Can you recommend a good explanation of the proof of the Fourier Transform?

I'm reading Steven Smith's "Scientist and Engineer's Guide to Digital
Signal Processing," which is excellent imo, and I've read Lyons'
"Understanding Digital Signal Processing," great as well.

However, neither of these books gets into the proof of the FT.  Smith goes
through a brief qualitative description of correlation, and points out
that
the Real DFT operates on this principle.  This is evident in the equation
describing the Real DFT.  However, I'd like to understand why correlation
works for this application..

I've googled and searched this forum, and haven't found a satisfactory
explanation.  Can anyone recommend a good one?