Proof of The Gaussian Sobolev Space: Linear Structure, Closedness, Hermite Polynomials and the Number-Operator Form of the Dirichlet Form, and the Log-Sobolev and Poincare Inequalities

Each result cited is universally quantified over the data in its own statement. Elementary real arithmetic and order (The Real Numbers: Standing Notation and Background §background) are used without citation; this covers the limit laws for sums, products and real multiples of convergent real sequences, the preservation of non-strict inequalities under limits, the inequality $(p+q)^{2}\le2p^{2}+2q^{2}$, and manipulations of finite sums and products. Integrals and integrability are those of Measure Spaces and the Lebesgue Integral: Standing Notation; linearity and monotonicity of the integral are Linearity and Monotonicity of the Lebesgue Integral §integrable and Linearity and Monotonicity of the Lebesgue Integral §nonnegative; bounded Borel functions are integrable with respect to the probability measure $\gamma_{c}$ by claim 6(b) of Borel Measurability and Bounded Integration on a Metric Space. In $L^{2}(\gamma_{c})$ we write $\langle\cdot,\cdot\rangle$ and $\lVert\cdot\rVert$ for the inner product and norm; by The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §inner-product, $\langle f,g\rangle=\int_{X}fg\,d\gamma_{c}$ for $2$-integrable $f,g$ and $\lVert f\rVert=\lVert f\rVert_{2}$. $L^{2}(\gamma_{c};X^{a})$ is a real Hilbert space whose norm $\lVert\cdot\rVert_{\gamma_{c}}$ is that of its inner product (The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §hilbert). In any real inner product space with norm $|\cdot|$: $|tx|=|t|\,|x|$ by Elementary Identities in a Real Inner Product Space §homogeneity; the triangle inequality $|x+y|\le|x|+|y|$ holds by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §triangle; the norm distance is a metric by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §metric; if $x_{j}\to x$ and $y_{j}\to y$, then $x_{j}+ty_{j}\to x+ty$ for real $t$ by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §linear-limits; and if $x_{j}\to x$, then $|x_{j}|\to|x|$ by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §continuity, hence $|x_{j}|^{2}\to|x|^{2}$. A convergent sequence in a metric space is Cauchy in the sense of Cauchy Sequence in a Metric Space, since if every term from some index on lies within $\varepsilon/2$ of the limit, any two such terms lie within $\varepsilon$ of each other by the triangle inequality of the metric; and its limit is unique by Uniqueness of Limits in a Metric Space.

Step 1 (Calculus of profiles). For $N\in\mathbb{N}$ the set $\mathbb{R}^{N}$ is open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, and the class $C^{1}$ and the partial derivatives on it are those of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §derivatives, that is, of C^k Maps on a Euclidean Open Set, which are the notions of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set and A Composition of $C^k$ Maps Between Euclidean Open Sets is of Class $C^k$.

(a) Let $f,g:\mathbb{R}^{N}\to\mathbb{R}$ be of class $C^{1}$ on $\mathbb{R}^{N}$ and $t\in\mathbb{R}$. By claim 3 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, applied to $g$ and the scalar $t$, then to $f$ and $tg$, and to $f$ and $g$, the functions $f+tg$ and $fg$ are of class $C^{1}$ on $\mathbb{R}^{N}$. The partial derivatives of $f$ and $g$ exist at every point by clause 1 of C^k Maps on a Euclidean Open Set, read through its clause 3, so claim 1 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, applied in the same way, gives, for every $i\in[N]$,

$$\partial_{i}(f+tg)=\partial_{i}f+t\,\partial_{i}g,\qquad\partial_{i}(fg)=\partial_{i}f\,g+f\,\partial_{i}g$$

pointwise on $\mathbb{R}^{N}$. If moreover $f$, $g$ and all $\partial_{i}f$, $\partial_{i}g$ are bounded, then so are $f+tg$ and its partial derivatives, by the displayed formula; thus the set $C^{1}_{b}(\mathbb{R}^{N})$ of Bounded Continuously Differentiable Functions with Bounded Partial Derivatives on Euclidean Space §bounded is closed under sums and real multiples.

