On Inductive Limits of Topological Algebraic Structures in relation to the Product Topologies

Abstract

In infinite-dimensional harmonic analysis, we encounter naturally inductive limits of certain topological algebraic objects, such as Lie groups, Banach algebras, topological semigroups and so on. In such cases, the inductive limit algebraic structures are not necessarily consistent with the inductive limit topologies, contrary to the affirmative statement in [Enc, Article 210]. This phenomenon is studied in [TSH] in the case of topological groups. We study in this paper similar situations for other categories of topological algebraic structures. Further, in relation to this, we study certain properties of general topological spaces for the 'commutativity' of (1) taking direct products and (2) taking inductive limits.