Axiom:Norm Axioms

Definition

Multiplicative Norm Axioms

Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$.

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

The multiplicative norm axioms are the conditions on $\norm {\, \cdot \,}$ which are satisfied for all elements of $R$ in order for $\norm {\, \cdot \,}$ to be a multiplicative norm:

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

When the concept of norm axioms is raised without qualification, it is usually the case that multiplicative norm axioms are under discussion.

Submultiplicative Norm Axioms

Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$.

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

The submultiplicative norm axioms are the conditions on $\norm {\, \cdot \,}$ which are satisfied for all elements of $R$ in order for $\norm {\, \cdot \,}$ to b a submultiplicative norm:

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

Norm Axioms (Vector Space)

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