# Schauder's fixed point theorem for metric linear space

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?

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What do you mean by a "compact map" in this context? –  Robert Israel May 18 '12 at 15:42
@RobertIsrael By that I mean $f(C)$ is a compact subset of $C$. Thanks for the Wikipedia link. The book I had presented this result only for Banach spaces. I must look more carefully to prove this general result –  Mayank May 18 '12 at 19:32

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 $C$.