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On perfectly homogeneous bases in quasi-Banach spaces. (English) Zbl 1184.46003

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 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.

MSC:

46A16 Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.)
46B10 Duality and reflexivity in normed linear and Banach spaces

Citations:

Zbl 0148.11202
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References:

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