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 ${\displaystyle f^{**}\leq f}$. 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).

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

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