Distribution Theory/Distributions

Preliminaries, convergence, TVS

Definition:

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

Construction:

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

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

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

Operations on distributions

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