Subset of Normed Vector Space is Everywhere Dense iff Closure is Normed Vector Space

Theorem
Let $\struct {X, \norm {\, \cdot \,}}$ is a normed vector space.

Let $D \subseteq X$ be a subset of $X$.

Let $D^-$ be the closure of $D$.

Then $D$ is dense iff $D^- = X$.