Axiom:Norm Axioms (Vector Space)

This page is about the norm axioms on a vector space. For other uses, see Norm Axioms.

Definition

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

Let $V$ be a vector space over $R$, with zero $\mathbf 0_V$.

Let $\norm {\,\cdot\,}: V \to \R_{\ge 0}$ be a norm on $V$.

The norm axioms are the following conditions on $\norm {\,\cdot\,}$ which define $\norm {\,\cdot\,}$ as being a norm:

\((\text N 1)\)	$:$		Positive Definiteness:	\(\ds \forall x \in V:\)		\(\ds \norm x = 0 \)	\(\ds \iff \)	\(\ds x = \mathbf 0_V \)
\((\text N 2)\)	$:$		Positive Homogeneity:	\(\ds \forall x \in V, \lambda \in R:\)		\(\ds \norm {\lambda x} \)	\(\ds = \)	\(\ds \norm {\lambda}_R \times \norm x \)
\((\text N 3)\)	$:$		Triangle Inequality:	\(\ds \forall x, y \in V:\)		\(\ds \norm {x + y} \)	\(\ds \le \)	\(\ds \norm x + \norm y \)

Also see

• Axiom:Multiplicative Norm Axioms
• Axiom:Submultiplicative Norm Axioms