Is there an infinitedimensional 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?
1 Answer
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I believe the James space is an example. It is isomorphic to its double dual (but not by the canonical embedding).

1$\begingroup$ Yes, James space has $X^{**}/X$ of dimension $1$. For example en.wikipedia.org/wiki/James%27_space Also mathoverflow.net/a/43987/454 $\endgroup$ Feb 8, 2015 at 13:40

2$\begingroup$ Also, for each n James has constructed a space X whose nth dual is the first nonseparable space. $\endgroup$ Feb 9, 2015 at 15:08