Countably-Normed Spaces, Their Dual, and the Gaussian Measure

Here we present an overview of countably-normed spaces. We discuss the main topologies–weak, strong, and inductive–placed on the dual of a countably-normed space and discuss the sigma-fields generated by these topologies. In particular, we show that under certain conditions the strong and inductive topologies coincide and the sigma-fields generated by the weak, strong, and inductive topologies are equal. With these sigma-fields, we develop a Gaussian measure on the dual of a nuclear space. The purpose in mind is to provide the background material for many of the results used is White Noise Analysis.