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 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 continuous linear functional on B attains a maximum on C.

The hypothesis of completeness in the theorem cannot be dropped (James 1971).

See also

• Banach–Alaoglu theorem
• Bishop–Phelps theorem
• Eberlein–Šmulian theorem
• Mazur's lemma
• Goldstine theorem

References

• James, Robert C. (1957), "Reflexivity and the supremum of linear functionals", Ann. of Math. 66 (1): 159–169, JSTOR 1970122, MR 0090019*
• James, Robert C. (1964), "Weakly compact sets", Trans. Amer. Math. Soc. (American Mathematical Society) 113 (1): 129–140, doi:10.2307/1994094, JSTOR 1994094, MR 165344.
• James, Robert C. (1971), "A counterexample for a sup theorem in normed space", Israel J. Math. 9 (4): 511–512, doi:10.1007/BF02771466.
• James, Robert C. (1972), "Reflexivity and the sup of linear functionals", Israel J. Math. 13 (3–4): 289–300, doi:10.1007/BF02762803, MR 338742.
• Megginson, Robert E. (1998), An introduction to Banach space theory, Graduate Texts in Mathematics 183, Springer-Verlag, ISBN 0-387-98431-3