Proof of Lifting Euclidean Tangent Fields of the Rescaled Head to Noise Tangent Fields on a Hilbert Space

Each result cited is universally quantified over the data in its own statement. Elementary arithmetic and order of real numbers, including finite sums, squares and square roots, is used through The Real Numbers: Standing Notation and Background §background without further citation. On $\mathbb{R}^{n}$ we use $d_{E}(u,v)=\lVert u-v\rVert$ by Euclidean Space and Lebesgue Measure: Standing Notation §space, $\lVert u\rVert^{2}=\sum_{k=1}^{n}u_{k}^{2}$ by Euclidean Norm on $\mathbb{R}^n$, and $u\cdot v=\sum_{k=1}^{n}u_{k}v_{k}$ by Difference, Dot Product, and Orthogonality in $\mathbb{R}^n$; on $X$, $d(x,x')=|x-x'|$ by Real Inner Product Space §distance.

Notation. Every $a_{k}$ is positive by Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §weights, and $a_{k}\le\bar{a}$ with $\bar{a}$ positive. By Existence and Uniqueness of the Nonnegative Square Root, $a_{k}^{1/2}$ is positive with $(a_{k}^{1/2})^{2}=a_{k}$, so $a_{k}^{-1}(a_{k}^{1/2})^{2}=1$, $(a_{k}^{-1/2})^{2}=a_{k}^{-1}$ and $a_{k}\,a_{k}^{-1/2}=a_{k}^{1/2}$. Put $m=\sum_{k=1}^{n}a_{k}^{-1}$; as every $a_{k}^{-1}$ is positive, $a_{k}^{-1}\le m$ for $k\le n$. Define $R,D_{+}:\mathbb{R}^{n}\to\mathbb{R}^{n}$ by

$$R(u)=\bigl(a_{1}^{-1/2}u_{1},\dots,a_{n}^{-1/2}u_{n}\bigr),\qquad D_{+}(v)=\bigl(a_{1}^{1/2}v_{1},\dots,a_{n}^{1/2}v_{n}\bigr),$$

so that $r_{n}=R\circ p_{n}$ with $p_{n}$ of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates. For $k\in\mathbb{N}$ let $f_{k}=a_{k}^{1/2}e_{k}$. For a Borel $\eta:\mathbb{R}^{n}\to\mathbb{R}^{n}$ and $x\in X$, the definition of $p_{n}^{*}$ in Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates and the identity $a_{k}^{1/2}\eta_{k}(r_{n}(x))\,e_{k}=\eta_{k}(r_{n}(x))\,f_{k}$ give

$$\Lambda_{n}\eta(x)=p_{n}^{*}\Bigl(D_{+}\bigl(\eta(r_{n}(x))\bigr)\Bigr)=\sum_{k=1}^{n}\eta_{k}\bigl(r_{n}(x)\bigr)\,f_{k}.\qquad(0)$$

Step 1 (Claim 1). For $u,u'\in\mathbb{R}^{n}$,

$$\lVert R(u)-R(u')\rVert^{2}=\sum_{k=1}^{n}a_{k}^{-1}(u_{k}-u'_{k})^{2}\le m\,\lVert u-u'\rVert^{2},\qquad \lVert D_{+}(u)-D_{+}(u')\rVert^{2}=\sum_{k=1}^{n}a_{k}(u_{k}-u'_{k})^{2}\le\bar{a}\,\lVert u-u'\rVert^{2},$$

and in particular $\lVert R(u)\rVert^{2}\le m\lVert u\rVert^{2}$. By Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity, $\lVert p_{n}(x)-p_{n}(x')\rVert\le|x-x'|$ for $x,x'\in X$, hence $\lVert r_{n}(x)-r_{n}(x')\rVert\le\sqrt{m}\,|x-x'|$. Thus $R$, $D_{+}$ and $r_{n}$ are Lipschitz with constants $\sqrt{m}$, $\sqrt{\bar{a}}$ and $\sqrt{m}$, hence continuous by A Lipschitz Map is Uniformly Continuous and Borel by claim 3 of Borel Measurability and Bounded Integration on a Metric Space. So $\tilde{\mu}$ is defined; here and below, every fact about push-forward measures is read through Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pushforward. By claim 1 of Image Measures, Measures with Densities, and Change of Variables, $\tilde{\mu}$ is a probability measure on the Borel $\sigma$-algebra of $(\mathbb{R}^{n},d_{E})$, which is $\mathcal{B}(\mathbb{R}^{n})$ by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §spaces; so $\tilde{\mu}\in\mathcal{P}(\mathbb{R}^{n})$ by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures. For $x\in X$, Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity gives

$$\lVert r_{n}(x)\rVert^{2}\le m\,\lVert p_{n}(x)\rVert^{2}=m\,|P_{n}x|^{2}\le m\bigl(|P_{n}x|^{2}+|Q_{n}x|^{2}\bigr)=m\,|x|^{2}.$$

