I have a problem understanding something

This is a snapshot of a book I am reading

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

The first part says

\(\displaystyle \sum_{j}V_j = {L^2(R)}\) which, to me, looks completely equivavalent to

\(\displaystyle \lim_{j \rightarrow \infty}V_j = {L^2(R)}\)

given the nested nature of these subspaces.

However, the paper says

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

*dense*in \(\displaystyle {L^2(R)}\)?

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

Much obliged!