# Distributions

Submitted by Structure on Sun, 12/16/2007 - 16:53.

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.