Proof of Basic Properties of an Admissible Cylindrical Potential: Continuity, Growth under Noise Translations, Integrability, the Tangent Inequality and Tangency of the Noise Gradient

Each result cited is universally quantified over the data in its own statement. Elementary real-number order and arithmetic, including manipulations of finite sums, the monotonicity of squares and of nonnegative square roots on the nonnegative reals, and the limit laws for real sequences, are carried by The Real Numbers: Standing Notation and Background §background and are not cited step by step.

Throughout, $x_{k}=\langle x,e_{k}\rangle$ and $p_{d}(x)=(x_{1},\dots,x_{d})$ as in Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates, $\lVert\cdot\rVert$ is the Euclidean norm of $\mathbb{R}^{d}$, and $\bar{a}$ is the bound $a_{k}\le\bar{a}$ of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §weights. For $w\in\mathbb{R}^{d}$ let $q(w)$ be the nonnegative square root of $\sum_{k=1}^{d}w_{k}^{2}/a_{k}$, so that $q(p_{d}(x))=|p_{d}(x)|_{a}$ in the notation of the statement.

Step 0 (Preliminaries). (i) Coordinates. The inner product of $X$ is linear in its first argument (Real Inner Product Space §inner-product), so $(x+z)_{k}=x_{k}+z_{k}$ and $(tx)_{k}=t\,x_{k}$ for $x,z\in X$, $t\in\mathbb{R}$; hence $p_{d}(x+z)=p_{d}(x)+p_{d}(z)$ and $p_{d}(0_{X})=0$. By Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity, $\lVert p_{d}(x)-p_{d}(x')\rVert\le|x-x'|$ for $x,x'\in X$; in particular $\lVert p_{d}(z)\rVert\le|z|$.

(ii) Finite combinations of basis vectors. Let $c_{1},\dots,c_{d}\in\mathbb{R}$ and $y=\sum_{j=1}^{d}c_{j}e_{j}$. Since $(e_{k})$ is orthonormal (Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §space), bilinearity (Real Inner Product Space §inner-product, Elementary Identities in a Real Inner Product Space §bilinear) gives $y_{k}=c_{k}$ for $k\le d$ and $y_{k}=0$ for $k>d$. Each $e_{k}=a_{k}^{-1/2}f_{k}$ lies in $X^{a}$ by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §basis, and $X^{a}$ is a linear subspace by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §hilbert, so $y\in X^{a}$. For $z\in X^{a}$, the terms $a_{k}^{-1}y_{k}z_{k}$ of the series defining $\langle y,z\rangle_{a}$ in The Noise Space of a Weight Sequence on a Hilbert Space with an Orthonormal Basis §inner-product vanish for $k>d$, so its partial sums are constant from the $d$-th on and its sum is the finite sum:

$$\langle y,z\rangle_{a}=\sum_{k=1}^{d}a_{k}^{-1}c_{k}z_{k},\qquad |y|_{a}^{2}=\sum_{k=1}^{d}a_{k}^{-1}c_{k}^{2}.\tag{P}$$

(iii) Heads of the noise norm. For $h\in X^{a}$, $q(p_{d}(h))^{2}=\sum_{k=1}^{d}a_{k}^{-1}h_{k}^{2}=S_{d}(h)\le|h|_{a}^{2}$ by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §partial-sums, so $q(p_{d}(h))\le|h|_{a}$.

