18.5. THE TYCHONOFF AND SCHAUDER FIXED POINT THEOREMS 497

18.5 The Tychonoff And Schauder Fixed Point Theo-rems

First 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. After this, we give a generalization to locally convex topological vector spaceswhere the theorem is sometimes called Tychonoff’s theorem. First here is an interestingexample [55].

Exercise 18.5.1 Let B be the closed unit ball in a separable Hilbert space H which isinfinite dimensional. Then there exists continuous f : B→ B which has no fixed point.

Let {ek}∞

k=1 be a complete orthonormal set in H. Let L ∈L (H,H) be defined as fol-lows. Lek = ek+1 and then extend linearly. Then in particular,

L

(∑

ixiei

)= ∑

ixiei+1

Then it is clear that L preserves norms and so it is linear and continuous. Note how thiswould not work at all if the Hilbert space were finite dimensional. Then define f (x) =12 (1−∥x∥H)e1 +Lx. Then if ∥x∥ ≤ 1,

∥ f (x)∥= 12(1−∥x∥)2 +∥Lx∥2 =

12(1−∥x∥)2 +∥x∥2 =

12∥x∥2 +

12≤ 1

and so f : B→ B yet has no fixed point because if it did, you would need to have

x =12(1−∥x∥H)e1 +Lx

and so

∥x∥2 =14(1−∥x∥)2 +∥Lx∥2 =

14(1−∥x∥)2 +∥Lx∥2

=14+

54∥x∥2− 1

2∥x∥

12∥x∥= 1

4+

14∥x∥2

this requires ∥x∥= 1. But then you would need to have x = Lx which is not so because if xis in the closure of the span of {ei}∞

i=m , such that the first nonzero Fourier coefficient is themth, then Lx is in the closure of the span of {ei}∞

i=m+1.This shows you need something other than continuity if you want to get a fixed point.

This also shows that there is a retraction of B onto ∂B in any infinite dimensional separableHilbert space. You get it the usual way. Take the line from x to f (x) and the retraction willbe the function which gives the point on ∂B which is obtained by extending this line tillit hits the boundary of B. Thus for Hilbert spaces, those which have ∂B a retraction of B