Apologies for the embarrassingly bad pun in the title.
Distributions in general
First, it’s necessary to talk about distributions on an arbitrary open set , which are not necessarily tempered. In particular, they may “grow arbitrarily” as one approaches the boundary. So, instead of requiring a functional on a Schwarz space, we consider functionals on the space of smooth functions compactly supported in . However, we need some notion of continuity, which would require a topology on . There is now the tricky question of how we would require completeness of the topological vector space , which we of course desire. We can get such a topology by talking about “strict inductive limits” and whatnot, but since I don’t really find that particularly fun, I’ll sidestep it (but not really—most of the ideas will still remain).
Anyway, the idea here will be to consider auxiliary spaces for compact. This is the space of smooth functions which are supported in . We give the space a Frechet topology by the family of seminorms