(iv) Slope. The constant $C$ of (c) is nonnegative: at any $u\in\mathbb{R}^{d}$ the left-hand side of (c) in Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §slope is a sum of nonnegative terms and $(1+|v(u)|)^{2}$ is positive. For $u,w\in\mathbb{R}^{d}$, the points $\alpha=(a_{k}^{1/2}\partial_{k}v(u))_{k\le d}$ and $\omega=(a_{k}^{-1/2}w_{k})_{k\le d}$ of $\mathbb{R}^{d}$ have dot product $\sum_{k=1}^{d}\partial_{k}v(u)w_{k}$, $\lVert\alpha\rVert^{2}=\sum_{k=1}^{d}a_{k}\partial_{k}v(u)^{2}$ and $\lVert\omega\rVert^{2}=q(w)^{2}$ by Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §square. By the Cauchy-Schwarz inequality for the dot product of $\mathbb{R}^{d}$ (Cauchy-Schwarz Inequality for the Euclidean Dot Product) and then (c) with monotonicity of square roots,

$$\Bigl|\sum_{k=1}^{d}\partial_{k}v(u)\,w_{k}\Bigr|\le\Bigl(\sum_{k=1}^{d}a_{k}\partial_{k}v(u)^{2}\Bigr)^{1/2}q(w)\le C^{1/2}\bigl(1+|v(u)|\bigr)\,q(w).\tag{S}$$

Taking for $w$ the $i$-th unit vector ($i\le d$), for which $q(w)=a_{i}^{-1/2}$, gives $|\partial_{i}v(u)|\le a_{i}^{-1/2}C^{1/2}(1+|v(u)|)$.

(v) Regularity of $v$. Since $v$ is of class $C^{2}$, clause 2 of C^k Maps on a Euclidean Open Set makes $v$ and each $\partial_{k}v$ ($k\le d$) of class $C^{1}$ on $\mathbb{R}^{d}$, with the iterated partial derivatives $\partial_{j}\partial_{k}v$ of clause 4 there; by clause 1 there, $v$ and each $\partial_{k}v$ are continuous at every point, which, as $\sum_{i}(u'_{i}-u_{i})^{2}=\lVert u'-u\rVert^{2}$ (Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §square), says: for $u\in\mathbb{R}^{d}$ and $\varepsilon>0$ there is $\delta>0$ with $|f(u')-f(u)|<\varepsilon$ whenever $\lVert u'-u\rVert<\delta$ ($f=v$ or $f=\partial_{k}v$). By A Real-Valued C^1 Function is Differentiable at Every Point, $v$ and each $\partial_{k}v$ are differentiable at every point of $\mathbb{R}^{d}$, with derivative matrix the row of their partial derivatives.

(vi) Composition with $p_{d}$. If $f:\mathbb{R}^{d}\to\mathbb{R}$ has the continuity property of (v), then $f\circ p_{d}$ is continuous on $(X,d)$: given $x$ and $\varepsilon$, take $\delta$ for $f$ at $p_{d}(x)$; then $|x'-x|<\delta$ gives $\lVert p_{d}(x')-p_{d}(x)\rVert<\delta$ by (i), so $|f(p_{d}(x'))-f(p_{d}(x))|<\varepsilon$. A continuous function $X\to\mathbb{R}$ is Borel by claims 2 and 3 of Borel Measurability and Bounded Integration on a Metric Space.

(vii) Affine paths. For $u,w\in\mathbb{R}^{d}$, Chain Rule Along an Affine Path (with $U=\mathbb{R}^{d}$, $J=\mathbb{R}$, every real number being an interior point of $\mathbb{R}$ by The Real Line: Standing Notation and Background for Calculus §intervals) and (v) show that $t\mapsto v(u+tw)$ is differentiable at every $t\in\mathbb{R}$ with derivative $\sum_{k=1}^{d}\partial_{k}v(u+tw)\,w_{k}$, and that for $k\le d$ the function $t\mapsto\partial_{k}v(u+tw)$ is differentiable at every $t$ with derivative $\sum_{j=1}^{d}\partial_{j}\partial_{k}v(u+tw)\,w_{j}$. Differentiable functions on $\mathbb{R}$ are continuous on $\mathbb{R}$ by Differentiability at an Interior Point Implies Continuity There, in the sense of The Real Line: Standing Notation and Background for Calculus §continuity.

