Given a topological vector space $$X$$ and its dual $$X^*$$, as well as a function $$f:X\to \mathbb{R}$$, the Fenchel conjugate $$f^* : X^*\to \mathbb{R}\cup\{\infty\}$$ is defined as $$f^*(x^*) := \sup\left\{ \langle x, x^* \rangle - f(x) : x \in X\right\}.$$