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.