$\{f_n\}_{n\geq 1}$ with its $\dot H^s({\mathbb{R}}^d)$-limit $f\in {\mathcal{S}}'_{H}({\mathbb{R}}^d)$.

We can characterize $\dot H^s({\mathbb{R}}^d)$ in another way which will be useful for the following section. Note that if $(\phi_n)_{n \in {\mathbb{N}}}$ is an $\dot
H^s({\mathbb{R}}^d)$-Cauchy sequence of Schwartz functions converging to $f$ in ${\mathcal{S}}'_{H}({\mathbb{R}}^d)$, then $({\widehat}{\phi}_n)_{n\in {\mathbb{N}}}$ is Cauchy in $L^2({\mathbb{R}}^d,|\xi|^{2s}\,d\xi)$, where $|\xi|^{2s}\,d\xi$ denotes the measure whose density with respect to Lebesgue measure is $\xi\mapsto
|\xi|^{2s}$. Therefore, there exists $g\in L^2({\mathbb{R}}^d,|\xi|^{2s}\,d\xi)$ to which ${\widehat}{\phi}_n$ converges with respect to the $L^2({\mathbb{R}}^d,|\xi|^{2s}\,d\xi)$ norm. Furthermore, it is straightforward to verify using the Cauchy-Schwarz inequality that $g={\widehat}{f} \in
{\widetilde}{{\mathcal{S}}}_H'({\mathbb{R}}^d)$. Therefore, $$\dot H^s({\mathbb{R}}^d) =
\left\{
f \in {\mathcal{S}}_H'({\mathbb{R}}^d) \,:\,
{\widehat}{f} \in L^2(|\xi|^{2s}\,d\xi)
\right\}, \label{not:SobAlt}$$ where ${\widehat}{f} \in L^2(|\xi|^{2s}\,d\xi)$ means that there exists $g \in
L^2(|\xi|^{2s}\,d\xi)$ such that $({\widehat}{f},\phi) = \int_{{\mathbb{R}}^d} g(x) \phi(x)
\,dx$ for all $\phi\in {\widetilde}{{\mathcal{S}}}_H({\mathbb{R}}^d)$.

For $x\in {\mathbb{R}}^d$, we define ${\langle x \rangle}\colonequals (1+|x|^2)^{1/2}$. For $s\in
{\mathbb{R}}$, let $H^s({\mathbb{R}}^d)$ be the space ${\mathcal{S}}'({\mathbb{R}}^d)$ of tempered distributions $f$ on ${\mathbb{R}}^d$ satisfying ${\langle \xi \rangle}^s{\widehat}{f}(\xi) \in
L^2({\mathbb{R}}^d)$, where ${\widehat}{f}$ denotes the Fourier transform of $f$. Equipped with the inner product $$\label{eq:ft}
(f,g)_{H^s({\mathbb{R}}^d)} \colonequals \int_{{\mathbb{R}}^d} {\langle \xi \rangle}^{2s}{\widehat}{f}(\xi) \overline{{\widehat}{g}(\xi)} \,d\xi,$$ $H^s({\mathbb{R}}^d)$ is a Hilbert space.

If $h\in {\mathcal{S}}'({\mathbb{R}}^d)$ and $\psi\in C_c^\infty({\mathbb{R}}^d)$, we define the product $\psi h 