$L^2$ convergence of Schwartz functions to a Schwartz function and possibility of extracting a "nicer" subsequence?

Question

Let $\{ f_n \}$ be a sequence of Schwartz functions on $\mathbb{R}^n$ converging to some Schwartz function $f$ in the $L^2(\mathbb{R}^n)$ norm.

Then, it is an elementary fact that we can extract a subsequence converging almost everywhere to $f$.

However, I wonder if we can extract "nicer" subsequences as well. For example, is it possible to extract a subsequence of $\{ f_n \}$ that converges to $f$ in the Schwartz class topology(=the Frechet topology on the Schwartz space)?

Could anyone please provide any information?

Answer 1

To get an example: consider non-negative smooth functions $\chi_n$ with compact support in the intervals $[0, 1/n]$ with maximum $1$ at some point in this interval. Note that such functions indeed exist and belong to the Schwartz space (being smooth with compact support). Their $L^2$-norm is then at most $1/n$, thus this sequence converges to the zero function in the $L^2$-sense. But it does not converge to zero uniformly, let alone in the Schwartz sense.