Interpreting the Space of Square-Integrable Functions I know that to construct the space $L^2( [-a,a) ) $, and to appreciate its richness, we need the machinery of lebesgue integration. However, I would like to work and talk about this space without ever having to invoke results from the lebesgue theory. What then is the best way to interpret $L^2( [-a,a) )$? Is it correct to say that $L^2( [-a,a) ) $ is obtained by completing the space of continuous functions on $[-a,a)$ (w.r.t. to the $L^2$ norm)? Can I say the same about $L^2(\mathbb{R})$? One result that I would really like to use is that any $L^2$ function can be approximated in norm by a step function, and this can be easily proved for a continuous function. My point is, I would like to work with $L^2$ without being hand-wavy with Lebesgue theory, whose machinery I don't really need to develop for my purposes.  
 A: For any (good enough) measure space that is also locally compact Hausdorff space $X$, the set $C_c(X)$ (compactly supported continuous functions) is dense in $L^2(X)$. The space $\mathbb{R}^n$ and any of its open or closed subsets are like that. 
Also, any bounded function is a uniform limit of step functions. Unbounded functions are $L^2$ limits of step functions (i.e. step functions are dense).
Whether you can avoid Lebesgue theory or not, depends on what you want to do with your $L^2$. 
A: $L^2[-a,a]$ is the smallest possible space whose elements (ie. functions) can be treated as geometric objects.


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*$L^1[-a,a]$ (integrable functions) is only a Banach space, with no basis. It is so packed with so many kinds of functions that it is not a very useful space to work on.

*In contrast, $L^2[-a,a]$ is a Hilbert space. Because $L^2$ supports an inner product, $L^2$ is automatically endowed with the five basic axioms of the euclidian geometry. 
Hence, $L^2$ and its functions behave similarly to the euclidian space we are all familiar with : functions of $L^2$ can be
"orthogonal" to one another, $L^2$ can be endowed with a basis ($L^1$ cannot), in $L^2$ function operations such as Fourier transforms behave similar to rotations or reflections (isometries), Pythagoras' theorem can be applied and much more.
This richness makes $L^2$ the ideal framework for functional analysis.
