So far nobody mentioned the most important application of square integrable
functions. Well, the most important one I know of at least ...
Wave functions in quantum mechanics have to be square integrable.
That's because the probability of a particle being observed
at a given point in space is the square of the value of the wave function at that point. And of course the integral over all points must give a probability of one - the particle has to be
somewhere after all. Therefore we have to normalize a potential wave function so that the integral over the squared function is one, and that can only work if the integral is finite.
This has some consequences, most importantly all wave functions that do not approach zero for x->∞ may be dismissed straight away as
unphysical. On the other hand,
the
physicists' most beloved wave functions, namely
plane waves, are
not square integrable either. A plane wave has a sharp
momentum and because of the
uncertainity relation the spatial
probability density is constant everywhere. But as long as one keeps in mind that they don't actually exist it's ok to use them :)