Abstract
We consider the notion of smallness and its applications to the characterization of isomorphically polyhedral Banach spaces and the existence of different coverings of either the unit sphere or the unit ball of a Banach space.
Similar content being viewed by others
References
Appell J.: Insiemi ed operatori “piccoli” in Analisi funzionale. Rend. Ist. Mat. Univ. Trieste 33, 127–199 (2001)
Ariasde Reyna J.: Hausdorff dimension of Banach spaces. Proc. Edinburgh Math. Soc. 31, 217–229 (1988)
B. Bagchi, G. Misra and N.S.N. Sastry, Coverup revealed in Hilbert space, Problem 6577 in Amer. Math. Monthly 97 (1990), 436–437.
Bandt C.: Spaces of largest Hausdorff dimension. Mathematika 28, 206–210 (1981)
Behrends E.: The small ball property in Banach spaces – Quantitative results. Rend. Circolo Mat. Palermo (II) Suppl. 76, 3–16 (2005)
E. Behrends, Acknowledgement of priority (previous paper), Rend. Circolo Mat. Palermo (II) 55 (2006), 449.
Behrends E., Kadets V. M.: Metric spaces with the small ball property. Studia Math. 148, 275–287 (2001)
M. Borkowski, D. Bugajewski and H. Przybycień, Some remarks on hyperconvex metric spaces and the small ball property, Function spaces, World Sci. Publ., River Edge, NJ, 2003, 90–99.
Bourgain J., Delbaen F.: Quotient maps onto C(K). Bull. Soc. Math. Belg. 30, 111–119 (1978)
Connett J.: On covering the unit ball in a Banach space. J. London Math. Soc. 7, 291–294 (1973)
J. Diestel, Sequences and series in Banach spaces, GTM 92, Springer.
C. Donnini and A. Martellotti, The small ball drop property, Australian J. Math. Anal. Appl. 5 (2008), art.8.
Duda R., Telgárski R.: On some covering properties of metric spaces. Czech. Math. J. 18((93), 66–82 (1968)
Dunkl F., Williams K. S.: A simple norm inequality. Amer. Math. Monthly 71, 53–54 (1964)
Elton J.: Extremely weakly unconditionally convergent series. Israel J. Math. 40, 255–258 (1981)
Fonf V.P.: Massiveness of the set of extreme points of the dual ball of a Banach spaces and polyhedral spaces. Funct. Anal. Appl. 12, 237–239 (1978)
Fonf V.P.: On the boundary of a polyhedral Banach space. Extracta Math. 15, 145–154 (2000)
Fonf V.P.: Polyhedral Banach spaces. Math. Notes Acad. Sci. USSR 30, 809–813 (1981)
Fonf V., Vesely L.: Infinite dimensional polyhedrality. Canad. J. Math. 56, 472–494 (2004)
Fonf V., Zanco C.: Almost–flat locally finite covers of the sphere. Positivity 6, 269–281 (2004)
Frericks L., Peris A.: Small ball properties for Fréchet spaces/. RACSAM 97, 257–261 (2003)
Furi M., Vignoli A.: On a property of the unit sphere in a linear normed space.Bull. Acad. Polon. Sci. Ser. Math. 18, 333–334 (1970)
P.R. Goodey, Generalized Hausdorff dimension, Mathematika 17 (1970), 324–327.
S. Heinrich. Ultraproducts in Banach space theory. J. Reine Angew. Math. 313 (1980), 72-104.
V. Kadets, V. Shepelska and D. Werner, Thickness of the unit sphere, ℓ 1–types and the almost Daugavet’s property, Houston J. Math. 37, (2011), 867–878.
W. K¨orner, Some covering theorems for infinite dimensional Banach spaces, J. London Math. Soc. 2 (1970), 643–646.
R. Livni, On extreme points of the dual ball of a polyhedral space, Extracta Math. 24 (2009), 243–249.
L. Lyusternik and L. Š Snirel’man, Topological methods in variational problems and their application to the differential geometry of surfaces. (Russian) Uspehi Matem. Nauk (N.S.) 2, (1947), no. 1(17), 166-217.
E. Maluta and P.L. Papini, Relative centers and finite nets for the unit ball and its finite subsets, Boll. Un. Mat. Ital. (7) 7–B (1993), 451–472
Papini P.L.: Covering the sphere and the ball in Banach spaces. Communic. Appl. Anal. 13, 579–586 (2009)
B. Sims. “Ultra;;–techniques in Banach space theory. Queen’s Papers in Pure and Applied Mathematics, 60. Kingston, ON, 1982.
L. Veselý, Boundary of polyhedral spaces: an alternative proof, Extracta Math. 15 (2000), 213-217.
Whitley R.: The size of the unit sphere. Canad. J. Math. 20, 450–455 (1968)
Author information
Authors and Affiliations
Corresponding author
Additional information
The research of the first author was realized during visits to the University of Bologna, supported in part by the project MTM2010–20190–C02–01, Junta de Extremadura (Spain) and the program Junta de Extremadura GR10113 IV Plan Regional I+D+i, Ayudas a Grupos de Investigaci´on. The topic of this research was presented during a lecture at the Mathematics Department of the University of Milano. The authors thank Prof. Zanco for his warm welcome.
Rights and permissions
About this article
Cite this article
Castillo, J.M.F., Papini, P.L. Smallness and the Covering of a Banach Space. Milan J. Math. 80, 251–263 (2012). https://doi.org/10.1007/s00032-012-0188-5
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00032-012-0188-5