Weak* derived sets

The weak* derived set of a subset A of a dual Banach space X* is the set of weak* limits of bounded nets in A. It is known that a convex subset of a dual Banach space is weak* closed if and only if it equals its weak* derived set. But this does not mean that the weak* closure of a convex set coincides with its weak* derived set. This inspires the definition of iterated weak* derived sets. The order of A is then the least ordinal for which the iteration stabilizes. M. Ostrovskii provided the complete description of possible orders of subspaces of duals of separable non-quasi-reflexive spaces. Later, M. Ostrovskii and the author provided analogous description of possible orders of convex sets in duals of non-reflexive spaces. In this talk, we will introduce these results.