@0celo7
Excerpt from Sakurai
https://scontent-lhr3-1.xx.fbcdn.net/v/t1.0-9/13658962_1219084281500640_4238653674961260481_n.jpg?oh=d42a97cca1b35483f523e71dd2f479d3&oe=57F82FCA
I wonder if I am overthinking this a bit, cause I found one case as follows:
Start with 1.3.3.
(a'-a"*)<a'|a">=0 ------1.3.3.
If we take (a'-a"*)=0, we obtain a'=a"*-----(2), thus this tell us that a' and a" are complex conjugates of each other
Now <a'|a"> becomes <a"*|a">=(a"1*)* a"1+(a"2*)* a"2 + (a"3*)* a"3 = a"1 a"1+a"2 a"2 + a"3 a"3 which is not only nonzero in general, it is in fact a complex number