Axiom:Norm Axioms

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:

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:

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: