Closed graph and open mapping theorems for topological $\wt{\C}$-modules and applications

Abstract

We present closed graph and open mapping theorems for $\wt{\C}$-linear maps acting between suitable classes of topological and locally convex topological $\wt{\C}$-modules. This is done by adaptation of De Wilde's theory of webbed spaces and Adasch's theory of barrelled spaces to the context of locally convex and topological $\wt{\C}$-modules respectively. We give applications of the previous theorems to Colombeau theory as well to the theory of Banach $\wt{\C}$-modules. In particular we obtain a necessary condition for $\Ginf$-hypoellipticity on the symbol of a partial differential operator with generalized constant coefficients.