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

By Norm of Negative of Unity then:

$\norm{-1_R} = 1$.


Then:

\(\displaystyle \norm{-x}\) \(=\) \(\displaystyle \norm{-1_R \circ x}\) Product with Ring Negative
\(\displaystyle \) \(=\) \(\displaystyle \norm{-1_R} \norm{x}\) Norm axiom (N2) (Multiplicativity)
\(\displaystyle \) \(=\) \(\displaystyle \norm{x}\) Norm of Negative of Unity

as desired.

$\blacksquare$


Sources