Schwartz space is Fréchet

The series converges since its terms are bounded by $2^{-j-1}$. Symmetry and translation invariance follow termwise; $\min (1,a+b)\le \min (1,a)+\min (1,b)$ proves the triangle inequality. Vanishing distance forces $q_0(f-g)=0$, hence $f=g$. To make $d(f,0)<\epsilon$, choose $J$ with ${\sum }_{j>J}{2}^{-j-1}<\epsilon \mathrm{/}2$ and require $q_J(f)<\epsilon \mathrm{/}2$. Conversely, to ensure $q_J(f)<\eta$, put ${\eta }^{\mathrm{\prime }}=\min (\eta \mathrm{/}2,1\mathrm{/}2)$ and require $d(f,0)<2^{-J-1}{\eta }^{\mathrm{\prime }}$. Then $\min (1,q_J(f))<{\eta }^{\mathrm{\prime }}<1$, as required. Every finite seminorm neighbourhood contains such a $q_J$ ball and each $q_J$ ball is a finite seminorm neighbourhood, proving equality of topologies.

Let $(f_k)$ be $d$-Cauchy. The converse estimate in step 1.1 makes it Cauchy in every $q_J$. Each ${\mathrm{\partial }}^{\beta }{f}_k$ therefore has a unique uniform complex limit $g_{\beta }$, by applying [F1] to its real and imaginary parts, and this limit is continuous by [F2]. On any fixed coordinate segment $x+t{e}_j$, $a\le t\le b$ with $a<b$, [F4] gives derivative ${\mathrm{\partial }}^{\beta +e_j}{f}_k$. Apply [F3] componentwise to these restricted functions: their values converge at $a$ and their derivatives converge uniformly. Consequently ${\mathrm{\partial }}_j{g}_{\beta }=g_{\beta +e_j}$ along every segment. Induction on the length of an ordered derivative now gives $f=g_0\in C^{\mathrm{\infty }}$ and ${\mathrm{\partial }}^{\beta }f=g_{\beta }$. Limits are unique specified values, so forming this family requires no choice.

Fix $\alpha ,\beta$ and $\epsilon >0$. The seminorm Cauchy property gives $N$ with $\mathrm{\mid }{x}^{\alpha }{\mathrm{\partial }}^{\beta }(f_k-f_l)(x)\mathrm{\mid }<\epsilon \mathrm{/}2$ for every $x$ and $k,l\ge N$. For fixed $x,k$, let $l\to \mathrm{\infty }$ using step 2.1. Then $p_{\alpha \beta }(f_k-f)\le \epsilon \mathrm{/}2<\epsilon$. Taking $k=N$ also shows $p_{\alpha \beta }(f)\le p_{\alpha \beta }(f_N)+\epsilon \mathrm{/}2<\mathrm{\infty }$. Thus $f\in \mathcal{S}$ and $f_k\to f$ in every seminorm and hence in $d$. Combined with local convexity, this proves the claimed Fréchet property.