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
- 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction ... (previous) ... (next): $\S 2.3$: Topology: Lemma $2.3.2$