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

 

$ p$-Convergent sequences and Banach spaces in which $ p$-compact sets are $ q$-compact


Authors: Cándido Piñeiro and Juan Manuel Delgado
Journal: Proc. Amer. Math. Soc. 139 (2011), 957-967
MSC (2010): Primary 46B50, 47B07; Secondary 47B10
Published electronically: July 28, 2010
MathSciNet review: 2745647
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Abstract: We introduce and investigate the notion of $ p$-convergence in a Banach space. Among others, a Grothendieck-like result is obtained; namely, a subset of a Banach space is relatively $ p$-compact if and only if it is contained in the closed convex hull of a $ p$-null sequence. We give a description of the topological dual of the space of all $ p$-null sequences which is used to characterize the Banach spaces enjoying the property that every relatively $ p$-compact subset is relatively $ q$-compact ($ 1\leq q<p$). As an application, Banach spaces satisfying that every relatively $ p$-compact set lies inside the range of a vector measure of bounded variation are characterized.


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Additional Information

Cándido Piñeiro
Affiliation: Department of Mathematics (Faculty of Experimental Sciences), Campus Universitario de El Carmen, Avenida de las Fuerzas Armadas s/n, 21071 Huelva, Spain
Email: candido@uhu.es

Juan Manuel Delgado
Affiliation: Department of Mathematics (Faculty of Experimental Sciences), Campus Universitario de El Carmen, Avenida de las Fuerzas Armadas s/n, 21071 Huelva, Spain
Address at time of publication: Departamento de Matemática Aplicada I (Escuela Técnica Superior de Arquitectura), Avenida de Reina Mercedes, 2, 41012 Sevilla, Spain
Email: jmdelga@us.es

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10508-7
PII: S 0002-9939(2010)10508-7
Keywords: $p$-compact set, $p$-convergent sequence, $p$-nuclear operator, $p$-summing operator, cotype
Received by editor(s): January 21, 2010
Received by editor(s) in revised form: March 6, 2010, and March 22, 2010
Published electronically: July 28, 2010
Additional Notes: This research was supported by MTM2009-14483-C02-01 project (Spain)
Communicated by: Nigel J. Kalton
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.