Properties of Norm on Division Ring/Norm of Difference

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 $1.6 \ \text {(d)}$