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
is weakly compact if and only if the dual norm each continuous linear functional on
The hypothesis of completeness in the theorem cannot be dropped.
considered can be a real or complex Banach space.
Its continuous dual space is denoted by
) James compactness criterion — Let
a weakly closed nonempty subset of
The following conditions are equivalent: A Banach space being reflexive if and only if its closed unit ball is weakly compact one deduces from this, since the norm of a continuous linear form is the upper bound of its modulus on this ball: James' theorem — A Banach space
Historically, these sentences were proved in reverse order.
In 1957, James had proved the reflexivity criterion for separable Banach spaces[2] and 1964 for general Banach spaces.
[3] Since the reflexivity is equivalent to the weak compactness of the unit sphere, Victor L. Klee reformulated this as a compactness criterion for the unit sphere in 1962 and assumes that this criterion characterizes any weakly compact quantities.