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 \(f^*(x^*) := \sup\left\{ \langle x, x^* \rangle - f(x) : x \in X\right\}.\)