204 CHAPTER 8. POSITIVE LINEAR FUNCTIONALS

Then if x ∈ BR, it follows

|gk (x)| ≤ |gk (x)− fk (x)|+ |fk (x)|

<R

1+ k+

kR1+ k

= R

and so gk maps BR to BR. By Lemma 8.7.4 each of these gk has a fixed point xk such thatgk (xk) = xk. The sequence of points {xk} is contained in the compact set BR and so thereexists a convergent subsequence still denoted by {xk} which converges to a point x ∈ BR.Then from uniform convergence of gk to f,

f(x) = limk→∞

f(xk) = limk→∞

gk (xk) = limk→∞

xk = x ■

It is not surprising that the ball does not need to be centered at 0.

Corollary 8.7.6 Let f : B(a,R)→ B(a,R) be continuous. Then there exists x ∈ B(a,R)such that f(x) = x.

Proof: Let g : BR→ BR be defined by g(y)≡ f(y+a)−a. Then g is a continuous mapfrom BR to BR. Therefore, there exists y ∈ BR such that g(y) = y. Therefore, f(y+a)−a = y and so letting x = y+a, f also has a fixed point as claimed. ■

Definition 8.7.7 A set A is a retract of a set B if A ⊆ B, and there is a continuousmap h : B→ A such that h(x) = x for all x∈ A and h is onto. B has the fixed point propertymeans that whenever g is continuous and g : B→ B, it follows that g has a fixed point.

Proposition 8.7.8 Let A be a retract of B and suppose B has the fixed point property.Then so does A.

Proof: Suppose f : A→ A. Let h be the retract of B onto A. Then f◦h : B→ B iscontinuous. Thus, it has a fixed point x ∈ B so f(h(x)) = x. However, h(x) ∈ A andf : A→ A so in fact, x ∈ A. Now h(x) = x and so f(x) = x. ■

Recall that every convex compact subset K of Rp is a retract of all of Rp obtained byusing the projection map. See Problems beginning with 22 on Page 77. In particular, Kis a retract of a large closed ball containing K, which ball has the fixed point property.Therefore, K also has the fixed point property. This shows the following which is a conve-nient formulation of the Brouwer fixed point theorem. However, Proposition 8.7.8 is moregeneral. You can probably imagine lots of sets which are retracts of some larger ball.

Theorem 8.7.9 Every convex closed and bounded subset of Rp has the fixed pointproperty.

8.7.2 Invariance of DomainAs an application of the inverse function theorem is a simple proof of the important invari-ance of domain theorem which says that continuous and one to one functions defined on anopen set in Rn with values in Rn take open sets to open sets. You know that this is true forfunctions of one variable because a one to one continuous function must be either strictlyincreasing or strictly decreasing. This will be used when considering orientations of curveslater. However, the n dimensional version isn’t at all obvious but is just as important if you