Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
    
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

1 Answer 1

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 $C$.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.