Closed graph and open mapping theorems for topological ℂ̃-modules and applications

We present closed graph and open mapping theorems for ℂ̃-linear maps acting between suitable classes of topological and locally convex topological ℂ̃-modules. This is done by adaptation of De Wilde's theory of webbed spaces and Adasch-Ernst-Keim's theory of barrelled spaces to the context of locally convex and topological ℂ̃-modules, respectively. We give applications of the previous theorems to Colombeau theory as well to the theory of Banach ℂ̃-modules. In particular we obtain a necessary condition for G{script}ς-hypoellipticity on the symbol of a partial differential operator with generalized constant coefficients. © 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.