TITLE: Separable bidual but nonseparable third dual
QUESTION [11 upvotes]: Does there exist a Banach space $X$ such that $X^{**}$ is separable but $X^{***}$ is non-separable?
More generally, for every natural $n$ can someone construct an example of Banach space $X$ such that $X^{n}$ is separable but not $X^{n+1}$?

REPLY [10 votes]: Yes to both.  Lindenstrauss extended James' construction to show that for any separable $X$ there is a separable $Y$ s.t. $Y^{**}/Y$ is isometrically isomorphic to $X$. Induct on that.  Spaces built that way are called James-Lindenstrauss spaces. Another proof is contained in my "Factoring Weakly Compact Operator" paper with Davis, Figiel and Pelczynski.