Proceedings of the Edinburgh Mathematical Society (Series 2)

Research Article

An Analogue of the Radon-Nikodym Property for Non-Locally Convex Quasi-Banach Spaces

N. J. Kaltona1a2

a1 Department of Pure Mathematics, University College of Swansea, Singleton Park, Swansea SA2 8PP

a2 Department of Mathematics, Michigan State University, East Lansing, Michigan 48824, U.S.A.

In recent years there has been considerable interest in Banach spaces with the Radon-Nikodym Property; see (1) for a summary of the main known results on this class of spaces.We may define this property as follows: a Banach space X has the Radon-Nikodym Property if whenever T xs2208 xs2112 (L1, X)(where L1 = L1(0, 1)) then T is differentiable i.e.

S0013091500027802_eqnU1

where g:(0, 1)→X is an essentially bounded strongly measurable function. In this paper we examine analogues of the Radon-Nikodym Property for quasi-Banach 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.

(Received November 01 1977)