 Solution

Positivity of the linear functional gives monotonicity. For any $g\in C(K)$, the pointwise bounds $-\Vert g{\Vert }_{\infty }1\le g\le \Vert g{\Vert }_{\infty }1$ therefore implyThus the positive linear functional is bounded, hence continuous, and its operator norm is at most $\varphi (1)$. Evaluating on the constant function $1$, which has supremum norm one when $K$ is nonempty, gives the reverse inequality. HenceFor empty $K$ the function space and functional are zero, so the same conclusion holds. This is the norm of a positive functional on C(K) identity.