The Variety of Topological Groups Generated by the Class of All Banach Spaces

A variety of topological groups is a class of (not necessarily Hausdorff) topological groups closed under the operations of forming subgroups, quotient groups and arbitrary products. The variety of topological groups generated by a class of topological groups is the smallest variety containing the class. The class of all topological groups underlying Banach spaces is considered. It is shown that the variety generated by this class equals the variety of all abelian topological groups.