Definition:Equivalent Division Ring Norms/Topologically Equivalent

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 topologically equivalent metrics.

That is:

Let $\tau_1$ and $\tau_2$ be the topologies induced by the norms $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ respectively.

By the definition of a topology induced by a norm, $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ are equivalent if and only if $\tau_1 = \tau_2$.

Sources

• 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction ... (previous) ... (next): $\S 3.1$: Absolute Values on $\Q$: Definition $3.1.1$