What does energy of the difference between two DFT's $X$ and $Y$ being 0 tell us about their corresponding signals $x$ and $y$?

Question

I think that would mean that the energy of difference between the signals $x$ and $y$ should also be 0, but how can I prove this?

Does this also mean $x$ = $y$?

Answer 1

1. If the energy of the difference is zero, than the difference is zero as well
2. If the difference is zero, than the signals are identical
3. Two signals with the same power spectral density can be completely different. A simple example: both white noise and a delta impulse have the same power spectral density