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 | References | Similar Articles | Additional Information

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 -closed subsets, then it is -saturated, thus answering a question posed by V. Klee concerning locally finite coverings of 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