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.