Generalized inductive limit topologies and barrelledness properties

Abstract

For a locally convex space (X, τ) and an increasing sequence (Aν)ν ∈ N of convex, circled subsets of X the generalized inductive limit topology related to (X, τ) and (Aν)ν ∈ N is defined to be the finest locally convex topology on X agreeing with τ on the sets Aνν ∈ N. Several results on the classification and the inheritance properties of various types of barrelledness and their evaluable analogs are shown to be consequences only of a few basic properties of such an inductive limit topology and, in this way, are deduced and extended in a unified manner. © 1976 by Pacific Journal of Mathematics.