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arXiv · 1905.09061

Octahedral norms in duals and biduals of Lipschitz-free spaces

Johann Langemets·Abraham Rueda Zoca
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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.

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Johann Langemets, Abraham Rueda Zoca. 2019-05-22. Octahedral norms in duals and biduals of Lipschitz-free spaces. https://arxiv.org/abs/1905.09061

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