A remark on bases in quotients of $ell_p$ when $0 < p < 1$

Abstract

In [9] Stiles showed that if $0 < p < 1, \ell_p$ has an infinite-dimensional closed subspace which contains no complemented copy of $\ell_p$; this contrasts with the well-known result of Pelczynski [5] for $1≤ p < ∈fty$. The following curious theorem is the main result of this note: Theorem 1. Let M be an infinite-dimensional closed subspace of $\ell_p$ where $0 < p < 1$. Suppose $\ell_p/M$ has a basis. Then M contains a subspace isomorphic to $\ell_p$ and complemented in $\ell_p$.