Definition:Norm/Algebra
< Definition:Norm(Redirected from Definition:Norm on Algebra)
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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 property And 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 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:
- $\norm e = 1$
Also see
Sources
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