18.5. THE TYCHONOFF AND SCHAUDER FIXED POINT THEOREMS 503

and let h : Σn−1→ Σn−1 be given by

h(a) j ≡ ψ j

(f

(n

∑i=1

aiyi

)).

Since h is continuous, the Brouwer fixed point theorem applies and we see there exists afixed point for h which is a solution to 18.5.23.

Theorem 18.5.8 Let K be a closed and convex subset of X, a locally convex topologicalvector space in which every point is closed. Let f : K → K be continuous and supposef (K) is compact. Then f has a fixed point.

Proof: First consider the following claim which will yield a candidate for the fixedpoint. Recall that f (xU )− fU (xU ) ∈U and fU (xU ) = xU with xU ∈ convex hull of f (K)⊆K.

Claim: There exists x ∈ f (K) with the property that if V ∈B0, there exists U ⊆ V ,U ∈B0, such that

f (xU ) ∈ x+V.

Proof of the claim: If no such x exists, then for each x ∈ f (K), there exists Vx ∈B0such that whenever U ⊆Vx, with U ∈B0,

f (xU ) /∈ x+Vx.

Since f (K) is compact, there exist x1, · · · ,xn ∈ f (K) such that

{xi +Vxi}ni=1

cover f (K). LetU ∈B0, U ⊆ ∩n

i=1Vxi

and consider xU .f (xU ) ∈ xi +Vxi

for some i because these sets cover f (K) and f (xU ) is something in f (K). But U ⊆Vxi , acontradiction. This shows the claim.

Now I show x is the desired fixed point. Let W ∈B0 and let V ∈B0 with

V +V +V ⊆W .

Since f is continuous at x, there exists V0 ∈B0 such that

V0 +V0 ⊆V

and ify− x ∈V0 +V0,

thenf (x)− f (y) ∈V.