My professor made an analogy between Fourier series and orthogonal projections and I was hoping someone could explain that someone more. Basically, as I understand it:
$$ a_n = \frac1L \int_L^L f(x)cos(\frac { n\pi x}{L}) dx <---> c_1= \frac {v\cdot b_i}{b_i \cdot b_i} $$
Where $ \frac 1L $ can be thought of as the 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 elucidate this for me?