An Analogue of the Radon-nikodym Property for Non-locally Convex Quasi-banach Spaces

where g:(0, 1)-»X is an essentially bounded strongly measurable function. In this paper we examine analogues of the Radon-Nikodym Property for quasiBanach spaces. If 0 < p < 1, there are several possible ways of defining "differentiable" operators on Lp, but they inevitably lead to the conclusion that the only differentiable operator is zero. For example, a differentiable operator on L\ has the Dunford-Pettis property; operators on L\ with the Dunford-Pettis property map the unit ball of L«, to a compact set (cf (12)). However any operator on Lp (p < 1) with this property is zero (4). Thus we define a quasi-Banach space X to be p-trivial if J£(LP, X) = {0}. The concept of p-triviality is then hoped to be an analogue of the Radon-Nikodym property amongst locally p-convex quasi-Banach spaces. It turns out that this hope is fulfilled to some extent. Our main results in Sections 4 and 5 demonstrate an analogue of Edgar's theorem (2) and of the Phelps characterisation of the Radon-Nikodym Property ((1), (9)) to this setting. Precisely we show that a locally p-convex quasiBanach space is p-trivial if and only if every closed bounded p-convex set is the closed p-convex hull of its "strongly p-extreme points". Our analogue of Edgar's theorem is that if C is a bounded closed p-convex subset of a p-trivial quasi-Banach space then every x E C may be represented in the form