The function $u\mapsto\lVert u\rVert^{2}$ is nonnegative and Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pairs, and $x\mapsto\lVert r_{n}(x)\rVert^{2}$ is Borel as a composition of Borel maps by claim 4 of Borel Measurability and Bounded Integration on a Metric Space, so the change of variables of claim 2 of Image Measures, Measures with Densities, and Change of Variables, the monotonicity and homogeneity of Linearity and Monotonicity of the Lebesgue Integral §nonnegative, and The Second Moment of a Borel Probability Measure on a Hilbert Space and the Probability Measures with Finite Second Moment §space give

$$\int_{\mathbb{R}^{n}}\lVert u\rVert^{2}\,\tilde{\mu}(du)=\int_{X}\lVert r_{n}(x)\rVert^{2}\,\mu(dx)\le m\int_{X}|x|^{2}\,\mu(dx)=m\,M_{2}(\mu)<\infty .$$

By The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §moment and The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §space, $\tilde{\mu}\in\mathcal{P}_{2}(\mathbb{R}^{n})$.

Step 2 (Pointwise structure of the lift). Let $\eta:\mathbb{R}^{n}\to\mathbb{R}^{n}$ be Borel, $x\in X$ and $y=D_{+}(\eta(r_{n}(x)))$, so $\Lambda_{n}\eta(x)=p_{n}^{*}(y)$ by (0). This is a linear combination of $e_{1},\dots,e_{n}$, so $\Lambda_{n}\eta(x)\in X_{n}$ by Span of a Finite Family of Vectors and Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates. By Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity, $p_{n}(\Lambda_{n}\eta(x))=p_{n}(p_{n}^{*}(y))=y$: for $k\le n$ the $k$-th coordinate of $\Lambda_{n}\eta(x)$ is $a_{k}^{1/2}\eta_{k}(r_{n}(x))$; and $P_{n}\Lambda_{n}\eta(x)=p_{n}^{*}(p_{n}(p_{n}^{*}(y)))=p_{n}^{*}(y)$, so $Q_{n}\Lambda_{n}\eta(x)=0_{X}$. By The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §hilbert, $X^{a}$ is a linear subspace of $X$ carrying the restricted operations, and by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §basis each $f_{k}$ lies in $X^{a}$ and $(f_{k})_{k\in\mathbb{N}}$ is an orthonormal basis of $X^{a}$, in particular an orthonormal sequence. Hence $\Lambda_{n}\eta(x)\in X^{a}$ by (0). For Borel $\xi,\eta$, additivity, homogeneity and symmetry of $\langle\cdot,\cdot\rangle_{a}$ (Real Inner Product Space §inner-product) and orthonormality give, from (0),

$$\bigl\langle\Lambda_{n}\xi(x),\Lambda_{n}\eta(x)\bigr\rangle_{a}=\sum_{k=1}^{n}\sum_{j=1}^{n}\xi_{k}\bigl(r_{n}(x)\bigr)\eta_{j}\bigl(r_{n}(x)\bigr)\langle f_{k},f_{j}\rangle_{a}=\xi\bigl(r_{n}(x)\bigr)\cdot\eta\bigl(r_{n}(x)\bigr),\qquad(1)$$

and taking $\xi=\eta$, by Real Inner Product Space §norm, $|\Lambda_{n}\eta(x)|_{a}^{2}=\lVert\eta(r_{n}(x))\rVert^{2}$.

Step 3 (Claim 3). Let $\eta$ be Borel and $x\in X$, and write $z=\Lambda_{n}\eta(x)$. Additivity of $\langle\cdot,\cdot\rangle$ (Real Inner Product Space §inner-product) gives $\langle x+z,e_{k}\rangle=x_{k}+z_{k}$, so by Step 2, for $k\le n$, the $k$-th coordinate of $r_{n}(x+z)$ is $a_{k}^{-1/2}\bigl(x_{k}+a_{k}^{1/2}\eta_{k}(r_{n}(x))\bigr)=a_{k}^{-1/2}x_{k}+\eta_{k}(r_{n}(x))$, which is the $k$-th coordinate of $r_{n}(x)+\eta(r_{n}(x))$. Further $p_{n}(x+z)=p_{n}(x)+p_{n}(z)$ by the same additivity, and $p_{n}^{*}$ is additive by its definition, so $P_{n}(x+z)=P_{n}x+P_{n}z$ and $Q_{n}(x+z)=Q_{n}x+Q_{n}z=Q_{n}x$, using $Q_{n}z=0_{X}$ from Step 2.

