NDLI: Weak Compactness in Spaces of Bochner Integrable Functions and Applications

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Weak Compactness in Spaces of Bochner Integrable Functions and Applications

Content Provider	Semantic Scholar
Copyright Year	2005
Abstract	In this paper we shall give the necessary and sufficient conditions for weak compactness of sets in the space of Bochner integrable functions Lel(/z). Roughly speaking, the necessary and sufficient conditions for bounded sets are: (a) uniform ~ additivity of the indefinite integrals; (b) the action of the indefinite integrals on each measurable set is a weakly compact subset in E, the range space of the functions; (e) weak uniform convergence off~ to f, for f belonging to the subset of LEI(p.) in question, where f~ is the conditional expectation of f with respect to a partition 7r of measurable sets. The precise statements are given in Theorem 1, part I. In part I I of the same theorem, criteria of weak compactness for subsets ofLeP(/,), 1 < p < oo, are given. In the following sections, additional results are presented, including: weak compactness and weak sequential completeness in le% 1 ~< p < oe; weak compactness in the space of finitely additive vector measures; applications to the structure of weakly compact operators between certain function spaces. The task of establishing weak compactness criteria in Lebesgue spaces has been a long one. This is not surprising, since not only is the weak topology of fundamental importance in the study of Banach spaces, but in classical measure theory the study of weak compactness is equivalent to the study of setwise convergence. In 1909 Lebesgue essentially gave criteria of weak convergence of indefinite integrals in terms of setwise convergence of the integrals. The Vital i -Hahn-Saks and Nikodym theorems are results in weak compactness
File Format	PDF HTM / HTML
Alternate Webpage(s)	https://core.ac.uk/download/pdf/82810622.pdf
Language	English
Access Restriction	Open
Content Type	Text
Resource Type	Article

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