Definition:Equivalent Division Ring Norms/Convergently Equivalent
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Definition
Let $R$ be a division ring.
Let $\norm {\,\cdot\,}_1: R \to \R_{\ge 0}$ and $\norm {\,\cdot\,}_2: R \to \R_{\ge 0}$ be norms on $R$.
Let $d_1$ and $d_2$ be the metrics induced by the norms $\norm {\,\cdot\,}_1$ and $\norm {\,\cdot\,}_2$ respectively.
$\norm {\,\cdot\,}_1$ and $\norm {\,\cdot\,}_2$ are equivalent if and only if $d_1$ and $d_2$ are convergently equivalent metrics
That is, $\norm {\,\cdot\,}_1$ and $\norm {\,\cdot\,}_2$ are equivalent if and only if for all sequences $\sequence {x_n}$ in $R$:
- $\sequence {x_n}$ converges to $l$ in $\norm{\,\cdot\,}_1 \iff \sequence {x_n}$ converges to $l$ in $\norm {\,\cdot\,}_2$
Sources
- 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction: $\S 3.1$ Absolute Values on $\Q$, Lemma 3.1.2 and Problem 66