(b) Let $n\le N$, $\pi:\mathbb{R}^{N}\to\mathbb{R}^{n}$, $\pi(u)=(u_{1},\dots,u_{n})$, and let $\psi\in C^{1}_{b}(\mathbb{R}^{n})$. For $u\in\mathbb{R}^{N}$, the slice function of $\psi\circ\pi$ at $u$ in the $i$-th variable is that of $\psi$ at $\pi(u)$ if $i\le n$ (both taken on a common interval, any positive radius being admissible in claim 1 of Slice Function and the Partial Derivative, the domains being whole Euclidean spaces), and is constant if $n<i\le N$; so by claim 2 of Slice Function and the Partial Derivative and claim 1 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives, $\partial_{i}(\psi\circ\pi)=(\partial_{i}\psi)\circ\pi$ for $i\le n$ and $\partial_{i}(\psi\circ\pi)=0$ for $n<i\le N$. The coordinate functions of $\pi$ are the coordinate functions $u\mapsto u_{l}$ ($l\in[n]$) of $\mathbb{R}^{N}$, which are smooth, hence of class $C^{1}$, on $\mathbb{R}^{N}$ by claim 2 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set and Smooth Map on a Euclidean Open Set; so $\pi$ is of class $C^{1}$ on $\mathbb{R}^{N}$ by clauses 1 and 3 of C^k Maps on a Euclidean Open Set, and $\psi\circ\pi$ is of class $C^{1}$ on $\mathbb{R}^{N}$ by claim 2 of A Composition of $C^k$ Maps Between Euclidean Open Sets is of Class $C^k$, applied with $F=\pi$, the open set $\mathbb{R}^{n}$ in place of $V$, and $G=\psi$. By the formulas just obtained, $\psi\circ\pi$ and its partial derivatives are bounded, so $\psi\circ\pi\in C^{1}_{b}(\mathbb{R}^{N})$. Since $p_{n}=\pi\circ p_{N}$ by Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates, every $\varphi\in\mathcal{F}C^{1}_{b}(X)$ with representation $(n,\psi)$ has the representation $(N,\psi\circ\pi)$ for every $N\ge n$.

(c) Let $F,F'\in\mathcal{F}C^{1}_{b}(X)$ and $t\in\mathbb{R}$. By (b) they have representations $(N,\psi)$ and $(N,\psi')$ with a common $N$, and by (a) $(N,\psi+t\psi')$ is a representation of $F+tF'$. 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 (a), $\partial_{k}(F+tF')=\partial_{k}F+t\,\partial_{k}F'$ for every $k\in\mathbb{N}$ (all three vanish for $k>N$), and hence, by The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §gradient computed with these representations, $\nabla_{a}(F+tF')(x)=\nabla_{a}F(x)+t\,\nabla_{a}F'(x)$ for every $x\in X$. By The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §operations the same identity holds for the classes in $L^{2}(\gamma_{c};X^{a})$.

(d) Coordinates of noise gradients. Let $\varphi\in\mathcal{F}C^{1}_{b}(X)$ with representation $(n,\psi)$, $x\in X$ and $k\in\mathbb{N}$. By The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §gradient and Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §partial, the $j$-th coordinate of $\nabla_{a}\varphi(x)$ is $a_{j}\,\partial_{j}\varphi(x)$ for every $j\in\mathbb{N}$; $f_{k}=a_{k}^{1/2}e_{k}$ by Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §noise-space, with $j$-th coordinate $a_{k}^{1/2}$ for $j=k$ and $0$ otherwise. So the series defining the noise pairing in The Noise Space of a Weight Sequence on a Hilbert Space with an Orthonormal Basis §inner-product has the single nonzero term $a_{k}^{-1}\,a_{k}\partial_{k}\varphi(x)\,a_{k}^{1/2}$, and $\langle\nabla_{a}\varphi(x),f_{k}\rangle_{a}=a_{k}^{1/2}\partial_{k}\varphi(x)$. Hence the coordinate along $f_{k}$ (The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates) of the class $\nabla_{a}\varphi$ is the class of $a_{k}^{1/2}\partial_{k}\varphi$. For $\varphi\in\mathcal{F}C^{1}_{b}(X)$ the noise gradient of The Gaussian Sobolev Space of the Noise Gradient §gradient is this class, as recorded in that clause, and so $\mathcal{E}_{a}(\varphi,\varphi)=\lVert\nabla_{a}\varphi\rVert_{\gamma_{c}}^{2}=\int_{X}|\nabla_{a}\varphi|_{a}^{2}\,d\gamma_{c}$ by The Dirichlet Form of the Noise Gradient on the Gaussian Sobolev Space §form and The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §operations.

