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

Full-text PDF Free Access

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. Chapters 1–5*, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1987. Translated from the French by H. G. Eggleston and S. Madan. MR**910295****2.**Mahlon M. Day,*Normed linear spaces*, 3rd ed., Springer-Verlag, New York-Heidelberg, 1973. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 21. MR**0344849****3.**C. Ward Henson and L. C. Moore Jr.,*Nonstandard analysis and the theory of Banach spaces*, Nonstandard analysis—recent developments (Victoria, B.C., 1980) Lecture Notes in Math., vol. 983, Springer, Berlin, 1983, pp. 27–112. MR**698954**, 10.1007/BFb0065334**4.**Robert C. James,*The Eberlein-Šmulian theorem*, Functional analysis, Narosa, New Delhi, 1998, pp. 47–49. MR**1668790****5.**Stefan Kremp,*An elementary proof of the Eberlein-Šmulian theorem and the double limit criterion*, Arch. Math. (Basel)**47**(1986), no. 1, 66–69. MR**855139**, 10.1007/BF01202501**6.**Albert E. Hurd and Peter A. Loeb,*An introduction to nonstandard real analysis*, Pure and Applied Mathematics, vol. 118, Academic Press, Inc., Orlando, FL, 1985. MR**806135****7.**W. A. J. Luxemburg,*A general theory of monads*, Applications of Model Theory to Algebra, Analysis, and Probability (Inte rnat. Sympos., Pasadena, Calif., 1967) Holt, Rinehart and Winston, New York, 1969, pp. 18–86. MR**0244931****8.**Robert E. Megginson,*An introduction to Banach space theory*, Graduate Texts in Mathematics, vol. 183, Springer-Verlag, New York, 1998. MR**1650235****9.**Helmut H. Schaefer,*Topological vector spaces*, Springer-Verlag, New York-Berlin, 1971. Third printing corrected; Graduate Texts in Mathematics, Vol. 3. MR**0342978**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
46B04,
46B10,
46B08

Retrieve articles in all journals with MSC (2000): 46B04, 46B10, 46B08

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:
http://dx.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