Proof of Noise Gradients of Bounded $C^2$ Cylindrical Functions Are Dense in the Noise Tangent Space

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

Conventions. For $\mu\in\mathcal{P}(X)$, the space $L^{2}(\mu;X^{a})$ of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §fields is a real Hilbert space by The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §hilbert, applied with $E=X^{a}$, the basis $(f_{k})_{k\in\mathbb{N}}$ and the measure space $(X,\mathcal{B}(X),\mu)$, and $\lVert\cdot\rVert_{\mu}$ is the norm of its inner product. In particular $L^{2}(\mu;X^{a})$ is a real inner product space and a real vector space (Real Hilbert Space §hilbert); we write $d_{\mu}(g,g')=\lVert g-g'\rVert_{\mu}$ for its distance, which is a metric by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §metric, and by Real Hilbert Space §topology the closure in the definition of $T^{a}_{\mu}$ and convergence in $L^{2}(\mu;X^{a})$ refer to the metric space $(L^{2}(\mu;X^{a}),d_{\mu})$.

Step 1 (Claim 1). Let $\psi\in\mathcal{F}C^{2}_{b}(X)$. By Bounded C^2 Cylindrical Functions on a Hilbert Space with an Orthonormal Basis §cylindrical there are $n\in\mathbb{N}$ and $\chi\in C^{2}_{b}(\mathbb{R}^{n})$ with $\psi=\chi\circ p_{n}$. By Bounded Twice Continuously Differentiable Functions with Bounded First and Second Partial Derivatives on Euclidean Space §bounded, taken with $q=n$, $\chi$ is of class $C^{2}$ on $\mathbb{R}^{n}$ and $\chi$ and $\partial_{i}\chi$ are bounded for every $i\in[n]$. By clause 2 of C^k Maps on a Euclidean Open Set, taken with $k=1$ and read through its clause 3, a function of class $C^{2}$ on $\mathbb{R}^{n}$ is of class $C^{1}$ there; so $\chi$ is of class $C^{1}$ on $\mathbb{R}^{n}$, and $\chi\in C^{1}_{b}(\mathbb{R}^{n})$ by Bounded Continuously Differentiable Functions with Bounded Partial Derivatives on Euclidean Space §bounded. Hence $\psi\in\mathcal{F}C^{1}_{b}(X)$ by Bounded C^1 Cylindrical Functions on a Hilbert Space with an Orthonormal Basis §cylindrical, and $(n,\chi)$ is a representation of $\psi$. This proves $\mathcal{F}C^{2}_{b}(X)\subseteq\mathcal{F}C^{1}_{b}(X)$; we record for later use that the argument shows: if $\chi\in C^{2}_{b}(\mathbb{R}^{n})$, then $\chi\in C^{1}_{b}(\mathbb{R}^{n})$ and $(n,\chi)$ is a representation of $\chi\circ p_{n}\in\mathcal{F}C^{1}_{b}(X)$.

Now let $\mu\in\mathcal{P}(X)$. Since $\psi\in\mathcal{F}C^{1}_{b}(X)$, its noise gradient $\nabla_{a}\psi:X\to X^{a}$ is defined, measurable into $X^{a}$ and square-integrable with respect to $\mu$ by The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §gradient, and its class $g$ belongs to $G^{a}_{\mu}$ by The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §gradients. The constant sequence $(g_{m})_{m\in\mathbb{N}}$ with $g_{m}=g$ for every $m$ lies in $G^{a}_{\mu}$ and converges to $g$ in $(L^{2}(\mu;X^{a}),d_{\mu})$: by condition 2 of Metric Space we have $d_{\mu}(g_{m},g)=0<\varepsilon$ for every $m$ and every positive $\varepsilon$, so the requirement of Convergent Sequence in a Metric Space holds with $N=1$. By Sequential Characterization of the Closure in a Metric Space, $g$ lies in the closure of $G^{a}_{\mu}$, which is $T^{a}_{\mu}$ by The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §tangent. This proves claim 1.

