An Elementary Proof for the Structure of Wasserstein Derivatives

Let $F: \mathbb{L}^2(\Omega, \mathbb{R}) \to \mathbb{R}$ be a law invariant and continuously Fr\'echet differentiable mapping. Based on Lions \cite{Lions}, Cardaliaguet \cite{Cardaliaguet} (Theorem 6.2 and 6.5) proved that: \bea \label{Derivative} D F (\xi) = g(\xi), \eea where $g: \mathbb{R} \to \mathbb{R}$ is a deterministic function which depends only on the law of $\xi$. See also Carmona \& Delarue \cite{CD} Section 5.2. In this short note we provide an elementary proof for this well known result. This note is part of our accompanying paper \cite{WZ}, which deals with a more general situation.