Proof of Plan Jets of a Series of Squared Wasserstein Distances at Optimal Couplings of Bounded Noncommutative Laws

Each result cited is universally quantified over the data in its own statement. Throughout, $j$ ranges over $[d]$ and $k$ over $\mathbb{N}$. Write $W_{k}=W_{2}(\mu,\nu_{k})$, $C_{n}=\sum_{k=1}^{n}c_{k}$, $C=\sum_{k=1}^{\infty}c_{k}$ and $\rho=2R\sqrt{d}$, with $\sqrt{d}$ the nonnegative square root of $d$. By Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §dominates, $0\le C_{n}\le C$ for every $n\in\mathbb{N}$.

Step 1 (The couplings). Fix $k$. By Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §coupling, $(s,t^{k})$ is a self-adjoint $2d$-tuple in $M$, $\gamma_{k}=\lambda_{(s,t^{k})}\in\Pi(\lambda_{s},\lambda_{t^{k}})$ and $I(\gamma_{k})=\sum_{j=1}^{d}\lVert s_{j}\Omega-t^{k}_{j}\Omega\rVert^{2}$. The vacuum tuples $s\Omega$ and $t^{k}\Omega$ are $L^{2}$ $d$-tuples by Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §bounded, so this sum is $\lVert s\Omega-t^{k}\Omega\rVert_{2}^{2}$ by Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §tuples; since $\gamma_{k}$ is optimal, $I(\gamma_{k})=W_{k}^{2}$ by The Noncommutative Quadratic Wasserstein Distance and Optimal Couplings §optimal. Both $\lVert s\Omega-t^{k}\Omega\rVert_{2}$ and $W_{k}$ are nonnegative, so claim 3 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives

$$\lVert s\Omega-t^{k}\Omega\rVert_{2}=W_{k}.\tag{1}$$

As $\gamma_{k}\in\Pi(\mu,\nu_{k})$ and $\lambda_{(s,t^{k})}\in\Pi(\lambda_{s},\lambda_{t^{k}})$, Couplings of Two Noncommutative Laws and Their Quadratic Cost §coupling gives $\lambda_{t^{k}}=\gamma_{k}\circ\iota^{2}=\nu_{k}$, hence by Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §bounded

$$\mathrm{law}(t^{k}\Omega)=\kappa_{d}(\nu_{k}).\tag{2}$$

Finally, as $\mu,\nu_{k}\in\Sigma_{d,R}$, The Noncommutative Laws with a Norm Bound Form a Complete Bounded Metric Space with Interpolation Points §bounded gives $W_{k}^{2}\le4dR^{2}=\rho^{2}$, the equality because $\rho=2R\sqrt{d}$ and $(\sqrt{d})^{2}=d$. As $W_{k}$ and $\rho$ are nonnegative, $0\le W_{k}\le\rho$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field.

Step 2 (Claim 1). By Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §tail-bound, applied to the nonnegative $c_{k}$ and to $w_{k}=W_{k}\in[0,\rho]$, the series $\sum_{k=1}^{\infty}c_{k}W_{k}$ converges; write $S_{n}=\sum_{k=1}^{n}c_{k}W_{k}$ and $S=\sum_{k=1}^{\infty}c_{k}W_{k}$. Its terms are nonnegative, so $0\le S_{n}\le S$ for every $n$ by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §dominates, and $S_{n}\to S$ by Series of Real Numbers §convergent.

For $n\in\mathbb{N}$ let $B^{n}_{j}=\sum_{k=1}^{n}2c_{k}(s_{j}-t^{k}_{j})$, a finite sum in the complex vector space $\mathcal{L}(H)$. By Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §operations, $B^{n}_{j}\in\mathcal{L}(H)$ and $\lVert s_{j}-t^{k}_{j}\rVert_{\mathrm{op}}\le\lVert s_{j}\rVert_{\mathrm{op}}+\lVert t^{k}_{j}\rVert_{\mathrm{op}}\le2R$.

Let $n<m$, and write $\sum_{k=n+1}^{m}a_{k}$ for $\sum_{i=1}^{m-n}a_{n+i}$, as in Splitting a Finite Sum at an Index. By that lemma in $\mathbb{R}$, $C_{m}-C_{n}=\sum_{k=n+1}^{m}c_{k}$, which is nonnegative by claim 5 of Properties of Finite Sums. A finite sum of vectors splits at the index $n$ in the same way, by induction on $m$ from the recursion in claim 1 of Properties of Finite Sums of Vectors; so $B^{m}_{j}-B^{n}_{j}=\sum_{k=n+1}^{m}2c_{k}(s_{j}-t^{k}_{j})$ in $\mathcal{L}(H)$. Let $\xi\in H$. Sums and multiples in $\mathcal{L}(H)$ are taken pointwise, so $T\mapsto T\xi$ is a linear map from $\mathcal{L}(H)$ to $H$, and claim 4 of Properties of Finite Sums of Vectors gives $(B^{m}_{j}-B^{n}_{j})\xi=\sum_{k=n+1}^{m}2c_{k}(s_{j}-t^{k}_{j})\xi$. The norm of $H$ is a norm by claim 2 of The Induced Norm is a Norm, and Induces a Metric. So Triangle Inequality for Finite Sums of Vectors in $H$, absolute homogeneity with $|2c_{k}|=2c_{k}$, the bound $\lVert(s_{j}-t^{k}_{j})\xi\rVert\le\lVert s_{j}-t^{k}_{j}\rVert_{\mathrm{op}}\lVert\xi\rVert\le2R\lVert\xi\rVert$ (Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §least-bound), claim 1 of Comparison and Absolute Value Bounds for Finite Sums of Real Numbers and claim 3 of Properties of Finite Sums give

$$\lVert(B^{m}_{j}-B^{n}_{j})\xi\rVert=\Bigl\lVert\sum_{k=n+1}^{m}2c_{k}(s_{j}-t^{k}_{j})\xi\Bigr\rVert\le\sum_{k=n+1}^{m}2c_{k}\lVert(s_{j}-t^{k}_{j})\xi\rVert\le4R\,(C_{m}-C_{n})\lVert\xi\rVert.$$

