  If a separable Banach space $X$ is such that for some nonquasireflexive
Banach space $Y$ there exists a surjective strictly singular operator $T:X\to
Y$ then for every countable ordinal $\alpha $ the dual of $X$ contains a
subspace whose weak$^*$ sequential closures of orders less than $\alpha $ are
not norming over any infinite-dimensional subspace of $X$ and whose weak$^*$
sequential closure of order $\alpha +1$ coincides with $X^*$
