James's theorem
In mathematics, particularly functional analysis, James's theorem, named for Robert C. James, states that a Banach space X {\displaystyle X} is reflexive if and only if every continuous linear functional's norm on X {\displaystyle X} attains its supremum on the closed unit ball in X . {\displaystyle X.} A stronger version of the theorem states that a weakly closed subset C {\displaystyle C} of a Banach space X {\displaystyle X} is weakly compact if and only if the dual norm each continuous linear functional on X {\displaystyle X} attains a maximum on C .