Definition:Sphere/Normed Vector Space/Center

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 $x$ is referred to as the center of the $\epsilon$-sphere.

Linguistic Note

The British English spelling of center is centre.

The convention on $\mathsf{Pr} \infty \mathsf{fWiki}$ is to use the American English spelling center, but it is appreciated that there may be lapses.