Solution
= Solution
Put $c=\|\phi\|=\phi(1)\geq0$. If $c=0$, the <linear functional> is zero and is positive. Otherwise, for a nonzero $g\geq0$ let $h=g/\|g\|_\infty$. Then $0\leq h\leq1$ and $\|1-h\|_\infty\leq1$, so boundedness gives
$$
\phi(h)=c-\phi(1-h)\geq c-|\phi(1-h)|\geq0.
$$
Positive rescaling gives $\phi(g)\geq0$. Thus \b[$\phi$ is a <positive linear functional>]. This is the scalar form of the <unital contraction positivity criterion>.