In functional analysis, there is a concept of a self dual space (Hilbert) and a self double dual space (reflexive). I am curious as to whether a generalization exists or not (and if it exists, whether or not it is actually useful). Say a space is not reflexive, but if we keep on taking the dual many times, say $X$, $X^*$, $X^{**}$, $X^{***}$, $X^{****}$, etc, do we eventually get a cycle, i.e. the $n$th dual is isomorphic to the $m$th dual for some $m<n$?

Such an answer or a theory of the structure of the chain of duals would be a theory of the dual as a functor in the cateogry of Banach spaces, e.g. in the category of Hilbert space, the dual would be a contravariant automorphism.