Proof of The Projection onto One-Particle Tangent Fields: Pairing with Product Fields, Contraction, and Product Fields of Tangent Fields

Each result cited is universally quantified over the data in its own statement. Write $\mu=P^{[1]}\in\mathcal{P}_{2}(\mathbb{R}^{d})$. The spaces $L^{2}(\mu;\mathbb{R}^{d})$ and $L^{2}(P;\mathbb{R}^{dN})$ of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §fields are real Hilbert spaces, in particular real inner product spaces, so The Cauchy-Schwarz Inequality in a Real Inner Product Space applies in each; their elements are classes of Borel maps, with the operations of The Space of Square-Integrable Random Vectors §classes applied as in that clause. Block maps $\mathfrak{p}_{k}$, configurations and product maps are those of Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points with $q=p=d$, and $\sqrt{t}$ is the nonnegative square root of Existence and Uniqueness of the Nonnegative Square Root.

Step 1 (Norms and differences of product fields). Let $g,h\in L^{2}(\mu;\mathbb{R}^{d})$. By Product Fields and the Projection onto One-Particle Tangent Fields §product-field, $\lVert g^{\oplus}\rVert_{P}^{2}=N\lVert g\rVert_{\mu}^{2}=(\sqrt{N}\lVert g\rVert_{\mu})^{2}$, and both $\lVert g^{\oplus}\rVert_{P}$ and $\sqrt{N}\lVert g\rVert_{\mu}$ are nonnegative, so $\lVert g^{\oplus}\rVert_{P}=\sqrt{N}\lVert g\rVert_{\mu}$ by the uniqueness in Existence and Uniqueness of the Nonnegative Square Root. Let $\tilde g,\tilde h$ be Borel representatives of $g,h$; then $\tilde g+(-1)\tilde h$ is a Borel representative of $g-h$ by The Space of Square-Integrable Random Vectors §classes. For $x\in\mathbb{R}^{dN}$, the product map of Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §product-map and Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §linear (with $t=-1$) give

Taking classes, $(g-h)^{\oplus}=g^{\oplus}-h^{\oplus}$ by Product Fields and the Projection onto One-Particle Tangent Fields §product-field and The Space of Square-Integrable Random Vectors §classes; hence $\lVert g^{\oplus}-h^{\oplus}\rVert_{P}=\sqrt{N}\lVert g-h\rVert_{\mu}$.

Step 2 (Pairing against gradients). Let $D\in L^{2}(P;\mathbb{R}^{dN})$ and $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$. Multiplying the identity of Product Fields and the Projection onto One-Particle Tangent Fields §projection by $N$ gives $\langle D,(\nabla\psi)^{\oplus}\rangle_{P}=N\langle\Pi_{P}(D),\nabla\psi\rangle_{\mu}$. Thus claim 1 holds for every $g$ in the set $G_{\mu}$ of The Tangent Space of the Wasserstein Space at a Probability Measure §gradients.

Step 3 (Claim 1). Let $D\in L^{2}(P;\mathbb{R}^{dN})$ and $g\in T_{\mu}$, and put $\Delta=\langle D,g^{\oplus}\rangle_{P}-N\langle\Pi_{P}(D),g\rangle_{\mu}$ and $C=\sqrt{N}\lVert D\rVert_{P}+N\lVert\Pi_{P}(D)\rVert_{\mu}\ge0$. Let $\varepsilon>0$. Since $T_{\mu}$ is the closure of $G_{\mu}$ in $L^{2}(\mu;\mathbb{R}^{d})$ (The Tangent Space of the Wasserstein Space at a Probability Measure §tangent), Characterization of the Closure in a Metric Space by Open Balls (claim 1 implies claim 3) gives $h\in G_{\mu}$ with $\lVert g-h\rVert_{\mu}<\varepsilon$. By Step 2 for $h$, bilinearity of the inner products and Step 1,

so by The Cauchy-Schwarz Inequality in a Real Inner Product Space in both spaces and Step 1, $|\Delta|\le\lVert D\rVert_{P}\sqrt{N}\lVert g-h\rVert_{\mu}+N\lVert\Pi_{P}(D)\rVert_{\mu}\lVert g-h\rVert_{\mu}\le C\varepsilon$. If $\Delta\ne0$, the choice $\varepsilon=|\Delta|(C+1)^{-1}>0$ would give $|\Delta|\le C(C+1)^{-1}|\Delta|<|\Delta|$ (Elementary Order Arithmetic in an Ordered Field, claims 7 and 10), which is impossible. Hence $\Delta=0$, which is claim 1.

