# Definition:Multiplicative Norm Axioms

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*This page is about the multiplicative norm axioms. For other uses, see Norm Axioms.*

## Definition

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: | \(\displaystyle \forall x \in R:\) | \(\displaystyle \norm x = 0 \) | \(\displaystyle \iff \) | \(\displaystyle x = 0_R \) | ||

\((\text N 2)\) | $:$ | Multiplicativity: | \(\displaystyle \forall x, y \in R:\) | \(\displaystyle \norm {x \circ y} \) | \(\displaystyle = \) | \(\displaystyle \norm x \times \norm y \) | ||

\((\text N 3)\) | $:$ | Triangle Inequality: | \(\displaystyle \forall x, y \in R:\) | \(\displaystyle \norm {x + y} \) | \(\displaystyle \le \) | \(\displaystyle \norm x + \norm y \) |

## Also known as

In the discussion of **norms**, it is usually the case that **multiplicative norms** are under consideration.

Hence it is common to speak directly of **norm axioms** without qualification as to their exact variety.