Proof 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

Each result cited is universally quantified over the data in its own statement, and so are the claims of the present lemma, which is stated for an arbitrary $m\in\mathbb{N}$; claim 5 is applied below with $m+1$ in place of $m$. Write $\Lambda=\Lambda_{m}$. Two coefficient families are equal exactly when their values agree at every point of $\mathbb{Z}^{n}$. By The Negative-Order Sobolev Spaces of the Torus (i), a family $c$ lies in $H^{-m}(\mathbb{T}^{n})$ exactly when some class $W$ has $\hat{W}(k)=\rho_{k}^{m}c(k)$ for every $k$, and then $\Lambda c$ is that class; by Properties of the Fourier Coefficients on the Torus, and the Realisation of Weighted Coefficient Families (vi), $\Lambda$ is a linear bijection from the real vector space $H^{-m}(\mathbb{T}^{n})$ onto $L^{2}(\mathbb{T}^{n})$, and by The Negative-Order Sobolev Spaces of the Torus (ii), $\langle c,d\rangle_{H^{-m}}=\langle\Lambda c,\Lambda d\rangle_{L^{2}}$. Hence Transport of an Inner Product and of Hilbert Space Structure along a Linear Bijection applies with $X=H^{-m}(\mathbb{T}^{n})$, $E=L^{2}(\mathbb{T}^{n})$ and $T=\Lambda$, its transported inner product being $\langle\,\cdot\,,\cdot\,\rangle_{H^{-m}}$, and its $T^{-1}$ the inverse $\Lambda^{-1}$, which sends $W\in L^{2}(\mathbb{T}^{n})$ to the unique $c$ with $\Lambda c=W$, by Bijection of Sets and claim 1 of that lemma. Squares of real numbers satisfy $(ab)^{2}=a^{2}b^{2}$ by claim 3 of Properties of Natural Number Powers in a Field, and $0\le t^{2}$ for every real $t$ by claim 2 of Nonnegativity of Squares in an Ordered Field; the multiplicative inverse of a positive real number exists and is positive by claim 7 of Elementary Order Arithmetic in an Ordered Field.

Claim 1. Fix $k\in\mathbb{Z}^{n}$. The number $\rho_{k}$ is positive, as recorded in Properties of the Fourier Coefficients on the Torus, and the Realisation of Weighted Coefficient Families. Since $1\le\mu_{k}$ by Summability of the Negative Powers of the Fourier Weights of the Torus (i) and $0\le\mu_{k}^{-1}$ by claim 7 of Elementary Order Arithmetic in an Ordered Field, claim 5 of Elementary Arithmetic in an Ordered Field gives $\rho_{k}^{2}=\mu_{k}^{-1}\cdot1\le\mu_{k}^{-1}\mu_{k}=1=1^{2}$; as $0\le\rho_{k}$ and $0\le1$, claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $\rho_{k}\le1$. By claim 5 of Properties of Natural Number Powers in a Field, $0\le\rho_{k}^{m}$ and $\rho_{k}^{m}\le1^{m}$, and $1^{m}=1$ by claim 2 of that lemma; by claim 4 of that lemma $\rho_{k}^{m}\ne0$, so $0<\rho_{k}^{m}$. By claim 1 of that lemma, with $S(m)=m+1$ from Natural Numbers, $\rho_{k}^{m+1}=\rho_{k}^{m}\rho_{k}$. Finally, by claim 3 of Properties of Natural Number Powers in a Field applied twice, $(\rho_{k}^{m})^{2}=\rho_{k}^{m}\rho_{k}^{m}=(\rho_{k}\rho_{k})^{m}=(\mu_{k}^{-1})^{m}$ and $(\mu_{k}^{-1})^{m}\mu_{k}^{m}=(\mu_{k}^{-1}\mu_{k})^{m}=1^{m}=1$; since $\mu_{k}^{m}\ne0$ by claim 4 of that lemma, $(\rho_{k}^{m})^{2}$ is the multiplicative inverse of $\mu_{k}^{m}$, that is, $(\rho_{k}^{m})^{2}=\tfrac{1}{\mu_{k}^{m}}$.