Step 1 (Claim 1). $V=v\circ p_{d}$ is continuous and Borel by (v) and (vi), and $-b\le v(p_{d}(x))=V(x)$ by (a). For $k\le d$, $\partial_{k}V=\partial_{k}v\circ p_{d}$ is continuous by (v), (vi); for $k>d$ it is the constant $0$.

Coordinates and formulas. By (ii) with $c_{k}=a_{k}\partial_{k}v(p_{d}(x))$, $\nabla_{a}V(x)$ has $k$-th coordinate $a_{k}\partial_{k}V(x)$ for every $k\in\mathbb{N}$ (both sides vanish for $k>d$), and (P) gives, for $h\in X^{a}$,

$$\langle\nabla_{a}V(x),h\rangle_{a}=\sum_{k=1}^{d}a_{k}^{-1}a_{k}\partial_{k}V(x)h_{k}=\sum_{k=1}^{d}\partial_{k}V(x)h_{k},\qquad|\nabla_{a}V(x)|_{a}^{2}=\sum_{k=1}^{d}a_{k}\,\partial_{k}V(x)^{2}.$$

Continuity and measurability of $\nabla_{a}V$. For $x,x'\in X$, $\nabla_{a}V(x')-\nabla_{a}V(x)=\sum_{k=1}^{d}a_{k}\bigl(\partial_{k}V(x')-\partial_{k}V(x)\bigr)e_{k}$, so by (P) $|\nabla_{a}V(x')-\nabla_{a}V(x)|_{a}^{2}=\sum_{k=1}^{d}a_{k}(\partial_{k}V(x')-\partial_{k}V(x))^{2}$. Given $x$ and $\varepsilon>0$, put $\eta=\varepsilon/(d\bar{a})^{1/2}$ and choose by continuity of each $\partial_{k}V$ ($k\le d$) a $\delta_{k}>0$ with $|\partial_{k}V(x')-\partial_{k}V(x)|<\eta$ when $|x'-x|<\delta_{k}$; with $\delta=\min_{k\le d}\delta_{k}$, $|x'-x|<\delta$ gives $|\nabla_{a}V(x')-\nabla_{a}V(x)|_{a}^{2}<d\bar{a}\eta^{2}=\varepsilon^{2}$. So $\nabla_{a}V$ is continuous from $(X,d)$ into $X^{a}$ with the distance of $|\cdot|_{a}$. Its coordinate functions $x\mapsto\langle\nabla_{a}V(x),e_{k}\rangle=a_{k}\partial_{k}V(x)$ are continuous, hence Borel by (vi), so $\nabla_{a}V$ is measurable into $X^{a}$ by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §measurable. Moreover $x\mapsto|\nabla_{a}V(x)|_{a}$ is continuous, since $\bigl||\nabla_{a}V(x')|_{a}-|\nabla_{a}V(x)|_{a}\bigr|\le|\nabla_{a}V(x')-\nabla_{a}V(x)|_{a}$ by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §reverse-triangle in the inner product space $X^{a}$ (The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §hilbert); hence it is Borel.

