Why do you need to understand distribution, dual space and functionals for fourier transform

I'm currently learning about treatment of dirac delta as a "tempered distribution", but the notation in this book is no short of unbelievable (Walter & Shen, Wavelets), explanations are very terse, and I'm having a hard time wrapping my head about the purpose of defining functionals on dual spaces and how functionals are related to distributions.

But the goal here is to understand fourier transform as a mapping between a space S and its dual space S'. Can someone please elucidate as to why a fourier transform requires a rigorous understanding of distribution and functionals and how the concept of dual space relates to frequency and time?

Thanks!

• The Fourier transform does not map things from S to S': it is an endomorphism of S'. Jul 31 '14 at 5:53
• There are lots of very simple functions whose Fourier transforms aren't functions. Like 1, for example. If you want to be able to take the Fourier transform of 1, you have to understand distributions. Jul 31 '14 at 6:02

In principle, understanding Fourier transform doesn't require understanding distributions and functionals, it's understanding of distributions that requires understanding of functionals, and the space of distributions $S'$ is a convenient place to study Fourier transform because it is invariant under it. It's the Fourier transform itself that relates to frequency and time, informally it decomposes "signals" in time into single frequency "waves", so you can think of argument of the original function as time, and of the Fourier transform's as frequency.
Another space invariant under Fourier transform is $L^2$, Fourier transform even preserves the $L^2$ norm and you can understand it there without distributions and functionals. But it is much smaller than $S'$, in particular it does not contain Dirac's delta functions or pure harmonics $e^{i\omega t}$. Interestingly enough on $S'$ Fourier transform maps them into each other, which makes sense since frequency of a pure harmonic should be concentrated at a single point. Distributions like delta functions are most often interpreted as functionals over spaces of "nice" test functions like $S$, it makes it easier to transfer constructions and properties from test functions to distributions. For example, that $S'$ is invariant under Fourier transform immediately follows from the fact that $S$ is. So it pays off to understand how functionals work, perhaps with a different book.