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EDIT - I was wrong, it turned out that all I need was the Reisz Representation theorem.

I am looking for a way to understand how $L^{\infty}$ is a realization of the strong dual of $L^1$, but I am only interested in the case when the measure is Lebesgue measure. Most of the proofs that I've (in Rudin and in Treves: TVS, Distributions and Kernels) seem involve the general case, and introduce complexities (such as appeals to the Radon-Nikodym theorem) that I haven't studied yet. I've looked at this section in Rudin's real and complex, but there is a lot of measure theory (I only know the very basic stuff) that I would have to learn before I can understand it. However, since I am only interested in one particular case of $L^1$ (Lebesgue integration) I thought it might be possible that there was a simpler proof out there (especially since, after glancing at the Radon-Nikodym theorem, it seems trivially true if the measure is just the Lebesgue measure throughout - but I'm not really sure if I'm thinking about it in the right way). If any body could direct me towards a resource where I could find such a proof I would be very happy.

This question comes out of trying to understand what Rudin means (in his Functional Analysis) when he shows that a distribution is locally the derivative of a continuous function (6.26): he extends a bounded linear functional $\Lambda_1$ on a subspace of $L^1(K)$ (K compact) by the Hahn-Banach theorem, and then says there there must be some bounded and measurable function g on K so that $\Lambda_1(\phi) = \int g(x) \phi(x) dx$ $(\phi \in D(K)$). I would like to understand this connection more thoroughly, the rest of the proof I understand (I think). I am assuming that this a reference to the duality relationship I asked about my question about above. If I am wrong, then understanding this move in the proof is more important to me than understanding the duality relationship (though that is also interesting to me).

Thanks in advance for your help.

EDIT - I was wrong, it turned out that all I need was the Reisz Representation theorem.

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