Step 2 (Smoothing a bounded $C^{1}$ cylindrical function). For the rest of the proof fix $\mu\in\mathcal{P}(X)$ and $v\in T^{a}_{\mu}$ as in claim 2. In Steps 2 to 4 fix moreover $\varphi\in\mathcal{F}C^{1}_{b}(X)$ and, by Bounded C^1 Cylindrical Functions on a Hilbert Space with an Orthonormal Basis §cylindrical, a representation $(n,h)$ of it, so that $h\in C^{1}_{b}(\mathbb{R}^{n})$ and $\varphi=h\circ p_{n}$. By Bounded Continuously Differentiable Functions with Bounded Partial Derivatives on Euclidean Space §bounded the function $h$ is of class $C^{1}$ on $\mathbb{R}^{n}$ and $h$ and every $\partial_{i}h$, $i\in[n]$, are bounded; fix nonnegative real numbers $M_{0}$ and $B_{i}$ ($i\in[n]$) with $|h(y)|\le M_{0}$ and $|\partial_{i}h(y)|\le B_{i}$ for every $y\in\mathbb{R}^{n}$.

The function $h$ is Borel. Indeed, by claim 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, applied with $U=\mathbb{R}^{n}$, $m=1$, $k=1$ and $F=h$, the function $h$ is continuous relative to $\mathbb{R}^{n}$ at every point, as a map from $(\mathbb{R}^{n},d_{E})$ into $(\mathbb{R},d_{\mathbb{R}})$, that is, continuous on $\mathbb{R}^{n}$; the metric $d_{\mathbb{R}}(s,t)=|s-t|$ is the Euclidean distance $\sqrt{(s-t)^{2}}$ of $\mathbb{R}^{1}$. So claim 3(a) of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets shows that $h$ is measurable with respect to $\mathcal{B}_{n}$ and $\mathcal{B}(\mathbb{R})$, and $\mathcal{B}_{n}=\mathcal{B}(\mathbb{R}^{n})$ by claim 5 there.

We apply Gaussian Smoothing of a Bounded Borel Function on Euclidean Space: the Two Forms, Derivatives through the Kernel and through the Function, and Pointwise Convergence with $q=n$, first with $H=h$ and $M=M_{0}$. For $0<s\le1$ the function $h_{s}$ is defined, with $|h_{s}(y)|\le M_{0}$ for every $y$, by Gaussian Smoothing of a Bounded Borel Function on Euclidean Space: the Two Forms, Derivatives through the Kernel and through the Function, and Pointwise Convergence §smoothing, and $h_{s}$ is of class $C^{1}$ on $\mathbb{R}^{n}$ by Gaussian Smoothing of a Bounded Borel Function on Euclidean Space: the Two Forms, Derivatives through the Kernel and through the Function, and Pointwise Convergence §kernel-derivative. Since $h\in C^{1}_{b}(\mathbb{R}^{n})$, Gaussian Smoothing of a Bounded Borel Function on Euclidean Space: the Two Forms, Derivatives through the Kernel and through the Function, and Pointwise Convergence §function-derivative shows that for every $i\in[n]$ the function $\partial_{i}h$ is Borel and bounded, so that its smoothing $(\partial_{i}h)_{s}$ is defined by Gaussian Smoothing of a Bounded Borel Function on Euclidean Space: the Two Forms, Derivatives through the Kernel and through the Function, and Pointwise Convergence §smoothing applied with $H=\partial_{i}h$ and its bound $M=B_{i}$, and

$$\partial_{i}(h_{s})=(\partial_{i}h)_{s}\qquad(0<s\le1,\ i\in[n]).$$

Next, for $i\in[n]$ we apply the same lemma with $H=\partial_{i}h$ and $M=B_{i}$: by Gaussian Smoothing of a Bounded Borel Function on Euclidean Space: the Two Forms, Derivatives through the Kernel and through the Function, and Pointwise Convergence §smoothing, $|(\partial_{i}h)_{s}(y)|\le B_{i}$ for every $y\in\mathbb{R}^{n}$, and by Gaussian Smoothing of a Bounded Borel Function on Euclidean Space: the Two Forms, Derivatives through the Kernel and through the Function, and Pointwise Convergence §kernel-derivative, $(\partial_{i}h)_{s}$ is of class $C^{1}$ on $\mathbb{R}^{n}$ with $|\partial_{j}((\partial_{i}h)_{s})(y)|\le B_{i}\sqrt{n}\,s^{-1/2}$ for every $j\in[n]$ and every $y\in\mathbb{R}^{n}$.

