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

TL;DR

This paper investigates the structure of subspaces in dual Banach spaces through iterated weak* derived sets, revealing that certain complex chains of these sets can be constructed in a broad class of non-quasi-reflexive Banach spaces.

Contribution

It extends Ostrovskii's result by showing the existence of subspaces with prescribed properties of weak* derived sets in duals of non-quasi-reflexive Banach spaces containing an infinite-dimensional subspace with separable dual.

Findings

Existence of subspaces with proper and norm dense weak* derived sets of any countable successor ordinal.

Extension of Ostrovskii's result to a wider class of Banach spaces.

Construction of long chains of iterated weak* derived sets in dual Banach spaces.

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.