Let $X$ be a Banach space with an unconditional basis $(x\_{n})\_{n}$.


Question. If $X$ contains a subspace isomorphic to $l\_{1}$, does $(x\_{n})\_{n}$ admit a block basic sequence equivalent to the unit vector basis of $l\_{1}$ ?


I do not know whether the question has already existed as a known result. But a self-contained proof is preferred.


Thank you.



$l_{1}$-block basic sequences in Banach spaces with an unconditional basis
Yes. This result of R. C. James can be found in standard references. See, for example, Theorem 3.3.1 in the book by Albiac and Kalton.


