Properties of Norm on Division Ring

Theorem

Let $\struct {R, +, \circ}$ be a division ring with zero $0_R$ and unity $1_R$.

Let $\norm{\,\cdot\,}$ be a norm on $R$.

Let $x, y \in R$.

Then the following hold:

Norm of Unity

$\norm {1_R} = 1$.

Norm of Negative of Unity

$\norm{-1_R} = 1$

Norm of Negative

$\norm {-x} = \norm {x}$

Norm of Difference

$\norm {x - y} \le \norm x + \norm y$

Norm of Inverse

$x \ne 0_R \implies \norm {x^{-1} } = \dfrac 1 {\norm x}$

Norm of Quotient

$y \ne 0_R \implies \norm{xy^{-1}} = \norm{y^{-1}x} = \dfrac {\norm{x}}{\norm{y}}$

Norm of Power Equals Unity

$\forall n \in \N_{\gt 0}: \norm {x^n} = 1 \implies \norm x = 1$

Norm of Integer

For all $n \in \N_{\gt 0}$, let $n \cdot 1_R$ denote the sum of $1_R$ with itself $n$-times. That is:

$n \cdot 1_R = \underbrace {1_R + 1_R + \dots + 1_R}_{n \, times}$

Then:

$\norm {n \cdot 1_R} \le n$.