# On the non-embedding of $\ell_1$ in the James Tree Space

## TL;DR

This paper provides a direct proof that the James Tree Space does not contain an isomorphic copy of , using classical theorems and measure theory, clarifying a fundamental property of this Banach space.

## Contribution

It offers a new, direct proof of the non-embedding of  in James Tree Space, simplifying previous arguments and employing measure-theoretic techniques.

## Key findings

- Confirmed  does not embed in  space
- Used Rosenthal's  Theorem and Riesz's Representation Theorem
- Provided a clearer, more direct proof of a known property

## Abstract

James Tree Space ($\mathcal{JT}$), introduced by R. James, is the first Banach space constructed having non-separable conjugate and not containing $\ell^1$. James actually proved that every infinite dimensional subspace of $\mathcal{JT}$ contains a Hilbert space, which implies the $\ell^1$ non-embedding. In this expository article, we present a direct proof of the $\ell^1$ non-embedding, using Rosenthal's $\ell^1$- Theorem and some measure theoretic arguments, namely Riesz's Representation Theorem.