James's theorem

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In mathematics, particularly functional analysis, the 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 maximum 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 continous linear functional on B attains a maximum on C.

References

  • James, Robert C., Weakly compact sets. Trans. Amer. Math. Soc. 113, 1964, 129-140. MR165344
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