TITLE: Topologies on the space $\mathcal D'(U)$ of distributions
QUESTION [7 upvotes]: In my analysis lecture I am given a topology on the space of distributions as follows:
Let $u_k$ be a sequence in $\mathcal D'(u)$, $u \in \mathcal D'(u)$. We say $u_k \rightarrow u$, if $\forall \phi \in \mathcal D(u) : u_k(\phi) \rightarrow u(\phi)$.
This is the weak-$*$-topology on $\mathcal D'(u)$. It seems lecturers don't care too much about the topology of $\mathcal D'(u)$, hence I wonder whether there are stronger topologies on $\mathcal D'(u)$.

REPLY [3 votes]: Some remarks:
$D(U)$ is reflexive, even a Montel space, so the dual of $D'(U)$ with weak or strong topology is again $D(U)$ [This is in contrast to Brian's remark].
A linear functional on $D(U)$ is continuous (i.e. a distribution) if and only if it is sequentially continuous. This is remarkable, as the space of test functions $D(U)$ is not metrizable, so sequential continuity is usually not sufficient.
A sequence of distributions is weakly convergent if and only if it is strongly convergent [i.e. uniformly on bounded subsets of $D(U)$].
The last remark is why usually only the weak topology is known. And Schwartz proved and mentioned this consequence quite often in his book.