Norm of a positive functional on C(K)
= Norm of a positive functional on C(K)
{title2=$\|\phi\|=\phi(1)$}
For a real <positive linear functional> on the <space of continuous functions on a compact space>, $-\|f\|_\infty1\le f\le\|f\|_\infty1$ gives $|\phi(f)|\le\phi(1)\|f\|_\infty$. Evaluation at $1$ gives equality of the <operator norm> and $\phi(1)$. Thus positivity alone already implies continuity in the <supremum norm>.