Is a normed $\mathbb R $ vectorspace complete in general?

Question

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.

FInite dimensional normed spaces over $\mathbb{R}$ (or $\mathbb{C}$) are complete, because all norms over them are equivalent; see Understanding of the theorem that all norms are equivalent in finite dimensional spaces. Completeness may fail for infinite dimensional space. — egreg, Mar 28 '14 at 11:24
@egreg Thanks for the comment, I just realised that the question was actually quite stupid since in my functionalanlysis course we are just considering $\mathbb R$ and $\mathbb C$ Vectorspaces. And assuming that every VS over those two fields was complete would make all proves that show that some space is a Banach-Space sensless :D — Thorben, Mar 28 '14 at 13:03

score 4 · Accepted Answer · answered Mar 28 '14 at 11:00

4

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.

answered Mar 28 '14 at 11:00

TZakrevskiy

22,980

@Thorben you are welcome! – TZakrevskiy Mar 28 '14 at 17:23

Is a normed $\mathbb R $ vectorspace complete in general?

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