Weakly compact sets

Abstract

It has been conjectured that a closed convex subset C of a Banach space B is weakly compact if and only if each continuous linear functional on B attains a maximum on C [5]. This reduces easily to the case in which C is bounded, and will be answered in the affirmative [Theorem 4] after some preliminary results are established. Following suggestions by Namioka and Peck, the result is then generalized, first to weakly closed subsets of Banach spaces and then to weakly closed subsets of complete locally convex linear spaces. The original motivation for this conjecture was the knowledge that it is true when C is the unit sphere of a separable Banach space [2], which was later extended to arbitrary Banach spaces [3, Theorem 5]. Additional support was given the conjecture when V. L. Klee [5] proved a seemingly related theorem-namely, that if C is a bounded closed non-weakly-compact convex subset of a Banach space, then there is a decreasing sequence {K1} of nonempty closed convex subsets of C such that, for each x E C and each m E [0, 1), the set x + m(C x) meets only finitely many of the sets {Ki}. Iff is a continuous linear functional whose sup on C is M and if there is no x in C for which f(x) = M, then a suitable choice for Klee's sequence {Ki} is to let Kn = C r1 {x:f (x) > M l/n} for each n. It is interesting to note that the conjecture can be verified easily for a bounded closed convex set that is symmetric about an interior point x, since if x is translated to 0, then the convex set as a unit sphere induces a norm for which the new Banach space is isomorphic to the original space. More generally, if 0 is an interior point of a bounded convex set C and K is the closed convex span of Cu ( C), then K is symmetric about 0 and the sup on K of a continuous linear functional is the larger of its sups on C and C. Therefore for convex bodies the conjecture can be established by using the known theorem for unit spheres [3, Theorem 5, p. 215]. The following theorem is a generalization of a characterization of weak compactness of the unit sphere that was useful in [2]. In the proof of this theorem and thereafter, we shall use the convention that a sequence of nonoverlapping