# Octahedral norms in duals and biduals of Lipschitz-free spaces

## TL;DR

This paper investigates conditions under which the norms of Lipschitz-free spaces and their duals are octahedral, providing new results that resolve several open questions in the field.

## Contribution

It establishes that the norm of the bidual of Lipschitz-free spaces is octahedral for unbounded or non-uniformly discrete metric spaces, and extends this to Lipschitz spaces with Banach space targets.

## Key findings

- The norm of $\
- $ ext{Lip}_0(M)$ is octahedral when $M$ has certain properties.
- The norm of $ ext{Lip}_0(M,X^*)$ is octahedral if $ ext{Lip}_0(M)$ has this property and $X$ is arbitrary.

## Abstract

We continue with the study of octahedral norms in the context of spaces of Lipschitz functions and in their duals. First, we prove that the norm of $\mathcal F(M)^{**}$ is octahedral as soon as $M$ is unbounded or is not uniformly discrete. Further, we prove that a concrete sequence of uniformly discrete and bounded metric spaces $(K_m)$ satisfies that the norm of $\mathcal F(K_m)^{**}$ is octahedral for every $m$. Finally, we prove that if $X$ is an arbitrary Banach space and the norm of $\operatorname{Lip}_0(M)$ is octahedral, then the norm of $\operatorname{Lip}_0(M,X^\ast)$ is octahedral. These results solve several open problems from the literature.