On subspaces whose weak$^*$ derived sets are proper and norm dense

Abstract

We study long chains of iterated weak∗ derived sets, that is sets of all weak∗ limits of bounded nets, of subspaces with the additional property that the penultimate weak∗ derived set is a proper norm dense subspace of the dual. We extend the result of Ostrovskii and show, that in the dual of any nonquasi-reflexive Banach space containing an infinite-dimensional subspace with separable dual, we can find for any countable successor ordinal α a subspace, whose weak∗ derived set of order α is proper and norm dense.