I am reading 'Ordinary differential equations, transport theory and Sobolev spaces' by DiPerna and Lions. I am stuck at the following step in page $533$ in the paper:
$\frac{\partial}{\partial t}(\beta_0(X_{\epsilon}))$ is bounded in $L^{\infty}(\mathbb R, L^1+L^{\infty})$ and belongs to a relatively compact set of $L^{\infty}(-T,T;L^1(B_R))$ (for every $R,T<\infty$)
Is it true that if a set $X\in L^{\infty}(\mathbb R, L^1+L^{\infty})$ is bounded, then it belongs to a relatively compact set $L^{\infty}([-T,T];L^1(B_R))$ (for every $R,T<\infty$)?
It should be enough to show that the closure of the set $X$, $cl(X)$, will be complete and totally bounded. Because it is a closed subset of a Banach space, it will be complete, but I am having some trouble proving totally boundedness or boundedness.