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
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis: Chapter $1$: Normed and Banach spaces