Compactness and countable compactness in weak topologies

W. Kirk

Studia Mathematica (1995)

  • Volume: 112, Issue: 3, page 243-250
  • ISSN: 0039-3223

Abstract

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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.

How to cite

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Kirk, W.. "Compactness and countable compactness in weak topologies." Studia Mathematica 112.3 (1995): 243-250. <http://eudml.org/doc/216151>.

@article{Kirk1995,
abstract = {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.},
author = {Kirk, W.},
journal = {Studia Mathematica},
keywords = {weak topologies; compactness; countable compactness; quasi-normal structure; convexity structures; convexity structure},
language = {eng},
number = {3},
pages = {243-250},
title = {Compactness and countable compactness in weak topologies},
url = {http://eudml.org/doc/216151},
volume = {112},
year = {1995},
}

TY - JOUR
AU - Kirk, W.
TI - Compactness and countable compactness in weak topologies
JO - Studia Mathematica
PY - 1995
VL - 112
IS - 3
SP - 243
EP - 250
AB - 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.
LA - eng
KW - weak topologies; compactness; countable compactness; quasi-normal structure; convexity structures; convexity structure
UR - http://eudml.org/doc/216151
ER -

References

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  1. [1] T. Büber and W. A. Kirk, Constructive aspects of fixed point theory for nonexpansive mappings, to appear. Zbl0844.47031
  2. [2] T. Büber and W. A. Kirk, Convexity structures and the existence of minimal sets, preprint. Zbl0866.54003
  3. [3] H. H. Corson and J. Lindenstrauss, On weakly compact subsets of Banach spaces, Proc. Amer. Math. Soc. 17 (1966), 407-412. Zbl0186.44703
  4. [4] M. M. Day, R. C. James and S. Swaminathan, Normed linear spaces that are uniformly convex in every direction, Canad. J. Math. 23 (1971), 1051-1059. Zbl0215.48202
  5. [5] R. Deville, G. Godefroy and V. Zizler, Smoothness and Renormings in Banach Spaces, Pitman Monographs Surveys Pure Appl. Math. 63, Longman, Essex, 1993. Zbl0782.46019
  6. [6] D. van Dulst, Equivalent norms and the fixed point property for nonexpansive mappings, J. London Math. Soc. (2) 25 (1982), 139-144. Zbl0453.46017
  7. [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. Zbl0674.46010
  8. [8] G. Godefroy and N. Kalton, The ball topology and its applications, ibid., 195-237. 
  9. [9] K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge Univ. Press, Cambridge, 1990. Zbl0708.47031
  10. [10] J. L. Kelley, General Topology, van Nostrand, Princeton, 1955. 
  11. [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. Zbl0611.46018
  12. [12] M. A. Khamsi, On metric spaces with uniform normal structure, Proc. Amer. Math. Soc. 106 (1989), 723-726. Zbl0671.47052
  13. [13] M. A. Khamsi and D. Misane, Compactness of convexity structures in metric spaces, to appear. 
  14. [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. [15] W. A. Kirk, Nonexpansive mappings in metric and Banach spaces, Rend. Sem. Mat. Fis. Milano 61 (1981), 133-144. Zbl0519.54029
  16. [16] J. P. Penot, Fixed point theorems without convexity, Bull. Soc. Math. France Mém. 60 (1979), 129-152. Zbl0454.47044
  17. [17] P. Soardi, Struttura quasi normale e teoremi di punto unito, Rend. Istit. Mat. Univ. Trieste 4 (1972), 105-114. Zbl0246.47066
  18. [18] S. Troyanski, On locally uniformly convex and differentiable norms in certain nonseparable Banach spaces, Studia Math. 37 (1971), 173-180. Zbl0214.12701
  19. [19] C. S. Wong, Close-to-normal structure and its applications, J. Funct. Anal. 16 (1974), 353-358. Zbl0281.46015
  20. [20] V. Zizler, On some rotundity and smoothness properties of Banach spaces, Dissertationes Math. (Rozprawy Mat.) 87 (1971). Zbl0231.46036

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