The Trigonometric System is Continuously Differentiable and Orthogonal in the Sobolev Space of the Torus

We work in the setting of The Flat Torus: Standing Notation, used here with a natural number $n$ satisfying $1\le n$; the initial segments $[n]$, Euclidean space $\mathbb{R}^{n}$ with its norm $\lVert\,\cdot\,\rVert$, the integer lattice $\mathbb{Z}^{n}$, the half-open unit cell $Q$, the periodic classes $C_{\mathrm{per}}$ and $C^{1}_{\mathrm{per}}$, the space $L^{2}(\mathbb{T}^{n})$ with the class map $[\,\cdot\,]$, the restriction $u|_{Q}$, and the partial derivatives $\partial_{i}$ and the classes $C^{k}$ on a Euclidean open set are the ones fixed there; $\mathbb{R}^{n}$ is open in itself by claim 1 of Euclidean Space Is Open in Itself, so those calculus notions apply to maps on all of $\mathbb{R}^{n}$. Let $\mathbb{Z}$ be the set of integers, let $\pi$ be the real number of The Number Pi (i) (the wrapping map of The Flat Torus: Standing Notation (ii) is not used here), and let $2=1+1$ and $4=2+2$. Let $e_{k}$ for $k\in\mathbb{Z}^{n}$ be the trigonometric system on $\mathbb{T}^{n}$ and $E_{k}$ its classes as introduced in The Trigonometric System on the Torus is Orthonormal (i). Let $H^{1}(\mathbb{T}^{n})$ be the Sobolev space of the torus with the inner product $\langle\,\cdot\,,\cdot\,\rangle_{H^{1}}$ of The Sobolev Space of Once Weakly Differentiable Square-Integrable Classes on the Torus (ii), and let $\partial_{j}U$ denote the weak partial derivative of a class $U$, as there.

For $k\in\mathbb{Z}^{n}$ and $i\in[n]$ let $k^{(i)}$ denote the point of $\mathbb{R}^{n}$ whose $i$th coordinate is $-k_{i}$ and whose $j$th coordinate is $k_{j}$ for every $j\in[n]$ with $j\ne i$. Then the following hold.

1. (Continuous differentiability and the classical derivative)(i) Let $k\in\mathbb{Z}^{n}$ and $i\in[n]$. Then $k^{(i)}\in\mathbb{Z}^{n}$, the map $e_{k}$ is of class $C^{1}$ on $\mathbb{R}^{n}$ and belongs to $C^{1}_{\mathrm{per}}$, and

$$\partial_{i}e_{k}(x)=-2\pi k_{i}\,e_{k^{(i)}}(x)\qquad\text{for every }x\in\mathbb{R}^{n}.$$

2. (Membership in the Sobolev space, and the weak derivative)(ii) Let $k\in\mathbb{Z}^{n}$. Then $E_{k}\in H^{1}(\mathbb{T}^{n})$, and for every $i\in[n]$

$$\partial_{i}E_{k}=-2\pi k_{i}\,E_{k^{(i)}} .$$

3. (Orthogonality in the Sobolev space)(iii) Let $k,m\in\mathbb{Z}^{n}$. Then $\langle E_{k},E_{m}\rangle_{H^{1}}$ equals $1+4\pi^{2}\lVert k\rVert^{2}$ if $k=m$, and equals $0$ if $k\ne m$.