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:


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

Also see

 * Definition:Open Ball in Normed Vector Space


 * Definition:Closed Ball in Normed Vector Space