= Minimality of the supremum norm on C(K)
{title2=$\lVert f\rVert_\infty\leq\lVert f\rVert_1$}
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.
Back to article page
