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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Covering a Banach space


Authors: Vladimir P. Fonf and Clemente Zanco
Journal: Proc. Amer. Math. Soc. 134 (2006), 2607-2611
MSC (2000): Primary 46B20; Secondary 54E52
DOI: https://doi.org/10.1090/S0002-9939-06-08254-2
Published electronically: February 17, 2006
MathSciNet review: 2213739
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Abstract: A well-known theorem by H. Corson states that if a Banach space admits a locally finite covering by bounded closed convex subsets, then it contains no infinite-dimensional reflexive subspace. We strengthen this result proving that if an infinite-dimensional Banach space admits a locally finite covering by bounded $ w$-closed subsets, then it is $ c_0$-saturated, thus answering a question posed by V. Klee concerning locally finite coverings of $ l_1$ spaces. Moreover, we provide information about massiveness of the set of singular points in (PC) spaces.


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Additional Information

Vladimir P. Fonf
Affiliation: Department of Mathematics, Ben-Gurion University of the Negev, 84105 Beer-Sheva, Israel
Email: fonf@math.bgu.ac.il

Clemente Zanco
Affiliation: Dipartimento di Matematica, Università degli Studi, via C. Saldini 50, 20133 Milano MI, Italy
Email: zanco@mat.unimi.it

DOI: https://doi.org/10.1090/S0002-9939-06-08254-2
Keywords: Covering, locally finite covering, space $c_0$, (PC) property
Received by editor(s): October 20, 2004
Received by editor(s) in revised form: March 22, 2005
Published electronically: February 17, 2006
Additional Notes: The first author was supported in part by Israel Science Foundation, Grant #139/03.
The second author was supported in part by the Ministero dell’Università e della Ricerca Scientifica e Tecnologica of Italy
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2006 American Mathematical Society