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AN ELEMENTARY PROOF OF JAMES’ CHARACTERIZATION OF WEAK COMPACTNESS

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

Published online by Cambridge University Press:  03 June 2011

WARREN B. MOORS
WARREN B. MOORS*
Affiliation:
Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland, New Zealand (email: [email protected])
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Abstract

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In this paper we provide an elementary proof of James’ characterization of weak compactness in separable Banach spaces. The proof of the theorem does not rely upon either Simons’ inequality or any integral representation theorems. In fact the proof only requires the Krein–Milman theorem, Milman’s theorem and the Bishop–Phelps theorem.

Keywords

compact convex setsboundariesRainwater’s theoremKrein–Milman theorem

MSC classification

Secondary: 46B20: Geometry and structure of normed linear spaces
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 84 , Issue 1 , August 2011 , pp. 98 - 102
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

References

[1]Arkhangel’skii, A. V., Topological Function Space, Mathematics and its Applications (Soviet Series), 78 (Kluwer Academic Publishers Group, Dordrecht, 1992).CrossRefGoogle Scholar
[2]Bishop, E. and Phelps, R., The Support Functionals of a Convex Set, Proceedings of Symposia in Pure Mathematics, VII (American Mathematical Society, Providence, RI, 1963), pp. 27–35.Google Scholar
[3]Fonf, V. and Lindenstrauss, J., ‘Boundaries and generation of convex sets’, Israel J. Math. 136 (2003), 157–172.Google Scholar
[4]Fonf, V., Lindenstrauss, J. and Phelps, R., ‘Infinite dimensional convexity’, in: Handbook of the Geometry of Banach Spaces, Vol. I (North-Holland, Amsterdam, 2001), pp. 599–670.Google Scholar
[5]Godefroy, G., ‘Boundaries of convex sets and interpolation sets’, Math. Ann. 277 (1987), 173–184.Google Scholar
[6]James, R. C., ‘Weakly compact sets’, Trans. Amer. Math. Soc. 113 (1964), 129–140.CrossRefGoogle Scholar
[7]Kalenda, O., ‘(I)-envelopes of the unit balls and James’ characterisation of reflexivity’, Studia Math. 182 (2007), 29–40.CrossRefGoogle Scholar
[8]Krein, M. and Milman, D., ‘On the extreme points of regular convex sets’, Studia Math. 9 (1940), 133–138.Google Scholar
[9]Milman, D., ‘Characteristics of extremal points or regularly convex sets’, Dokl. Akad. Nauk SSSR (N.S.) 57 (1947), 119–122.Google Scholar
[10]Moors, W. B., ‘A characterisation of weak compactness in Banach spaces’, Bull. Aust. Math. Soc. 55 (1997), 497–501.Google Scholar
[11]Moors, W. B. and Reznichenko, E. A., ‘Separable subspaces of affine function spaces on convex compact set’, Topology Appl. 155 (2008), 1306–1322.Google Scholar
[12]Moors, W. B. and Spurný, J., ‘On the topology of pointwise convergence on the boundaries of L 1-preduals’, Proc. Amer. Math. Soc. 137 (2009), 1421–1429.CrossRefGoogle Scholar
[13]Rainwater, J., ‘Weak convergence of bounded sequences’, Proc. Amer. Math. Soc. 14 (1963), 999.Google Scholar
[14]Simons, S., ‘A convergence theorem with boundary’, Pacific J. Math. 40 (1972), 703–708.Google Scholar