Properties of Norm on Division Ring

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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

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


Norm of Power Equals Unity

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


Norm of Integer

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

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

Then:

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


Sources