Strict Topologies and Vector-Measures on non-Archimedean Spaces

A. K. KATSARAS

University of Ioannina, Greece

Abstract ,5cm Let C_b(X,E) be the space of all bounded continuous functions from a zero-dimensional Hausdorff topological space X to a non-Archimedean Hausdorff locally convex space E. In this paper, we study several of the properties of the strict topologies $\beta , \beta', \beta_{1}$ and $\beta_{1}'$ on C_b(X,E) and we show that the corresponding dual spaces are certain subspaces of a space M(X,E') of finitely-additive E'-valued measures on the algebra of all clopen subsets of X. In case E is polar, it is proved that the topology $\beta_{o}$ coincides with the polar topology associated with $\beta'$. Also two new topologies $\beta_{e}$ and $\beta_{e}'$ on C_b(X,E) are introduced. These topologies are defined as inductive topologies by using the family of all continuous ultra-pseudometrics on X. Some of the properties of these topologies are investigated. When E is metrizable, it is shown that $\beta_{e}$ is coarser than $\beta_{1}$ and coincides with the topology of simple convergence on uniformly-bounded equicontinuous subsets of C_b(X,E). In the same case, it is proved that $\beta_{e}'$ yields as dual space the space of the so called separable members of M(X,E').