It is known that the closed unit ball of $L\_{\infty}(\mu)$ is weakly compact in $L\_{1}(\mu)$. A natural question arises in the case of spaces of Bochner integral functions:


Question. Let $X$ be a Banach space. In what cases the closed unit ball of $L\_{\infty}(\mu,X)$ is weakly compact in $L\_{1}(\mu,X)$ ?


I am not sure that this question is interesting or remains open.



Weak compactness of the closed unit ball of $L_{\infty}(\mu,X)$ in $L_{1}(\mu,X)$
As Jochen commented, you need $X$ to be reflexive, and this is sufficient. It is enough to show that the unit ball of $L\_\infty(X)$ is closed in the reflexive space $L\_2(X)$. But the injection from $L\_\infty(X)$ into $L\_2(X)$ is the adjoint of the injection from $L\_2(X)$ into $L\_1(X)$, so this is automatic. (Recall that $L\_p(X)^\* = L\_{p'}(X^\*)$ when $X$ is reflexive and $p <\infty$. See Diestel-Uhl section IV.1.)


