If we equip the Schwartz space $\mathcal{S}$ with its usual Fréchet space topology, then the space of continuous linear functionals $\mathcal{S}^\ast$ is known as the space of Schwartz distributions or tempered distributions.  If we equip this space with the [strong topology][1], is there anything we can say about the resulting topological vector space?  Evidently, the resulting space [will not be a Fréchet space][2], but perhaps it will have other nice properties?  In particular, I am interested in the space of continous linear operators on $\mathcal{S}^\ast$.  Is there anything interesting we can say about this space?

Unfortunately, a quick google search did not turn up many sources that dealt with the particulars of the topology on $\mathcal{S}^\ast$, much less the topology on the space of continuous linear operators on $\mathcal{S}^\ast$, so a point in the right direction to a reference would also be greatly appreciated.


  [1]: http://en.wikipedia.org/wiki/Strong_topology_(polar_topology)
  [2]: http://mathoverflow.net/questions/63383/which-frechet-spaces-have-a-dual-that-is-a-frechet-space