Abstract and Applied Analysis

On Perfectly Homogeneous Bases in Quasi-Banach Spaces

F. Albiac and C. Leránoz

Full-text: Open access


For 0 < p < the unit vector basis of p has the property of perfect homogeneity: it is equivalent to all its normalized block basic sequences, that is, perfectly homogeneous bases are a special case of symmetric bases. For Banach spaces, a classical result of Zippin (1966) proved that perfectly homogeneous bases are equivalent to either the canonical c 0 -basis or the canonical p -basis for some 1 p < . In this note, we show that (a relaxed form of) perfect homogeneity characterizes the unit vector bases of p for 0 < p < 1 as well amongst bases in nonlocally convex quasi-Banach spaces.

Article information

Abstr. Appl. Anal., Volume 2009 (2009), Article ID 865371, 7 pages.

First available in Project Euclid: 16 March 2010

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Albiac, F.; Leránoz, C. On Perfectly Homogeneous Bases in Quasi-Banach Spaces. Abstr. Appl. Anal. 2009 (2009), Article ID 865371, 7 pages. doi:10.1155/2009/865371. https://projecteuclid.org/euclid.aaa/1268745577

Export citation


  • N. J. Kalton, N. T. Peck, and J. W. Rogers, An F-Space Sampler, vol. 89 of London Mathematical Society Lecture Note, Cambridge University Press, Cambridge, UK, 1985.
  • S. Rolewicz, Metric Linear Spaces, vol. 20 of Mathematics and Its Applications (East European Series), D. Reidel, Dordrecht, The Netherlands, 2nd edition, 1985.
  • N. J. Kalton, ``Quasi-Banach spaces,'' in Handbook of the Geometry of Banach Spaces, Vol. 2, pp. 1099--1130, North-Holland, Amsterdam, The Netherlands, 2003.
  • T. Aoki, ``Locally bounded linear topological spaces,'' Proceedings of the Imperial Academy, Tokyo, vol. 18, pp. 588--594, 1942.
  • S. Rolewicz, ``On a certain class of linear metric spaces,'' Bulletin de L'Académie Polonaise des Sciences, vol. 5, pp. 471--473, 1957.
  • N. J. Kalton, ``Orlicz sequence spaces without local convexity,'' Mathematical Proceedings of the Cambridge Philosophical Society, vol. 81, no. 2, pp. 253--277, 1977.
  • J. Lindenstrauss and L. Tzafriri, ``On Orlicz sequence spaces,'' Israel Journal of Mathematics, vol. 10, pp. 379--390, 1971.
  • F. Albiac and C. Leránoz, ``Uniqueness of symmetric basis in quasi-Banach spaces,'' Journal of Mathematical Analysis and Applications, vol. 348, no. 1, pp. 51--54, 2008.
  • F. Albiac and C. Leránoz, ``Uniqueness of unconditional basis in Lorentz sequence spaces,'' Proceedings of the American Mathematical Society, vol. 136, no. 5, pp. 1643--1647, 2008.
  • N. J. Kalton, C. Leránoz, and P. Wojtaszczyk, ``Uniqueness of unconditional bases in quasi-Banach spaces with applications to Hardy spaces,'' Israel Journal of Mathematics, vol. 72, no. 3, pp. 299--311, 1990.
  • F Albiac and N. J. Kalton, Topics in Banach Space Theory, vol. 233 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 2006.
  • J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces. I: Sequence Spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 9, Springer, Berlin, Germany, 1977.
  • M. Zippin, ``On perfectly homogeneous bases in Banach spaces,'' Israel Journal of Mathematics, vol. 4, pp. 265--272, 1966.