Reflective and coreflective hulls in the category of locally convex spaces

Abstract

A new method of characterizing epi-reflective and mono-coreflective hulls in certain subcategories of the category HAUS of Hausdorff spaces and continuous maps is investigated. For comparison, the basic results are first presented briefly for HAUS. The remainder of the paper is concerned with the category LCS of Hausdorff locally convex topological vector spaces and continuous linear transformations. In LCS, reflective means closed under the taking of products and closed subspaces, and coreflective has a similar interpretation involving direct sums and quotients. Theorems analogous to those for HAUS are proved and then applied to study the distribution of reflective and coreflective subcategories in LCS and the hulls of certain classes of spaces. Subcategories of LCS that are both reflective and coreflective are also investigated, and some consequences of results are the representation theorems that every space is a quotient of a complete space, a closed subspace of a barrelled space, and, assuming the nonexistence of a strongly inaccessible cardinal, a closed subspace of a bornological space.