Normal functionals on Lipschitz spaces are weak${}^{\ast }$ continuous

Abstract

Let ${\mathrm{Lip}}_0(M)$ be the space of Lipschitz functions on a complete metric space $M$ that vanish at a base point. We show that every normal functional in ${\mathrm{Lip}}_0(M{)}^{\ast }$ is weak${}^{\ast }$ continuous, answering a question by N. Weaver.

arXiv commentv2: Revised version

References in corpus (4)

• Supports in Lipschitz-free spaces and applications to extremal structure
• On strongly norm attaining Lipschitz maps
• Norm attaining Lipschitz functionals
• Finitely additive measures and complementability of Lipschitz-free spaces

Cited by in corpus (2)

• Convex integrals of molecules in Lipschitz-free spaces
• Pełczyński's property (V${}^{\ast }$) in Lipschitz-free spaces