Axiom:Submultiplicative Norm Axioms

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This page is about the submultiplicative norm axioms. For other uses, see 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:      \(\ds \forall x, y \in R:\)    \(\ds \norm {x + y} \)   \(\ds \le \)   \(\ds \norm x + \norm y \)      

Also see