Fenchel–Moreau theorem

Statement

Let ${\displaystyle (X,\tau )}$ be a Hausdorff locally convex space, for any extended real valued function ${\displaystyle f:X\to \mathbb{R}\cup \{\pm \mathrm{\infty }\}}$ it follows that ${\displaystyle f=f^{\ast \ast }}$ if and only if one of the following is true

1. ${\displaystyle f}$ is a proper, lower semi-continuous, and convex function,
2. ${\displaystyle f\equiv +\mathrm{\infty }}$, or
3. ${\displaystyle f\equiv -\mathrm{\infty }}$.