TITLE: Topology on the space of Schwartz Distributions
QUESTION [14 upvotes]: 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, is there anything we can say about the resulting topological vector space?  Evidently, the resulting space will not be a Fréchet space, 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.
EDIT:  After thinking about this more deeply, I realize that I am interested in a specific type of operator on $\mathcal{S}^\ast$.  $\mathcal{S}$ occurs naturally inside of $L^2$, so after identifying the dual of $L^2$ with itself via the Riesz Representation Theorem, we can in turn regard $\mathcal{S}$ as a subspace of $\mathcal{S}^\ast$.  With this in mind, I am interested in the operators on $\mathcal{S}^\ast$ that restrict to operators on $\mathcal{S}$.
The motivation for this question comes from quantum mechanics, where I have in mind the position and momentum operators acting on $\mathcal{S}^\ast$.  I am thus interested in the operator algebra they generate.  Furthermore, these of course restrict to operators on $\mathcal{S}$, and so I am likewise interseted in the operator algebra of operators on $\mathcal{S}^\ast$ that restrict to operators on $\mathcal{S}$.  In particular, I would like to abstractly characterize this space.
As this is the natural space for observables in quantum mechanics, there has to be at least something known about this space. . .

REPLY [3 votes]: Just as a variation on the other answers: the Schwartz space can easily be written as a (projective) limit of Hilbert spaces $V_s$, basically requiring that both a function and its Fourier transform be in a Levi-Sobolev space. The inclusions for $s>t$ are compact (in fact, trace-class, seen by proving that they're composites of Hilbert-Schmidt) inclusions. This adds a bit to the assertion that it is Frechet, which would indeed give a proj lim of Banach spaces, but if we can have Hilbert spaces, it's even better. Then, as in the question, identifying $L^2(\mathbb R)$ with its own dual (up to complex conjugation, anyway), the dual of $V_s$ for positive $s$ is $V_{-s}$... and we can give it the strong (=Hilbert-space) topology. It is not completely formal-categorical, but easy enough, to show that the dual of the limit is the colimit of the duals, however we topologize them. With the Hilbert-space topologies we obtain (yet another presentation of) the strong topology on tempered distributions.
It is completely formal that the dual of a colimit is the limit of duals, and a virtue of this set-up, with Hilbert spaces, is that the reflexivity is trivial.
Also, operators on $V_{-\infty}=\bigcup_s V_s$ are easy to understand in this presentation. The operators that stabilize $V_{+\infty}=\bigcap_s V_s$ must stabilize each Hilbert space $V_s$, etc.
Edit: in light of the additions/edits to the question: the very tangible instance of operators as in the question on $\mathbb R$ uses the "Dirac factorization" $-\Delta+x^2=RL+1=LR-1$, with raising and lowering operators $R=i({\partial\over \partial x}-x)$ and $L=i({\partial\over \partial x}+x)$. That "Schr\"odinger/Hamiltonian" $S=-\Delta+x^2$ can be used to give a family of semi-norms $|f|^2_k = \langle S^k, f\rangle$, with respective Hilbert spaces $V_s$, and $V_{s+1}\to V_s$ is trace-class. The projective limit is a nuclear Frechet space, and exhibits the Schwartz space as such. Likewise, the colimit of the Hilbert space duals $V_{-s}$ of $V_s$'s exhibit tempered distributions as dual-of-nuclear-Frechet.
This Hilbert-space case of more general constructions, with fairly obvious generalizations, suffices for many purposes.