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Some examples concerning normal and uniform normal structure in Banach spaces

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

Published online by Cambridge University Press:  09 April 2009

Mark A. Smith  and
Barry Turett
Mark A. Smith
Affiliation:
Miami University, Oxford, Ohio 45056, U.S.A.
Barry Turett
Affiliation:
Oakland University, Rochester, Michigan 48309, U.S.A.
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Abstract

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Examples are given that show the following: (1) normal structure need not be inherited by quotient spaces; (2) uniform normal structure is not a self-dual property; and (3) no degree of k–uniform rotundity need be present in a space with uniform normal structure.

MSC classification

Secondary: 46B20: Geometry and structure of normed linear spaces
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 48 , Issue 2 , April 1990 , pp. 223 - 234
Copyright
Copyright © Australian Mathematical Society 1990

References

[1]Amir, D., ‘On Jung's constant and related constants in normed linear spaces’, (Longhorn Notes, The University of Texas at Austin, 1982–1983, pp. 143–159).Google Scholar
[2]Banach, S. and Mazur, S., ‘Zur theorie der linearen dimension’, Studia Math. 4 (1933), 100–112.CrossRefGoogle Scholar
[3]Brodskii, M. S. and Milman, D. P., ‘On the center of a convex set’, Dokl. Akad. Nauk SSSR 59 (1948), 837–840.Google Scholar
[4]Bynum, W. L., ‘A class of spaces lacking normal structure’, Compositio Math. 25 (1972), 233–236.Google Scholar
[5]Clarkson, J. A., ‘Uniformly convex spaces’, Trans. Amer. Math. Soc. 40 (1936), 396–414.CrossRefGoogle Scholar
[6]Day, M. M., Normed linear spaces (3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 21, Springer-Verlag, New York, 1973).CrossRefGoogle Scholar
[7]Downing, D. J. and Turett, B., ‘Some properties of the characteristic of convexity relating to fixed point theory’, Pacific J. Math. 104 (1983), 343–350.CrossRefGoogle Scholar
[8]Geremia, R. and Sullivan, F., ‘Multi-dimensional volumes and moduli of convexity in Banach spaces’, Ann. Mat. Pura Appl. (4) 127 (1981), 231–251.CrossRefGoogle Scholar
[9]Gillespie, A. A. and Williams, B. B., ‘Fixed points for nonexpansive mappings on Banach spaces with uniformly normal structure’, Applicable Anal. 9 (1979), 121–124.CrossRefGoogle Scholar
[10]Sims, B., The existence question for fixed points of nonexpansive maps (Kent State Seminar Notes, Spring 1986).Google Scholar
[11]Sullivan, F., ‘A generalization of uniformly rotund Banach spaces’, Canad. J. Math. 31 (1979), 628–636.CrossRefGoogle Scholar
[12]Zizler, V., ‘One theorem on rotundity and smoothness of separable Banach spaces’, Comment. Math. Univ. Carolin. 9 (1968), 637–640.Google Scholar
[13]Zizler, V., ‘On some rotundity and smoothness properties of Banach space’, Dissertationes Math. (Rozprawy Mat.) 87 (1971).Google Scholar