Closed Ball in Normed Division Ring is Closed Ball 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 $\map {{B_\epsilon}^-} {a; \norm {\,\cdot\,} }$ denote the closed ball in the normed division ring $\struct {R, \norm {\,\cdot\,} }$.
Let $\map {{B_\epsilon}^-} {a; d }$ denote the closed ball in the metric space $\struct {R, d}$.
Then:
- $\map {{B_\epsilon}^-} {a; \norm {\,\cdot\,} }$ = $\map {{B_\epsilon}^-} {a; d }$
Proof
\(\ds x\) | \(\in\) | \(\ds \map { {B_\epsilon}^-} {a; \norm {\,\cdot\,} }\) | ||||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \norm {x - a}\) | \(\le\) | \(\ds \epsilon\) | Definition of Closed Ball of Normed Division Ring | ||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \map d {x, a}\) | \(\le\) | \(\ds \epsilon\) | Definition of Metric Induced by Norm on Division Ring | ||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds x\) | \(\in\) | \(\ds \map { {B_\epsilon}^-} {a; d }\) | Definition of Closed Ball |
The result follows from Equality of Sets.
$\blacksquare$