Equivalence of Definitions of Equivalent Division Ring Norms
Theorem
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.
The following definitions of the concept of Equivalent Division Ring Norms are equivalent:
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 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 Power of 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 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 $\norm{\,\cdot\,}_2$
Proof
Topologically Equivalent implies Convergently Equivalent
Let $d_1$ and $d_2$ be topologically equivalent metrics.
Then:
- $d_1$ and $d_2$ are convergently equivalent metrics.
$\Box$
Convergently Equivalent implies Null Sequence Equivalent
Let $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ satisfy:
- for all sequences $\sequence {x_n}$ in $R:\sequence {x_n}$ converges to $l$ in $\norm{\, \cdot \,}_1 \iff \sequence {x_n}$ is a converges to $l$ in $\norm {\, \cdot \,}_2$
Then 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 $\norm {\, \cdot \,}_2$
$\Box$
Null Sequence Equivalent implies Open Unit Ball Equivalent
Let $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ satisfy:
- 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 $\norm {\, \cdot \,}_2$
Then $\forall x \in R$:
- $\norm x_1 < 1 \iff \norm x_2 < 1$
$\Box$
Open Unit Ball Equivalent implies Norm is Power of Other Norm
Let $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ satisfy:
- $\forall x \in R: \norm x_1 < 1 \iff \norm x_2 < 1$
Then:
- $\exists \alpha \in \R_{> 0}: \forall x \in R: \norm x_1 = \norm x_2^\alpha$
$\Box$
Norm is Power of Other Norm implies Topologically Equivalent
Let $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ satisfy:
- $\exists \alpha \in \R_{\gt 0}: \forall x \in R: \norm x_1 = \norm x_2^\alpha$
Then $d_1$ and $d_2$ are topologically equivalent metrics.
$\Box$
Norm is Power of Other Norm implies Cauchy Sequence Equivalent
Let $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ satisfy:
- $\exists \alpha \in \R_{> 0}: \forall x \in R: \norm x_1 = \norm x_2^\alpha$
Then for all sequences $\sequence {x_n}$ in $R$:
- $\sequence {x_n}$ is a Cauchy sequence in $\norm {\, \cdot \,}_1$ if and only if $\sequence {x_n}$ is a Cauchy sequence in $\norm {\, \cdot \,}_2$
$\Box$
Cauchy Sequence Equivalent implies Open Unit Ball Equivalent
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 $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ satisfy:
- for all sequences $\sequence {x_n}$ in $R$: $\sequence {x_n}$ is a Cauchy sequence in $\norm {\, \cdot \,}_1$ if and only if $\sequence {x_n}$ is a Cauchy sequence in $\norm {\, \cdot \,}_2$
Then $\forall x \in R$:
- $\norm x_1 < 1 \iff \norm x_2 < 1$
$\blacksquare$
Sources
- 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction ... (previous) ... (next): $\S 3.1$ Absolute Values on $\Q$: Lemma $3.1.2$ and Problem $66$
- 2007: Svetlana Katok: p-adic Analysis Compared with Real ... (previous) ... (next): $\S 1.2$ Normed Fields: Proposition $1.10$ and Exercise $13$