# a function in the unit ball of C(X) where X is a compact space is a limit of convex combinations of extreme points

Suppose $f(x)$ is a function in $C(X)$ such that $\|f\|<1-\frac{2}{n}$. Then there exist n extreme points of the unit ball of $C(X)$, $g_1,\ldots,g_n$ such that $f(x)=\frac{1}{n}(g_1(x)+\cdots+g_n(x))$.

I know that $g$ is an extreme point of the unit ball of $C(X)$ if and only if $|g(x)|=1$ for all $x\in X$.

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In the accepted answer here I give references for that fact which is true for all unital $C^\ast$-algebras (Russo-Dye-Gardner). The proof given in Pedersen, Analysis Now, 3.2.23 can easily be read for continuous functions in place of operators. – Martin Jan 3 '13 at 18:18
It is important to work with complex-valued functions here (Norbert pointed this out in the other question). – Martin Jan 3 '13 at 18:31
I got it to work. Thank you – john Jan 4 '13 at 0:36
Maybe you could post your solution as an answer? – Martin Jan 5 '13 at 6:43