The Weak and Weak* Topologies: An Introduction

Abstract

As we saw in our brief study of compactness in normed linear spaces, the norm topology is too strong to allow any widely applicable subsequential extraction principles. Indeed, in order that each bounded sequence in X have a norm convergent subsequence, it is necessary and sufficient that X be finite dimensional. This fact leads us to consider other, weaker topologies on normed linear spaces which are related to the linear structure of the spaces and to search for subsequential extraction principles therein. As so often happens in such ventures, the roles of these topologies are not restricted to the situations initially responsible for their introduction. Rather, they play center court in many aspects of Banach space theory.