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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?2012-05-18
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    @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 result2012-05-18

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