The space of tempered distributions as a k-space

Abstract

In this paper, we investigate the roles of compact sets in the space of tempered distributions $\mathscr{S}^{\prime}$. The key notion is "k-spaces", which constitute a fairly general class of topological spaces. In a k-space, the system of compact sets controls continuous functions and Borel measures. Focusing on the k-space structure of $\mathscr{S}^{\prime}$, we prove some theorems which seem to be fundamental for infinite dimensional harmonic analysis from a new and unified view point. For example, the invariance principle of Donsker for the white noise measure is shown in terms of infinite dimansional characteristic functions.