Weak$^*$ closures and derived sets for convex sets in dual Banach spaces

Abstract

The paper is devoted to the convex-set counterpart of the theory of weak * derived sets initiated by Banach and Mazurkiewicz for subspaces. The main result is the following: For every nonreflexive Banach space X and every countable successor ordinal α, there exists a convex subset A in X * such that α is the least ordinal for which the weak * derived set of order α coincides with the weak * closure of A. This result extends the previously known results on weak * derived sets by Ostrovskii (2011) and Silber (2021).1. Introduction. Let X be a Banach space. For a subset A of the dual Banach space X * , we denote the weak * closure of A by A * . The weak * derived set of A is defined aswhere B X * is the unit ball of X * . That is, A (1) is the set of all limits of weak * convergent bounded nets in A. If X is separable, A (1) coincides with the set of all limits of weak * convergent sequences from A, called the weak * sequential closure. The strong closure of a set A in a Banach space is denoted by A. We set A (0) := A.It was noticed in the early days of Banach space theory by Mazurkiewicz [23] that A (1) does not have to coincide with A * even for a subspace A, and (A (1) ) (1) can be different from A (1) . In this connection, it is natural to introduce derived sets for all ordinals: (1) if A (α) has already been defined, then A (α+1) := (A (α) ) (1) ; (2) if α is a limit ordinal and A (β) has already been defined for all β < α, then(1.1) β) .