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

Each result cited is universally quantified over the data in its own statement, and is applied here to the data named below. We use silently that the order of $\mathbb{R}$ is reflexive, transitive and antisymmetric, that equal real numbers satisfy $\le$ in both directions, and that $x<y$ entails $x\le y$. We use repeatedly the following consequence of claims 2 and 3 of Elementary Arithmetic in an Ordered Field: if $a\le b$ and $c\le d$ then $a+c\le b+d$, since $0\le b-a$ and $0\le d-c$ give $0\le(b+d)-(a+c)$.

(N0) The absolute value of a nonnegative real. Let $x$ be a real number with $0\le x$. Then $|x|=x$. Indeed $x\le|x|$ by claim 3 of Properties of the Absolute Value in an Ordered Field. Also $-x\le -0$ by claim 4 of Elementary Order Arithmetic in an Ordered Field, and $-0=0$ by claim 4 of Additive Cancellation and Elementary Additive Identities in a Field, so $-x\le 0$ and hence $-x\le x$ by transitivity, while $x\le x$; hence $|x|\le x$ by claim 6 of Properties of the Absolute Value in an Ordered Field applied with $c=x$. Antisymmetry gives $|x|=x$.

(N1) Norms, squares and square roots. Let $E$ be a real inner product space with inner product $\langle\,\cdot\,,\cdot\,\rangle_{E}$, norm $\lVert\,\cdot\,\rVert_{E}$ and distance $d_{E}$. By Real Inner Product Spaces (ii), $0\le\lVert X\rVert_{E}$ and the product $\lVert X\rVert_{E}\cdot\lVert X\rVert_{E}$ equals $\langle X,X\rangle_{E}$; by Properties of Real Powers (ii) that product is the power $\lVert X\rVert_{E}^{2}$ of Real Powers (i), so $\lVert X\rVert_{E}^{2}=\langle X,X\rangle_{E}$. By Properties of Real Powers (vi), for nonnegative reals $s$ and $t$ we have $(t^{2})^{1/2}=t$, $(t^{1/2})^{2}=t$, and $s^{2}\le t$ if and only if $s\le t^{1/2}$; consequently $s^{2}\le t^{2}$ implies $s\le t$. By Properties of Real Powers (i) a power of a nonnegative real is nonnegative, and $0^{1/2}=0$. By Real Inner Product Spaces (iii), $d_{E}(X,Y)=\lVert X-Y\rVert_{E}$. We apply all of this to $L^{2}(\mathbb{T}^{n})$, which is a real inner product space by L²(μ) Is a Real Hilbert Space (i), and to $H^{1}(\mathbb{T}^{n})$, which is one by The Sobolev Space of Once Weakly Differentiable Square-Integrable Classes on the Torus (ii), without further mention.

(N2) The decomposition of the Sobolev norm and distance. Let $U,U'\in H^{1}(\mathbb{T}^{n})$. By The Sobolev Space of Once Weakly Differentiable Square-Integrable Classes on the Torus (ii) and (N1),

$$\lVert U\rVert_{H^{1}}^{2}=\lVert U\rVert_{L^{2}}^{2}+\sum_{j=1}^{n}\lVert\partial_{j}U\rVert_{L^{2}}^{2}.$$

Moreover $U-U'\in H^{1}(\mathbb{T}^{n})$, $H^{1}(\mathbb{T}^{n})$ being a linear subspace of $L^{2}(\mathbb{T}^{n})$ by The Sobolev Space of Once Weakly Differentiable Square-Integrable Classes on the Torus (i), and $\partial_{j}(U-U')=\partial_{j}U-\partial_{j}U'$ for every $j\in[n]$ by Elementary Properties of the Weak Partial Derivative on the Torus (ii) applied with $c=-1$. Applying the display to $U-U'$ and using (N1) therefore gives

$$d_{H^{1}}(U,U')^{2}=d_{L^{2}}(U,U')^{2}+\sum_{j=1}^{n}d_{L^{2}}(\partial_{j}U,\partial_{j}U')^{2}.$$

