Duality in hypercomplex function theory

AbstractLet A be the Clifford algebra constructed over a quadratic n-dimensional real vector space with orthogonal basis {e1,…, en}, and e0 be the identity of A. Furthermore, let Mk(Ω;A) be the set of A-valued functions defined in an open subset Ω of Rm+1 (1 ⩽ m ⩽ n) which satisfy Dkf = 0 in Ω, where D is the generalized Cauchy-Riemann operator D = ∑i = 0m ei(∂∂xi) and k ϵ N. The aim of this paper is to characterize the dual and bidual of Mk(Ω;A). It is proved that, if Mk(Ω;A) is provided with the topology of uniform compact convergence, then its strong dual is topologically isomorphic to an inductive limit space of Fréchet modules, which in its turn admits Mk(Ω;A) as its dual. In this way, classical results about the spaces of holomorphic functions and analytic functionals are generalized