\begin{definition}[Definition:Closed Unit Ball]
Let $\struct {X, \norm {\, \cdot \,}}$ be a normed vector space.
Let $a \in X$.
The '''closed unit ball''' of $X$, denoted $\operatorname{ball} X$, is the set:
:$\map {B_1^-} a := \set {x \in X: \norm {x - a} \le 1}$
\end{definition}