Weak sequential completeness It is obvious that reflexive spaces are weakly sequentially complete. Can we have a kind of a converse to this fact? Is there a non-reflexive Banach space $X$ such that both $X$ and $X^*$ are weakly sequentially complete? Note that $X$ cannot be a Banach lattice.

On page 12 of this paper, it's stated that the Bourgain-Delbaen space (citation in the linked paper) is an infinite dimensional Schur space whose dual is weakly sequentially complete. Schur spaces are weakly sequentially complete and non-reflexive if infinite dimensional (see here for a proof of these facts).

(It should be noted that by Rosenthal's $\ell_1$-Theorem, a weakly sequentially complete Banach space is either reflexive or contains a copy of $\ell_1$.)