The Tangent Space of the Wasserstein Space at a Probability Measure

In the setting of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation, let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, and let $L^{2}(\mu;\mathbb{R}^{d})$ be the space of square-integrable vector fields with respect to $\mu$.

1. (Gradients of test functions)(i) $G_{\mu}$ denotes the set of the classes in $L^{2}(\mu;\mathbb{R}^{d})$ of the gradients $\nabla\psi$ of the test functions $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$.

2. (Tangent space)(ii) The tangent space to $\mathcal{P}_{2}(\mathbb{R}^{d})$ at $\mu$ is the closure of $G_{\mu}$ in $L^{2}(\mu;\mathbb{R}^{d})$,

$$T_{\mu}=\overline{G_{\mu}} .$$