James' theorem

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 the completeness in the theorem cannot be dropped harv|James|1971.

See also

*Banach–Alaoglu theorem

References

*citation|last=James|first=Robert C.|title=Weakly compact sets|journal=Trans. Amer. Math. Soc.|volume=113|year=1964|pages=129-140|id=MR|id=165344|url=http://links.jstor.org/sici?sici=0002-9947(196410)113%3A1%3C129%3AWCS%3E2.0.CO%3B2-8.
*citation|last=James|first=Robert C.|title=A counterexample for a sup theorem in normed space|journal=Israel J. Math.|volume=9|number=4|year=1971|pages=511-512.
*citation|last=James|first=Robert C.|title=Reflexivity and the sup of linear functionals|journal=Israel J. Math.|volume=13|year=1972|pages=289-300|id=MR|id=338742 .


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