Weak convergence of probability measures: a topological vector space point of view

Weak convergence of probability measures is one of the most important topics in the field probability and statistics. In this survey paper, we look at weak convergence of probability measures from the topological vector space point of view. We start from the key concepts and results about weak topology and weak convergence under the general framework of topological vector spaces. Then we restrict our attention to the space of probability measures and see how the general results specialize to those in probability theory. Finally, we will review some important facts about the metrizability of weak topology. We hope the general approach reviewed in this paper can provide an alternative view and some insights.