The Sobolev Space of Once Weakly Differentiable Square-Integrable Classes on the Torus

We work in the setting of The Flat Torus: Standing Notation, used here with a natural number $n$ satisfying $1\le n$; the measure space $(Q,\mathcal{B}_{Q},\lambda_{Q})$, the class $\mathcal{L}^{2}(\mathbb{T}^{n})$, the space $L^{2}(\mathbb{T}^{n})$ with the class map $[\,\cdot\,]$, the periodic class $C^{1}_{\mathrm{per}}$ and the restriction $u|_{Q}$ are the ones fixed there. Weak partial derivatives of classes and the notation $\partial_{j}U$ and $\nabla U$ are those of The Weak Partial Derivative on the Torus, $\langle\,\cdot\,,\cdot\,\rangle_{L^{2}}$ is the inner product of $L^{2}(\mathbb{T}^{n})$, and a sum $\sum_{j=1}^{n}$ is that of Finite Sum Notation in a Field.

1. (The Sobolev space)(i) $H^{1}(\mathbb{T}^{n})$ denotes the set of those $U\in L^{2}(\mathbb{T}^{n})$ for which, for every $j\in[n]$, the $j$-th weak partial derivative of $U$ exists in $L^{2}(\mathbb{T}^{n})$; it is a linear subspace of $L^{2}(\mathbb{T}^{n})$, and contains $[u|_{Q}]$ for every $u\in C^{1}_{\mathrm{per}}$, by Elementary Properties of the Weak Partial Derivative on the Torus (vi).

2. (The inner product, norm and distance)(ii) $\langle\,\cdot\,,\cdot\,\rangle_{H^{1}}$ denotes the map sending $(U,V)\in H^{1}(\mathbb{T}^{n})\times H^{1}(\mathbb{T}^{n})$ to

$$\langle U,V\rangle_{H^{1}}=\langle U,V\rangle_{L^{2}}+\sum_{j=1}^{n}\langle\partial_{j}U,\partial_{j}V\rangle_{L^{2}} ,$$

which is an inner product on $H^{1}(\mathbb{T}^{n})$ by Elementary Properties of the Weak Partial Derivative on the Torus (vi), so that $H^{1}(\mathbb{T}^{n})$ together with it is a real inner product space. Its norm is written $\lVert\,\cdot\,\rVert_{H^{1}}$ and its distance $d_{H^{1}}$; every metric and topological notion applied to $H^{1}(\mathbb{T}^{n})$ refers to these.

This clause and the previous one introduce no object beyond the names just fixed.