I'm reading Signals and Systems by Oppenheim, and in the section 10.7.2 about stability there's a conclusion I don't understand:
For the impulse response h[n] of a discrete time system he summarizes:
BIBO stability $\iff$ h[n] absolutely integrable $\Rightarrow$ Fourier-Transform of h[n] exists $\iff$ Unit circle is within the ROC of the Z-Transform of h[n].
I didn't find a proof of it in the book or online, but he then concludes
ROC contains the unit circle $\iff$ BIBO stability.
As far as I know, absolute summability is sufficient for the Fourier-Transform to exist, but not necessary, so it can't be because of that. Where does the equivalence relationship come from?
Edit The part with the Fourier-Transform is a bit ambiguous. I think the equivalence with "ROC contains unit circle" only holds if the Fourier-Transform converges absolutely, in which case it is also equivalent with absolute summability, giving full equivalence for all statements.