Weak topology

Summary

Let $(X, Y)$ be a dual pair. The weak topology $\sigma(X, Y)$ is the coarsest topology on $X$ making every map $x \mapsto \langle x, y \rangle$ continuous. For a topological vector space $E$, the weak topology is $\sigma(E, E')$, defined by its continuous linear forms, and the weak-* topology on $E'$ is $\sigma(E', E)$. Weak topologies have fewer open sets than norm topologies, so more sets are compact; this is the content of the Banach–Alaoglu theorem.