6 - Weak Topologies

Summary

6.1 Weak Convergence

In Chapter 4, we saw a number of interesting ways in which a normed linear space interacts with its dual space. Notably, the Hahn–Banach Theorem gave us a powerful way to construct bounded linear functionals. In this chapter, we explore this relationship further and once again the Hahn–Banach Theorem plays a central role.

We will begin by exploring the notion of weak convergence. The ideas developed in this section will, together with a geometric version of the Hahn–Banach Theorem, help us define new topologies on a normed linear space and its dual space. These topologies are weaker (andmore forgiving) than the norm topology, which allows us to prove some powerful compactness theorems.