Proof of Properties of a Mode-Restriction Link: Isometric Embedding, Contraction of Noise Norms and Noise Wasserstein Distances, and Pull-Back of Cylindrical Functions and Tangent Fields

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

Preliminaries. For $x\in X_{1}$ and $y\in X_{2}$ the coordinates $x_{k}=\langle x,e^{(1)}_{k}\rangle_{1}$ and $y_{m}=\langle y,e^{(2)}_{m}\rangle_{2}$ are those of Two Noise Levels over Hilbert Spaces: a Fine and a Coarse Reading of the Noise Framework and Level-Indexed Notation §coordinates; for $i\in\{1,2\}$ and $k\in\mathbb{N}$ put $f^{(i)}_{k}=(a^{(i)}_{k})^{1/2}e^{(i)}_{k}$. We record four facts.

(P1) Let $x\in X_{1}$. By Orthonormal Expansions in a Real Hilbert Space §riesz-fischer, applied in $X_{2}$ to $(e^{(2)}_{m})_{m\in\mathbb{N}}$ with $c_{m}=x_{\kappa(m)}$, the series $\sum_{m=1}^{\infty}x_{\kappa(m)}^{2}$ being convergent as shown in Mode-Restriction Links between a Fine and a Coarse Noise Level §restriction,

$$p(x)_{m}=x_{\kappa(m)}\quad(m\in\mathbb{N}),\qquad|p(x)|_{2}^{2}=\sum_{m=1}^{\infty}x_{\kappa(m)}^{2}.$$

(P2) Let $y\in X_{2}$. By the same claim, applied in $X_{1}$ to the orthonormal sequence $(e^{(1)}_{\kappa(m)})_{m\in\mathbb{N}}$ of Mode-Restriction Links between a Fine and a Coarse Noise Level §orthonormal with $c_{m}=y_{m}$, the series $\sum_{m=1}^{\infty}y_{m}^{2}$ being convergent as shown in Mode-Restriction Links between a Fine and a Coarse Noise Level §embedding, one has $\langle j(y),e^{(1)}_{\kappa(m)}\rangle_{1}=y_{m}$ for every $m$ and $|j(y)|_{1}^{2}=\sum_{m=1}^{\infty}y_{m}^{2}$. Let $k\in\mathbb{N}$ not be a value of $\kappa$. By Elementary Properties of Series in a Real Inner Product Space §inner-product, applied in $X_{1}$ with $w=e^{(1)}_{k}$, the number $\langle j(y),e^{(1)}_{k}\rangle_{1}$ is the sum of the series of real numbers with terms $\langle y_{m}e^{(1)}_{\kappa(m)},e^{(1)}_{k}\rangle_{1}$, each of which is $0$ because $\kappa(m)\neq k$ and $(e^{(1)}_{k})_{k\in\mathbb{N}}$ is orthonormal; its partial sums are $0$ by claim 7 of Properties of Finite Sums, applied in the field of real numbers with $i=1$ (every summand, the first included, being $0$), so the sum is $0$. Since $\kappa$ is injective, each $k$ equals $\kappa(m)$ for at most one $m$; thus $j(y)_{k}=y_{m}$ if $k=\kappa(m)$, and $j(y)_{k}=0$ if $k$ is not a value of $\kappa$.

(P3) For $i\in\{1,2\}$, two vectors of $X_{i}$ with the same coordinates are equal: by the additivity and homogeneity of the inner product (Real Inner Product Space §inner-product) their difference has every coordinate $0$, so it is the zero vector by Orthonormal Basis of a Real Hilbert Space §basis.

