From wikipedia, ''homogeneity'' for vector spaces means $d(\alpha x, \alpha y) = |\alpha| d(x,y)$. An infinite dimensional Banach space has this property, and also the unit ball is not compact; therefore the space 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.

From wikipedia, ''homogeneity'' means $d(\alpha x, \alpha y) = |\alpha| d(x,y)$. An infinite dimensional Banach space has this property, and also the unit ball is not compact; therefore the space is not locally compact.

From wikipedia, ''homogeneity'' for vector spaces means $d(\alpha x, \alpha y) = |\alpha| d(x,y)$. An infinite dimensional Banach space has this property, and also the unit ball is not compact; therefore the space is not locally compact.