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?



Сomplete homogeneous 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.