Step 4 (Claim 2). Let $\eta$ be Borel with $\int_{\mathbb{R}^{n}}\lVert\eta\rVert^{2}\,d\tilde{\mu}<\infty$. By (0), $\Lambda_{n}\eta=p_{n}^{*}\circ D_{+}\circ\eta\circ r_{n}$, a composition of Borel maps ($p_{n}^{*}$ by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity, $D_{+}$ and $r_{n}$ by Step 1), hence Borel from $X$ to $X$ by claim 4 of Borel Measurability and Bounded Integration on a Metric Space. Its values lie in $X_{n}$ and in $X^{a}$ by Step 2, so it is measurable as a map into $X^{a}$ by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §measurable, and $|\Lambda_{n}\eta(x)|_{a}^{2}=\lVert\eta(r_{n}(x))\rVert^{2}$ by Step 2. The function $\lVert\eta\rVert^{2}$ is nonnegative and Borel, as the composition of $\eta$ with the Borel function $u\mapsto\lVert u\rVert^{2}$ of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pairs (claim 4 of Borel Measurability and Bounded Integration on a Metric Space), so claim 2 of Image Measures, Measures with Densities, and Change of Variables gives

$$\int_{X}|\Lambda_{n}\eta|_{a}^{2}\,d\mu=\int_{X}\lVert\eta(r_{n}(x))\rVert^{2}\,\mu(dx)=\int_{\mathbb{R}^{n}}\lVert\eta\rVert^{2}\,d\tilde{\mu}<\infty ,$$

so $\Lambda_{n}\eta$ is square-integrable with respect to $\mu$ in the sense of The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §space, with $E=X^{a}$ as in Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §fields.

Dependence on the class. Let $\eta'$ be a Borel map with $\eta'\sim_{\tilde{\mu}}\eta$ in the sense of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §l2mu; it is square-integrable by The Space of Square-Integrable Random Vectors §classes. The set $\{\eta=\eta'\}$ is Borel, as presupposed by the relation $\tilde{\mu}(\{\eta=\eta'\})=1$ of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §l2mu, so its complement $N=\{u:\eta(u)\ne\eta'(u)\}$ is Borel and satisfies $\tilde{\mu}(N)=1-1=0$ by Basic Properties of a Measure §differences. Since $a_{k}^{1/2}\ne0$, $D_{+}$ is injective, and $p_{n}^{*}$ is injective because $p_{n}\circ p_{n}^{*}$ is the identity (Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity); so by (0), $\Lambda_{n}\eta(x)\ne\Lambda_{n}\eta'(x)$ exactly when $r_{n}(x)\in N$. Thus $\{x:\Lambda_{n}\eta(x)\ne\Lambda_{n}\eta'(x)\}=r_{n}^{-1}(N)$, whose $\mu$-measure is $\tilde{\mu}(N)=0$ by Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pushforward. Hence $\Lambda_{n}\eta\sim_{\mu}\Lambda_{n}\eta'$ in the sense of 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 §almost-everywhere with $E=X^{a}$, and the two have the same class by The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §classes. So $\Lambda_{n}$ is well defined on $L^{2}(\tilde{\mu};\mathbb{R}^{n})$.

Linearity. For square-integrable Borel $\xi,\eta$ and $s,t\in\mathbb{R}$, $(s\xi+t\eta)_{k}=s\xi_{k}+t\eta_{k}$, so $\Lambda_{n}(s\xi+t\eta)(x)=s\,\Lambda_{n}\xi(x)+t\,\Lambda_{n}\eta(x)$ for every $x\in X$. Since the operations on classes are $[\xi]+[\eta]=[\xi+\eta]$ and $s[\xi]=[s\xi]$ on both sides (The Space of Square-Integrable Random Vectors §classes through Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §l2mu, and The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §operations), $\Lambda_{n}(s\xi+t\eta)=s\Lambda_{n}\xi+t\Lambda_{n}\eta$ in $L^{2}(\mu;X^{a})$.

