As we know the inhomogeneous Sobolev space (we only consider $s>0$) $${H}^{s}\left(\mathbb{R}^{n}\right)=\left\{f \in L^2(\mathbb{R}^n):\int_{\mathbb{R}^{n}}|\xi|^{2 s}|\hat{f}(\xi)|^{2} \mathrm{d} \xi<\infty\right\}$$ I am quite confused by the homogeneous one $\dot H^s$ which consists of functions with the following quantity is bounded $$ \|f\|_{\dot{H}^{s}}=\left(\int_{\mathbb{R}^{n}}|\xi|^{2 s}|\hat{f}(\xi)|^{2} \mathrm{d} \xi\right)^{1 / 2}\label{1}\tag{*}$$
There are several definitions of $\dot H^s$ (I use a subscript to distinguish them).
In L. Grafakos, Modern Fourier analysis, he defines $$\dot{H}_G^{s}\left(\mathbb{R}^{n}\right)=\left\{f \in \mathscr{S}^{\prime}/\mathscr{P}: \int_{\mathbb{R}^{n}}|\xi|^{2 s}|\hat{f}(\xi)|^{2} \mathrm{d} \xi<\infty\right\}$$ Here $\mathscr{S}^{\prime}/\mathscr{P}$ is the equivalent class of distributions modulo polynomials (that is, we identify two distributions whose difference is a polynomial). Then \eqref{1} is a norm.
Maybe there is a more natural one by
$$\dot H^s_N(\mathbb{R}^n)=\text{completion of }\left\{f\in \mathscr{S}:\int_{\mathbb{R}^{n}}|\xi|^{2 s}|\hat{f}(\xi)|^{2} \mathrm{d} \xi<\infty\right\}\text{ under the norm \eqref{1}}$$
My questions are
Is $\dot H_G^s$ complete under \eqref{1}?
Is $\dot H^s_N=\dot H_G^s$?
Add after Michael's comment: There is another version in H. Bahouri, J.-Y. Chemin, R. Danchin, Fourier analysis and nonlinear partial differential equations. They define $$ \dot{H}_B^{s}\left(\mathbb{R}^{n}\right)=\left\{f \in \mathscr{S}^{\prime}: \hat{f} \in L_{\operatorname{loc}}^{1}\left(\mathbb{R}^{n}\right) \text { and } \int_{\mathbb{R}^{n}}|\xi|^{2 s}|\hat{f}(\xi)|^{2} \mathrm{d} \xi<\infty\right\}$$ The reason why they include $ \hat{f} \in L_{\operatorname{loc}}^{1}\left(\mathbb{R}^{n}\right)$ arise from problems with understanding the meaning of $|\xi|^s \hat f$ if one only knows that $\hat f\in \mathscr{S}'$
This definition makes more sense by restricting $\hat f\in L_{loc}$. However, it is known that for $s < n/2$, $\dot H^s_B$ is a Hilbert space with $(*)$, while, when $\dot H^{s}_B$ is not complete when $s\geq n/2$. I am trying to avoid this definition because it is not complete.
