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MEASURES OF NONCOMPACTNESS IN ULTRAPRODUCTS

Part of: Other (nonclassical) types of functional analysis Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  08 June 2009

WIESŁAWA KACZOR ,
ADAM STACHURA ,
JUSTYNA WALCZUK  and
MAGDALENA ZOŁA
WIESŁAWA KACZOR*
Affiliation:
Instytut Matematyki UMCS, 20-031 Lublin, Poland (email: [email protected])
ADAM STACHURA
Affiliation:
Institute of Mathematics and Computer Science, John Paul II Catholic University of Lublin, 20-708 Lublin, Poland (email: [email protected])
JUSTYNA WALCZUK
Affiliation:
Instytut Matematyki PWSZ, 22-400 Zamość, Poland (email: [email protected])
MAGDALENA ZOŁA
Affiliation:
Institute of Mathematics and Computer Science, John Paul II Catholic University of Lublin, 20-708 Lublin, Poland (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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We investigate the connection between measures of noncompactness of a bounded subset of a given Banach space and the corresponding measures of noncompactness of an ultrapower of this subset. The Kuratowski, Hausdorff and separation measures of noncompactness are considered. We prove that in the first two cases the measures of a subset are equal to the respective measures of ultrapowers of this subset. In the case of separation measure of noncompactness, the equality is not necessarily fulfilled.

Keywords

measures of noncompactnessultrafilters and ultraproducts

MSC classification

Secondary: 46B08: Ultraproduct techniques in Banach space theory 46S20: Nonstandard functional analysis
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 80 , Issue 1 , August 2009 , pp. 165 - 172
Copyright
Copyright © Australian Mathematical Society 2009

References

[1] Aksoy, A. G. and Khamsi, M. A., Nonstandard Methods in Fixed Point Theory (Springer, New York, 1990).CrossRefGoogle Scholar
[2] Ayerbe Toledano, J. M., Dominguez Benavides, T. and López Acedo, G., Measures of Noncompactness in Metric Fixed Point Theory (Birkhäuser, Basel, 1997).CrossRefGoogle Scholar
[3] Goebel, K. and Kirk, W. A., Zagadnienia metrycznej teorii punktów stałych (Wydawnictwo UMCS, Lublin, 1999).Google Scholar
[4] Prus, S., Personal communication.Google Scholar
[5] Sims, B., ‘Ultra’-techniques in Banach Space Theory, Queens Papers in Pure and Applied Mathematics, 60 (Queen’s University, Kingston, ON, 1982).Google Scholar