(P4) For $i\in\{1,2\}$, $X^{a}_{i}$ is a linear subspace of $X_{i}$ which, with the operations of $X_{i}$ and the inner product $\langle\cdot,\cdot\rangle_{a,i}$, is a real Hilbert space by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §hilbert, and $(f^{(i)}_{k})_{k\in\mathbb{N}}$ is an orthonormal basis of it with $\langle x,f^{(i)}_{k}\rangle_{a,i}=(a^{(i)}_{k})^{-1/2}x_{k}$ for $x\in X^{a}_{i}$ by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §basis, both taken at level $i$. Analogously to Mode-Restriction Links between a Fine and a Coarse Noise Level §orthonormal, the injectivity of $\kappa$ and the orthonormality of $(f^{(1)}_{k})_{k\in\mathbb{N}}$ in $X^{a}_{1}$ (The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §basis at level 1) make $(f^{(1)}_{\kappa(m)})_{m\in\mathbb{N}}$ an orthonormal sequence in $X^{a}_{1}$. Finally $a^{(1)}_{\kappa(m)}=a^{(2)}_{m}$ for every $m$ by Mode-Restriction Links between a Fine and a Coarse Noise Level §link.

Claim (linear). Let $x,x'\in X_{1}$, $y,y'\in X_{2}$ and $t\in\mathbb{R}$. By (P1) and the linearity of the inner product (Real Inner Product Space §inner-product), $p(x+tx')$ and $p(x)+t\,p(x')$ both have $m$-th coordinate $x_{\kappa(m)}+t\,x'_{\kappa(m)}$, so they are equal by (P3). By (P2) and the same linearity, $j(y+ty')$ and $j(y)+t\,j(y')$ both have $k$-th coordinate $y_{m}+t\,y'_{m}$ if $k=\kappa(m)$ and $0$ if $k$ is not a value of $\kappa$, so they are equal by (P3).

Claim (inverse). Let $y\in X_{2}$. By (P1) applied to $j(y)$ and then (P2), $p(j(y))_{m}=j(y)_{\kappa(m)}=y_{m}$ for every $m$, so $p(j(y))=y$ by (P3).

Claim (norms). Let $x\in X_{1}$. By Orthonormal Expansions in a Real Hilbert Space §bessel, applied in $X_{1}$ to the orthonormal sequence $(e^{(1)}_{\kappa(m)})_{m\in\mathbb{N}}$, $\sum_{m}x_{\kappa(m)}^{2}\le|x|_{1}^{2}$; so $|p(x)|_{2}^{2}\le|x|_{1}^{2}$ by (P1), and $|p(x)|_{2}\le|x|_{1}$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, both norms being nonnegative. Let $y\in X_{2}$. By (P2) and Orthonormal Expansions in a Real Hilbert Space §parseval in $X_{2}$, $|j(y)|_{1}^{2}=\sum_{m}y_{m}^{2}=|y|_{2}^{2}$, so $|j(y)|_{1}=|y|_{2}$ by claim 3 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field.

Claim (noise). Let $x\in X^{a}_{1}$. By Orthonormal Expansions in a Real Hilbert Space §bessel, applied in the Hilbert space $X^{a}_{1}$ to the orthonormal sequence $(f^{(1)}_{\kappa(m)})_{m\in\mathbb{N}}$ of (P4), the series $\sum_{m}\langle x,f^{(1)}_{\kappa(m)}\rangle_{a,1}^{2}$ converges with sum at most $|x|_{a,1}^{2}$. By (P4) and (P1) its $m$-th term is $(a^{(1)}_{\kappa(m)})^{-1}x_{\kappa(m)}^{2}=(a^{(2)}_{m})^{-1}p(x)_{m}^{2}$. Hence $p(x)\in X^{a}_{2}$ by The Noise Space of a Weight Sequence on a Hilbert Space with an Orthonormal Basis §space, and $|p(x)|_{a,2}^{2}\le|x|_{a,1}^{2}$ by The Noise Space of a Weight Sequence on a Hilbert Space with an Orthonormal Basis §inner-product, both taken at level 2; claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $|p(x)|_{a,2}\le|x|_{a,1}$.