Thus the nonnegative real $4R\,(C_{m}-C_{n})$ is a bound for $B^{m}_{j}-B^{n}_{j}$ in the sense of Bounded Linear Maps between Complex Normed Spaces and the Operator Norm §bounded. By the least-bound clause Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §least-bound,

$$\lVert B^{m}_{j}-B^{n}_{j}\rVert_{\mathrm{op}}\le4R\,(C_{m}-C_{n})\qquad(n<m).$$

Given a real $\varepsilon>0$, Elementary Properties of Series of Real Numbers §cauchy for the convergent series $\sum_{k}c_{k}$ gives $N\in\mathbb{N}$ with $|C_{m}-C_{n}|<\varepsilon/(4R)$ for all $m,n\ge N$, hence $\lVert B^{m}_{j}-B^{n}_{j}\rVert_{\mathrm{op}}\le4R\,|C_{m}-C_{n}|<\varepsilon$ for all $m>n\ge N$. The case $m<n$ follows, since $B^{m}_{j}-B^{n}_{j}=(-1)(B^{n}_{j}-B^{m}_{j})$ and $\lVert(-1)T\rVert_{\mathrm{op}}=\lVert T\rVert_{\mathrm{op}}$ by Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §operations. The case $m=n$ is trivial: the difference is the zero map, for which $0$ is a bound, so its operator norm is at most $0<\varepsilon$ by Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §least-bound. Hence $\lVert B^{m}_{j}-B^{n}_{j}\rVert_{\mathrm{op}}<\varepsilon$ for all $m,n\ge N$. By Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §complete (with $V=W=H$) there is $P_{j}\in\mathcal{L}(H)$ with $\lVert B^{n}_{j}-P_{j}\rVert_{\mathrm{op}}\to0$, that is, the partial sums converge to $P_{j}$ in operator norm, and

$$B^{n}_{j}\xi\to P_{j}\xi\ \text{in }H\text{ as }n\to\infty,\ \text{for every }\xi\in H.\tag{3}$$

Membership. By Tracial W*-Probability Spaces §space, $M=M''$ is the commutant of $M'$. As $s_{j},t^{k}_{j}\in M$, Basic Properties of Commutants: Unital Algebras Closed under Weak Limits, Order Reversal, the Triple Commutant, and Conjugation §algebra (for the set $M'$) gives $B^{n}_{j}\in M$, and Basic Properties of Commutants: Unital Algebras Closed under Weak Limits, Order Reversal, the Triple Commutant, and Conjugation §closed then gives $P_{j}\in M$, the $B^{n}_{j}$ converging to $P_{j}$ in operator norm; also $-P_{j}=(-1)P_{j}\in M$ by Basic Properties of Commutants: Unital Algebras Closed under Weak Limits, Order Reversal, the Triple Commutant, and Conjugation §algebra.

Self-adjointness. Let $u,v\in H$. Each $s_{j}$ and $t^{k}_{j}$ is self-adjoint in the sense of Self-Adjoint Operator and each $2c_{k}$ is real, so $\langle B^{n}_{j}u,v\rangle=\langle u,B^{n}_{j}v\rangle$ by sesquilinearity of the inner product. Hence, by claim 1 of The Induced Norm is a Norm, and Induces a Metric, for every $n$,

$$0\le\bigl|\langle P_{j}u,v\rangle-\langle u,P_{j}v\rangle\bigr|\le\lVert P_{j}u-B^{n}_{j}u\rVert\,\lVert v\rVert+\lVert u\rVert\,\lVert B^{n}_{j}v-P_{j}v\rVert,$$

and the right side tends to $0$ by (3) and Arithmetic of Limits of Real Sequences; claim 1 of Order Properties of Limits of Real Sequences gives $\langle P_{j}u,v\rangle=\langle u,P_{j}v\rangle$. So $P_{j}$, and likewise $-P_{j}$, is self-adjoint, and $P$ and $-P$ are self-adjoint $d$-tuples in $M$ by Self-Adjoint Tuples in a Tracial W*-Probability Space and Their Laws §tuple.

Estimates. By claim 4 of Properties of Finite Sums of Vectors, for the linear map $T\mapsto T\Omega$ from $\mathcal{L}(H)$ to $H$, $B^{n}_{j}\Omega=\sum_{k=1}^{n}2c_{k}(s_{j}\Omega-t^{k}_{j}\Omega)$. By the same claim for the linear coordinate maps from $H^{d}$ to $H$, the $d$-tuple $B^{n}\Omega=(B^{n}_{1}\Omega,\dots,B^{n}_{d}\Omega)$ is the $L^{2}$ $d$-tuple $\sum_{k=1}^{n}2c_{k}(s\Omega-t^{k}\Omega)$, sums and real multiples being those of $H^{d}$ (Sums, Real Multiples and the Pairing of Square-Integrable Tuples in a Tracial W*-Probability Space §operations). Now $\lVert\cdot\rVert_{2}$ is the norm of $H^{d}$ (Sums, Real Multiples and the Pairing of Square-Integrable Tuples in a Tracial W*-Probability Space §pairing), a norm by claim 2 of The Induced Norm is a Norm, and Induces a Metric. Take $n<m$. The splitting of finite sums of vectors used above, now in $H^{d}$, gives $B^{m}\Omega-B^{n}\Omega=\sum_{k=n+1}^{m}2c_{k}(s\Omega-t^{k}\Omega)$ in $H^{d}$. Then Triangle Inequality for Finite Sums of Vectors in $H^{d}$, absolute homogeneity with $2c_{k}\ge0$, (1), claim 3 of Properties of Finite Sums and Splitting a Finite Sum at an Index in $\mathbb{R}$ give

