Closed graph and open mapping theorems for topological $ \tilde \lc $‐modules and applications

We present closed graph and open mapping theorems for ‐linear maps acting between suitable classes of topological and locally convex topological ‐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 ‐modules, respectively. We give applications of the previous theorems to Colombeau theory as well to the theory of Banach ‐modules. In particular we obtain a necessary condition for ??^∞‐hypoellipticity on the symbol of a partial differential operator with generalized constant coefficients (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)