A continuity property related to an index of non-separability and its applications
1992 ◽
Vol 46
(1)
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pp. 67-79
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Keyword(s):
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.
2004 ◽
Vol 77
(3)
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pp. 357-364
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2004 ◽
Vol 70
(3)
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pp. 463-468
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Keyword(s):
1982 ◽
Vol 32
(1)
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pp. 134-144
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1979 ◽
Vol 28
(2)
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pp. 205-213
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1997 ◽
Vol 63
(2)
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pp. 238-262
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2014 ◽
Vol 68
(1)
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Keyword(s):
Keyword(s):
2017 ◽
Vol 165
(3)
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pp. 467-473
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