Expansions onto "bases"...? When we consider expanding functions into fourier series, or taylor series, or onto the spherical harmonics-are these projections onto a basis? Are these bases complete? How can we show this? I know that in physics they talk about fourier as expanding onto vibrational modes, and the spherical harmonics as expansions onto vibrations on a sphere, but I don't understand how we know that we're guaranteed that these expansions will be accurate. 
 A: Well...there's the problem that the bases aren't exactly bases in the sense you know and love from your first linear algebra course. For instance, if you expand a square wave as a fourier series, and then sum the series, you'll get back a function that's equal to your square wave...except perhaps at the break-points. A a break point, the FS will sum to the mid-value at the break, while your function might have taken on the upper or lower value. So not every periodic function is equal to the sum of a Fourier series, no. But once you say "equal to the sum of a Fourier Series almost everywhere", then you're OK (as long as you restrict to functions that are square-summable). 
How do you know this? I know it's not very satisfying as an answer, but ... well, you read the proofs of the theorems. Dym and McKean's Fourier Series and Integrals gives a great many of these proofs in very little space, but unfortunately they also are a bit short on motivation/explanation for my taste. But if you want the proofs, they're all there, in a compact little volume. 