Let $y\in X^{a}_{2}$ and $c_{m}=(a^{(2)}_{m})^{-1/2}y_{m}$. The series $\sum_{m}c_{m}^{2}=\sum_{m}(a^{(2)}_{m})^{-1}y_{m}^{2}$ converges with sum $|y|_{a,2}^{2}$ by The Noise Space of a Weight Sequence on a Hilbert Space with an Orthonormal Basis §space and The Noise Space of a Weight Sequence on a Hilbert Space with an Orthonormal Basis §inner-product at level 2. By Orthonormal Expansions in a Real Hilbert Space §riesz-fischer, applied in $X^{a}_{1}$ to $(f^{(1)}_{\kappa(m)})_{m\in\mathbb{N}}$, the series $\sum_{m}c_{m}f^{(1)}_{\kappa(m)}$ converges in $X^{a}_{1}$ to some $z$ with $|z|_{a,1}^{2}=|y|_{a,2}^{2}$. By (P4) its $m$-th term is $(a^{(2)}_{m})^{-1/2}(a^{(1)}_{\kappa(m)})^{1/2}y_{m}e^{(1)}_{\kappa(m)}=y_{m}e^{(1)}_{\kappa(m)}$, the $m$-th term of the series defining $j(y)$. The inclusion $\iota:X^{a}_{1}\to X_{1}$ is linear, the operations of $X^{a}_{1}$ being those of $X_{1}$ by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §hilbert, and, $\bar{a}^{(1)}$ being positive by Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §weights at level 1 (Two Noise Levels over Hilbert Spaces: a Fine and a Coarse Reading of the Noise Framework and Level-Indexed Notation §levels), so that its square root is defined, $|\iota(v)|_{1}\le(\bar{a}^{(1)})^{1/2}|v|_{a,1}$ for $v\in X^{a}_{1}$ by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §embedding at level 1 and claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field; so $\iota$ is a bounded linear map in the sense of Bounded Linear Maps and Bounded Linear Functionals on Real Inner Product Spaces, and the Operator Norm §bounded. By Elementary Properties of Series in a Real Inner Product Space §linear-maps, the series $\sum_{m}y_{m}e^{(1)}_{\kappa(m)}$ converges in $X_{1}$ with sum $\iota(z)=z$. Its sum is $j(y)$, and the sum of a convergent series is unique by Series in a Real Inner Product Space §convergent. Hence $j(y)=z\in X^{a}_{1}$, and we record the representation

$$j(y)=\sum_{m=1}^{\infty}c_{m}f^{(1)}_{\kappa(m)}\qquad\text{(convergence in }X^{a}_{1}),$$

called (R) below. Moreover $|j(y)|_{a,1}^{2}=|y|_{a,2}^{2}$, whence $|j(y)|_{a,1}=|y|_{a,2}$ by claim 3 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field.

Claim (adjoint). Let $y\in X^{a}_{2}$ and $x\in X^{a}_{1}$, and let $c_{m}$ be as in the previous claim. By (R) and Elementary Properties of Series in a Real Inner Product Space §inner-product, applied in $X^{a}_{1}$ with $w=x$, together with the homogeneity and symmetry of the inner product (Real Inner Product Space §inner-product),

$$\langle j(y),x\rangle_{a,1}=\sum_{m=1}^{\infty}c_{m}\,\langle x,f^{(1)}_{\kappa(m)}\rangle_{a,1}.$$

By (P4) and (P1) the $m$-th term equals $(a^{(2)}_{m})^{-1/2}y_{m}\,(a^{(1)}_{\kappa(m)})^{-1/2}x_{\kappa(m)}=(a^{(2)}_{m})^{-1}y_{m}\,p(x)_{m}$. Since $p(x)\in X^{a}_{2}$ by claim 4, the sum of this series is $\langle y,p(x)\rangle_{a,2}$ by The Noise Space of a Weight Sequence on a Hilbert Space with an Orthonormal Basis §inner-product at level 2.

Claim (borel). For $x,x'\in X_{1}$, claims 1 and 3 give $d_{2}(p(x),p(x'))=|p(x-x')|_{2}\le|x-x'|_{1}=d_{1}(x,x')$, so $p$ is Lipschitz with constant $1$ in the sense of Lipschitz Map Between Metric Spaces, continuous by A Lipschitz Map is Uniformly Continuous, and Borel by claim 3 of Borel Measurability and Bounded Integration on a Metric Space. For $z=(x,y)$ and $z'=(x',y')$ in $X_{1}\times X_{1}$, Properties of the Product of Two Real Inner Product Spaces §metric, applied to both products, gives

$$d\bigl((p\times p)(z),(p\times p)(z')\bigr)^{2}=d_{2}(p(x),p(x'))^{2}+d_{2}(p(y),p(y'))^{2}\le d_{1}(x,x')^{2}+d_{1}(y,y')^{2}=d(z,z')^{2},$$

so $p\times p$ is Lipschitz with constant $1$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, hence continuous and Borel by the same two results.

