# Distribution Theory/Distributions

## 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 ${\displaystyle U}$ .

## Operations on distributions

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