Normed Algebras and Seminormed Algebras

Normed Algebras and Seminormed Algebras

Algebra Norms and Algebra Seminorms

Recall that if $X$ is a linear space then a norm on $X$ is a function $\| \cdot \| : X \to [0, \infty)$ that satisfies the following properties:

• 1. $\| x \| = 0$ if and only if $x = 0$.
• 2. $\| \alpha x \| = |\alpha| \| x \|$ for all $x \in X$ and for all $\alpha \in \mathbf{F}$.
• 3. $\| x + y \| \leq \| x \| + \| y \|$ for all $x, y \in X$.

The pair $(X, \| \cdot \|)$ is called a normed linear space. If a function $p : X \to [0, \infty)$ satisfies only properties 2 and 3 then $p$ is said to be a seminorm on $X$ and the pair $(X, p)$ is said to be a seminormed linear space.

We have recently defined what an algebra is - a linear space $\mathfrak{A}$ with an additional operation $\cdot : \mathfrak{A} \times \mathfrak{A} \to \mathfrak{A}$ which satisfies:

• 1. $a \cdot (b \cdot c) = (a \cdot b) \cdot c$ for all $a, b, c \in \mathfrak{A}$.
• 2. $a \cdot (b + c) = a \cdot b + a \cdot c$ for all $a, b, c \in \mathfrak{A}$.
• 3. $(\alpha a ) \cdot b = \alpha (a \cdot b) = a \cdot (\alpha b)$, for all $a, b, c \in X$ and all $\alpha \in \mathbf{F}$.

We will now extend the concept of a norm and seminorm on a linear space to a norm and seminorm on an algebra. We will case these algebra norms and algebra seminorms.

 Definition: Let $\mathfrak{A}$ be a be an algebra. A function $\| \cdot \| : \mathfrak{A} \to [0, \infty)$ is said to be an Algebra Norm (or simply Norm) on $\mathfrak{A}$ if $p$ satisfies the following properties: 1) $\| a \| = 0$ if and only if $a = 0$. 2) $\| \alpha a \| = |\alpha| \| a \|$ for all $a \in \mathfrak{A}$ and for all $\alpha \in \mathbf{F}$. 3) $\| a + b \| \leq \| a \| + \| b \|$ for all $a, b \in \mathfrak{A}$. 4) $\| a \cdot b \| \leq \| a \| \| b \|$ for all $a, b \in \mathfrak{A}$. A function $\| \cdot \| : \mathfrak{A} \to [0. \infty)$ is said to be an Algebra Seminorm on $\mathfrak{A}$ if $\| \cdot \|$ satisfies properties 2, 3, and 4.

Normed Algebras and Seminormed Algebras

 Definition: A Normed Algebra is a pair consisting of an algebra $\mathfrak{A}$ and an algebra norm $\| \cdot \|$ on $\mathfrak{A}$. A Seminormed Algebra is a pair consisting of an algebra $\mathfrak{A}$ and a seminormed algebra on $\mathfrak{A}$.

Let $\mathfrak{A}$ be a normed algebra. Then $\mathfrak{A}$ can be viewed as a metric space with the metric defined by:

(1)
\begin{align} \quad d(a, b) = \| a - b \| \end{align}
 Definition: A Complete Normed Algebra or Banach Algebra is a normed algebra $\mathfrak{A}$ for which $X$ is complete as a metric space with metric $d(a, b) = \| a - b \|$.