On thickness and thinness of Banach spaces

TL;DR

This paper explores the behavior of Whitley's indices of thinness and thickness in Banach spaces, especially under $ ext{l}_p$-sums, subspace relations, and renormings involving $c_0$, revealing their distinct properties and potential for normalization.

Contribution

It provides new formulas for indices under $ ext{l}_p$-sums, examines their relation between spaces and subspaces, and shows how to renorm spaces containing $c_0$ to achieve indices equal to 1.

Findings

Indices behave differently under $ ext{l}_p$-sums.

Relation of indices between a space and its subspace is characterized.

Spaces containing $c_0$ can be renormed so that indices equal 1.

Abstract

The aim of this note is to complement and extend some recent results on Whitley's indices of thinness and thickness in three main directions. Firstly, we investigate both the indices when forming $\ell_p$-sums of Banach spaces, and obtain formulas which show that they behave rather differently. Secondly, we consider the relation of the indices of the space and a subspace. Finally, every Banach space $X$ containing a copy of $c_0$ can be equivalently renormed so that in the new norm $c_0$ is an M-ideal in $X$ and both the thickness and thinness index of $X$ equal 1.