Give an example of connected space but is not locally compact ?
Is it true the rational number space ?
Take any connected space which is Hausdorff but not (i.e. Tychonoff space). If were locally compact then it would be embeddable in (i.e. the Alexandroff Compactification) and thus . Any normed vector space is path connected (and thus connected) but if you have an infinite dimensional Banach space it won't be
That said, any example like Opalg said would work. To see this note that in infinite dimensional Banach spaces the closure of the unit ball is not compact. Use this to show that any neighborhood of is not precompact.