Operator representation theorems

Abstract

We represent certain types of bounded linear operators from a Banach space into a space BC(S) where S is an arbitrary topological space and BC(S) is the space of bounded continuous scalar-valued functions on S with the sup norm. A comprehensive treatment of work to date may be found in [1] and [7]. We represent the completely continuous (not to be confused with compact) operator, which has not been considered previously. This yields a sufficient condition for T: X-+BC(S) to be strictly singular. DEFINITIONS. (a) A set UCX is said to be absolutely convex if it is convex and XUC U for IXI ?1. (b) We denote the smallest closed absolutely convex set containing W by aco(W). Let UCX. By Uo (the polar of U) we shall mean {x*| Ix*(u)| I1 for all uEU}. (c) If UCX* we define U0= {x |u(x) I 5 1 for all u E U }. We write o for (U0)o. The Mackey topology on X* is that topology generated by the polars of all absolutely convex, weakly compact sets of X [6, p. 173]. A bounded linear operator T: X-Y is said to be completely continuous if it maps weakly convergent sequences into norm convergent sequences. (This is the original definition of Hilbert, which coincides with compactness when X is reflexive.)