\begin{definition}[Definition:Norm/Algebra]
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 $a, b \in A: \norm {a b} \le \norm a \cdot \norm b$
\end{definition}