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.
Categories: math.FA