Extreme functionals on an upper semicontinuous function space

Abstract

A representation theorem is given for the extreme points of the dual ball of a vector valued function space Xwith upper semicontinuous norm defined on a compact Hausdorff space (?. This generalizes the Arens-Kelley theorem which is the case X=C(Q). One of the most powerful tools in the abstract geometric description of Banach spaces Xwhich are given by concrete representations is knowing explicitly what the extreme points of the unit ball in X* are. The prototype theorem of this type is the theorem of Arens-Kelley to the effect that if X=C(Q), Q compact Hausdorff, the extreme points of its dual ball are exactly the evaluations at points of Q up to multiplication by scalars of absolute value 1. The purpose of this note is to generalize this result in two directions: (1) replace continuity of the functions by upper semicontinuity of their absolute values, and (2) allow the functions to be vector valued. See Theorem I below. The significance of these particular hypotheses is to be found in [1]. Briefly, they make the theorem applicable to arbitrary Banach spaces, viewed from the uniform norm point of view. One of us (Roy) will exploit this generality elsewhere along the lines already indicated. Our second theorem is an application of Theorem 1 to quotient modules to show how our method can be used to obtain a related result announced by W. J. Strobele [3]. Let Q be a compact Hausdorff space, and for each t E S2 let Xt be a normed linear space. We are interested in linear spaces X of functions x defined on Q2 with x(t) E Xt for each t, and satisfying at least the first, and usually both of the following: (i) (Upper semicontinuity) For each x E X the norm function tb-+ Ijx(t)IJ (norm in Xt) is upper semicontinuous. (ii) (Module property) For x E X and fE C(-Q) the function fx defined by multiplication (fx(t)=f(t)x(t) for all t) belongs to X. Presented to the Society, September 1, 1972; received by the editors September 18, 1972 and, in revised form, April 23, 1973. AMS (MOS) subject class{fications (1970). Primary 46E40,46B99; Secondary 54C35.