Confused about a version of Schauder's fixed point theorem 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.
 A: I will answer your last question ("Can someone give me a good definition of weakly compact (in terms of sequences and boundedness)?"), and I will try to edit the post later to see if I can answer the rest.
Let us denote by $X^*$ the space of all linear continuous functional on $X$, a normed vector space. Then, the weak topology $\sigma(X,X^*)$ on $X$ is the final topology on $X$ with respect to $X^*$. Then, a set $A$ is weakly compact if it is compact with respect to the weak topology $\sigma(X,X^*)$. Thus, $A$ is weak sequentially compact, by the Eberlien-Smulian theorem (see Proof of Eberlein–Smulian Theorem for a reflexive Banach spaces). So,
"Theorem": $A$ is weakly compact if every sequence in $A$ has a convergent subsequence whose limit is in $A$.
A: I think what is used here is the following version of the Schauder theorem:
Let $V$ be a separable reflexive Banach space. Let $K \subset V$ be nonempty, convex, bounded, and compact for the weak topology. Let $S : K \to K$ be weak-to-weak continuous. Then $S$ has a fixed point.
I couldn't find a reference for this. However, it seems to me that the usual proof of the Schauder theorem carries over using metrizability of the weak topology (on bounded sets of a reflexive Banach space).
Edit: Cf. Tikhonov fixed point theorem.
