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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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1 Answer

up vote 1 down vote accepted

Hint: use the Hahn-Banach theorem.

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Of course! thanks. –  newbie Mar 6 '12 at 2:22
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