James's theorem

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

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

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

Statements

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

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

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

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

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

• Banach–Alaoglu theorem – Theorem in functional analysis
• Bishop–Phelps theorem
• Dual norm – Measurement on a normed vector space
• Eberlein–Šmulian theorem – Relates three different kinds of weak compactness in a Banach space
• Goldstine theorem
• Mazur's lemma – On strongly convergent combinations of a weakly convergent sequence in a Banach space
• Operator norm – Measure of the "size" of linear operators

Notes

1. James (1971)
2. James (1957)
3. James (1964)
4. Klee (1962)
5. James (1964)

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