Definition:Open Ball/Real Analysis
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Definition
Let $n \ge 1$ be a natural number.
Let $\R^n$ denote the real Euclidean space of dimension $n$.
Let $\norm {\, \cdot \,}$ denote the Euclidean norm.
Let $a \in \R^n$.
Let $\epsilon \in \R_{>0}$ be a strictly positive real number.
The open (Euclidean) ball of center $a$ and radius $\epsilon$ is the subset:
- $\map {B_\epsilon} a = \set {x \in \R^n : \norm {x - a} < \epsilon}$