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Homenormed algebra
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normed algebra
A ring $A$ is said to be a normed ring if $A$ possesses a norm $\\cdot\$, that is, a nonnegative realvalued function $\\cdot\:A\to\mathbb{R}$ such that for any $a,b\in A$,
1. $\a\=0$ iff $a=0$,
2. $\a+b\\leq\a\+\b\$,
3. $\a\=\a\$, and
4. $\ab\\leq\a\\b\$.
Remarks.

If $A$ contains the multiplicative identity $1$, then $0<\1\\leq\1\\1\$ and so $1\leq\1\$.

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

$\\cdot\$ defines a metric $d$ on $A$ given by $d(a,b)=\ab\$, 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 $B(a,r):=\{x\in A\mid d(a,x)<r\}$ called open balls for any $a\in A$ and $r>0$. With this definition, it is easy to see that $\\cdot\$ is continuous.

Given a sequence $\{a_{n}\}$ of elements in $A$, we say that $a$ is a limit point of $\{a_{n}\}$, if
$\lim_{{n\to\infty}}\a_{n}a\=0.$ By the triangle inequality, $a$, if it exists, is unique, and so we also write
$a=\lim_{{n\to\infty}}a_{n}.$ 
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 $\\cdot\$,
2. $k$ is equipped with a valuation $\cdot$, and
3. $\\alpha a\=\alpha\a\$ 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 $\\cdot\$.

Typically, $k$ is either the reals $\mathbb{R}$ or the complex numbers $\mathbb{C}$, and $A$ 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.
References
 1 M. A. Naimark: Normed Rings, Noordhoff, (1959).
 2 C. E. Rickart: General Theory of Banach Algebras, Van Nostrand, 1960.
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