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

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


Then $\forall x \in R$:

$\norm{x}_1 \lt 1 \iff \norm{x}_2 \lt 1$

Proof

The contrapositive is proved.


Let there exist $x \in R$ such that $\norm{x}_1 \lt 1$ and $\norm{x}_2 \ge 1$.

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

By Sequence of Powers of Number less than One in Normed Division Ring then $\sequence {x_n}$ is a null sequence in $\norm{\,\cdot\,}_1$.

By convergent sequence in normed division ring is a Cauchy sequence then $\sequence {x_n}$ is a Cauchy sequence in $\norm{\,\cdot\,}_1$.


Let $0_R$ be the zero of $R$ and $1_R$ be the unit of $R$.

By norm of unity and the assumption $\norm{x}_1 \lt 1$ then $x \neq 1_R$.

Then $x - 1_R \neq 0_R$.

By norm axiom (N1) (Positive Definiteness) then $\norm {x - 1_R}_2 \gt 0$.


Let $\epsilon = \dfrac {\norm {x - 1_R}_2} 2$.


Then $\norm {x - 1_R}_2 \gt \epsilon$.


Since $\norm{x}_2 \ge 1$ then for all $n \in \N$ then:

\(\displaystyle \norm{x_n}_2\) \(=\) \(\displaystyle \norm{x^n}_2\) $\quad$ Definition of $x_n$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \norm{x}_2^n\) $\quad$ norm axiom (N2) (Multiplicativity) $\quad$
\(\displaystyle \) \(\ge\) \(\displaystyle 1\) $\quad$ $\quad$


For all $n \in \N$ then:

\(\displaystyle \norm {x_{n+1} - x_n}_2\) \(=\) \(\displaystyle \norm {x^{n+1} - x^n}_2\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \norm {x^n x - x^n}_2\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \norm {x^n \paren {x - 1_R} }_2\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \norm {x^n}_2 \norm {x - 1_R}_2\) $\quad$ norm axiom (N2) (Multiplicativity) $\quad$
\(\displaystyle \) \(\gt\) \(\displaystyle \epsilon\) $\quad$ $\quad$

So $\sequence {x_n}$ is not a Cauchy sequence in $\norm{\,\cdot\,}_2$.

The theorem now follows by the Rule of Transposition.


$\blacksquare$

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