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 on $A$ as a vector space:
 * $\left\Vert{\cdot}\right\Vert: A \to \R_{\ge 0}$

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

Norm on unital algebra
Let $A$ be an unital algebra over $R$ with unity $e$.

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

such that:
 * 1) for all $a,b \in A$: $\left\Vert ab \right\Vert \leq \left\Vert a \right\Vert \cdot \left\Vert b\right\Vert$
 * 2) $\left\Vert e \right\Vert = 1$

Also see

 * Definition:Normed Algebra
 * Definition:Normed Unital Algebra