Proof of The Negative-Order Sobolev Triple of the Torus: a Diagonal Hilbert Triple whose Form Operator Is One Minus the Laplacian

Each result cited is universally quantified over the data in its own statement. Coefficient families are compared pointwise: two elements of $\mathrm{Map}(\mathbb{Z}^{n},\mathbb{R})$ are equal exactly when their values agree at every $k\in\mathbb{Z}^{n}$, and the operations of the Sobolev spaces are the pointwise operations of The Real Vector Space of Real-Valued Functions on a Set (i), as recorded in The Negative-Order Sobolev Spaces of the Torus (i). Squares of real numbers satisfy $(ab)^{2}=a^{2}b^{2}$ by claim 3 of Properties of Natural Number Powers in a Field. The clauses of The Negative-Order Sobolev Spaces of the Torus are Hilbert Spaces: the Embedding of the Square-Integrable Classes, the Rescaled Trigonometric Basis, the Series Form of the Inner Product and the Inclusion of the Scale are applied with its $m$ equal to $s$ and to $s+1$; by its clause The Negative-Order Sobolev Spaces of the Torus are Hilbert Spaces: the Embedding of the Square-Integrable Classes, the Rescaled Trigonometric Basis, the Series Form of the Inner Product and the Inclusion of the Scale (i), $\rho_{k}^{s+1}=\rho_{k}^{s}\rho_{k}$, and $\rho_{k}^{2}\mu_{k}=1$ because $\rho_{k}^{2}=\tfrac{1}{\mu_{k}}$. Hence, for every $k\in\mathbb{Z}^{n}$,

$$\mu_{k}\,(\rho_{k}^{s+1})^{2}=\mu_{k}\,(\rho_{k}^{s})^{2}\rho_{k}^{2}=(\rho_{k}^{s})^{2}\,(\rho_{k}^{2}\mu_{k})=(\rho_{k}^{s})^{2},$$

by the commutativity and associativity of multiplication; this identity is referred to as (I) below. An enumeration exists by The Integer Lattice Admits an Enumeration by the Natural Numbers (ii).

Claim 1. Let $\kappa$ be an enumeration. By The Negative-Order Sobolev Spaces of the Torus are Hilbert Spaces: the Embedding of the Square-Integrable Classes, the Rescaled Trigonometric Basis, the Series Form of the Inner Product and the Inclusion of the Scale (ii) and The Negative-Order Sobolev Spaces of the Torus are Hilbert Spaces: the Embedding of the Square-Integrable Classes, the Rescaled Trigonometric Basis, the Series Form of the Inner Product and the Inclusion of the Scale (iv), applied with $m=s+1$, the space $H$ is a real Hilbert space and $e_{j}=\zeta_{s+1,\kappa(j)}$ defines an orthonormal basis $(e_{j})_{j\in\mathbb{N}}$ of $H$; and $\lambda_{j}=\mu_{\kappa(j)}$ defines a sequence of real numbers with $1\le\lambda_{j}$ by Summability of the Negative Powers of the Fourier Weights of the Torus (i). For $x\in H$ and $j\in\mathbb{N}$ write $x_{j}=\langle x,e_{j}\rangle_{H}$; by The Negative-Order Sobolev Spaces of the Torus are Hilbert Spaces: the Embedding of the Square-Integrable Classes, the Rescaled Trigonometric Basis, the Series Form of the Inner Product and the Inclusion of the Scale (iv) with $m=s+1$,

$$x_{j}=\rho_{\kappa(j)}^{s+1}\,x(\kappa(j)),\qquad\text{hence}\qquad\lambda_{j}x_{j}^{2}=\mu_{\kappa(j)}(\rho_{\kappa(j)}^{s+1})^{2}x(\kappa(j))^{2}=(\rho_{\kappa(j)}^{s})^{2}x(\kappa(j))^{2}$$

