Proposition 17

The spaces $\mathscr{D}(\mathbf{R}^n)$ and $\mathscr{S}(\mathbf{R}^n)$ are dense in $\mathscr{S}'(\mathbf{R}^n)$.

It is enough to prove that $\mathscr{D}(\mathbf{R}^n)$ is dense in $\mathscr{S}'(\mathbf{R}^n)$. Let $\lambda$ be a continuous linear form on $\mathscr{S}'(\mathbf{R}^n)$ which is zero on $\mathscr{D}(\mathbf{R}^n)$. Since the space $\mathscr{S}(\mathbf{R}^n)$ is reflexive, there exists a function $\varphi \in \mathscr{S}(\mathbf{R}^n)$ such that $\lambda (f) =\langle f, \varphi \rangle$ for every $f\in \mathscr{S}'(\mathbf{R}^n)$. Hence

for every $\psi \in \mathscr{D}(\mathbf{R}^n)$. The measure $\varphi \cdot \mu$ on $\mathbf{R}^n$ is therefore zero (prop. 6 of IV, p. 206), whence $\varphi = 0$. The proposition then follows from the Hahn–Banach theorem (EVT, II, p. 46, cor. 1).