Non-Archimedean Norm iff Non-Archimedean Metric/Necessary Condition
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Theorem
Let $\struct {R, \norm {\,\cdot\,}}$ be a non-Archimedean normed division ring.
Let $d$ be the metric induced by $\norm {\,\cdot\,}$.
Then $d$ is a non-Archimedean metric.
Proof
By Metric Induced by Norm on Normed Division Ring is Metric then $d$ satisfies the metric space axioms $(\text M 1)$ to $(\text M 4)$.
To complete the proof, all that remains is to show that $d$ is non-Archimedean.
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 $\norm {\,\cdot\,}$ |
$\blacksquare$
Sources
- 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction: $\S 2.3$: Topology