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).