\begin{definition}[Definition:Everywhere Dense/Normed Vector Space]
Let $M = \struct {X, \norm {\, \cdot \,} }$ be a normed vector space.
Let $Y \subseteq X$ be a subset of $X$.
Suppose:
:$\forall x \in X: \forall \epsilon \in \R_{>0}: \exists y \in Y: \norm {x - y} < \epsilon$
Then $Y$ is '''(everywhere) dense''' in $X$.
\end{definition}