# Definition:Norm/Algebra

< Definition:Norm(Redirected from Definition:Norm on Algebra)

Jump to navigation
Jump to search
*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 $a, b \in A: \norm {a b} \le \norm a \cdot \norm b$

This article, or a section of it, needs explaining.In particular: Should this not be called submulticative norm? I have never expected this propertyAnd why is the norm of ring submultiplicative, while the norm of field is multiplicative? Are these common? You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Explain}}` from the code. |

### Norm on Unital Algebra

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

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

- $\norm e = 1$

## Also see

## Sources

There are no source works cited for this page.Source citations are highly desirable, and mandatory for all definition pages.Definition pages whose content is wholly or partly unsourced are in danger of having such content deleted. To discuss this page in more detail, feel free to use the talk page. |