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.