What use is $L^2$-convergence for Fourier series? I'm working through some notes for my signal processing class and there's something elementary that baffles me. We spent lots of hours and dozens of pages setting up the entire theory of Hilbert spaces in order to define the Fourier series of a square integrable periodic function in terms of the orthogonal basis of exponentials $ e_n(t):[0,2\pi] \to \mathbb{C}: t \to e^{int} $.
Then all of the sudden, out of the blue, the notes assault me with a seemingly unrelated theorem about pointwise convergence of the series, and I find out that pointwise convergence isn't guaranteed by $L^2$-convergence. So my (probably naive) question is: what was all that work good for? What use is $L^2$-convergence if it doesn't guarantee pointwise convergence?

In addition to paul's answer, one thing to keep in mind is that Hilbert spaces are typically meaningless when it comes to function values at individual points. An often glanced over fact is that if you take a function $f\in L^2$ and modify it at a single point $x_0$ calling the result $g$, then $\|f-g\|_2=0$. More generally, the elements of $L^2$ are representatives of the equivalence relation $f\equiv g$ defined by $\|f-g\|_2=0$. This can also be expressed in terms of measure theory, in that $\|f-g\|_2=0$ iff $f$ and $g$ differ on a set of measure 0. So, when speaking about convergence in these spaces, typically the best you can do is almost-everywhere convergence, and not point-wise.
