Minimality of the supremum norm on C(K) ($\Vert f{\Vert }_{\infty }\le \Vert f{\Vert }_1$)

Any submultiplicative algebra norm on all of $C(K)$ dominates its supremum norm, even when the new norm is incomplete. Every evaluation character extends continuously to the completion: the restrictions of the completion's character space form a closed subset of $K$, and an omitted point would give a nonzero function annihilated by an invertible function in the completion.