Locally solid topologies on spaces of vector-valued continuous functions

summary:Let $X$ be a completely regular Hausdorff space and $E$ a real normed space. We examine the general properties of locally solid topologies on the space $C_b(X,E)$ of all $E$-valued continuous and bounded functions from $X$ into $E$. The mutual relationship between locally solid topologies on $C_b(X,E)$ and $C_b(X)$ $(=C_b(X,\Bbb R))$ is considered. In particular, the mutual relationship between strict topologies on $C_b(X)$ and $C_b(X,E)$ is established. It is shown that the strict topology $\beta _\sigma(X,E)$ (respectively $\beta _\tau(X,E)$) is the finest $\sigma $-Dini topology (respectively Dini topology) on $C_b(X,E)$. A characterization of $\sigma $-Dini and Dini topologies on $C_b(X,E)$ in terms of their topological duals is given