Proof of Lifted Test Data from Second-Order Data on the Space of Square-Integrable Random Vectors: Compression along the Constant Tuple

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

Preliminaries. By The Constant Tuple on $L^{2}(\Omega;\mathbb{R}^{d})$: Coordinates, Projection, Tail Form, and Translation-Closed Preimages (ii) the tuple $\gamma$ is orthonormal, so Coordinate Maps of a Finite Orthonormal Tuple: Forms, the Tail Form, and Quadratic Test Functions and Compression of a Bounded Symmetric Bilinear Form along an Orthonormal Tuple apply to it as described in the statement. By The Constant Tuple on $L^{2}(\Omega;\mathbb{R}^{d})$: Coordinates, Projection, Tail Form, and Translation-Closed Preimages (iii) the map $\Lambda^{\sharp}$ determined by $\gamma$ satisfies $\Lambda^{\sharp}a=c_{a}$ for every $a\in\mathbb{R}^{d}$. By Translations on a Space of Square-Integrable Random Vectors: Constant Classes, Law Invariance, the Translation Derivative and the Translation Hessian (i) we have, for all $a,b\in\mathbb{R}^{d}$ and every $t\in\mathbb{R}$,

$$c_{a}+c_{b}=c_{a+b},\qquad t\,c_{a}=c_{ta},\qquad\lVert c_{a}\rVert_{L^{2}}=\lVert a\rVert,$$

and for each $Z\in L^{2}(\Omega;\mathbb{R}^{d})$ the map $J_{Z}:\mathbb{R}^{d}\to L^{2}(\Omega;\mathbb{R}^{d})$ with $J_{Z}(a)=Z+c_{a}$ is continuous and satisfies $J_{Z}(0_{\mathbb{R}^{d}})=Z$. Taking $t=-1$ in the second identity and then using the first, $c_{z}-c_{w}=c_{z}+c_{-w}=c_{z-w}$ for all $z,w\in\mathbb{R}^{d}$, the identity $z-w=z+(-1)w$ in $\mathbb{R}^{d}$ being that of Euclidean Space $\mathbb{R}^n$ is a Real Vector Space; hence $\lVert c_{z}-c_{w}\rVert_{L^{2}}=\lVert z-w\rVert$. Finally $\lVert e_{i}\rVert=1$ for $i\in[d]$ by Euclidean Space and Real Matrices: Standing Notation (vi), so $\lVert c_{e_{i}}\rVert_{L^{2}}=1$.

Proof of clause 1. Let $\Phi\in C^{2}(L^{2}(\Omega;\mathbb{R}^{d}))$. By The Classes $C^1$ and $C^2$ on an Open Subset of a Real Inner Product Space (ii) the function $\Phi$ is of class $C^{1}$, has a second derivative at every point, and its Hessian map is continuous into $(\mathrm{Sym},d_{\mathrm{Sym}})$. The first of these is property (a) of Lifted Test Functions on the Space of Square-Integrable Random Vectors and Their Translation Hessians (i).

Fix $Z\in L^{2}(\Omega;\mathbb{R}^{d})$ and write $\phi=\phi_{Z}$ for the function $\phi(a)=\Phi(Z+c_{a})$ on $\mathbb{R}^{d}$. By Translations on a Space of Square-Integrable Random Vectors: Constant Classes, Law Invariance, the Translation Derivative and the Translation Hessian (iii) the function $\phi$ is of class $C^{1}$ on $\mathbb{R}^{d}$ and

$$\partial_{i}\phi(a)=\bigl\langle D\Phi(Z+c_{a}),c_{e_{i}}\bigr\rangle_{L^{2}}\qquad(a\in\mathbb{R}^{d},\ i\in[d]).$$

The second partial derivatives. Fix $i,j\in[d]$ and $a\in\mathbb{R}^{d}$, and write $x=Z+c_{a}$ and $b=D^{2}\Phi(x)\in\mathrm{Sym}$. Let $\varepsilon\in\mathbb{R}$ be positive. Since $L^{2}(\Omega;\mathbb{R}^{d})$ is open in itself and $\Phi$ is differentiable at every one of its points, The Second Derivative and the Hessian on an Open Subset of a Real Inner Product Space (i) provides a positive $\rho\in\mathbb{R}$ such that all $W,Y\in L^{2}(\Omega;\mathbb{R}^{d})$ with $\lVert W\rVert_{L^{2}}<\rho$ satisfy

$$\bigl|\bigl\langle D\Phi(x+W)-D\Phi(x),Y\bigr\rangle_{L^{2}}-b(W,Y)\bigr|\ \le\ \varepsilon\,\lVert W\rVert_{L^{2}}\lVert Y\rVert_{L^{2}} .$$

