Let $f : X \to \mathbb{R}$ be a D4364: Real function such that

(i)	$X \subseteq \mathbb{R}^N$ is a D78: Subset of $\mathbb{R}^N$
(ii)	\begin{equation} Y : = \left\{ y \in \mathbb{R}^N : \sup_{x \in X} (y^T x - f(x)) < \infty \right\} \end{equation}

The convex conjugate of $f$ is the D4364: Real function \begin{equation} Y \to \mathbb{R}, \quad y \mapsto \sup_{x \in \mathbb{R}^N} (y^T x - f(x)) \end{equation}