The Negative-Order Sobolev Spaces of the Torus

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 its inner product $\langle\,\cdot\,,\cdot\,\rangle_{L^{2}}$ are the ones fixed there. The real vector space $\mathrm{Map}(\mathbb{Z}^{n},\mathbb{R})$ of coefficient families, the Fourier coefficient family $\hat{W}$ of a class $W$, the Fourier weights $\mu_{k}$ and the positive real numbers $\rho_{k}$ with $\rho_{k}^{2}=\tfrac{1}{\mu_{k}}$, and their natural powers $\rho_{k}^{m}$ are as in Properties of the Fourier Coefficients on the Torus, and the Realisation of Weighted Coefficient Families. Let $m\in\mathbb{N}$.

1. (The space)(i) The Sobolev space of order $-m$ on $\mathbb{T}^{n}$, written $H^{-m}(\mathbb{T}^{n})$ and realised concretely as a space of coefficient families, is the set of those $c\in\mathrm{Map}(\mathbb{Z}^{n},\mathbb{R})$ for which there is a class $W\in L^{2}(\mathbb{T}^{n})$ with

$$\hat{W}(k)=\rho_{k}^{m}\,c(k)\qquad\text{for every }k\in\mathbb{Z}^{n};$$

it is the set $\mathcal{H}_{m}$ of Properties of the Fourier Coefficients on the Torus, and the Realisation of Weighted Coefficient Families (vi), a linear subspace of $\mathrm{Map}(\mathbb{Z}^{n},\mathbb{R})$ and a real vector space under the pointwise operations by that clause. For $c\in H^{-m}(\mathbb{T}^{n})$ the class $W$ above is unique by Properties of the Fourier Coefficients on the Torus, and the Realisation of Weighted Coefficient Families (vi), which writes it $\Lambda_{m}c$; it is called the realisation of $c$.

2. (The inner product)(ii) The map sending $(c,d)\in H^{-m}(\mathbb{T}^{n})\times H^{-m}(\mathbb{T}^{n})$ to

$$\langle c,d\rangle_{H^{-m}}=\langle\Lambda_{m}c,\Lambda_{m}d\rangle_{L^{2}}$$

is an inner product on $H^{-m}(\mathbb{T}^{n})$, by Transport of an Inner Product and of Hilbert Space Structure along a Linear Bijection (i) applied to the linear bijection $\Lambda_{m}$ of Properties of the Fourier Coefficients on the Torus, and the Realisation of Weighted Coefficient Families (vi); its norm and distance are written $|\cdot|_{H^{-m}}$ and $d_{H^{-m}}$, the latter a metric by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity (iii), and every metric and topological notion applied to $H^{-m}(\mathbb{T}^{n})$ refers to them.

These two clauses introduce no object beyond the names just fixed.