We started by considering the projection of a 3-dimensional vector onto a 2-dimensional sub-space, and found that the best approximation of the 3-D vector in the 2-D subspace is given in terms of inner products. By generalizing this analogy into an infinite-continuous-dimensional function space, we find that the minimum error in approximating a function in terms of a subspace of the basis functions is also given in terms of inner products, ie. generalized Fourier coefficients. We found that the partial Fourier series gives the best approximation in the mean to a given function.
This led directly Bessel's Inequality, which in turn implied that the limit of the Fourier coefficient "N" is zero as "N" goes to infinity. Convergence in the Mean was defined, and this led to Parseval's Equation, which is really just a generalized Pythagorus theorem.