by (I). Let $V_{\kappa}$ be the weighted coefficient subspace of $H$ determined by $(e_{j})$ and $(\lambda_{j})$ as in The Weighted Coefficient Subspace Determined by an Orthonormal Basis and a Sequence of Weights: the set of $x\in H$ for which $\sum_{j=1}^{\infty}\lambda_{j}x_{j}^{2}$ converges, with inner product $\langle x,y\rangle_{V_{\kappa}}=\sum_{j=1}^{\infty}\lambda_{j}x_{j}y_{j}$. By the display and The Negative-Order Sobolev Spaces of the Torus are Hilbert Spaces: the Embedding of the Square-Integrable Classes, the Rescaled Trigonometric Basis, the Series Form of the Inner Product and the Inclusion of the Scale (v) with $m=s$, a family $x\in H$ lies in $V_{\kappa}$ exactly when $\sum_{j=1}^{\infty}(\rho_{\kappa(j)}^{s})^{2}x(\kappa(j))^{2}$ converges, that is, exactly when $x\in H^{-s}(\mathbb{T}^{n})=V$; since $V\subseteq H$ by The Negative-Order Sobolev Spaces of the Torus are Hilbert Spaces: the Embedding of the Square-Integrable Classes, the Rescaled Trigonometric Basis, the Series Form of the Inner Product and the Inclusion of the Scale (vi) with $m=s$, this gives $V_{\kappa}=V$ as sets. For $x,y\in V$, the same computation, with $x_{j}=\rho_{\kappa(j)}^{s+1}x(\kappa(j))$ and $y_{j}=\rho_{\kappa(j)}^{s+1}y(\kappa(j))$ and then (I), gives $\lambda_{j}x_{j}y_{j}=(\rho_{\kappa(j)}^{s})^{2}x(\kappa(j))y(\kappa(j))$ for every $j$, so $\langle x,y\rangle_{V_{\kappa}}=\langle x,y\rangle_{H^{-s}}=\langle x,y\rangle_{V}$ by The Negative-Order Sobolev Spaces of the Torus are Hilbert Spaces: the Embedding of the Square-Integrable Classes, the Rescaled Trigonometric Basis, the Series Form of the Inner Product and the Inclusion of the Scale (v) with $m=s$. Thus $V_{\kappa}$ with its inner product is $V$ with $\langle\,\cdot\,,\cdot\,\rangle_{V}$.

By The Diagonal Hilbert Triple Determined by an Orthonormal Basis and a Sequence of Weights (i), the diagonal Hilbert triple determined by $(e_{j})$ and $(\lambda_{j})$ is the Hilbert triple formed by $H$, by $V_{\kappa}=V$ with its inner product, and by the form operator these two spaces determine; that definition records, by The Weighted Coefficient Subspace Determined by an Orthonormal Basis and a Sequence of Weights (i), The Weighted Coefficient Subspace Determined by an Orthonormal Basis and a Sequence of Weights (iii) and The Weighted Coefficient Subspace Determined by an Orthonormal Basis and a Sequence of Weights (iv), that $V$ is a linear subspace of $H$ with $|x|_{H}\le|x|_{V}$ for $x\in V$, is a real Hilbert space with $\langle\,\cdot\,,\cdot\,\rangle_{V}$, and is dense in $H$, which are the requirements of Hilbert Triple: a Densely and Continuously Embedded Hilbert Space and Its Form Operator (i). Since the form operator is determined by $H$ and by $V$ with its inner product alone, the triple $(H,V,A)$ of the statement is a Hilbert triple and coincides with that diagonal triple. The metric space $(V,d_{V})$ is separable by The Negative-Order Sobolev Spaces of the Torus are Hilbert Spaces: the Embedding of the Square-Integrable Classes, the Rescaled Trigonometric Basis, the Series Form of the Inner Product and the Inclusion of the Scale (ii) with $m=s$, so Hilbert Triples: Standing Notation and Background (ii) holds for $(H,V,A)$. As $\kappa$ was an arbitrary enumeration, claim 1 is proved.

