Definition:Norm/Algebra

This page is about Norm on Algebra over Division Ring. For other uses, see Norm.

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 $a, b \in A: \norm {a b} \le \norm a \cdot \norm b$

Let $A$ be an unital algebra over $R$ with unit $e$.

A norm on $A$ is an algebra norm $\norm{\,\cdot\,}: A \to \R_{\ge 0}$ such that:

$\norm e = 1$