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Let $X$ be an open set in $\mathbb{R}^n$ and $M(X)$ be the space of finite signed measures defined on $X$. $L(p)$ is a lower-semicontinuous convex functional defined on $M(X)$. My question is: (1) What is the dual space of $M(X)$? (2) Suppose $E^*$ is the dual space of $M(X)$, is the following identity true: $$ L(p)=\sup_{f \in E^*} \langle f,p \rangle-L^*(f), $$ where $L^*(f)=\sup_{p\in M(X)}\langle p,f \rangle-L(p)$ is the Legendre transform of $L(p)$.

Although this question is trivial when $M(X)$ is a finite-dimensional space, there seem to be critical gaps to generalize to infinite-dimensional spaces.

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  • $\begingroup$ Answers to the first question are here: math.stackexchange.com/questions/74875/…. $\endgroup$ Oct 29, 2018 at 0:07
  • $\begingroup$ Since this looks like you want to work with duality, I'd like to remark that in most cases it is more appropriate to take the predual, i.e. using that that measure space is the dual of continuous functions and finding some $M$ such that $M^*=L$. $\endgroup$
    – Dirk
    Oct 29, 2018 at 5:54

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