Proposition 11

The space $\mathscr{S}(\mathbf{R}^n)$ is a Fréchet space and a Montel space.

Since the space $\mathscr{C}^{\infty}(\mathbf{R}^n)$ is complete (EVT, III, p. 9, example b)), the corollary to proposition 10 implies that every Cauchy sequence in $\mathscr{S}(\mathbf{R}^n)$ converges in $\mathscr{S}(\mathbf{R}^n)$ since it is bounded and converges in $\mathscr{C}^{\infty}(\mathbf{R}^n)$.

The space $\mathscr{S}(\mathbf{R}^n)$ is therefore a Fréchet space; in particular, it is barrelled (EVT, III, p. 25, cor. to prop. 2). Let B be a bounded subset of $\mathscr{S}(\mathbf{R}^n)$ and $(\varphi_m)_{m\in\mathbf{N}}$ a sequence with values in B. Since $\mathscr{C}^{\infty}(\mathbf{R}^n)$ is a Montel space (EVT, IV, p. 18, example (4)), there exists a subsequence of $(\varphi_m)_{m\in\mathbf{N}}$ which converges in $\mathscr{C}^{\infty}(\mathbf{R}^n)$, hence in $\mathscr{S}(\mathbf{R}^n)$ (proposition 10). Therefore B is relatively compact in $\mathscr{S}(\mathbf{R}^n)$. It follows that $\mathscr{S}(\mathbf{R}^n)$ is a Montel space.