Unconditional almost squareness and applications to spaces of Lipschitz functions

TL;DR

This paper introduces an unconditional form of almost squareness in Banach spaces, offering insights into the non-duality of certain Lipschitz function subspaces and addressing longstanding questions in Banach space theory.

Contribution

It proposes a new unconditional concept of almost squareness and applies it to establish criteria for non-duality in Lipschitz function spaces.

Findings

Partial negative answer to dual almost square Banach space existence

Criteria for non-duality of Lipschitz function subspaces

Introduction of an unconditional almost squareness concept

Abstract

We introduce an unconditional concept of almost squareness in order to provide a partial negative answer to the problem of existence of any dual almost square Banach space. We also take advantage of this notion to provide some criterion of non-duality of some subspaces of scalar as well as vector valued Lipschitz functions.