$$\lVert B^{m}\Omega-B^{n}\Omega\rVert_{2}\le\sum_{k=n+1}^{m}2c_{k}W_{k}=2(S_{m}-S_{n})\le2(S-S_{n}),\qquad\lVert B^{m}\Omega\rVert_{2}\le2S_{m}\le2S.$$

Let $e_{m}=\sum_{j=1}^{d}\lVert P_{j}\Omega-B^{m}_{j}\Omega\rVert$; then $e_{m}\to0$ by (3) and Arithmetic of Limits of Real Sequences, and $\lVert P\Omega-B^{m}\Omega\rVert_{2}^{2}=\sum_{j=1}^{d}\lVert P_{j}\Omega-B^{m}_{j}\Omega\rVert^{2}\le e_{m}^{2}$; the inequality holds because a sum of squares of nonnegative reals is at most the square of their sum: with $a_{j}=\lVert P_{j}\Omega-B^{m}_{j}\Omega\rVert\ge0$, claim 6 of Properties of Finite Sums gives $a_{j}\le e_{m}$, so $a_{j}^{2}\le a_{j}e_{m}$, and summing over $j$ (claim 1 of Comparison and Absolute Value Bounds for Finite Sums of Real Numbers and claim 3 of Properties of Finite Sums) gives $\sum_{j=1}^{d}a_{j}^{2}\le e_{m}\sum_{j=1}^{d}a_{j}=e_{m}^{2}$. Hence $\lVert P\Omega-B^{m}\Omega\rVert_{2}\le e_{m}$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. By the triangle inequality in $H^{d}$, for all $m>n$,

$$\Bigl\lVert P\Omega-\sum_{k=1}^{n}2c_{k}(s\Omega-t^{k}\Omega)\Bigr\rVert_{2}\le e_{m}+2(S-S_{n}),\qquad\lVert P\Omega\rVert_{2}\le e_{m}+2S.$$

Letting $m\to\infty$ (claim 1 of Order Properties of Limits of Real Sequences, applied to the sequences indexed by $m\ge n+1$) gives both inequalities of claim 1.

Step 3 (Claim 2). By Step 2 and Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §coupling, $(s,P)$ and $(s,-P)$ are self-adjoint $2d$-tuples in $M$ with $\pi\in\Pi(\lambda_{s},\lambda_{P})$ and $\pi^{-}\in\Pi(\lambda_{s},\lambda_{-P})$; by Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §law, $\pi,\pi^{-}\in\Sigma_{2d}$ (with $R'$ the larger of $R$ and $\max_{j}\lVert P_{j}\rVert_{\mathrm{op}}$, finite as $P_{j}\in\mathcal{L}(H)$; $R'$ bounds the operator norms of all components of $(s,P)$ and $(s,-P)$, as $\lVert-P_{j}\rVert_{\mathrm{op}}=\lVert P_{j}\rVert_{\mathrm{op}}$ by Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §operations), and by Couplings of Two Noncommutative Laws and Their Quadratic Cost §coupling, $\pi\circ\iota^{1}=\pi^{-}\circ\iota^{1}=\lambda_{s}=\mu$. So $\pi$ and $\pi^{-}$ are bounded plans at $\mu$ by Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §plans. By Calculus of Square-Integrable Noncommutative Laws: Agreement on Bounded Laws, Lipschitz Estimates, Functoriality of Push-Forwards, Moment Formulas, Positivity, the Cost and the Diagonal Coupling §bounded for the datum $\mathrm{pr}^{1}$ and Affine Substitutions of Noncommutative Laws: Self-Adjointness, Composition, Moment Formulas and Positivity, and the Coordinate Data §coordinate,

$$\mathrm{pr}^{1}_{\#}\kappa_{2d}(\pi)=\kappa_{d}(\pi\circ\sigma_{\mathrm{pr}^{1}})=\kappa_{d}(\pi\circ\iota^{1})=\kappa_{d}(\mu),$$

and likewise for $\pi^{-}$; so $\kappa_{2d}(\pi)$ and $\kappa_{2d}(\pi^{-})$ are plans at $\kappa_{d}(\mu)$ by Plans at a Square-Integrable Noncommutative Law §plan. By Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §moments for $(s,P)$ and $(s,-P)$ at the index $d+j$, $\pi(x_{d+j}x_{d+j})=\langle P_{j}\Omega,P_{j}\Omega\rangle$ and $\pi^{-}(x_{d+j}x_{d+j})=\langle-P_{j}\Omega,-P_{j}\Omega\rangle=\lVert P_{j}\Omega\rVert^{2}$; summing over $j$, both momentum sums equal $\lVert P\Omega\rVert_{2}^{2}$ (Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §tuples). As $|\pi|_{\mathrm{mom}}$, $|\pi^{-}|_{\mathrm{mom}}$ and $\lVert P\Omega\rVert_{2}$ are nonnegative, claim 3 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $|\pi|_{\mathrm{mom}}=|\pi^{-}|_{\mathrm{mom}}=\lVert P\Omega\rVert_{2}$.

