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Possible Duplicate:
The Duals of $l^\infty$ and $L^{\infty}$

In learning real analysis, I do understand that the dual of $L^\infty$ cannot be $L^1$ because the latter is separable, whereas the former is not separable. So the double dual of $L^1$ is strictly larger than $L^1$ (non-reflexivity). See for example this question:

Dual of $\ell_{\infty}$ is not $\ell_1$

However, I am not able to find in my textbooks etc. the precise space which is the dual of $L^\infty(\Bbb R)$.

I want to know exactly what it is. I would very much appreciate help.

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    The short answer is that the nontrivial elements of the dual aren't describable without some form of the axiom of choice. See http://mathoverflow.net/questions/22661/explicit-element-of-ell-infty-ell1 .2012-06-05
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    It's described in IV.8.16 of Dunford and Schwartz' *Linear Operators, Part I* as a certain space of bounded, scalar-valued set functions.2012-06-05
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    See [here](http://math.stackexchange.com/questions/47395/the-duals-of-l-infty-and-l-infty).2012-06-05
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    weaker statement than what quaochu said : it contains elements that cannot reasonnably be represented as functions.2012-06-05
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    Thank you very much for all these helpful answers.2012-06-06

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