PlanetMath: normed algebra

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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.