Definition:Seminorm

Definition

Let $\struct {K, +, \circ}$ be a division ring with norm $\norm {\,\cdot\,}_K$.

Let $V$ be a vector space over $\struct {K, \norm {\,\cdot\,}_K}$, with zero vector $0_V$.

Let $\norm {\, \cdot \,}: V \to \R_{\ge 0}$ be a mapping from $V$ to the positive reals $\R_{\ge 0}$.

The mapping $\norm {\, \cdot \,}$ is a seminorm if and only if $\norm {\, \cdot \,}$ satisfies the seminorm axioms:

$(\text N 2)$	$:$		Positive Homogeneity	$\ds \forall x \in V, \lambda \in K:$		$\ds \norm {\lambda x} $	$\ds = $	$\ds \norm \lambda_K \times \norm x $
$(\text N 3)$	$:$		Triangle Inequality	$\ds \forall x, y \in V:$		$\ds \norm {x + y} $	$\ds \le $	$\ds \norm x + \norm y $

It is usual to define a seminorm when $K$ is $\R$ or $\C$.

In this context, $\norm {\,\cdot\,}_\R$ is the absolute value and $\norm {\,\cdot\,}_\C$ is the modulus.

\((\text N 2)\)	$:$		Positive Homogeneity	\(\ds \forall x \in V, \lambda \in K:\)		\(\ds \norm {\lambda x} \)	\(\ds = \)	\(\ds \norm \lambda_K \times \norm x \)
\((\text N 3)\)	$:$		Triangle Inequality	\(\ds \forall x, y \in V:\)		\(\ds \norm {x + y} \)	\(\ds \le \)	\(\ds \norm x + \norm y \)