Submultiplicativity
= Submultiplicativity
{title2=$\|ab\|\le\|a\|\|b\|$}
An <algebra norm> is submultiplicative when it satisfies the displayed product inequality. Induction gives $\|a_1\cdots a_n\|\le\prod_j\|a_j\|$, and in particular $\|a^n\|\le\|a\|^n$. The <operator norm> has this property because $\|STv\|\le\|S\|\|T\|\|v\|$. The <supremum norm> for bounded functions with pointwise multiplication is another example. This estimate supports the <Neumann series> and the <spectral radius formula>.