= Entire function algebra admits no complete algebra norm
{title2=$\|f\|=\sup_{|z|\le1}|f(z)|$}
The displayed formula is a submultiplicative norm on the algebra of <entire functions>: definiteness follows from the <identity theorem>. No submultiplicative norm on the whole algebra can be complete. The coordinate function $Z(z)=z$ has every complex number in its algebraic <spectrum of an element>, because $Z-\lambda$ vanishes somewhere and has no entire inverse. This contradicts compactness of spectra in a <Banach algebra>.
Back to article page
