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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?

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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.

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What is $\alpha x$ for a metric space? I think that "homogeneous" means that the isometry group is transitive. Anyway, Banach spaces are homogeneous. –  Sergei Ivanov Aug 3 '10 at 14:01
    
@Sergei Ivanov: The wikipedia article gives the definition only for vector spaces. I have edited the answer to include this. –  Anweshi Aug 3 '10 at 14:03
    
I mean Definition: A metric space $(X,d)$ is called homogeneous if the group of its isometries acts transitively on $(X,d)$. –  Ivan Gundyrev Aug 3 '10 at 14:05
    
@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. –  Anweshi Aug 3 '10 at 14:08
    
I have edited the answer. –  Anweshi Aug 3 '10 at 14:22

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