I am stuck on where to go next for this problem. I would also like to know if my steps so far are correct.
If $c_n$ are Fourier coefficients of $f$ and $f_n$ is an orthonormal set, show that the inner product $$\left<\sum_{n=1}^{N}c_nf_n,f-\sum_{n=1}^{N}c_nf_n\right> = 0$$
This is what I have so far:
$$\left<\sum_{n=1}^{N}c_nf_n,f-\sum_{n=1}^{N}c_nf_n\right> = \left<\sum_{n=1}^{N}c_nf_n,f\right>+\left<\sum_{n=1}^{N}c_nf_n,-\sum_{n=1}^{N}c_nf_n\right> =$$ $$\left<\sum_{n=1}^{N}c_nf_n,f\right>-\left<\sum_{n=1}^{N}c_nf_n,\sum_{n=1}^{N}c_nf_n\right> = \left<\sum_{n=1}^{N}c_nf_n,f\right>-\sum_{n=1}^{N}\sum_{m=1}^{N}c_nc_m\left<f_n,f_m\right>=\sum_{n=1}^{N}c_n\left<f,f_n\right>-\sum_{n=1}^{N}\sum_{m=1}^{N}c_nc_m(1)=\sum_{n=1}^{N}c_nc_n-\sum_{n=1}^{N}\sum_{m=1}^{N}c_nc_m.$$
EDIT: I have redone my work above to show the steps of where I currently am at. I unselected the previous answer, as I did not see how both summations ended up with $c_n$ becoming $c_n^*$.
The last step is wrong. It should be $$\bigg\langle\sum_{n=1}^Nc_nf_n,f\bigg\rangle-\sum_{n=1}^N\sum_{m=1}^Nc_n^*c_m\langle f_n,f_m\rangle$$
And then $$\sum_{n-1}^N c_n^*\langle f_n,f\rangle-\sum_{n=1}^N\sum_{m=1}^Nc_n^*c_m\delta_{nm}=\sum_{n=1}^N c_n^* c_n-\sum_{n=1}^N c_n^*c_n=0$$