Let $(X,\Sigma,\mu)$ be a Borel probability space. Suppose we have a Banach space of functions on $X$ $(B,\|.\|)$ such that the unit ball of $B$ is compact in $L^p(X)$ for all $p>1$. Can we say that the unit ball of $B$ is compact in $L^\infty(X)$? (Note there is no compactness condition on $X$).
Edit: I probably should mention that $B\subset L^\infty$ and the norm $\|\cdot\|$ satisfies $\|f\|\leq \|f\|_p$ for all $p>1$ and all $f\in B$.
This relates to a problem I am looking at and had me stumped for a while. Any help would be appreciated!