Proof of Linearity of the Noise Gradient, and the Noise Tangent Space is a Closed Linear Subspace

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

The vector operations of $X^{a}$ are those of $X$ restricted to $X^{a}$, and its zero vector is $0_{X}$, by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §hilbert; so pointwise sums and multiples of maps into $X^{a}$ are computed in $X$.

Claim (linear). Let $\varphi,\psi\in\mathcal{F}C^{1}_{b}(X)$, $s,t\in\mathbb{R}$, and $\chi=s\varphi+t\psi$, which lies 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. Choose representations $(n,\alpha)$, $(m,\beta)$ and $(p,\gamma)$ of $\varphi$, $\psi$ and $\chi$ (Bounded C^1 Cylindrical Functions on a Hilbert Space with an Orthonormal Basis §representation), and let $q$ be the largest of $n,m,p$.

Step 1 (partial derivatives). 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, $\varphi$ and $\psi$ are differentiable on $X$. Applying Constants, Sums, Scalar Multiples and Differences of Differentiable Functions on an Open Subset of a Real Inner Product Space §scalar and Constants, Sums, Scalar Multiples and Differences of Differentiable Functions on an Open Subset of a Real Inner Product Space §sum with $E=U=X$, $\chi$ is differentiable at every $x\in X$ with $D\chi(x)=s\,D\varphi(x)+t\,D\psi(x)$. By Partial Derivatives along an Orthonormal Basis of a Function Differentiable on a Hilbert Space §partial and additivity and homogeneity of $\langle\cdot,\cdot\rangle$ in the first argument (Real Inner Product Space §inner-product),

$$\partial_{k}\chi(x)=\langle s\,D\varphi(x)+t\,D\psi(x),e_{k}\rangle=s\,\partial_{k}\varphi(x)+t\,\partial_{k}\psi(x)\qquad(x\in X,\ k\in\mathbb{N}).$$

Step 2 (a common index). 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, $\partial_{k}\varphi(x)=0$ for $n<k\le q$, so the formula of The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §gradient gives $\nabla_{a}\varphi(x)=\sum_{k=1}^{q}a_{k}\,\partial_{k}\varphi(x)\,e_{k}$, the added terms being $0_{X}$ by claim 3 of Elementary Identities in a Vector Space, so that the sum is unchanged by the recursion of claim 1 of Properties of Finite Sums of Vectors (each added summand $0_{X}$ leaving the partial sum unchanged); in the same way, using the representations $(m,\beta)$ and $(p,\gamma)$, $\nabla_{a}\psi(x)=\sum_{k=1}^{q}a_{k}\,\partial_{k}\psi(x)\,e_{k}$ and $\nabla_{a}\chi(x)=\sum_{k=1}^{q}a_{k}\,\partial_{k}\chi(x)\,e_{k}$. Inserting Step 1 and using the rules for finite sums in a vector space (Properties of Finite Sums of Vectors),

$$\nabla_{a}\chi(x)=\sum_{k=1}^{q}a_{k}\bigl(s\,\partial_{k}\varphi(x)+t\,\partial_{k}\psi(x)\bigr)e_{k}=s\,\nabla_{a}\varphi(x)+t\,\nabla_{a}\psi(x)\qquad(x\in X),$$

which is the claim.

Claim (subspace). First, $G^{a}_{\mu}$ is a linear subspace of $L^{2}(\mu;X^{a})$, whose operations are those of The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §operations (Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §fields). The constant function $0$ lies 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, its gradient is $0_{X}$ at every point by Constants, Sums, Scalar Multiples and Differences of Differentiable Functions on an Open Subset of a Real Inner Product Space §constant, so all its partial derivatives vanish (Partial Derivatives along an Orthonormal Basis of a Function Differentiable on a Hilbert Space §partial) and $\nabla_{a}0$ is the constant map with value $0_{X}$, each summand $a_{k}\cdot0\cdot e_{k}$ being $0_{X}$ by claim 3 of Elementary Identities in a Vector Space and a finite sum of zero vectors being $0_{X}$ by claim 7 of Properties of Finite Sums of Vectors; its class is the zero vector of $L^{2}(\mu;X^{a})$ by The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §hilbert. So the zero vector lies in $G^{a}_{\mu}$. For $\varphi,\psi\in\mathcal{F}C^{1}_{b}(X)$ and $t\in\mathbb{R}$, Claim (linear) (with coefficients $1,1$, respectively $t,0$ and $\psi=\varphi$) and The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §operations give

$$[\nabla_{a}\varphi]+[\nabla_{a}\psi]=[\nabla_{a}\varphi+\nabla_{a}\psi]=[\nabla_{a}(\varphi+\psi)],\qquad t[\nabla_{a}\varphi]=[t\nabla_{a}\varphi]=[\nabla_{a}(t\varphi)],$$

and $\varphi+\psi$, $t\varphi$ lie 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; so both classes lie in $G^{a}_{\mu}$, and $G^{a}_{\mu}$ is a linear subspace by Linear Subspace.

Next, $T^{a}_{\mu}$ is the closure of $G^{a}_{\mu}$ in the metric space of $L^{2}(\mu;X^{a})$ (The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §tangent, Real Hilbert Space §topology), a real Hilbert space by The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §hilbert. It is closed by claim 2 of The Closure is the Smallest Closed Superset, and it contains $G^{a}_{\mu}$, hence the zero vector, by claim 1 of that theorem. Let $u,w\in T^{a}_{\mu}$ and $t\in\mathbb{R}$. By Sequential Characterization of the Closure in a Metric Space there are sequences $(u_{m})$ and $(w_{m})$ in $G^{a}_{\mu}$ converging to $u$ and to $w$. Then $u_{m}+w_{m}$ and $t\,u_{m}$ lie in $G^{a}_{\mu}$ by the first part, and they converge to $u+w$ and to $t\,u$ by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §linear-limits. By Sequential Characterization of the Closure in a Metric Space again, $u+w$ and $t\,u$ lie in $T^{a}_{\mu}$. Thus $T^{a}_{\mu}$ is a linear subspace (Linear Subspace) that is closed, that is, a closed linear subspace of $L^{2}(\mu;X^{a})$ in the sense of Real Hilbert Space §closed-subspace.