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 $$ \|f\|_{\dot{H}^{s}}=\left(\int_{\mathbb{R}^{n}}|\xi|^{2 s}|\hat{f}(\xi)|^{2} \mathrm{d} \xi\right)^{1 / 2}\quad (*)$$

There are several definitions of $\dot H^s$ (I use a subscript to distinguish them).

(1) 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 $(*)$ is a norm.

(2) Maybe there is a more natural one by

$$\dot H^s_N(\mathbb{R}^n)=\text{completion of }\{f\in \mathscr{S}:\int_{\mathbb{R}^{n}}|\xi|^{2 s}|\hat{f}(\xi)|^{2} \mathrm{d} \xi<\infty\}\text{ under the norm }(*)$$

My questions are

(1) Is $\dot H_G^s$ complete under $(*)$?

(2) Is $\dot H^s_N=\dot H_G^s$?