Abstract:It is known that a Banach lattice with order continuous norm contains a copy of $\ell _1$ if and only if it contains a lattice copy of $\ell _1$. The purpose of this note is to present a more direct proof of this useful fact, which extends a similar theorem due to R.C. James for Banach spaces with unconditional bases, and complements the $c_0$- and $\ell _{\infty }$-cases considered by Lozanovskii, Mekler and Meyer-Nieberg.
Keywords: Banach lattice, order continuous norm, embedding of $\ell _1$
AMS Subject Classification: 46B42, 46B45