Definition:Sphere/Normed Vector Space

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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.


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

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


Radius

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


Center

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


Also see


Sources