Octahedral norms in duals and biduals of Lipschitz-free spaces

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 $\mathcal{F}(M{)}^{\ast \ast }$ 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 $\mathcal{F}(K_m{)}^{\ast \ast }$ is octahedral for every $m$. Finally, we prove that if $X$ is an arbitrary Banach space and the norm of ${\mathrm{Lip}}_0(M)$ is octahedral, then the norm of ${\mathrm{Lip}}_0(M,X^{\ast })$ is octahedral. These results solve several open problems from the literature.

17 pages