James's theorem
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In mathematics, particularly functional analysis, James' theorem, named for Robert C. James, states that a Banach space is reflexive if and only if every continuous linear functional's norm on attains its supremum on the closed unit ball in
A stronger version of the theorem states that a weakly closed subset of a Banach space is weakly compact if and only if the dual norm each continuous linear functional on attains a maximum on
The hypothesis of completeness in the theorem cannot be dropped.[1]