On certain subspaces of $${\ell _p}$$ for $${0<p\le 1}$$ and their applications to conditional quasi-greedy bases in p-Banach spaces

Abstract

We construct for each $$0<p\le 1$$ an infinite collection of subspaces of $$\ell _p$$ that extend the example of Lindenstrauss (Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 12, 539–542, 1964) of a subspace of $$\ell _{1}$$ with no unconditional basis. The structure of this new class of p-Banach spaces is analyzed and some applications to the general theory of $${\mathcal {L}}_{p}$$ -spaces for $$0<p<1$$ are provided. The introduction of these spaces serves the purpose to develop the theory of conditional quasi-greedy bases in p-Banach spaces for $$p<1$$ . Among the topics we consider are the existence of infinitely many conditional quasi-greedy bases in the spaces $$\ell _{p}$$ for $$p\le 1$$ and the careful examination of the conditionality constants of the “natural basis” of these spaces.