Proof of The Galerkin Solutions of the Wick-Square Problem Converge to the Renormalised Viscosity Solution for Every Bounded Lipschitz Running Cost

Each result cited below is universally quantified over the data in its own statement. Elementary facts about real numbers (manipulation of inequalities, absolute values, square roots of nonnegative numbers, and that $a\le b+c\eta$ for every real $\eta>0$, with $c\ge0$ fixed, implies $a\le b$) are those of The Real Numbers: Standing Notation and Background. Throughout, $(H,V,A)$ is the Sobolev triple of order $2$ of The Wick-Square Problem on the Sobolev Triple of Order Two: Standing Notation §triple, and we work in the setting of Hilbert Triples: Standing Notation and Background for it. By The Sobolev Triple of Order Two for the Wick-Square Problem: the Domain Is the Sobolev Space of Order -1, Unit Families Are Square-Summable, the Gradient Form and the Riccati Drift §domain, $D(A)=H^{-1}$; by Hilbert Triples: Standing Notation and Background §operator, $D(A)\subseteq V$; and by Hilbert Triples: Standing Notation and Background §triple, $V\subseteq H$ with $|x|_{H}\le|x|_{V}$ for $x\in V$. Thus $H^{-1}\subseteq V\subseteq H$, and the distance $d_{H}$ on $H^{-1}$ and on $V$ is the restriction of the distance of $H$, a metric by Real Hilbert Spaces: Standing Notation and Background §space and Real Hilbert Spaces: Standing Notation and Background §topology. By The Wick-Square Problem on the Torus: Standing Notation §parameters (in force through The Wick-Square Problem on the Sobolev Triple of Order Two: Standing Notation §problem), $\nu$ and $\gamma$ are positive. Write $L=\frac{\ell_{g}}{\gamma}+\frac{2C_{g}}{\gamma}+1$, a nonnegative real number.

Step 1 (Extension from $H^{-1}$ to $H$). We show: if $C,\ell\in\mathbb{R}$ are nonnegative and $f:H^{-1}\to\mathbb{R}$ satisfies $|f(x)|\le C$ and $|f(x)-f(y)|\le\ell\,|x-y|_{H}$ for all $x,y\in H^{-1}$, then there is a function $\hat{f}:H\to\mathbb{R}$, continuous on $H$, with $\hat{f}(x)=f(x)$ for $x\in H^{-1}$, $|\hat{f}(x)|\le C$ for $x\in H$, and $|\hat{f}(x)-\hat{f}(y)|\le\ell\,|x-y|_{H}$ for $x,y\in H$.

First, $H^{-1}$ is nonempty and dense in $(H,d_{H})$. The set $H$ is open in $H$ (Hilbert Triples: Standing Notation and Background §open-sets) and nonempty (it contains $0_{H}$), so by The Penalty Function $h=\tfrac12|\cdot|_V^2$ of a Hilbert Triple: Expansion Identities, Closed Sublevel Sets, Density and Local Bounds §dense, applied to this triple with $U=H$, the set $D(A)=H^{-1}$ is nonempty and for every $x\in H$ and real $\varepsilon>0$ there is $y\in H^{-1}$ with $|y-x|_{H}<\varepsilon$. Let $x\in H$ and let $O$ be an open set of $(H,d_{H})$ containing $x$. By Open Subset of a Metric Space there is a real $r>0$ with $B_{d_{H}}(x,r)\subseteq O$, and the point $y\in H^{-1}$ with $|y-x|_{H}<r$ lies in this open ball, so $O\cap H^{-1}\neq\varnothing$. Hence every $x\in H$ lies in the closure of $H^{-1}$, which is therefore $H$, so $H^{-1}$ is dense in $(H,\mathcal{T}_{d_{H}})$.

