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

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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

$\blacksquare$


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