Weak compactness characterization of reflexivity
= Weak compactness characterization of reflexivity
A <Banach space> is <reflexive Banach space>[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.