Definition:Norm/Algebra

Definition
Let $R$ be a division ring with norm $\norm {\,\cdot\,}_R$.

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

A norm on $A$ is a vector space norm $\norm{\,\cdot\,}: A \to \R_{\ge 0}$ on $A$ as a vector space such that:
 * for all $a, b \in A: \norm {a b} \le \norm a \cdot \norm b$

Also see

 * Definition:Norm on Division Ring
 * Definition:Norm on Vector Space
 * Definition:Norm on Unital Algebra