Step 2 (Claim 1). The constant function $0$ 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, so $0\in\mathbb{D}^{1,2}_{a}$ (constant approximating sequence). Let $F,F'\in\mathbb{D}^{1,2}_{a}$ with approximating sequences $(F_{j})$, $(F'_{j})$ (The Gaussian Sobolev Space of the Noise Gradient §space), and $t\in\mathbb{R}$. Then $F_{j}+tF'_{j}\in\mathcal{F}C^{1}_{b}(X)$ 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, the class of $F_{j}+tF'_{j}$ being $F_{j}+tF'_{j}$ in $L^{2}(\gamma_{c})$ (The Lebesgue Space of Power-Integrable Functions §space),

$$\lVert F_{j}+tF'_{j}-(F+tF')\rVert_{2}\le\lVert F_{j}-F\rVert_{2}+|t|\,\lVert F'_{j}-F'\rVert_{2}\to0$$

by the norm properties of The Riesz-Fischer Theorem: the Lebesgue Space is a Real Banach Space §normed. By Step 1(c), $\nabla_{a}(F_{j}+tF'_{j})=\nabla_{a}F_{j}+t\,\nabla_{a}F'_{j}$; by The Gaussian Sobolev Space of the Noise Gradient §gradient, $\nabla_{a}F_{j}\to\nabla_{a}F$ and $\nabla_{a}F'_{j}\to\nabla_{a}F'$ in $L^{2}(\gamma_{c};X^{a})$, so $\nabla_{a}(F_{j}+tF'_{j})\to\nabla_{a}F+t\,\nabla_{a}F'$ by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §linear-limits, and this sequence is Cauchy by the preamble. Hence $(F_{j}+tF'_{j})$ approximates $F+tF'$, so $F+tF'\in\mathbb{D}^{1,2}_{a}$, and $\nabla_{a}(F+tF')=\nabla_{a}F+t\,\nabla_{a}F'$ by The Gaussian Sobolev Space of the Noise Gradient §gradient. So $\mathbb{D}^{1,2}_{a}$ is a linear subspace of $L^{2}(\gamma_{c})$ and $\nabla_{a}$ is linear on it.

Step 3 (Claim 2). Let $(F_{j})$, $F$, $G$ be as in claim 2. For each $j$ fix an approximating sequence $(F_{j,l})_{l\in\mathbb{N}}$ of $F_{j}$; then $\lVert F_{j,l}-F_{j}\rVert_{2}\to0$ and $\nabla_{a}F_{j,l}\to\nabla_{a}F_{j}$ as $l\to\infty$ by The Gaussian Sobolev Space of the Noise Gradient §gradient, so there is $l_{j}\in\mathbb{N}$ with $\lVert F_{j,l_{j}}-F_{j}\rVert_{2}<1/j$ and $\lVert\nabla_{a}F_{j,l_{j}}-\nabla_{a}F_{j}\rVert_{\gamma_{c}}<1/j$. Put $F'_{j}=F_{j,l_{j}}\in\mathcal{F}C^{1}_{b}(X)$. By the triangle inequality,

$$\lVert F'_{j}-F\rVert_{2}<\frac{1}{j}+\lVert F_{j}-F\rVert_{2}\to0,\qquad\lVert\nabla_{a}F'_{j}-G\rVert_{\gamma_{c}}<\frac{1}{j}+\lVert\nabla_{a}F_{j}-G\rVert_{\gamma_{c}}\to0 .$$

So $(\nabla_{a}F'_{j})$ converges, hence is Cauchy, and $(F'_{j})$ approximates $F$. Thus $F\in\mathbb{D}^{1,2}_{a}$, and $\nabla_{a}F$, the limit of $(\nabla_{a}F'_{j})$ by The Gaussian Sobolev Space of the Noise Gradient §gradient, equals $G$.

