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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

[itex]\sum_{j}V_j = \mathbb{L^2(R)}[/itex] which, to me, looks completely equivavalent to

[itex]\lim_{j \rightarrow \infty}V_j = \mathbb{L^2(R)}[/itex]

given the nested nature of these subspaces.

However, the paper says

so what troubles me is this: is this countable union [itex]\sum_{j}V_j[/itex] equal to [itex]\mathbb{L^2(R)}[/itex] or is it onlydensein [itex]\mathbb{L^2(R)}[/itex]?

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

Much obliged!

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

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