Definition:Norm on Algebra

Definition
Let $R$ be a division ring with absolute value $|\cdot|$.

Let $A$ be an algebra over $R$.

A norm on $A$ is a norm:
 * $\left\Vert{\cdot}\right\Vert: A \to \R_{\ge 0}$

on $A$ as a vector space such that for all $a,b \in A$:
 * $\left\Vert ab \right\Vert \leq \left\Vert a \right\Vert \cdot \left\Vert b\right\Vert$

Also see

 * Definition:Normed Algebra