Step 4 (Claim 3: approximation of $H_{\alpha}$). Fix $\alpha\in\mathcal{A}$ with a length bound $n$, and let $h_{\alpha}$ be as in Cylindrical Hermite Polynomials: Growth and Integrability, Partial Derivatives, Orthogonality, and Monomials as Hermite Combinations §cylindrical, so $h_{\alpha}$ is of class $C^{2}$, hence of class $C^{1}$ (clause 2 of C^k Maps on a Euclidean Open Set), and $H_{\alpha}=h_{\alpha}\circ p_{n}$. For $i\in[n]$ with $\alpha_{i}\ge1$, $n$ is a length bound of $\alpha-\varepsilon_{i}$ (Finitely Supported Multi-Indices: Order, Factorial, Powers and Weighted Order §unit), and we write $h_{\alpha-\varepsilon_{i}}$ for the corresponding function, so $H_{\alpha-\varepsilon_{i}}=h_{\alpha-\varepsilon_{i}}\circ p_{n}$ by the same claim.

(a) Partial derivatives of $h_{\alpha}$. Let $y\in\mathbb{R}^{n}$, $i\in[n]$, and $P=\prod_{k\in[n],k\ne i}H^{c_{k}}_{\alpha_{k}}(y_{k})$. The slice function of $h_{\alpha}$ at $y$ in the $i$-th variable is $s\mapsto P\,H^{c_{i}}_{\alpha_{i}}(s)$. The function $H^{c_{i}}_{\alpha_{i}}$ is of class $C^{2}$ on $\mathbb{R}^{1}$ by Hermite Polynomials: Low Orders, Derivatives, Recursion, Scaling, the Addition Formula, Gaussian Orthogonality and the Mehler Identity §polynomial, and by claim 2 of Slice Function and the Partial Derivative (with $n=1$, where a slice is the function itself) it is differentiable at $y_{i}$ with derivative $(H^{c_{i}}_{\alpha_{i}})'(y_{i})$, which is $\alpha_{i}H^{c_{i}}_{\alpha_{i}-1}(y_{i})$ if $\alpha_{i}\ge1$ and $0$ if $\alpha_{i}=0$, by Hermite Polynomials: Low Orders, Derivatives, Recursion, Scaling, the Addition Formula, Gaussian Orthogonality and the Mehler Identity §derivative. By claim 2 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives and claim 2 of Slice Function and the Partial Derivative,

$$\partial_{i}h_{\alpha}(y)=\alpha_{i}\,h_{\alpha-\varepsilon_{i}}(y)\ \text{ if }\alpha_{i}\ge1,\qquad\partial_{i}h_{\alpha}(y)=0\ \text{ if }\alpha_{i}=0 .$$

For uniform notation put $\eta_{i}=\alpha_{i}H_{\alpha-\varepsilon_{i}}$ if $\alpha_{i}\ge1$ and $\eta_{i}=0$ if $\alpha_{i}=0$ ($i\in\mathbb{N}$); then $(\partial_{i}h_{\alpha})\circ p_{n}=\eta_{i}$ for $i\in[n]$, and $\eta_{i}=0$ for $i>n$. By Cylindrical Hermite Polynomials: Growth and Integrability, Partial Derivatives, Orthogonality, and Monomials as Hermite Combinations §cylindrical, applied to $\alpha$ and to each $\alpha-\varepsilon_{i}$ with $\alpha_{i}\ge1$ (with $p=2$, and taking the largest of the finitely many growth constants and multiplying by $\alpha_{i}$), $H_{\alpha}$ and each $\eta_{i}$ are Borel with $\int_{X}H_{\alpha}^{2}\,d\gamma_{c}<\infty$ and $\int_{X}\eta_{i}^{2}\,d\gamma_{c}<\infty$, and there is $M_{0}\ge0$ such that, with $q_{i}=|\alpha-\varepsilon_{i}|$ if $\alpha_{i}\ge1$ and $q_{i}=0$ if $\alpha_{i}=0$, the growth bounds of that claim give $|h_{\alpha}(y)|\le M_{0}(1+\lVert y\rVert^{|\alpha|})$ and $|\partial_{i}h_{\alpha}(y)|\le M_{0}(1+\lVert y\rVert^{q_{i}})$ for all $y\in\mathbb{R}^{n}$ and $i\in[n]$.