Fix $s$ with $0<s\le1$. Then $h_{s}$ is of class $C^{1}$ on $\mathbb{R}^{n}$ and each $\partial_{i}(h_{s})=(\partial_{i}h)_{s}$ is of class $C^{1}$ on $\mathbb{R}^{n}$, so $h_{s}$ is of class $C^{2}$ on $\mathbb{R}^{n}$ by clause 2 of C^k Maps on a Euclidean Open Set, taken with $k=1$ and read through its clause 3. Its iterated partial derivatives in the sense of clause 4 there are $\partial_{j}\partial_{i}(h_{s})=\partial_{j}((\partial_{i}h)_{s})$, so $h_{s}$, $\partial_{i}(h_{s})$ and $\partial_{j}\partial_{i}(h_{s})$ are bounded, by $M_{0}$, $B_{i}$ and $B_{i}\sqrt{n}\,s^{-1/2}$ respectively, for all $i,j\in[n]$. Hence $h_{s}\in C^{2}_{b}(\mathbb{R}^{n})$ by Bounded Twice Continuously Differentiable Functions with Bounded First and Second Partial Derivatives on Euclidean Space §bounded, taken with $q=n$, and the function

$$\varphi_{s}=h_{s}\circ p_{n}$$

belongs to $\mathcal{F}C^{2}_{b}(X)$ by Bounded C^2 Cylindrical Functions on a Hilbert Space with an Orthonormal Basis §cylindrical. By the fact recorded in Step 1, $(n,h_{s})$ is a representation of $\varphi_{s}\in\mathcal{F}C^{1}_{b}(X)$.

Step 3 (The pointwise discrepancy of the noise gradients). Let $0<s\le1$. For $x\in X$ and $k\in[n]$ put

$$c^{s}_{k}(x)=(\partial_{k}h)_{s}\bigl(p_{n}(x)\bigr)-\partial_{k}h\bigl(p_{n}(x)\bigr).$$

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, applied to the representations $(n,h_{s})$ of $\varphi_{s}$ and $(n,h)$ of $\varphi$, together with Step 2, for $k\le n$ and $x\in X$ we have $\partial_{k}(\varphi_{s})(x)=\partial_{k}(h_{s})(p_{n}(x))=(\partial_{k}h)_{s}(p_{n}(x))$ and $\partial_{k}\varphi(x)=\partial_{k}h(p_{n}(x))$. Write $u_{s}=\nabla_{a}\varphi_{s}$ and $u=\nabla_{a}\varphi$. By The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §gradient, both are measurable maps $X\to X^{a}$, square-integrable with respect to $\mu$, and for $x\in X$ the $l$-th coordinates $\langle u_{s}(x),e_{l}\rangle$ and $\langle u(x),e_{l}\rangle$ are $a_{l}\,\partial_{l}(\varphi_{s})(x)$ and $a_{l}\,\partial_{l}\varphi(x)$ for $l\le n$ and $0$ for $l>n$.

Fix $x\in X$ and put $w=u_{s}(x)+(-1)u(x)$, which is $u_{s}(x)-u(x)$ by claim 5 of Elementary Identities in a Vector Space and lies in $X^{a}$, a linear subspace of $X$ carrying the restricted vector operations, by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §hilbert. By the additivity and homogeneity (b) and (c) of Real Inner Product Space §inner-product for the inner product of $X$, $\langle w,e_{l}\rangle=\langle u_{s}(x),e_{l}\rangle-\langle u(x),e_{l}\rangle$, which equals $a_{l}\,c^{s}_{l}(x)$ for $l\le n$ and $0$ for $l>n$. By The Noise Space of a Weight Sequence on a Hilbert Space with an Orthonormal Basis §inner-product, $|w|_{a}^{2}=\sum_{l=1}^{\infty}a_{l}^{-1}\langle w,e_{l}\rangle^{2}$; the terms with $l>n$ vanish, so every partial sum with index at least $n$ equals $\sum_{l=1}^{n}a_{l}^{-1}a_{l}^{2}\,c^{s}_{l}(x)^{2}$, and this common value is the sum of the series by Series of Real Numbers §convergent. Hence

$$|u_{s}(x)-u(x)|_{a}^{2}=\sum_{k=1}^{n}a_{k}\,c^{s}_{k}(x)^{2}\qquad(x\in X).\tag{1}$$

