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A nonstandard proof of the Eberlein-Smulian theorem
Author(s):
Stefano
Baratella;
Siu-Ah
Ng
Journal:
Proc. Amer. Math. Soc.
131
(2003),
3177-3180.
MSC (2000):
Primary 46B04;
Secondary 46B10, 46B08
Posted:
January 28, 2003
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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:
-
- 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:
10.1090/S0002-9939-03-06894-1
PII:
S 0002-9939(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
Posted:
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
Copyright of article:
Copyright
2003,
American Mathematical Society
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