Relationship between the notions of completeness for a metric space, a vector space and a field

Question

My question is motivated by the varying notions of 'completeness' one attaches to these objects.

1. Cauchy completeness: Pertaining to metric spaces. R with the Euclidean metric is Cauchy complete.
2. LUB-completeness: Pertaining to fields. The field of reals is LUB-complete.
3. Completeness of a vector space: V is complete if there exists some set S such that span(S)=V and $\sum_{k=1}^{n} |{\phi_k} \rangle \langle \phi_k|=I$, with $\phi_k$ in S.

I have also seen that the Cauchy completeness of a metric space, when coupled with the Archimedean property, leads to the LUB-completeness of it as a field. Furthermore, we build up vector spaces from fields. So can a vector space be complete only if the underlying field is complete? If so, in what sense must the underlying field by complete?