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

### Volume 268 / 2023

Studia Mathematica 268 (2023), 319-332
MSC: Primary 46B10; Secondary 46B20.
DOI: 10.4064/sm220303-29-4
Published online: 22 September 2022

#### 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.