Continuous Periodic Functions are Power-Integrable and Dense on the Torus

In the setting of The Flat Torus: Standing Notation, let $p$ be a real number with $1\le p$. Recall from clause 3 there the measure space $(Q,\mathcal{B}_{Q},\lambda_{Q})$ and the notation $\int_{\mathbb{T}^{n}}v\,dx$, from clause 4 the spaces $\mathcal{L}^{p}(\mathbb{T}^{n})$ and $L^{p}(\mathbb{T}^{n})$ with the norm $\lVert\,\cdot\,\rVert_{L^{p}(\mathbb{T}^{n})}$ and the metric and topology it induces, and from clause 5 the class $C_{\mathrm{per}}$. Write $\mathbf{1}_{A}$ for the indicator of $A\subseteq\mathbb{R}^{n}$. Then the following hold.

1. (Power-integrability)(i) Let $u\in C_{\mathrm{per}}$ and let $M$ be a nonnegative real number with $|u(x)|\le M$ for every $x\in\mathbb{R}^{n}$, such a number existing by Elementary Properties of Lattice-Periodic Functions (iii). Then $u|_{Q}$ is measurable with respect to $\mathcal{B}_{Q}$, belongs to $\mathcal{L}^{p}(\mathbb{T}^{n})$, and satisfies

$$\bigl\lVert[\,u|_{Q}\,]\bigr\rVert_{L^{p}(\mathbb{T}^{n})}\le M .$$

2. (The integral over the cell)(ii) Let $u\in C_{\mathrm{per}}$. Then $\mathbf{1}_{Q}u$ is measurable with respect to $\mathcal{B}(\mathbb{R}^{n})$ and integrable with respect to $\lambda_{n}$, and

$$\int_{\mathbb{T}^{n}}u|_{Q}\,dx=\int_{\mathbb{R}^{n}}\mathbf{1}_{Q}\,u\,d\lambda_{n}.$$

3. (Linearity of the restriction map)(iii) Let $u,v\in C_{\mathrm{per}}$ and $c\in\mathbb{R}$. Then $u+v$ and $cu$ lie in $C_{\mathrm{per}}$ by Elementary Properties of Lattice-Periodic Functions (i), and in $L^{p}(\mathbb{T}^{n})$

$$[\,(u+v)|_{Q}\,]=[\,u|_{Q}\,]+[\,v|_{Q}\,],\qquad [\,(cu)|_{Q}\,]=c\,[\,u|_{Q}\,].$$

4. (Density)(iv) The set

$$\mathcal{C}=\bigl\{\,[\,u|_{Q}\,]\ :\ u\in C_{\mathrm{per}}\,\bigr\}$$

is a dense subset of $L^{p}(\mathbb{T}^{n})$; equivalently, for every $f\in\mathcal{L}^{p}(\mathbb{T}^{n})$ and every real $\varepsilon$ with $0<\varepsilon$ there is $u\in C_{\mathrm{per}}$ with $\lVert[f]-[\,u|_{Q}\,]\rVert_{L^{p}(\mathbb{T}^{n})}\le\varepsilon$.