Let us consider the Fréchet space $C^\infty\Bigl([0,1],\mathbb{R} \Bigr)$ of real-valued, periodic smooth functions.

That is, $f_n \to f$ in $C^\infty\Bigl([0,1],\mathbb{R} \Bigr)$ if $f^{(m)}_n$ converges uniformly on $[0,1]$ to $f^{(m)}$ as $n \to \infty$ and for each $m \in \mathbb{N} \cup \{0\}$.

Now, suppose that there exist two sequences $\{g_n\},\{K_m\} \subset C^\infty\Bigl([0,1],\mathbb{R} \Bigr)$ such that

$K_m(x) \to \sum_{k \in \mathbb{Z}} \delta(x-k)$ as $m \to \infty$ in the sense of distributions.

$\{ g_n * K_m \}$ is convergent in $C^\infty\Bigl([0,1],\mathbb{R} \Bigr)$ as $n \to \infty$ for each fixed $m$. Here $(g_n * K_m)(x):=\int_0^1 g_n(x-y)K_m(y)dy$.

Then, I wonder if $g_n$ itself is convergent in $C^\infty\Bigl([0,1],\mathbb{R} \Bigr)$ as well. Or at least, is there a convergent subsequence of $g_n$?

Edition : I am grateful for the counterexample provided by Iosif Pinelis. It certainly seems that I need further conditions. So, I assume that $g_n$ converges almost everywhere on $[0,1]$ to some $L^1$ function $g$. In this case, is $g$ in fact smooth and $g_n$ converges to $g$ in the Frechet topology defined above?

I may write a new post if this is too much of an addition.

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