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vector-valued continuous functions; strict topologies; locally solid topologies; Dini-topologies; strong Mackey space; $\sigma $-additive operators; $\tau $-additive operators
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 $(Y,\|\cdot\|_Y)$. In particular, we derive Banach-Steinhaus type theorems for $(\beta_z,\|\cdot\|_Y)$ 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 $Y$.
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