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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 non-quasi-reflexive Banach space containing an infinite-dimensional subspace with separable dual, we can find for any countable successor ordinal $\alpha $ a subspace whose weak$^*$ derived set of order $\alpha $ is proper and norm dense.