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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.

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Hint: use the Hahn-Banach theorem.