Equivalence of Definitions of Equivalent Division Ring Norms/Null Sequence Equivalent implies Open Unit Ball Equivalent

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 $\norm {\, \cdot \,}_1$ 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 $\forall x \in R$:

$\norm x_1 < 1 \iff \norm x_2 < 1$

Proof

Let $x \in R$.

Let $\sequence {x_n}$ be the sequence defined by: $\forall n: x_n = x^n$.

\(\ds \norm x_1 < 1 \quad\)

\(\leadstoandfrom\)

\(\ds \)

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

\(\quad\) Sequence of Powers of Number less than One in Normed Division Ring

\(\ds \)

\(\leadstoandfrom\)

\(\ds \)

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

\(\quad\) Assumption

\(\ds \)

\(\leadstoandfrom\)

\(\ds \)

$\norm x_2 < 1$

\(\quad\) Sequence of Powers of Number less than One in Normed Division Ring

$\blacksquare$

Sources

• 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction: $\S 3.1$ Absolute Values on $\Q$, Lemma 3.1.2 and Problem 66