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

 

Weak compactness in $ L\sp 1(\mu,X)$


Author: A. Ülger
Journal: Proc. Amer. Math. Soc. 113 (1991), 143-149
MSC: Primary 46E40; Secondary 46B20
MathSciNet review: 1070533
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Abstract: Let $ (\Omega ,\Sigma ,\mu )$ be a probability space, $ X$ a Banach space, and $ {L^1}(\mu ,X)$ the Banach space of Bochner integrable functions $ f:\Omega \to X$. Let $ W = \{ f \in {L^1}(\mu ,X):{\text{ for a}}{\text{.e}}{\text{. }}\omega {\text{ in }}\Omega ,\vert\vert f(\omega )\vert\vert \leq 1\} $. In this paper we characterize the rwc (relatively weakly compact) subsets of $ {L^1}(\mu ,X)$. The main results are as follows:

Theorem A. A subset $ H$ of $ W$ is rwc iff given any sequence $ ({f_n})$ in $ H$ there exists a sequence $ ({\tilde f_n})$, with $ {\tilde f_n} \in \operatorname{Co}({f_n},{f_{n + 1}}, \ldots )$ such that, for a.e. $ \omega $ in $ \Omega $, the sequence $ ({\tilde f_n}(\omega ))$ converges weakly in $ X$.

Theorem B. A subset $ A$ of $ {L^1}(\mu ,X)$ is rwc iff given any $ \varepsilon > 0$ there exist an integer $ N$ and a rwc subset $ H$ of NW such that $ A \subseteq H + \varepsilon B(0)$, where $ B(0)$ is the unit ball of $ {L^1}(\mu ,X)$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1991-1070533-0
PII: S 0002-9939(1991)1070533-0
Keywords: Bochner integrable functions, weak compactness
Article copyright: © Copyright 1991 American Mathematical Society