Abstract: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 $conv (S(H))$ 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.
Keywords: vector-valued continuous functions, strict topologies, locally solid topologies, weak-star compactness, vector measures
AMS Subject Classification: 46E10, 46E15, 46E40, 46G10