Hostname: page-component-77c89778f8-5wvtr Total loading time: 0 Render date: 2024-07-16T21:44:32.388Z Has data issue: false hasContentIssue false
  • English
  • Français

A Banach Space in Which a Ball is Contained in the Range of Some Countably Additive Measure is Superreflexive

Published online by Cambridge University Press:  20 November 2018

Yeneng Sun
Yeneng Sun*
Affiliation:
Department of Mathematics, University of Illinois, Urbana, Illinois 61801 USA
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A nonstandard proof of the fact that a Banach space in which a ball is contained in the range of a countably additive measure is superreflexive is given. The proof is an application of a general method in which we first transfer certain standard objects to the nonstandard hull of a Banach space and then, using the quite well developed theory of nonstandard hulls, derive results about the objects in the original Banach space. It also provides us with an example of the applications of the theory of nonstandard hull valued measures.

Keywords

28E0546G1003H0546B20
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 33 , Issue 1 , 01 March 1990 , pp. 45 - 49
Copyright
Copyright © Canadian Mathematical Society 1990

References

1. Bretagnolle, J., Dacunha-Castelle, D. and Krivine, J. L., Lois stables et espaces LP , Ann. Inst. H. Poincare Sect. B (N.S.) 2 (1965/66), 231-259.Google Scholar
2. Diestel, J. and Seifert, C. J., An averaging property of the range of a vector measure, Bull. A.M.S. 82 (1976), 907–909.Google Scholar
3. Diestel, J. and Uhl, J. J., “Vector Measures” Mathematical Surveys Number 15, A.M.S. Providence, Rhode Island, 1977.Google Scholar
4. Henson, C. W. and Moore, L. C., Nonstandard analysis and theory of Banach spaces. In “Nonstandard Analysis - Recent Developments”, Lect. Notes in Math., Vol. 983, Springer-Verlag, Berlin, 1983.Google Scholar
5. Hurd, A. E. and Loeb, P. A., “A« Introduction to Nonstandard Real Analysis”, Academic Press, Orlando, Florida, 1985.Google Scholar
6. James, R. C., Characterizations of reflexivity, Studia Math. 23 (1963/64), 205-216.Google Scholar
7. Kaczmarz, S. and Steinhaus, H., Théorie der Orthogonalreihen; reprint, Chelsea, New York, 1951.Google Scholar
8. Loeb, P. A. and Osswald, H., Nonstandard integration theory in solid Riesz spaces, in preparation.Google Scholar
9. Maurey, B., Théorèmes de factorization pour les operateurs linéaires a valeurs dans un espace Lp, Astérisque 11 (1974).Google Scholar
10. Pisier, G., Factorization of linear operators and geometry of Banach spaces, C.B.M.S. 60 (1986).Google Scholar
11. Osswald, H. and Sun, Y. N., On the extensions of vector valued Loeb measures, submitted to Proc. A.M.S.Google Scholar
12. Rosenthal, H. P., On subspaces of If, Ann. of Math. 97 (1973), 344–373.Google Scholar
13. Sun, Y. N., On the theory of vector valued Loeb measures and integration, in preparation.Google Scholar
14. Sun, Y. N., A nonstandard proof of the Riesz representation theorem for weakly compact operators on C(Q), Math. Proc. Camb. Phil. Soc. (to appear).Google Scholar
15. Van Dulst, D., “Reflexive and Superreflexive Banach Spaces”, Mathematical Centre Tracts 102, Amsterdam, 1978.Google Scholar