TITLE: Engineer: Unable to appreciate importance of L2 space
QUESTION [15 upvotes]: Can someone provide me with some material to make clear of the importance of $L^2$ space in engineering/physics? 
Having a background in all the introductory mathematics courses offered in engineering, I find myself completely unable to appreciate the usefulness in knowing the definition of a $L^2$ space. Why bother define $L^2$? All physical construct I've seen so far, whether it is a signal or the wave function, naturally satisfies of square integrability. I have literally never encountered anything that is useful in engineering that do not satisfy this condition.
Therefore you can see why I am so curious as to why so many people are incline to say "suppose $f$ is $L^2$" before they prove something. That is like saying "suppose $x$ is in ${\mathbb R}$", which begs the question, what functional space would $f$ be a part of if not $L^2$? 
Can someone show me some actual example on how the definition of a L2 space is invoked to prove something practical (i.e. calc, linear alg, ODE). If possible, can someone also show what would have been different if $f$ belonged to $L^1$ or other $L^p$s or some other spaces?
Many thanks.

REPLY [6 votes]: The Fourier transform
$$
           \hat{f}(s)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(t)e^{-ist}\,dt
$$
is built from functions $e^{-ist}$ which are not in $L^{2}$ on the real line $\mathbb{R}$. This is an import example physically and mathematically. It tells you have "wave packets" that are in $L^{2}$ even though the components of the pack are not! The differentiation operator $D=\frac{1}{i}\frac{d}{dt}$ has no $L^{2}$ eigenfunctions on $\mathbb{R}$, but it has a wealth of non-$L^{2}$ eigenfunctions $e^{ist}$ from which general functions can be built. These non-$L^{2}$ eigenfunctions aren't physical, but they can used to approximate anything that is.
Wherever there is continuous spectrum in Quantum Mechanics found in unbound states, you have this situation. And you cannot escape continuous spectrum in Quantum Mechanics; continuous spectrum occurs in any system where any value in an interval can be an expected measurement. It's this interplay between non-$L^{2}$ eigenfunctions and $L^{2}$ packets that is critical, and it is related to uncertainty. Very important in nearly every free space problem, including atomic phenomena.