(b) Cutoffs. Fix $\chi$ as in Scaled Cutoffs and the Second-Moment Test Functions: Uniform Derivative Bounds and Agreement on a Ball with $q=n$ (it exists by Existence of a Smooth Plateau Function on Euclidean Space, as recorded there), with the cutoffs $\chi_{R}$ and the constant $M_{1}\ge0$ of Scaled Cutoffs and the Second-Moment Test Functions: Uniform Derivative Bounds and Agreement on a Ball §cutoff. For $m\in\mathbb{N}$, $\chi_{m}$ is smooth, hence of class $C^{1}$ (Smooth Map on a Euclidean Open Set), $0\le\chi_{m}\le1$, $\chi_{m}(y)=1$ for $\lVert y\rVert\le m$, $\chi_{m}(y)=0$ for $\lVert y\rVert\ge2m$, and $|\partial_{i}\chi_{m}|\le M_{1}/m$. If $\lVert y\rVert>2m$, put $\delta=\lVert y\rVert-2m>0$; for $|s-y_{i}|<\delta$ the point $y[s]$ (the $i$-th coordinate of $y$ replaced by $s$) satisfies $\lVert y[s]\rVert\ge\lVert y\rVert-|s-y_{i}|>2m$ by Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §triangle, so the slice function of $\chi_{m}$ at $y$ on the interval of radius $\delta$ is $0$, and $\partial_{i}\chi_{m}(y)=0$ by claim 2 of Slice Function and the Partial Derivative and claim 1 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives.

Put $\psi_{m}=\chi_{m}h_{\alpha}$. By Step 1(a), $\psi_{m}$ is of class $C^{1}$ with $\partial_{i}\psi_{m}=\partial_{i}\chi_{m}\,h_{\alpha}+\chi_{m}\,\partial_{i}h_{\alpha}$. For $\lVert y\rVert>2m$ both $\psi_{m}(y)$ and $\partial_{i}\psi_{m}(y)$ vanish; for $\lVert y\rVert\le2m$, (a) gives $|\psi_{m}(y)|\le M_{0}(1+(2m)^{|\alpha|})$ and $|\partial_{i}\psi_{m}(y)|\le(M_{1}/m)M_{0}(1+(2m)^{|\alpha|})+M_{0}(1+(2m)^{q_{i}})$. So $\psi_{m}\in C^{1}_{b}(\mathbb{R}^{n})$, and $F_{m}=\psi_{m}\circ p_{n}\in\mathcal{F}C^{1}_{b}(X)$ with representation $(n,\psi_{m})$. Writing $\varrho_{m}=\chi_{m}\circ p_{n}$ and $\varrho_{m,i}=(\partial_{i}\chi_{m})\circ p_{n}$ (Borel, by Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, claims 3 and 4 of Borel Measurability and Bounded Integration on a Metric Space and Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity), 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 (a) give

$$F_{m}=\varrho_{m}H_{\alpha},\qquad\partial_{i}F_{m}=\varrho_{m,i}H_{\alpha}+\varrho_{m}\,\eta_{i}\ (i\le n),\qquad\partial_{i}F_{m}=0\ (i>n).$$

(c) Convergence of the functions. For $x\in X$ and $m\ge\lVert p_{n}(x)\rVert$ one has $\varrho_{m}(x)=1$, so $(F_{m}(x)-H_{\alpha}(x))^{2}=(1-\varrho_{m}(x))^{2}H_{\alpha}(x)^{2}\to0$ as $m\to\infty$, and $(F_{m}-H_{\alpha})^{2}\le H_{\alpha}^{2}$, which is integrable. By Dominated Convergence Theorem, $\lVert F_{m}-H_{\alpha}\rVert_{2}^{2}=\int_{X}(F_{m}-H_{\alpha})^{2}\,d\gamma_{c}\to0$ (the $2$-seminorm of Power-Integrable Functions and the p-Seminorm §seminorm, with $|f|^{2}=f^{2}$ by Properties of Real Powers of Nonnegative Real Numbers §agreement).

