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

We work in the setting of The Flat Torus: Standing Notation, used here with a natural number $n$ satisfying $1\le n$, and in the setting of Real Hilbert Spaces: Standing Notation and Background, whose standing space $H$ is not used here; the integer lattice $\mathbb{Z}^{n}$ and the real Hilbert space $L^{2}(\mathbb{T}^{n})$, with inner product $\langle\,\cdot\,,\cdot\,\rangle_{L^{2}}$, norm $\lVert\,\cdot\,\rVert_{L^{2}}$ and distance $d_{L^{2}}$, are the ones fixed there. The real vector space $\mathrm{Map}(\mathbb{Z}^{n},\mathbb{R})$ of coefficient families, the Fourier coefficient family $\hat{U}$ of a class $U$, the classes $E_{k}$, the enumerations $\kappa$ of the lattice, the Fourier weights $\mu_{k}$, the positive numbers $\rho_{k}$ with $\rho_{k}^{2}=\tfrac{1}{\mu_{k}}$, and the natural powers $t^{l}$ of real numbers, in particular $\rho_{k}^{l}$ and $\mu_{k}^{l}$ for $l\in\mathbb{N}$, with $t^{2}=tt$, are as in Properties of the Fourier Coefficients on the Torus, and the Realisation of Weighted Coefficient Families. For $l\in\mathbb{N}$, $H^{-l}(\mathbb{T}^{n})$ is the Sobolev space of order $-l$ with its realisation map $\Lambda_{l}$ and its inner product $\langle\,\cdot\,,\cdot\,\rangle_{H^{-l}}$, norm $|\cdot|_{H^{-l}}$ and distance $d_{H^{-l}}$; in claim 6, $H^{-(m+1)}(\mathbb{T}^{n})$ and $|\cdot|_{H^{-(m+1)}}$ are these for $l=m+1$. Series of real numbers and series in a real inner product space are as defined there. Let $m\in\mathbb{N}$. Then the following hold.

1. (The weights)(i) For every $k\in\mathbb{Z}^{n}$: $\rho_{k}\le1$, $0<\rho_{k}^{m}\le1$, $\rho_{k}^{m+1}=\rho_{k}^{m}\rho_{k}$, and $(\rho_{k}^{m})^{2}\,\mu_{k}^{m}=1$, so that $(\rho_{k}^{m})^{2}=\tfrac{1}{\mu_{k}^{m}}$.

2. (Hilbert space)(ii) $H^{-m}(\mathbb{T}^{n})$ with $\langle\,\cdot\,,\cdot\,\rangle_{H^{-m}}$ is a real Hilbert space, and the metric space $(H^{-m}(\mathbb{T}^{n}),d_{H^{-m}})$ is separable. The realisation map $\Lambda_{m}$ is a linear bijection from $H^{-m}(\mathbb{T}^{n})$ onto $L^{2}(\mathbb{T}^{n})$, and

$$|c|_{H^{-m}}=\lVert\Lambda_{m}c\rVert_{L^{2}},\qquad d_{H^{-m}}(c,d)=d_{L^{2}}(\Lambda_{m}c,\Lambda_{m}d)\qquad\text{for all }c,d\in H^{-m}(\mathbb{T}^{n}).$$

3. (The square-integrable classes embed)(iii) Let $U\in L^{2}(\mathbb{T}^{n})$. Then $\hat{U}\in H^{-m}(\mathbb{T}^{n})$ and $|\hat{U}|_{H^{-m}}\le\lVert U\rVert_{L^{2}}$; for every enumeration $\kappa$ the series $\sum_{j=1}^{\infty}\rho_{\kappa(j)}^{m}\,\hat{U}(\kappa(j))\,E_{\kappa(j)}$ converges in $L^{2}(\mathbb{T}^{n})$ with sum $\Lambda_{m}\hat{U}$. The map $U\mapsto\hat{U}$ from $L^{2}(\mathbb{T}^{n})$ to $H^{-m}(\mathbb{T}^{n})$ is linear and injective: if $U,U'\in L^{2}(\mathbb{T}^{n})$ satisfy $\hat{U}=\hat{U}'$, then $U=U'$. Moreover, for every $k\in\mathbb{Z}^{n}$,

$$\Lambda_{m}\hat{E}_{k}=\rho_{k}^{m}\,E_{k},\qquad |\hat{E}_{k}|_{H^{-m}}=\rho_{k}^{m}.$$

4. (The rescaled trigonometric basis)(iv) For $k\in\mathbb{Z}^{n}$ let $\zeta_{m,k}$ be the unique element of $H^{-m}(\mathbb{T}^{n})$ with $\Lambda_{m}\zeta_{m,k}=E_{k}$. Then $\zeta_{m,k}(k')$ equals $\tfrac{1}{\rho_{k}^{m}}$ if $k'=k$ and equals $0$ if $k'\ne k$; for every $c\in H^{-m}(\mathbb{T}^{n})$,

$$\langle c,\zeta_{m,k}\rangle_{H^{-m}}=\rho_{k}^{m}\,c(k);$$

and for every enumeration $\kappa$ the sequence $(\zeta_{m,\kappa(j)})_{j\in\mathbb{N}}$ is an orthonormal basis of $H^{-m}(\mathbb{T}^{n})$.

5. (Membership and the inner product as series)(v) Let $\kappa$ be an enumeration. A coefficient family $c\in\mathrm{Map}(\mathbb{Z}^{n},\mathbb{R})$ belongs to $H^{-m}(\mathbb{T}^{n})$ if and only if the series $\sum_{j=1}^{\infty}(\rho_{\kappa(j)}^{m})^{2}\,c(\kappa(j))^{2}$ converges. For $c,d\in H^{-m}(\mathbb{T}^{n})$ the series $\sum_{j=1}^{\infty}(\rho_{\kappa(j)}^{m})^{2}\,c(\kappa(j))\,d(\kappa(j))$ converges with sum $\langle c,d\rangle_{H^{-m}}$; in particular $(|c|_{H^{-m}})^{2}=\sum_{j=1}^{\infty}(\rho_{\kappa(j)}^{m})^{2}\,c(\kappa(j))^{2}$.

6. (The scale)(vi) $H^{-m}(\mathbb{T}^{n})\subseteq H^{-(m+1)}(\mathbb{T}^{n})$, and $|c|_{H^{-(m+1)}}\le|c|_{H^{-m}}$ for every $c\in H^{-m}(\mathbb{T}^{n})$.