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

which, to me, looks completely equivavalent to

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 equal to or is it only *dense* in ?

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

Much obliged!

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

It is equal.

What the author wants to state by saying dense, is that for every element of , there exists a series with .

Enter the wavelets!

That is, a Schauder basis for with exactly one element in each .