(N3) Convergence and the sequence of distances. Let $(Y,d)$ be a metric space, let $(X_{m})_{m\in\mathbb{N}}$ be a sequence in $Y$ and let $X\in Y$. Then $(X_{m})$ converges to $X$ in $(Y,d)$ if and only if the sequence of real numbers $\bigl(d(X_{m},X)\bigr)_{m\in\mathbb{N}}$ converges to $0$. Indeed the two conditions are the same assertion about $\varepsilon$ and $N$, because $0\le d(X_{m},X)$ by Metric Space and hence $|d(X_{m},X)-0|=d(X_{m},X)$ by (N0), the difference $d(X_{m},X)-0$ being $d(X_{m},X)$ by claim 4 of Additive Cancellation and Elementary Additive Identities in a Field.

(N4) A convergence criterion in the Sobolev space. Let $U$ and, for every $m\in\mathbb{N}$, $U_{m}$ belong to $H^{1}(\mathbb{T}^{n})$, and suppose that $(U_{m})$ converges to $U$ in $L^{2}(\mathbb{T}^{n})$ and that, for every $j\in[n]$, $(\partial_{j}U_{m})$ converges to $\partial_{j}U$ in $L^{2}(\mathbb{T}^{n})$. Then $(U_{m})$ converges to $U$ in $H^{1}(\mathbb{T}^{n})$.

To see this, put $b_{m}=d_{L^{2}}(U_{m},U)$ and $b_{j,m}=d_{L^{2}}(\partial_{j}U_{m},\partial_{j}U)$ for $j\in[n]$. By (N3) the real sequences $(b_{m})_{m}$ and $(b_{j,m})_{m}$ converge to $0$, so by claim 2 of Arithmetic of Limits of Real Sequences the sequences $(b_{m}^{2})_{m}$ and $(b_{j,m}^{2})_{m}$ converge to $0\cdot 0=0$, the squares being products by Properties of Real Powers (ii). By Finite Sums of Real Numbers: Nonnegativity, Domination by the Sum, and Limits (iii), applied with $N=n$ and the constant limits $0$, the sequence whose $m$th term is $\sum_{j=1}^{n}b_{j,m}^{2}$ converges to $\sum_{j=1}^{n}0$, which is $0$ by the second assertion of Finite Sums of Real Numbers: Nonnegativity, Domination by the Sum, and Limits (i). By claim 1 of Arithmetic of Limits of Real Sequences and (N2), the sequence whose $m$th term is $d_{H^{1}}(U_{m},U)^{2}$ converges to $0+0=0$. Applying Properties of Real Powers (vii) with exponent $1/2$ to this sequence of nonnegative reals, the sequence whose $m$th term is $\bigl(d_{H^{1}}(U_{m},U)^{2}\bigr)^{1/2}$ converges to $0^{1/2}$; by (N1) that $m$th term is $d_{H^{1}}(U_{m},U)$ and $0^{1/2}=0$. By (N3) again, $(U_{m})$ converges to $U$ in $H^{1}(\mathbb{T}^{n})$.

Proof of claim 1. Let $U,U'\in H^{1}(\mathbb{T}^{n})$ and let $j\in[n]$. For $k\in[n]$ put $t_{k}=\lVert\partial_{k}U\rVert_{L^{2}}^{2}$, a nonnegative real by (N1), and put $\Sigma=\sum_{k=1}^{n}t_{k}$.

By Finite Sums of Real Numbers: Nonnegativity, Domination by the Sum, and Limits (i), $0\le\Sigma$. By (N2), $\lVert U\rVert_{H^{1}}^{2}-\lVert U\rVert_{L^{2}}^{2}=\Sigma$, so claim 3 of Elementary Arithmetic in an Ordered Field gives $\lVert U\rVert_{L^{2}}^{2}\le\lVert U\rVert_{H^{1}}^{2}$, and (N1) gives $\lVert U\rVert_{L^{2}}\le\lVert U\rVert_{H^{1}}$.

