TITLE: Non-reflexive Banach space s.t. X,X*,X**,... are separable
QUESTION [5 upvotes]: 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?

REPLY [7 votes]: I believe the James space is an example.  It is isomorphic to its double dual (but not by the canonical embedding).