The right way to study this question is to learn the Hilbert space terminology and technique.  The theorem you are looking for can be reformulated as follows: "Let f_n be an orthonormal system in a Hilbert space H, a_n be scalars.  If the series \sum |a_n|^2  converges then the series \sum a_n f_n converges in H". The key of the proof is the completeness of the space. See, for example, Theorem 3.8.3 of Debnath L., Mikusinski P. "Introduction to Hilbert spaces with applications", or almost every textbook in Functional Analysis.

The right way to study this question is to learn the Hilbert space terminology and technique.  The theorem you are looking for can be reformulated as follows: "Let f_n be an orthonormal system in a Hilbert space H, a_n be scalars.  If the series \sum |a_n|^2  converges then the series \sum a_n f_n converges in H". The key of the proof is the completeness of the space. See, for example, Theorem 3.8.3 of Debnath L., Mikusinski P. "Introduction to Hilbert spaces with applications", or almost every textbook in Functional Analysis.

Thanks for the help Vladimir. I appreciate it.

First of all, please identify on which set the $L^2$ space is given. From your question, I suspect that you are talking about $L^2(0,1)$. Right? And, the theorem that you are looking at is called "Plancherel Theorem". Plancherel Theorem asserts that the Fourier transform is a unitary operator defined on $L^2(0,1)$ onto $l^2(Z)$ (Z here is the set of integer numbers). A unitary map is isometry and onto. So, the all image of the the Fourier transform will  be $l^2$. I hope this answer helps.