After reading several books on distribution theory, I got a strange feeling. Why do they all begin with the theory of distributions and then move on to tempered distributions? Why can't we just start with Schwartz class and tempered distributions since they are the ones we use most often?
I think the theory of Schwartz class and tempered distributions can pretty well live on their own, and this theory would be much easier to develop since schwartz class is metrizable and everything works well with tempered distributions.
So why do we begin with test functions and distributions, not schwartz class and tempered distributions?