Let $\Omega$ be a domain with smooth boundary. Let $S\subset W^{1,1}(\Omega)$ be a relatively weakly compact set.
Is it true that $(S,w)$ is metrizable?
Since $S$ is relatively weakly compact, it is norm-bounded. If I recall correctly, a norm-bounded subset of a Banach space $X$ is metrizable (with respect to the weak topology) if $X^*$ is separable. However, $W^{1,1}(\Omega)^*$ is not separable so the result do not apply here.
I apologize in advance if the question is too simple here.
