Definition:Non-Archimedean/Norm (Division Ring)/Archimedean
< Definition:Non-Archimedean | Norm (Division Ring)(Redirected from Definition:Archimedean Division Ring Norm)
Jump to navigation
Jump to search
Definition
Let $\norm {\, \cdot \,}$ be a norm on a division ring $R$ satisfying the norm axioms:
\((\text N 1)\) | $:$ | Positive Definiteness: | \(\ds \forall x \in R:\) | \(\ds \norm x = 0 \) | \(\ds \iff \) | \(\ds x = 0_R \) | |||
\((\text N 2)\) | $:$ | Multiplicativity: | \(\ds \forall x, y \in R:\) | \(\ds \norm {x \circ y} \) | \(\ds = \) | \(\ds \norm x \times \norm y \) | |||
\((\text N 3)\) | $:$ | Triangle Inequality: | \(\ds \forall x, y \in R:\) | \(\ds \norm {x + y} \) | \(\ds \le \) | \(\ds \norm x + \norm y \) |
A norm $\norm {\, \cdot \,}$ is said to be Archimedean if and only if it does not satisfy the Non-Archimedean Norm Axiom $\text N 4$: Ultrametric Inequality.
That is, $\norm {\, \cdot \,}$ is Archimedean if and only if:
\((\text N 5)\) | $:$ | \(\ds \exists x, y \in R:\) | \(\ds \norm {x + y} \) | \(\ds > \) | \(\ds \max \set {\norm x, \norm y} \) |
Sources
- 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction: $\S 2.1$: Foundations