By Measurable Maps into a Hilbert Space with an Orthonormal Basis: Norms, Inner Products and Linear Combinations, Synthesis from Coordinates, Square-Integrability, and Almost-Everywhere Equality §operations, applied with $E=X^{a}$, the basis $(f_{k})_{k\in\mathbb{N}}$ and $(S,\mathcal{S})=(X,\mathcal{B}(X))$, the map $u_{s}+(-1)u$ is measurable and the function $F_{s}=|u_{s}+(-1)u|_{a}^{2}$ on $X$ is measurable; it is nonnegative. In the real vector space $L^{2}(\mu;X^{a})$, claim 5 of Elementary Identities in a Vector Space and The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §operations give $[u_{s}]-[u]=[u_{s}]+(-1)[u]=[u_{s}+(-1)u]$ and

$$\lVert\nabla_{a}\varphi_{s}-\nabla_{a}\varphi\rVert_{\mu}^{2}=\int_{X}F_{s}\,d\mu .\tag{2}$$

Finally, the bounds of Step 2 give $|c^{s}_{k}(x)|\le|(\partial_{k}h)_{s}(p_{n}(x))|+|\partial_{k}h(p_{n}(x))|\le2B_{k}$, and each $a_{k}$ is positive (The Noise Space of a Weight Sequence on a Hilbert Space with an Orthonormal Basis), so by (1)

$$0\le F_{s}(x)\le C\qquad(x\in X),\qquad\text{where }C=4\sum_{k=1}^{n}a_{k}B_{k}^{2}.\tag{3}$$

Step 4 (Approximating one noise gradient). For $j\in\mathbb{N}$ we have $0<1/j\le1$; put $f_{j}=F_{1/j}$.

Pointwise convergence. Fix $x\in X$ and put $y=p_{n}(x)$. For $k\in[n]$, since $h$ is of class $C^{1}$ on $\mathbb{R}^{n}$, clause 1 of C^k Maps on a Euclidean Open Set, read through its clause 3, shows that $\partial_{k}h$ is continuous at every point of $\mathbb{R}^{n}$, in particular at $y$. This is exactly the hypothesis of Gaussian Smoothing of a Bounded Borel Function on Euclidean Space: the Two Forms, Derivatives through the Kernel and through the Function, and Pointwise Convergence §convergence for $H=\partial_{k}h$, which is Borel and bounded by $B_{k}$ (Step 2), at the point $y$; so that claim applies. Let $\varepsilon>0$, and let $\bar{a}$ be the positive bound of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §weights, with $a_{k}\le\bar{a}$ for every $k$. The choices are made in the following order. First put $\delta=\min\{1,\varepsilon/(1+n\bar{a})\}$, which is positive. Second, for each $k\in[n]$, Gaussian Smoothing of a Bounded Borel Function on Euclidean Space: the Two Forms, Derivatives through the Kernel and through the Function, and Pointwise Convergence §convergence with $H=\partial_{k}h$, the point $y$ and the tolerance $\delta$ gives $s_{k}$ with $0<s_{k}\le1$ and $|c^{s}_{k}(x)|<\delta$ for $0<s\le s_{k}$. Third, put $s_{*}=\min\{s_{1},\dots,s_{n}\}>0$ and choose $N\in\mathbb{N}$ with $N s_{*}\ge1$. For $j\ge N$ we have $0<1/j\le1/N\le s_{*}\le s_{k}$, hence $|c^{1/j}_{k}(x)|<\delta\le1$ and so $c^{1/j}_{k}(x)^{2}\le|c^{1/j}_{k}(x)|<\delta$ for every $k\in[n]$; by (1), (3) and $0<a_{k}\le\bar{a}$,

$$0\le f_{j}(x)=\sum_{k=1}^{n}a_{k}\,c^{1/j}_{k}(x)^{2}<n\bar{a}\,\delta\le\frac{n\bar{a}\,\varepsilon}{1+n\bar{a}}<\varepsilon .$$

Thus $|f_{j}(x)-0|<\varepsilon$ for $j\ge N$, and $(f_{j}(x))_{j\in\mathbb{N}}$ converges to $0$ by Limit of a Sequence of Real Numbers.

