Definition:Open Ball/Normed Division Ring

Definition
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.

The open $\epsilon$-ball of $a$ in $\struct {R, \norm {\,\cdot\,} }$ is defined as:


 * $\map {B_\epsilon} a = \set {x \in R: \norm{x - a} < \epsilon}$

Let $d$ be the metric induced by the norm $\norm {\,\cdot\,}$.

By the definition of the metric induced by the norm, the open $\epsilon$-ball of $a$ in $\struct {R, \norm {\,\cdot\,} }$ is the open $\epsilon$-ball of $a$ in $\struct {R, d}$.

If it is necessary to show the norm itself, then the notation $\map {B_\epsilon} {a; \norm {\,\cdot\,} }$ can be used.

Also see

 * Definition:Closed Ball of Normed Division Ring