1. To cheer up
2. To improve in prospects or conditions
3. To search for or find out by looking; as, to look up a word in the dictionary.
4. To search for in or as if in a reference work
5. To seek out especially for a brief visit
6. When a situation is looking up or starting to look up, it is improving.

When you get information from a reference book, such as a word from a dictionary or a telephone number from a telephone book, you look the word or number up.
When you locate and visit people you have not seen for a long time, you look them up.