The Tangent Space of the Torus Wasserstein Space at a Probability Measure

In the setting of Optimal Transport on the Flat Torus: Standing Notation, let $\mu\in\mathcal{P}(\mathbb{T}^{d})$, let $L^{2}(\mu;\mathbb{R}^{d})$ be the space of square-integrable vector fields against $\mu$, and let $C^{\infty}_{\mathrm{per}}$ be the smooth periodic functions, whose gradients have classes in $L^{2}(\mu;\mathbb{R}^{d})$ by Optimal Transport on the Flat Torus: Standing Notation §calculus.

1. (Gradients of periodic functions)¶ $G_{\mu}$ denotes the set of the classes in $L^{2}(\mu;\mathbb{R}^{d})$ of the gradients $\nabla f$ of the functions $f\in C^{\infty}_{\mathrm{per}}$.

2. (Tangent space)¶ The tangent space at $\mu$ is the closure $T_{\mu}=\overline{G_{\mu}}$ of $G_{\mu}$ in $L^{2}(\mu;\mathbb{R}^{d})$.