Next, $f$ is uniformly continuous on $H^{-1}$: given a real $\varepsilon>0$ put $\delta=\varepsilon/(\ell+1)>0$; for $x,y\in H^{-1}$ with $d_{H}(x,y)<\delta$ we get $|f(x)-f(y)|\le\ell\,d_{H}(x,y)\le\ell\delta=\varepsilon\ell/(\ell+1)<\varepsilon$, and $|f(x)-f(y)|$ is $d_{\mathbb{R}}(f(x),f(y))$. By Extension of a Uniformly Continuous Real Function from a Dense Subset §existence, with $(X,d)=(H,d_{H})$ and $S=H^{-1}$, there is $\hat{f}:H\to\mathbb{R}$, continuous on $H$, with $\hat{f}=f$ on $H^{-1}$. Since $|\hat{f}(x)|=|f(x)|\le C$ for $x\in H^{-1}$, Extension of a Uniformly Continuous Real Function from a Dense Subset §bounds gives $|\hat{f}(x)|\le C$ for all $x\in H$. For the Lipschitz bound fix $x,y\in H$ and a real $\eta>0$, and put $\theta=|x-y|_{H}+\eta>0$ and $\varepsilon=\ell\theta+\eta>0$. All $x',y'\in H^{-1}$ with $d_{H}(x',y')\le\theta$ satisfy $|\hat{f}(x')-\hat{f}(y')|=|f(x')-f(y')|\le\ell\,d_{H}(x',y')\le\ell\theta\le\varepsilon$. Since $d_{H}(x,y)<\theta$, Extension of a Uniformly Continuous Real Function from a Dense Subset §modulus gives $|\hat{f}(x)-\hat{f}(y)|\le\varepsilon=\ell\,|x-y|_{H}+(\ell+1)\eta$. As $\eta>0$ was arbitrary, $|\hat{f}(x)-\hat{f}(y)|\le\ell\,|x-y|_{H}$.

Apply this to $g$ with $C=C_{g}$, $\ell=\ell_{g}$, obtaining $\hat{g}:H\to\mathbb{R}$, and let $\bar{g}:V\to\mathbb{R}$ be the restriction of $\hat{g}$ to $V$. Then $|\bar{g}(x)|\le C_{g}$ and $|\bar{g}(x)-\bar{g}(y)|\le\ell_{g}|x-y|_{H}$ for all $x,y\in V$, and $\bar{g}|_{H^{-1}}=g$ because $H^{-1}\subseteq V$.

Step 2 (Clause galerkin). Fix $N\in\mathbb{N}$, with $m$, $\kappa_{N}$, $\iota_{N}$, $\mathrm{pr}_{N}$ and $\Pi_{N}$ as in The Galerkin Problems of the Wick-Square Problem: Standing Notation for Cubes of Modes, Coordinates, the Free Galerkin Quadratic and the Riccati Potential §cutoff and The Galerkin Problems of the Wick-Square Problem: Standing Notation for Cubes of Modes, Coordinates, the Free Galerkin Quadratic and the Riccati Potential §coordinates. We show that $g\circ\iota_{N}$ is continuous from $(\mathbb{R}^{m},d_{E})$ to $\mathbb{R}$. Let $z,z'\in\mathbb{R}^{m}$ and $t=z-z'$. Since the operations of $\mathrm{Map}(\mathbb{Z}^{n},\mathbb{R})$ are pointwise, the family $y=\iota_{N}z-\iota_{N}z'$ has value $t_{i}$ at $\kappa_{N}(i)$ for $i\in[m]$ and value $0$ at every mode outside $\Gamma_{N}$, so $y=\iota_{N}t=\sum_{i=1}^{m}t_{i}e_{\kappa_{N}(i)}$ by The Galerkin Problems of the Wick-Square Problem: Standing Notation for Cubes of Modes, Coordinates, the Free Galerkin Quadratic and the Riccati Potential §coordinates; it lies in $H^{-1}\subseteq H$. By linearity of the inner product in its second argument over finite sums (Real Hilbert Spaces: Standing Notation and Background §background, which puts Inner Products Against Finite Sums, and Orthonormal Families, in a Real Inner Product Space in force) and by The Sobolev Triple of Order Two for the Wick-Square Problem: the Domain Is the Sobolev Space of Order -1, Unit Families Are Square-Summable, the Gradient Form and the Riccati Drift §pairings, which gives $\langle y,e_{k}\rangle_{H}=y(k)/\mu_{k}^{3}$,

$$|y|_{H}^{2}=\sum_{i=1}^{m}t_{i}\,\langle y,e_{\kappa_{N}(i)}\rangle_{H}=\sum_{i=1}^{m}\frac{t_{i}^{2}}{\mu_{\kappa_{N}(i)}^{3}}\le\sum_{i=1}^{m}t_{i}^{2}=\lVert z-z'\rVert^{2},$$

using $1\le\mu_{k}$, hence $1\le\mu_{k}^{3}$, from The Wick-Square Problem on the Torus: Standing Notation §modes, and the Euclidean norm of Second-Order Equations on Euclidean Open Sets §space. Hence $|\iota_{N}z-\iota_{N}z'|_{H}\le d_{E}(z,z')$, and the hypothesis on $g$ gives $|g(\iota_{N}z)-g(\iota_{N}z')|\le\ell_{g}\,d_{E}(z,z')$. Given a real $\varepsilon>0$, the number $\delta=\varepsilon/(\ell_{g}+1)>0$ therefore satisfies $|g(\iota_{N}z)-g(\iota_{N}z')|<\varepsilon$ whenever $d_{E}(z,z')<\delta$, so $g\circ\iota_{N}$ is continuous on $\mathbb{R}^{m}$.

