Why the Sobolev norm $\left\|u\right\|_{H^s}=\left\|(1+\xi^2)^{s/2}\widehat{u}\right\|_{L^2(\mathbb{R}^n)}$ is equivalent to the graph norm of the operator $(-\Delta)^{s/2}$, $$\left\|u\right\|_{(-\Delta)^{s/2}}:=\left\|u\right\|_{L^2(\mathbb{R})}+\left\|(-\Delta)^{s/2}u\right\|_{L^2(\mathbb{R})}$$ with $(-\Delta)^{s/2}u=\mathcal{F}^{-1}\left(|\xi|^{s}\widehat{u}\right)$ ?
My attempt: I think that this is right because $(1+|\xi|^2)^{s}\leq C(s) (1+(|\xi|^2)^s)$ and $(1+(|\xi|^2)^s)\leq (1+|\xi|^2)^s$ (for all $s\geq 0$) or not?