Let $s\in\mathbb{R}$ satisfy $0<|s|<\rho$ and put $W=c_{s e_{j}}$ and $Y=c_{e_{i}}$. Then $\lVert W\rVert_{L^{2}}=\lVert s e_{j}\rVert=|s|\,\lVert e_{j}\rVert=|s|<\rho$ by claim 5 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square, and $\lVert Y\rVert_{L^{2}}=1$. Moreover $x+W=Z+c_{a}+c_{se_{j}}=Z+c_{a+se_{j}}$, so that by the formula for $\partial_{i}\phi$ displayed above and by Elementary Identities in a Real Inner Product Space (i),

$$\bigl\langle D\Phi(x+W)-D\Phi(x),c_{e_{i}}\bigr\rangle_{L^{2}}=\partial_{i}\phi(a+se_{j})-\partial_{i}\phi(a).$$

Also $W=c_{se_{j}}=s\,c_{e_{j}}$, so $b(W,Y)=s\,b(c_{e_{j}},c_{e_{i}})$ by the homogeneity of $b$ in its first argument, Bounded Symmetric Bilinear Forms on a Real Inner Product Space: Norm, Order, Identity Form and Restriction (i). The displayed estimate therefore reads

$$\bigl|\partial_{i}\phi(a+se_{j})-\partial_{i}\phi(a)-s\,b(c_{e_{j}},c_{e_{i}})\bigr|\ \le\ \varepsilon\,|s| ,$$

and dividing by the positive number $|s|$ (claims 4 and 5 of Elementary Arithmetic in an Ordered Field) and using claim 4 of Properties of the Absolute Value in an Ordered Field,

$$\left|\frac{\partial_{i}\phi(a+se_{j})-\partial_{i}\phi(a)}{s}-b(c_{e_{j}},c_{e_{i}})\right|\ \le\ \varepsilon\qquad\text{whenever }0<|s|<\rho .$$

Here $\varepsilon$ was an arbitrary positive real number; applying what precedes to $\tfrac{\varepsilon}{2}$, which is positive and satisfies $\tfrac{\varepsilon}{2}<\varepsilon$ by claim 8 of Elementary Order Arithmetic in an Ordered Field, we obtain for every positive $\varepsilon$ a positive $\rho$ such that the left-hand side above is strictly less than $\varepsilon$ whenever $0<|s|<\rho$. That is the statement that the limit defining the partial derivative of $\partial_{i}\phi$ with respect to the $j$th variable at $a$ exists and equals $b(c_{e_{j}},c_{e_{i}})$. In the notation of clause 4 of Functions of Class Cᵏ on an Open Subset of ℝⁿ,

$$\partial_{j}\partial_{i}\phi(a)=\bigl(D^{2}\Phi(Z+c_{a})\bigr)(c_{e_{j}},c_{e_{i}})=\Bigl(\bigl(D^{2}\Phi(Z+c_{a})\bigr)^{\flat}\Bigr)_{ji}$$

by Compression of a Bounded Symmetric Bilinear Form along an Orthonormal Tuple (i), the $j$th and $i$th components of $\gamma$ being $c_{e_{j}}$ and $c_{e_{i}}$.

Continuity. The map $a\mapsto Z+c_{a}$ is continuous, being $J_{Z}$; the Hessian map $D^{2}\Phi$ is continuous into $(\mathrm{Sym},d_{\mathrm{Sym}})$; and the map $\mathrm{Sym}\to\mathbb{R}$ with value $c(c_{e_{j}},c_{e_{i}})$ at $c$ is continuous, since $|c(c_{e_{j}},c_{e_{i}})-c'(c_{e_{j}},c_{e_{i}})|\le\lVert c-c'\rVert$ for $c,c'\in\mathrm{Sym}$ by Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity (ix) together with $\lVert c_{e_{j}}\rVert_{L^{2}}=\lVert c_{e_{i}}\rVert_{L^{2}}=1$. By claim 3 of Semicontinuity and Continuity Under Composition with a Continuous Map, applied twice, the composite $a\mapsto\partial_{j}\partial_{i}\phi(a)$ is continuous at every point of $\mathbb{R}^{d}$.