Step 4 (Tools for $L^{2}$ tuples). (E) For $L^{2}$ $d$-tuples $U,V$ of a tracial W*-probability space, $\langle U,V\rangle_{2}$ is the inner product of $H^{d}$ and is real (Sums, Real Multiples and the Pairing of Square-Integrable Tuples in a Tracial W*-Probability Space §pairing), so $\langle V,U\rangle_{2}=\overline{\langle U,V\rangle_{2}}=\langle U,V\rangle_{2}$, and

$$\lVert U+V\rVert_{2}^{2}=\lVert U\rVert_{2}^{2}+2\langle U,V\rangle_{2}+\lVert V\rVert_{2}^{2},\qquad|\langle U,V\rangle_{2}|\le\lVert U\rVert_{2}\lVert V\rVert_{2},$$

the second by claim 1 of The Induced Norm is a Norm, and Induces a Metric in $H^{d}$.

(S) For $q\in\mathbb{N}$ and $a\in[q]$ let $G^{q,a}=(E^{q,a},0)$ be the affine datum from $qd$ to $d$ variables with $E^{q,a}_{il}=1$ if $l=(a-1)d+i$ and $E^{q,a}_{il}=0$ otherwise. For $L^{2}$ $d$-tuples $U^{1},\dots,U^{q}$ write $(U^{1},\dots,U^{q})=((\dots(U^{1},U^{2}),\dots),U^{q})$ for the $L^{2}$ $qd$-tuple formed by iterated pairs of Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §operations; its components are those of $U^{1},\dots,U^{q}$ in this order. The same convention applies to tuples of operators: for self-adjoint $d$-tuples $u^{1},\dots,u^{q}$ in $M$, $(u^{1},\dots,u^{q})$ is the $qd$-tuple listing their components in order, a self-adjoint $qd$-tuple in $M$ by Self-Adjoint Tuples in a Tracial W*-Probability Space and Their Laws §tuple, and its vacuum tuple is $(u^{1}\Omega,\dots,u^{q}\Omega)$. By Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §operations, $G^{q,a}(U^{1},\dots,U^{q})=U^{a}$, so $\mathrm{law}(U^{a})=G^{q,a}_{\#}\mathrm{law}(U^{1},\dots,U^{q})$ by Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §push-forward.

(G) Let $U^{1},\dots,U^{q}$ be $L^{2}$ $d$-tuples of one tracial W*-probability space and $V^{1},\dots,V^{q}$ $L^{2}$ $d$-tuples of another, with $\mathrm{law}(U^{1},\dots,U^{q})=\mathrm{law}(V^{1},\dots,V^{q})$. By Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §moments, $\langle U^{a}_{i},U^{b}_{i'}\rangle$ and $\langle V^{a}_{i},V^{b}_{i'}\rangle$ both equal the quadratic moment of this law at the indices $(a-1)d+i$ and $(b-1)d+i'$, for all $a,b\in[q]$ and $i,i'\in[d]$. Hence, for real $\alpha_{1},\dots,\alpha_{q},\beta_{1},\dots,\beta_{q}$, sesquilinearity with real coefficients gives

$$\Bigl\langle\sum_{a=1}^{q}\alpha_{a}U^{a},\sum_{b=1}^{q}\beta_{b}U^{b}\Bigr\rangle_{2}=\sum_{a=1}^{q}\sum_{b=1}^{q}\alpha_{a}\beta_{b}\sum_{i=1}^{d}\langle U^{a}_{i},U^{b}_{i}\rangle=\Bigl\langle\sum_{a=1}^{q}\alpha_{a}V^{a},\sum_{b=1}^{q}\beta_{b}V^{b}\Bigr\rangle_{2};$$

in particular corresponding real combinations have equal $\lVert\cdot\rVert_{2}$ (nonnegative numbers with equal squares), and $\mathrm{law}(U^{a})=\mathrm{law}(V^{a})$ for every $a$ by (S).

Step 5 (Claim 3). Convergence. Let $\lambda\in\Sigma^{2}_{d}$ and $\alpha=\widehat{W}_{2}(\lambda,\kappa_{d}(\mu))$. As $\Sigma^{2}_{d}$ is the metric completion of $(\Sigma_{d},W_{2})$ (Square-Integrable Noncommutative Laws: the Wasserstein Completion of the Laws, Affine Push-Forwards, Moments, Couplings and Cost §laws), The Metric Completion is a Complete Metric Space with a Dense Isometric Copy of the Space, and Maps Preserving Cauchy Sequences Extend to It §metric and The Metric Completion is a Complete Metric Space with a Dense Isometric Copy of the Space, and Maps Preserving Cauchy Sequences Extend to It §isometry give

$$0\le\widehat{W}_{2}(\lambda,\kappa_{d}(\nu_{k}))\le\alpha+\widehat{W}_{2}(\kappa_{d}(\mu),\kappa_{d}(\nu_{k}))=\alpha+W_{k}\le\alpha+\rho,$$

so $0\le\widehat{W}_{2}(\lambda,\kappa_{d}(\nu_{k}))^{2}\le(\alpha+\rho)^{2}$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, and Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §tail-bound (weights $c_{k}$, bound $(\alpha+\rho)^{2}$) shows that $\sum_{k}c_{k}\widehat{W}_{2}(\lambda,\kappa_{d}(\nu_{k}))^{2}$ converges. By The Metric Completion is a Complete Metric Space with a Dense Isometric Copy of the Space, and Maps Preserving Cauchy Sequences Extend to It §isometry, $\varphi(\kappa_{d}(\mu))=\sum_{k=1}^{\infty}c_{k}W_{k}^{2}$, and $\sum_{k=1}^{n}c_{k}W_{k}^{2}\le\varphi(\kappa_{d}(\mu))$ for every $n$ by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §dominates.

