Convergence in $\mathcal{S}'$

Question

In a course I am taking, we discussed the Schwartz space $\mathcal{S}=\mathcal{S}(\mathbb{R}^d)$ and its (topological) dual $\mathcal{S}'$. When it came to the discussion of the topology on $\mathcal{S}'$, however, the professor only wrote

1. We say that a sequence $\{u_j\}_{j=1}^{\infty}$ converges to $u\in\mathcal S'$ if $\langle u_j,\varphi\rangle\to\langle u,\varphi\rangle$ for all $\varphi\in\mathcal S$;
2. A map $\mathcal{S}'\to\mathcal{S}'$ is said to be continuous if it takes every convergent sequence to a convergent sequence;

and did not give a precise description of the topology on $\mathcal{S}'$. Judging from the definition 1, I believe that he was giving $\mathcal{S}'$ the weak* topology. However, I am having a hard time proving that the definition of continuity defined as above matches the continuity in terms of the weak$^*$ topology. If the weak* topology were first countable, the answer would be yes, but unfortunately this is not the case. I asked the professor this question but he said he didn't know the answer. So my question is:

• What topology, if any, do we usually give $\mathcal{S}'$?
• Is the continuity defined as above the same as the weak* continuity; that is, does sequential continuity of a map $F:\mathcal{S}'\to \mathcal{S}'$ imply the continuity of $F$ when we endow $\mathcal{S}'$ with the weak* topology?

Any help is appreciated. Thanks in advance!

Answer 1

This is a rather tedious question. Usually, in the theory of locally convex spaces, you give dual spaces the strong topology, i.e. the topology of uniform convergence on bounded sets (this coincides with the norm topologies in case of normed spaces). This topology is strictly finer as the weak(-star) topology, which is the topology of uniform onvergence on finite sets.

But now it comes. $\mathcal{S}$ and and its strong dual $\mathcal{S}'$ are Montel spaces, so on bounded sets, sequential and normal convergence coincide. As a convergent sequence is bounded, a sequence of distributions converges (strongly) iff it converges pointwise (weakly). This carries on to linear maps: They are continuous iff they are weakly continuous.