Conclusion of clause 1. Each $\partial_{i}\phi$ is continuous at every point of $\mathbb{R}^{d}$, because $\phi$ is of class $C^{1}$ (clause 1 of Functions of Class Cᵏ on an Open Subset of ℝⁿ), and by the two paragraphs above each of its partial derivatives exists at every point and is continuous there. Hence each $\partial_{i}\phi$ is of class $C^{1}$ on $\mathbb{R}^{d}$, and $\phi$ is of class $C^{2}$ on $\mathbb{R}^{d}$ by clauses 2 and 3 of Functions of Class Cᵏ on an Open Subset of ℝⁿ. As $Z$ was arbitrary, $\Phi$ is twice continuously differentiable along translations at every point, that is twice continuously differentiable along translations, which is property (b) of Lifted Test Functions on the Space of Square-Integrable Random Vectors and Their Translation Hessians (i). So $\Phi$ is a lifted test function.

Finally, by Lifted Test Functions on the Space of Square-Integrable Random Vectors and Their Translation Hessians (ii) and The Hessian Matrix of a Function of Class C² the entry of $H_{\Phi}(Z)=D^{2}\phi_{Z}(0_{\mathbb{R}^{d}})$ in row $i$ and column $j$ is $\partial_{i}\partial_{j}\phi(0_{\mathbb{R}^{d}})$. Applying the computation above with the roles of $i$ and $j$ exchanged and with $a=0_{\mathbb{R}^{d}}$, so that $Z+c_{0_{\mathbb{R}^{d}}}=Z$ by Translations on a Space of Square-Integrable Random Vectors: Constant Classes, Law Invariance, the Translation Derivative and the Translation Hessian (i), this entry equals $\bigl((D^{2}\Phi(Z))^{\flat}\bigr)_{ij}$. Two real $d\times d$ matrices with the same entries are equal by Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations, so $H_{\Phi}(Z)=(D^{2}\Phi(Z))^{\flat}$.

Proof of clause 2. Assume the quadruple is approximable by test data from above and let $\varepsilon\in\mathbb{R}$ be positive. By Quadruple Approximable by Test Data on a Subset of a Real Inner Product Space (i) there are $Y\in A$ and $\varphi\in C^{2}(L^{2}(\Omega;\mathbb{R}^{d}))$ such that $w-\varphi$ has a local maximum at $Y$ relative to $A$ and

$$\lVert Y-\bar{X}\rVert_{L^{2}}<\varepsilon,\quad\bigl|w(Y)-w(\bar{X})\bigr|<\varepsilon,\quad\lVert D\varphi(Y)-p\rVert_{L^{2}}<\varepsilon,\quad\bigl\lVert D^{2}\varphi(Y)-\mathbb{B}\bigr\rVert<\varepsilon .$$

By clause 1 the function $\Phi=\varphi$ is a lifted test function with $H_{\varphi}(Y)=(D^{2}\varphi(Y))^{\flat}$. By Compression of a Bounded Symmetric Bilinear Form along an Orthonormal Tuple (ii), applied to the sum of $D^{2}\varphi(Y)-\mathbb{B}$ and $\mathbb{B}$ and to the scalar multiple of $\mathbb{B}$ by $-1$,

$$\bigl(D^{2}\varphi(Y)\bigr)^{\flat}-\mathbb{B}^{\flat}=\bigl(D^{2}\varphi(Y)-\mathbb{B}\bigr)^{\flat}, \qquad \bigl\lVert\bigl(D^{2}\varphi(Y)-\mathbb{B}\bigr)^{\flat}\bigr\rVert\le\bigl\lVert D^{2}\varphi(Y)-\mathbb{B}\bigr\rVert<\varepsilon ,$$

the differences of matrices and of forms being those of Euclidean Space and Real Matrices: Standing Notation (ii) and Bounded Symmetric Bilinear Forms on a Real Inner Product Space: Norm, Order, Identity Form and Restriction (iv). The four displayed inequalities of the clause therefore hold with $\Phi=\varphi$. The case from below is the same argument with Quadruple Approximable by Test Data on a Subset of a Real Inner Product Space (ii) and a local minimum in place of a local maximum.

