Why are distributions only defined on maps with compact support?

Question

Why do we restrict ourselves to maps of compact support, what’s wrong with merely smooth functions or some more general space.

Answer 1

Re: "Why do we restrict ourselves to maps of compact support", we don't. The space of smooth functions $\mathscr{E}(\Omega)$ is perfectly fine as a space of test functions. It has a natural Frechet space topology which is the locally convex topology define by the collection of seminorms $$ f\longmapsto\ \sup_{x\in K} \max_{|\alpha|\le n} |\partial^{\alpha}f(x)| $$ where $K$ ranges over compact subsets of $\Omega$ and $n$ ranges over nonnegative integers. The topological dual $\mathscr{E}'(\Omega)$ is the space of compactly supported distributions which can be seen as a subset of $\mathscr{D}'(\Omega)$.

Elementary introductions to the theory of distributions don't mention these spaces much but you can find more information in more substantial treatments like the book by Schwartz or the one by Horváth.