Distribution Theory/Distributions

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Preliminaries, convergence, TVS[edit]

Definition:

A distribution is a linear and continuous map from to for an open .

Construction:

We now construct the LCTVS of distributions on , denoted by . Indeed, to induce the locally convex topology, we use a family of seminorms given by

for ,

where ranges over the natural numbers and over all compact subsets of .


Stability properties[edit]

Distributions and functions[edit]

Operations on distributions[edit]

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