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NORMAL FUNCTIONALS ON LIPSCHITZ SPACES ARE WEAK* CONTINUOUS

Part of: Normed linear spaces and Banach spaces; Banach lattices Linear function spaces and their duals

Published online by Cambridge University Press:  08 April 2021

Ramón J. Aliaga  and
Eva Pernecká Open the ORCID record for Eva Pernecká [Opens in a new window]
Ramón J. Aliaga
Affiliation:
Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, Camino de Vera S/N, 46022 Valencia, Spain (raalva@upvnet.upv.es)
Eva Pernecká
Affiliation:
Faculty of Information Technology, Czech Technical University in Prague, Thákurova 9, 160 00, Prague 6, Czech Republic (perneeva@fit.cvut.cz)
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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 prove that every normal functional in ${\mathrm {Lip}_0(M)}^*$ is weak* continuous; that is, in order to verify weak* continuity it suffices to do so for bounded monotone nets of Lipschitz functions. This solves a problem posed by N. Weaver. As an auxiliary result, we show that the series decomposition developed by N. J. Kalton for functionals in the predual of $\mathrm {Lip}_0(M)$ can be partially extended to ${\mathrm {Lip}_0(M)}^*$ .

Keywords

Lipschitz-free spaceLipschitz functionLipschitz spacenormal functional

MSC classification

Primary: 46B20: Geometry and structure of normed linear spaces
Secondary: 46E15: Banach spaces of continuous, differentiable or analytic functions
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 21 , Issue 6 , November 2022 , pp. 2093 - 2102
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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