TITLE: Banach spaces with no reflexive complemented subspaces
QUESTION [7 upvotes]: If $X$ is a Banach space with the Dunford Pettis Property (DPP), then no infinite reflexive subspace can be complemented. Suppose now that the Banach space has the property, that no infinite reflexive subspace is complemented. Is it true that $X$ has DPP?

REPLY [5 votes]: No. The Schreier space $S$ is $c_0$-saturated (every closed infinite-dimensional subspace of $S$ contains a copy of $c_0$) as it embeds into $C[0,\omega^\omega]$ which has this property, yet if fails the Dunford-Pettis Property (you will find an easy proof here (p. 168)).