Schauder fixed point theorem states that
If $A$ is a compact convex subset of a Banach space $B$ and $T$ is a continuous mapping from $A$ to $A$, then $T$ has fixed point in $A$.
I would like to know if $A$ is compactly embedding into $B$, can we still use this theorem? For example, $A= W^{1,2}(B_1)$ and $B=L^2(B_1)$. I think $A$ is compact in $B$ means that for any sequence in $A$, we have a convergent subsequence with its limit in $A$. However $A$ is compactly embedded into $B$ means that any bounded sequence in $A$ has a subsequence which is convergent in $B$.
Let $f_0 \in W^{1,2} (B_1)$ be any non-zero element and let $$ T: A\to A,\ \ T(f) = f+ f_0.$$ Then $T$ is continuous and has no fixed point. Compactness of $A$ is crucial in the theorem.