Show that $(1+|\xi|^2)^{-s/2}\widehat{u} \in L^2(\mathbb{R}^n)$ where $u$ is a tempered distribution ($u \in S^{\prime}(\mathbb{R}^n)$) and is a linear functional on $H^{s}(\mathbb{R}^n)$ for $s>0$. $H^{s}\left(\mathbb{R}^{n}\right)=\left\{u \in L^2\left(\mathbb{R}^{n}\right) \mid\left(1+|\xi|^{2}\right)^{s/2} \hat{u}(\xi) \in L^{2}\left(\mathbb{R}^{n}\right)\right\}$ for $s>0$.
We need to find a $g \in L^2(\mathbb{R}^n)$ such that $$\left\langle\left(1+|\xi|^{2}\right)^{-s / 2} \hat{u}, \phi\right\rangle=\langle g, \phi\rangle=\int_{\mathbb{R}^{n}} g(x) \phi(x) d x$$ for all $\phi \in C_c^{\infty}(\mathbb{R}^n)$. We are basically trying to classify the elements in the dual of the sobolev space $H^s(\mathbb{R}^n)$ which is $H^{-s}(\mathbb{R}^n)$. This is being used a standard result in almost all textbooks I have seen on sobolev spaces. However, I don't see how this follows.