Differential Forms Are Dual to a Differential Coalgebra

There is a fundamental asymmetry between algebras and their dual objects, coalgebras, namely that the dual of a coalgebra is an algebra, but the converse is only true in finite dimensions. We prove that there exists a differential graded coalgebra whose continuous dual is the differential graded algebra of differential forms. This coalgebra will be constructed as an explicit subspace of de Rham currents. A general theory of topological duality between coalgebras and algebras will also be developed. 1. Algebraic and Topological Preliminaries A coalgebra over R is a triple pC,∆, q, where C is a vector space over R, ∆ : C Ñ C b C and : C Ñ R are linear, and the following diagrams commute: C ∆ // ∆ C b C Idb∆ C b C ∆bId // C b C b C , C ∆ // ∆ Id (( C b C Idb C b C bId // Rb C » C » C b R. The map ∆ is called a coproduct and is the counit. A topological vector space is a vector space C over R equipped with a topology τ such that addition C ˆ C Ñ C and scalar multiplication R ˆ C Ñ C are continuous, where C ˆ C and R ˆ C are equipped with the product topology. Such a topology τ is called a vector space topology. A vector space topology τ on C is locally convex if there exists a neighborhood base of 0 consisting of convex sets U such that ́U Ă U . Equivalently, τ is locally convex if it is generated by a family of semi-norms tρiuiPI , that is, if τ is the coarsest vector space topology on C such that ρi : CÑ R is continuous for all i P I. If a vector space C is equipped with a locally convex topology, we say that C is a locally convex space, or LCS. If C and D are LCS’s, the tensor product C b D can be equipped with the finest locally convex topology π such that the natural bilinear map θ : C ˆD Ñ C bD is continuous. This topology π on CbD is called the projective tensor product topology, and the topological space pCbD, πq is denoted by Cbπ D. If the topology on C is generated by a family of semi-norms tρiuiPI , and the topology on D is generated by a family of semi-norms tκjujPJ , then π is generated by the family of semi-norms tρi b κjuiPI,jPJ , where ρi b κjpxq ” inf ! ÿ ρipckqκjpdkq : x “ ÿ