I am trying to show that $C_c^{\infty}\cap{\cal F}C_c^{\infty}=\{0\}$, where $C_c^{\infty}$ denotes the space of all $C^\infty$ functions on $\mathbb{R}^n$ whose support is compact, and ${\cal F}C_c^{\infty}$ denotes the space of all Schwartz functions whose Fourier transform is in $C_c^{\infty}$.
My thought is the following: it is obvious that $C_c^{\infty}\cap{\cal F}C_c^{\infty}\supset\{0\}$. So I need to only show that $C_c^{\infty}\cap{\cal F}C_c^{\infty}\subset\{0\}$. Let $f\in C_c^{\infty}\cap{\cal F}C_c^{\infty}$. Then there exists a compact set $K\subset\mathbb{R}^n$ such that $K={\rm supp}\ {\cal F}f$. And I want to show that $f=0$, but I could not do that.
Any advice will be greatly appreciated. Thank you in advance.