Differentiability. Fix $x\in X$, put $u=p_{d}(x)$ and $p=\sum_{k=1}^{d}\partial_{k}v(u)e_{k}\in X$. Let $\varepsilon>0$. By (v) and Differentiability at a Point for Maps Between Euclidean Spaces (with $m=1$, where the Euclidean norm of $\mathbb{R}^{1}$ is the absolute value by Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §square) there is $\delta>0$ such that $0<\lVert w\rVert<\delta$ implies $|v(u+w)-v(u)-\sum_{k=1}^{d}\partial_{k}v(u)w_{k}|\le\varepsilon\lVert w\rVert$. Let $z\in X$ with $|z|<\delta$ and $w=p_{d}(z)$. By (i), $p_{d}(x+z)=u+w$ and $\lVert w\rVert\le|z|<\delta$, and by linearity of the inner product in its first argument $\langle p,z\rangle=\sum_{k=1}^{d}\partial_{k}v(u)z_{k}=\sum_{k=1}^{d}\partial_{k}v(u)w_{k}$. If $w=0$ then $|V(x+z)-V(x)-\langle p,z\rangle|=0$; otherwise it is at most $\varepsilon\lVert w\rVert\le\varepsilon|z|$. Hence $V$ is differentiable at $x$ with gradient $p$, so $DV(x)=p$ by Fréchet Differentiability and the Gradient on an Open Subset of a Real Inner Product Space §gradient, $V$ is differentiable on $X$ (Fréchet Differentiability and the Gradient on an Open Subset of a Real Inner Product Space §on-set), and by Partial Derivatives along an Orthonormal Basis of a Function Differentiable on a Hilbert Space §partial and (ii) its $k$-th partial derivative is $\langle p,e_{k}\rangle$, which is $\partial_{k}v(u)$ for $k\le d$ and $0$ for $k>d$, that is, $\partial_{k}V(x)$.

Step 2 (Claim 2). $C\ge0$ was shown in (iv), so $L=C^{1/2}(|b+1|+2)\ge0$. Put $g(u)=v(u)+b+1$ for $u\in\mathbb{R}^{d}$; by (a), $g(u)\ge1$. As $v(u)=g(u)-(b+1)$, $|v(u)|\le g(u)+|b+1|$, and using $1\le g(u)$, $1+|v(u)|\le g(u)+g(u)+|b+1|g(u)=(|b+1|+2)g(u)$. With (S),

$$\Bigl|\sum_{k=1}^{d}\partial_{k}v(u)\,w_{k}\Bigr|\le L\,g(u)\,q(w)\qquad(u,w\in\mathbb{R}^{d}).\tag{G}$$

Fix $x\in X$ and $h\in X^{a}$; let $u=p_{d}(x)$, $w=p_{d}(h)$ and $s=q(w)$, so $s\le|h|_{a}$ by (iii). Let $\phi(t)=g(u+tw)=v(u+tw)+b+1$ for $t\in\mathbb{R}$. By (vii) and claims 1 and 2 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives, $\phi$ is differentiable at every $t$ with $\phi'(t)=\sum_{k=1}^{d}\partial_{k}v(u+tw)w_{k}$, so $\phi'(t)\le Ls\,\phi(t)$ by (G). Let $E(t)=\exp(-Ls\,t)$; by Derivative and Continuity of the Scaled Exponential Function (with $c=-Ls$) it is differentiable with $E'(t)=-Ls\,E(t)$, and $E(t)>0$ by claim 2 of Basic Properties of the Exponential Function. By claim 3 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives, $\psi=\phi E$ is differentiable at every $t$ with $\psi'(t)=(\phi'(t)-Ls\,\phi(t))E(t)\le0$. $\psi$ is continuous on $\mathbb{R}$ by (vii), so The Sign of the Derivative and Monotonicity §nonincreasing (with $I=\mathbb{R}$) gives $\psi(1)\le\psi(0)$, that is, $\phi(1)\exp(-Ls)\le\phi(0)$. Multiplying by $\exp(Ls)>0$ and using $\exp(Ls)\exp(-Ls)=\exp(0)=1$ (claim 1 of Basic Properties of the Exponential Function) gives $\phi(1)\le\phi(0)\exp(Ls)$. Since $L\ge0$ and $s\le|h|_{a}$, $Ls\le L|h|_{a}$, and $\exp$ is increasing (claim 4 there), while $\phi(0)\ge1>0$; hence $\phi(1)\le\phi(0)\exp(L|h|_{a})$. Finally $\phi(0)=V(x)+b+1$ and, as $u+w=p_{d}(x+h)$ by (i), $\phi(1)=V(x+h)+b+1$. This is claim 2.

