Axiom:Non-Archimedean Norm Axioms
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Definition
Non-Archimedean Norm Axioms (Division Ring)
Let $\struct {R, +, \circ}$ be a division ring whose zero is $0_R$.
Let $\norm {\, \cdot \,}: R \to \R_{\ge 0}$ be a non-Archimedean norm on $R$.
The non-Archimedean norm axioms are the conditions on $\norm {\, \cdot \,}$ which are satisfied for all elements of $R$ in order for $\norm {\, \cdot \,}$ to be a non-Archimedean 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: | \(\ds \forall x, y \in R:\) | \(\ds \norm {x \circ y} \) | \(\ds = \) | \(\ds \norm x \times \norm y \) | |||
\((\text N 4)\) | $:$ | Ultrametric Inequality: | \(\ds \forall x, y \in R:\) | \(\ds \norm {x + y} \) | \(\ds \le \) | \(\ds \max \set {\norm x, \norm y} \) |
Non-Archimedean Norm Axioms (Vector Space)
Let $\struct {R, +, \circ}$ be a division ring with norm $\norm {\,\cdot\,}_R$.
Let $X$ be a vector space over $R$, with zero $0_X$.
Let $\norm {\, \cdot \,}: X \to \R_{\ge 0}$ be a mapping from $X$ to the non-negative reals.
$\norm {\, \cdot \,}$ is satisfies the non-Archimedean norm axioms if and only if $\norm {\, \cdot \,}$ satisfies the following contitions:
\((\text N 1)\) | $:$ | Positive Definiteness: | \(\ds \forall x \in X:\) | \(\ds \norm x = 0 \) | \(\ds \iff \) | \(\ds x = 0_R \) | |||
\((\text N 2)\) | $:$ | Positive Homogeneity: | \(\ds \forall x \in X, \lambda \in R:\) | \(\ds \norm {\lambda x} \) | \(\ds = \) | \(\ds \norm {\lambda}_R \times \norm x \) | |||
\((\text N 4)\) | $:$ | Ultrametric Inequality: | \(\ds \forall x, y \in X:\) | \(\ds \norm {x + y} \) | \(\ds \le \) | \(\ds \max \set {\norm x, \norm y} \) |