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

Authors

  • Charles Swartz New Mexico State University.

DOI:

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

Keywords:

Weak convergence, Weak compactness, Integrable functions, Measure and integration

Abstract

We consider weak convergence and weak compactness in the space L1(m) of real valued integrable functions with respect to a Banach space calued measure 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.

References

G. Curbera, “Banach space properties of L1 of a vector measure”, Proceedings of the American mathematical society, vol. 123, no. 12 pp. 3797-3806, Dec. 1995, doi: 10.2307/2161909.

N. Dunford and J. T. Schwartz, Linear operators, vol. 1. New York, NY: Interscience, 1958.

P. Halmos, Measure theory, Princeton, NJ: D. Van Nostrand Company, 1950.

J. Kelley, General topology, Princeton, NJ: D. Van Nostrand Company, 1955.

I. Kluvanek and G. Knowles, Vector measures and control systems, Amsterdam: North Holland, 1975.

D. Lewis, “Integration with respect to vector measures”, Pacific journal of mathematics, vol. 33, no. 1. pp. 157-165, 1970. [On line]. Available: https://bit.ly/3138dbT

S. Okada, “The dual space of L1(μ) for a vector measure μ”, Journal of mathematical analysis and applications, vol. 177, no. 2, pp. 583-599, Aug. 1993, doi: 10.1006/jmaa.1993.1279.

S. Okada, W. J. Ricker, E. A. Sánchez-Pérez,” Lattice copies of c0 and l∞ in spaces of integrable functions for a vector measure”, Dissertationes mathematicae, vol. 500, 2014, doi: 10.4064/dm500-0-1.

C. Swartz, Measure, integration, and function spaces. Singapore: World Scientific, 1994.

C. Swartz, Multiplier convergent series, Singapore: World Scientific, 2009.

C. Swartz, Abstract duality pairs in analysis, Singapore: World Scientific, 2018.

B. L. Thorp, “Sequential evaluation convergence”, Journal of the London mathematical society, vol. s1-44, no. 1pp. 201-209, Jan. 1969, doi: 10.1112/jlms/s1-44.1.201.

Published

2020-02-04

How to Cite

[1]
C. Swartz, “Weak convergence and weak compactness in the space of integrable functions with respect to a vector meansure”, Proyecciones (Antofagasta, On line), vol. 39, no. 1, pp. 123-133, Feb. 2020.

Issue

Section

Artículos