# Characterisation of Non-Archimedean Division Ring Norms/Sufficient Condition/Lemma 1

## Theorem

Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with unity $1_R$.

Let $x, y \in R$.

Let $y \ne 0_R$ where $0_R$ is the zero of $R$.

Then:

$\norm {x + y} \le \max \set {\norm x, \norm y} \iff \norm {x y^{-1} + 1_R} \le \max \set {\norm {x y^{-1} }, 1}$

## Proof

 $\displaystyle \norm {x + y}$ $\le$ $\displaystyle \max \set {\norm x, \norm y}$ $\displaystyle \leadstoandfrom \ \$ $\displaystyle \norm {x + y} \norm {y^{-1} }$ $\le$ $\displaystyle \max \set {\norm x \norm {y^{-1} }, \norm y \norm {y^{-1} } }$ Multiply through by $\norm{y^{-1} }$ $\displaystyle \leadstoandfrom \ \$ $\displaystyle \norm {\paren {x + y} y^{-1} }$ $\le$ $\displaystyle \max \set {\norm {x y^{-1} }, \norm {y y^{-1} } }$ Norm Axiom $\text N 2$: Multiplicativity $\displaystyle \leadstoandfrom \ \$ $\displaystyle \norm {\paren {x y^{-1} + y y^{-1} } }$ $\le$ $\displaystyle \max \set {\norm {x y^{-1} }, \norm {y y^{-1} } }$ Ring axiom $\text D$: Product is Distributive over Addition $\displaystyle \leadstoandfrom \ \$ $\displaystyle \norm {\paren {x y^{-1} + 1_R } }$ $\le$ $\displaystyle \max \set {\norm {x y^{-1} }, \norm {1_R } }$ Product with Inverse is Unit $\displaystyle \leadstoandfrom \ \$ $\displaystyle \norm {\paren {x y^{-1} + 1_R } }$ $\le$ $\displaystyle \max \set {\norm {x y^{-1} }, 1 }$ Norm of Unity

$\blacksquare$