= Continuous functions on the complex plane admit no algebra norm
Take disjoint disks escaping to infinity and continuous cutoff functions $h_n$ supported in them and equal to one on smaller disks. The locally finite function $f=\sum_n n h_n$ is continuous. Choose nonzero $g_n$ supported in the smaller disks; then $fg_n=ng_n$. Any <algebra norm> would give $n\|g_n\|=\|fg_n\|\le\|f\|\|g_n\|$, forcing $n\le\|f\|$ for every positive integer. This is impossible even without a completeness assumption.
Back to article page
