On perfectly homogeneous bases in quasi-Banach spaces (Q1035082)

Summary: 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 \textit{M. Zippin} [Isr. J. Math. 4, 265--272 (1966; Zbl 0148.11202)] proved that perfectly homogeneous bases are equivalent to either the canonical \(c_{0}\)-basis or the canonical \(\ell _{p}\)-basis for some \(1\leq p<\infty \). 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.