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Give an example of a function sequence in the Schwartz space $\mathcal S(\Bbb R)$ which does not converge.
That is, for any $a,b \in \Bbb Z_+$, $ \|f_n\|_{a,b} < \infty, $ but $ \|f\|_{u,v}=\infty, $ for some $u,v \in \Bbb Z_+$, when $\{f_n\} \subset \mathcal S$ converges to $f$ pointwise, that is $ \lim_n f_n(x)=f(x), \forall x \in \Bbb R. $
Use the standard $\mathcal S$-norms $ \|f\|_{a,b}=\sup_{x \in \Bbb R} \left| x^a f^{(b)}(x) \right|, \, a,b \in \Bbb Z_+. $

The following function does not belongs to $\mathcal S$ \begin{align} f(x)&=e^{-x^2}sin\left(e^{x^2}\right), \\ f(x)&=\frac{1}{1+x^2}. \end{align} But I don't see how to make an $\mathcal S$-sequence converging to them pointwise.

This question arised when considering this post.

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    @sos440 Your can post your comment as an answer.2012-11-16

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