On weak compactness in Lebesgue-Bochner spaces

Rodríguez, José

doi:10.1090/proc/12846

On weak compactness in Lebesgue-Bochner spaces
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Proc. Amer. Math. Soc. 144 (2016), 103-108 Request permission

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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