I'm studying for a qualifying exam and in our study group someone asked the question whether the delta function is in $L^2$ spaces.
My argument is that it is; since the delta function function can be approximated by a sequence of $L^2$ functions (say, Gaussian curves with decreasing spread), and $L^2$ spaces are complete, the delta function must be included in them as well, even though the delta function is not a proper function.
This is still a question of controversy in our group, though. Is my thinking correct?
This reasoning is not correct. If you want to use completeness, you would need that your sequence is Cauchy in $L^2$ (which it isn't). Furthermore, as you already mentioned yourself, the delta function is not a function (it is a distribution) and is therefore not in $L^2$.