Since $|g(x)|\le C_{g}$ on $H^{-1}$, The Riccati Shift of the Galerkin Equations of the Wick-Square Problem: a Penalty-Drift Equation with Bounded Cost, and Well-Posedness at Each Cutoff §well-posed applies: there is exactly one function $U_{N}:\mathbb{R}^{m}\to\mathbb{R}$ that is a viscosity subsolution and a viscosity supersolution of $\mathcal{F}_{N}$ on $\mathbb{R}^{m}$ (that is, a viscosity solution in the sense of The Galerkin Hamilton-Jacobi-Bellman Equations of the Wick-Square Problem in the Coordinates of a Cube of Modes §equation) with $U_{N}-\tilde{u}_{N}$ bounded; moreover $W_{N}=U_{N}-\tilde{u}_{N}$ is continuous on $\mathbb{R}^{m}$ and $|W_{N}(z)|\le C_{g}/\gamma$ for every $z\in\mathbb{R}^{m}$. This is clause galerkin.

Step 3 (Lifting the Galerkin solutions). Fix $N\in\mathbb{N}$. Applying The Riccati Shift of the Galerkin Equations of the Wick-Square Problem: a Penalty-Drift Equation with Bounded Cost, and Well-Posedness at Each Cutoff §solutions to $U=U_{N}$, which is a viscosity subsolution and a viscosity supersolution of $\mathcal{F}_{N}$, shows that $W_{N}$ is a viscosity subsolution and a viscosity supersolution of the shifted Galerkin operator $\mathcal{F}^{\sharp}_{N}$ for $g$ on $\mathbb{R}^{m}$. Now apply Lifting Shifted Galerkin Solutions of the Wick-Square Problem to Viscosity Solutions of the Cut-Off Shifted Equation on the Sobolev Triple with the running cost $\bar{g}:V\to\mathbb{R}$ and $Z=W_{N}$, which is bounded; the shifted Galerkin operator there is the one for $\bar{g}|_{H^{-1}}=g$, that is, exactly the operator $\mathcal{F}^{\sharp}_{N}$ just used. Put $w_{N}(x)=W_{N}(\mathrm{pr}_{N}x)$ for $x\in H$. By Lifting Shifted Galerkin Solutions of the Wick-Square Problem to Viscosity Solutions of the Cut-Off Shifted Equation on the Sobolev Triple §continuity, $w_{N}$ is continuous on $H$; by Lifting Shifted Galerkin Solutions of the Wick-Square Problem to Viscosity Solutions of the Cut-Off Shifted Equation on the Sobolev Triple §subsolution and Lifting Shifted Galerkin Solutions of the Wick-Square Problem to Viscosity Solutions of the Cut-Off Shifted Equation on the Sobolev Triple §supersolution it is a viscosity subsolution and a viscosity supersolution of the cut-off shifted operator $F^{\sharp}_{N}$ for $\bar{g}$ on $H$. Moreover $|w_{N}(x)|\le C_{g}/\gamma$ for every $x\in H$, so $w_{N}$ is bounded above and below near each point of $H$, and $w_{N}$ is a viscosity solution of $F^{\sharp}_{N}$ on $H$. For $x\in H^{-1}$,

$$w_{N}(x)=U_{N}(\mathrm{pr}_{N}x)-\tilde{u}_{N}(\mathrm{pr}_{N}x)=u_{N}(x)-u_{0,N}(x).$$

In particular $|u_{N}(x)-u_{0,N}(x)|\le C_{g}/\gamma$ for every $x\in H^{-1}$, the first inequality of clause bounds.

