normed algebra
A ring $A$ is said to be a normed ring if $A$ possesses a norm $\parallel \cdot \parallel $, that is, a nonnegative realvalued function $\parallel \cdot \parallel :A\to \mathbb{R}$ such that for any $a,b\in A$,

1.
$\parallel a\parallel =0$ iff $a=0$,

2.
$\parallel a+b\parallel \le \parallel a\parallel +\parallel b\parallel $,

3.
$\parallel a\parallel =\parallel a\parallel $, and

4.
$\parallel ab\parallel \le \parallel a\parallel \parallel b\parallel $.
Remarks.

•
If $A$ contains the multiplicative identity^{} $1$, then $$ and so $1\le \parallel 1\parallel $.

•
However, it is usually required that in a normed ring, $\parallel 1\parallel =1$.

•
$\parallel \cdot \parallel $ defines a metric $d$ on $A$ given by $d(a,b)=\parallel ab\parallel $, so that $A$ with $d$ is a metric space and one can set up a topology^{} on $A$ by defining its subbasis a collection of $$ called open balls for any $a\in A$ and $r>0$. With this definition, it is easy to see that $\parallel \cdot \parallel $ is continuous.

•
Given a sequence $\{{a}_{n}\}$ of elements in $A$, we say that $a$ is a limit point^{} of $\{{a}_{n}\}$, if
$$\underset{n\to \mathrm{\infty}}{lim}\parallel {a}_{n}a\parallel =0.$$ By the triangle inequality^{}, $a$, if it exists, is unique, and so we also write
$$a=\underset{n\to \mathrm{\infty}}{lim}{a}_{n}.$$ 
•
In addition, the last condition ensures that the ring multiplication is continuous.
An algebra $A$ over a field $k$ is said to be a normed algebra if

1.
$A$ is a normed ring with norm $\parallel \cdot \parallel $,

2.
$k$ is equipped with a valuation^{} $\cdot $, and

3.
$\parallel \alpha a\parallel =\alpha \parallel a\parallel $ for any $\alpha \in k$ and $a\in A$.
Remarks.

•
Alternatively, a normed algebra $A$ can be defined as a normed vector space^{} with a multiplication defined on $A$ such that multiplication is continuous with respect to the norm $\parallel \cdot \parallel $.

•
Typically, $k$ is either the reals $\mathbb{R}$ or the complex numbers^{} $\u2102$, and $A$ is called a real normed algebra or a complex normed algebra correspondingly.

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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.

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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.
References
 1 M. A. Naimark: Normed Rings, Noordhoff, (1959).
 2 C. E. Rickart: General Theory of Banach Algebras, Van Nostrand, 1960.
Title  normed algebra 
Canonical name  NormedAlgebra 
Date of creation  20130322 16:11:38 
Last modified on  20130322 16:11:38 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  13 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 46H05 
Related topic  GelfandTornheimTheorem 
Related topic  SuperfieldsSuperspace 
Defines  normed ring 
Defines  topological algebra 
Defines  real normed algebra 
Defines  complex normed algebra 