TITLE: Are there non-reflexive vector spaces isomorphic to their bi-dual?
QUESTION [32 upvotes]: Let $V$ be an infinite dimensional topological vector space and consider the natural application $\iota\colon V\to V^{**}$. The space $V$ is said to be reflexive if $\iota$ is an isomorphism.
Are there examples where $\iota$ fails to be an isomorphism but $V$ and $V^{**}$ are nevertheless isomorphic?
Can one find an example where $V$ is a Banach space and the isomorphism is actually an isometry?

REPLY [4 votes]: James' construction can be iterated, in order to produce a countable family of pairwise non-isomorphic Banach spaces with the same property.
Look at the following paper:
Marek Wójtowicz - "Finitely Nonreflexive Banach Spaces"
Proceedings of the American Mathematical Society
Vol. 106, No. 4 (Aug., 1989), pp. 961-965. 
EDIT. The family $Z_n$ of Banach spaces constructed by Wójtowicz has the following properties:

$Z_n$ is isomorphic to $Z_n^{**}$;
$Z_n$ is $n$-reflexive;
if $n < m$ then $Z_n$ is not isomorphic in $Z_m$ and $1$-complemented in $Z_m$.