Suppose $X$ is a reflexive space (possibly non-separable) which is not super-reflexive. Then (by definition) there exists a non-reflexive Banach space $Y$ which is non-reflexive 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. non-reflexive) which is separable? In this spirit, what are examples of reflexive but not super-reflexive spaces in which neither $\ell\_1$ nor $c\_0$ is finitely representable?



Non-super reflexive space
The first question is easy: Every non reflexive space has a separable non reflexive subspace (e.g. by the Eberlein-Smulian 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 proof--you can find their paper by using MathSciNet. Their approach is more conceptual and uses interpolation theory but is not easy.


