  The definition of classical holomorphic function spaces such as the Hardy
space or the Dirichlet space on the Hartogs triangle is not canonical. In this
paper we introduce a natural family of holomorphic function spaces on the
Hartogs triangle which includes some weighted Bergman spaces, a candidate Hardy
space and a candidate Dirichlet space. For the weighted Bergman spaces and the
Hardy space we study the $L^p$ mapping properties of Bergman and Szeg\H{o}
projection respectively, whereas for the Dirichlet space we prove it is
isometric to the Dirichlet space on the bidisc.
