Copyright © 2009 F. Albiac and C. Leránoz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

For 0<p<∞ the unit vector basis of ℓp 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 c0-basis or the canonical ℓp-basis for some 1≤p<∞. In this note, we show that (a relaxed form of) perfect homogeneity characterizes the unit vector bases of ℓp for 0<p<1 as well amongst bases in nonlocally convex quasi-Banach spaces.