The inequality. Let $\eta>0$ be real and $r=\eta/(1+C)>0$. Let $(H_{1},M_{1},\Omega_{1})$ be a tracial W*-probability space and $X,Q,X'$ $L^{2}$ $d$-tuples of it with $\mathrm{law}(X,Q)=\kappa_{2d}(\pi)$ and $\lVert X'-X\rVert_{2}<r$; put $\Delta=X'-X$. We show

$$\varphi_{M_{1}}(X')\le\varphi(\kappa_{d}(\mu))+\langle Q,X'-X\rangle_{2}+(0+\eta)\lVert X'-X\rVert_{2}.\tag{4}$$

Fix $n\in\mathbb{N}$ and let $T^{n}=(t^{1}\Omega,\dots,t^{n}\Omega)$ (concatenation as in (S)); it is the vacuum tuple of the self-adjoint $nd$-tuple $(t^{1},\dots,t^{n})$ in $M$, hence an $L^{2}$ $nd$-tuple of $(H,M,\Omega)$ by Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §bounded. Likewise $(s,P)\Omega=(s\Omega,P\Omega)$ is the vacuum tuple of the self-adjoint $2d$-tuple $(s,P)$ of Step 3, hence an $L^{2}$ $2d$-tuple, and $((s,P)\Omega,T^{n})$ is the vacuum tuple of the self-adjoint $(2d+nd)$-tuple $(s,P,t^{1},\dots,t^{n})$. Put $\pi_{1}=\mathrm{law}(X,Q,X')\in\Sigma^{2}_{2d+d}$ and $\pi_{2}=\mathrm{law}((s,P)\Omega,T^{n})\in\Sigma^{2}_{2d+nd}$. With the data $F^{l}$ of Gluing Two Square-Integrable Noncommutative Laws along a Common Marginal, taking $2d$ in place of the theorem's $k$, Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §operations gives $F^{d}(X,Q,X')=(X,Q)$ and $F^{nd}((s,P)\Omega,T^{n})=(s,P)\Omega$, so by Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §push-forward, Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §bounded and Step 3,

$$F^{d}_{\#}\pi_{1}=\mathrm{law}(X,Q)=\kappa_{2d}(\pi)=\kappa_{2d}(\lambda_{(s,P)})=\mathrm{law}((s,P)\Omega)=F^{nd}_{\#}\pi_{2}.$$

By Gluing Two Square-Integrable Noncommutative Laws along a Common Marginal §glue (with $2d$, $d$ and $nd$ in place of the theorem's $k$, $m$ and $n$) there are a tracial W*-probability space $(K,N,\Psi)$, an $L^{2}$ $2d$-tuple $(\widetilde{X},\widetilde{Q})$, an $L^{2}$ $d$-tuple $\widetilde{X}'$ and an $L^{2}$ $nd$-tuple $(\widetilde{Z}^{1},\dots,\widetilde{Z}^{n})$ of it. Here $\widetilde{X}$ and $\widetilde{Q}$ are the $d$-tuples of the first $d$ and the last $d$ components of the $2d$-tuple, and $\widetilde{Z}^{1},\dots,\widetilde{Z}^{n}$ are the consecutive blocks of $d$ components of the $nd$-tuple; each is an $L^{2}$ $d$-tuple of $(K,N,\Psi)$ by Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §tuples, its components being components of an $L^{2}$ tuple. These tuples satisfy

$$\mathrm{law}(\widetilde{X},\widetilde{Q},\widetilde{X}')=\mathrm{law}(X,Q,X'),\qquad\mathrm{law}(\widetilde{X},\widetilde{Q},\widetilde{Z}^{1},\dots,\widetilde{Z}^{n})=\mathrm{law}(s\Omega,P\Omega,t^{1}\Omega,\dots,t^{n}\Omega).$$

Put $\widetilde{\Delta}=\widetilde{X}'-\widetilde{X}$ and $\widetilde{E}^{n}=\widetilde{Q}-\sum_{k=1}^{n}2c_{k}(\widetilde{X}-\widetilde{Z}^{k})$. By (G) for the first equality, $\lVert\widetilde{\Delta}\rVert_{2}=\lVert\Delta\rVert_{2}$, $\langle\widetilde{\Delta},\widetilde{Q}\rangle_{2}=\langle\Delta,Q\rangle_{2}$ and $\mathrm{law}(\widetilde{X}')=\mathrm{law}(X')$. By (G) for the second equality, (1), (2) and claim 1, for $k\le n$,

$$\lVert\widetilde{X}-\widetilde{Z}^{k}\rVert_{2}=\lVert s\Omega-t^{k}\Omega\rVert_{2}=W_{k},\qquad\mathrm{law}(\widetilde{Z}^{k})=\mathrm{law}(t^{k}\Omega)=\kappa_{d}(\nu_{k}),\qquad\lVert\widetilde{E}^{n}\rVert_{2}=\Bigl\lVert P\Omega-\sum_{k=1}^{n}2c_{k}(s\Omega-t^{k}\Omega)\Bigr\rVert_{2}\le2(S-S_{n}).$$

