# Сomplete homogeneous space which is not locally compact

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?

• 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
• 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