One way to think is: what is the definition of space space? It is a way to write the wave function such that:
And the, what is the definition of momentum space? It is of course a way to write the wave function such that:
- the position operator is the multiplication by
- the momentum operator is the derivative by
- the momentum operator is the multiplication by
- https://physics.stackexchange.com/questions/39442/intuitive-explanation-of-why-momentum-is-the-fourier-transform-variable-of-posit/39508#39508 gives the best idea: the Fourier transform writes a function as a (continuous) sum of plane waves, and each plane wave has a fixed momentum.