Definition:Closed Ball/Normed Division Ring/Radius

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.

Let $\map { {B_\epsilon}^-} a$ be the closed $\epsilon$-ball of $a$.

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

Linguistic Note

The plural of radius is radii, pronounced ray-dee-eye.

This irregular plural form stems from the Latin origin of the word radius, meaning ray.

The ugly incorrect form radiuses can apparently be found, but rarely in a mathematical context.

The adjectival form radial usually means directed along a radius.

Sources

• 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction: $\S 2.3$ Topology, Proposition 2.3.5