Continuous convergence and functional analysis

Abstract

The usual setting for Functional Analysis is the category LCS of locally convex topological vector spaces. There are, however, advantages in working in a larger setting, the category CVS of convergence vector spaces—even if one's interest is restricted to LCS. In CVS, one has access to a dual structure, continuous convergence, unavailable in LCS. We show that theorems such as Grothendieck's completion theorem, Ptak's closed graph and open mapping theorems and the Banach-Steinhaus theorem are transformed from technical results in LCS to transparent and elegant results when examined in CVS with continuous convergence. In the theory of distributions, important bilinear mappings such as evaluations, multiplication and convolution, which are separately continuous when viewed in LCS, become jointly continuous in CVS.