Definition:Equivalent Division Ring Norms/Null Sequence 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 $R$.


$\norm {\,\cdot\,}_1$ and $\norm {\,\cdot\,}_2$ are equivalent if and only if for all sequences $\sequence {x_n}$ in $R$:

$\sequence {x_n}$ is a null sequence in $\norm{\,\cdot\,}_1 \iff \sequence {x_n}$ is a null sequence in $\norm {\,\cdot\,}_2$


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