Abstract

We study Banach spaces having smooth faces in their unit ball. In particular, we show that if the unit ball of a finite dimensional Banach space has an infinite number of smooth faces then their interiors relative to the unit sphere approach the empty set in a certain way. We also show that this situation does not hold in infinite dimensions since we prove that every infinite dimensional Banach space can be equivalently renormed to have infinitely many smooth faces with interior relative to the unit sphere of the same "size". This fact characterizes having infinite algebraic dimension.

Author information

Contact details are reproduced from the original publication and may be historical.

Francisco J. García-Pacheco

Dept. of Mathematical Sciences, Kent State University, Kent, OH 44242, U.S.A.

fgarcia@math.kent.edu

Keywords

  • Smooth face
  • interior relative to the unit sphere
  • Hausdorff metric

Mathematics Subject Classification

46B20, 46B07, 46A35

References

  1. M. D. Acosta, A. Aizpuru, R. M. Aron, F. J. García-Pacheco: Functionals that do not attain their norm, Bull. Belg. Math. Soc. Simon Stevin 14 (2007) 407–418.https://doi.org/10.36045/bbms/1190994202Google Scholar
  2. A. Aizpuru, F. J. García–Pacheco: Some questions about rotundity and renormings in Banach spaces, J. Austral. Math. Soc. 79 (2005) 131–140.https://doi.org/10.1017/S144678870000937XGoogle Scholar
  3. G. Beer: Topologies on Closed and Closed and Convex Sets, Mathematics and its Applications 268, Kluwer Academic Publishers, Dordrecht (1993).https://doi.org/10.1007/978-94-015-8149-3Google Scholar
  4. F. J. García-Pacheco, A. Miralles: Real renormings on complex Banach spaces, Chinese Ann. Math., Ser. B., to appear.Google Scholar

Suggested citation

F. J. García-Pacheco. “Banach Spaces with an Infinite Number of Smooth Faces in their Unit Ball.” Journal of Convex Analysis 15 (2008), No. 2, 215–218. https://doi.org/10.68381/jca15015

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2008. Rights now held by Banach Press LLC.