Finite Linear Combinations of Continuous Periodic Functions and Their 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 natural numbers $\mathbb{N}$ and the initial segments $[m]$, Euclidean space $\mathbb{R}^{n}$, the cell $Q$, the class $\mathcal{L}^{2}(\mathbb{T}^{n})$, the space $L^{2}(\mathbb{T}^{n})$ with the class map $[\,\cdot\,]$, the periodic class $C_{\mathrm{per}}$ and the restriction $u|_{Q}$ are the ones fixed there, and $\langle\,\cdot\,,\cdot\,\rangle_{L^{2}}$ is the inner product of $L^{2}(\mathbb{T}^{n})$. For maps $v,w:\mathbb{R}^{n}\to\mathbb{R}$ and $c\in\mathbb{R}$, the maps $v+w$ and $cv$ are the pointwise sum and scalar multiple of Constants, Affine Functions, Sums, Products and Quotients of Cᵏ Functions. Finite sums of real numbers are those of The Real Numbers: Standing Notation and Background (ii), and finite sums in $L^{2}(\mathbb{T}^{n})$ are the finite sums of that vector space.

Let $M\in\mathbb{N}$, let $c:[M]\to\mathbb{R}$ be a map with values written $c_{j}$, and for every $j\in[M]$ let $g_{j}\in C_{\mathrm{per}}$. Let $u:\mathbb{R}^{n}\to\mathbb{R}$ be the map

$$u(x)=\sum_{j=1}^{M}c_{j}\,g_{j}(x)\qquad(x\in\mathbb{R}^{n}).$$

Then the following hold.

1. (Membership)(i) $u\in C_{\mathrm{per}}$; moreover $u|_{Q}$, and each $g_{j}|_{Q}$, belong to $\mathcal{L}^{2}(\mathbb{T}^{n})$, by Continuous Periodic Functions are Power-Integrable and Dense on the Torus (i) used with the exponent $2$.

2. (The class)(ii) In $L^{2}(\mathbb{T}^{n})$,

$$[\,u|_{Q}\,]=\sum_{j=1}^{M}c_{j}\,[\,g_{j}|_{Q}\,].$$

3. (Pairing)(iii) For every $U\in L^{2}(\mathbb{T}^{n})$,

$$\bigl\langle U,[\,u|_{Q}\,]\bigr\rangle_{L^{2}}=\sum_{j=1}^{M}c_{j}\,\bigl\langle U,[\,g_{j}|_{Q}\,]\bigr\rangle_{L^{2}} .$$