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vector-valued continuous functions; strict topologies; locally solid topologies; weak-star compactness; vector measures
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.
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