Isometry. For square-integrable Borel $\xi,\eta$, the function $\xi\cdot\eta$ is Borel and integrable with respect to $\tilde{\mu}$ by Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form §operations, applied on the probability space $(\mathbb{R}^{n},\mathcal{B}(\mathbb{R}^{n}),\tilde{\mu})$ as in Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §l2mu. By The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §operations, (1), claim 2 of Image Measures, Measures with Densities, and Change of Variables in its integrable form, and Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §l2mu,

$$\langle\Lambda_{n}\xi,\Lambda_{n}\eta\rangle_{\mu}=\int_{X}\bigl\langle\Lambda_{n}\xi(x),\Lambda_{n}\eta(x)\bigr\rangle_{a}\,\mu(dx)=\int_{X}\xi\bigl(r_{n}(x)\bigr)\cdot\eta\bigl(r_{n}(x)\bigr)\,\mu(dx)=\int_{\mathbb{R}^{n}}\xi\cdot\eta\,d\tilde{\mu}=\langle\xi,\eta\rangle_{\tilde{\mu}} .$$

Step 5 (Claim 4). Let $\psi\in C_{c}^{\infty}(\mathbb{R}^{n})$. For $l\le n$ write $\mathrm{pr}_{l}(u)=u_{l}$ for the $l$th coordinate function on $\mathbb{R}^{n}$ of claim 2 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set. Each coordinate $R_{l}=a_{l}^{-1/2}\mathrm{pr}_{l}$ of $R$ is smooth on $\mathbb{R}^{n}$ by claims 2 and 3 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, so $R$ is smooth by claim 1 of Coordinate Functions, the $C^k$ Hierarchy, and Partial Derivatives of a Smooth Map, hence of class $C^{1}$ by Smooth Map on a Euclidean Open Set. For $i,l\le n$ and $u\in\mathbb{R}^{n}$ the difference quotient of $\mathrm{pr}_{l}$ in the $i$th variable at $u$ equals $1$ if $i=l$ and $0$ otherwise, for every $h\ne0$, so every positive $\delta$ serves for every positive $\varepsilon$; thus by Partial Derivative on a Euclidean Open Set and Uniqueness of the Partial Derivative on a Euclidean Open Set, $\partial_{i}\mathrm{pr}_{l}(u)$ is $1$ if $i=l$ and $0$ otherwise, and by claim 1 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, $\partial_{i}R_{l}(u)=a_{l}^{-1/2}\partial_{i}\mathrm{pr}_{l}(u)$. The test function $\psi$ is of class $C^{1}$ on $\mathbb{R}^{n}$ (Test Functions on Euclidean Space, Their Gradient Maps and Laplacians §gradient), which is open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous. By claims 1 and 2 of A Composition of $C^k$ Maps Between Euclidean Open Sets is of Class $C^k$ (with $k=1$), $\chi=\psi\circ R$ is of class $C^{1}$ on $\mathbb{R}^{n}$ and, for $i\le n$ and $u\in\mathbb{R}^{n}$,

$$\partial_{i}\chi(u)=\sum_{l=1}^{n}\partial_{l}\psi\bigl(R(u)\bigr)\,\partial_{i}R_{l}(u)=a_{i}^{-1/2}\,\partial_{i}\psi\bigl(R(u)\bigr).\qquad(2)$$

The function $\psi$ is continuous from $(\mathbb{R}^{n},d_{E})$ to $(\mathbb{R},d_{\mathbb{R}})$ by claim 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, and compactly supported for the topology of $d_{E}$ by Test Functions on Euclidean Space, Their Gradient Maps and Laplacians §space; so $\psi$ is bounded by claim 1 of A Continuous Compactly Supported Function on $\mathbb{R}^n$ is Bounded and Integrable, and then so is $\chi$, whose values are values of $\psi$. Each $\partial_{i}\psi$ is bounded by The Gradient of a Test Function is Bounded and Square-Integrable, and Its Laplacian Bounded and Integrable, Against Every Probability Measure §gradient, so each $\partial_{i}\chi$ is bounded by (2). Hence $\chi\in C^{1}_{b}(\mathbb{R}^{n})$ by Bounded Continuously Differentiable Functions with Bounded Partial Derivatives on Euclidean Space §bounded. Since $r_{n}=R\circ p_{n}$, $\varphi=\psi\circ r_{n}=\chi\circ p_{n}$, so $\varphi\in\mathcal{F}C^{1}_{b}(X)$ with representation $(n,\chi)$ by Bounded C^1 Cylindrical Functions on a Hilbert Space with an Orthonormal Basis §cylindrical and Bounded C^1 Cylindrical Functions on a Hilbert Space with an Orthonormal Basis §representation. 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 (2), $\partial_{k}\varphi(x)=\partial_{k}\chi(p_{n}(x))=a_{k}^{-1/2}\partial_{k}\psi(r_{n}(x))$ for $k\le n$ and $x\in X$. The gradient map $\nabla\psi$ is Borel by Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §test, and its $k$-th coordinate is $\partial_{k}\psi$ by Test Functions on Euclidean Space, Their Gradient Maps and Laplacians §gradient. Hence, by The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §gradient with the representation $(n,\chi)$,

