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