I asked the question
Why is multiplication on the space of smooth functions with compact support continuous? on M.SE
sometime ago but I didn't receive a satisfactory answer.
I was reading this post of Terence Tao
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.
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.
Best Answer
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" by Horváth on p.171.
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.