Almost square dual Banach spaces

TL;DR

This paper investigates the properties of almost square Banach spaces, demonstrating limitations in finite dimensions, constructing new norms on $\, ext{ell}_ ext{infinity}$, and characterizing separable spaces through their duals.

Contribution

It shows finite dimensional Banach spaces are not uniformly non locally almost square and constructs an almost square bidual norm on $\, ext{ell}_ ext{infinity}$, advancing the understanding of dual space structures.

Findings

Finite dimensional Banach spaces are not uniformly non locally almost square.

An almost square bidual norm on $\, ext{ell}_ ext{infinity}$ is constructed.

Separable real almost square spaces are characterized via their fourth duals.

Abstract

We show that finite dimensional Banach spaces fail to be uniformly non locally almost square. Moreover, we construct an equivalent almost square bidual norm on $\ell_\infty.$ As a consequence we get that every dual Banach space containing $c_0$ has an equivalent almost square dual norm. Finally we characterize separable real almost square spaces in terms of their position in their fourth duals.