Proof of The Relative Score on a Hilbert Space: the Gaussian Measure Has Score Zero, and the Score as a Square-Integrable Field in the Weighted Sequence Space

Each result cited is universally quantified over the data in its own statement.

Claim 1. By Moments of a Diagonal Gaussian Measure on a Hilbert Space: Coordinate Covariances, Finite Second Moment, and Exponential Moments of the Squared Norm §moment, $\gamma_{c}\in\mathcal{P}_{2}(X)$, so The Relative Score with Respect to a Diagonal Gaussian Measure on a Hilbert Space §score applies with $\mu=\gamma_{c}$. Fix $k\in\mathbb{N}$ and $\varphi\in\mathcal{F}C^{1}_{b}(X)$, and let $(n,\psi)$ be a representation of $\varphi$, so that $\varphi=\psi\circ p_{n}$ and $\psi\in C^{1}_{b}(\mathbb{R}^{n})$ in the sense of Bounded Continuously Differentiable Functions with Bounded Partial Derivatives on Euclidean Space §bounded with $q=n$. By Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §gradient, $\varphi$ is differentiable on $X$, so its partial derivative $\partial_{k}\varphi$ is defined by Partial Derivatives along an Orthonormal Basis of a Function Differentiable on a Hilbert Space §partial. The functions $x\mapsto x_{k}\varphi(x)$ and $\partial_{k}\varphi$ are integrable with respect to $\gamma_{c}$ by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §coordinate-integrable and Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §square-integrable, and $c_{k}>0$ by Variance Sequences and Their Truncations §variances; so Linearity and Monotonicity of the Lebesgue Integral §integrable, with coefficients $\frac{1}{c_{k}}$ and $-1$, gives

$$\int_{X}\Bigl(\frac{x_{k}}{c_{k}}\varphi(x)-\partial_{k}\varphi(x)\Bigr)\gamma_{c}(dx)=\frac{1}{c_{k}}\int_{X}x_{k}\varphi(x)\,\gamma_{c}(dx)-\int_{X}\partial_{k}\varphi\,d\gamma_{c}.\tag{1}$$

For each $i\in[n]$ choose a bound $M_{i}\ge0$ for $\partial_{i}\psi$ and put $M=\sum_{i=1}^{n}M_{i}$; then $0\le M_{i}\le M$ and $|\partial_{i}\psi(y)|\le M\le M(1+\lVert y\rVert)$ for all $y\in\mathbb{R}^{n}$ and $i\in[n]$, since $\lVert y\rVert\ge0$. As $\psi$ is of class $C^{1}$ on $\mathbb{R}^{n}$, Gaussian Integration by Parts for Functions of Finitely Many Coordinates Relative to a Diagonal Gaussian Measure on a Hilbert Space applies to $n$, $M$ and $\psi$, with its $\varphi=\psi\circ p_{n}$ our $\varphi$. If $k\le n$, then $\partial_{k}\varphi=(\partial_{k}\psi)\circ p_{n}$ by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §partial, and Gaussian Integration by Parts for Functions of Finitely Many Coordinates Relative to a Diagonal Gaussian Measure on a Hilbert Space §coordinate with $i=k$ gives $\int_{X}\partial_{k}\varphi\,d\gamma_{c}=\frac{1}{c_{k}}\int_{X}x_{k}\varphi(x)\,\gamma_{c}(dx)$, so the right side of (1) is $0$. If $k>n$, then $\int_{X}x_{k}\varphi(x)\,\gamma_{c}(dx)=0$ by Gaussian Integration by Parts for Functions of Finitely Many Coordinates Relative to a Diagonal Gaussian Measure on a Hilbert Space §orthogonal, and $\partial_{k}\varphi(x)=0$ for every $x$ by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §partial, so $\partial_{k}\varphi=0\cdot\partial_{k}\varphi$ and $\int_{X}\partial_{k}\varphi\,d\gamma_{c}=0$ by Linearity and Monotonicity of the Lebesgue Integral §integrable, applied to the integrable function $\partial_{k}\varphi$ twice with both coefficients $0$; again the right side of (1) is $0$. Let $0_{L}$ be the zero vector of the real Hilbert space $L^{2}(\gamma_{c})$ of The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §hilbert. By Elementary Identities in a Real Inner Product Space §zero, $\langle0_{L},\varphi\rangle_{L^{2}(\gamma_{c})}=0$, which equals the left side of (1). As $k$ and $\varphi$ were arbitrary, the constant sequence with every term $0_{L}$ satisfies the identity of The Relative Score with Respect to a Diagonal Gaussian Measure on a Hilbert Space §score; so $\gamma_{c}$ has a relative score with respect to $\gamma_{c}$, and by the uniqueness of each component recorded there, its $k$-th component is the zero vector for every $k$. Now let $a$ be a weight sequence. Since $\lVert0_{L}\rVert_{L^{2}(\gamma_{c})}=0$ by Elementary Identities in a Real Inner Product Space §zero, $a_{k}\lVert0_{L}\rVert_{L^{2}(\gamma_{c})}^{2}=0$ for every $k$, so all partial sums of $\sum_{k}a_{k}\lVert0_{L}\rVert_{L^{2}(\gamma_{c})}^{2}$ are $0$; the series converges with sum $0$ in the sense of Series of Real Numbers §convergent. By Weight Sequences and the Weighted Fisher Information Relative to a Diagonal Gaussian Measure on a Hilbert Space §information, $\gamma_{c}$ has finite Fisher information relative to $\gamma_{c}$ with weights $a$, and $\mathcal{I}_{a}(\gamma_{c}\,|\,\gamma_{c})=0$.

