Non-Archimedean Norm iff Non-Archimedean Metric/Sufficient Condition
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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 $\norm {\, \cdot \,}$.
Let $d$ be non-Archimedean.
Then:
- $\norm {\, \cdot \,}$ is a non-Archimedean norm.
Proof
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 $\norm {\, \cdot \,}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \max \set {\norm x, \norm y}\) |
Hence $\norm {\,\cdot\,}$ is non-Archimedean.
$\blacksquare$
Sources
- 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction: $\S 2.3$: Topology