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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 $X^{n+1}$?

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    $\begingroup$ Why do not you accept the answer? $\endgroup$
    – August Cleaner
    Commented Nov 19, 2017 at 18:49

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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.

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    $\begingroup$ Is $Y^{***}$ nonseparable easy? $\endgroup$
    – Nik Weaver
    Commented Nov 12, 2017 at 19:02
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    $\begingroup$ @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?) $\endgroup$
    – Philip Brooker
    Commented Nov 13, 2017 at 10:31
  • $\begingroup$ Nice! Thank you. $\endgroup$
    – Nik Weaver
    Commented Nov 13, 2017 at 12:27

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