By Finite Sums of Real Numbers: Nonnegativity, Domination by the Sum, and Limits (ii), applied with $N=n$, the index $n$ in place of its $j$ and the index $j$ in place of its $i$, we get $t_{j}\le\Sigma$. Also $\lVert U\rVert_{H^{1}}^{2}-\Sigma=\lVert U\rVert_{L^{2}}^{2}$ by (N2), which is nonnegative by (N1), so $\Sigma\le\lVert U\rVert_{H^{1}}^{2}$ by claim 3 of Elementary Arithmetic in an Ordered Field. By transitivity $\lVert\partial_{j}U\rVert_{L^{2}}^{2}\le\lVert U\rVert_{H^{1}}^{2}$, and (N1) gives $\lVert\partial_{j}U\rVert_{L^{2}}\le\lVert U\rVert_{H^{1}}$.

For the distance inequalities, apply the two norm inequalities just proved to $U-U'$, which lies in $H^{1}(\mathbb{T}^{n})$ by (N2). Using (N1) and the identity $\partial_{j}(U-U')=\partial_{j}U-\partial_{j}U'$ of (N2),

$$d_{L^{2}}(U,U')=\lVert U-U'\rVert_{L^{2}}\le\lVert U-U'\rVert_{H^{1}}=d_{H^{1}}(U,U'),$$ $$d_{L^{2}}(\partial_{j}U,\partial_{j}U')=\lVert\partial_{j}(U-U')\rVert_{L^{2}}\le\lVert U-U'\rVert_{H^{1}}=d_{H^{1}}(U,U').$$

Proof of claim 2. By The Sobolev Space of Once Weakly Differentiable Square-Integrable Classes on the Torus (ii), $H^{1}(\mathbb{T}^{n})$ with $\langle\,\cdot\,,\cdot\,\rangle_{H^{1}}$ is a real inner product space whose distance is $d_{H^{1}}$, and $d_{H^{1}}$ is a metric on it by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity (iii). By Real Hilbert Spaces (i) it therefore suffices to prove that the metric space $(H^{1}(\mathbb{T}^{n}),d_{H^{1}})$ is complete.

Let $(U_{m})_{m\in\mathbb{N}}$ be a Cauchy sequence in $(H^{1}(\mathbb{T}^{n}),d_{H^{1}})$ and let $j\in[n]$ be arbitrary; the argument that follows uses nothing about $j$ beyond $j\in[n]$, so its conclusion holds for every $j\in[n]$. Given a real $\varepsilon>0$, choose $N\in\mathbb{N}$ with $d_{H^{1}}(U_{m},U_{\ell})<\varepsilon$ for all $m,\ell\in\mathbb{N}$ with $m\ge N$ and $\ell\ge N$. By claim 1, for such $m$ and $\ell$ we have $d_{L^{2}}(U_{m},U_{\ell})\le d_{H^{1}}(U_{m},U_{\ell})<\varepsilon$ and $d_{L^{2}}(\partial_{j}U_{m},\partial_{j}U_{\ell})\le d_{H^{1}}(U_{m},U_{\ell})<\varepsilon$. Hence $(U_{m})$ and $(\partial_{j}U_{m})$ are Cauchy sequences in $L^{2}(\mathbb{T}^{n})$ with the distance $d_{L^{2}}$.

By The Flat Torus: Standing Notation (iv) the space $L^{2}(\mathbb{T}^{n})$ is a real Banach space, so by that clause its metric space is complete; by the preamble that metric is $d_{L^{2}}$. Hence there is $U\in L^{2}(\mathbb{T}^{n})$ to which $(U_{m})$ converges, and for each $j\in[n]$ there is $G_{j}\in L^{2}(\mathbb{T}^{n})$ to which $(\partial_{j}U_{m})$ converges; the $n$ choices of $G_{j}$ are finite in number and require no appeal to a choice principle beyond selecting one object for each of finitely many indices.