Claim (pushforward-measure). Let $\mu\in\mathcal{P}(X_{1})$. By claim 1 of Image Measures, Measures with Densities, and Change of Variables, as in Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pushforward, $p_{\#}\mu$ is a Borel measure on $X_{2}$ with total mass $\mu(X_{1})=1$, so $p_{\#}\mu\in\mathcal{P}(X_{2})$.

Claim (pushforward-moment). Suppose $\mu\in\mathcal{P}_{2}(X_{1})$. The functions $w\mapsto|w|_{2}^{2}$ on $X_{2}$ and $x\mapsto|x|_{1}^{2}$ on $X_{1}$ are Borel and nonnegative, as recorded in The Second Moment of a Borel Probability Measure on a Hilbert Space and the Probability Measures with Finite Second Moment at the levels concerned, and $x\mapsto|p(x)|_{2}^{2}$ is Borel by claim 4 of Borel Measurability and Bounded Integration on a Metric Space. By claim 3 and claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, both norms being nonnegative, $|p(x)|_{2}^{2}\le|x|_{1}^{2}$ for every $x\in X_{1}$. By claim 2 of Image Measures, Measures with Densities, and Change of Variables, this inequality, and the monotonicity in Linearity and Monotonicity of the Lebesgue Integral §nonnegative,

$$M_{2}(p_{\#}\mu)=\int_{X_{2}}|w|_{2}^{2}\,(p_{\#}\mu)(dw)=\int_{X_{1}}|p(x)|_{2}^{2}\,\mu(dx)\le\int_{X_{1}}|x|_{1}^{2}\,\mu(dx)=M_{2}(\mu)<\infty,$$

with $M_{2}$ the second moment of The Second Moment of a Borel Probability Measure on a Hilbert Space and the Probability Measures with Finite Second Moment §moment; so $p_{\#}\mu\in\mathcal{P}_{2}(X_{2})$ by The Second Moment of a Borel Probability Measure on a Hilbert Space and the Probability Measures with Finite Second Moment §space.

Claim (couplings). Let $\pi\in\Pi^{a,1}(\mu,\nu)$ and $\pi'=(p\times p)_{\#}\pi$, a Borel probability measure on $X_{2}\times X_{2}$ by claim 1 of Image Measures, Measures with Densities, and Change of Variables. Write $\pi_{1},\pi_{2}$ for the coordinate maps of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pairs at either level; then $\pi_{1}\circ(p\times p)=p\circ\pi_{1}$, so for $B\in\mathcal{B}(X_{2})$ the defining formula of claim 1 of Image Measures, Measures with Densities, and Change of Variables gives $((\pi_{1})_{\#}\pi')(B)=\pi'(\pi_{1}^{-1}(B))=\pi((p\times p)^{-1}(\pi_{1}^{-1}(B)))=\pi(\pi_{1}^{-1}(p^{-1}(B)))=\mu(p^{-1}(B))=(p_{\#}\mu)(B)$, using $(\pi_{1})_{\#}\pi=\mu$ from Couplings of Two Borel Probability Measures on a Hilbert Space and Their Quadratic Cost §coupling; likewise $(\pi_{2})_{\#}\pi'=p_{\#}\nu$. Thus $\pi'\in\Pi(p_{\#}\mu,p_{\#}\nu)$ by Couplings of Two Borel Probability Measures on a Hilbert Space and Their Quadratic Cost §coupling. Let $D_{a,i}$, $c_{a,i}$ and $n_{a,i}$ be the set and functions of The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §pairs and The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §borel at level $i$. If $z=(x,y)\in D_{a,1}$, then $y-x\in X^{a}_{1}$, so $p(y)-p(x)=p(y-x)\in X^{a}_{2}$ by claims 1 and 4; that is, $(p\times p)(z)\in D_{a,2}$, and

