What is the relationship between angular frequency and normalized angular frequency
Again there is no wrong or right here. In the Alan Oppenheim's Discrete-Time Signal Processing book, the notation is as follows: 


when there are only continuous-time signals we use $\omega$ for radians per second frequency.
when there are only discrete-time signals we use $\omega$ for radians per sample frequency
when both types of signals are present, (as in sampling), we use $\Omega$ for the continuous-time radians per second frequency, and $\omega$ for the discrete-time radians per sample frequency.


Furthermore the relationship between the two frequencies because of sampling normalization is:

$$ \Omega = \frac{\omega}{T_s} $$ 

or equivalently

$$ \omega = \Omega ~ T_s $$

However, your instructor seems to prefer the opposite notation for frequencies. That's why most literature seems to be reciprocal of your instructor's.