Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
I think you should clarify which mode of convergence is involved in your question. For example, if the convergence is induced by the $\mathcal{S}$-norms, then $f$ also lie in $\mathcal{S}$ (since the Schwartz space is a Fréchet-Montel space.) – Sangchul Lee Nov 13 '12 at 0:26
@sos440 Good point. I choosed pointwise convergence of $f_n$ to $f$. – Nicolas Essis-Breton Nov 13 '12 at 0:58
Then how about a sequence of Gaussian kernels? For example $$ f_n(x) = \exp\left( -\frac{x^2}{n} \right) $$ is a sequence of Schwartz functions while converging to $1$ pointeise. – Sangchul Lee Nov 13 '12 at 1:30
@sos440 A perfectly fine example. Thank you. – Nicolas Essis-Breton Nov 13 '12 at 1:45
@sos440 Your can post your comment as an answer. – Davide Giraudo Nov 16 '12 at 21:51

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.