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.