Fourier transform on the test function space.

Hi All,

I am reading many books on the Fourier transform. They all state that the Fourier transform is not a mapping from test function space ($\displaystyle C^\infty$ functions of compact support) to the test function space. In one of the books I read, they explain that if $\displaystyle \phi$ is a test function, and F($\displaystyle \phi$) is a test function, then $\displaystyle \phi$ is 0. Does anyone know how to start on proving this? Or can anyone give me a test function (by above, any non zero one will do!) that I can compute the FT of? I have tried, and end up with having to integrate something that even Mathematica can't do.

Thanks.