Subtlety with showing continuity of the embedding $i : C^\infty(\mathbb{R}^n) \to \mathcal{D}'(\mathbb{R}^n)$

Question

I am trying to show that the embedding $i : C^\infty(\mathbb{R}^n) \to \mathcal{D}'(\mathbb{R}^n)$ is continuous and injective.

Here $C^\infty(\mathbb{R}^n)$ is the space of continuous functions on $\mathbb{R}^n$, equipped with the Frechet topology of uniform convergence of all partial derivatives on compact subsets of $\mathbb{R}^n$.

$\mathcal{D}'(\mathbb{R}^n)$ is the usual space of distributions, equipped with the strong dual topology. That is, $\mathcal{D}(\mathbb{R}^n)$ is the space of smooth, compactly supported functions on $\mathbb{R}^n$.

The problem is that $C^\infty(\mathbb{R}^n)$ is a Frechet space and it is enough to work with sequences. Moreover, any sequence in $\mathcal{D}'(\mathbb{R}^n)$ converges in its strong dual topology if and only if it converges pointwise.

However, $\mathcal{D}'(\mathbb{R}^n)$ is known to be non-metrizable and NOT sequential. And the above property for sequences in $\mathcal{D}'(\mathbb{R}^n)$ does NOT hold for arbitrary nets.

So, I am kind of stuck at how to show continuity of the embedding $i : C^\infty(\mathbb{R}^n) \to \mathcal{D}'(\mathbb{R}^n)$. Could anyone please help me?

Answer 1

You are correct that it is not true that $\mathcal{D}'(\mathbb{R}^n)$ is not a sequential space when equipped with the strong topology.

However, it is at least a locally convex topological vector space. A (non-obvious) fact is that if $X$ is a sufficiently nice (Bornological) space and $Y$ is a locally convex space then every sequentially continuous linear map $T: X \to Y$ is continuous (this is e.g. Exercise 13.114 of the book "topological vector spaces" by Narici and Beckenstein). Since all Frechet spaces are in particular bornological, this tells you that it is enough to check sequential continuity of your embedding which in turn makes the problem straightforward.

Answer 2

First read

Doubt in understanding Space $\mathscr D(\Omega)$

Now let $\iota$ be the map $C^{\infty}(\mathbb{R}^n)\rightarrow D'(\mathbb{R}^n)$ which sends a smooth function $f$ to the distribution $\iota(f)$ defined by $$ \iota(f)(g):=\int_{\mathbb{R}^n}f(x)g(x)\ {\rm d}^nx $$ for all $g$ in $D(\mathbb{R}^n)$, the space of smooth compactly supported functions on $\mathbb{R}^n$. The strong dual locally convex topology of $D'(\mathbb{R}^n)$ is defined by the following collection of seminorms $\|\cdot\|_A$ indexed by bounded subsets of $D(\mathbb{R}^n)$. Namely, for every distribution $\phi$, $$ \|\phi\|_A:=\sup_{g\in A}|\phi(g)|\ . $$ The continuity of the map $\iota$ amounts to showing that for each such (defining) seminorm $\|\cdot\|_A$ there exists a continuous seminorm $\rho$ on $C^{\infty}(\mathbb{R}^n)$ such that for all $f\in C^{\infty}(\mathbb{R}^n)$, $$ \|\iota(f)\|_A\le \rho(f)\ . $$ Now $A$ being bounded, we have the existence of a compact subset $K$ in $\mathbb{R}^n$ such that all functions in $A$ have support contained in $K$. Moreover, for all (derivation) multiindex $\alpha$, we have $$ C_{\alpha}:=\sup_{g\in A}\sup_{x\in K}\ |\partial^{\alpha}g(x)|<\infty $$ So for $g\in A$ and $f\in C^{\infty}(\mathbb{R}^n)$, we have $$ |\iota(f)(g)|=\left|\int_{K}fg\right|\le \rho(f) $$ with $$ \rho(f):=C_0\times{\rm Vol}(K)\times \sup_{x\in K}|f(x)| $$ which is a continuous seminorm for the Fréchet topology of $C^{\infty}(\mathbb{R}^n)$.