From what I understand, the Fourier transform decomposes a function into sines and cosines; and it is up to me to assign original domain and the transformed domain. But from my little experience, I noticed that only domains that are inverse of the other $$[Time]=\frac{1}{[Frequency]}; [Space]=\frac{1}{[wavenumber]}$$ uses Fourier transform.
I see no reason why that would be the case. Or am I misunderstanding something? Why did this happen, or is it just a coincidence? Can I use Fourier transform, say from the space domain to the frequency domain? Thank you!
I believe thanks to @reuns I have found the answer. Let's say you transform from $x$ domain to $y$ domain. Physically, $e^{2i \pi xy}$ is a phase, so it must be dimensionless. Therefore, it can only be that $[x]=\frac{1}{[y]}$.