Claim 2. By Weighted Square-Summable Sequence Spaces are Real Hilbert Spaces: the Standard Orthonormal Basis and the Embedding of a Hilbert Space under Bounded Weights §hilbert and Weighted Square-Summable Sequence Spaces are Real Hilbert Spaces: the Standard Orthonormal Basis and the Embedding of a Hilbert Space under Bounded Weights §basis, $X_{a}$ is a real Hilbert space with orthonormal basis $(f^{a}_{k})_{k}$; the triple $(X,\mathcal{B}(X),\mu)$ is a measure space, $\mu$ being a Borel measure on $(X,d)$ by Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §measures. So The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis applies with $(S,\mathcal{S},\mu)=(X,\mathcal{B}(X),\mu)$, $E=X_{a}$ and $f_{k}=f^{a}_{k}$, and its $L^{2}(\mu)$ is the real Hilbert space $L^{2}(\mu)$ of The Relative Score with Respect to a Diagonal Gaussian Measure on a Hilbert Space §score. Since $a_{k}>0$ by Weight Sequences and the Weighted Fisher Information Relative to a Diagonal Gaussian Measure on a Hilbert Space §weights, the real number $a_{k}^{1/2}$ is the power of $a_{k}$ with exponent $\tfrac12$, which by Properties of Real Powers of Nonnegative Real Numbers §agreement is the nonnegative square root of $a_{k}$, so $(a_{k}^{1/2})^{2}=a_{k}$. Put $g_{k}=a_{k}^{1/2}\zeta_{k}\in L^{2}(\mu)$. By conditions (a) and (c) of Real Inner Product Space §inner-product and Real Inner Product Space §norm,

$$\lVert g_{k}\rVert_{L^{2}(\mu)}^{2}=\langle a_{k}^{1/2}\zeta_{k},a_{k}^{1/2}\zeta_{k}\rangle_{L^{2}(\mu)}=\bigl(a_{k}^{1/2}\bigr)^{2}\langle\zeta_{k},\zeta_{k}\rangle_{L^{2}(\mu)}=a_{k}\lVert\zeta_{k}\rVert_{L^{2}(\mu)}^{2}\qquad(k\in\mathbb{N}).$$

Since $\mu$ has finite Fisher information relative to $\gamma_{c}$ with weights $a$, by Weight Sequences and the Weighted Fisher Information Relative to a Diagonal Gaussian Measure on a Hilbert Space §information the series $\sum_{k}\lVert g_{k}\rVert_{L^{2}(\mu)}^{2}=\sum_{k}a_{k}\lVert\zeta_{k}\rVert_{L^{2}(\mu)}^{2}$ converges with sum $\mathcal{I}_{a}(\mu\,|\,\gamma_{c})$. By The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §synthesis there is exactly one $Z\in L^{2}(\mu;X_{a})$ with $Z_{k}=g_{k}=a_{k}^{1/2}\zeta_{k}$ for every $k\in\mathbb{N}$, and by The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates

$$\lVert Z\rVert_{L^{2}(\mu;X_{a})}^{2}=\sum_{k=1}^{\infty}\lVert Z_{k}\rVert_{L^{2}(\mu)}^{2}=\sum_{k=1}^{\infty}a_{k}\lVert\zeta_{k}\rVert_{L^{2}(\mu)}^{2}=\mathcal{I}_{a}(\mu\,|\,\gamma_{c}).$$