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

Full-text PDF Free Access

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.

**[C]**H.H. Corson,*Collections of convex sets which cover a Banach space,*Fund. Math.**49**(1961), 143-145. MR**0125430 (23:A2732)****[EW]**G.A. Edgar and R.F. Wheeler,*Topological properties of Banach spaces,*Pacific J. Math.**115**(1984), 317-350. MR**0765190 (86e:46013)****[F1]**V.P. Fonf,*Polyhedral Banach spaces,*Math. Notes USSR**30**(1981), 809-813. MR**0638435 (84j:46018)****[F2]**V.P. Fonf,*Three characterizations of polyhedral Banach spaces,*Ukrainian Math. J.**42**(1990), 1286-1290. MR**1093646 (92e:46028)****[F3]**V.P. Fonf,*Boundedly complete basic sequences, -subspaces and injections of Banach spaces,*Isarel J. Math.**89**(1995), 173-188. MR**1324460 (96a:46021)****[FPZ]**V.P. Fonf, A. Pezzotta and C. Zanco,*Singular points for tilings of normed spaces,*Rocky Mountain J. Math.**30**(2000), 857-868. MR**1797820 (2001k:46020)****[K]**V. Klee,*Dispersed Chebyshev sets and covering by balls,*Math. Ann.**257**(1981), 251-260. MR**0634466 (84e:41036)****[L]**D.H. Leung,*Some isomorphically polyhedral Orlicz sequence spaces,*Israel J. Math.**87**(1994), 117-128. MR**1286820 (95f:46033)****[Z]**C. Zanco,*Even infinite-dimensional Banach spaces can enjoy carpeting and tiling,*Proc. of the 13th Seminar on Analysis and its Applications, Isfahan Univ. Press, Isfahan, 2003. MR**2114509**

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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