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:

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

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

In $\map {B_\epsilon} a$, the value $\epsilon$ is referred to as the radius of the open $\epsilon$-ball.

In $\map {B_\epsilon} a$, the value $a$ is referred to as the center of the open $\epsilon$-ball.

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

From Open Ball in Normed Division Ring is Open Ball in Induced Metric, the open $\epsilon$-ball of $a$ in $\struct {R, \norm {\,\cdot\,} }$ is the open $\epsilon$-ball of $a$ in $\struct {R, d}$.