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

  1. Dec 12, 2012 #1
    Hi,

    I have a problem understanding something

    This is a snapshot of a book I am reading

    NfBL7.png

    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
    1F4KF.png

    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 only dense in [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!
     
  2. jcsd
  3. Dec 15, 2012 #2

    lavinia

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  4. Dec 15, 2012 #3
    Yes, it does. I had trouble understanding it, but I think I make some progress. I actually think now I may got it wrong from the very beginning.

    E.g., if you have a family of functions {f_n} that has a limit f, I think it's okay to say that "lim(n→∞)=f" as well as "family {f_n} can approach arbitrarily close to f", because it's basically the same statement.

    In the same manner, I now think it's equivalent to say "lim(n→∞) V_n = L2 (R)" or "union V_n can approach arbitrarily close to L2 (R)".
     
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