Proof of clause 3. We prove (a); part (b) is the same argument with $-g$ in place of $g$, with Quadruple Approximable by Test Data on a Subset of a Real Inner Product Space (ii) in place of Quadruple Approximable by Test Data on a Subset of a Real Inner Product Space (i), and with local minima in place of local maxima.

By Affine and Quadratic Functions on a Real Hilbert Space are of Class $C^2$ (iii), read with $2\mu$ in place of the number written $\alpha$ there and with $q$ in place of $y_{0}$, the function $g$, whose value at $Z$ is $\tfrac{2\mu}{2}\lVert Z-q\rVert_{L^{2}}^{2}$, belongs to $C^{2}(L^{2}(\Omega;\mathbb{R}^{d}))$ with $Dg(Z)=2\mu(Z-q)$ and $D^{2}g(Z)=2\mu I$ for every $Z$.

Let $\varepsilon\in\mathbb{R}$ be positive and choose a positive $\varepsilon'\in\mathbb{R}$ with

$$\varepsilon'\le1,\qquad\varepsilon'\le\varepsilon,\qquad(1+2|\mu|)\,\varepsilon'\le\varepsilon,\qquad\varepsilon'+|\mu|\,\varepsilon'\bigl(2\lVert\bar{X}-q\rVert_{L^{2}}+1\bigr)\le\varepsilon ,$$

which is possible because each of the four right-hand sides is positive and each left-hand side is a nonnegative multiple of $\varepsilon'$ (claim 9 of Elementary Order Arithmetic in an Ordered Field selects the least of four positive candidates). By Quadruple Approximable by Test Data on a Subset of a Real Inner Product Space (i) applied to $w-g$ with $\varepsilon'$ there are $Y\in A$ and $\varphi\in C^{2}(L^{2}(\Omega;\mathbb{R}^{d}))$ such that $(w-g)-\varphi$ has a local maximum at $Y$ relative to $A$ and

$$\lVert Y-\bar{X}\rVert_{L^{2}}<\varepsilon',\quad\bigl|(w-g)(Y)-(w-g)(\bar{X})\bigr|<\varepsilon',\quad\lVert D\varphi(Y)-p\rVert_{L^{2}}<\varepsilon',\quad\bigl\lVert D^{2}\varphi(Y)-\mathbb{B}\bigr\rVert<\varepsilon' .$$

Put $\psi=\varphi+g$. By Constants, Sums, Scalar Multiples and Differences of Differentiable Functions on an Open Subset of a Real Inner Product Space (ii), $\psi\in C^{2}(L^{2}(\Omega;\mathbb{R}^{d}))$ with $D\psi(Y)=D\varphi(Y)+2\mu(Y-q)$ and $D^{2}\psi(Y)=D^{2}\varphi(Y)+2\mu I$. The value of $w-\psi$ at $Z\in A$ is $(w(Z)-g(Z))-\varphi(Z)$, that is the value of $(w-g)-\varphi$ at $Z$; hence $w-\psi$ has a local maximum at $Y$ relative to $A$.

It remains to bound the four quantities. First $\lVert Y-\bar{X}\rVert_{L^{2}}<\varepsilon'\le\varepsilon$. Secondly, write $s=\lVert Y-q\rVert_{L^{2}}$ and $t=\lVert\bar{X}-q\rVert_{L^{2}}$. By The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity (ii), $|s-t|\le\lVert Y-\bar{X}\rVert_{L^{2}}<\varepsilon'$, so $s\le t+\varepsilon'$ and $s+t\le2t+\varepsilon'\le2t+1$ by claim 3 of Properties of the Absolute Value in an Ordered Field and $\varepsilon'\le1$; therefore, by claim 4 of Properties of the Absolute Value in an Ordered Field and the identity $s^{2}-t^{2}=(s-t)(s+t)$ of claim 4 of Zero Products and Elementary Identities in a Field,

$$\bigl|g(Y)-g(\bar{X})\bigr|=|\mu|\,|s-t|\,(s+t)\le|\mu|\,\varepsilon'\,(2t+1),$$

the factors $|s-t|$ and $s+t$ being nonnegative (claim 5 of Elementary Arithmetic in an Ordered Field). Since $w(Z)=(w-g)(Z)+g(Z)$ for $Z\in A$, claim 5 of Properties of the Absolute Value in an Ordered Field gives

$$\bigl|w(Y)-w(\bar{X})\bigr|\le\bigl|(w-g)(Y)-(w-g)(\bar{X})\bigr|+\bigl|g(Y)-g(\bar{X})\bigr|<\varepsilon'+|\mu|\,\varepsilon'\,\bigl(2\lVert\bar{X}-q\rVert_{L^{2}}+1\bigr)\le\varepsilon .$$

