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

\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

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!