# 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: $\ds \forall x, y \in V:$ $\ds \norm {x + y}$ $\ds \le$ $\ds \norm x + \norm y$