6.4. THE SCHAUDER FIXED POINT THEOREM 163

k→ ∞ and so 0 ≤ si ≤ ti this for each i and these si, ti pertain to the single point x. Butthese si add to 1 as do the ti and so in fact, si = ti for each i and so f (x) = x. This provesthe following theorem which is the Brouwer fixed point theorem.

Theorem 6.3.1 Let S be a simplex [x0, · · · ,xn] such that {xi−x0}ni=1 are indepen-

dent. Also let f : S→ S be continuous. Then there exists x ∈ S such that f (x) = x.

Corollary 6.3.2 Let K be a closed convex bounded subset of Rn. Let f : K → K becontinuous. Then there exists x ∈ K such that f (x) = x.

Proof: Let S be a large simplex containing K and let P be the projection map onto K.See Problem 10 on Page 152 for the necessary properties of this projection map. Considerg (x) ≡ f (Px) . Then g satisfies the necessary conditions for Theorem 6.3.1 and so thereexists x ∈ S such that g (x) = x. But this says x ∈ K and so g (x) = f (x). ■

Definition 6.3.3 A set B has the fixed point property if whenever f : B→ B for fcontinuous, it follows that f has a fixed point.

The proof of this corollary is pretty significant. By a homework problem, a closedconvex set is a retract of Rn. This is what it means when you say there is this continuousprojection map which maps onto the closed convex set but does not change any point inthe closed convex set. When you have a set A which is a subset of a set B which has theproperty that continuous functions f : B→ B have fixed points, and there is a continuousmap P from B to A which leaves points of A unchanged, then it follows that A will have thesame “fixed point property”. You can probably imagine all sorts of sets which are retractsof closed convex bounded sets. Also, if you have a compact set B which has the fixed pointproperty and h : B→ h(B) with h one to one and continuous, it will follow that h−1 iscontinuous and that h(B) will also have the fixed point property. This is very easy to show.This will allow further extensions of this theorem. This says that the fixed point propertyis topological.

Several of the following theorems are generalizations of the Brouwer fixed point theo-rem.

6.4 The Schauder Fixed Point TheoremFirst we give a proof of the Schauder fixed point theorem which is an infinite dimensionalgeneralization of the Brouwer fixed point theorem. This is a theorem which lives in Banachspace. Recal that one of these is a complete normed vector space. There is also a versionof this theorem valid in locally convex topological vector spaces where the theorem issometimes called Tychonoff’s theorem. In infinite dimensions, the closed unit ball fails tohave the fixed point property. Thus something more is needed to get a fixed point.

We let K be a closed convex subset of X a Banach space and let

f be continuous, f : K→ K, and f (K) is compact.

Lemma 6.4.1 For each r > 0 there exists a finite set of points

{y1, · · · ,yn} ⊆ f (K)