(X,E))$-compact, then the set $\operatorname{conv} (S(H))$ is still relatively $\sigma(M_z(X,E'), C_b(X,E))$-compact ($S(H)=$ the solid hull of $H$ in $M_z(X,E')$). A Mackey-Arens type theorem for locally convex-solid topologies on $C_b(X,E)$ is obtained.

Détail

Classification : 46E10, 46E15, 46E40, 46G10
Keywords: vector-valued continuous functions; strict topologies; locally solid topologies; weak-star compactness; vector measures