# Properties of Norm on Division Ring/Norm of Inverse

## 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 $x \in R$

Then:

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

## Proof

Let $x \ne 0_R$.

$\norm x \ne 0$

So:

 $\displaystyle \norm x \norm {x^{-1} }$ $=$ $\displaystyle \norm {x \circ x^{-1} }$ Norm axiom (N2) (Multiplicativity) $\displaystyle$ $=$ $\displaystyle \norm {1_R}$ Definition of Product Inverse $\displaystyle$ $=$ $\displaystyle 1$ Norm of Unity $\displaystyle \leadsto \ \$ $\displaystyle \norm {x^{-1} }$ $=$ $\displaystyle \dfrac 1 {\norm x}$

$\blacksquare$