Center is Element of Open Ball/Normed Division Ring
< Center is Element of Open Ball(Redirected from Center is Element of Open Ball in Normed Division Ring)
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Theorem
Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring.
Let $a \in R$.
Let $\epsilon \in \R_{>0}$ be a strictly positive real number.
Let $\map {B_\epsilon} a$ be the open $\epsilon$-ball of $a$ in $\struct{R, \norm {\,\cdot\,} }$.
Then:
- $a \in \map {B_\epsilon} a$
Proof
Let $d$ be the metric induced by the norm $\norm {\,\cdot\,}$.
From Open Ball in Normed Division Ring is Open Ball in Induced Metric, $\map {B_\epsilon} a$ is the open $\epsilon$-ball of $a$ in the metric space $\struct{R,d}$.
From Center is Element of Open Ball:
- $a \in \map {B_\epsilon} a$
$\blacksquare$