Claim 2. By Transport of an Inner Product and of Hilbert Space Structure along a Linear Bijection (i), $|c|_{H^{-m}}=\lVert\Lambda c\rVert_{L^{2}}$ and $d_{H^{-m}}(c,d)=d_{L^{2}}(\Lambda c,\Lambda d)$, the norm and distance of $L^{2}(\mathbb{T}^{n})$ being those of its inner product by L²(μ) Is a Real Hilbert Space (i). Since $L^{2}(\mathbb{T}^{n})$ is a real Hilbert space by L²(μ) Is a Real Hilbert Space (ii), Transport of an Inner Product and of Hilbert Space Structure along a Linear Bijection (iii) shows that $H^{-m}(\mathbb{T}^{n})$ with $\langle\,\cdot\,,\cdot\,\rangle_{H^{-m}}$ is a real Hilbert space. That $\Lambda$ is a linear bijection is Properties of the Fourier Coefficients on the Torus, and the Realisation of Weighted Coefficient Families (vi). Separability is proved after claim 4.

Claim 3. Let $U\in L^{2}(\mathbb{T}^{n})$ and let $\kappa$ be an enumeration. Define the family $\tilde{c}\in\mathrm{Map}(\mathbb{Z}^{n},\mathbb{R})$ by $\tilde{c}(k)=\rho_{k}^{m}\hat{U}(k)$. By Properties of the Fourier Coefficients on the Torus, and the Realisation of Weighted Coefficient Families (iv) the series $\sum_{j=1}^{\infty}\hat{U}(\kappa(j))^{2}$ converges with sum $(\lVert U\rVert_{L^{2}})^{2}$. For every $j$, $\tilde{c}(\kappa(j))^{2}=(\rho_{\kappa(j)}^{m})^{2}\hat{U}(\kappa(j))^{2}$, where $0\le\hat{U}(\kappa(j))^{2}$ and $0\le\tilde{c}(\kappa(j))^{2}$ by claim 2 of Nonnegativity of Squares in an Ordered Field, and $(\rho_{\kappa(j)}^{m})^{2}\le1=1^{2}$ by claim 1 and claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field; hence $\tilde{c}(\kappa(j))^{2}\le\hat{U}(\kappa(j))^{2}$ by claim 5 of Elementary Arithmetic in an Ordered Field, the multiplier $\hat{U}(\kappa(j))^{2}$ being nonnegative. By Series of Nonnegative Real Numbers, Comparison, and the Geometric Series (iii), $\sum_{j=1}^{\infty}\tilde{c}(\kappa(j))^{2}$ converges with sum at most $(\lVert U\rVert_{L^{2}})^{2}$. By Properties of the Fourier Coefficients on the Torus, and the Realisation of Weighted Coefficient Families (v) there is exactly one $W\in L^{2}(\mathbb{T}^{n})$ with $\hat{W}=\tilde{c}$, namely the sum of the convergent series $\sum_{j=1}^{\infty}\tilde{c}(\kappa(j))E_{\kappa(j)}=\sum_{j=1}^{\infty}\rho_{\kappa(j)}^{m}\hat{U}(\kappa(j))E_{\kappa(j)}$, and $(\lVert W\rVert_{L^{2}})^{2}=\sum_{j=1}^{\infty}\tilde{c}(\kappa(j))^{2}\le(\lVert U\rVert_{L^{2}})^{2}$. Since $\hat{W}(k)=\rho_{k}^{m}\hat{U}(k)$ for every $k$, the family $\hat{U}$ lies in $H^{-m}(\mathbb{T}^{n})$ with $\Lambda\hat{U}=W$; this gives the series representation of $\Lambda\hat{U}$, and $(|\hat{U}|_{H^{-m}})^{2}=(\lVert W\rVert_{L^{2}})^{2}\le(\lVert U\rVert_{L^{2}})^{2}$ by claim 2, whence $|\hat{U}|_{H^{-m}}\le\lVert U\rVert_{L^{2}}$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, both norms being nonnegative by Real Inner Product Spaces (ii). The map $U\mapsto\hat{U}$, now regarded as a map into the vector space $H^{-m}(\mathbb{T}^{n})$, whose operations are those of $\mathrm{Map}(\mathbb{Z}^{n},\mathbb{R})$ by The Negative-Order Sobolev Spaces of the Torus (i), is linear by Properties of the Fourier Coefficients on the Torus, and the Realisation of Weighted Coefficient Families (i) and satisfies $\hat{U}=\hat{U}'$ only if $U=U'$ by Properties of the Fourier Coefficients on the Torus, and the Realisation of Weighted Coefficient Families (ii).

