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. 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). The space X considered can be a real or complex Banach space. Its topological dual is denoted by X ' . The topological dual of ℝ-Banach space deduced from X by any restriction scalar will be denoted X ' ℝ . (It is of interest only if X is a space car because if X is a ℝ-space then X ' ℝ = X' .)