The Sobolev Space on the Torus is a Separable Hilbert Space Densely Embedded in the Square-Integrable Space

We work in the setting of The Flat Torus: Standing Notation, used here with a natural number $n$ satisfying $1\le n$; the cell $Q$, the measure space $(Q,\mathcal{B}_{Q},\lambda_{Q})$, the class $\mathcal{L}^{2}(\mathbb{T}^{n})$ and the space $L^{2}(\mathbb{T}^{n})$ with the class map $[\,\cdot\,]$, the periodic classes $C_{\mathrm{per}}$, $C^{1}_{\mathrm{per}}$ and $C^{\infty}_{\mathrm{per}}$, the restriction $u|_{Q}$, and the partial derivatives $\partial_{i}$ are the ones fixed there. Weak partial derivatives of classes and the notation $\partial_{j}U$ are those of The Weak Partial Derivative on the Torus, and $H^{1}(\mathbb{T}^{n})$, together with its inner product $\langle\,\cdot\,,\cdot\,\rangle_{H^{1}}$, norm $\lVert\,\cdot\,\rVert_{H^{1}}$ and distance $d_{H^{1}}$, is the Sobolev space fixed there. We write $\langle\,\cdot\,,\cdot\,\rangle_{L^{2}}$, $\lVert\,\cdot\,\rVert_{L^{2}}$ and $d_{L^{2}}$ for the inner product of $L^{2}(\mathbb{T}^{n})$ and its norm and distance, which by that clause are the norm $\lVert\,\cdot\,\rVert_{L^{2}(\mathbb{T}^{n})}$ and the distance fixed in The Flat Torus: Standing Notation (iv). Let $\psi\star u$ denote the periodic convolution and let a sum $\sum_{j=1}^{n}$ be that of Finite Sum Notation in a Field. Density, convergence and separability in $L^{2}(\mathbb{T}^{n})$ and in $H^{1}(\mathbb{T}^{n})$ refer to the respective distances, as in Real Hilbert Spaces (ii).

Then the following hold.

1. (Comparison of the two norms)(i) Let $U,U'\in H^{1}(\mathbb{T}^{n})$ and let $j\in[n]$. Then

$$\lVert U\rVert_{L^{2}}\le\lVert U\rVert_{H^{1}},\qquad \lVert\partial_{j}U\rVert_{L^{2}}\le\lVert U\rVert_{H^{1}},$$

and consequently

$$d_{L^{2}}(U,U')\le d_{H^{1}}(U,U'),\qquad d_{L^{2}}(\partial_{j}U,\partial_{j}U')\le d_{H^{1}}(U,U').$$

2. (Completeness)(ii) $H^{1}(\mathbb{T}^{n})$, with the inner product $\langle\,\cdot\,,\cdot\,\rangle_{H^{1}}$, is a real Hilbert space.

3. (Mollification in the Sobolev space)(iii) Let $\delta$ be a real number with $0<\delta$, let $\rho$ be a mollifier kernel of radius $\delta$ on $\mathbb{R}^{n}$, and for a natural number $k$ let $\rho_{1/k}$ be the rescaling of $\rho$ with parameter $1/k$, where $1/k$ denotes the multiplicative inverse of $k$, a positive real number by claim 7 of Elementary Order Arithmetic in an Ordered Field; thus $\rho_{1/k}$ is a mollifier kernel of radius $(1/k)\,\delta$ on $\mathbb{R}^{n}$. Let $U\in H^{1}(\mathbb{T}^{n})$, let $u$ be a representative of $U$, and for $j\in[n]$ let $g_{j}$ be a representative of $\partial_{j}U$. For a natural number $k$ write

$$U_{k}=\bigl[(\rho_{1/k}\star u)|_{Q}\bigr],$$

the class of the restriction to $Q$ of the mollification of $u$ of parameter $1/k$ by $\rho$. Then for every natural number $k$ the following hold.

(a) (iv) $\rho_{1/k}\star u\in C^{\infty}_{\mathrm{per}}$, and $U_{k}\in H^{1}(\mathbb{T}^{n})$ with $\partial_{j}U_{k}=[(\rho_{1/k}\star g_{j})|_{Q}]$ for every $j\in[n]$.

(b) (v) $\lVert U_{k}\rVert_{H^{1}}\le\lVert U\rVert_{H^{1}}$.

Moreover, (vi) the sequence $(U_{k})_{k\in\mathbb{N}}$ converges to $U$ in $H^{1}(\mathbb{T}^{n})$.

4. (The smooth periodic classes are dense)(vii) The set $\{[w|_{Q}]:w\in C^{\infty}_{\mathrm{per}}\}$ is a subset of $H^{1}(\mathbb{T}^{n})$ and is dense in $H^{1}(\mathbb{T}^{n})$.

5. (Separability)(viii) The metric space $(H^{1}(\mathbb{T}^{n}),d_{H^{1}})$ is separable.

6. (Density in the square-integrable space)(ix) $H^{1}(\mathbb{T}^{n})$ is a dense subset of $L^{2}(\mathbb{T}^{n})$.