On Perfectly Homogeneous Bases in Quasi-Banach Spaces

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.