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 {\Omega \subset \mathbb{R}^n}, 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 {C_0^{\infty}(\Omega),} the space of smooth functions compactly supported in {\Omega}. However, we need some notion of continuity, which would require a topology on {C_0^{\infty}(\Omega)}. There is now the tricky question of how we would require completeness of the topological vector space {C_0^{\infty}(\Omega)}, 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 {C^{\infty}(K)} for {K \subset \Omega} compact. This is the space of smooth functions {f: \Omega \rightarrow \mathbb{R}} which are supported in {K}. We give the space a Frechet topology by the family of seminorms

\displaystyle ||f||_a := \sup_K |D^a f|. (more…)