$$c_{a,2}\bigl((p\times p)(z)\bigr)=|p(y-x)|_{a,2}^{2}\le|y-x|_{a,1}^{2}=c_{a,1}(z)$$

by claim 4 and claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. Since $D_{a,1}\subseteq(p\times p)^{-1}(D_{a,2})$ and $\pi(D_{a,1})=1$ by Couplings of Finite Noise Cost and Their Noise Cost §finite, the monotonicity of the measure $\pi$, a consequence of the countable additivity in Measure, Measure Space, and Probability Measure, gives $1\le\pi((p\times p)^{-1}(D_{a,2}))=\pi'(D_{a,2})\le1$. By the same additivity the complement of $D_{a,1}$ is $\pi$-null, so the inequality of costs holds for $\pi$-almost every $z$; the function $c_{a,2}\circ(p\times p)$ is Borel by claim 4 of Borel Measurability and Bounded Integration on a Metric Space. By claim 2 of Image Measures, Measures with Densities, and Change of Variables, applied to the nonnegative Borel function $c_{a,2}$, and by The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §comparison,

$$\int_{X_{2}\times X_{2}}c_{a,2}\,d\pi'=\int_{X_{1}\times X_{1}}c_{a,2}\circ(p\times p)\,d\pi\le\int_{X_{1}\times X_{1}}c_{a,1}\,d\pi=I^{a,1}(\pi)<\infty .$$

Hence $\pi'\in\Pi^{a,2}(p_{\#}\mu,p_{\#}\nu)$ by Couplings of Finite Noise Cost and Their Noise Cost §finite and Couplings of Finite Noise Cost and Their Noise Cost §couplings, and $I^{a,2}(\pi')\le I^{a,1}(\pi)$ by Couplings of Finite Noise Cost and Their Noise Cost §cost.

Claim (lipschitz). Suppose $(\mu,\nu)$ is noise-connected at level 1. Then $\Pi^{a,1}(\mu,\nu)$ has a member $\pi$ by Couplings of Finite Noise Cost and Their Noise Cost §connected, and $(p\times p)_{\#}\pi\in\Pi^{a,2}(p_{\#}\mu,p_{\#}\nu)$ by the previous claim, so $(p_{\#}\mu,p_{\#}\nu)$ is noise-connected at level 2 by the same definition. For every $\pi\in\Pi^{a,1}(\mu,\nu)$, The Noise Wasserstein Distance §distance at level 2 and the previous claim give

$$W_{a,2}(p_{\#}\mu,p_{\#}\nu)^{2}\le I^{a,2}\bigl((p\times p)_{\#}\pi\bigr)\le I^{a,1}(\pi).$$

So $W_{a,2}(p_{\#}\mu,p_{\#}\nu)^{2}$ is a lower bound of the set $\{I^{a,1}(\pi):\pi\in\Pi^{a,1}(\mu,\nu)\}$, whose greatest lower bound is $W_{a,1}(\mu,\nu)^{2}$ by The Noise Wasserstein Distance §distance at level 1; hence $W_{a,2}(p_{\#}\mu,p_{\#}\nu)^{2}\le W_{a,1}(\mu,\nu)^{2}$, and claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives the asserted inequality, both distances being nonnegative. Any $\mu,\nu\in\mathcal{P}^{a}_{\rho_{1}}$ are noise-connected at level 1 by The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §connected, so this applies to them.