$$\nabla_{a}\varphi(x)=\sum_{k=1}^{n}a_{k}\,a_{k}^{-1/2}\,\partial_{k}\psi\bigl(r_{n}(x)\bigr)\,e_{k}=\sum_{k=1}^{n}a_{k}^{1/2}\,(\nabla\psi)_{k}\bigl(r_{n}(x)\bigr)\,e_{k}=\Lambda_{n}(\nabla\psi)(x)\qquad(x\in X).$$

Step 6 (Claim 5). By Step 1, $\tilde{\mu}\in\mathcal{P}_{2}(\mathbb{R}^{n})$, so $G_{\tilde{\mu}}$ and $T_{\tilde{\mu}}$ are defined. For $\psi\in C_{c}^{\infty}(\mathbb{R}^{n})$, $\nabla\psi$ is Borel and square-integrable against $\tilde{\mu}$ (Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §test), and by Step 4 and Step 5 the class $\Lambda_{n}(\nabla\psi)$ is the class of $\nabla_{a}(\psi\circ r_{n})$ with $\psi\circ r_{n}\in\mathcal{F}C^{1}_{b}(X)$, an element 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 §gradients. By The Tangent Space of the Wasserstein Space at a Probability Measure §gradients, $\Lambda_{n}$ thus maps $G_{\tilde{\mu}}$ into $G^{a}_{\mu}$.

Let $\eta\in T_{\tilde{\mu}}$, the closure of $G_{\tilde{\mu}}$ in $L^{2}(\tilde{\mu};\mathbb{R}^{n})$ (The Tangent Space of the Wasserstein Space at a Probability Measure §tangent), taken in the topology of the distance $d_{\tilde{\mu}}$ (Real Hilbert Space §topology, Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §l2mu). By Sequential Characterization of the Closure in a Metric Space there is a sequence $(\zeta_{j})_{j\in\mathbb{N}}$ in $G_{\tilde{\mu}}$ converging to $\eta$ in $(L^{2}(\tilde{\mu};\mathbb{R}^{n}),d_{\tilde{\mu}})$. By Step 4 (linearity, then the isometry with both arguments equal to $\zeta_{j}-\eta$), and since in both spaces the norm is that of the inner product (The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §operations with Real Inner Product Space §norm, and The Space of Square-Integrable Random Vectors §inner-product),

$$\lVert\Lambda_{n}\zeta_{j}-\Lambda_{n}\eta\rVert_{\mu}^{2}=\bigl\langle\Lambda_{n}(\zeta_{j}-\eta),\Lambda_{n}(\zeta_{j}-\eta)\bigr\rangle_{\mu}=\langle\zeta_{j}-\eta,\zeta_{j}-\eta\rangle_{\tilde{\mu}}=\lVert\zeta_{j}-\eta\rVert_{\tilde{\mu}}^{2},$$

so the distance of $L^{2}(\mu;X^{a})$ (Real Inner Product Space §distance) from $\Lambda_{n}\zeta_{j}$ to $\Lambda_{n}\eta$ equals $d_{\tilde{\mu}}(\zeta_{j},\eta)$ for every $j$. By Convergent Sequence in a Metric Space, $(\Lambda_{n}\zeta_{j})_{j\in\mathbb{N}}$ converges to $\Lambda_{n}\eta$ in $L^{2}(\mu;X^{a})$, and each $\Lambda_{n}\zeta_{j}$ lies in $G^{a}_{\mu}$. By Sequential Characterization of the Closure in a Metric Space again, $\Lambda_{n}\eta$ lies in the closure of $G^{a}_{\mu}$ in $L^{2}(\mu;X^{a})$ in the sense of Real Hilbert Space §topology, 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.