L-Differentiability of a Function on the Wasserstein Space via the Fréchet Derivative of Its Lift

In the setting of The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation, let $L^{2}(\Omega;\mathbb{R}^{d})$ be the space of classes of square-integrable random vectors and $\mathcal{P}_{2}(\mathbb{R}^{d})$ the Wasserstein space, and let $u:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$ with lift $U$. That $U$ is differentiable on $L^{2}(\Omega;\mathbb{R}^{d})$ with gradient $DU(X)\in L^{2}(\Omega;\mathbb{R}^{d})$ at a point $X$ of $L^{2}(\Omega;\mathbb{R}^{d})$, and the class $C^{1}(L^{2}(\Omega;\mathbb{R}^{d}))$, are as fixed in The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation (iii) for the open set $L^{2}(\Omega;\mathbb{R}^{d})$; thus the open set written $U$ in Real Hilbert Spaces: Series, Products, Orthonormal Bases and Differential Calculus (iv) is $L^{2}(\Omega;\mathbb{R}^{d})$ here, and $U$ denotes the function on it.

1. ($L$-differentiability)(i) The function $u$ is $L$-differentiable if $U$ is differentiable on $L^{2}(\Omega;\mathbb{R}^{d})$. For $X\in L^{2}(\Omega;\mathbb{R}^{d})$ the gradient $DU(X)$ is then called the $L$-gradient of $u$ at $X$.

2. (Continuous $L$-differentiability)(ii) The function $u$ is continuously $L$-differentiable if $U\in C^{1}(L^{2}(\Omega;\mathbb{R}^{d}))$.