Source: cirosantilli/the-fourier-transform-is-a-bijection-in-l-2
= The Fourier transform is a bijection in $L^2$
As mentioned at <Plancherel theorem>{full}, some people call this part of <Plancherel theorem>, while others say it is just a corollary.
This is an important fact in <quantum mechanics>, since it is because of this that it makes sense to talk about <position and momentum space> as two dual representations of the <wave function> that contain the exact same amount of information.