element of ${\widetilde}{{\mathcal{S}}}_r'({\mathbb{R}}^d)$ via $({\widehat}{f},\phi)
\colonequals (f,{\widehat}{\phi})$ whenever $f\in {\mathcal{S}}_r'({\mathbb{R}}^d)$ and $\phi \in
{\widetilde}{{\mathcal{S}}}_r({\mathbb{R}}^d)$.

Define the space $$\mathring{H}^s({\mathbb{R}}^d) \colonequals \left\{ f \in {\mathcal{S}}({\mathbb{R}}^d)\,:\xi\mapsto |\xi|^s
    {\widehat}{f}(\xi) \in L^2({\mathbb{R}}^d)\right\} \label{not:uSob}$$ and equip $\mathring{H}^s({\mathbb{R}}^d)$ with the inner product $$(f,g)_{\dot H^s({\mathbb{R}}^d)}\colonequals \left(\xi\mapsto |\xi|^s
  {\widehat}{f}(\xi),\xi\mapsto |\xi|^s \overline{{\widehat}{g}(\xi)}\right)_{L^2({\mathbb{R}}^d)}.$$ We define the Sobolev space $\dot H^s({\mathbb{R}}^d)$ to be the Hilbert space completion of $\mathring{H}^s({\mathbb{R}}^d)$\[not:Sob\], which we continuously embed in ${\mathcal{S}}_{H}'({\mathbb{R}}^d)$ as follows. If $\{f_n\}_{n\geq 1}$ is a Cauchy sequence in $\mathring{H}^s({\mathbb{R}}^d)$ and $\phi \in {\mathcal{S}}_{H}({\mathbb{R}}^d)$, then by Plancherel and Cauchy-Schwarz we have $$\begin{aligned}
 \label{eq:cauchy} 
  \lefteqn{|(f_m - f_n,\phi)_{L^2({\mathbb{R}}^d)}| \leq} \\ \nonumber &\left(\int
    |{\widehat}{f}_m(\xi) - {\widehat}{f}_n(\xi)|^2|\xi|^{2s}
    \,d\xi\right)^{1/2}\left(\int|{\widehat}{\phi}(\xi)||\xi|^{-2s}\,d\xi\right)^{1/2}.\end{aligned}$$ The first factor on the right-hand side tends to 0 as $\min(m,n) \to\infty$ and the second factor is finite since $\phi \in {\mathcal{S}}_{H}({\mathbb{R}}^d)$. It follows that $(f_m - f_n,\phi)_{L^2({\mathbb{R}}^d)}$ is Cauchy in ${\mathbb{R}}$, which implies that we can define a linear map $f:{\mathcal{S}}_{H}({\mathbb{R}}^d) \to {\mathbb{R}}$ by $(f,\phi) \colonequals
\lim_{n\to\infty} (f_n,\phi)$ for all $\phi \in {\mathcal{S}}_{H}({\mathbb{R}}^d)$. Observing that $\phi_k \to 0$ in ${\mathcal{S}}_{H}({\mathbb{R}}^d)$ implies $$\begin{aligned}
\limsup_{k\to\infty} |(f,\phi_k)|^2 &\leq \\
\limsup_{k\to\infty}&\limsup_{n\to\infty} \int|{\widehat}{f}_n(\xi)|^2|\xi|^{2s}
  \,d\xi \times \int|{\widehat}{\phi}_k(\xi)|^2|\xi|^{-2s}\,d\xi = 0,\end{aligned}$$ we conclude that $f$ is a continuous functional on ${\mathcal{S}}_{H}({\mathbb{R}}^d)$. Therefore, we may realize $\dot H^s({\mathbb{R}}^d)$ as a subset of ${\mathcal{S}}'_{H} ({\mathbb{R}}^d)$ by identifying each Cauchy sequence 