Basic Properties of the Tangent Space: Closed Subspace, the Identity Map Belongs to It, Second-Moment Limits, and Representation of Bounded Functionals on Gradients

In the setting of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation, let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, let $L^{2}(\mu;\mathbb{R}^{d})$ be the space of square-integrable vector fields, and let $G_{\mu}$ and $T_{\mu}$ be the gradients of test functions and the tangent space at $\mu$. Let $P_{T_{\mu}}$ be the orthogonal projection onto $T_{\mu}$ (a closed linear subspace by claim 1 below), let $\mathrm{id}:\mathbb{R}^{d}\to\mathbb{R}^{d}$ be the identity map, and let $M_{2}(\mu)$ be the second moment of $\mu$. Fix a function $\chi$ as in Scaled Cutoffs and the Second-Moment Test Functions: Uniform Derivative Bounds and Agreement on a Ball for the dimension $d$, and for $n\in\mathbb{N}$ let $\psi_{n}$ be the function $\psi_{R}$ of that lemma with $R=n$, the natural number $n$ read as a positive real number through the canonical map of The Canonical Map from the Natural Numbers to a Field (claim 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field); by Scaled Cutoffs and the Second-Moment Test Functions: Uniform Derivative Bounds and Agreement on a Ball (ii) each $\psi_{n}$ is smooth and compactly supported, hence a test function, so that $\nabla\psi_{n}$ and $\Delta\psi_{n}$ are as fixed in Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation (iii) and $\nabla\psi_{n}\in G_{\mu}$. The natural number $d$ is likewise read as a real number where a real number is required.

1. (Closed subspace)(i) $G_{\mu}$ is a linear subspace of $L^{2}(\mu;\mathbb{R}^{d})$, $T_{\mu}$ is a closed linear subspace of $L^{2}(\mu;\mathbb{R}^{d})$ containing $G_{\mu}$, and $G_{\mu}$ is dense in the metric space $(T_{\mu},d_{\mu})$, where $d_{\mu}$ is the distance of $L^{2}(\mu;\mathbb{R}^{d})$ restricted to $T_{\mu}$.

2. (The identity map)(ii) The map $\mathrm{id}$ is Borel with $\int_{\mathbb{R}^{d}}\lVert\mathrm{id}\rVert^{2}\,d\mu=M_{2}(\mu)$, so its class belongs to $L^{2}(\mu;\mathbb{R}^{d})$ and is again written $\mathrm{id}$; and $\lVert\nabla\psi_{n}-\mathrm{id}\rVert_{\mu}\to0$ as $n\to\infty$. In particular $\mathrm{id}\in T_{\mu}$.

3. (Second-moment limits)(iii) $\int_{\mathbb{R}^{d}}\Delta\psi_{n}\,d\mu\to d$ as $n\to\infty$.

4. (Representation)(iv) Let $\ell:C_{c}^{\infty}(\mathbb{R}^{d})\to\mathbb{R}$ satisfy $\ell(a\psi+b\phi)=a\,\ell(\psi)+b\,\ell(\phi)$ for all $\psi,\phi\in C_{c}^{\infty}(\mathbb{R}^{d})$ and $a,b\in\mathbb{R}$, and let $C\ge0$ be a real number with $|\ell(\psi)|\le C\,\lVert\nabla\psi\rVert_{\mu}$ for every $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$. Then there is exactly one $\xi\in T_{\mu}$ with

$$\langle\xi,\nabla\psi\rangle_{\mu}=\ell(\psi)\qquad\text{for every }\psi\in C_{c}^{\infty}(\mathbb{R}^{d}),$$

and it satisfies $\lVert\xi\rVert_{\mu}\le C$.

5. (Projection)(v) In the situation of claim 4, let $\eta\in L^{2}(\mu;\mathbb{R}^{d})$ satisfy $\langle\eta,\nabla\psi\rangle_{\mu}=\ell(\psi)$ for every $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$. Then $\xi=P_{T_{\mu}}\eta$ and $\lVert\xi\rVert_{\mu}\le\lVert\eta\rVert_{\mu}$; and if moreover $\eta\in T_{\mu}$, then $\xi=\eta$.