What is established at the end of Act 1 in Hamlet?
By the end of Act 1 of Hamlet, the following facts have been clearly established.
First, there is a suspicion that Hamlet's father did not die of natural causes and that he was killed by Claudius, Hamlet's uncle, who has now seized the throne and married his former sister-in-law.
Second, we know that Ophelia loves Hamlet, but we are not yet sure of his affections toward her.
Third, we know that Hamlet has pledged to avenge his father's ghost.
In a sense, however, we do have more questions than solid answers or facts at the end of Act 1. There is the suspicion that the ghost might not be what he appears to be, and this question plagues Hamlet for the rest of the play. This casts great doubt on the role of Claudius in the death of King Hamlet, which Prince Hamlet sets to clear up before he can progress any further with his plans. The initial conflict--Hamlet having to decide if, when, and how to take revenge--is introduced, and we are left to see how the rest of the play will develop and resolve that conflict.