Octahedral norms in duals and biduals of Lipschitz-free spaces

2019/05/22 by Langemets, Johann, Zoca, Abraham Rueda · 1 citation
#46B28 #FOS: Mathematics #Functional Analysis (math.FA) #Primary 46B04 #Secondary 46B20

paper · doi:10.48550/arxiv.1905.09061

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 Lip₀(M) is octahedral, then the norm of Lip₀(M,X^∗) is octahedral. These results solve several open problems from the literature.

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