# Extreme points

1. Oct 28, 2009

### Ivah

1. The problem statement, all variables and given/known data

Show that if X is a locally compact space but not compact,

then $B(C_0 (X))$ has no extreme points,

in which $B(X)=\{ x | \; ||x|| \leq 1 \}$ and

$C_0(X)$ = all continuous function $f: X \rightarrow \mathbb{F}$ ( with $\mathbb{F}$ the complex plain or the real line) such that for all $\epsilon>0$, $\{x \in X : \; |f(x)| \geq \epsilon \}$ is compact.

2. Relevant equations

3. The attempt at a solution

One desires to proof that every $z \in B(C_0(X))$

can be written as a convex combination of $y_1 , y_2 \in B(C_0(X))$, i.e. $y_1+y_2=z$. The tricky part is finding such $y_1,y_2$.

The only thing I have proved so far is the existence of a converging net $x_i \rightarrow x$ such that $f(x_i) \rightarrow 0$, anybody got an idea how I can use this to prove the above.