The paradigm of an American dream can also refer to the "working as much as you earn"/ You work hard, you win hard. You do not work, you do not get. It is called the American Dream because that was the very idea behind the drafting of our Constitution and our Bill of Rights - that we all will have the same opportunities to succeed and that success does not come pre-ordained, but by hard work to earn all the things that you want.
Death of a Salesman is the antithesis of the American dream in a way. Sure Willy Loman was able to achieve such dream in the beginning, but the dream came crashing down and he is taking mental risks to bring it back, leading to his death. Therefore, the play shows you how the American Dream would look like in a sad scenario where one cannot achieve what one wants- or lost it forever.
I moved your question to the "Death of a Salesman" group since I didn't see any relationship to "The Scarlet Letter."
The American Dream has several interpretations. One of the most common deals with economic success --- the house, the yard, the car. The Dream says that we all the "right" to this dream, although I think the dream, if it has economic interpretations, only refers to the fact that these things are possible for anyone who wishes to work for them. For Willy, the dream is related to this. He has the home and the car, and dreams of being the success that the old salesman on the train was. He thinks that the recognition and the things will make him happy.
The other possible interpretation of the American Dream is that we all are free here to seek the life that we wish. We have all the freedoms of the Bill or Rights. It does not necessarily entail economic success, just the freedom to be whatever we wish. This is not the sense in which it is used in "Death," but is an important interpretation of the Dream.