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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]

Statements

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The space considered can be a real or complex Banach space. Its continuous dual space is denoted by The topological dual of -Banach space deduced from by any restriction scalar will be denoted (It is of interest only if is a complex space because if is a -space then )

James compactness criterion — Let be a Banach space and a weakly closed nonempty subset of The following conditions are equivalent:

  • is weakly compact.
  • For every there exists an element such that
  • For any there exists an element such that
  • For any there exists an element such that

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 is reflexive if and only if for all there exists an element of norm such that

History

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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.[4] This was then actually proved by James in 1964.[5]

See also

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Notes

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References

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  • James, Robert C. (1957), "Reflexivity and the supremum of linear functionals", Annals of Mathematics, 66 (1): 159–169, doi:10.2307/1970122, JSTOR 1970122, MR 0090019
  • Klee, Victor (1962), "A conjecture on weak compactness", Transactions of the American Mathematical Society, 104 (3): 398–402, doi:10.1090/S0002-9947-1962-0139918-7, MR 0139918.
  • James, Robert C. (1964), "Weakly compact sets", Transactions of the American Mathematical Society, 113 (1): 129–140, doi:10.2307/1994094, JSTOR 1994094, MR 0165344.
  • James, Robert C. (1971), "A counterexample for a sup theorem in normed space", Israel Journal of Mathematics, 9 (4): 511–512, doi:10.1007/BF02771466, MR 0279565.
  • James, Robert C. (1972), "Reflexivity and the sup of linear functionals", Israel Journal of Mathematics, 13 (3–4): 289–300, doi:10.1007/BF02762803, MR 0338742.
  • Megginson, Robert E. (1998), An introduction to Banach space theory, Graduate Texts in Mathematics, vol. 183, Springer-Verlag, ISBN 0-387-98431-3