Fenchel–Moreau theorem



In convex analysis, the Fenchel–Moreau theorem (named after Werner Fenchel and Jean Jacques Moreau) or Fenchel biconjugation theorem (or just biconjugation theorem) is a theorem which gives necessary and sufficient conditions for a function to be equal to its biconjugate. This is in contrast to the general property that for any function $$f^{**} \leq f$$. This can be seen as a generalization of the bipolar theorem. It is used in duality theory to prove strong duality (via the perturbation function).

Statement
Let $$(X,\tau)$$ be a Hausdorff locally convex space, for any extended real valued function $$f: X \to \mathbb{R} \cup \{\pm \infty\}$$ it follows that $$f = f^{**}$$ if and only if one of the following is true
 * 1) $$f$$ is a proper, lower semi-continuous, and convex function,
 * 2) $$f \equiv +\infty$$, or
 * 3) $$f \equiv -\infty$$.