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Fixed point theorem

Fermat

Active member
Nov 3, 2013
188
This question is about theorem 4.1 in the link
Schauder Fixed Point Theorem

In particular, where it restricts g to con(F). I don't understand how we know con(F) is a subset of C. C is convex so if C contained F, then certainly con(F) would be a subset of C. But how do we know F is a subset of C.

Thanks
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,703
This question is about theorem 4.1 in the link
Schauder Fixed Point Theorem

In particular, where it restricts g to con(F). I don't understand how we know con(F) is a subset of C. C is convex so if C contained F, then certainly con(F) would be a subset of C. But how do we know F is a subset of C.
That is a good point, and it looks to me as though that link skates over this bit of the proof. What it should say is that, because $K = \overline{f(C)}$ is compact, for each $\varepsilon>0$ there exists a finite $\varepsilon$-net $F = \{x_1,\ldots,x_n\}$ for $K$, whose elements $x_1,\ldots,x_n$ are all in $f(C)$. It is easy to show that such an $\varepsilon$-net exists (because the set of $\varepsilon$-neighbourhoods of all points in $f(C)$ is an open cover of $K$ and therefore has a finite subcover). Since $f$ maps $C$ into $C$, it then follows that $\mathrm{con}\,F\subset C.$