My professor made an analogy between Fourier series and orthogonal projections, and I was hoping someone could explain that somewhat more. Basically, as I understand it:
$$a_n = \frac{1}{L} \int_L^L f(x) \cos\left(\frac{ n\pi x}{L}\right) \ dx \longleftrightarrow c_1 = \frac{v \cdot b_i}{b_i \cdot b_i},$$
where $\dfrac{1}{L}$ can be thought of as normalizing the projection, and the integrand is the inner product (equivalent to the dot product on the right side).
Am I understanding this correctly? And can someone clarify this for me?