A question about multiresolution analysis (from a topological point of view)

Hi,

I have a problem understanding something

This is a snapshot of a book I am reading

http://i.imgur.com/NfBL7.png

Point no. 2 concerns me, because it looks to me like it contradicts itself, with "this or this"

The first part says

$\sum_{j}V_j = {L^2(R)}$ which, to me, looks completely equivavalent to
$\lim_{j \rightarrow \infty}V_j = {L^2(R)}$
given the nested nature of these subspaces.

However, the paper says
http://i.imgur.com/1F4KF.png

so what troubles me is this: is this countable union $\sum_{j}V_j$ equal to ${L^2(R)}$ or is it only dense in ${L^2(R)}$ ?

I personally think it's the former, and I don't understand this "dense" part. Could someone perhaps clarify this for me?

Much obliged!