The extension of norms on subgroups of free topological groups

Abstract

A norm on a group G is a nonnegative real-valued function N which is zero at the identity and satisfies N ( x y − 1 ) ⩽ N ( x ) + N ( y ) N(x{y^{ - 1}}) \leqslant N(x) + N(y) , for x , y ∈ G x,y \in G . Let F ( X ) F(X) be the free topological group on a space X. Bicknell and Morris have shown that any norm on a subgroup of F ( X ) F(X) generated by a finite subset of X may be extended to a continuous norm on the whole of F ( X ) F(X) . In this note a very direct and simple proof of this theorem is given.