The following result is a corollary of the Hahn-Banach theorem.

Theorem - Let $X$ be a normed vector space. Given a linearly independent set $\{x_1,\dots ,x_n\} \subset X$ there exist continuous linear functionals $f_1, \dots , f_n \in X'$ such that

If $x \in span\{x_1, \dots , x_n\}$ , then $\displaystyle x= \sum_{j=1}^n f_j(x)x_j$ .

The above theorem shows that if $f(x)=f(y)$ for every continuous linear functional $f$ then $x=y$ , therefore the dual space $X'$ separates the points of $X$ .