For $k\le n$, Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §lipschitz in $(K,N,\Psi)$ gives $0\le\widehat{W}_{2}(\mathrm{law}(X'),\kappa_{d}(\nu_{k}))=\widehat{W}_{2}(\mathrm{law}(\widetilde{X}'),\mathrm{law}(\widetilde{Z}^{k}))\le\lVert\widetilde{X}'-\widetilde{Z}^{k}\rVert_{2}$; as $\widetilde{X}'-\widetilde{Z}^{k}=\widetilde{\Delta}+(\widetilde{X}-\widetilde{Z}^{k})$, claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field and (E) give

$$\widehat{W}_{2}(\mathrm{law}(X'),\kappa_{d}(\nu_{k}))^{2}\le\lVert\Delta\rVert_{2}^{2}+2\langle\widetilde{\Delta},\widetilde{X}-\widetilde{Z}^{k}\rangle_{2}+W_{k}^{2}.$$

Multiply by $c_{k}\ge0$ and sum over $k\le n$ (claim 1 of Comparison and Absolute Value Bounds for Finite Sums of Real Numbers). The sum of the right sides splits by claims 2 and 3 of Properties of Finite Sums into $C_{n}\lVert\Delta\rVert_{2}^{2}$, $\sum_{k=1}^{n}2c_{k}\langle\widetilde{\Delta},\widetilde{X}-\widetilde{Z}^{k}\rangle_{2}$ and $\sum_{k=1}^{n}c_{k}W_{k}^{2}$. The pairing is the inner product of $H^{d}$ (Sums, Real Multiples and the Pairing of Square-Integrable Tuples in a Tracial W*-Probability Space §pairing) and the $2c_{k}$ are real, so claim 6 of Properties of Finite Sums of Vectors in $H^{d}$ gives $\sum_{k=1}^{n}2c_{k}\langle\widetilde{\Delta},\widetilde{X}-\widetilde{Z}^{k}\rangle_{2}=\langle\widetilde{\Delta},\sum_{k=1}^{n}2c_{k}(\widetilde{X}-\widetilde{Z}^{k})\rangle_{2}$. Here $\sum_{k=1}^{n}2c_{k}(\widetilde{X}-\widetilde{Z}^{k})=\widetilde{Q}-\widetilde{E}^{n}$ by the definition of $\widetilde{E}^{n}$, and the inner product is additive in its second argument. So the partial sum $\Phi_{n}=\sum_{k=1}^{n}c_{k}\widehat{W}_{2}(\mathrm{law}(X'),\kappa_{d}(\nu_{k}))^{2}$ satisfies

$$\Phi_{n}\le C_{n}\lVert\Delta\rVert_{2}^{2}+\langle\widetilde{\Delta},\widetilde{Q}\rangle_{2}-\langle\widetilde{\Delta},\widetilde{E}^{n}\rangle_{2}+\sum_{k=1}^{n}c_{k}W_{k}^{2}\le\varphi(\kappa_{d}(\mu))+\langle Q,\Delta\rangle_{2}+C\lVert\Delta\rVert_{2}^{2}+2\lVert\Delta\rVert_{2}(S-S_{n}),$$

using $C_{n}\le C$, the bound on $\sum_{k\le n}c_{k}W_{k}^{2}$ above, $\langle\widetilde{\Delta},\widetilde{Q}\rangle_{2}=\langle\Delta,Q\rangle_{2}=\langle Q,\Delta\rangle_{2}$ and (E) with $\lVert\widetilde{E}^{n}\rVert_{2}\le2(S-S_{n})$. As $n\to\infty$, $\Phi_{n}\to\varphi(\mathrm{law}(X'))$ by the definition of $\varphi$ and Series of Real Numbers §convergent, while the right side tends to $\varphi(\kappa_{d}(\mu))+\langle Q,\Delta\rangle_{2}+C\lVert\Delta\rVert_{2}^{2}$ because $S_{n}\to S$ (Arithmetic of Limits of Real Sequences); claim 1 of Order Properties of Limits of Real Sequences gives $\varphi(\mathrm{law}(X'))\le\varphi(\kappa_{d}(\mu))+\langle Q,\Delta\rangle_{2}+C\lVert\Delta\rVert_{2}^{2}$. As $0\le\lVert\Delta\rVert_{2}<r$, $C\lVert\Delta\rVert_{2}^{2}\le Cr\lVert\Delta\rVert_{2}\le\eta\lVert\Delta\rVert_{2}$, since $Cr=C\eta/(1+C)\le\eta$. Finally $\varphi_{M_{1}}(X')=\varphi(\mathrm{law}(X'))$ by Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §lifts and Lifts of Functions on Square-Integrable Noncommutative Laws to Square-Integrable Tuples §lift, which proves (4). Since $\kappa_{2d}(\pi)$ is a plan at $\kappa_{d}(\mu)$ (Step 3), it is a plan superdifferential of $\varphi$ at $\kappa_{d}(\mu)$ with slack $0$ by Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §super, and $\kappa_{2d}(\pi)\in J^{+}\varphi(\kappa_{d}(\mu))$ by Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §superjet.

