Spanning Criterion of Normed Vector Space

Theorem
Let $\GF \in \set {\R, \C}$.

Let $\struct {X, \norm {\, \cdot \,}_X}$ be a normed vector space over $\GF$.

Let $X^\ast$ be the vector space of bounded linear functionals on $X$.

Let $A \subseteq X$ be a subset.

Let $\vee A$ be the closed linear span of $A$, i.e. the closure of the linear span of $A$.

Then $z \in \vee A$ :
 * $\forall \ell \in X^\ast : \ell \restriction_A = 0 \implies \map \ell z = 0$

where $\ell \restriction_A$ denotes the restriction of $\ell$ to $A$.