Concerning things, one usually says:
"I will/ won't give up this thing. What about persons?"
If a man for example is in love with a woman but his family opposes their marriage, what would the man say?
Would he say: "I won't give up her."
Or very simply: "I won't leave her."
***neither a teacher nor a native-speaker***
And all in all, both ("I won't give her up." and "I won't leave her.") mean that you don't give in your family's demand.
Thank you very much
"I won't give up on her" is another alternative.
We do say 'give something up', meaning 'to part with something that one would prefer to keep'. But I have never heard 'to give someone up' before in the same sense (you can 'give someone up to the police', but that has a different meaning), and I can find no such definition in the Oxford English Dictionary. As IHIVG has noted, we do, however, say 'to give up on someone'.
-deliver an unwanted person to authority
-stop hoping that someone is going to arrive
-pronounce a sick person incurable
I would say 'I'll never give up loving her' (to give up something) or 'I'll never give up on her' but not 'I'll never give her up.'
Thank you all for your help.
Never mind the different points of view! It is alright.
I'll use both.