In an earlier paper we
studied the Radon-Nikodym property (RNP) for Fréchet spaces. D. Gilliam
continued the study by examining the RNP for locally convex spaces with the strict
Mackey convergence property. The aim of this paper is to take one more step by
studying the RNP for the class of locally convex spaces in which every bounded
subset is metrizable. Although this class strictly includes the class of spaces with the
strict Mackey convergence property, our goal is not a generalization for the
sake of generalization. Indeed, we shall prove a theorem that reduces the
study of the RNP for this class of spaces directly to the study of the RNP
for Banach spaces. This will provide a quick and simultaneous extension
of many of the basic Radon-Nikodym theorems in Banach spaces to this
class of locally convex spaces. We hope that our technique will eliminate
some of the mystery that seems to surround the RNP for locally convex
spaces.
