Non-Archimedean Norm iff Non-Archimedean Metric

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Theorem

Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring with zero $0$.

Let $d$ be the metric induced by $\norm {\,\cdot\,}$.


Then:

$\norm {\, \cdot \,}$ is a non-Archimedean norm if and only if $d$ is a non-Archimedean metric.


Proof

Necessary Condition

Let $x, y, z \in R$.

\(\ds \map d {x, y}\) \(=\) \(\ds \norm {x - y}\) Definition of Metric Induced by $\norm {\,\cdot\,}$
\(\ds \) \(=\) \(\ds \norm {\paren {x - z} + \paren {z - y} }\)
\(\ds \) \(\le\) \(\ds \max \set {\norm {x - z}, \norm {z - y} }\) Definition of Non-Archimedean Division Ring Norm
\(\ds \) \(=\) \(\ds \max \set {\map d {x, z}, \map d {z, y} }\) Definition of Metric Induced by $\norm {\,\cdot\,}$

$\Box$


Sufficient Condition

Let $x, y \in R$.

\(\ds \norm {x + y}\) \(=\) \(\ds \norm {x - \paren {-y} }\)
\(\ds \) \(=\) \(\ds \map d {x, - y}\) Definition of Metric Induced by $\norm {\, \cdot \,}$
\(\ds \) \(\le\) \(\ds \max \set {\map d {x, 0}, \map d {0, -y} }\) Definition of Non-Archimedean Metric
\(\ds \) \(=\) \(\ds \max \set {\norm {x - 0 }, \norm {0 - \paren {-y} } }\) Definition of Metric Induced by $\norm {\, \cdot \,}$
\(\ds \) \(=\) \(\ds \max \set {\norm x, \norm y}\)

$\blacksquare$


Sources

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