Definition:Open Ball/Real Analysis

Definition
Let $n \ge 1$ be a natural number.

Let $\R^n$ denote a real Euclidean space

Let $\norm \cdot$ denote the Euclidean norm.

Let $a \in \R^n$.

Let $R > 0$ be a strictly positive real number.

The open ball of center $a$ and radius $R$ is the subset:
 * $\map B {a, R} = \set {x \in \R^n : \norm {x - a} < R}$

Also see

 * Definition:Closed Ball (Real Analysis)
 * Definition:Open Set (Real Analysis)