Vol. 75, No. 1, 1978

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ISSN: 0030-8730
On the Radon-Nikodým property in a class of locally convex spaces

Elias Saab

Vol. 75 (1978), No. 1, 281–291
Abstract

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.

Mathematical Subject Classification 2000
Primary: 46G10
Secondary: 46B22
Milestones
Received: 16 August 1976
Revised: 20 April 1977
Published: 1 March 1978
Authors
Elias Saab