James's theorem

In mathematics, particularly functional analysis, James' theorem, named for Robert C. James, states that a Banach space X {\displaystyle X} is reflexive if and only if every continuous linear functional's norm on X {\displaystyle X} attains its supremum on the closed unit ball in X . {\displaystyle X.}

A stronger version of the theorem states that a weakly closed subset C {\displaystyle C} of a Banach space X {\displaystyle X} is weakly compact if and only if the dual norm each continuous linear functional on X {\displaystyle X} attains a maximum on C . {\displaystyle C.}

The hypothesis of completeness in the theorem cannot be dropped.[1]

Statements

The space X {\displaystyle X} considered can be a real or complex Banach space. Its continuous dual space is denoted by X ′ . {\displaystyle X^{\prime }.} The topological dual of R {\displaystyle \mathbb {R} } -Banach space deduced from X {\displaystyle X} by any restriction scalar will be denoted X R ′ . {\displaystyle X_{\mathbb {R} }^{\prime }.} (It is of interest only if X {\displaystyle X} is a complex space because if X {\displaystyle X} is a R {\displaystyle \mathbb {R} } -space then X R ′ = X ′ . {\displaystyle X_{\mathbb {R} }^{\prime }=X^{\prime }.} )

James compactness criterion — Let X {\displaystyle X} be a Banach space and A {\displaystyle A} a weakly closed nonempty subset of X . {\displaystyle X.} The following conditions are equivalent:

  • A {\displaystyle A} is weakly compact.
  • For every f ∈ X ′ , {\displaystyle f\in X^{\prime },} there exists an element a 0 ∈ A {\displaystyle a_{0}\in A} such that | f ( a 0 ) | = sup a ∈ A | f ( a ) | . {\displaystyle \left|f\left(a_{0}\right)\right|=\sup _{a\in A}|f(a)|.}
  • For any f ∈ X R ′ , {\displaystyle f\in X_{\mathbb {R} }^{\prime },} there exists an element a 0 ∈ A {\displaystyle a_{0}\in A} such that f ( a 0 ) = sup a ∈ A | f ( a ) | . {\displaystyle f\left(a_{0}\right)=\sup _{a\in A}|f(a)|.}
  • For any f ∈ X R ′ , {\displaystyle f\in X_{\mathbb {R} }^{\prime },} there exists an element a 0 ∈ A {\displaystyle a_{0}\in A} such that f ( a 0 ) = sup a ∈ A f ( a ) . {\displaystyle f\left(a_{0}\right)=\sup _{a\in A}f(a).}

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 X {\displaystyle X} is reflexive if and only if for all f ∈ X ′ , {\displaystyle f\in X^{\prime },} there exists an element a ∈ X {\displaystyle a\in X} of norm ‖ a ‖ ≤ 1 {\displaystyle \|a\|\leq 1} such that f ( a ) = ‖ f ‖ . {\displaystyle f(a)=\|f\|.}

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
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