Hostname: page-component-f554764f5-wjqwx Total loading time: 0 Render date: 2025-04-21T17:42:16.836Z Has data issue: false hasContentIssue false
  • English
  • Français

AN ELEMENTARY PROOF OF JAMES’ CHARACTERISATION OF WEAK COMPACTNESS. II

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

Published online by Cambridge University Press:  26 September 2016

WARREN B. MOORS  and
SAMUEL J. WHITE
WARREN B. MOORS*
Affiliation:
Department of Mathematics, University of Auckland, Private Bag 92019 Auckland, New Zealand email moors@math.auckland.ac.nz
SAMUEL J. WHITE
Affiliation:
Department of Mathematics, University of Auckland, Private Bag 92019 Auckland, New Zealand email sjwhite93@gmail.com
*
✉ moors@math.auckland.ac.nz
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.

In this paper we provide an elementary proof of James’ characterisation of weak compactness for Banach spaces whose dual ball is weak∗ sequentially compact.

Keywords

weakly compact setsJames’ theoremboundaries

MSC classification

Primary: 46B20: Geometry and structure of normed linear spaces
Secondary: 46B22: Radon-Nikodým, Kreĭn-Milman and related properties
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 95 , Issue 1 , February 2017 , pp. 133 - 137
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

References

Deville, R., Godefroy, G. and Zizler, V., Smoothness and Renormings in Banach Spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics, 64 (Longman Scientific and Technical, Harlow, 1993), (copublished in the United States with John Wiley, New York).Google Scholar
Hagler, J. and Sullivan, F., ‘Smoothness and weak* sequential compactness’, Proc. Amer. Math. Soc. 78 (1980), 497–503.Google Scholar
James, R. C., ‘Weakly compact sets’, Trans. Amer. Math. Soc. 113 (1964), 129–140.Google Scholar
Larman, D. and Phelps, R. R., ‘Gateaux differentiability of convex functions on Banach spaces’, J. Lond. Math. Soc. (2) 20 (1979), 115–127.Google Scholar
Moors, W. B., ‘An elementary proof of James’ characterisation of weak compactness’, Bull. Aust. Math. Soc. 84 (2011), 98–102.Google Scholar
Pryce, J. D., ‘Weak compactness in locally convex spaces’, Proc. Amer. Math. Soc. 17 (1966), 148–155.Google Scholar