The fourier Transform essentially means that every L^2 (the square of the function has a finite integral) is completely "described" by an l^2 series (a series o

The fourier Transform essentially means that every L^2 (the square of the function has a finite integral) is completely "described" by an l^2 series (a series of numbers whose sum of squares converges), which is about the greatest piece of magic in mathematics. One very important piece here is that the term "function" is somewhat of a lie (since the result couldn't be true if it weren't). The result for measurable functions (not almost functions) shouldn't be true (I think). I am not even sure it is true for L^1 almost functions.