Fix $j\in[n]$. The classes $U_{m}$ lie in $L^{2}(\mathbb{T}^{n})$ and $\partial_{j}U_{m}$ is the $j$-th weak partial derivative of $U_{m}$ for every $m$, and $(U_{m})$ and $(\partial_{j}U_{m})$ converge in $L^{2}(\mathbb{T}^{n})$ to $U$ and to $G_{j}$; so Elementary Properties of the Weak Partial Derivative on the Torus (v) shows that $G_{j}$ is the $j$-th weak partial derivative of $U$. As this holds for every $j\in[n]$, The Sobolev Space of Once Weakly Differentiable Square-Integrable Classes on the Torus (i) gives $U\in H^{1}(\mathbb{T}^{n})$, and $\partial_{j}U=G_{j}$ by the uniqueness recorded in The Weak Partial Derivative on the Torus (ii). By (N4), $(U_{m})$ converges to $U$ in $H^{1}(\mathbb{T}^{n})$. Every Cauchy sequence therefore converges, as required.

Proof of claim 3. Let $k$ be a natural number. By Rescaling a Mollifier Kernel, $\rho_{1/k}$ is a mollifier kernel of radius $R=(1/k)\,\delta$ on $\mathbb{R}^{n}$, and $0<R$ by claim 5 of Elementary Order Arithmetic in an Ordered Field. By Mollifier Kernel of Radius $\delta$ on $\mathbb{R}^n$ the map $\rho_{1/k}$ is smooth, satisfies $0\le\rho_{1/k}(y)$ for every $y\in\mathbb{R}^{n}$, vanishes at every $y$ with $R<\lVert y\rVert$, and has $\int_{\mathbb{R}^{n}}\rho_{1/k}\,d\lambda_{n}=1$; being smooth it is continuous by claim 3 of Euclidean Space Is Open in Itself. So $\rho_{1/k}$ meets the standing hypotheses imposed on the kernel in Properties of Periodic Convolution on the Torus and in The Lebesgue Bound for Periodic Convolution and Convergence of Mollifications, with this $R$.

We record once that $C^{\infty}_{\mathrm{per}}\subseteq C^{1}_{\mathrm{per}}$: a smooth map on $\mathbb{R}^{n}$ is of class $C^{1}$ by claim 2 of Euclidean Space Is Open in Itself, and periodicity is the same condition for the two classes by Lattice-Periodic Functions and the Periodic Function Classes (iii).

Part (a). Since $u\in\mathcal{L}^{2}(\mathbb{T}^{n})$ and $\rho_{1/k}$ is smooth, Properties of Periodic Convolution on the Torus (i) gives $\rho_{1/k}\star u\in C^{\infty}_{\mathrm{per}}$, hence $\rho_{1/k}\star u\in C^{1}_{\mathrm{per}}$. By The Sobolev Space of Once Weakly Differentiable Square-Integrable Classes on the Torus (i), $U_{k}\in H^{1}(\mathbb{T}^{n})$, and by Elementary Properties of the Weak Partial Derivative on the Torus (i) the $j$-th weak partial derivative of $U_{k}$ is the class of $\bigl(\partial_{j}(\rho_{1/k}\star u)\bigr)|_{Q}$ for every $j\in[n]$. By Elementary Properties of the Weak Partial Derivative on the Torus (iv), applied to $U$, to $\partial_{j}U$, to the representatives $u$ and $g_{j}$, and to the kernel $\rho$ with parameter $1/k$, we have $\partial_{j}(\rho_{1/k}\star u)=\rho_{1/k}\star g_{j}$ as maps on $\mathbb{R}^{n}$. Hence $\partial_{j}U_{k}=[(\rho_{1/k}\star g_{j})|_{Q}]$.

