Krein–Smulian theorem

Remarks

The weak* topology is the weak topology on $X^*$ given by the evaluations $f \mapsto f(x)$ for $x \in X$. The proof introduces the bounded weak* topology, the finest topology that agrees with the weak* topology on each ball $B_r$. It shows that this topology is locally convex and that its continuous linear functionals are again the evaluations at points of $X$; this step uses completeness of $X$. A convex set closed in the bounded weak* topology is then weak*-closed by the Hahn–Banach separation theorem.

A main corollary, the Banach–Dieudonné theorem, says that a linear subspace $M \subseteq X^*$ is weak*-closed when $M \cap B_1$ is weak*-closed. Convexity is needed: the theorem is a statement about convex sets, and a nonconvex set can meet each ball in a weak*-closed set without being weak*-closed. For instance, in $\ell^2$ (identified with its own dual), the set $\set{\sqrt{n}\, e_n : n \ge 1}$ meets each ball in a finite set but has $0$ in its weak closure. The name Krein–Smulian theorem is also used for a second result: in a Banach space, the closed convex hull of a weakly compact set is weakly compact.

Going deeper

For a student

The theorem is used to check weak* closedness of subspaces and convex sets, for example to show that a subspace of a dual space is itself a dual space. When $X$ is separable, each ball $B_r$ with the weak* topology is metrizable, so the test reduces to weak* convergent sequences in bounded sets. It is a companion of the Banach–Alaoglu theorem, which gives weak* compactness of the balls $B_r$.