dual space separates points

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

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

$$f_j(x_k)={\delta }_{jk}\mathit{\hspace{1em}},1\le j,k\le n$$

If $x\in span\{x_1,\mathrm{\dots },x_n\}$, then $x={\displaystyle \sum _{j=1}^n}{f}_j(x)x_j$.

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