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

May 2011

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!
Jul 2005
My house.
It is equal.
What the author wants to state by saying dense, is that for every element \(\displaystyle u\) of \(\displaystyle L^2\), there exists a series \(\displaystyle (u_n)\subset \cup_j V_j\) with \(\displaystyle u=\sum_j u_j\).

Enter the wavelets!
That is, a Schauder basis for \(\displaystyle L^2\) with exactly one element in each \(\displaystyle V_j\).
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