Is there any proof for this formula? $$f(x)=a_{0}+\sum_{n=1}^{\infty} \Big[a_{n}\cos\Big(\frac{n \pi x}{L}\Big)+b_{n}\sin\Big(\frac{n\pi x}{L}\Big)\Big]$$ It seems no matter how hard I look, this is just a given in any papers on the Fourier series.
Can anyone direct me to a proof of this formula or show how it is derived please.
Thank you.
Hint: Defined $\sin$ and $\cos$ are orthonormal (each pair dot product should be zero). Hence, they can be the basis of the space.