 Solution

Put $c=\Vert \varphi \Vert =\varphi (1)\ge 0$. If $c=0$, the linear functional is zero and is positive. Otherwise, for a nonzero $g\ge 0$ let $h=g/\Vert g{\Vert }_{\infty }$. Then $0\le h\le 1$ and $\Vert 1-h{\Vert }_{\infty }\le 1$, so boundedness givesPositive rescaling gives $\varphi (g)\ge 0$. Thus $\varphi$ is a positive linear functional. This is the scalar form of the unital contraction positivity criterion.