Claim 2. Let $\kappa$ be an enumeration. By claim 1, $(H,V,A)$ is the diagonal Hilbert triple determined by $(\zeta_{s+1,\kappa(j)})_{j\in\mathbb{N}}$ and $(\mu_{\kappa(j)})_{j\in\mathbb{N}}$, and the standing hypothesis of Hilbert Triples: Standing Notation and Background holds for it. Hence The Form Operator and the Riesz Map of a Diagonal Hilbert Triple (i) gives that $D(A)$ is the set of $x\in H$ for which $\sum_{j=1}^{\infty}\mu_{\kappa(j)}^{2}x_{j}^{2}$ converges and that $\sum_{j=1}^{\infty}\mu_{\kappa(j)}x_{j}\zeta_{s+1,\kappa(j)}$ converges in $H$ with sum $Ax$ for such $x$, where $x_{j}=\langle x,\zeta_{s+1,\kappa(j)}\rangle_{H}=\rho_{\kappa(j)}^{s+1}x(\kappa(j))$ by The Negative-Order Sobolev Spaces of the Torus are Hilbert Spaces: the Embedding of the Square-Integrable Classes, the Rescaled Trigonometric Basis, the Series Form of the Inner Product and the Inclusion of the Scale (iv) with $m=s+1$.

Claim 3. Let $k\in\mathbb{Z}^{n}$. By The Negative-Order Sobolev Spaces of the Torus are Hilbert Spaces: the Embedding of the Square-Integrable Classes, the Rescaled Trigonometric Basis, the Series Form of the Inner Product and the Inclusion of the Scale (iv), applied with $m=s+1$ and with $m=s$, the value $\zeta_{s+1,k}(k')$ equals $\tfrac{1}{\rho_{k}^{s+1}}$ if $k'=k$ and $0$ otherwise, and likewise $\zeta_{s,k}(k')$ equals $\tfrac{1}{\rho_{k}^{s}}$ if $k'=k$ and $0$ otherwise; while $\hat{E}_{k}(k')$ equals $1$ if $k'=k$ and $0$ otherwise by Properties of the Fourier Coefficients on the Torus, and the Realisation of Weighted Coefficient Families (iii). Comparing values pointwise, using $\rho_{k}^{l}\cdot\tfrac{1}{\rho_{k}^{l}}=1$ and $t\cdot0=0$,

$$\hat{E}_{k}=\rho_{k}^{s+1}\,\zeta_{s+1,k}\qquad\text{and}\qquad\hat{E}_{k}=\rho_{k}^{s}\,\zeta_{s,k}.$$

In particular $\hat{E}_{k}\in V$ and $\hat{E}_{k}\in H$, by The Negative-Order Sobolev Spaces of the Torus are Hilbert Spaces: the Embedding of the Square-Integrable Classes, the Rescaled Trigonometric Basis, the Series Form of the Inner Product and the Inclusion of the Scale (iii), and $\zeta_{s+1,k}=\tfrac{1}{\rho_{k}^{s+1}}\hat{E}_{k}\in V$, the space $V$ being a linear subspace of $H$.

Let $y\in V$. By condition (c) and condition (a) of Real Inner Product Spaces (i) and by The Negative-Order Sobolev Spaces of the Torus are Hilbert Spaces: the Embedding of the Square-Integrable Classes, the Rescaled Trigonometric Basis, the Series Form of the Inner Product and the Inclusion of the Scale (iv) with $m=s$,

$$\langle\hat{E}_{k},y\rangle_{V}=\rho_{k}^{s}\langle\zeta_{s,k},y\rangle_{V}=\rho_{k}^{s}\langle y,\zeta_{s,k}\rangle_{H^{-s}}=\rho_{k}^{s}\rho_{k}^{s}\,y(k)=(\rho_{k}^{s})^{2}y(k),$$

and likewise, by condition (c) and condition (a), by The Negative-Order Sobolev Spaces of the Torus are Hilbert Spaces: the Embedding of the Square-Integrable Classes, the Rescaled Trigonometric Basis, the Series Form of the Inner Product and the Inclusion of the Scale (iv) with $m=s+1$, and then by (I),

$$\langle\mu_{k}\hat{E}_{k},y\rangle_{H}=\mu_{k}\rho_{k}^{s+1}\langle\zeta_{s+1,k},y\rangle_{H}=\mu_{k}\rho_{k}^{s+1}\rho_{k}^{s+1}\,y(k)=\mu_{k}(\rho_{k}^{s+1})^{2}y(k)=(\rho_{k}^{s})^{2}y(k).$$

