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Let $X$ be any locally compact Hausdorff space and assume that it is not compact. I've heard that the Banach space $(C_0(X),\|\!\cdot\!\|_\infty)$ is not isometrically isomorphic to the (norm) dual of a Banach space. Is there a good book where I can find a proof this result?

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    Hint: Show that the unit ball has no extreme points. Use [Alaoglu](http://en.wikipedia.org/wiki/Banach-Alaoglu_theorem) and [Krein-Milman](http://en.wikipedia.org/wiki/Krein-Milman_theorem) to derive a contradiction.2012-10-09
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    @commenter whu don't you arite this comment as an answer?2012-10-10
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    @Norbert: Because I couldn't think of a reference containing that argument. I posted an expanded version as an answer.2012-10-10

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