Compact subsets of $L^\infty$

Question

The Riesz Frechet Kolmogorov theorem gives a necessary and sufficient condition for a subset of $L^p(\Omega)$ spaces for $1\leq p<\infty$ and equipped with Lebesgue measure to be relatively compact for the Banach space topology.

Is there any similar result known in case of $L^\infty$? Thanks

Answer 1

For the case of finite measure space see chapter IV section 8 theorem 18 in Linear Operators. General theory. Volume 1 , N.Dunford, J. T. Schwartz

For general case you should recall that $L_\infty$ is a $C(K)$-space for some weird Hausdorff compact space $K$ and apply Arzela-Ascoli theorem.