(d) Convergence of the gradients. For $k\in\mathbb{N}$ put $\xi_{k}=a_{k}^{1/2}\eta_{k}\in L^{2}(\gamma_{c})$; only finitely many $\xi_{k}$ are nonzero, so $\sum_{k}\lVert \xi_{k}\rVert^{2}$ converges, and by The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §synthesis there is exactly one $G_{\alpha}\in L^{2}(\gamma_{c};X^{a})$ with coordinate $\xi_{k}$ along $f_{k}$ for every $k$. By Step 1(d), (b) and The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates, the coordinate of $\nabla_{a}F_{m}-G_{\alpha}$ along $f_{k}$ is $d_{m,k}=a_{k}^{1/2}(\varrho_{m,k}H_{\alpha}+(\varrho_{m}-1)\eta_{k})$ for $k\le n$ and $0$ for $k>n$, and

$$\lVert\nabla_{a}F_{m}-G_{\alpha}\rVert_{\gamma_{c}}^{2}=\sum_{k=1}^{n}\int_{X}d_{m,k}^{2}\,d\gamma_{c}.$$

For fixed $k\le n$ and $x\in X$: $|\varrho_{m,k}(x)H_{\alpha}(x)|\le(M_{1}/m)|H_{\alpha}(x)|\to0$, and $(\varrho_{m}(x)-1)\eta_{k}(x)=0$ once $m\ge\lVert p_{n}(x)\rVert$; so $d_{m,k}(x)^{2}\to0$, while $d_{m,k}^{2}\le2a_{k}(M_{1}^{2}H_{\alpha}^{2}+\eta_{k}^{2})$ for $m\ge1$, an integrable function. By Dominated Convergence Theorem each integral tends to $0$, so $\nabla_{a}F_{m}\to G_{\alpha}$ in $L^{2}(\gamma_{c};X^{a})$, and $(\nabla_{a}F_{m})$ is Cauchy.

(e) By (c) and (d), $(F_{m})$ approximates $H_{\alpha}$ (The Gaussian Sobolev Space of the Noise Gradient §space), so $H_{\alpha}\in\mathbb{D}^{1,2}_{a}$ and $\nabla_{a}H_{\alpha}=G_{\alpha}$ by The Gaussian Sobolev Space of the Noise Gradient §gradient. Its coordinate along $f_{k}$ is $\xi_{k}=a_{k}^{1/2}\eta_{k}$, which is $a_{k}^{1/2}\alpha_{k}H_{\alpha-\varepsilon_{k}}$ if $\alpha_{k}\ge1$ and $0$ if $\alpha_{k}=0$. (On $X$ this is $a_{k}^{1/2}\partial_{k}H_{\alpha}$, in agreement with Cylindrical Hermite Polynomials: Growth and Integrability, Partial Derivatives, Orthogonality, and Monomials as Hermite Combinations §derivative.) This proves claim 3.

Step 5 (Claim 4). Let $\alpha,\beta\in\mathcal{A}$, and let $n$ be a common length bound (the larger of two length bounds). By The Dirichlet Form of the Noise Gradient on the Gaussian Sobolev Space §form, The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates and claim 3,

$$\mathcal{E}_{a}(H_{\alpha},H_{\beta})=\langle\nabla_{a}H_{\alpha},\nabla_{a}H_{\beta}\rangle_{\gamma_{c}}=\sum_{k=1}^{\infty}t_{k},\qquad t_{k}=\langle(\nabla_{a}H_{\alpha})_{k},(\nabla_{a}H_{\beta})_{k}\rangle .$$

If $\alpha_{k}=0$ or $\beta_{k}=0$ (in particular if $k>n$), one coordinate is $0$ and $t_{k}=0$ by Elementary Identities in a Real Inner Product Space §zero. If $\alpha_{k},\beta_{k}\ge1$, then $t_{k}=a_{k}\alpha_{k}\beta_{k}\langle H_{\alpha-\varepsilon_{k}},H_{\beta-\varepsilon_{k}}\rangle$ by bilinearity, and $\alpha-\varepsilon_{k}=\beta-\varepsilon_{k}$ exactly when $\alpha=\beta$. Hence, by Cylindrical Hermite Polynomials: Growth and Integrability, Partial Derivatives, Orthogonality, and Monomials as Hermite Combinations §orthogonality: if $\alpha\ne\beta$, every $t_{k}$ is $0$ and $\mathcal{E}_{a}(H_{\alpha},H_{\beta})=0$. If $\alpha=\beta$ and $\alpha_{k}\ge1$, then $t_{k}=a_{k}\alpha_{k}^{2}\,(\alpha-\varepsilon_{k})!\,c^{\alpha-\varepsilon_{k}}$, with the factorial and power of Finitely Supported Multi-Indices: Order, Factorial, Powers and Weighted Order §order computed with the length bound $n$. The two products differ from $\alpha!$ and $c^{\alpha}$ only in the $k$-th factor, which is $(\alpha_{k}-1)!$ instead of $\alpha_{k}!=\alpha_{k}(\alpha_{k}-1)!$, and $c_{k}^{\alpha_{k}-1}$ instead of $c_{k}^{\alpha_{k}}=c_{k}\,c_{k}^{\alpha_{k}-1}$; so $\alpha_{k}(\alpha-\varepsilon_{k})!=\alpha!$ and $c_{k}\,c^{\alpha-\varepsilon_{k}}=c^{\alpha}$, and with $\theta_{k}=a_{k}/c_{k}$ (Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation §rates)

