completeness and the Baire category theorem

Question

I am studying the baire category theorem and trying to find a counterexample. This theorem says that a non-empty complete metric space can not be the countable union of nowhere-dense closed subsets

In particular, i'm trying to find a normed space that is the union of countably many closed nowhere-dense subsets. Obviously this means this set cannot be complete.

Answer 1

For $n\in\Bbb Z^+$ let

$$A_n=\{f\in C[0,1]:|f(x)|\le n\text{ for all }x\in[0,1]\}\;,$$

and give $C[0,1]$ the $L^1$ norm.

Every open ball contains functions with very narrow, very tall spikes.

Answer 2

Any normed vector space having countable dimension (over $\mathbb{C}$, for instance) will do. The finite-dimensional subspaces are closed and nowhere dense.