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On weak compactness in Lebesgue-Bochner spaces


Author: José Rodríguez
Journal: Proc. Amer. Math. Soc. 144 (2016), 103-108
MSC (2010): Primary 46B50, 46G10
DOI: https://doi.org/10.1090/proc/12846
Published electronically: August 5, 2015
MathSciNet review: 3415580
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Abstract: Let $ X$ be a Banach space, $ (\Omega ,\Sigma ,\mu )$ a probability space and $ K$ a weakly compact subset of $ L^p(\mu ,X)$, $ 1\leq p<\infty $. The following question was posed by J. Diestel: is there a weakly compactly generated subspace $ Y \subset X$ such that $ K \subset L^p(\mu ,Y)$? We show that, in general, the answer is negative. We also prove that the answer is affirmative if either $ \mu $ is separable or $ X$ is weakly sequentially complete.


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Additional Information

José Rodríguez
Affiliation: Departamento de Matemática Aplicada, Facultad de Informática, Universidad de Murcia, 30100 Espinardo (Murcia), Spain
Email: joserr@um.es

DOI: https://doi.org/10.1090/proc/12846
Keywords: Weakly compact set, weakly compactly generated Banach space, Lebesgue-Bochner space, separable measure
Received by editor(s): October 28, 2014
Published electronically: August 5, 2015
Additional Notes: This research was supported by MINECO and FEDER under project MTM2011-25377
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2015 American Mathematical Society

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