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.