It is well-known theorem that every locally compact, homogeneous, metric space is complete. Does anybody know example of complete, homogeneous, metric space which is not locally compact?


A Banach space is homogeneous since the metric is arising from a norm. An infinite dimensional Banach space has the property that its unit ball is not compact; therefore the space is not locally compact.

For a concrete example, take the space of continuous real functions on an interval with the supremum norm.

  • $\begingroup$ What is $\alpha x$ for a metric space? I think that "homogeneous" means that the isometry group is transitive. Anyway, Banach spaces are homogeneous. $\endgroup$ – Sergei Ivanov Aug 3 '10 at 14:01
  • $\begingroup$ @Sergei Ivanov: The wikipedia article gives the definition only for vector spaces. I have edited the answer to include this. $\endgroup$ – Anweshi Aug 3 '10 at 14:03
  • $\begingroup$ I mean Definition: A metric space $(X,d)$ is called homogeneous if the group of its isometries acts transitively on $(X,d)$. $\endgroup$ – Ivan Gundyrev Aug 3 '10 at 14:05
  • $\begingroup$ @Ivan: In any case, Sergei Ivanov assures us that Banach spaces are homogeneous. Infinite dimensional Banach spaces are complete and not locally compact. Therefore it would be an example. If you want a concrete example you can take the space of continuous functions on an interval, with supremum norm. $\endgroup$ – Anweshi Aug 3 '10 at 14:08
  • $\begingroup$ I have edited the answer. $\endgroup$ – Anweshi Aug 3 '10 at 14:22

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