Step 4 (The limit as a solution of the shifted equation). The hypotheses on $g$ are those of Well-Posedness of the Renormalised Wick-Square Problem: Existence, Comparison and Uniqueness of Renormalised Viscosity Solutions Differing from the Free Solution by a Bounded Lipschitz Function, with the same $C_{g}$, $\ell_{g}$ and the same class $\mathcal{C}$ of functions $v:H^{-1}\to\mathbb{R}$ with $v-u_{0}$ bounded and Lipschitz from $(H^{-1},d_{H})$ to $\mathbb{R}$. By Well-Posedness of the Renormalised Wick-Square Problem: Existence, Comparison and Uniqueness of Renormalised Viscosity Solutions Differing from the Free Solution by a Bounded Lipschitz Function §existence there is a renormalised viscosity solution $u^{*}\in\mathcal{C}$ with $|u^{*}(x)-u_{0}(x)|\le C_{g}/\gamma$ and $|(u^{*}-u_{0})(x)-(u^{*}-u_{0})(y)|\le L\,|x-y|_{H}$ for $x,y\in H^{-1}$. The function $u$ of the statement is a renormalised viscosity solution in $\mathcal{C}$, so by Well-Posedness of the Renormalised Wick-Square Problem: Existence, Comparison and Uniqueness of Renormalised Viscosity Solutions Differing from the Free Solution by a Bounded Lipschitz Function §uniqueness it equals $u^{*}$, and the two bounds hold for $u$.

Apply Step 1 to $f=u-u_{0}$ with $C=C_{g}/\gamma$ and $\ell=L$, obtaining $w:H\to\mathbb{R}$, continuous on $H$, with $w=u-u_{0}$ on $H^{-1}$, $|w(x)|\le C_{g}/\gamma$ for $x\in H$, and $|w(x)-w(y)|\le L\,|x-y|_{H}$ for $x,y\in H$; so $w$ is bounded and Lipschitz (with constant $L$) from $(H,d_{H})$ to $\mathbb{R}$, and $u=u_{0}+w|_{H^{-1}}$. Now apply Renormalised Viscosity Solutions versus Viscosity Solutions of the Shifted Equation on the Sobolev Triple of Order Two with the running cost $\bar{g}:V\to\mathbb{R}$, which satisfies its hypotheses with the constants $C_{g}$, $\ell_{g}$ by Step 1; its renormalised viscosity sub- and supersolutions are those for $\bar{g}|_{H^{-1}}=g$. Since $u$ is a renormalised viscosity solution for $g$, it is a renormalised viscosity subsolution and supersolution (Renormalised Viscosity Subsolutions, Supersolutions and Solutions of the Wick-Square Problem, Tested by the Gaussian Penalty plus a Regular Function §solution), so by Renormalised Viscosity Solutions versus Viscosity Solutions of the Shifted Equation on the Sobolev Triple of Order Two §to-shifted $w$ is a viscosity subsolution and a viscosity supersolution of the shifted operator $F^{\sharp}$ for $\bar{g}$ on $H$. Being bounded, $w$ is bounded above and below near each point of $H$, hence a viscosity solution of $F^{\sharp}$ on $H$ by Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on a Hilbert Triple §solution.

