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A nonstandard proof of the Eberlein-Smulian theorem

Authors: Stefano Baratella and Siu-Ah Ng
Journal: Proc. Amer. Math. Soc. 131 (2003), 3177-3180
MSC (2000): Primary 46B04; Secondary 46B10, 46B08
Published electronically: January 28, 2003
MathSciNet review: 1992858
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Abstract: The Eberlein-Smulian theorem on the equivalence of weak compactness and the finite intersection property of bounded closed convex sets is given a short elementary proof by applying Abraham Robinson's nonstandard characterization of compactness.

References [Enhancements On Off] (What's this?)

  • 1. N. Bourbaki, Topological Vector Spaces, Chapter 1-5, translated from French by H.G. Eggleston and S. Madan, Elements of Mathematics, Springer, 1987. MR 88g:46002
  • 2. M. M. Day, Normed Linear Spaces, 3rd ed., Springer, Berlin, 1973. MR 49:9588
  • 3. C.W. Henson and L.C. Moore Jr., Nonstandard analysis and the theory of Banach spaces. Nonstandard Analysis-recent developments, Lecture Notes in Mathematics, Springer, 27-112, 1983. MR 85f:46033
  • 4. R. C. James, The Eberlein-Smulian theorem. Functional analysis, 47-49, Narosa, New Delhi, 1998. MR 99m:46035
  • 5. S. Kremp, An elementary proof of the Eberlein-Smulian theorem and the double limit criterion, Arch. Math. 47, no. 1, 66-69, 1986. MR 88a:46007
  • 6. A.E. Hurd & P.A. Loeb, An Introduction to Nonstandard Real Analysis, Academic Press, New York, 1985. MR 87d:03184
  • 7. W.A.J. Luxemburg, A general theory of monads, Applications of model theory to algebra, analysis and probability (Internat. Sympos., Pasadena CA, 1967), 18-86, Holt, Rinehart and Winston, New York, 1969. MR 39:6244
  • 8. R.E. Megginson, An Introduction to Banach Space Theory, Springer, 1998. MR 99k:46002
  • 9. H. H. Schaefer, Topological Vector Spaces, Springer, Berlin, 1971. MR 49:7722

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Additional Information

Stefano Baratella
Affiliation: Dipartimento di Matematica, Università di Trento, I-38050 Povo (TN), Italy

Siu-Ah Ng
Affiliation: School of Mathematics, Statistics and Information Technology, University of Natal, Pietermaritzburg, 3209 South Africa

Keywords: Nonstandard analysis, Eberlein-\v Smulian theorem, weak compactness
Received by editor(s): September 11, 2001
Received by editor(s) in revised form: May 13, 2002
Published electronically: January 28, 2003
Additional Notes: The authors were supported by South African NRF 2039556 and Progetto Tematico GNSAGA “Teoria dei Modelli ed Applicazioni”
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2003 American Mathematical Society

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