Approximation of a distribution from $\mathcal{D}'$ by functions from $\mathcal{D}$

Question

I need to prove that for any generalized function ( they are also called distributions) $f \in \mathcal{D}'$ there exists generalized functions $f_n \in \mathcal{D} $, given by normal functions from $\mathcal{D},$ for which $$\langle f_n, \phi\rangle \rightarrow \langle f, \phi\rangle \quad \forall \phi \in \mathcal{D}.$$

1. My idea was the following: I am able to prove that $\mathcal{D}$ is dense in $\mathcal{D}'$ relatively *-weak convergence; after this it seemed obvious to me that we can simply take $f_n$ from this dense $\mathcal{D}$ with the condition $$|\langle f_n, \phi\rangle - \langle f, \phi\rangle| \leq 1/n,$$ and it will be the required sequence. But at this point I have a problem: this won't work because topology in $\mathcal{D}'$ can't be made metric;

2. Moreover, my teacher told me that for given $\phi$ we can find such $f_n$ that $$|\langle f_n, \phi\rangle - \langle f, \phi\rangle| \leq 1/n,$$ but it can't be done for all $\phi.$

3. Then I also tried to approximate $f$ by a finite function $g$ since they are also dense in $\mathcal{D}'$ but I faced the same problem: we can find the sequence for $g,$ but not for $f.$

4. I would be glad for any help!

Answer 1

Take $\psi \in \mathcal{D}$ such that $\int \psi = 1$ and let $\psi_n(x) = n \psi(nx).$ Then $\psi_n \to \delta$ is a mollifier.

Also take $\rho \in \mathcal{D}$ such that $\rho(0)=1$ and let $\rho_n(x) = \rho(x/n).$ Then $\rho_n \to 1$ pointwise and in $\mathcal{E} = C^\infty.$

Let $F_n = \rho_n (\psi_n*F).$

Does $F_n \in \mathcal{D}$? Does $F_n \to F$ in $\mathcal{D}'$?