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: