Is there an infinite-dimensional Banach space $X$, which is not reflexive, such that all the spaces $X,X^{\ast},X^{\ast\ast}, X^{\ast\ast\ast},\dots$ are separable?



Non-reflexive Banach space s.t. X,X*,X**,... are separable
I believe the James space is an example. It is isomorphic to its double dual (but not by the canonical embedding).


