Non-separable tree-like Banach spaces and Rosenthal's $\ell_1$-theorem

TL;DR

This paper introduces a class of non-separable tree-like Banach spaces and demonstrates the limitations of extending Rosenthal's -theorem to uncountably infinite-dimensional -spaces, highlighting fundamental structural differences.

Contribution

It defines and studies non-separable tree-like Banach spaces and shows the impossibility of extending Rosenthal's -theorem to certain uncountable -spaces.

Findings

Non-separable tree-like Banach spaces are constructed and analyzed.

Rosenthal's -theorem cannot be extended to () spaces with uncountable .

Structural limitations of uncountable -spaces are established.

Abstract

We introduce and investigate a class of non-separable tree-like Banach spaces. As a consequence, we prove that we can not achieve a satisfactory extension of Rosenthal's $\ell_1$-theorem to spaces of the type $\ell_1(\kappa)$, for $\kappa$ an uncountable cardinal.