Equivalence of ratios of transforms

Please help me out on this, The evaluvation of IFFT(x)/IFFT(y) and FFT(x)/FFT(y), are giving the same answer regardless of the size or nature (real or complex) of x or y. Is this any special property of FFT/IFFT transforms? Please let me know if there are any references regarding this.

[Star Strider]

I would not expect them to be different.

The Fourier and inverse Fourier transforms are linear transforms:

and:

so the energies of the transforms of the functions remain the same.

Answer 1

[JOB]

Thank you, for sparing your valuable time for clearing the doubt. Are there any references where this type of identies are given with proof?