One-sided James' compactness theorem

Abstract

We present some extensions of classical results that involve elements of the dual of Banach spaces, such as Bishop-Phelp's theorem and James' compactness theorem, but restricting to sets of functionals determined by geometrical properties. The main result, which answers a question posed by F. Delbaen, is the following: Let E be a Banach space such that (B E * , ω * ) is convex block compact. Let A and B be bounded, closed and convex sets withattains its infimum on A and its supremum on B, then A and B are both weakly compact.We obtain new characterizations of weakly compact sets and reflexive spaces, as well as a result concerning a variational problem in dual Banach spaces.2010 Mathematics Subject Classification. 46A50, 46B50.