Claim (space). Let $\mu\in\mathcal{P}^{a}_{\rho_{1}}$. Since $\rho_{1}\in\mathcal{P}^{a}_{\rho_{1}}$ by The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §reference at level 1, the previous claim shows that $(p_{\#}\mu,p_{\#}\rho_{1})$ is noise-connected at level 2, and $(p_{\#}\rho_{1},\rho_{2})$ is noise-connected at level 2 by the hypothesis $p_{\#}\rho_{1}\in\mathcal{P}^{a}_{\rho_{2}}$ and The Measures Noise-Connected to the Reference Measure §space. Choose $\pi_{12}\in\Pi^{a,2}(p_{\#}\mu,p_{\#}\rho_{1})$ and $\pi_{23}\in\Pi^{a,2}(p_{\#}\rho_{1},\rho_{2})$. The three measures lie in $\mathcal{P}_{2}(X_{2})$: $\mu,\rho_{1}\in\mathcal{P}_{2}(X_{1})$ by The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §moments at level 1, so $p_{\#}\mu,p_{\#}\rho_{1}\in\mathcal{P}_{2}(X_{2})$ by claim 6, and $\rho_{2}\in\mathcal{P}_{2}(X_{2})$ by Two Noise Levels over Hilbert Spaces: a Fine and a Coarse Reading of the Noise Framework and Level-Indexed Notation §levels. Hence Gluing Two Couplings on a Hilbert Space over a Common Middle Marginal, and the Triangle Inequalities for the Quadratic and Noise Costs §glued, at level 2, provides a gluing $\sigma$ of $\pi_{12}$ and $\pi_{23}$, and Gluing Two Couplings on a Hilbert Space over a Common Middle Marginal, and the Triangle Inequalities for the Quadratic and Noise Costs §noise-triangle gives $(q_{1},q_{3})_{\#}\sigma\in\Pi^{a,2}(p_{\#}\mu,\rho_{2})$. Therefore $(p_{\#}\mu,\rho_{2})$ is noise-connected at level 2 and $p_{\#}\mu\in\mathcal{P}^{a}_{\rho_{2}}$ by The Measures Noise-Connected to the Reference Measure §space. Now $p_{\#}\mu$, $p_{\#}\rho_{1}$ and $\rho_{2}$ belong to $\mathcal{P}^{a}_{\rho_{2}}$, the last by The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §reference at level 2, so the triangle inequality The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §triangle at level 2 and the previous claim, applied to $\mu$ and $\rho_{1}$, give

$$W_{a,2}(p_{\#}\mu,\rho_{2})\le W_{a,2}(p_{\#}\mu,p_{\#}\rho_{1})+W_{a,2}(p_{\#}\rho_{1},\rho_{2})\le W_{a,1}(\mu,\rho_{1})+W_{a,2}(p_{\#}\rho_{1},\rho_{2}).$$

Claim (fields). Let $g$ be as stated. For $x\in X_{1}$, $g(p(x))\in X^{a}_{2}$, so $j(g(p(x)))\in X^{a}_{1}$ and $|j(g(p(x)))|_{a,1}=|g(p(x))|_{a,2}$ by claim 4. For $k\in\mathbb{N}$, by (P2) the function $x\mapsto\langle j(g(p(x))),e^{(1)}_{k}\rangle_{1}$ is $x\mapsto\langle g(p(x)),e^{(2)}_{m}\rangle_{2}$ if $k=\kappa(m)$, and the constant $0$ if $k$ is not a value of $\kappa$. The function $w\mapsto\langle g(w),e^{(2)}_{m}\rangle_{2}$ on $X_{2}$ is measurable by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §measurable at level 2, $g$ being measurable into $X^{a}_{2}$, and its composite with the Borel map $p$ of claim 6 is measurable by claim 4 of Borel Measurability and Bounded Integration on a Metric Space; constant functions are measurable. Hence $j\circ g\circ p$ is measurable as a map into $X^{a}_{1}$ by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §measurable at level 1. The function $|g|_{a,2}^{2}$ is nonnegative and measurable 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, so claim 2 of Image Measures, Measures with Densities, and Change of Variables gives

$$\int_{X_{1}}\bigl|j(g(p(x)))\bigr|_{a,1}^{2}\,\mu(dx)=\int_{X_{1}}|g(p(x))|_{a,2}^{2}\,\mu(dx)=\int_{X_{2}}|g(y)|_{a,2}^{2}\,(p_{\#}\mu)(dy)<\infty,$$

the finiteness by The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §space; so $j\circ g\circ p$ is square-integrable with respect to $\mu$, and the asserted identity holds. The map $j$ is injective by claim 2, since $j(w)=j(w')$ implies $w=p(j(w))=p(j(w'))=w'$. Let $g'$ be measurable with $g'\sim_{p_{\#}\mu}g$. Then $\{x:j(g(p(x)))\neq j(g'(p(x)))\}=p^{-1}(\{y:g(y)\neq g'(y)\})$, and its $\mu$-measure is the $p_{\#}\mu$-measure of $\{y:g(y)\neq g'(y)\}$, which is $0$ (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); so $j\circ g'\circ p\sim_{\mu}j\circ g\circ p$, and the two classes coincide by The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §classes.

