Definition:Equivalent Division Ring Norms

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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.


Topologically Equivalent

$\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ are equivalent if and only if $d_1$ and $d_2$ are topologically equivalent metrics.


Convergently Equivalent

$\norm {\,\cdot\,}_1$ 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 $\norm {\,\cdot\,}_2$


Null Sequence Equivalent

$\norm {\,\cdot\,}_1$ 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 $\norm {\,\cdot\,}_2$


Open Unit Ball Equivalent

$\norm {\,\cdot\,}_1$ and $\norm {\,\cdot\,}_2$ are equivalent if and only if $\forall x \in R: \norm x_1 < 1 \iff \norm x_2 < 1$


Norm is the Power of the Other Norm

$\norm{\,\cdot\,}_1$ and $\norm{\,\cdot\,}_2$ are equivalent if and only if $\exists \alpha \in \R_{\gt 0}: \forall x \in R: \norm{x}_1 = \norm{x}_2^\alpha$


Cauchy Sequence Equivalent

$\norm {\,\cdot\,}_1$ and $\norm {\,\cdot\,}_2$ are equivalent if and only if $d_1$ and $d_2$ are Cauchy equivalent metrics.


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

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


Also see


Sources

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