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 $\text N 1$: Positive Definiteness:
- $\norm x \ne 0$
So:
\(\ds \norm x \norm {x^{-1} }\) | \(=\) | \(\ds \norm {x \circ x^{-1} }\) | Norm Axiom $\text N 2$: Multiplicativity | |||||||||||
\(\ds \) | \(=\) | \(\ds \norm {1_R}\) | Definition of Product Inverse | |||||||||||
\(\ds \) | \(=\) | \(\ds 1\) | Norm of Unity of Division Ring | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \norm {x^{-1} }\) | \(=\) | \(\ds \dfrac 1 {\norm x}\) |
$\blacksquare$
Sources
- 2007: Svetlana Katok: p-adic Analysis Compared with Real ... (previous) ... (next): $\S 1.2$: Normed Fields, Theorem $1.6 \ \text{(e)}$