Thirdly, $D\psi(Y)-\bigl(p+2\mu(\bar{X}-q)\bigr)=\bigl(D\varphi(Y)-p\bigr)+2\mu\,(Y-\bar{X})$, so by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity (i) and Elementary Identities in a Real Inner Product Space (iv),

$$\bigl\lVert D\psi(Y)-\bigl(p+2\mu(\bar{X}-q)\bigr)\bigr\rVert_{L^{2}}\le\lVert D\varphi(Y)-p\rVert_{L^{2}}+2|\mu|\,\lVert Y-\bar{X}\rVert_{L^{2}}<(1+2|\mu|)\,\varepsilon'\le\varepsilon .$$

Fourthly, $D^{2}\psi(Y)-(\mathbb{B}+2\mu I)=D^{2}\varphi(Y)-\mathbb{B}$ by Bounded Symmetric Bilinear Forms on a Real Inner Product Space: Norm, Order, Identity Form and Restriction (iv), so its norm is less than $\varepsilon'\le\varepsilon$. Finally the value recorded by the quadruple is correct, since $(w-g)(\bar{X})+g(\bar{X})=w(\bar{X})$. As $\varepsilon$ was an arbitrary positive real number, $\bigl(\bar{X},w(\bar{X}),p+2\mu(\bar{X}-q),\mathbb{B}+2\mu I\bigr)$ is approximable by test data from above for $w$ on $A$.

Proof of clause 4. Let $z,w\in\mathbb{R}^{d}$. By Compression of a Bounded Symmetric Bilinear Form along an Orthonormal Tuple (i) and $\Lambda^{\sharp}a=c_{a}$,

$$z\cdot\bigl(\mathbb{X}^{\flat}z\bigr)=\mathbb{X}(c_{z},c_{z}),\qquad w\cdot\bigl(\mathbb{Y}^{\flat}w\bigr)=\mathbb{Y}(c_{w},c_{w}).$$

Applying the hypothesis with $Z=c_{z}$ and $W=c_{w}$ and using $\lVert c_{z}\rVert_{L^{2}}=\lVert z\rVert$, $\lVert c_{w}\rVert_{L^{2}}=\lVert w\rVert$ and $\lVert c_{z}-c_{w}\rVert_{L^{2}}=\lVert z-w\rVert$ from the preliminaries,

$$-3\alpha\bigl(\lVert z\rVert^{2}+\lVert w\rVert^{2}\bigr)\ \le\ z\cdot\bigl(\mathbb{X}^{\flat}z\bigr)-w\cdot\bigl(\mathbb{Y}^{\flat}w\bigr)\ \le\ 3\alpha\lVert z-w\rVert^{2}.$$

By Compression of a Bounded Symmetric Bilinear Form along an Orthonormal Tuple (ii) the compression is order-preserving, so $\mathbb{X}\preceq\mathbb{Y}$ gives $\mathbb{X}^{\flat}\preceq\mathbb{Y}^{\flat}$, and it does not increase the norm, so $\lVert\mathbb{X}^{\flat}\rVert\le\lVert\mathbb{X}\rVert\le6\alpha$ and $\lVert\mathbb{Y}^{\flat}\rVert\le6\alpha$ by claim 2 of Elementary Order Arithmetic in an Ordered Field. These are exactly the four requirements of The Second-Order Structure Condition at Optimally Coupled Pairs on the Lift of the Wasserstein Space (i), so $(\mathbb{X}^{\flat},\mathbb{Y}^{\flat})$ is admitted at $\alpha$. The two identities $(\mathbb{X}+tN)^{\flat}=\mathbb{X}^{\flat}$ and $(\mathbb{Y}+tN)^{\flat}=\mathbb{Y}^{\flat}$ are Compression of a Bounded Symmetric Bilinear Form along an Orthonormal Tuple (iii).

