James' theorem

In mathematics, particularly functional analysis, James' theorem, named for Robert C. James, states that a Banach space B is reflexive if and only if every continuous linear functional on B attains its supremum on the closed unit ball in B.

A stronger version of the theorem states that a weakly closed subset C of a Banach space B is weakly compact if and only if each continuous linear functional on B attains a maximum on C.

The hypothesis of completeness in the theorem cannot be dropped (James 1971).