Now let $k\in\mathbb{Z}^{n}$ and put $W=\rho_{k}^{m}E_{k}$. For $k'\in\mathbb{Z}^{n}$, Properties of the Fourier Coefficients on the Torus, and the Realisation of Weighted Coefficient Families (i) and Properties of the Fourier Coefficients on the Torus, and the Realisation of Weighted Coefficient Families (iii) give $\hat{W}(k')=\rho_{k}^{m}\hat{E}_{k}(k')$, which equals $\rho_{k}^{m}$ if $k'=k$ and $0$ otherwise; and $\rho_{k'}^{m}\hat{E}_{k}(k')$ takes the same two values. So $\hat{W}(k')=\rho_{k'}^{m}\hat{E}_{k}(k')$ for every $k'$, and by uniqueness $\Lambda\hat{E}_{k}=W=\rho_{k}^{m}E_{k}$. Hence, by claim 2, Elementary Identities in a Real Inner Product Space (iv), Absolute Value in an Ordered Field with $0<\rho_{k}^{m}$, and $\lVert E_{k}\rVert_{L^{2}}=1$ from The Trigonometric System on the Torus is Orthonormal (iii) and Real Inner Product Spaces (ii), $|\hat{E}_{k}|_{H^{-m}}=\lVert\rho_{k}^{m}E_{k}\rVert_{L^{2}}=\rho_{k}^{m}\lVert E_{k}\rVert_{L^{2}}=\rho_{k}^{m}$.

Claim 4. Let $k\in\mathbb{Z}^{n}$. Since $\Lambda$ is a bijection, there is exactly one $\zeta_{m,k}\in H^{-m}(\mathbb{T}^{n})$ with $\Lambda\zeta_{m,k}=E_{k}$, and it is the value at $E_{k}$ of the map $T^{-1}$ of Transport of an Inner Product and of Hilbert Space Structure along a Linear Bijection. By The Negative-Order Sobolev Spaces of the Torus (i), $\hat{E}_{k}(k')=\rho_{k'}^{m}\zeta_{m,k}(k')$ for every $k'$; multiplying by $(\rho_{k'}^{m})^{-1}$, which exists because $0<\rho_{k'}^{m}$ by claim 1, and using Properties of the Fourier Coefficients on the Torus, and the Realisation of Weighted Coefficient Families (iii), $\zeta_{m,k}(k')$ equals $(\rho_{k}^{m})^{-1}=\tfrac{1}{\rho_{k}^{m}}$ if $k'=k$ and $0$ otherwise. For $c\in H^{-m}(\mathbb{T}^{n})$, by The Negative-Order Sobolev Spaces of the Torus (ii) and The Fourier Coefficients of a Square-Integrable Class on the Torus (i), $\langle c,\zeta_{m,k}\rangle_{H^{-m}}=\langle\Lambda c,E_{k}\rangle_{L^{2}}=\widehat{\Lambda c}(k)=\rho_{k}^{m}c(k)$, the last by The Negative-Order Sobolev Spaces of the Torus (i). Finally let $\kappa$ be an enumeration. By The Trigonometric System is an Orthonormal Basis of the Square-Integrable Space of the Torus (ii), $(E_{\kappa(j)})_{j\in\mathbb{N}}$ is an orthonormal basis of $L^{2}(\mathbb{T}^{n})$, so Transport of an Inner Product and of Hilbert Space Structure along a Linear Bijection (iv) shows that $(T^{-1}E_{\kappa(j)})_{j\in\mathbb{N}}=(\zeta_{m,\kappa(j)})_{j\in\mathbb{N}}$ is an orthonormal basis of $H^{-m}(\mathbb{T}^{n})$.

Separability. An enumeration exists by The Integer Lattice Admits an Enumeration by the Natural Numbers (ii), so claim 4 furnishes an orthonormal basis of the real Hilbert space $H^{-m}(\mathbb{T}^{n})$, and A Real Hilbert Space with an Orthonormal Basis is Separable (ii) shows that $(H^{-m}(\mathbb{T}^{n}),d_{H^{-m}})$ is separable. This completes claim 2.

