Suppose $X$ is a reflexive space (possibly nonseparable) which is not superreflexive. Then (by definition) there exists a nonreflexive Banach space $Y$ which is nonreflexive but is finitely representable in $X$, meaning that for each $\lambda >1$, every finite dimensional subspace of $Y$ is $\lambda$isomorphic to a subspace of $X$. Can we always find such $Y$ (i.e. nonreflexive) which is separable? In this spirit, what are examples of reflexive but not superreflexive spaces in which neither $\ell_1$ nor $c_0$ is finitely representable?
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The first question is easy: Every non reflexive space has a separable non reflexive subspace (e.g. by the EberleinSmulian theorem or by R. C. James' characterization of non reflexivity). The second question was a longstanding open problem that was solved by James in the 1970s. Pisier and Xu gave another proofyou can find their paper by using MathSciNet. Their approach is more conceptual and uses interpolation theory but is not easy. 

