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