Axiom:Submultiplicative Norm Axioms
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This page is about the submultiplicative 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 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: | \(\ds \forall x, y \in R:\) | \(\ds \norm {x + y} \) | \(\ds \le \) | \(\ds \norm x + \norm y \) |