Weak $*$ derived sets of sets of linear functionals

For a Banach space $X$ the $w^*$–sequential closure operator in the adjoint space is, in general, not the topological closure operator. That is, it may happen that the $w^*$–sequential closure of a subspace $\Gamma$ of $X^*$ is not $w^*$–sequentially closed. The possible length of the chain of repeated $w^*$–sequential closures of a subspace of $X^*$ in dependence on the dimension of $X^{**}/X$ is investigated.