# Completeness of normed spaces

As earlier, I have received an answer from this site that Bolzano Weierstrass' theorem is true for finite dimensional normed spaces, but not for infinite dimensional spaces. This, in particular => all finite dim. normed spaces are complete(in the sense that every Cauchy sequence converges(w.r.t. norm)). However, is it true that every normed vector space is complete?

Let $C[0,1]$ be the space of continuous, real-valued functions on $[0,1]$. Then $$\|f\|=\int_0^1 |f(x)|dx$$ defines a norm on this space. The sequence of functions defined by $$f_n(x) = \left\{ \begin{array}{rl} 0 & \text{if } x \leq 1/2,\\ 1 & \text{if } x \geq 1/2+1/n,\\ n(x-1/2) & \text{if } 1/2\leq x\leq 1/2+1/n. \end{array} \right.$$
Ok, here we go: take the one dimensional vector space over $\mathbb Q$ with the usual norm $|q|$. Then you can find a Cauchy sequence that converges to an irrational, hence the space is not complete.
Or: sequences that are non-zero only in finitely many places (over $\mathbb R$) and you can take the norm to be the $\ell^1$ norm, i.e. $\|x\| := \sum_i |x_i|$.