Part (b). Let $K$ be the integral of $|\rho_{1/k}|$ over $\mathbb{R}^{n}$. Since $0\le\rho_{1/k}(y)$ for every $y$, (N0) gives $|\rho_{1/k}(y)|=\rho_{1/k}(y)$ for every $y$, so $|\rho_{1/k}|$ and $\rho_{1/k}$ are the same map and $K=1$. Applying The Lebesgue Bound for Periodic Convolution and Convergence of Mollifications (i) with $p=2$ to $u$ and to each $g_{j}$, and using $1\cdot t=t$,

$$\bigl\lVert(\rho_{1/k}\star u)|_{Q}\bigr\rVert_{2}\le\lVert u\rVert_{2},\qquad\bigl\lVert(\rho_{1/k}\star g_{j})|_{Q}\bigr\rVert_{2}\le\lVert g_{j}\rVert_{2}\quad(j\in[n]).$$

By The Lebesgue Space Lᵖ(μ) (iii) and L²(μ) Is a Real Hilbert Space (i) these are the inequalities $\lVert U_{k}\rVert_{L^{2}}\le\lVert U\rVert_{L^{2}}$ and $\lVert\partial_{j}U_{k}\rVert_{L^{2}}\le\lVert\partial_{j}U\rVert_{L^{2}}$, by part (a). Squaring, which preserves $\le$ between nonnegative reals by Properties of Real Powers (v), and comparing the two sums term by term by claim 1 of Comparison and Absolute Value Bounds for Finite Sums of Real Numbers,

$$\lVert U_{k}\rVert_{L^{2}}^{2}\le\lVert U\rVert_{L^{2}}^{2},\qquad\sum_{j=1}^{n}\lVert\partial_{j}U_{k}\rVert_{L^{2}}^{2}\le\sum_{j=1}^{n}\lVert\partial_{j}U\rVert_{L^{2}}^{2}.$$

Adding these two inequalities and using (N2) gives $\lVert U_{k}\rVert_{H^{1}}^{2}\le\lVert U\rVert_{H^{1}}^{2}$, whence $\lVert U_{k}\rVert_{H^{1}}\le\lVert U\rVert_{H^{1}}$ by (N1).

Convergence. By The Lebesgue Bound for Periodic Convolution and Convergence of Mollifications (ii) with $p=2$, applied to $u$, the sequence whose $k$th term is the class of $(\rho_{1/k}\star u)|_{Q}$, that is $U_{k}$, converges to $[u]=U$ in $L^{2}(\mathbb{T}^{n})$; applied to $g_{j}$, the sequence whose $k$th term is the class of $(\rho_{1/k}\star g_{j})|_{Q}$, that is $\partial_{j}U_{k}$ by part (a), converges to $[g_{j}]=\partial_{j}U$. By (N4), $(U_{k})_{k\in\mathbb{N}}$ converges to $U$ in $H^{1}(\mathbb{T}^{n})$.

Proof of claim 4. Write $\mathcal{S}_{0}=\{[w|_{Q}]:w\in C^{\infty}_{\mathrm{per}}\}$. Since $C^{\infty}_{\mathrm{per}}\subseteq C^{1}_{\mathrm{per}}$, as recorded in the proof of claim 3, The Sobolev Space of Once Weakly Differentiable Square-Integrable Classes on the Torus (i) gives $\mathcal{S}_{0}\subseteq H^{1}(\mathbb{T}^{n})$.

Let $U\in H^{1}(\mathbb{T}^{n})$ and let $\varepsilon$ be a real number with $0<\varepsilon$. By Existence of Mollifier Kernels of Every Radius there is a mollifier kernel $\rho$ of radius $1$ on $\mathbb{R}^{n}$. Choose a representative $u$ of $U$ and, for each $j\in[n]$, a representative $g_{j}$ of $\partial_{j}U$. By claim 3, applied with $\delta=1$, the sequence $(U_{k})_{k\in\mathbb{N}}$ converges to $U$ in $H^{1}(\mathbb{T}^{n})$, so by Convergent Sequence in a Metric Space there is $k\in\mathbb{N}$ with $d_{H^{1}}(U_{k},U)<\varepsilon$. By part (a) of claim 3, $\rho_{1/k}\star u\in C^{\infty}_{\mathrm{per}}$, so $U_{k}\in\mathcal{S}_{0}$.

