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Uniqueness of unconditional basis in Lorentz sequence spaces. (English) Zbl 1140.46002

Summary: We show that the Lorentz sequence spaces \( d(\omega,p)\) with \( 0<p<1\) and \( \inf\frac{\omega_1+\cdots+\omega_n}{n^p}>0\) have unique unconditional basis. This completely settles the question of uniqueness of the unconditional basis in Lorentz sequence spaces, and solves a problem raised by Popa in 1981 and Nawrocki and Ortyński in 1985.

MSC:

46A16 Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.)
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
46A35 Summability and bases in topological vector spaces
46A40 Ordered topological linear spaces, vector lattices
46A45 Sequence spaces (including Köthe sequence spaces)
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