Properties of Norm on Division Ring/Norm of Inverse

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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 $x \ne 0_R$.

By Norm axiom (N1) (Positive Definiteness) then:

$\norm x \ne 0$

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
\(\displaystyle \leadsto \ \ \) \(\displaystyle \norm {x^{-1} }\) \(=\) \(\displaystyle \dfrac 1 {\norm x}\)

$\blacksquare$


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