I am learning about Fourier Series in Carother's Real Analysis. We have just learned the Fourier Partial Sum $S_{n}(f)$ is the closest function to $f$ in the set of trigonometric polynomials with at most degree n, $T_{n}$, with respect to the $L_{2}$ norm. i.e \begin{align} \underset{T \in T_{n}}{\inf}||f-T|| =||f-S_n(f)|| \end{align}
The author then writes the following: \begin{align} ||f - S_{n}||^{2} &= ||f||^{2} - ||S_{n}(f)||^{2}\\ &= \frac{1}{\pi}\int_{-\pi}^{\pi}f^{2}(x)dx - \frac{\alpha_{0}^{2}}{2} - \sum_{k=1}^{n}\alpha_{k}^{2}+\beta_{k}^{2} \end{align}
I don't understand how the author gets this result.
I have shown $||S_{n}(f)||^{2} = \frac{\alpha_{0}^{2}}{2}+\sum_{k=1}^{n}\alpha_{k}^{2}+\beta_{k}^{2}$, then have tried multiplying out $||f- S_{n}(f)||^{2} = ||f||^{2} - 2||fS_{n}(f)||+ ||S_{n}(f)||^2$ and simplifying it with what I have shown. However, no luck. If anyone can point me into the right direction that would be greatly appreciated.