A property of reflexive spaces

Let $X$ be a reflexive normed space and let $\phi$ be a continuous linear functional on $X$ . Show that there exists $x \in B_X$ such that $\|\phi\|=\phi(x)$ . In other words, show that every continuous linear functional on $X$ attains its norm.

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Solution by La chouette aveugleFR

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Par le théorème de Hahn-Banach, si $\phi \in X^{*}$ , il existe un élément $\psi$ de $B_{X^{**}}$ tel que $\psi(\phi) = \lVert \phi \rVert$ . Comme $X$ est réflexif il existe $x \in B_{X}$ tel que $\psi$ est de la forme $\phi \mapsto \phi(x)$ . Ainsi, $\lVert \phi \rVert = \psi(\phi) = \phi(x)$ , d’où le résultat.

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