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:

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

$\ds \norm {x + y}$

$\le$

$\ds \max \set {\norm x, \norm y}$

$\ds \leadstoandfrom \ \ $

$\ds \norm {x + y} \norm {y^{-1} }$

$\le$

$\ds \max \set {\norm x \norm {y^{-1} }, \norm y \norm {y^{-1} } }$

Multiply through by $\norm{y^{-1} }$

$\ds \leadstoandfrom \ \ $

$\ds \norm {\paren {x + y} y^{-1} }$

$\le$

$\ds \max \set {\norm {x y^{-1} }, \norm {y y^{-1} } }$

$\ds \leadstoandfrom \ \ $

$\ds \norm {\paren {x y^{-1} + y y^{-1} } }$

$\le$

$\ds \max \set {\norm {x y^{-1} }, \norm {y y^{-1} } }$

$\ds \leadstoandfrom \ \ $

$\ds \norm {\paren {x y^{-1} + 1_R } }$

$\le$

$\ds \max \set {\norm {x y^{-1} }, \norm {1_R } }$

$\ds \leadstoandfrom \ \ $

$\ds \norm {\paren {x y^{-1} + 1_R } }$

$\le$

$\ds \max \set {\norm {x y^{-1} }, 1 }$

$\blacksquare$

2007: Svetlana Katok: p-adic Analysis Compared with Real: $\S 1.2$ Normed Fields, Proposition $1.14$
1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction: $\S 2.2$ Basic Properties, Theorem $2.2.2$