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Lipschitz slices versus linear slices in Banach spaces


Authors: Julio Becerra Guerrero, Ginés López-Pérez and Abraham Rueda Zoca
Journal: Proc. Amer. Math. Soc. 145 (2017), 1699-1708
MSC (2010): Primary 46B20, 46B22
DOI: https://doi.org/10.1090/proc/13372
Published electronically: October 13, 2016
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Abstract: The aim of this note is to study the topology generated by Lipschitz slices in the unit sphere of a Banach space. We prove that the above topology agrees with the weak topology in the unit sphere and, as a consequence, we obtain easy Lipschitz characterizations of classical linear topics in Banach spaces as the Daugavet property, Radon-Nikodym property, convex point of continuity property and strong regularity, which shows that the above classical linear properties depend only on the natural uniformity in the Banach space given by the metric and the linear structure.


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Additional Information

Julio Becerra Guerrero
Affiliation: Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071-Granada, Spain
Email: 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)
Email: glopezp@ugr.es

Abraham Rueda Zoca
Affiliation: Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071-Granada, Spain
Email: arz0001@correo.ugr.es

DOI: https://doi.org/10.1090/proc/13372
Keywords: Radon-Nikodym property, Lipschitz slices, Lipchitz topology
Received by editor(s): April 7, 2016
Received by editor(s) in revised form: June 17, 2016
Published electronically: October 13, 2016
Additional Notes: The first author was partially supported by MEC (Spain) grant MTM2014-58984-P and Junta de Andalucía grants FQM-0199, FQM-1215.
The second author was partially supported by MINECO (Spain) grant MTM2015-65020-P and Junta de Andalucía grant FQM-185.
The third author was partially supported by Junta de Andalucía grants FQM-0199.
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2016 American Mathematical Society