Abstract: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~$Y$.
Keywords: vector-valued continuous functions, strict topologies, locally solid topologies, Dini-topologies, strong Mackey space, $\sigma $-additive operators, $\tau $-additive operators
AMS Subject Classification: 47A70 47B38 46E10