On the Completion of Uniform Convergence Spaces and an Application to Nonlinear Pdes

We obtain a characterization of the completion of an initial uniform convergence structure, which includes, among others, subspaces and projective limits of uniform convergence spaces. In this regard, there seems to be a gap in the literature. This characterization is obtained in a surprisingly straightforward way, and this should be viewed as a consequence of the fact that the category UCS of all uniform convergence spaces is Cartesian closed [6], as apposed to the category UTS consisting of all uniform topological spaces. As an application, we give an existence and regularity result for the solutions of nonlinear PDEs.