Axiom:Seminorm Axioms

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 on $V$ if and only if $\norm {\, \cdot \,}$ satisfies the following 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 $

These criteria are called the seminorm axioms.

The $(\text N 2)$ and $(\text N 3)$ labels originate from the fact that these axioms are also used in defining norms.

\((\text N 2)\)	$:$		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:	\(\ds \forall x, y \in V:\)		\(\ds \norm {x + y} \)	\(\ds \le \)	\(\ds \norm x + \norm y \)