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Octahedrality in Lipschitz-free Banach spaces

Part of: Normed linear spaces and Banach spaces; Banach lattices General convexity

Published online by Cambridge University Press:  28 December 2017

Julio Becerra Guerrero ,
Ginés López-Pérez  and
Abraham Rueda Zoca
Julio Becerra Guerrero
Affiliation:
Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain (juliobg@ugr.es)
Ginés López-Pérez
Affiliation:
Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain and Instituto de Matemáticas de la Universidad de Granada (IEMath-GR), Calle Ventanilla, 11, 18001 Granada, Spain (glopezp@ugr.es)
Abraham Rueda Zoca
Affiliation:
Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain (arz0001@correo.ugr.es)
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Abstract

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The aim of this note is to study octahedrality in vector-valued Lipschitz-free Banach spaces on a metric space, under topological hypotheses on it, by analysing the weak-star strong diameter 2 property in Lipschitz function spaces. Also, we show an example that proves that our results are optimal and that octahedrality in vector-valued Lipschitz-free Banach spaces actually relies on the underlying metric space as well as on the Banach one.

Keywords

diameter 2 propertiesoctahedral normsslices

MSC classification

Secondary: 46B20: Geometry and structure of normed linear spaces 46B22: Radon-Nikodým, Kreĭn-Milman and related properties 52A10: Convex sets in $2$ dimensions (including convex curves)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 148 , Issue 3 , June 2018 , pp. 447 - 460
Copyright
Copyright © Royal Society of Edinburgh 2018