Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-17T16:23:56.072Z Has data issue: false hasContentIssue false
  • English
  • Français

Completeness of normed linear spaces admitting centers

Part of: Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  09 April 2009

Ali Ansari Astaneh
Ali Ansari Astaneh
Affiliation:
Department of Mathematics, University of Mashhad, Mashhad, Iran
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

It is shown that a normed linear space admitting (Chebyshev) centers is complete. Then the ideas in the proof of this fact are used to show that every incomplete CLUR (compactly locally uniformly rotund) normed linear space contains a closed bounded convex subset B with the following properties: (a) B does not contain any farthest point; (b) B does not contain any nearest point (to the elements of its complement).

MSC classification

Secondary: 46B20: Geometry and structure of normed linear spaces
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 39 , Issue 3 , December 1985 , pp. 360 - 366
Copyright
Copyright © Australian Mathematical Society 1985

References

Astaneh, A. A. (1983), ‘A characterization of local uniform convexity of the norm’, Indian J. Pure Appl. Math. 14 (10), 1217–1219.Google Scholar
Garkavi, A. L. (1964), ‘The best possible net and the best possible cross section of a set in a normed linear space’, Amer. Math. Soc. Transl. Ser. 2, 39, 111–132.Google Scholar
Holmes, R. B. (1972) A course in optimization and best approximation, (Lecture Notes in Math., vol. 257, Springer-Verlag, New York, London).CrossRefGoogle Scholar
Panda, B. B. and Kapoor, O. P. (1975), ‘A generalization of local uniform convexity of the norm’, J. Math. Anal. Appl. 52, 300–308.CrossRefGoogle Scholar