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).

A function that is not lower semi-continuous. By the Fenchel-Moreau theorem, this function is not equal to its biconjugate.

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. $f\equiv -\infty$.