Definition:Sphere/Normed Vector Space/Radius

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Definition

Let $\struct{X, \norm{\,\cdot\,}}$ be a normed vector space.

Let $x \in X$.

Let $\epsilon \in \R_{>0}$ be a strictly positive real number.

Let $\map {S_\epsilon} x$ be the $\epsilon$-sphere of $x$.


In $\map {S_\epsilon} x$, the value $\epsilon$ is referred to as the radius of the $\epsilon$-sphere.


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.