Why begin with distributions and then move to tempered ones?

Question

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?

Thanks!

Answer 1

Tempered distribution is a ideal place for fourier transform,since it is one-to-one and surjective on $\mathcal{S'}(\mathbb{R}^n)$,so it's extremely useful when you do fourier analysis.

On the other hand,the distributional space is essentially the smallest extention of the space of continuous functions where differentiation is always well defined. And this is perhaps a major purpose of the distribution theory

Answer 2

Some problems with Schwartz class:

1) $\exp$ doesn't belong to $\mathcal{S}'(\mathbb{R})$. Do you like the idea of building a notion of "generalized function" which does not include the exponential?

2) If $\Omega$ is a proper open subset of $\mathbb{R}^n$, you do not have a "Schwartz space" on $\Omega$.

I think that this is enough to start with general distributions and then specialize to proper subclasses.