I would like to find out if a normed $\mathbb R$-Vectorspace is complete in general. Or even in a more general case if a normed $K$-Vectorpace, where K is a close field is complete?
I somehow think that this is not true, however I am not able to construct a counter example since by intuition the abelian group over $\mathbb R$ must be at least as "large" as $\mathbb R$ since otherwise i dont see how the abelian group will be closed under multiplication by elements of $\mathbb R$.
I would be happy about some hints, thanks!
It would be even sufficent if someone could tell me if the statement is true or not such that I could work on a prove.
The answer to the first question is no. Consider a space of continuous functions on the interval $[0,1]$ with the $L^1$-norm. It's a normed vector space, but it's not complete.