we have $\mathscr{S}(\mathbb{R}^n)$ schwartz space and $C_{c}^{\infty}(\mathbb{R}^n)$, and for $s\in \mathbb{R}$ the Sobolev norm
$||u||^2_s={\int_{\mathbb{R}^n}^{}}{(1+|\xi|)^{2s}\widehat{u}(\xi)\overline{\widehat{u}(\xi)} d\xi}$.
so let
$H^0_s(\mathbb{R}^n)$=$\overline{C_c^\infty (\mathbb{R}^n)}^{ || \ ||_s}$ and $H_s (\mathbb{R}^n)=\overline{\mathscr{S}(\mathbb{R}^n)}^{ || \ ||_s}$
since $C_{c}^{\infty}(\mathbb{R}^n) \subset \mathscr{S}(\mathbb{R}^n)$ we have
$$H_s^0(\mathbb{R}^n) \subset H_s(\mathbb{R}^n)$$
but my question is, is inclusion strict for all $s$? what function could be in?
I imagine a function with infinites bumps in a plane
thanks.
edit: if $s=0$ is an equal because $C_{c}^{\infty}(\mathbb{R}^n)$ are dense in $L^2(\mathbb{R}^n)$.