Is there an analogue of Schauder type fixed point theorems that can be used over a metric linear space. So, here $(X,d)$ is a complete vector space with metric $d$. If $C\subseteq X$ and $f:C\rightarrow C$ is a continuous and compact map. Then does $f$ have a fixed point?
A general form of Schauder's fixed point theorem (see http://en.wikipedia.org/wiki/Schauder_fixed_point_theorem ) says if $f$ is a continuous map of a convex subset $C$ of a topological vector space into itself and $f(C)$ is contained in a compact subset of $C$, then $f$ has a fixed point in $C$.