Are there any vector spaces that cannot be given a norm that makes the vector space a complete metric space

Question

And if so, how can one prove that there is no such norm? I suppose one can use the form of Baire Category Theorem which states that a complete metric space cannot be written as a countable union of nowhere dense subsets to show that the metric space defined by the given vector space and norm is not complete but I have no idea on how to generalise this idea to all possible norms.

Answer 1

If a space admits an enumerable non finite basis $(e_i)_{i\in \bf N}$ it cannot be complete, as it is the enuramble union of finite dimensional vector spaces $V_n= vect (e_1,...e_n)$.