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Center Points Of Nets

Published online by Cambridge University Press:  20 November 2018

C. L. Anderson ,
W. H. Hyams  and
C. K. McKnight
C. L. Anderson
Affiliation:
The University of Southern Louisiana, Lafayette, Louisiana
W. H. Hyams
Affiliation:
The University of Southern Louisiana, Lafayette, Louisiana
C. K. McKnight
Affiliation:
The University of Southern Louisiana, Lafayette, Louisiana
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Suppose x = (x∝) is a net with values in a metric space X having metric ρ. If a point z in X can be found to minimize

then z is called a center point (c.p.) of x. The space X is (netwise) c.p. complete if every bounded net has at least one c.p.; it is sequentially c.p. complete if every bounded sequence has a c.p. Netwise c.p. completeness implies sequential c.p. completeness, and the latter implies completeness since any c.p. of a Cauchy sequence will necessarily be a limit point of that sequence.

These notions are related to the set centers of Calder et al. [2].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 27 , Issue 2 , 01 April 1975 , pp. 418 - 422
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Anderson, C. L., Hyams, W. H., and McKnight, C. K., Generalization of a fixed point theorem of M. Edelstein (to appear).Google Scholar
2. Calder, J. R., Coleman, W. P., and Harris, R. L., Centers of infinite bounded sets in a normed space, Can. J. Math. 25 (1973), 986–999.Google Scholar
3. Edelstein, M., The construction of an asymptotic center with a fixed point property, Bull. Amer. Math. Soc. 78 (1972), 206–208.Google Scholar
4. Garkavi, A. L., The best possible net and best possible cross-section of a set in a normed space, Amer. Math. Soc. Transi. Ser. 2, 39 (1964), 111–132.Google Scholar
5. Hyams, W. H., Sequentially center point complete spaces, Ph.D. Thesis, University of Southwestern Louisiana, 1973.Google Scholar
6. Mizel, V. and Sundaresun, K., Banach sequence spaces, Arch. Math. (Basel) 19 (1968), 59–69.Google Scholar
7. Ruckle, W., On perfect symmetric BK spaces, Math. Ann. 175 (1968), 121–126.Google Scholar