Thus for every real $\varepsilon>0$ there is an element of $\mathcal{S}_{0}$ at $d_{H^{1}}$-distance less than $\varepsilon$ from $U$; by condition 3 of Characterization of the Closure in a Metric Space by Open Balls, $U$ belongs to the closure of $\mathcal{S}_{0}$ in $(H^{1}(\mathbb{T}^{n}),d_{H^{1}})$. As $U$ was arbitrary, that closure is $H^{1}(\mathbb{T}^{n})$, so $\mathcal{S}_{0}$ is dense in $H^{1}(\mathbb{T}^{n})$.

Proof of claim 5. By Existence of Mollifier Kernels of Every Radius fix a mollifier kernel $\rho$ of radius $1$ on $\mathbb{R}^{n}$, and for $k\in\mathbb{N}$ let $\rho_{1/k}$ be its rescaling, a mollifier kernel of radius $R_{k}=1/k$ on $\mathbb{R}^{n}$ by Rescaling a Mollifier Kernel, with $0<R_{k}$. Let $D$ be the set of maps furnished by Separability of the Lebesgue Spaces of the Torus for $p=2$; by Separability of the Lebesgue Spaces of the Torus (ii) every $s\in D$ lies in $\mathcal{L}^{2}(\mathbb{T}^{n})$ and the set $\{[s]:s\in D\}$ is countable. Put

$$\mathcal{S}=\bigl\{\,[(\rho_{1/k}\star s)|_{Q}]\ :\ s\in D,\ k\in\mathbb{N}\,\bigr\}.$$

$\mathcal{S}$ is a subset of $H^{1}(\mathbb{T}^{n})$. As in part (a) of the proof of claim 3, $\rho_{1/k}\star s\in C^{\infty}_{\mathrm{per}}\subseteq C^{1}_{\mathrm{per}}$ for every $s\in D$ and every $k$, so The Sobolev Space of Once Weakly Differentiable Square-Integrable Classes on the Torus (i) applies.

$\mathcal{S}$ is countable. By Properties of Periodic Convolution on the Torus (ii) the map $\rho_{1/k}\star s$ depends only on the class $[s]$, so there is a well-defined map $\Phi$ from $\{[s]:s\in D\}\times\mathbb{N}$ to $\mathcal{S}$ with $\Phi([s],k)=[(\rho_{1/k}\star s)|_{Q}]$, and $\mathcal{S}$ is exactly the set of its values. The set $\mathbb{N}$ is countable by claim 1 of Basic Properties of Countable Sets, so the Cartesian product $\{[s]:s\in D\}\times\mathbb{N}$ is countable by claim 1 of Products and Powers of Countable Sets, and $\mathcal{S}$ is countable by claim 4 of Basic Properties of Countable Sets.

$\mathcal{S}$ is dense. Let $U\in H^{1}(\mathbb{T}^{n})$ and let $\varepsilon$ be a real number with $0<\varepsilon$. Choose a representative $u$ of $U$ and, for each $j\in[n]$, a representative $g_{j}$ of $\partial_{j}U$. By claim 3 applied with $\delta=1$, the sequence $(U_{k})_{k\in\mathbb{N}}$ converges to $U$ in $H^{1}(\mathbb{T}^{n})$, so there is $k\in\mathbb{N}$ with $d_{H^{1}}(U_{k},U)<\varepsilon/2$, where $U_{k}=[(\rho_{1/k}\star u)|_{Q}]$. Fix this $k$ for the rest of the argument.

