Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

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

Let $\mathcal{D}'(\Omega)$ be the space of distributions on an open set $\Omega$, and $\mathcal{E}'(\Omega)$ the compactly supported ones.

When you have a linear operator $T:\mathcal{D}'(\Omega)\rightarrow\mathcal{D}'(\Omega)$ or $T:\mathcal{E}'(\Omega)\rightarrow\mathcal{E}'(\Omega)$, I often find it easy to prove that convergent sequences get mapped to convergent sequences (in the weak topology), but proving continuity with respect to the weak topology is much harder. Since these spaces are not first-countable (I think), one must work with a local basis around 0.

For example, if you have a pseudodifferential operator $P:\mathcal{E}'(\Omega)\rightarrow\mathcal{D}'(\Omega)$ which is properly supported, then it continuously extends to a (necessarily unique) operator $P:\mathcal{D}'(\Omega)\rightarrow\mathcal{D}'(\Omega)$ and maps $\mathcal{E}'(\Omega)$ continuously into $\mathcal{E}'(\Omega)$. I was able to prove the first statement, but I was able to prove the second statement only in terms of sequential continuity (and even this was quite difficult).

In many textbooks involving distributions, authors don't seem to be careful about this. They give an argument proving sequential continuity and claim that it is continuous in the more general sense. Is there a general scheme for converting sequential continuity arguments to actual continuity arguments for these spaces? Or can you point me to a detailed proof of the aforementioned fact about proper PDOs?

share|cite|improve this question

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.