Let $X$ be a compact Hausdorff space, $C(X)$ denote the set of all complex-valued continuous functions from $X$. Show that the smallest closed convex set containing the extreme points of the unit ball of $C(X)$ is the unit ball. The problem is from Douglas's book Banach Algebra Techniques in Operator Theory, problem 1.8. For the case $X=[0,1]$, I have shown that the set of all extreme functions are those that have modulus value $1$, but still have no clue how to proceed, even for the case $X=[0,1]$.
Please help.
The extreme points of the unit ball are exactly unitary elements. Now apply the Russo-Dye theorem.