Step 6 (Claim 4). Let $\eta>0$, $r=\eta/(1+C)$, and let $(H_{1},M_{1},\Omega_{1})$ be a tracial W*-probability space and $X,Q,X'$ $L^{2}$ $d$-tuples of it with $\mathrm{law}(X,Q)=\kappa_{2d}(\pi^{-})$ and $\lVert X'-X\rVert_{2}<r$. Let $\mathcal{N}=(A,0)$ be the affine datum from $2d$ to $2d$ variables with $A_{jl}=1$ if $l=j$, $A_{d+j,l}=-1$ if $l=d+j$, and all other entries $0$. By Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §operations, $\mathcal{N}(X,Q)=(X,-Q)$ with $-Q=(-1)Q$; by Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §affine for the self-adjoint $2d$-tuple $(s,-P)$, the tuple with components $s_{j}$ and $-(-P_{j})=P_{j}$ is $(s,P)$, so $\pi^{-}\circ\sigma_{\mathcal{N}}=\lambda_{(s,P)}=\pi$. Hence Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §push-forward and Calculus of Square-Integrable Noncommutative Laws: Agreement on Bounded Laws, Lipschitz Estimates, Functoriality of Push-Forwards, Moment Formulas, Positivity, the Cost and the Diagonal Coupling §bounded give

$$\mathrm{law}(X,-Q)=\mathcal{N}_{\#}\kappa_{2d}(\pi^{-})=\kappa_{2d}(\pi^{-}\circ\sigma_{\mathcal{N}})=\kappa_{2d}(\pi).$$

So (4), which Step 5 proved for every $\eta>0$ with this $r$ and for all such data, applied to $X,-Q,X'$, gives $\varphi_{M_{1}}(X')\le\varphi(\kappa_{d}(\mu))-\langle Q,X'-X\rangle_{2}+\eta\lVert X'-X\rVert_{2}$, the pairing being real-linear in its first argument. With the lift $(-\varphi)_{M_{1}}(X')=-\varphi(\mathrm{law}(X'))=-\varphi_{M_{1}}(X')$ of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §lifts and Lifts of Functions on Square-Integrable Noncommutative Laws to Square-Integrable Tuples §lift,

$$(-\varphi)_{M_{1}}(X')\ge(-\varphi)(\kappa_{d}(\mu))+\langle Q,X'-X\rangle_{2}-(0+\eta)\lVert X'-X\rVert_{2}.$$

As $\kappa_{2d}(\pi^{-})$ is a plan at $\kappa_{d}(\mu)$ (Step 3), Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §sub and Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §subjet give $\kappa_{2d}(\pi^{-})\in J^{-}(-\varphi)(\kappa_{d}(\mu))$.

Step 7 (GNS isometries into $H$). Let $m\in\mathbb{N}$, let $u$ be a self-adjoint $m$-tuple in $M$ and $\lambda=\lambda_{u}$, which lies in $\Sigma_{m}$ by Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §law, with $R'$ the larger of $1$ and $\max_{i}\lVert u_{i}\rVert_{\mathrm{op}}$, finite as each $u_{i}\in M$ lies in $\mathcal{L}(H)$; let $h_{\lambda}(p,q)=\lambda(p^{*}q)$ as in The Complex GNS Space of a Tracial State on Noncommutative Polynomials. The map $p\mapsto p(u)\Omega$ from $\mathcal{P}_{m}$ to $H$ is complex-linear, evaluation being linear (Evaluation of Noncommutative Polynomials at a Tuple of Bounded Operators §evaluation). For $p,q\in\mathcal{P}_{m}$, the adjoint $p(u)^{*}$ exists by Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §adjoint, and by Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §adjoint, Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §homomorphism and Self-Adjoint Tuples in a Tracial W*-Probability Space and Their Laws §law,

$$\langle p(u)\Omega,q(u)\Omega\rangle=\langle\Omega,p(u)^{*}q(u)\Omega\rangle=\langle\Omega,(p^{*}q)(u)\Omega\rangle=\lambda(p^{*}q)=h_{\lambda}(p,q).$$

Hence The Complex Hilbert Completion is a Complex Hilbert Space Containing a Dense Isometric Image, and Bounded Complex-Linear Maps Extend to It §extension-linear (with $K=H$ and bound $1$), together with The Complex GNS Space of a Tracial State on Noncommutative Polynomials §gns and The Complex GNS Space of a Tracial State on Noncommutative Polynomials §classes, gives exactly one continuous map $U_{u}:\mathcal{H}_{\lambda}\to H$ with $U_{u}\widehat{p}^{\,\lambda}=p(u)\Omega$ for every $p\in\mathcal{P}_{m}$; moreover $U_{u}\in\mathcal{L}(\mathcal{H}_{\lambda},H)$ and

$$\langle U_{u}\xi,U_{u}\xi'\rangle=\langle\xi,\xi'\rangle_{\mathcal{H}_{\lambda}}\qquad(\xi,\xi'\in\mathcal{H}_{\lambda}).\tag{5}$$

Write $U_{\mu}=U_{s}$, $U_{\pi}=U_{(s,P)}$, $U_{\pi^{-}}=U_{(s,-P)}$ and $U_{k}=U_{(s,t^{k})}$, on $\mathcal{H}_{\mu}$, $\mathcal{H}_{\pi}$, $\mathcal{H}_{\pi^{-}}$ and $\mathcal{H}_{\gamma_{k}}$.

Let $\varpi$ be one of $\pi,\pi^{-},\gamma_{k}$, with tuple $w=(s,y)$ where $y$ is $P$, $-P$ or $t^{k}$. Then $\varpi\circ\iota^{1}=\mu$ (Steps 1 and 3), so Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §isometries gives $V^{1}_{\varpi}\in\mathcal{L}(\mathcal{H}_{\mu},\mathcal{H}_{\varpi})$ with $V^{1}_{\varpi}\widehat{q}^{\,\mu}=\widehat{\iota^{1}(q)}^{\,\varpi}$ for $q\in\mathcal{P}_{d}$. The composite $U_{w}V^{1}_{\varpi}$ lies in $\mathcal{L}(\mathcal{H}_{\mu},H)$ by Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §operations, and is continuous since $\lVert U_{w}V^{1}_{\varpi}(\xi-\xi')\rVert\le\lVert U_{w}V^{1}_{\varpi}\rVert_{\mathrm{op}}\lVert\xi-\xi'\rVert$ by Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §least-bound. As $\iota^{1}=\sigma_{(x_{1},\dots,x_{d})}$ (Couplings of Two Noncommutative Laws and Their Quadratic Cost §marginals), Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §substitution and Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §values give $(\iota^{1}(q))(w)=q(x_{1}(w),\dots,x_{d}(w))=q(s)$, so $U_{w}V^{1}_{\varpi}\widehat{q}^{\,\mu}=q(s)\Omega$ for every $q\in\mathcal{P}_{d}$. By the uniqueness in The Complex Hilbert Completion is a Complex Hilbert Space Containing a Dense Isometric Image, and Bounded Complex-Linear Maps Extend to It §extension-linear for $u=s$,

