proof of Schauder fixed point theorem
The idea of the proof is to reduce to the finite dimensional case where we can apply the Brouwer fixed point theorem.
Define the functions by
It is clear that each is continuous, and for every .
Thus we can define a function in by
The above function is a continuous function from to the convex hull of . Moreover one can easily prove the following
Now, define the function . The restriction of to provides a continuous function .
Since is compact convex subset of a finite dimensional vector space, we can apply the Brouwer fixed point theorem to assure the existence of such that
Therefore and we have the inequality
Summarizing, for each there exists such that . Then
As is in the compact space , there is a subsequence such that , for some .
We then have
which means that .
As is continuous we have . Both limits of must coincide, so we conclude that
i.e. has a fixed point.
|Title||proof of Schauder fixed point theorem|
|Date of creation||2013-03-22 13:45:22|
|Last modified on||2013-03-22 13:45:22|
|Last modified by||asteroid (17536)|