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The extension of norms on subgroups of free topological groups

Authors: Sidney A. Morris and Peter Nickolas
Journal: Proc. Amer. Math. Soc. 80 (1980), 185-188
MSC: Primary 22A05
MathSciNet review: 574533
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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}}) \leqslant N(x) + N(y)$, for $ x,y \in G$. Let $ 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)$ generated by a finite subset of X may be extended to a continuous norm on the whole of $ F(X)$. In this note a very direct and simple proof of this theorem is given.

References [Enhancements On Off] (What's this?)

  • [1] Kevin Bicknell and Sidney A. Morris, Norms on free topological groups, Bull. London Math. Soc. 10 (1978), 280-284. MR 519909 (80b:22007)
  • [2] Ronald Brown and Sidney A. Morris, Embeddings in contractible or compact objects, Colloq. Math. 38 (1978), 213-222. MR 0578534 (58:28249)
  • [3] S. Hartman and J. Mycielski, On the imbedding of topological groups into connected topological groups, Colloq. Math. 5 (1958), 167-169. MR 0100044 (20:6480)
  • [4] Sidney A. Morris, Varieties of topological groups. II, Bull. Austral. Math. Soc. 2 (1970), 1-13. MR 0259011 (41:3655b)

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Keywords: Norm on a topological group, free topological group
Article copyright: © Copyright 1980 American Mathematical Society

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