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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On a difference between quantitative weak sequential completeness and the quantitative Schur property


Authors: O. F. K. Kalenda and J. Spurný
Journal: Proc. Amer. Math. Soc. 140 (2012), 3435-3444
MSC (2010): Primary 46B20, 46B25
Published electronically: February 6, 2012
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Abstract: We study quantitative versions of the Schur property and weak sequential completeness, proceeding with investigations started by G. Godefroy, N. Kalton and D. Li and continued by H. Pfitzner and the authors. We show that the Schur property of $ \ell _1$ holds quantitatively in the strongest possible way and construct an example of a Banach space which is quantitatively weakly sequentially complete, has the Schur property, but fails the quantitative form of the Schur property.


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Additional Information

O. F. K. Kalenda
Affiliation: Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75, Praha 8, Czech Republic
Email: kalenda@karlin.mff.cuni.cz

J. Spurný
Affiliation: Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75, Praha 8, Czech Republic
Email: spurny@karlin.mff.cuni.cz

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11175-X
PII: S 0002-9939(2012)11175-X
Keywords: Weakly sequentially complete Banach space, Schur property, quantitative versions of weak sequential completeness, quantitative versions of the Schur property, $L$-embedded Banach space
Received by editor(s): March 15, 2011
Received by editor(s) in revised form: April 5, 2011
Published electronically: February 6, 2012
Additional Notes: The authors were supported by the Research Project MSM 0021620839 from the Czech Ministry of Education
The first author was additionally supported in part by grant GAAV IAA 100190901
The second author was partly supported by grant GAČR 201/07/0388.
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.