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

Published online by Cambridge University Press:  20 January 2009

N. J. Kalton
Affiliation:
Department of Pure Mathematics, University College of Swansea, Singleton Park, Swansea SA2 8PP Department of Mathematics, Michigan State University, East Lansing, Michigan 48824, U.S.A.
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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 ∈ ℒ (L1, X)(where L1 = L1(0, 1)) then T is differentiable i.e.

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.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 1979

References

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