# 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.

### Radius

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

### Center

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

## Also known as

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 $\struct {R, d}$.

## Also see

## Sources

- 1997: Fernando Q. Gouvea:
*p-adic Numbers: An Introduction*... (previous) ... (next): $\S 2.3$ Topology: Proposition $2.3.5$ - 2013: Francis Clarke:
*Functional Analysis, Calculus of Variations and Optimal Control*... (previous) ... (next): $1.1$: Basic Definitions