If a landlord administers a tresspassing notice to someone in a apartment complex is that person totally prohibited to visit family that live there. Or is they prohibited from being in the community part of the complex.

Well considering they have to cross the common area to get into the apartment, I would say they are effectively barred.

However, I'm curious about this trespassing notice. Is it a court order or just something the landlord wrote up? What is the story behind it?

There was a violation of the girl staying in the apartment withou prior notice. She was relocating from another state. Had only been there a couple of weeks. There was a verbal confortation between the landlord and the relocator. Can the landlord really punish the tenants and not allow them to have the girl come visit there home?

As said above, it would depend how legal this document is, if the landlord wrote it it's not really anything that can be enforced by law, but he could call the police and say he gave a written warning to her to stay out of the building.

I don't understand, was she staying illegally in an empty apartment?

This is still not clear. This girl was staying as a guest in someone's apt for a few weeks while she found a place of her own? During that time she got into an argument with the landlord so he banned her?

Doesn't sound like anything that would be held up in a court of law. If he called the police I doubt if they would do anything.

Your side is of course she was just "visting" but to the landlord, she had moved into the apartment and was living there, not just visiting.
Since she had no other place of residence, it could appear that way.

Next of course if the apartment lease says no one can live there without notice they knew she should have notified the landlord someone was staying there. I have had a lot of "friends" who were suppose to stay 3 weeks, be there 3 months latter, and since rent is often based on how many live in a apartment, it is justified.

Next there was a "run in" depending on what that was.

A tresspass warning is based an apartments right to restrict access to a complex for trouble makers, often done for possible drug dealers, possible gang members or trouble makers.

Normally what happens before the police will arrest someone there has to be a record where the manager or a agent for the landlord has given proper notice to the person, So the landlord or agent warns the person and documents it. In some places, the first time the police are called they also give an official warning. ( that is what we did in GA)

But upon a formal complaint of the landlord and proper tresspass warning the person will be arrested, The police don't argue law and tennants rights, they arrest upon complaint of the landlord.

So yes they will be arrested, now the DA may drop the charge, or they may have to prove it in court.

So remember the term not hold up in court still means hand cuffs, police car ride, fingerprnited and a court hearing