# A simple estimation

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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