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 $X$ with an additional operation $\cdot : X \times X \to X$ which satisfies $x \cdot (y \cdot z) = (x \cdot y) \cdot z$, $x \cdot (y + z) = x \cdot y + x \cdot z$, and $(\alpha x ) \cdot y = \alpha (x \cdot y) = x \cdot (\alpha y)$, for all $x, y, z \in X$ and all $\alpha \in \mathbf{F}$. We can now look at more advanced types of norms and seminorms - in particular, algebra norms and algebra seminorms.

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

Later, we may use the frequently adopted notation $\| \cdot \|$ instead to "$p$" for algebra norms and algebra seminorms defined on an algebra.

Clearly every algebra norm is a norm and every algebra seminorm is a seminorm.

Normed Algebras and Seminormed Algebras

Definition: A Normed Algebra is a pair consisting of a linear space $X$ and an algebra norm on $X$. A Seminormed Algebra is a pair consisting of a linear space $X$ and a seminormed algebra on $X$.

Suppose that $(X, \| \cdot \|)$ is a normed algebra. Recall that $X$ can be viewed as a metric space with metric $d : X \times X \to [0, \infty)$ defined for all $x, y \in X$ by:

(1)
\begin{align} \quad d(x, y) = \| x - y \| \end{align}

When $X$ as viewed as a metric space is complete, then we say that $X$ is a complete normed algebra or Banach algebra.

Definition: A Complete Normed Algebra or Banach Algebra is a normed algebra $(X, \| \cdot \|)$ for which $X$ is complete as a metric space.
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