James's theorem

Statements

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 \mathbb{R}-Banach space deduced from X by any restriction scalar will be denoted X^{\prime}{\R}. (It is of interest only if X is a complex space because if X is a \R-space then X^{\prime}{\R} = X^{\prime}.)

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^{\prime}{\R}, 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^{\prime}{\R}, 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:

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 and 1964 for general Banach spaces. 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. This was then actually proved by James in 1964.

Notes

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References

1. {{harvtxt. James. 1971
2. {{harvtxt. James. 1957
3. {{harvtxt. James. 1964
4. {{harvtxt. Klee. 1962
5. {{harvtxt. James. 1964