I have a certain functional $E : W_2 \rightarrow \mathbb{R}$, where $W_2$ is the 2-Wasserstein space (metric and separable). Such functional is convex.
Now, can I state that if $E$ is strongly lower semicontinuous + convex then $E$ is weakly lower semicontinuous?
From Villani's book (Topics in optimal transportation) Theorem 7.12 we know that Wasserstein distances metrize weak convergence, so if the above statement is true I can conclude that the functional $E$ is lower semicontinuous in the Wasserstein topology.
Another way to state the problem would be: is it true that a convex subset $C$ of a metric, separable space $X$ is closed in strong topology iff $C$ is closed in the weak topology?