Step 3 (Claim 3). Let $x\in X$, $h=\sum_{k=1}^{d}x_{k}e_{k}$ and $x'=x-h$. By (ii), $h\in X^{a}$, $h_{k}=x_{k}$ for $k\le d$, so $p_{d}(h)=p_{d}(x)$, and by (P) $|h|_{a}^{2}=\sum_{k=1}^{d}x_{k}^{2}/a_{k}$, so $|h|_{a}=|p_{d}(x)|_{a}$. By (i), $p_{d}(x')=p_{d}(x)-p_{d}(h)=0$, so $V(x')=v(0)$. Claim 2 applied at $x'$ with this $h$ gives $V(x)+b+1=V(x'+h)+b+1\le(v(0)+b+1)\exp(L|p_{d}(x)|_{a})$.

Step 4 (Claim 4). Let $\mu\in\mathcal{P}(X)$ with $V$ integrable. For $x\in X$, (c) at $u=p_{d}(x)$, the formula of Step 1 and monotonicity of square roots give $|\nabla_{a}V(x)|_{a}\le C^{1/2}(1+|V(x)|)$; and for $k\le d$, $a_{k}\partial_{k}V(x)^{2}\le|\nabla_{a}V(x)|_{a}^{2}$, so $|\partial_{k}V(x)|\le a_{k}^{-1/2}C^{1/2}(1+|V(x)|)$, while $\partial_{k}V=0$ for $k>d$. The functions $|\nabla_{a}V|_{a}$ and $|\partial_{k}V|$ are Borel by Step 1 and claim 4 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. Since $V$ is integrable, $\int_{X}|V|\,d\mu<\infty$ (Integrable Function and the Lebesgue Integral), and the constant $1$ is integrable by claim 6(a) of Borel Measurability and Bounded Integration on a Metric Space; so $\int_{X}C^{1/2}(1+|V|)\,d\mu<\infty$ by Linearity and Monotonicity of the Lebesgue Integral §nonnegative, and by the monotonicity there the nonnegative Borel functions $|\nabla_{a}V|_{a}$ and $a_{k}^{1/2}|\partial_{k}V|$ have finite integrals. By the criterion of Integrable Function and the Lebesgue Integral (and Linearity and Monotonicity of the Lebesgue Integral §integrable for the factor $a_{k}^{-1/2}$), $|\nabla_{a}V|_{a}$ and each $\partial_{k}V$ are integrable.

Step 5 (Claim 5). Let $\bar{c}=\sum_{k=1}^{\infty}c_{k}$, a convergent series of positive terms (Variance Sequences and Their Truncations §variances); then $c_{k}\le\sum_{j=1}^{k}c_{j}\le\bar{c}$ for every $k$ by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §dominates. Let $m=\min_{k\le d}a_{k}$, positive. Put $t=4\bar{c}/m+1$, positive, and $\alpha=1/(tm)$, positive; then $2\alpha c_{k}\le2\bar{c}/(4\bar{c}+m)\le\tfrac12$ for every $k$, so by Moments of a Diagonal Gaussian Measure on a Hilbert Space: Coordinate Covariances, Finite Second Moment, and Exponential Moments of the Squared Norm §exponential (with $\theta=\tfrac12$) the function $x\mapsto\exp(\alpha|x|^{2})$ is Borel and integrable with respect to $\gamma_{c}$.

For $x\in X$ let $s=|p_{d}(x)|_{a}$. Then $s^{2}=\sum_{k=1}^{d}x_{k}^{2}/a_{k}\le m^{-1}\lVert p_{d}(x)\rVert^{2}\le m^{-1}|x|^{2}$ by Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §square and (i). Since $0\le(Lt-s)^{2}/t=L^{2}t-2Ls+s^{2}/t$, we have $2Ls\le L^{2}t+s^{2}/t\le L^{2}t+\alpha|x|^{2}$. Let $G_{0}=v(0)+b+1\ge1$. By claim 3 and $V(x)+b+1\ge1$, squaring nonnegative numbers and using claims 1 and 4 of Basic Properties of the Exponential Function,

$$(V(x)+b+1)^{2}\le G_{0}^{2}\exp(2Ls)\le G_{0}^{2}\exp(L^{2}t)\exp(\alpha|x|^{2}).$$

