Definition:Closed 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 closed $\epsilon$-ball of $a$ in $\struct {R, \norm {\,\cdot\,} }$ is defined as:

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

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 closed $\epsilon$-ball.

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

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

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