So $\langle\hat{E}_{k},y\rangle_{V}=\langle\mu_{k}\hat{E}_{k},y\rangle_{H}$ for every $y\in V$, with $\mu_{k}\hat{E}_{k}\in H$; by Hilbert Triple: a Densely and Continuously Embedded Hilbert Space and Its Form Operator (iv), $\hat{E}_{k}\in D(A)$ and $A\hat{E}_{k}=\mu_{k}\hat{E}_{k}$. The same computation for $\zeta_{s+1,k}=\tfrac{1}{\rho_{k}^{s+1}}\hat{E}_{k}$: by condition (c), $\langle\zeta_{s+1,k},y\rangle_{V}=\tfrac{1}{\rho_{k}^{s+1}}(\rho_{k}^{s})^{2}y(k)$ and $\langle\mu_{k}\zeta_{s+1,k},y\rangle_{H}=\tfrac{1}{\rho_{k}^{s+1}}\langle\mu_{k}\hat{E}_{k},y\rangle_{H}=\tfrac{1}{\rho_{k}^{s+1}}(\rho_{k}^{s})^{2}y(k)$, the scalars $\mu_{k}$ and $\tfrac{1}{\rho_{k}^{s+1}}$ being interchanged by condition 5 of Vector Space over a Field, applied twice, and the commutativity of multiplication in $\mathbb{R}$; so $\zeta_{s+1,k}\in D(A)$ and $A\zeta_{s+1,k}=\mu_{k}\zeta_{s+1,k}$ by Hilbert Triple: a Densely and Continuously Embedded Hilbert Space and Its Form Operator (iv).

Claim 4. Let $u\in C^{2}_{\mathrm{per}}$ and put $U=[\,u|_{Q}\,]$ and $D=[\,(\Delta u)|_{Q}\,]$. By The Laplacian of the Trigonometric System, and the Fourier Coefficients of a Laplacian on the Torus (iii), $u|_{Q}$ and $(\Delta u)|_{Q}$ lie in $\mathcal{L}^{2}(\mathbb{T}^{n})$ and, for every $k\in\mathbb{Z}^{n}$, $\hat{D}(k)=\langle D,E_{k}\rangle_{L^{2}}=-4\pi^{2}\lVert k\rVert^{2}\,\hat{U}(k)$, the Fourier coefficients being those of The Fourier Coefficients of a Square-Integrable Class on the Torus (i). By The Negative-Order Sobolev Spaces of the Torus are Hilbert Spaces: the Embedding of the Square-Integrable Classes, the Rescaled Trigonometric Basis, the Series Form of the Inner Product and the Inclusion of the Scale (iii), $\hat{U}\in V$, and both $\hat{U}$ and $\hat{D}$ lie in $H$; put $z=\hat{U}-\hat{D}\in H$, the difference being pointwise, so that for every $k$

$$z(k)=\hat{U}(k)+4\pi^{2}\lVert k\rVert^{2}\hat{U}(k)=\mu_{k}\,\hat{U}(k),$$

the first equality because $-(-t)=t$ for a real number $t$, and the second by distributivity. Let $y\in V$ and let $\kappa$ be an enumeration. By The Negative-Order Sobolev Spaces of the Torus are Hilbert Spaces: the Embedding of the Square-Integrable Classes, the Rescaled Trigonometric Basis, the Series Form of the Inner Product and the Inclusion of the Scale (v) with $m=s$ and with $m=s+1$, and by (I),

$$\langle\hat{U},y\rangle_{V}=\sum_{j=1}^{\infty}(\rho_{\kappa(j)}^{s})^{2}\hat{U}(\kappa(j))\,y(\kappa(j)),\qquad \langle z,y\rangle_{H}=\sum_{j=1}^{\infty}(\rho_{\kappa(j)}^{s+1})^{2}\mu_{\kappa(j)}\hat{U}(\kappa(j))\,y(\kappa(j))=\sum_{j=1}^{\infty}(\rho_{\kappa(j)}^{s})^{2}\hat{U}(\kappa(j))\,y(\kappa(j)),$$

both series converging by that clause; the two series have the same terms, hence the same sum. Thus $\langle\hat{U},y\rangle_{V}=\langle z,y\rangle_{H}$ for every $y\in V$, and Hilbert Triple: a Densely and Continuously Embedded Hilbert Space and Its Form Operator (iv) gives $\hat{U}\in D(A)$ with $A\hat{U}=z=\hat{U}-\hat{D}$, which is claim 4.