As $(r-r')^{2}\le2r^{2}+2r'^{2}$, $V(x)^{2}\le2(V(x)+b+1)^{2}+2(b+1)^{2}\le2G_{0}^{2}\exp(L^{2}t)\exp(\alpha|x|^{2})+2(b+1)^{2}$. The right-hand side is integrable with respect to $\gamma_{c}$ (Linearity and Monotonicity of the Lebesgue Integral §integrable, constants being integrable by claim 6(a) of Borel Measurability and Bounded Integration on a Metric Space), and $V^{2}$ is Borel (Step 1 and claim 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions) and nonnegative, so $V^{2}$ is integrable by Linearity and Monotonicity of the Lebesgue Integral §nonnegative and the criterion of Integrable Function and the Lebesgue Integral. Since $(|V|-1)^{2}\ge0$ gives $|V|\le\tfrac12(1+V^{2})$, $V$ (Borel) is integrable in the same way. Finally, by Step 4, $|\nabla_{a}V|_{a}^{2}\le C(1+|V|)^{2}\le2C(1+V^{2})$, and $|\nabla_{a}V|_{a}^{2}=\sum_{k=1}^{d}a_{k}(\partial_{k}V)^{2}$ is Borel (Step 1, claims 2 and 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions); so it is integrable with respect to $\gamma_{c}$ in the same way.

Step 6 (Claim 6). Let $x,y\in X$ with $h=y-x\in X^{a}$; let $u=p_{d}(x)$, $w=p_{d}(h)$, so $u+w=p_{d}(y)$ by (i), and $s=q(w)\le|h|_{a}$ by (iii). Let $F(t)=v(u+tw)$ and $F_{1}(t)=\sum_{k=1}^{d}\partial_{k}v(u+tw)w_{k}$ for $t\in\mathbb{R}$. By (vii) and claim 2 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives (applied to the finitely many summands), $F$ is differentiable with $F'=F_{1}$, and $F_{1}$ is differentiable with

$$F_{1}'(t)=\sum_{k=1}^{d}\sum_{j=1}^{d}\partial_{j}\partial_{k}v(u+tw)\,w_{j}w_{k}\ \ge\ -K\sum_{k=1}^{d}\frac{w_{k}^{2}}{a_{k}}=-Ks^{2}$$

by (b) in Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §semiconvex, at the point $u+tw$ with $h=w$ there. By claim 1 of Derivative of a Polynomial Function on the Real Line and claims 1, 2 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives, $Q(t)=\tfrac{K}{2}s^{2}t^{2}$ has derivative $Ks^{2}t$, and $t\mapsto Ks^{2}t$ has derivative $Ks^{2}$. So $G=F+Q$ and $G_{1}=F_{1}+Ks^{2}t$ satisfy $G'=G_{1}$ and $G_{1}'=F_{1}'+Ks^{2}\ge0$ on $\mathbb{R}$. By A Real Function with Nonnegative Second Derivative is Convex on an Interval with $J=\mathbb{R}$ (order-convex, each $t$ lying strictly between $t-1$ and $t+1$), applied with the points $0$ and $1$: $G(\theta)\le(1-\theta)G(0)+\theta G(1)$ for $0\le\theta\le1$, so $(G(\theta)-G(0))/\theta\le G(1)-G(0)$ for $0<\theta\le1$. Suppose $G_{1}(0)>G(1)-G(0)$; put $\varepsilon=G_{1}(0)-(G(1)-G(0))>0$ and let $\delta$ be given for $\varepsilon$ by differentiability of $G$ at $0$ with $G'(0)=G_{1}(0)$ (Single-Variable Calculus on an Interval §derivative); for $\theta=\min(\delta/2,1)$ we get $(G(\theta)-G(0))/\theta>G_{1}(0)-\varepsilon=G(1)-G(0)$, a contradiction. Hence $F_{1}(0)=G_{1}(0)\le G(1)-G(0)=F(1)+\tfrac{K}{2}s^{2}-F(0)$. Now $F(0)=V(x)$, $F(1)=v(p_{d}(y))=V(y)$, and $F_{1}(0)=\sum_{k=1}^{d}\partial_{k}V(x)h_{k}=\langle\nabla_{a}V(x),y-x\rangle_{a}$ by Step 1; and $\tfrac{K}{2}s^{2}\le\tfrac{K}{2}|y-x|_{a}^{2}$ as $K\ge0$. This gives claim 6.

