Proposition 12

The space $\mathscr{D}(\mathbf{R}^n)$ is contained in $\mathscr{S}(\mathbf{R}^n)$, and the inclusion of $\mathscr{D}(\mathbf{R}^n)$ in $\mathscr{S}(\mathbf{R}^n)$ is continuous with dense image.

Let $B\subset \mathscr{D}(\mathbf{R}^n)$ be a bounded subset, and let K be a compact subset of $\mathbf{R}^n$ such that $B\subset \mathscr{C}_K^{\infty}(\mathbf{R}^n)$. Let $\alpha \in \mathbf{N}^n$ and $k\in \mathbf{N}$. For every function $\varphi \in B$, one has

$\widetilde{q}_{\alpha ,k}(\varphi )\leqslant ($sup$_{x\in K}\|x\|^k)p_{\alpha ,K}(\varphi )$,

therefore B is bounded in $\mathscr{S}(\mathbf{R}^n)$. The continuity of the inclusion then follows from the fact that the spaces $\mathscr{S}(\mathbf{R}^n)$ and $\mathscr{D}(\mathbf{R}^n)$ are bornological.

Let us prove that $\mathscr{D}(\mathbf{R}^n)$ is dense in $\mathscr{S}(\mathbf{R}^n)$. Let B be the unit ball of $\mathbf{R}^n$ and let $\eta \in \mathscr{D}(\mathbf{R}^n)$ be a test function with support contained in 2B such that $0\leqslant \eta \leqslant 1$ and $\eta (x) = 1$ for every $x\in B$ (lemma 1, a) of IV, p. 196).

Let $\varphi \in \mathscr{S}(\mathbf{R}^n)$. For every integer $m\geqslant 1$ and every $x\in \mathbf{R}^n$, put $\eta_m(x) =\eta (x/m)$. Finally define $\varphi_m=\eta_m\varphi$; we have $\varphi_m\in \mathscr{D}(\mathbf{R}^n)$. Since $\partial^{\alpha}\eta_m=m^{-|\alpha|}(\partial^{\alpha}\eta )(x/m)$ for all $\alpha \in \mathbf{N}^n$ and $x\in \mathbf{R}^n$, one deduces from formula (2) of IV, p. 210 that the sequence $(\varphi_m)$ is bounded in $\mathscr{S}(\mathbf{R}^n)$.

Let C be a compact subset of $\mathbf{R}^n$. The sequence $(\varphi_m)_{m\geqslant 1}$ converges to $\varphi$ in $\mathscr{C}_K^{\infty}(\mathbf{R}^n)$ since $\varphi_m$ coincides with $\varphi$ on C for every sufficiently large $m$. Thus the sequence $(\varphi_m)$ converges to $\varphi$ in $\mathscr{C}^{\infty}(\mathbf{R}^n)$, and the corollary to proposition 10 of IV, p. 211 makes it possible to conclude that the sequence $(\varphi_m)$ converges to $\varphi$ in $\mathscr{S}(\mathbf{R}^n)$.