Let $\{u_n\}_{n \in \mathcal{N}}$ be an orthonormal base of an Hilbert space $H$. For every funcion $f \in H$ we define the $n$-th Fourier coefficient as $$\hat f (n) := (f, u_n)_H$$ and we define the Fourier series of $f$ as \begin{equation} Sf(x) := \sum_{n \in \mathcal{N}} \hat f (n) \ u_n. \end{equation}
My question is: while I was working with $L^2[0,1]$ I have encounthered different basis of this space but noone has never showed me that the Fourier series obtained for every basis was the same. (Since the calculations could be very long I have not tried to evaluate the Fourier series of the same function using two different basis)
In other words:
Is the Fourier series independent from the basis used in its calculation according to the formula written above?