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1994-1995 | 112 | 3 | 243-250
Tytuł artykułu

Compactness and countable compactness in weak topologies

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A bounded closed convex set K in a Banach space X is said to have quasi-normal structure if each bounded closed convex subset H of K for which diam(H) > 0 contains a point u for which ∥u-x∥ < diam(H) for each x ∈ H. It is shown that if the convex sets on the unit sphere in X satisfy this condition (which is much weaker than the assumption that convex sets on the unit sphere are separable), then relative to various weak topologies, the unit ball in X is compact whenever it is countably compact.
  • [1] T. Büber and W. A. Kirk, Constructive aspects of fixed point theory for nonexpansive mappings, to appear.
  • [2] T. Büber and W. A. Kirk, Convexity structures and the existence of minimal sets, preprint.
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  • [7] G. Godefroy, Existence and uniqueness of isometric preduals: A survey, in: Banach Space Theory, B. L. Lin (ed.), Contemp. Math. 85, Amer. Math. Soc., Providence, R.I., 1989, 131-194.
  • [8] G. Godefroy and N. Kalton, The ball topology and its applications, ibid., 195-237.
  • [9] K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge Univ. Press, Cambridge, 1990.
  • [10] J. L. Kelley, General Topology, van Nostrand, Princeton, 1955.
  • [11] M. A. Khamsi, Étude de la propriété du point fixe dans les espaces de Banach et les espaces métriques, thèse de doctorat de l'Université Paris VI, 1987.
  • [12] M. A. Khamsi, On metric spaces with uniform normal structure, Proc. Amer. Math. Soc. 106 (1989), 723-726.
  • [13] M. A. Khamsi and D. Misane, Compactness of convexity structures in metric spaces, to appear.
  • [14] W. A. Kirk, Nonexpansive mappings and normal structure in Banach spaces, in: Proc. Research Workshop on Banach Space Theory, B. L. Lin (ed.), Univ. of Iowa, 1981, 113-127.
  • [15] W. A. Kirk, Nonexpansive mappings in metric and Banach spaces, Rend. Sem. Mat. Fis. Milano 61 (1981), 133-144.
  • [16] J. P. Penot, Fixed point theorems without convexity, Bull. Soc. Math. France Mém. 60 (1979), 129-152.
  • [17] P. Soardi, Struttura quasi normale e teoremi di punto unito, Rend. Istit. Mat. Univ. Trieste 4 (1972), 105-114.
  • [18] S. Troyanski, On locally uniformly convex and differentiable norms in certain nonseparable Banach spaces, Studia Math. 37 (1971), 173-180.
  • [19] C. S. Wong, Close-to-normal structure and its applications, J. Funct. Anal. 16 (1974), 353-358.
  • [20] V. Zizler, On some rotundity and smoothness properties of Banach spaces, Dissertationes Math. (Rozprawy Mat.) 87 (1971).
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