Step 5 (Clauses bounds and convergence). We apply Convergence of Solutions of Viscous Hamilton-Jacobi Equations with Monotone Nonlinearities on a Hilbert Triple under Convergence of the Operators to the triple $(H,V,A)$, which is in the setting of Hilbert Triples: Standing Notation and Background by The Wick-Square Problem on the Sobolev Triple of Order Two: Standing Notation §triple. Its hypotheses are verified as follows. $H$ is not finite-dimensional by The Sobolev Triple of Order Two for the Wick-Square Problem: the Domain Is the Sobolev Space of Order -1, Unit Families Are Square-Summable, the Gradient Form and the Riccati Drift §infinite. Every sequence in $V$ bounded in $V$ has a subsequence converging in $H$ by The Cut-Off Shifted Operators of the Wick-Square Problem: the Hypotheses of the Convergence Theorem with Constants Independent of the Cutoff, Convergence on Bounded Test Data, and Compactness of the Triple §compact, applied with the running cost $\bar{g}$ (whose hypotheses hold by Step 1). Take $\lambda_{0}=\gamma>0$, the given $\nu>0$, the given nonnegative $C_{g}$ and $\ell_{g}$, and $C'=C_{g}/\gamma\ge0$. Take the quadruple $(B,f,\Gamma,g)$ of that theorem to be $(B_{q},f,G,\bar{g})$: $B_{q}$ is a monotone nonlinearity for $(H,V,A)$ by The Sobolev Triple of Order Two for the Wick-Square Problem: the Domain Is the Sobolev Space of Order -1, Unit Families Are Square-Summable, the Gradient Form and the Riccati Drift §drift, $f$ is square-summable in $V$ by The Sobolev Triple of Order Two for the Wick-Square Problem: the Domain Is the Sobolev Space of Order -1, Unit Families Are Square-Summable, the Gradient Form and the Riccati Drift §noise, $G\in\mathrm{Sym}(H)$ with $0_{\mathrm{Sym}}\preceq G\preceq I_{H}$ by The Sobolev Triple of Order Two for the Wick-Square Problem: the Domain Is the Sobolev Space of Order -1, Unit Families Are Square-Summable, the Gradient Form and the Riccati Drift §gradient-form (these being the data fixed in The Wick-Square Problem on the Sobolev Triple of Order Two: Standing Notation §noise and The Wick-Square Problem on the Sobolev Triple of Order Two: Standing Notation §riccati), and $\bar{g}$ is bounded by $C_{g}$ and $\ell_{g}$-Lipschitz for $d_{H}$ on $V$ by Step 1. For $N\in\mathbb{N}$ take the quadruple there written $(B_{N},f^{N},\Gamma_{N},g_{N})$ to be $(B^{N}_{q},f^{N},G_{N},g_{N})$ with the cut-off data and $g_{N}:V\to\mathbb{R}$, $g_{N}(x)=\bar{g}(\Pi_{N}x)$, which is defined because $\Pi_{N}$ maps $H$ into $V$ by The Shifted Hamilton-Jacobi-Bellman Equation of the Wick-Square Problem with Noise, Control and Cost Cut Off to a Cube of Modes §data: $f^{N}$ is square-summable in $V$ by that clause, and $G_{N}\in\mathrm{Sym}(H)$ with $0_{\mathrm{Sym}}\preceq G_{N}\preceq I_{H}$, $B^{N}_{q}$ is a monotone nonlinearity for $(H,V,A)$, and $|g_{N}(x)|\le C_{g}$, $|g_{N}(x)-g_{N}(y)|\le\ell_{g}|x-y|_{H}$ for $x,y\in V$, by The Cut-Off Shifted Operators of the Wick-Square Problem: the Hypotheses of the Convergence Theorem with Constants Independent of the Cutoff, Convergence on Bounded Test Data, and Compactness of the Triple §hypotheses for $\bar{g}$.

With these choices, the operator $F$ of the convergence theorem is, at every $(x,r,p,X)\in D(A)\times\mathbb{R}\times H\times\mathrm{Sym}(V)$,

$$\gamma\,r-\tfrac{\nu}{2}\,\mathrm{Tr}_{f}X+\tfrac12\,G(p,p)+\langle Ax+B_{q}x,p\rangle_{H}-\bar{g}(x),$$

which is the value of the shifted operator $F^{\sharp}$ for $\bar{g}$ of The Shifted Hamilton-Jacobi-Bellman Equation of the Wick-Square Problem on the Sobolev Triple of Order Two §operator; and $F_{N}$ is

$$\gamma\,r-\tfrac{\nu}{2}\,\mathrm{Tr}_{f^{N}}X+\tfrac12\,G_{N}(p,p)+\langle Ax+B^{N}_{q}x,p\rangle_{H}-\bar{g}(\Pi_{N}x),$$

which is the value of the cut-off shifted operator $F^{\sharp}_{N}$ for $\bar{g}$ of The Shifted Hamilton-Jacobi-Bellman Equation of the Wick-Square Problem with Noise, Control and Cost Cut Off to a Cube of Modes §operator. So $F=F^{\sharp}$ and $F_{N}=F^{\sharp}_{N}$ as functions on the same domain, and $(F^{\sharp}_{N})_{N\in\mathbb{N}}$ converges to $F^{\sharp}$ on bounded test data by The Cut-Off Shifted Operators of the Wick-Square Problem: the Hypotheses of the Convergence Theorem with Constants Independent of the Cutoff, Convergence on Bounded Test Data, and Compactness of the Triple §convergence for $\bar{g}$. Finally take the solutions of that theorem to be $w$ and $w_{N}$ ($N\in\mathbb{N}$): by Steps 3 and 4 they are continuous on $H$, bounded in absolute value by $C'=C_{g}/\gamma$, and viscosity solutions of $F^{\sharp}$ and of $F^{\sharp}_{N}$ on $H$.

