## Distributions

Posted December 16th, 2007 by Structure

Distributions are continuous linear applications on different test functions.

Usually test functions are in the space with topology defined by convergent sequence, as follows

A sequence is convergent to if and only if there is a compact such as for all and for all ,

We write for the space

There is a useful equivalent condition for a linear map on to be a distribution:

Theorem. A linear map u on is a distribution if and only if

, there are constants such as we have

The minimal integer k in this definition is called the order of distribution on K.

An important example of distribution is Dirac defined by

Dirac is a distribution of order zero.