\begin{definition}[Definition:Closed Convex Hull]
Let $\struct {X, \norm \cdot}$ be a normed vector space over $\R$. 
Let $U \subseteq X$. 
We define the '''closed convex hull''' of $U$ as the closure of the convex hull of $U$ in $\struct {X, \norm \cdot}$.
\end{definition}