The Trigonometric System is an Orthonormal Basis of the Square-Integrable 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 integer lattice $\mathbb{Z}^{n}$ and the real Hilbert space $L^{2}(\mathbb{T}^{n})$ are the ones fixed there. Let $\langle\,\cdot\,,\cdot\,\rangle_{L^{2}}$ be the inner product of $L^{2}(\mathbb{T}^{n})$, and let $0_{L^{2}}$ be the zero vector of $L^{2}(\mathbb{T}^{n})$. Let $E_{k}$ for $k\in\mathbb{Z}^{n}$ be the classes of the trigonometric system on $\mathbb{T}^{n}$ introduced in The Trigonometric System on the Torus is Orthonormal §classes. Then the following hold.

1. (Completeness)¶ Let $U\in L^{2}(\mathbb{T}^{n})$ satisfy $\langle U,E_{k}\rangle_{L^{2}}=0$ for every $k\in\mathbb{Z}^{n}$. Then $U=0_{L^{2}}$.

2. (Orthonormal bases)¶ Let $\kappa:\mathbb{N}\to\mathbb{Z}^{n}$ be a bijection, one of which exists by The Integer Lattice Admits an Enumeration by the Natural Numbers §enumeration. Then $(E_{\kappa(j)})_{j\in\mathbb{N}}$ is an orthonormal basis of $L^{2}(\mathbb{T}^{n})$.