Preduals and double preduals of some Banach spaces

TL;DR

This paper investigates conditions under which Banach spaces can be recognized as duals or double duals of other spaces, using relationships with classical spaces like  and  to derive properties of spaces such as BMO and BV.

Contribution

It provides new criteria for identifying dual and double dual Banach spaces by relating them to well-understood classical spaces.

Findings

Established criteria for duality and double duality of Banach spaces.

Applied results to characterize spaces like BMO and BV.

Connected properties of classical spaces to more abstract Banach spaces.

Abstract

In this paper we study ways to establish when a Banach space can be identified as the dual or the double dual of another Banach space. To obtain these results, we relate these spaces with other, concrete Banach spaces - tipically $\ell^1$ and $\ell^\infty$ - and show that under suitable assumptions we can transfer properties of these spaces to the space we consider. In particular, we show how these results can be used to obtain in a simple way interesting results about spaces such as $BMO$ and $BV$.