For $j\in[n]$, the map $\partial_{j}\rho_{1/k}$ is continuous and vanishes at every $y$ with $R_{k}<\lVert y\rVert$, by Convolution of a Locally Integrable Function with a Compactly Supported Kernel (iv) applied to the kernel $\rho_{1/k}$, which is of class $C^{1}$ by claim 2 of Euclidean Space Is Open in Itself. Let $K_{j}$ be the integral of $|\partial_{j}\rho_{1/k}|$ over $\mathbb{R}^{n}$, a nonnegative real number by that clause, and put

$$C=\Bigl(1+\sum_{j=1}^{n}K_{j}^{2}\Bigr)^{1/2}.$$

The number $1+\sum_{j=1}^{n}K_{j}^{2}$ is positive: the sum is nonnegative by Finite Sums of Real Numbers: Nonnegativity, Domination by the Sum, and Limits (i) and $0<1$ by claim 6 of Elementary Order Arithmetic in an Ordered Field, so claim 3 of Elementary Order Arithmetic in an Ordered Field applies. Hence $0<C$ by Properties of Real Powers (i), and $C^{2}=1+\sum_{j=1}^{n}K_{j}^{2}$ by (N1).

By Separability of the Lebesgue Spaces of the Torus (iii), applied with the positive real $\varepsilon/(4C)$, there is $s\in D$ with $\bigl\lVert[u]-[s]\bigr\rVert_{L^{2}(\mathbb{T}^{n})}\le\varepsilon/(4C)$. Put $w=u-s$, which lies in $\mathcal{L}^{2}(\mathbb{T}^{n})$ by Minkowski's Inequality (ii), and note $[w]=[u]-[s]$ by The Lebesgue Space Lᵖ(μ) (ii), so that $\lVert w\rVert_{2}\le\varepsilon/(4C)$ by The Lebesgue Space Lᵖ(μ) (iii).

Put $V=[(\rho_{1/k}\star s)|_{Q}]$, an element of $\mathcal{S}$, and $W=[(\rho_{1/k}\star w)|_{Q}]$. By Properties of Periodic Convolution on the Torus (iii) applied with $c=-1$ we have $\rho_{1/k}\star w=\rho_{1/k}\star u-\rho_{1/k}\star s$ as maps on $\mathbb{R}^{n}$; all three maps lie in $C^{\infty}_{\mathrm{per}}$ and hence in $C_{\mathrm{per}}$, so Continuous Periodic Functions are Power-Integrable and Dense on the Torus (iii) gives $W=U_{k}-V$ in $L^{2}(\mathbb{T}^{n})$. As $U_{k}$ and $V$ lie in $H^{1}(\mathbb{T}^{n})$, so does $W$, and $d_{H^{1}}(U_{k},V)=\lVert W\rVert_{H^{1}}$ by (N1).

We bound $\lVert W\rVert_{H^{1}}$. By Properties of Periodic Convolution on the Torus (i), $\partial_{j}(\rho_{1/k}\star w)=(\partial_{j}\rho_{1/k})\star w$ for every $j\in[n]$, so by Elementary Properties of the Weak Partial Derivative on the Torus (i) we have $\partial_{j}W=[((\partial_{j}\rho_{1/k})\star w)|_{Q}]$. Applying The Lebesgue Bound for Periodic Convolution and Convergence of Mollifications (i) with $p=2$, first with the kernel $\rho_{1/k}$, whose mass is $1$ as computed in part (b) of the proof of claim 3, and then with the kernel $\partial_{j}\rho_{1/k}$, whose mass is $K_{j}$,

$$\lVert W\rVert_{L^{2}}\le\lVert w\rVert_{2},\qquad\lVert\partial_{j}W\rVert_{L^{2}}\le K_{j}\lVert w\rVert_{2}\quad(j\in[n]).$$

Squaring by Properties of Real Powers (v), using Properties of Real Powers (iii) to write $\bigl(K_{j}\lVert w\rVert_{2}\bigr)^{2}=K_{j}^{2}\lVert w\rVert_{2}^{2}$, comparing the sums term by term by claim 1 of Comparison and Absolute Value Bounds for Finite Sums of Real Numbers, and adding the resulting inequalities, (N2) gives

$$\lVert W\rVert_{H^{1}}^{2}\le\lVert w\rVert_{2}^{2}+\sum_{j=1}^{n}K_{j}^{2}\,\lVert w\rVert_{2}^{2}.$$