$$U_{\pi}V^{1}_{\pi}=U_{\pi^{-}}V^{1}_{\pi^{-}}=U_{k}V^{1}_{\gamma_{k}}=U_{\mu}.\tag{6}$$

By Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §values and linearity of evaluation, with $h_{j}=x_{j}-x_{d+j}$ as in Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws,

$$U_{\pi}\widehat{x_{d+j}}^{\,\pi}=P_{j}\Omega,\qquad U_{\pi^{-}}\widehat{x_{d+j}}^{\,\pi^{-}}=-P_{j}\Omega,\qquad U_{k}\widehat{h_{j}}^{\,\gamma_{k}}=s_{j}\Omega-t^{k}_{j}\Omega.\tag{7}$$

Step 8 (Claim 5). Let $\zeta\in\mathcal{H}_{\mu}^{d}$ and $a_{j}=U_{\mu}\zeta_{j}\in H$. By (5), (6) and (7),

$$\bigl\langle V^{1}_{\pi}\zeta_{j},\widehat{x_{d+j}}\bigr\rangle_{\mathcal{H}_{\pi}}=\langle a_{j},P_{j}\Omega\rangle,\qquad\bigl\langle V^{1}_{\pi^{-}}\zeta_{j},\widehat{x_{d+j}}\bigr\rangle_{\mathcal{H}_{\pi^{-}}}=-\langle a_{j},P_{j}\Omega\rangle,\qquad\bigl\langle V^{1}_{\gamma_{k}}\zeta_{j},\widehat{h_{j}}\bigr\rangle_{\mathcal{H}_{\gamma_{k}}}=\langle a_{j},s_{j}\Omega-t^{k}_{j}\Omega\rangle.$$

Taking real parts and summing over $j$, Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §plan-pairing and Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §coupling-pairing give $\mathcal{J}(\zeta,\pi)=\sum_{j=1}^{d}\operatorname{Re}\langle a_{j},P_{j}\Omega\rangle$, $\mathcal{J}(\zeta,\pi^{-})=-\mathcal{J}(\zeta,\pi)$ and $\mathcal{J}^{1}_{\gamma_{k}}(\zeta)=\sum_{j=1}^{d}\operatorname{Re}\langle a_{j},s_{j}\Omega-t^{k}_{j}\Omega\rangle$. The $n$-th partial sum of $\sum_{k}2c_{k}\mathcal{J}^{1}_{\gamma_{k}}(\zeta)$ is $\sum_{j=1}^{d}\operatorname{Re}\langle a_{j},B^{n}_{j}\Omega\rangle$, with $B^{n}_{j}$ as in Step 2. Indeed, by claim 3 of Properties of Finite Sums, and by claim 2 of that lemma with induction on $d$, the finite sums over $k\le n$ and over $j$ may be exchanged. For fixed $j$, $\operatorname{Re}$ is a real-linear map from $\mathbb{C}$ to $\mathbb{R}$ with $\operatorname{Re}(2c_{k}z)=2c_{k}\operatorname{Re}z$ for real $2c_{k}$, so it commutes with finite sums by claim 4 of Properties of Finite Sums of Vectors over $\mathbb{R}$. Claim 6 of the same lemma in $H$ then gives $\sum_{k=1}^{n}2c_{k}\langle a_{j},s_{j}\Omega-t^{k}_{j}\Omega\rangle=\langle a_{j},\sum_{k=1}^{n}2c_{k}(s_{j}\Omega-t^{k}_{j}\Omega)\rangle=\langle a_{j},B^{n}_{j}\Omega\rangle$, the last equality by Step 2. By claim 1 of The Induced Norm is a Norm, and Induces a Metric,

$$\bigl|\operatorname{Re}\langle a_{j},B^{n}_{j}\Omega\rangle-\operatorname{Re}\langle a_{j},P_{j}\Omega\rangle\bigr|\le\bigl|\langle a_{j},B^{n}_{j}\Omega-P_{j}\Omega\rangle\bigr|\le\lVert a_{j}\rVert\,\lVert B^{n}_{j}\Omega-P_{j}\Omega\rVert,$$

which tends to $0$ by (3) and Arithmetic of Limits of Real Sequences; so $\operatorname{Re}\langle a_{j},B^{n}_{j}\Omega\rangle\to\operatorname{Re}\langle a_{j},P_{j}\Omega\rangle$ by claim 3 of Order Properties of Limits of Real Sequences, and summing over $j$ (Arithmetic of Limits of Real Sequences) the partial sums converge to $\mathcal{J}(\zeta,\pi)$. By Series of Real Numbers §convergent the series converges and $\mathcal{J}(\zeta,\pi)=\sum_{k=1}^{\infty}2c_{k}\mathcal{J}^{1}_{\gamma_{k}}(\zeta)$, which completes claim 5.