# normed algebra

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

1. 1.

$\|a\|=0$ iff $a=0$,

2. 2.

$\|a+b\|\leq\|a\|+\|b\|$,

3. 3.

$\|-a\|=\|a\|$, and

4. 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 $d(a,b)=\|a-b\|$, 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) 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. 1.

$A$ is a normed ring with norm $\|\cdot\|$,

2. 2.

$k$ is equipped with a valuation $|\cdot|$, and

3. 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.
 Title normed algebra Canonical name NormedAlgebra Date of creation 2013-03-22 16:11:38 Last modified on 2013-03-22 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