Dominated convergence. The triple $(X,\mathcal{B}(X),\mu)$ is a measure space (Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §measures). The functions $f_{j}$ are measurable (Step 3), converge to $0$ at every point of $X$, and satisfy $|f_{j}|\le C$ on $X$ by (3). The constant function on $X$ with value $C$ belongs to $\mathcal{F}C^{1}_{b}(X)$ by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §linear, and is therefore integrable with respect to $\mu$ by 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. Claim 2 of Dominated Convergence Theorem, applied with the sequence $(f_{j})_{j\in\mathbb{N}}$, the limit function $f=0$ and $g$ this constant function, gives $\int_{X}|f_{j}-0|\,d\mu\to0$, that is, $\int_{X}f_{j}\,d\mu\to0$ as $j\to\infty$, since $f_{j}\ge0$; for nonnegative $f_{j}$ this integral is the integral of the nonnegative function $f_{j}$ appearing in (2), the negative part of $f_{j}$ being $0$ (Integrable Function and the Lebesgue Integral).

Conclusion. Let $\varepsilon>0$. By Limit of a Sequence of Real Numbers choose $N\in\mathbb{N}$ with $\int_{X}f_{j}\,d\mu<\varepsilon^{2}$ for $j\ge N$, and put $\psi=\varphi_{1/N}$, which belongs to $\mathcal{F}C^{2}_{b}(X)$ by Step 2. By (2), $\lVert\nabla_{a}\psi-\nabla_{a}\varphi\rVert_{\mu}^{2}=\int_{X}f_{N}\,d\mu<\varepsilon^{2}$, and the norm is nonnegative, so $\lVert\nabla_{a}\psi-\nabla_{a}\varphi\rVert_{\mu}<\varepsilon$. Since $\varphi$ was an arbitrary element of $\mathcal{F}C^{1}_{b}(X)$, we have shown:

(P) for every $\varphi\in\mathcal{F}C^{1}_{b}(X)$ and every positive $\varepsilon\in\mathbb{R}$ there is $\psi\in\mathcal{F}C^{2}_{b}(X)$ with $\lVert\nabla_{a}\psi-\nabla_{a}\varphi\rVert_{\mu}<\varepsilon$.

Step 5 (Claim 2). Since $v\in T^{a}_{\mu}$, which is the closure of $G^{a}_{\mu}$ by The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §tangent, Sequential Characterization of the Closure in a Metric Space gives a sequence $(g_{m})_{m\in\mathbb{N}}$ with $g_{m}\in G^{a}_{\mu}$ for every $m$ that converges to $v$ in $(L^{2}(\mu;X^{a}),d_{\mu})$. For each $j\in\mathbb{N}$ the choices are made in the following order. First, by Convergent Sequence in a Metric Space with the tolerance $1/(2j)$, choose $N_{j}\in\mathbb{N}$ with $d_{\mu}(g_{m},v)<1/(2j)$ for every $m\ge N_{j}$. Second, since $g_{N_{j}}\in G^{a}_{\mu}$, choose by The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §gradients a function $\varphi_{(j)}\in\mathcal{F}C^{1}_{b}(X)$ whose noise gradient has class $g_{N_{j}}$. Third, by (P) with $\varphi=\varphi_{(j)}$ and $\varepsilon=1/(2j)$, choose $\psi_{j}\in\mathcal{F}C^{2}_{b}(X)$ with $\lVert\nabla_{a}\psi_{j}-\nabla_{a}\varphi_{(j)}\rVert_{\mu}<1/(2j)$. By the triangle inequality, condition 4 of Metric Space for the metric $d_{\mu}$,

$$\lVert\nabla_{a}\psi_{j}-v\rVert_{\mu}=d_{\mu}(\nabla_{a}\psi_{j},v)\le d_{\mu}(\nabla_{a}\psi_{j},g_{N_{j}})+d_{\mu}(g_{N_{j}},v)<\frac{1}{2j}+\frac{1}{2j}=\frac{1}{j}.$$

Let $\varepsilon>0$ and choose $N\in\mathbb{N}$ with $N\varepsilon>1$. For $j\ge N$ we get $0\le\lVert\nabla_{a}\psi_{j}-v\rVert_{\mu}<1/j\le1/N<\varepsilon$, so $\bigl|\lVert\nabla_{a}\psi_{j}-v\rVert_{\mu}-0\bigr|<\varepsilon$. By Limit of a Sequence of Real Numbers, $\lim_{j\to\infty}\lVert\nabla_{a}\psi_{j}-v\rVert_{\mu}=0$, and $(\psi_{j})_{j\in\mathbb{N}}$ is a sequence in $\mathcal{F}C^{2}_{b}(X)$. This proves claim 2.