Weak convergence and weak compactness in the space of integrable functions with respect to a vector meansure
AbstractWe consider weak convergence and weak compactness in the space L1(m) of real valued integrable functions with respect to a Banach space calued 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.
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 ????∞ 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.
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