Step 7 (Claim 7). Let $\mu\in\mathcal{P}(X)$ with $\int_{X}|\nabla_{a}V|_{a}^{2}\,d\mu<\infty$. By Step 1, $\nabla_{a}V$ is measurable into $X^{a}$, so it is square-integrable (The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §space) and its class lies in $L^{2}(\mu;X^{a})$ (The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §classes). Let $B=|b|$, so $\exp(B)\ge1$.

The truncations. For $n\in\mathbb{N}$ let $\chi_{n}(r)=n-n\exp(-r/n)$ ($r\in\mathbb{R}$). By Derivative and Continuity of the Scaled Exponential Function (with $c=-1/n$) and claims 1, 2 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives, $\chi_{n}$ is differentiable at every $r$ with $\chi_{n}'(r)=\exp(-r/n)$, and $r\mapsto\exp(-r/n)$ is continuous (same lemma); $\chi_{n}$ is continuous by Differentiability at an Interior Point Implies Continuity There. Reading $\mathbb{R}$ as $\mathbb{R}^{1}$, the partial derivative of Partial Derivative on a Euclidean Open Set is defined by the same difference quotients as the derivative of Single-Variable Calculus on an Interval §derivative, and continuity at a point in the sense of Continuity at a Point for Maps Between Euclidean Spaces for $n=m=1$ is the $\varepsilon$-$\delta$ continuity of The Real Line: Standing Notation and Background for Calculus §continuity (as $(r'-r)^{2}<\delta^{2}$ iff $|r'-r|<\delta$); so $\chi_{n}$ is of class $C^{1}$ on $\mathbb{R}^{1}$ by clauses 1 and 3 of C^k Maps on a Euclidean Open Set, $\mathbb{R}^{1}$ being open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous. By (v) and claims 1, 2 of A Composition of $C^k$ Maps Between Euclidean Open Sets is of Class $C^k$ (with $k=1$), $\psi_{n}=\chi_{n}\circ v$ is of class $C^{1}$ on $\mathbb{R}^{d}$ with $\partial_{i}\psi_{n}(u)=\exp(-v(u)/n)\,\partial_{i}v(u)$.

Elementary bounds. For $r\ge-b$ and $n\in\mathbb{N}$: (B1) $0<\exp(-r/n)\le\exp(B)$, since $-r/n\le B$ and $\exp$ is positive and increasing (claims 2, 4 of Basic Properties of the Exponential Function); hence $n(1-\exp(B))\le\chi_{n}(r)\le n$. (B2) $(1+|r|)\exp(-r/n)\le n+(1+B)\exp(B)$: if $r\ge0$, $\exp(r/n)\ge1+r/n$ (claim 4 there), so by claim 2 there $\exp(-r/n)\le n/(n+r)$ and $(1+r)n/(n+r)\le n$ as $n\ge1$; if $-b\le r<0$, then $|r|\le B$ and (B1) applies. (B3) $|1-\exp(-r/n)|\le\exp(B)$: for $r\ge0$, $0<\exp(-r/n)\le\exp(0)=1$; for $r<0$, $1\le\exp(-r/n)\le\exp(B)$.

