PlanetMath: normed algebra

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (205)
Orphanage
Unclass'd
Unproven (490)
Corrections (8)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

normed algebra

(Definition)

A ring is said to be a normed ring if possesses a norm $\Vert \cdot \Vert$ , that is, a non-negative real-valued function $\Vert\cdot \Vert:A\to \mathbb{R}$ such that for any $a,b\in A$ ,

$\Vert a\Vert=0$ iff ,
$\Vert a+b\Vert\le \Vert a\Vert+\Vert b\Vert$ ,
$\Vert-a\Vert=\Vert a\Vert$ , and
$\Vert ab\Vert\le \Vert a\Vert\Vert b\Vert$ .

Remarks.

If contains the multiplicative identity , then $0<\Vert 1\Vert \le\Vert 1\Vert\Vert 1\Vert$ and so $1\le \Vert 1\Vert$ .
However, it is usually required that in a normed ring, $\Vert 1\Vert=1$ .
$\Vert\cdot\Vert$ defines a metric on given by $d(a,b)=\Vert a-b\Vert$ , so that with is a metric space and one can set up a topology on by defining its subbasis a collection of $B(a,r):=\lbrace x\in A\mid d(a,x)< r\rbrace$ called open balls for any $a\in A$ and . With this definition, it is easy to see that $\Vert\cdot\Vert$ is continuous.
Given a sequence $\lbrace a_n\rbrace$ of elements in , we say that is a limit point of $\lbrace a_n\rbrace$ , if
$\displaystyle \lim_{n\to\infty}\Vert a_n-a\Vert=0.$
By the triangle inequality, , if it exists, is unique, and so we also write
$\displaystyle a=\lim_{n\to\infty}a_n.$
In addition, the last condition ensures that the ring multiplication is continuous.

An algebra over a field is said to be a normed algebra if

is a normed ring with norm $\Vert\cdot\Vert$ ,
is equipped with a valuation $\vert \cdot \vert$ , and
$\Vert\alpha a\Vert=\vert\alpha\vert\Vert a\Vert$ for any $\alpha \in k$ and $a\in A$ .

Remarks.

Alternatively, a normed algebra can be defined as a normed vector space with a multiplication defined on such that multiplication is continuous with respect to the norm $\Vert\cdot\Vert$ .
Typically, is either the reals $\mathbb{R}$ or the complex numbers $\mathbb{C}$ , and is called a real normed algebra or a complex normed algebra correspondingly.
A normed algebra that is complete with respect to the norm is called Banach algebra (the underlying field must be complete and algebraically closed), paralleling with the analogy with a Banach space versus a normed vector space.
Normed rings and normed algebras are special cases of the more general notions of a topological ring and a topological algebra, the latter of which is defined as a topological ring over a field such that the scalar multiplication is continuous.