Proof of clause 5. Throughout, $\lVert\mathbb{X}_{n}\rVert\le6\alpha$ and $\lVert\mathbb{Y}_{n}\rVert\le6\alpha$ for every $n$, by The Second-Order Structure Condition at Optimally Coupled Pairs on the Lift of the Wasserstein Space (i).

(a) By Limits and Bounded Sequences of Symmetric Real Matrices (i), read with $d$ in place of the dimension written $n$ there and with $R=6\alpha$, there are a strictly increasing $\varphi_{1}:\mathbb{N}\to\mathbb{N}$ and $\mathbb{X}\in\mathcal{S}(d)$ such that the sequence whose $k$-th term is $\mathbb{X}_{\varphi_{1}(k)}$ converges to $\mathbb{X}$. The sequence whose $k$-th term is $\mathbb{Y}_{\varphi_{1}(k)}$ lies in $\mathcal{S}(d)$ and is bounded in norm by $6\alpha$, so the same clause provides a strictly increasing $\varphi_{2}:\mathbb{N}\to\mathbb{N}$ and $\mathbb{Y}\in\mathcal{S}(d)$ such that the sequence whose $k$-th term is $\mathbb{Y}_{\varphi_{1}(\varphi_{2}(k))}$ converges to $\mathbb{Y}$. Put $\varphi=\varphi_{1}\circ\varphi_{2}$, which is strictly increasing and indexes a subsequence of the original sequences by A Subsequence of a Subsequence is a Subsequence; the sequence whose $k$-th term is $\mathbb{X}_{\varphi(k)}$ is a subsequence of a sequence converging to $\mathbb{X}$ and therefore converges to $\mathbb{X}$ by A Subsequence of a Convergent Sequence Has the Same Limit.

(b) Let $z,w\in\mathbb{R}^{d}$. By Limits and Bounded Sequences of Symmetric Real Matrices (ii) the sequences whose $n$-th terms are $z\cdot(\mathbb{X}_{n}z)$ and $w\cdot(\mathbb{Y}_{n}w)$ converge to $z\cdot(\mathbb{X}z)$ and to $w\cdot(\mathbb{Y}w)$, so by Arithmetic of Limits of Real Sequences the sequence of their differences converges to $z\cdot(\mathbb{X}z)-w\cdot(\mathbb{Y}w)$. Every term of that sequence lies between $-3\alpha(\lVert z\rVert^{2}+\lVert w\rVert^{2})$ and $3\alpha\lVert z-w\rVert^{2}$, so by Order Properties of Limits of Real Sequences, applied to the constant sequences with these two values, so does its limit. Next, $\mathbb{X}_{n}\preceq\mathbb{Y}_{n}$ for every $n$ gives $\mathbb{X}\preceq\mathbb{Y}$ by Limits and Bounded Sequences of Symmetric Real Matrices (iii). Finally let $\xi\in\mathbb{R}^{d}$ satisfy $\lVert\xi\rVert\le1$. By claim 2 of Properties of the Norm of a Symmetric Real Matrix and claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field,

$$\bigl|\xi\cdot(\mathbb{X}_{n}\xi)\bigr|\le\lVert\mathbb{X}_{n}\rVert\,\lVert\xi\rVert^{2}\le6\alpha\qquad(n\in\mathbb{N}).$$

The sequence whose $n$-th term is $\xi\cdot(\mathbb{X}_{n}\xi)$ converges to $\xi\cdot(\mathbb{X}\xi)$, and the sequence of absolute values converges to $|\xi\cdot(\mathbb{X}\xi)|$ because $\bigl||r|-|r'|\bigr|\le|r-r'|$ for real $r,r'$ by claim 7 of Properties of the Absolute Value in an Ordered Field; hence $|\xi\cdot(\mathbb{X}\xi)|\le6\alpha$ by Order Properties of Limits of Real Sequences. Thus $6\alpha$ is an upper bound of the set whose supremum is $\lVert\mathbb{X}\rVert$ in Norm of a Symmetric Real Matrix, so $\lVert\mathbb{X}\rVert\le6\alpha$; the same argument gives $\lVert\mathbb{Y}\rVert\le6\alpha$. The four requirements of The Second-Order Structure Condition at Optimally Coupled Pairs on the Lift of the Wasserstein Space (i) hold, so $(\mathbb{X},\mathbb{Y})$ is admitted at $\alpha$.