$$t_{k}=\frac{a_{k}}{c_{k}}\,\alpha_{k}\,\alpha!\,c^{\alpha}=\theta_{k}\alpha_{k}\,\alpha!\,c^{\alpha};$$

this also holds, both sides being $0$, if $\alpha_{k}=0$. The partial sums of $\sum_{k}t_{k}$ are constant from the $n$-th on, so by Series of Real Numbers §convergent and Finitely Supported Multi-Indices: Order, Factorial, Powers and Weighted Order §order

$$\mathcal{E}_{a}(H_{\alpha},H_{\alpha})=\sum_{k=1}^{n}\theta_{k}\alpha_{k}\,\alpha!\,c^{\alpha}=(\theta\cdot\alpha)\,\alpha!\,c^{\alpha}.$$

Step 6 (Approximants for claims 5 and 6). Let $\kappa$, $F$ and $f$ be as in claim 5, and let $(F_{j})$ approximate $F$; so $f$ is $2$-integrable, $\lVert F_{j}-f\rVert_{2}\to0$ and $\nabla_{a}F_{j}\to\nabla_{a}F$ in $L^{2}(\gamma_{c};X^{a})$ (The Gaussian Sobolev Space of the Noise Gradient §gradient). By the preamble, $\mathcal{E}_{a}(F_{j},F_{j})=\lVert\nabla_{a}F_{j}\rVert_{\gamma_{c}}^{2}\to\lVert\nabla_{a}F\rVert_{\gamma_{c}}^{2}=\mathcal{E}_{a}(F,F)$ and $\int_{X}F_{j}^{2}\,d\gamma_{c}=\lVert F_{j}\rVert^{2}\to\lVert f\rVert^{2}=\int_{X}f^{2}\,d\gamma_{c}$; in particular $f^{2}$ is integrable. The constant $1$ is $2$-integrable with $\lVert1\rVert=1$, so $f=f\cdot1$ is integrable by The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §inner-product, and by linearity and The Cauchy-Schwarz Inequality in a Real Inner Product Space

$$\Bigl|\int_{X}F_{j}\,d\gamma_{c}-\int_{X}f\,d\gamma_{c}\Bigr|=|\langle F_{j}-f,1\rangle|\le\lVert F_{j}-f\rVert_{2}\to0 .$$

By The Gaussian Log-Sobolev, HWI and Poincare Inequalities in the Noise Geometry, in Entropy Form and in Gross's Functional Form §gross and The Gaussian Log-Sobolev, HWI and Poincare Inequalities in the Noise Geometry, in Entropy Form and in Gross's Functional Form §poincare, with Step 1(d), for every $j$: $F_{j}^{2}$ and $\phi\circ F_{j}^{2}$ are integrable and

$$\operatorname{Ent}_{\gamma_{c}}(F_{j}^{2})\le2\kappa\,\mathcal{E}_{a}(F_{j},F_{j}),\qquad\int_{X}F_{j}^{2}\,d\gamma_{c}-\Bigl(\int_{X}F_{j}\,d\gamma_{c}\Bigr)^{2}\le\kappa\,\mathcal{E}_{a}(F_{j},F_{j}).$$

Step 7 (Claim 6). Letting $j\to\infty$ in the second inequality of Step 6, with the three limits established there, gives $\int_{X}f^{2}\,d\gamma_{c}-(\int_{X}f\,d\gamma_{c})^{2}\le\kappa\,\mathcal{E}_{a}(F,F)$; and $f$ is integrable by Step 6. This proves claim 6.

