Frechet spaces of non-archimedean valued continuous functions

Abstract

Let $X$ be an ultraregular space and let $K$ be a complete non-archimedean non-trivially valued field. Assume that the locally convex space $E$ = $C_c(X;K)$ of all continuous functions from $X$ to $K$ with the topology of uniform convergence on compact subsets of $X$ is a Frechet space. We shall prove that $E$ has an orthogonal basis consisting of $K$-valued characteristic functions of clopen (i.e. closed and open) subsets of $X$ and that it is isomorphic to the product of a countable family of Banach spaces with an orthonormal basis.