Properties of Norm on Division Ring
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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 $x, y \in R$.
Then the following hold:
Norm of Unity
- $\norm {1_R} = 1$.
Norm of Negative of Unity
- $\norm {-1_R} = 1$
Norm of Negative
- $\norm {-x} = \norm x$
Norm of Difference
- $\norm {x - y} \le \norm x + \norm y$
Norm of Inverse
- $x \ne 0_R \implies \norm {x^{-1} } = \dfrac 1 {\norm x}$
Norm of Quotient
- $y \ne 0_R \implies \norm {x y^{-1} } = \norm {y^{-1} x} = \dfrac {\norm x} {\norm y}$
Norm of Power Equals Unity
- $\forall n \in \N_{>0}: \norm {x^n} = 1 \implies \norm x = 1$
Norm of Integer
For all $n \in \N_{>0}$, let $n \cdot 1_R$ denote the sum of $1_R$ with itself $n$-times. That is:
- $n \cdot 1_R = \underbrace {1_R + 1_R + \dots + 1_R}_{\text {$n$ times} }$
Then:
- $\norm {n \cdot 1_R} \le n$.
Sources
- 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction ... (previous) ... (next): $\S 2.2$: Basic Properties: Lemma $2.2.1$
- 2007: Svetlana Katok: p-adic Analysis Compared with Real ... (previous) ... (next): $\S 1.2$: Normed Fields: Theorem $1.6$