Definition:Norm/Ring/Submultiplicative
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This page is about Submultiplicative Norm on Ring. For other uses, see Norm.
Definition
Let $\struct {R, +, \circ}$ be a ring whose zero is denoted $0_R$.
A submultiplicative norm on $R$ is a mapping from $R$ to the non-negative reals:
- $\norm {\,\cdot\,}: R \to \R_{\ge 0}$
satisfying the (ring) submultiplicative norm axioms:
\((\text N 1)\) | $:$ | 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: | \(\ds \forall x, y \in R:\) | \(\ds \norm {x + y} \) | \(\ds \le \) | \(\ds \norm x + \norm y \) |