Here is a naïve question: Let $K$ an absorbing, symmetric and convex set of a vector space $X$ that contains 0 that is bounded in the sense that for any direction $x\not=0$, there exists some $n$ such that $nx \not \in K$. Then the Minkowski functional $p\_K(x) = \inf \{ \lambda>0: \lambda^{-1} x \in K \}$ defines a norm on $X$. 


Is there some criterion that ensures completeness of $X$ with this norm?



Criteria for Minkowski functionals to induce a complete space?
There is a simple sufficient condition---that there exists a suitable locally convex topology on the space which is weaker than your norm topology and for which $K$ is complete. This is the Grothendieck completeness theorem.


