Existence of Fourier Series

Question

To prove the existence of a Fourier Series for a function obeying certain properties, is it sufficient to claim that if a function $\phi$ can be represented in Hilbert Space, it can also be represented as a Fourier Series, since the bases obey orthogonality when they are expressed as $e^{inx}$? $$\phi(x)=\sum^{\infty}\phi(x_n)\mid x_n \rangle=\sum^{\infty}a_{i}e^{inx}$$