give up on (someone or something)

1. To abandon some task or goal. Don't give up on your thesis—you'll get it done eventually. I know injuries have really taken a toll on him, but I'd hate to see him give up on becoming a major league baseball player. I'm gonna have to give up on this repair for now since I don't have the part that we need. I'll go out and get it tomorrow.

2. To lose faith in someone or something; to no longer believe in someone's or something's potential for betterment. I know I've failed a lot in the past year, but please, don't give up on me. Don't give up on this team just yet. When I was in the height of my addiction, I don't know how my wife didn't give up on me.