normed algebra

A ring $A$ is said to be a normed ring if $A$ possesses a norm $\parallel \cdot \parallel$, that is, a non-negative real-valued function $\parallel \cdot \parallel :A\to ℝ$ 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 $d(a,b)=\parallel a-b\parallel$, 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 $ℝ$ or the complex numbers $ℂ$, 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.

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