Distribution Theory/Distributions

Preliminaries, convergence, TVS

Definition:

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

Construction:

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

$\Vert T{\Vert }_{K,n}:=$ for $T\in {𝒟}^{\prime }(U)$,

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: