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 $X^{n+1}$?



Separable bidual but nonseparable third dual
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. 


