I asked the question 
[Why is multiplication on the space of smooth functions with compact support continuous?](https://math.stackexchange.com/questions/1680725/why-is-multiplication-on-the-space-of-smooth-functions-with-compact-support-cont) on M.SE 
sometime ago but I didn't receive a satisfactory answer. 


I was reading [this post of Terence Tao](https://terrytao.wordpress.com/2009/04/19/245c-notes-3-distributions/)
and I'm not able to prove the last item of exercise 4. 


I have a map $F:C\_c^{\infty}(\mathbb R^d)\times C\_c^{\infty}(\mathbb R^d)\to C\_c^{\infty}(\mathbb R^d)$ given by $F(f,g) = fg$. 


The question is: Why is $F$ continuous?


I proved that if a sequence $(f\_n,g\_n)$ converges to $(f,g)$ then $F(f\_n,g\_n) \to F(f,g)$, that is, $F$ is sequentially continuous. But, as far as i know, this does not implies that $F$ is continuous because [$C\_c^\infty (\mathbb R^d)$ is not first countable](https://math.stackexchange.com/questions/982556/why-is-the-topology-of-compactly-supported-smooth-function-in-mathbb-rd-not). 


The topology of $C\_c^{\infty}(\mathbb R^d)$ is given by seminorms $p:C\_c^{\infty}(\mathbb R^d) \to \mathbb R\_{\geq 0}$ such that $p\big|\_{C\_c^{\infty}( K)}:{C\_c^{\infty}( K)} \to \mathbb R\_{\geq 0}$ is continuous for every $K\subset \mathbb R^d$ compact; the topology of ${C\_c^{\infty}( K)}$ is given by the seminorms $ f\mapsto \sup\_{x\in K} |\partial^{\alpha} f(x)|$, $\alpha \in \mathbb N^d,$ and $C\_c^{\infty}( K)$ is a Fréchet space. 



Why is multiplication on the space of smooth functions with compact support continuous?
You can spare yourself the functional analytic abstract nonsense by using an explicit set of seminorms on $\mathcal{D}(\mathbb{R}^d)=C\_{c}^{\infty}(\mathbb{R}^d)$
which, unfortunately, are not well-known but can be found in the excellent book ["Topological Vector Spaces and Distributions"](https://books.google.com/books?id=gjkV05SN6CUC&source=gbs_navlinks_s) by Horváth [on p.171](https://books.google.com/books?id=gjkV05SN6CUC&pg=PA171). 


Let $\mathbb{N}=\{0,1,\ldots\}$, and denote the set of multiindices by $\mathbb{N}^d$. A locally finite family $\theta=(\theta\_{\alpha})\_{\alpha\in\mathbb{N}^d}$ of continuous functions $\mathbb{R}^d\rightarrow \mathbb{R}$ is one such that for all $x\in\mathbb{R}^d$ there is a neighborhood $V$ such that $V\cap {\rm Supp}\ \theta\_{\alpha}=\varnothing$ for all but finitely many $\alpha$'s.
Let
$$
\|f\|\_{\theta}=\sup\_{\alpha\in\mathbb{N}^d}\sup\_{x\in\mathbb{R}^d}
|\theta\_{\alpha}(x)D^{\alpha}f(x)|\ ,
$$
then the seminorms $\|\cdot\|\_{\theta}$ where $\theta$ runs over all such locally finite families define the topology of $\mathcal{D}(\mathbb{R}^d)$.


Continuity of the pointwise product follows once you show that for every $\theta$, there exists $\theta'$ and $\theta''$ such that
$$
\|fg\|\_{\theta}\le \|f\|\_{\theta'}\|g\|\_{\theta''}
$$
for all test functions $f$ and $g$, which one can do by hand.


For instance, you can use the Leibniz or product rule
$$
D^{\alpha}(fg)=\sum\_{\beta+\gamma=\alpha}\frac{\alpha!}{\beta!\gamma!} D^{\beta}f D^{\gamma}g\ ,
$$
and the brutal $l^1$-$l^{\infty}$ estimate
$$
|D^{\alpha}(fg)|\le \prod\_{i=1}^{d}(\alpha\_i+1)
\times\max\_{\beta+\gamma=\alpha} \frac{\alpha!}{\beta!\gamma!} |D^{\beta}f| |D^{\gamma}g|\ ,
$$
in order to see that $\theta'=\theta''$ works if it is defined by
$$
\theta'\_{\beta}(x):=\frac{1}{\beta!} \sup\_{\alpha\ge \beta}
\sqrt{\prod\_{i=1}^{d}(\alpha\_i+1)!}\times\sqrt{|\theta\_{\alpha}(x)|}\ .
$$




---


Brief Feb 2020 addendum:


@Martin Sleziak: Thank you for the edit. I didn't know one could link to a specific page as you did for the reference to Horváth. That's great!


Request for references: I attribute these explicit seminorms to Horváth because I only saw them in the book I mentioned. If you are aware of an earlier reference where these seminorms appeared, please let me know.


