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 $M_z(X,E^{\text{'}})$ for $z=\sigma ,\tau ,t$. It is shown that if a subset $H$ of $M_z(X,E^{\text{'}})$ is relatively $\sigma (M_z(X,E^{\text{'}}),C_b(X,E))$-compact, then the set $conv\left(S\right(H\left)\right)$ is still...

Let $(X,\parallel ·{\parallel }_X)$ be a real Banach space and let $E$ be an ideal of $L^0$ over a $\sigma$-finite measure space $(Ø,\Sigma ,\mu )$. Let $\left(X\right)$ be the space of all strongly $\Sigma$-measurable functions $fØ\to X$ such that the scalar function $\tilde{f}$, defined by $\tilde{f}\left(ø\right)={\parallel f\left(ø\right)\parallel }_X$ for $ø\in Ø$, belongs to $E$. The paper deals with strong topologies on $E\left(X\right)$. In particular, the strong topology $\beta (E\left(X\right),E{\left(X\right)}_n^{\sim })$ ($E{\left(X\right)}_n^{\sim }=$ the order continuous dual of $E\left(X\right)$) is examined. We generalize earlier results of [PC] and [FPS] concerning the strong topologies.

Let $X$ be a completely regular Hausdorff space, $E$ a real Banach space, and let $C_b(X,E)$ be the space of all $E$-valued bounded continuous functions on $X$. We study linear operators from $C_b(X,E)$ endowed with the strict topologies ${\beta }_z$ $(z=\sigma ,\tau ,\infty ,g)$ to a real Banach space $(Y,\parallel ·{\parallel }_Y)$. In particular, we derive Banach-Steinhaus type theorems for $({\beta }_z,\parallel ·{\parallel }_Y)$ continuous linear operators from $C_b(X,E)$ to $Y$. Moreover, we study $\sigma$-additive and $\tau$-additive operators from $C_b(X,E)$ to $Y$.