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

DOI:
https://doi.org/10.1090/S0002-9939-03-06894-1

Published electronically:
January 28, 2003

MathSciNet review:
1992858

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Abstract | References | Similar Articles | Additional Information

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.

**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

Email:
baratell@science.unitn.it

**Siu-Ah Ng**

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

Email:
ngs@nu.ac.za

DOI:
https://doi.org/10.1090/S0002-9939-03-06894-1

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