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).

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 \eqref{1} is a norm.

2. 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

1. Is $\dot H_G^s$ complete under \eqref{1}?

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