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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

[1]Edelstein, M., “On nearest points of sets in uniformly convex Banach spaces”, J. London Math. Soc. 43 (1968), 375–377.CrossRefGoogle Scholar
[2]Efimov, N.V. and Stečkin, S.B., “Approximative compactness and Čebyšev sets”, Soviet Math. Dokl. 2 (1961), 1226–1228.Google Scholar
[3]Fan, Ky and Glicksberg, Irving, “Some geometric properties of the spheres in a normed linear space”, Duke Math. J. 25 (1958), 553–568.CrossRefGoogle Scholar
[4]Holmes, Richard B., A course on optimization and best approximation (Lecture Notes in Mathematics, 257. Springer-Verlag, Berlin, Heidelberg, New York, 1972).CrossRefGoogle Scholar
[5]Klee, Victor, “Convexity of Chebyshev sets”, Math. Ann. 142 (1961), 292–304.CrossRefGoogle Scholar
[6]Klee, V., “Two renorming constructions related to a question of Anselone”, Studia Math. 33 (1969), 231–242.CrossRefGoogle Scholar
[7]Panda, B.B. and Kapoor, O.P., “A generalization of local uniform convexity of the norm”, J. Math. Anal. Appl, (to appear).Google Scholar
[8]Singer, Ivan, “Some remarks on approximative compactness”, Rev. Roumaine Math. Pures Appl. 9 (1964), 167–177.Google Scholar
[9]Vlasov, L.P., “On Čebyšev sets”, Soviet Math. Dokl. 8 (1967), 401–404.Google Scholar