Cylindrical approximants. Since $v\ge-b$, (B1) shows $\psi_{n}$ is bounded, and (iv) with (B2) gives $|\partial_{i}\psi_{n}(u)|\le a_{i}^{-1/2}C^{1/2}(1+|v(u)|)\exp(-v(u)/n)\le a_{i}^{-1/2}C^{1/2}(n+(1+B)\exp(B))$. So $\psi_{n}\in C^{1}_{b}(\mathbb{R}^{d})$ (Bounded Continuously Differentiable Functions with Bounded Partial Derivatives on Euclidean Space §bounded) and $\varphi_{n}=\psi_{n}\circ p_{d}\in\mathcal{F}C^{1}_{b}(X)$ with representation $(d,\psi_{n})$ (Bounded C^1 Cylindrical Functions on a Hilbert Space with an Orthonormal Basis §cylindrical, Bounded C^1 Cylindrical Functions on a Hilbert Space with an Orthonormal Basis §representation). 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,

$$\nabla_{a}\varphi_{n}(x)=\sum_{k=1}^{d}a_{k}\,\partial_{k}\psi_{n}(p_{d}(x))\,e_{k}=\exp(-V(x)/n)\,\nabla_{a}V(x)\qquad(x\in X),$$

and its class lies in $G^{a}_{\mu}$ (The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §gradients).

Convergence. By The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §operations, $[\nabla_{a}\varphi_{n}]-[\nabla_{a}V]$ is the class of $x\mapsto(\exp(-V(x)/n)-1)\nabla_{a}V(x)$, whose squared noise norm is $f_{n}(x)=(1-\exp(-V(x)/n))^{2}|\nabla_{a}V(x)|_{a}^{2}$ by Elementary Identities in a Real Inner Product Space §homogeneity, so $\lVert[\nabla_{a}\varphi_{n}]-[\nabla_{a}V]\rVert_{\mu}^{2}=\int_{X}f_{n}\,d\mu$. Each $f_{n}$ is Borel: $x\mapsto\exp(-V(x)/n)$ is the composite of the Borel $V$ with a continuous, hence Borel, function (claims 3, 4 of Borel Measurability and Bounded Integration on a Metric Space), and products of Borel functions are Borel (claims 2, 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions). For fixed $x$, $\exp(-V(x)/n)$ is the value at $1/n$ of $E(\tau)=\exp(-V(x)\tau)$, which is continuous at $0$ with $E(0)=1$ (Derivative and Continuity of the Scaled Exponential Function, claim 1 of Basic Properties of the Exponential Function), and $1/n\to0$; so $f_{n}(x)\to0$. By (B3) and $V\ge-b$, $0\le f_{n}\le\exp(2B)|\nabla_{a}V|_{a}^{2}$, which is integrable with respect to $\mu$: the function $|\nabla_{a}V|_{a}^{2}$ is Borel (as in Step 5) and nonnegative with finite integral, hence integrable by the criterion of Integrable Function and the Lebesgue Integral, and so its constant multiple $\exp(2B)|\nabla_{a}V|_{a}^{2}$ is integrable by Linearity and Monotonicity of the Lebesgue Integral §integrable. Claim 3 of Dominated Convergence Theorem (with limit the constant $0$) gives $\int_{X}f_{n}\,d\mu\to\int_{X}0\,d\mu=0$, hence $\lVert[\nabla_{a}\varphi_{n}]-[\nabla_{a}V]\rVert_{\mu}\to0$, i.e. $[\nabla_{a}\varphi_{n}]\to[\nabla_{a}V]$ in the metric of the real Hilbert space $L^{2}(\mu;X^{a})$ (Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §fields, Real Inner Product Space §distance). By Sequential Characterization of the Closure in a Metric Space, $[\nabla_{a}V]$ 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 and Real Hilbert Space §topology. This proves claim 7.