Definition:

Adistribution 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${\displaystyle {\mathcal{D}}^{\prime }(U)}$. Indeed, to induce the locally convex topology, we use a family of seminorms given by

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

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

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

• Book:Distribution Theory