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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A tree characterization of the point of continuity property in general Banach spaces


Authors: Ginés López Pérez and José Antonio Soler Arias
Journal: Proc. Amer. Math. Soc. 140 (2012), 4243-4245
MSC (2010): Primary 46B20, 46B22
Published electronically: April 17, 2012
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Abstract: We obtain a characterization of the point of continuity property for general bounded subsets in Banach spaces in terms of trees. For this we introduce the notion of a topologically weakly null tree and, as a consequence, we get that a general Banach space satisfies the point of continuity property if, and only if, every seminormalized topologically weakly null tree has a boundedly complete branch.


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Ginés López Pérez
Affiliation: Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071-Granada, Spain
Email: glopezp@ugr.es

José Antonio Soler Arias
Affiliation: Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071-Granada, Spain
Email: jasoler@ugr.es

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11281-X
PII: S 0002-9939(2012)11281-X
Keywords: Point of continuity property, trees, boundedly complete sequences
Received by editor(s): February 18, 2011
Received by editor(s) in revised form: March 9, 2011, April 11, 2011, May 23, 2011, and May 30, 2011
Published electronically: April 17, 2012
Additional Notes: This work was partially supported by MEC (Spain) Grant MTM2006-04837 and Junta de Andalucía Grants FQM-185 and Proyecto de Excelencia P06-FQM-01438.
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2012 American Mathematical Society