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

The space considered can be a real or complex Banach space. Its continuous dual space is denoted by The topological dual of ℝ-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

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

Notes

References

  • 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
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.