Step 8 (Claim 5). By The Riesz-Fischer Theorem: the Lebesgue Space is a Real Banach Space §subsequence with $p=2$, applied to the classes of the $F_{j}$ with representatives $F_{j}$ and to $F$ with representative $f$, there are natural numbers $j_{1}<j_{2}<\cdots$ and a $\gamma_{c}$-null set $\mathcal{N}$ such that $F_{j_{l}}(x)\to f(x)$ for every $x\in X\setminus\mathcal{N}$. The function $\phi$ is continuous on $[0,\infty)$ by The Function $s\log s$: Continuity, Young's Inequality and Lower Bounds, with the Elementary Bounds for the Exponential and the Logarithm §continuous, so, unwinding the $\varepsilon$-$\delta$ condition of Continuous Map Between Metric Spaces, $\phi(F_{j_{l}}(x)^{2})\to\phi(f(x)^{2})$ for $x\notin\mathcal{N}$, and $\phi\bigl(\int_{X}F_{j_{l}}^{2}\,d\gamma_{c}\bigr)\to\phi\bigl(\int_{X}f^{2}\,d\gamma_{c}\bigr)$ by Step 6.

Put $g_{l}=\phi\circ F_{j_{l}}^{2}+\exp(-1)$ and $g=\phi\circ f^{2}+\exp(-1)$. They are Borel by The Function $s\log s$: Continuity, Young's Inequality and Lower Bounds, with the Elementary Bounds for the Exponential and the Logarithm §continuous and claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, and nonnegative by The Function $s\log s$: Continuity, Young's Inequality and Lower Bounds, with the Elementary Bounds for the Exponential and the Logarithm §lower. For $x\notin\mathcal{N}$, $(g_{l}(x))$ converges to $g(x)$, so $\sup_{k}\inf_{l\ge k}g_{l}(x)=g(x)$. By Fatou's Lemma and The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §comparison (the two nonnegative measurable functions $g$ and $\liminf_{l}g_{l}$ agreeing outside $\mathcal{N}$),

$$\int_{X}g\,d\gamma_{c}=\int_{X}\Bigl(\liminf_{l}g_{l}\Bigr)\,d\gamma_{c}\le\liminf_{l}\int_{X}g_{l}\,d\gamma_{c}.$$

Each $g_{l}$ is integrable (Step 6 and claim 6(a) of Borel Measurability and Bounded Integration on a Metric Space), so its integral as a nonnegative function is its integral as an integrable function (Integrable Function and the Lebesgue Integral), and by linearity, The Entropy of a Nonnegative Function with Respect to a Probability Measure §entropy and Step 6

$$\int_{X}g_{l}\,d\gamma_{c}=\operatorname{Ent}_{\gamma_{c}}(F_{j_{l}}^{2})+\phi\Bigl(\int_{X}F_{j_{l}}^{2}\,d\gamma_{c}\Bigr)+\exp(-1)\le b_{l},\qquad b_{l}=2\kappa\,\mathcal{E}_{a}(F_{j_{l}},F_{j_{l}})+\phi\Bigl(\int_{X}F_{j_{l}}^{2}\,d\gamma_{c}\Bigr)+\exp(-1).$$

By Step 6 and the first paragraph, $b_{l}\to b=2\kappa\,\mathcal{E}_{a}(F,F)+\phi(\int_{X}f^{2}\,d\gamma_{c})+\exp(-1)$, so $\liminf_{l}\int_{X}g_{l}\,d\gamma_{c}\le b$. Hence $\int_{X}g\,d\gamma_{c}\le b<\infty$: the nonnegative Borel function $g$, whose positive part is $g$ and whose negative part is $0$, is integrable by Integrable Function and the Lebesgue Integral, so $\phi\circ f^{2}=g-\exp(-1)$ is integrable by linearity, and

$$\int_{X}\phi\circ f^{2}\,d\gamma_{c}\le2\kappa\,\mathcal{E}_{a}(F,F)+\phi\Bigl(\int_{X}f^{2}\,d\gamma_{c}\Bigr).$$

Since $f^{2}$ is Borel, nonnegative and integrable (Step 6) and $\phi\circ f^{2}$ is integrable, $\operatorname{Ent}_{\gamma_{c}}(f^{2})$ is defined by The Entropy of a Nonnegative Function with Respect to a Probability Measure §entropy, and the last display is $\operatorname{Ent}_{\gamma_{c}}(f^{2})\le2\kappa\,\mathcal{E}_{a}(F,F)$. This proves claim 5.