Isometric embedding of $\ell_1$ into Lipschitz-free spaces and $\ell_\infty$ into their duals

TL;DR

This paper investigates the structure of Lipschitz-free Banach spaces, showing their duals always contain an isometric copy of _, and explores the relationship between embeddings of _ and _ in Banach spaces.

Contribution

It proves that the duals of all infinite-dimensional Lipschitz-free Banach spaces contain _, and that these spaces often contain a complemented _, advancing understanding of their geometric properties.

Findings

Duals of infinite-dimensional Lipschitz-free spaces contain _

Lipschitz-free Banach spaces often contain complemented _

Lipschitz-free spaces are never rotund

Abstract

We show that the dual of every infinite-dimensional Lipschitz-free Banach space contains an isometric copy of $\ell_\infty$ and that it is often the case that a Lipschitz-free Banach space contains a $1$-complemented subspace isometric to $\ell_1$. Even though we do not know whether the latter is true for every infinite-dimensional Lipschitz-free Banach space, we show that the space is never rotund. Further, in the last section we survey the relations between "isometric embedding of~$\ell_\infty$ into the dual" and "containing as good copy of~$\ell_1$ as possible" in a general Banach space.