A convenient differential category

Abstract: In this paper, we show that the category of Mackey-complete, separated, topological convex bornological vector spaces and bornological linear maps is a differential category. Such spaces were introduced by Fr\"olicher and Kriegl, where they were called convenient vector spaces. While much of the structure necessary to demonstrate this observation is already contained in Fr\"olicher and Kriegl's book, we here give a new interpretation of the category of convenient vector spaces as a model of the differential linear logic of Ehrhard and Regnier. Rather than base our proof on the abstract categorical structure presented by Fr\"olicher and Kriegl, we prefer to focus on the bornological structure of convenient vector spaces. We believe bornological structures will ultimately yield a wide variety of models of differential logics.