Any idea for the following interesting question:

As we know we can calculate inner product of two wave functions A and B

as <A|B>. here both A and B are vector in hilbert space. here we may use

fourier transform to get momentum representation of A and B, and get same

result.

Then lets apply this idea in this way, if we have two real signals, such as X and Y and I wanna get their inner product because signals could be consider as vector in hilbert space also. Now I use fourier transform to get P = fft(X) and Q = fft(Y).

then get their inner product = conj(P)*Q.

now may I condiser this result as an inner product of signal X and Y?

Thanks for comments and I still don't have clear idea in this problem so just lets discuss it.