Definition:Open Ball/Normed Division Ring

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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.


Radius

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


Center

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


Also see


Sources