Weak compactness characterization of reflexivity

A Banach space is reflexive if and only if its closed unit ball is compact in the weak topology. One direction applies the Banach-Alaoglu theorem to the unit ball in the bidual; the converse follows because Goldstine theorem makes the canonical image of the unit ball weak-star dense, while weak compactness makes that image weak-star closed.

Every bounded sequence in a reflexive Banach space has a weakly convergent subsequence. This is the sequential form of weak compactness supplied by the Eberlein-Smulian theorem.