Those in functional analysis should read the book Beattie R, Butzmann H P : Convergence Strutures and Applications to Functional Analysis. Kluwer, Dordrecht, 2002, and see how incredibly useful is to get beyond the Hausdorff-Kuratowski-Bourbaki topology, that is, the usual topology. And the issue is not whether one uses sequences, nets, filters, or whatever else. No the issue is simply how one uses them. And with the rather slight relaxation og the concept of usual topology in the above book, one can do miracles. Among them, the duality theory of locally convex vector spaces becomes so clear and simple.

Those in functional analysis should read the book Beattie R, Butzmann H P : Convergence Structures and Applications to Functional Analysis. Kluwer, Dordrecht, 2002, and see how incredibly useful is to get beyond the Hausdorff-Kuratowski-Bourbaki topology, that is, the usual topology. And the issue is not whether one uses sequences, nets, filters, or whatever else. No the issue is simply how one uses them. And with the rather slight relaxation of the concept of usual topology in the above book, one can do miracles. Among them, the duality theory of locally convex vector spaces becomes so clear and simple.