Let $F:L^2(\Omega) \to L^2(\Omega)$ be continuous map. Let $D$ be a function space.
Since $F(L^2(\Omega)) \subset D$, and $D \subset L^2(\Omega)$ is a compact embedding, $F$ is a compact operator from $L^2(\Omega)$ into itself. So Schauder's fixed point theorem applies and $F$ has a fixed point.
(The source also mentions that $|F(w)|_{L^2(\Omega)} \leq R$ for all $w$, but I don't know whether this is for all $w \in D$ or $w \in L^2(\Omega).$)
Can someone explain to me how this works when looking at the classical statement of the Schauder fixed point theorem? They seem to use $L^2(\Omega)$ as the closed convex subset.. but shouldn't there be a bigger space involved too?
If $D$ is compact, then so is the closure of its convex closure $K=\overline{\mathrm{co}(D)}$.
Hence, you have a compact map $F: K\to D\subset K$, which according to Schauder FPT possesses a fixed point, as no $K$ is compact and convex.