Characterisation of Non-Archimedean Division Ring Norms/Corollary 5

From ProofWiki
Jump to: navigation, search

Theorem

Let $\struct{R, \norm{\,\cdot\,}}$ be a normed division ring with unity $1_R$.


If $\norm{\,\cdot\,}$ is non-Archimedean then:

$\sup \set {\norm{n \cdot 1_R}: n \in \Z} = 1$.


where $n \cdot 1_R = \begin{cases} \underbrace {1_R + 1_R + \dots + 1_R}_{n \, times} &\mbox{if } n \gt 0 \\ 0 &\mbox{if } n = 0 \\ \\ -\underbrace {\paren {1_R + 1_R + \dots + 1_R}}_{-n \, times} &\mbox{if } n \lt 0 \\ \end{cases}$


Proof

By Corollary 1 of Characterisation of Non-Archimedean Division Ring Norms then:

$\sup \set {\norm{n \cdot 1_R}: n \in \N_{\gt 0}} = 1$.

By Norm Axiom (N1) (Positive Definiteness) then:

$\norm{0 \cdot 1_R} = 0 \le 1$


Let $n \lt 0$ then:

\(\displaystyle \norm{n \cdot 1_R}\) \(=\) \(\displaystyle \norm{ -\underbrace {\paren {1_R + 1_R + \dots + 1_R} }_{-n \, times} }\)
\(\displaystyle \) \(\) \(\displaystyle \)
\(\displaystyle \) \(=\) \(\displaystyle \norm{\underbrace {1_R + 1_R + \dots + 1_R}_{-n \, times} }\) Norm of Negative
\(\displaystyle \) \(\) \(\displaystyle \)
\(\displaystyle \) \(=\) \(\displaystyle \norm{\paren {-n} \cdot 1_R}\)
\(\displaystyle \) \(\) \(\displaystyle \)
\(\displaystyle \) \(\le\) \(\displaystyle 1\) Characterisation of Non-Archimedean Division Ring Norms


The result follows.

$\blacksquare$

Sources