Properties of Norm on Division Ring/Norm of Difference
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Theorem
Let $\struct {R, +, \circ}$ be a division ring.
Let $\norm {\,\cdot\,}$ be a norm on $R$.
Let $x, y \in R$.
Then:
- $\norm {x - y} \le \norm x + \norm y$
Proof
Then:
\(\ds \norm {x - y}\) | \(=\) | \(\ds \norm {x + \paren {-y} }\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds \norm x + \norm {-y}\) | Norm Axiom $\text N 3$: Triangle Inequality | |||||||||||
\(\ds \) | \(=\) | \(\ds \norm x + \norm y\) | Norm of Ring Negative |
as desired.
$\blacksquare$
Sources
- 2007: Svetlana Katok: p-adic Analysis Compared with Real ... (previous) ... (next): $\S 1.2$: Normed Fields, Theorem $1.6 \ \text {(d)}$