= Compact-disk algebra norm on entire functions
{title2=$\|f\|=\sup_{|z|\leq1}|f(z)|$}
The displayed <supremum norm> is finite on every <entire function>, positive on every nonzero one by the <identity theorem>, and submultiplicative under pointwise multiplication. It is not complete: partial geometric sums $\sum_{j=0}^n(z/2)^j$ are Cauchy on the disk but their limit there extends meromorphically with a pole at two. More generally, no complete <algebra norm> exists on all <entire functions> because the coordinate function has algebraic <spectrum of an element> equal to the whole complex plane.
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