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