By Convergence of Solutions of Viscous Hamilton-Jacobi Equations with Monotone Nonlinearities on a Hilbert Triple under Convergence of the Operators §lipschitz, for every $N$ and $x,y\in H$, $|w_{N}(x)-w_{N}(y)|\le(\ell_{g}/\gamma+2C_{g}/\gamma+1)|x-y|_{H}=L\,|x-y|_{H}$. For $x,y\in H^{-1}\subseteq H$, Step 3 gives $w_{N}(x)-w_{N}(y)=(u_{N}-u_{0,N})(x)-(u_{N}-u_{0,N})(y)$, which proves the second inequality of clause bounds.

Let $R>0$ and $\varepsilon>0$ be real, and let $N_{0}\in\mathbb{N}$ be given by Convergence of Solutions of Viscous Hamilton-Jacobi Equations with Monotone Nonlinearities on a Hilbert Triple under Convergence of the Operators §uniform, so that $|w_{N}(x)-w(x)|\le\varepsilon$ for all $N\ge N_{0}$ and $x\in V$ with $|x|_{V}\le R$. Every $x\in H^{-1}$ with $|x|_{V}\le R$ lies in $V$, and for it $w_{N}(x)=(u_{N}-u_{0,N})(x)$ by Step 3 and $w(x)=(u-u_{0})(x)$ by Step 4. Hence $|(u_{N}-u_{0,N})(x)-(u-u_{0})(x)|\le\varepsilon$ for all such $N$ and $x$, which is clause convergence.

Step 6 (Clause pointwise). Fix $x\in H^{-1}$. By Convergence of Solutions of Viscous Hamilton-Jacobi Equations with Monotone Nonlinearities on a Hilbert Triple under Convergence of the Operators §pointwise, with the data of Step 5, the sequence $(w_{N}(x))_{N\in\mathbb{N}}$ converges to $w(x)=u(x)-u_{0}(x)$. Next, since $(\mathrm{pr}_{N}x)_{i}=x(\kappa_{N}(i))$ for $i\in[m]$ (The Galerkin Problems of the Wick-Square Problem: Standing Notation for Cubes of Modes, Coordinates, the Free Galerkin Quadratic and the Riccati Potential §coordinates),

$$u_{0,N}(x)=\tilde{u}_{N}(\mathrm{pr}_{N}x)=\sum_{i=1}^{m}\bigl(q_{\kappa_{N}(i)}\,x(\kappa_{N}(i))^{2}+a_{\kappa_{N}(i)}\bigr)=\sum_{k\in\Gamma_{N}}\bigl(q_{k}\,x(k)^{2}+a_{k}\bigr),$$

the first equality by The Galerkin Problems of the Wick-Square Problem: Standing Notation for Cubes of Modes, Coordinates, the Free Galerkin Quadratic and the Riccati Potential §functions and the last by Sum over a Finite Index Set, because $\Gamma_{N}$ has $m$ elements and $\kappa_{N}:[m]\to\Gamma_{N}$ is a bijection (The Galerkin Problems of the Wick-Square Problem: Standing Notation for Cubes of Modes, Coordinates, the Free Galerkin Quadratic and the Riccati Potential §cutoff). The right-hand side is the value at $x$ of the function written $u_{N}$ in The Galerkin Solutions of the Wick-Square Problem Converge to a Classical Solution of the Renormalised Equation, while the Bare Solutions Diverge (whose definition does not involve a running cost, the coefficients $q_{k}$, $a_{k}$ being the Riccati data of The Galerkin Problems of the Wick-Square Problem: Standing Notation for Cubes of Modes, Coordinates, the Free Galerkin Quadratic and the Riccati Potential §problem), so by The Galerkin Solutions of the Wick-Square Problem Converge to a Classical Solution of the Renormalised Equation, while the Bare Solutions Diverge §limit the sequence $(u_{0,N}(x))_{N\in\mathbb{N}}$ converges to $\sum_{k\in\mathbb{Z}^{n}}(q_{k}x(k)^{2}+a_{k})$, which is $u_{0}(x)$ by The Wick-Square Problem on the Sobolev Triple of Order Two: Standing Notation §free. Since $u_{N}(x)=w_{N}(x)+u_{0,N}(x)$ for every $N$ by Step 3, Arithmetic of Limits of Real Sequences §sums shows that $(u_{N}(x))_{N\in\mathbb{N}}$ converges to $(u(x)-u_{0}(x))+u_{0}(x)=u(x)$. This is clause pointwise.