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 $y \neq 0_R$ 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

\(\displaystyle \norm{x + y} \le \max \set{\norm{x}, \norm{y} }\) \(\iff\) \(\displaystyle \norm{x + y} \norm{y^{-1} } \le \max \set{\norm{x} \norm{y^{-1} }, \norm{y} \norm{y^{-1} } }\) Multiply through by $\norm{y^{-1} }$
\(\displaystyle \) \(\) \(\displaystyle \)
\(\displaystyle \) \(\iff\) \(\displaystyle \norm{\paren {x + y} y^{-1} } \le \max \set{\norm{x y^{-1} }, \norm{y y^{-1} } }\) Norm axiom (N2) (Multiplicativity)
\(\displaystyle \) \(\) \(\displaystyle \)
\(\displaystyle \) \(\iff\) \(\displaystyle \norm{\paren {x y^{-1}+ y y^{-1} } } \le \max \set{\norm{x y^{-1} }, \norm{y y^{-1} } }\) Ring axiom (D) (Product is distributive over addition)
\(\displaystyle \) \(\) \(\displaystyle \)
\(\displaystyle \) \(\iff\) \(\displaystyle \norm{\paren {x y^{-1}+ 1_R } } \le \max \set{\norm{x y^{-1} }, \norm{1_R } }\) Product with inverse is unit
\(\displaystyle \) \(\) \(\displaystyle \)
\(\displaystyle \) \(\iff\) \(\displaystyle \norm{\paren {x y^{-1}+ 1_R } } \le \max \set{\norm{x y^{-1} }, 1 }\) Norm of Unity

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