Step 4 (Linearity). Let $D,D'\in L^{2}(P;\mathbb{R}^{dN})$ and $a,b\in\mathbb{R}$, and put $\zeta=a\Pi_{P}(D)+b\Pi_{P}(D')$. Then $\zeta\in T_{\mu}$, since $T_{\mu}$ is a linear subspace by Basic Properties of the Tangent Space: Closed Subspace, the Identity Map Belongs to It, Second-Moment Limits, and Representation of Bounded Functionals on Gradients §closed, and for every $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$ the defining identity of Product Fields and the Projection onto One-Particle Tangent Fields §projection for $D$ and $D'$ and bilinearity give

By the uniqueness in Product Fields and the Projection onto One-Particle Tangent Fields §projection, $\zeta=\Pi_{P}(aD+bD')$. So $\Pi_{P}$ is linear.

Step 5 (Contraction). Let $D\in L^{2}(P;\mathbb{R}^{dN})$ and $a=\sqrt{N}\lVert\Pi_{P}(D)\rVert_{\mu}\ge0$, so that $a^{2}=N\lVert\Pi_{P}(D)\rVert_{\mu}^{2}$ and, by Step 1, $\lVert\Pi_{P}(D)^{\oplus}\rVert_{P}=a$. Claim 1 with $g=\Pi_{P}(D)\in T_{\mu}$ and The Cauchy-Schwarz Inequality in a Real Inner Product Space give

If $a=0$, then $N\lVert\Pi_{P}(D)\rVert_{\mu}^{2}=0\le\lVert D\rVert_{P}^{2}$. If $a>0$, then $a\le\lVert D\rVert_{P}$, since $\lVert D\rVert_{P}<a$ would give $\lVert D\rVert_{P}\,a<a^{2}$ by claim 10 of Elementary Order Arithmetic in an Ordered Field; hence $\lVert D\rVert_{P}>0$ and, by claim 5 of Elementary Arithmetic in an Ordered Field applied to $a\le\lVert D\rVert_{P}$ with the nonnegative factors $a$ and $\lVert D\rVert_{P}$, $a^{2}\le\lVert D\rVert_{P}\,a\le\lVert D\rVert_{P}^{2}$. In both cases $N\lVert\Pi_{P}(D)\rVert_{\mu}^{2}\le\lVert D\rVert_{P}^{2}$, which with Step 4 is claim 2.

Step 6 (Claim 3). Let $g\in T_{\mu}$ with Borel representative $\tilde g$, and let $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$. The function $f=\tilde g\cdot\nabla\psi:\mathbb{R}^{d}\to\mathbb{R}$ is Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions and integrable with respect to $\mu$ with $\int_{\mathbb{R}^{d}}f\,d\mu=\langle g,\nabla\psi\rangle_{\mu}$, by Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form §operations on the probability space $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}),\mu)$ as in Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §fields. Since $\mu$ is the measure $A_{P}$ of Existence and Uniqueness of Tensor Powers, and the Average of the Block Marginals §average (with $q=d$) by The One-Particle Marginal of a Probability Measure on the Configuration Space §marginal, that clause shows that each $f\circ\mathfrak{p}_{k}$ is integrable with respect to $P$ and $\sum_{k=1}^{N}\int_{\mathbb{R}^{dN}}f\circ\mathfrak{p}_{k}\,dP=N\int_{\mathbb{R}^{d}}f\,d\mu$. For $x\in\mathbb{R}^{dN}$, Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §inner-product (in $\mathbb{R}^{dN}$ with $q=d$) and Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §product-map give

Integrating against $P$ and using Linearity of the Lebesgue Integral over a Finite Sum of Integrable Functions §linear (all coefficients $1$),

Thus $g\in T_{\mu}$ satisfies $\langle g,\nabla\psi\rangle_{\mu}=\frac{1}{N}\langle g^{\oplus},(\nabla\psi)^{\oplus}\rangle_{P}$ for every $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$, and the uniqueness in Product Fields and the Projection onto One-Particle Tangent Fields §projection (with $D=g^{\oplus}$) gives $\Pi_{P}(g^{\oplus})=g$, which is claim 3.