Why are $L^2$ functions important? From reading around I have three hypotheses:

* they show up in QM (but, why?)
* they form an inner product space (but, is that a "tight bound" or is the class easily extended to a bigger inner-product space?)
* they represent "finite energy", which everything in the world obeys

All I really know is I keep seeing the term "square-integrable" tossed around as if it's obviously important, obviously the condition one would want to impose, etc. But why?