A question about multiresolution analysis (from a topological point of view)
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
which, to me, looks completely equivavalent to
given the nested nature of these subspaces.
However, the paper says
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?
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 .