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 theoremMathworldPlanetmath.

Given ϵ>0 notice that the family of open sets {Bϵ(x):xK} is an open covering of K. Being K compactPlanetmathPlanetmath there exists a finite subcover, i.e. there exists n points x1,,xn of K such that the balls Bϵ(xi) cover the whole set K.

Define the functions g1,,gn by

gi(x):={ϵ-x-xi,if x-xiϵ0,if x-xiϵ

It is clear that each gi is continuousMathworldPlanetmath, gi(x)0 and i=1ngi(x)>0 for every xK.

Thus we can define a function in K by


The above function g is a continuous function from K to the convex hull K0 of x1,,xn. Moreover one can easily prove the following

g(x)-xϵ  xK

Now, define the function B:=gf. The restriction B~ of B to K0 provides a continuous function K0K0.

Since K0 is compact convex subset of a finite dimensional vector space, we can apply the Brouwer fixed point theorem to assure the existence of zK0 such that


Therefore g(f(z))=z and we have the inequalityMathworldPlanetmath


Summarizing, for each ϵ>0 there exists z=z(ϵ)K such that f(z)-zϵ. Then

m  zmK  f(zm)-zm1m

As f(zm) is in the compact space K, there is a subsequence zmk such that f(zmk)x0, for some x0K.

We then have

zmk-x0 = zmk-f(zmk)+f(zmk)-x0

which means that zmkx0.

As f is continuous we have f(zmk)f(x0). Both limits of f(zmk) must coincide, so we conclude that


i.e. f has a fixed point.

Title proof of Schauder fixed point theorem
Canonical name ProofOfSchauderFixedPointTheorem
Date of creation 2013-03-22 13:45:22
Last modified on 2013-03-22 13:45:22
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 10
Author asteroid (17536)
Entry type Proof
Classification msc 47H10
Classification msc 46T99
Classification msc 46T20
Classification msc 46B50
Classification msc 54H25