In discussing the character of Willy Loman in Arthur Miller's "Death of a Salesman," the tragedy is overwhelming, the "hero" aspect less clear.
Willy Loman is one of the most tragic characters in modern theater history. He has lived his life in the thankless and not particularly prosperous role of a traveling salesman. He has clearly not been a financial success, as his family resides in a low-income house and, visually, the appearances of Willy and his wife, Linda, clearly indicates a lower-middle class lifestyle. As he approaches the twilight of his life, he has little to show for it but for his self-perception as a well-liked man. His oldest son, Biff, has been, in Willy's eyes, a failure, as is younger son Happy. Willy suffers in comparison with his friend and neighbor, Charlie, who is a successful businessman with an equally successful son. Willy's only consolation is that, in his mind, Charlie is not well-liked.
Willy's death -- a probable suicide intended to provide Linda financial support -- is the proverbial last act of a desperate man.
To the extent that Willy Loman can be considered heroic, it is in the very same manner in which he can be considered tragic. He is the quintessential American working man, the salt of the earth, providing for his family through hard work. He loves his family, but can't come to grips with the fact that his sons will likely never amount to much (although, Happy, at the end of the play, appears to be heading in the right direction). He lives an increasingly illusory life, with Linda acted as a loving facilitator. If Willy is heroic, it is not in the sense that he triumphed over adversity; it is solely in the way he accepted responsibility for himself and his family and tried to do well.