Definition:Open Ball/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 open $\epsilon$-ball of $x$ in $\struct {X, \norm {\,\cdot\,} }$ is defined as:

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


Also see

Sources