James's theorem

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

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

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

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

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

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

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

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]

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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