Let's say $X$ is a normed linear space, and $X^*$ is its dual space.
One can define the norm in $X$ in such a way
$$ \|x\| = \sup\{|\phi(x)|:\phi \in X^*,\|\phi\|\le 1\}. $$
The direction $$ \|x\| \ge \sup\{|\phi(x)|:\phi \in X^*,\|\phi\|\le 1\} $$ is obvious. How about the other direction?
Sol:
apply Hahn-Banach thm, there exists a functional $\psi:x\mapsto \|x\|$ with $\|\psi\| = 1$.
$$ \psi(x) = \|x\|\le \sup\{|\phi(x)|:\phi \in X^*,\|\phi\|\le 1\}. $$
Hence equality is achieved.