Claim 5. Let $\kappa$ be an enumeration, let $c\in\mathrm{Map}(\mathbb{Z}^{n},\mathbb{R})$, and define $\tilde{c}\in\mathrm{Map}(\mathbb{Z}^{n},\mathbb{R})$ by $\tilde{c}(k)=\rho_{k}^{m}c(k)$, so that $\tilde{c}(\kappa(j))^{2}=(\rho_{\kappa(j)}^{m})^{2}c(\kappa(j))^{2}$ for every $j$. If $c\in H^{-m}(\mathbb{T}^{n})$, then $\widehat{\Lambda c}=\tilde{c}$, and Properties of the Fourier Coefficients on the Torus, and the Realisation of Weighted Coefficient Families (iv) shows that $\sum_{j=1}^{\infty}\tilde{c}(\kappa(j))^{2}$ converges. Conversely, if that series converges, Properties of the Fourier Coefficients on the Torus, and the Realisation of Weighted Coefficient Families (v) furnishes $W\in L^{2}(\mathbb{T}^{n})$ with $\hat{W}=\tilde{c}$, so $c\in H^{-m}(\mathbb{T}^{n})$ by The Negative-Order Sobolev Spaces of the Torus (i). Now let $c,d\in H^{-m}(\mathbb{T}^{n})$. By The Negative-Order Sobolev Spaces of the Torus (ii) and Properties of the Fourier Coefficients on the Torus, and the Realisation of Weighted Coefficient Families (iv), applied to $\Lambda c$ and $\Lambda d$,

$$\langle c,d\rangle_{H^{-m}}=\langle\Lambda c,\Lambda d\rangle_{L^{2}}=\sum_{j=1}^{\infty}\widehat{\Lambda c}(\kappa(j))\,\widehat{\Lambda d}(\kappa(j))=\sum_{j=1}^{\infty}(\rho_{\kappa(j)}^{m})^{2}\,c(\kappa(j))\,d(\kappa(j)),$$

the series converging, since $\widehat{\Lambda c}(k)\widehat{\Lambda d}(k)=\rho_{k}^{m}c(k)\rho_{k}^{m}d(k)=(\rho_{k}^{m})^{2}c(k)d(k)$. Taking $d=c$ and using $(|c|_{H^{-m}})^{2}=\langle c,c\rangle_{H^{-m}}$ from Real Inner Product Spaces (ii) gives the last assertion.

Claim 6. Let $c\in H^{-m}(\mathbb{T}^{n})$ and let $\kappa$ be an enumeration, which exists by The Integer Lattice Admits an Enumeration by the Natural Numbers (ii). By claim 5, $\sum_{j=1}^{\infty}(\rho_{\kappa(j)}^{m})^{2}c(\kappa(j))^{2}$ converges with sum $(|c|_{H^{-m}})^{2}$. For every $k$, claim 1 gives $(\rho_{k}^{m+1})^{2}=(\rho_{k}^{m}\rho_{k})^{2}=(\rho_{k}^{m})^{2}\rho_{k}^{2}$ with $\rho_{k}^{2}\le1=1^{2}$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field applied to $0\le\rho_{k}\le1$; hence, since $0\le(\rho_{k}^{m+1})^{2}c(k)^{2}$ and $0\le(\rho_{k}^{m})^{2}c(k)^{2}$ by claim 2 of Nonnegativity of Squares in an Ordered Field and claim 3 of Properties of Natural Number Powers in a Field, claim 5 of Elementary Arithmetic in an Ordered Field with the nonnegative multiplier $(\rho_{k}^{m})^{2}c(k)^{2}$ gives

$$0\le(\rho_{k}^{m+1})^{2}c(k)^{2}=\rho_{k}^{2}\,(\rho_{k}^{m})^{2}c(k)^{2}\le(\rho_{k}^{m})^{2}c(k)^{2}.$$

By Series of Nonnegative Real Numbers, Comparison, and the Geometric Series (iii), $\sum_{j=1}^{\infty}(\rho_{\kappa(j)}^{m+1})^{2}c(\kappa(j))^{2}$ converges with sum at most $(|c|_{H^{-m}})^{2}$. By claim 5 applied with $m+1$ in place of $m$, $c\in H^{-(m+1)}(\mathbb{T}^{n})$ and $(|c|_{H^{-(m+1)}})^{2}$ is that sum, so $(|c|_{H^{-(m+1)}})^{2}\le(|c|_{H^{-m}})^{2}$, and $|c|_{H^{-(m+1)}}\le|c|_{H^{-m}}$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, both norms being nonnegative.