What is the importance of the play Hamlet put on by actors for the entertainment of the court
I'm assuming you mean the play within a play...What Hamlet tells Claudius is called "The Mousetrap", right?...basically, it's an old Italian play Hamlet knows as "the Murder of Gonzago", and he asks the players if they can perform it with a little inventive editing from Hamlet himself. Hamlet chooses the play because it's action closely resembles the tale told to Hamlet by the ghost - that a man murders his brother, a King, and marries with the widow queen. Hamlet's hope is that Claudius will see the action, recognize it and expose himself to the court that he (Claudius) is in fact the murderer of old King Hamlet. Why should Hamlet want to do this? Because he still is not convinced the ghost's tale is true, and he's looking for extra "proof" before he enacts his revenge. His hope is that Claudius will see the production, then break down and confess to his brother's murder. However, before the play within a play takes place, there is a dumb-show - a mimed performance of the upcoming featured play that shows all the action that is to come. As the dumb-show progresses, Hamlet notices that Claudius is NOT effected at all. This is where Shakespeare's genius for tension really rings out (although a lot of productions leave this out or don't pick this out of the text in the correct manner)...Hamlet suddenly becomes very agitated. Then, after the dumb-show, (when the play has speaking lines) Hamlet gets involved in the play, trying to goad Claudius into revealing himself instead of patiently waiting for the play to do the work on it's own. Hamlet starts adding commentary as the players read their lines, ensuring Claudius cannot ignore the message the action conveys to the court...that a King's brother can murder the King, and take his wife and his crown. Even if Claudius wasn't guilty, Hamlet's commentary and goading would drive the King to stop the play because of the doubt such a work would plant in the minds of those who saw the production. It allows Claudius to try and play the "victim" of Hamlet's mad antics...making Gertrude feel sorry for Claudius enough to demand Hamlet come to her room so that she may set him "straight".....sorry it's so wordy, but I love HAMLET...it's the greatest play ever written.