I have read this:
We have a map $S:W_0 \to W_0$. Moreover $W_0$ is not empty, convex, and weakly compact in $W$. Thus we can apply Schauder's fixed point theorem:
Schauder's fixed point theorem: If $E$ is convex compact subset of a Banach space and if $S:E \to E$ is continuous then there is a fixed point of $S$.
So we just need to prove that $S$ is weakly continuous (from $W_0 \to W_0$).
Why the author checks for weakly continuous instead of the stronger continuity? Can someone give me a good definition of weakly compact (in terms of sequences and boundedness)? Thanks.