Claim (fields-isometry). By The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §operations, $\lVert j\circ g\circ p\rVert_{\mu,1}$ and $\lVert g\rVert_{p_{\#}\mu,2}$ are the nonnegative square roots of the first and last integrals of the display, which are equal; so the norms are equal. Claim (fields-linear). For representatives $g,g'$ and $t\in\mathbb{R}$, claim 1 gives $j(g(p(x))+t\,g'(p(x)))=j(g(p(x)))+t\,j(g'(p(x)))$ for every $x\in X_{1}$, so by The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §operations the class of $j\circ(g+tg')\circ p$ is the class of $j\circ g\circ p$ plus $t$ times the class of $j\circ g'\circ p$.

Claim (cylindrical). Let $(n,\chi)$ be a representation of $\psi$ (Bounded C^1 Cylindrical Functions on a Hilbert Space with an Orthonormal Basis §representation, at level 2), so $\chi\in C^{1}_{b}(\mathbb{R}^{n})$ and $\psi=\chi\circ \varpi^{(2)}_{n}$. Let $N$ be the largest of $\kappa(1),\dots,\kappa(n)$, and let $L:\mathbb{R}^{N}\to\mathbb{R}^{n}$ be $L(t)=(t_{\kappa(1)},\dots,t_{\kappa(n)})$. By (P1), $\varpi^{(2)}_{n}(p(x))=(x_{\kappa(1)},\dots,x_{\kappa(n)})=L(\varpi^{(1)}_{N}(x))$ for $x\in X_{1}$, so $\psi\circ p=(\chi\circ L)\circ \varpi^{(1)}_{N}$.

We show $\chi\circ L\in C^{1}_{b}(\mathbb{R}^{N})$. Let $t\in\mathbb{R}^{N}$, $k\in[N]$ and $h\in\mathbb{R}$, and let $t^{h}$ be $t$ with its $k$-th entry replaced by $t_{k}+h$. If $k=\kappa(i)$ with $i\in[n]$ ($i$ unique by injectivity), then $L(t^{h})$ is $L(t)$ with its $i$-th entry replaced by $L(t)_{i}+h$; if $k$ is not among $\kappa(1),\dots,\kappa(n)$, then $L(t^{h})=L(t)$. Hence the difference quotients of $\chi\circ L$ at $t$ in the $k$-th variable are those of $\chi$ at $L(t)$ in the $i$-th variable, respectively $0$, and Partial Derivative on a Euclidean Open Set gives that $\partial_{k}(\chi\circ L)(t)$ exists and equals $\partial_{i}\chi(L(t))$, respectively $0$. For $t,t'\in\mathbb{R}^{N}$, $\sum_{i=1}^{n}(t'_{\kappa(i)}-t_{\kappa(i)})^{2}\le\sum_{k=1}^{N}(t'_{k}-t_{k})^{2}$, the left side being the sum of some of the nonnegative terms on the right since $\kappa(1),\dots,\kappa(n)$ are distinct elements of $[N]$; so $L$ is continuous at every point in the sense of Continuity at a Point for Maps Between Euclidean Spaces, with $\delta=\varepsilon$. The function $\chi$ and its partial derivatives $\partial_{i}\chi$ are continuous at every point of $\mathbb{R}^{n}$ by clauses 1 and 3 of C^k Maps on a Euclidean Open Set, so $\chi\circ L$ and $\partial_{i}\chi\circ L$ are continuous at every point by Composition of Continuous Euclidean Maps, and constant functions are continuous. Thus $\chi\circ L$ is of class $C^{1}$ on $\mathbb{R}^{N}$ by the same clauses, and it and its partial derivatives are bounded because $\chi$ and the $\partial_{i}\chi$ are (Bounded Continuously Differentiable Functions with Bounded Partial Derivatives on Euclidean Space §bounded). So $\chi\circ L\in C^{1}_{b}(\mathbb{R}^{N})$, and $\psi\circ p\in\mathcal{F}C^{1}_{b}(X_{1})$ with representation $(N,\chi\circ L)$ by Bounded C^1 Cylindrical Functions on a Hilbert Space with an Orthonormal Basis §cylindrical at level 1.

