Weak convergence and weak compactness in the space of integrable functions with respect to a vector measure

Swartz, Charles; Swartz, Charles

doi:10.22199/issn.0717-6279-2020-01-0008

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Proyecciones (Antofagasta)

Print version ISSN 0716-0917

Proyecciones (Antofagasta) vol.39 no.1 Antofagasta Feb. 2020

http://dx.doi.org/10.22199/issn.0717-6279-2020-01-0008

Artículos

Weak convergence and weak compactness in the space of integrable functions with respect to a vector measure

Charles Swartz¹

^¹New Mexico State University, Mathematics Dept, Las Cruces, NM, U. S. A. E-mail: cswartz@nmsu.edu

Abstract

We consider weak convergence and weak compactness in the space L¹(m) of real valued integrable functions with respect to a Banach space valued measure m equipped with its natural norm. We give necessary and sufficient conditions for a sequence in L¹(m) to be weak Cauchy, and we give necessary and sufficient conditions for a subset of L¹(m) to be conditionally sequentially weakly compact.

Keywords: Weak convergence; Weak compactness; Integrable functions; Measure and integration

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Acknowledgement

The author would like to thank Susumu Okada for his help.

References

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Received: December 2018; Accepted: August 2019

Article copyright: © 2020 Charles Swartz. This is an open access article distributed under the terms of the Creative Commons Licence, which permits unrestricted use and distribution provided the original author and source are credited

This is an open-access article distributed under the terms of the Creative Commons Attribution License