Sphere in Normed Division Ring is Sphere in Induced Metric
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Theorem
Let $\struct{R, \norm {\,\cdot\,} }$ be a normed division ring.
Let $d$ be the metric induced by the norm $\norm {\,\cdot\,}$.
Let $a \in R$.
Let $\epsilon \in \R_{>0}$ be a strictly positive real number.
Let $\map {S_\epsilon} {a; \norm {\,\cdot\,} }$ denote the sphere in the normed division ring $\struct {R, \norm {\,\cdot\,} }$.
Let $\map {S_\epsilon} {a; d }$ denote the sphere in the metric space $\struct {R, d}$.
Then:
- $\map {S_\epsilon} {a; \norm {\,\cdot\,} }$ = $\map {S_\epsilon} {a; d }$
Proof
\(\ds x\) | \(\in\) | \(\ds \map {S_\epsilon} {a; \norm {\,\cdot\,} }\) | ||||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \norm {x - a}\) | \(=\) | \(\ds \epsilon\) | Definition of Sphere in Normed Division Ring | ||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \map d {x, a}\) | \(=\) | \(\ds \epsilon\) | Definition of Metric Induced by Norm on Division Ring | ||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds x\) | \(\in\) | \(\ds \map {S_\epsilon} {a; d }\) | Definition of Sphere in $\struct {R, d}$ |
The result follows from Equality of Sets.
$\blacksquare$