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Approximative compactness and continuity of metric projections

Published online by Cambridge University Press:  17 April 2009

B.B. Panda  and
O.P. Kapoor
B.B. Panda
Affiliation:
Department of Mathematics, Regional Engineering College, Rourkela, Orissa, India.
O.P. Kapoor
Affiliation:
Department of Mathematics, Indian Institute of Technology, Kanpur, Kanpur, India.
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Abstract

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In the paper “Some remarks on approximative compactness”, Rev. Roumaine Math. Pures Appl. 9 (1964), Ivan Singer proved that if K is an approximatively compact Chebyshev set in a metric space, then the metric projection onto K is continuous. The object of this paper is to show that though, in general, the continuity of the metric projection supported by a Chebyshev set does not imply that the set is approximatively compact, it is indeed so in a large class of Banach spaces, including the locally uniformly convex spaces. It is also proved that in such a space X the metric projection onto a Chebyshev set is continuous on a set dense in X.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 11 , Issue 1 , August 1974 , pp. 47 - 55
Copyright
Copyright © Australian Mathematical Society 1974

References

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