# Definition:Sphere/Normed Vector Space

## 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.

The $\epsilon$-sphere of $x$ in $\struct {X, \norm {\,\cdot\,} }$ is defined as:

$\map {S_\epsilon} x = \set {y \in X: \norm {y - x} = \epsilon}$

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