By claims 1 and 4 of Properties of a Sum over a Finite Index Set, applied to the map $j\mapsto K_{j}^{2}$ on $[n]$ with the scalar $\lVert w\rVert_{2}^{2}$, the sum on the right equals $\lVert w\rVert_{2}^{2}\sum_{j=1}^{n}K_{j}^{2}$, so the right-hand side is $\bigl(1+\sum_{j=1}^{n}K_{j}^{2}\bigr)\lVert w\rVert_{2}^{2}=C^{2}\lVert w\rVert_{2}^{2}$, which equals $\bigl(C\lVert w\rVert_{2}\bigr)^{2}$ by Properties of Real Powers (iii). By (N1), $\lVert W\rVert_{H^{1}}\le C\lVert w\rVert_{2}$. Since $0\le C$ and $\lVert w\rVert_{2}\le\varepsilon/(4C)$, claim 5 of Elementary Arithmetic in an Ordered Field gives $C\lVert w\rVert_{2}\le C\cdot\varepsilon/(4C)=\varepsilon/4$.

Finally, $d_{H^{1}}$ is a metric by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity (iii), so by the triangle inequality of Metric Space,

$$d_{H^{1}}(U,V)\le d_{H^{1}}(U,U_{k})+d_{H^{1}}(U_{k},V)<\varepsilon/2+\varepsilon/4,$$

and $\varepsilon/2+\varepsilon/4<\varepsilon$ because $\varepsilon/4$ is positive. Hence $d_{H^{1}}(U,V)<\varepsilon$ with $V\in\mathcal{S}$.

Since $\varepsilon$ was an arbitrary positive real, condition 3 of Characterization of the Closure in a Metric Space by Open Balls shows that $U$ lies in the closure of $\mathcal{S}$; since $U$ was arbitrary, $\mathcal{S}$ is dense in $H^{1}(\mathbb{T}^{n})$ by Dense Subset of a Topological Space. Being also countable, it witnesses that $(H^{1}(\mathbb{T}^{n}),d_{H^{1}})$ is separable.

Proof of claim 6. Let $\mathcal{S}_{0}=\{[w|_{Q}]:w\in C^{\infty}_{\mathrm{per}}\}$ as in claim 4. By The Lebesgue Bound for Periodic Convolution and Convergence of Mollifications (iii) with $p=2$, $\mathcal{S}_{0}$ is dense in $L^{2}(\mathbb{T}^{n})$, so its closure in $L^{2}(\mathbb{T}^{n})$ is $L^{2}(\mathbb{T}^{n})$ by Dense Subset of a Topological Space. By claim 4, $\mathcal{S}_{0}\subseteq H^{1}(\mathbb{T}^{n})$, and $H^{1}(\mathbb{T}^{n})\subseteq L^{2}(\mathbb{T}^{n})$ by The Sobolev Space of Once Weakly Differentiable Square-Integrable Classes on the Torus (i). Hence claim 5 of The Closure is the Smallest Closed Superset, applied in the topological space $L^{2}(\mathbb{T}^{n})$, gives

$$L^{2}(\mathbb{T}^{n})=\operatorname{cl}(\mathcal{S}_{0})\subseteq\operatorname{cl}\bigl(H^{1}(\mathbb{T}^{n})\bigr)\subseteq L^{2}(\mathbb{T}^{n}),$$

the closures being taken in $L^{2}(\mathbb{T}^{n})$. Therefore the closure of $H^{1}(\mathbb{T}^{n})$ in $L^{2}(\mathbb{T}^{n})$ is $L^{2}(\mathbb{T}^{n})$, that is, $H^{1}(\mathbb{T}^{n})$ is dense in $L^{2}(\mathbb{T}^{n})$ by Dense Subset of a Topological Space.