Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF

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 $ 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.


References [Enhancements On Off] (What's this?)

  • [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, $ c_0$-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

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 46B20, 54E52

Retrieve articles in all journals with MSC (2000): 46B20, 54E52


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

American Mathematical Society