On Perfectly Homogeneous Bases in Quasi-Banach Spaces

Abstract

For $0<p<\infty$ the unit vector basis of ${\ell }_p$ 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 ${\ell }_p$-basis for some $1\le p<\infty$. In this note, we show that (a relaxed form of) perfect homogeneity characterizes the unit vector bases of ${\ell }_p$ for $0<p<1$ as well amongst bases in nonlocally convex quasi-Banach spaces.