On nested sequences of convex sets in a Banach space

TL;DR

This paper investigates how weak*-compact convex sets in the bidual of a separable Banach space can be represented through nested sequences of convex sets within the original space, enhancing understanding of dual space structures.

Contribution

It introduces a novel approach to representing weak*-compact convex sets in biduals using nested sequences in the original Banach space.

Findings

Representation of weak*-compact convex sets via nested sequences

Conditions for the existence of such representations

Implications for the structure of dual and bidual spaces

Abstract

In this paper we study different aspects of the representation of weak*-compact convex sets of the bidual $X^{**}$ of a separable Banach space $X$ via a nested sequence of closed convex bounded sets of $X$.