My f is given by
They are orthogonal to each other.
It follows that if you have a (nonzero) function such that for all n, then for all n.
I note belatedly that f is given by you in post #3, which also invokes Parseval's theorem.
The theorem that if ( , ) = 0 for all n and some not zero implies is incomplete, might help the novice.
Finally, I found many versions of Parsevals theorem, with various (formula, theorem, identity) interchanged names, including for Fourier Series and one that states the Fourier transform is bijective, but none with your version. Could you please give a source for your version, preferably internet?
Parseval's theorem - Wikipedia, the free encyclopedia (see especially the section headed Equivalence of the norm and inner product forms.