# Separable bidual but nonseparable third dual

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 $X^{n+1}$?

• Why do not you accept the answer? – August Cleaner Nov 19 '17 at 18:49

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.
• Is $Y^{***}$ nonseparable easy? – Nik Weaver Nov 12 '17 at 19:02
• @NikWeaver: Apply the Lindenstrauss construction with $X=\ell_1$. The lifting property of $\ell_1$ yields a (complemented) subspace of $Y^{\ast\ast}$ that is isomorphic to $\ell_1$ (not sure if this qualifies as easy though?) – Philip Brooker Nov 13 '17 at 10:31