Английская Википедия: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 <math>f^{**} \leq f</math>.[1][2] This can be seen as a generalization of the bipolar theorem.[1] It is used in duality theory to prove strong duality (via the perturbation function).
Statement
Let <math>(X,\tau)</math> be a Hausdorff locally convex space, for any extended real valued function <math>f: X \to \mathbb{R} \cup \{\pm \infty\}</math> it follows that <math>f = f^{**}</math> if and only if one of the following is true
- <math>f</math> is a proper, lower semi-continuous, and convex function,
- <math>f \equiv +\infty</math>, or
- <math>f \equiv -\infty</math>.[1][3][4]
References
Шаблон:Convex analysis and variational analysis Шаблон:Functional analysis Шаблон:Topological vector spaces
