A geometric characterization of Fréchet spaces with the Radon-Nikodým property

Abstract

Let F be a locally convex Frechet space. F is said to have the Radon?Nikodym property if for every positive finite measure space (Ql,E ,u ) and every s,t-continuous vector measure m Z; F of bounded variation, there exists an integrable function f: Q -F such that m(S) = fsf(co)d,u(co), for every S E Z. Maynard proved that a Banach space has the Radon-Nikodym property iff it is an s-dentable space. It is the purpose of this paper to give the following analogous characterization: A Frechet space F has the Radon-Nikodym property iff F is s-dentable. 0. Introduction. In [8], Maynard obtained some equivalent geometric conditions for the average range of a vector measure in the characterization of Rieffel's Radon-Nikodym theorem [11, Main theorem, p. 4661. Based on these results, Maynard [9, Theorem 2.2] recently extended Rieffel's [12, Theorem 1] condition on the dentability of the average range to s?dentabil= ity of the average range. It was shown in [2], [7] that all of these results can be extended to locally convex Frechet spaces; see ? 2. As a consequence, the geometric characterization of Frechet spaces having the RadonNikodym property will be proved in ?3 below. 1. Preliminaries. Let (Q, E, jx) be a positive finite measure space, where Q is an abstract set, E is a o-algebra of subsets of Q, and ji is a real-valued measure defined on E. Without loss of generality, one can assume that E is jx-complete. Let V S G l11(S) > O}. Throughout this paper, let F be a locally convex Frechet space, and I U I be a fundamental decreasing sequence of closed absolutely n n=1 Received by the editors June 14, 1973 and, in revised form, November 12, 1973. AMS (MOS) subject classifications (1970). Primary 28A45, 46G10; Secondary 46A05