Preliminaries, convergence, TVS

Definition:

A distribution is a linear and continuous map from ${\displaystyle {\mathcal {D}}(U)}$ to ${\displaystyle \mathbb {R} }$ for an open ${\displaystyle U\subseteq \mathbb {R} ^{d}}$.

Construction:

We now construct the LCTVS of distributions on ${\displaystyle {\mathcal {D}}(U)}$, denoted by ${\displaystyle {\mathcal {D}}'(U)}$. Indeed, to induce the locally convex topology, we use a family of seminorms given by

${\displaystyle \|T\|_{K,n}:=}$ for ${\displaystyle T\in {\mathcal {D}}'(U)}$,

where ${\displaystyle n}$ ranges over the natural numbers ${\displaystyle \mathbb {N} }$ and ${\displaystyle K}$ over all compact subsets of ${\displaystyle U}$.

Stability properties

Distributions and functions

Operations on distributions

When given a distribution, we can do several things with it. These include: