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A continuity property related to an index of non-separability and its applications

Published online by Cambridge University Press:  17 April 2009

Warren B. Moors
Warren B. Moors
Affiliation:
Department of Mathematics, The University of Newcastle, Newcastle NSW 2308, Australia
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Abstract

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For a set E in a metric space X the index of non-separability is β(E) = inf{r > 0: E is covered by a countable-family of balls of radius less than r}.

Now, for a set-valued mapping Φ from a topological space A into subsets of a metric space X we say that Φ is β upper semi-continuous at t ∈ A if given ε > 0 there exists a neighbourhood U of t such that β(Φ(U)) < ε. In this paper we show that if the subdifferential mapping of a continuous convex function Φ is β upper semi-continuous on a dense subset of its domain then Φ is Fréchet differentiable on a dense Gδ subset of its domain. We also show that a Banach space is Asplund if and only if every weak* compact subset has weak* slices whose index of non-separability is arbitrarily small.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 46 , Issue 1 , August 1992 , pp. 67 - 79
Copyright
Copyright © Australian Mathematical Society 1992

References

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