Properties of Norm on Division Ring/Norm of Negative
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Theorem
Let $\struct {R, +, \circ}$ be a division ring with unity $1_R$.
Let $\norm {\,\cdot\,}$ be a norm on $R$.
Let $x \in R$.
Then:
- $\norm {-x} = \norm x$
Proof
- $\norm {-1_R} = 1$
Then:
\(\ds \norm {-x}\) | \(=\) | \(\ds \norm {-1_R \circ x}\) | Product with Ring Negative | |||||||||||
\(\ds \) | \(=\) | \(\ds \norm {-1_R} \norm x\) | Norm Axiom $\text N 2$: Multiplicativity | |||||||||||
\(\ds \) | \(=\) | \(\ds \norm x\) | Norm of Negative of Unity |
as desired.
$\blacksquare$
Sources
- 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction ... (previous) ... (next): $\S 2.2$: Basic Properties, Lemma $2.2.1 \ \text {iv)}$
- 2007: Svetlana Katok: p-adic Analysis Compared with Real ... (previous) ... (next): $\S 1.2$: Normed Fields, Theorem $1.6 \ \text {(b)}$