Fix $x\in X_{1}$; we compare coordinates. 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 at level 1, $\partial_{k}(\psi\circ p)(x)=\partial_{k}(\chi\circ L)(\varpi^{(1)}_{N}(x))$ for $k\le N$, which by the above is $\partial_{i}\chi(\varpi^{(2)}_{n}(p(x)))=\partial_{i}\psi(p(x))$ if $k=\kappa(i)$ with $i\le n$, the last equality 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 at level 2, and $0$ otherwise. By The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §gradient, which identifies the coordinates of a noise gradient, at level 1, the $k$-th coordinate of $\nabla_{a,1}(\psi\circ p)(x)$ is therefore $a^{(1)}_{\kappa(i)}\partial_{i}\psi(p(x))=a^{(2)}_{i}\partial_{i}\psi(p(x))$ if $k=\kappa(i)$ with $i\le n$, and $0$ for every other $k$ (for $k>N$ by that clause, and $k>N$ is not among $\kappa(1),\dots,\kappa(n)$). By the same clause at level 2, the $m$-th coordinate of $\nabla_{a,2}\psi(p(x))$ is $a^{(2)}_{m}\partial_{m}\psi(p(x))$ for $m\le n$ and $0$ for $m>n$; so by (P2) the $k$-th coordinate of $j(\nabla_{a,2}\psi(p(x)))$ is $a^{(2)}_{i}\partial_{i}\psi(p(x))$ if $k=\kappa(i)$ with $i\le n$, and $0$ otherwise (if $k=\kappa(m)$ with $m>n$, or $k$ is not a value of $\kappa$). The two vectors have the same coordinates, so they are equal by (P3).

Claim (tangent). For $h\in L^{2}(p_{\#}\mu;X^{a}_{2})$ write $\Phi(h)=j\circ h\circ p\in L^{2}(\mu;X^{a}_{1})$. By claim 9, $\Phi$ is linear and norm-preserving, so $\lVert\Phi(h)-\Phi(h')\rVert_{\mu,1}=\lVert\Phi(h-h')\rVert_{\mu,1}=\lVert h-h'\rVert_{p_{\#}\mu,2}$. If $h\in G^{a,2}_{p_{\#}\mu}$, that is, $h$ is the class of $\nabla_{a,2}\psi$ for some $\psi\in\mathcal{F}C^{1}_{b}(X_{2})$ (The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §gradients at level 2), then by claim 10 $\Phi(h)$ is the class of $j\circ\nabla_{a,2}\psi\circ p=\nabla_{a,1}(\psi\circ p)$ with $\psi\circ p\in\mathcal{F}C^{1}_{b}(X_{1})$, so $\Phi(h)\in G^{a,1}_{\mu}$. Now $g\in T^{a,2}_{p_{\#}\mu}$, the closure of $G^{a,2}_{p_{\#}\mu}$ in $L^{2}(p_{\#}\mu;X^{a}_{2})$ (The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §tangent, with closure as in Real Hilbert Space §topology). By Sequential Characterization of the Closure in a Metric Space there is a sequence $(g_{n})_{n\in\mathbb{N}}$ in $G^{a,2}_{p_{\#}\mu}$ converging to $g$. Then $\Phi(g_{n})\in G^{a,1}_{\mu}$ and the distance from $\Phi(g_{n})$ to $\Phi(g)$ equals the distance from $g_{n}$ to $g$ for every $n$ (in both spaces the distance is the norm of the difference, by Real Inner Product Space §distance), so $(\Phi(g_{n}))_{n\in\mathbb{N}}$ converges to $\Phi(g)$ in the sense of Convergent Sequence in a Metric Space; by Sequential Characterization of the Closure in a Metric Space again, $\Phi(g)=j\circ g\circ p$ lies in the closure of $G^{a,1}_{\mu}$, which is $T^{a,1}_{\mu}$.