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Math Help - Quick help with sequences and orthogonal spaces

  1. #1
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    Quick help with sequences and orthogonal spaces

    Hello!

    We have the following space:

    <br />
Z=span \{z_r:r+2,3,...\}<br />
,

    considered as a subspace of the Hilbert space l^2 (the space of all sequences z_k such that \sum_{k=1}^{\infty} |z_k|^2 < \infty , with the norm \| (z_k)\|=(\sum_{k=1}^{\infty} |z_k|^2)^{\frac{1}{2}}), where z_r=(r^{-n}_{n \geq 1}\in L^2).
    I have to find its orthogonal complement, and decide whether the space is dense and closed in  l^2.


    Now, here's what I got:

    z_r>0 \forall r
    and
    r_0>r_1 \rightarrow z_{r_0}<z_{r_1}
    i.e. they are strictly monotone decreasing and positive.
    Hence, their linear combinations would again be either strictly decreasing or strictly increasing, and would either be all positive (in the former case) or all negative (in the latter).

    Hence, we cannot approximate, say, (-1,3,-10,5,4,3,2,1,0,0,0,0...) \in l^2, by linear combinations of sequences in Z. Therefore, we have that Z is not dense in l^2.
    From this (and a lemma, which says that a space is dense iff its orthogonal complement is {0}) we can conclude that the orthogonal complement is not \{0\}.

    But how do I go about finding the orthogonal complement? We haven't covered orthonormal bases of Hilbert spaces yet... My intuition is the following:

    Z^\perp \supseteq \{(\alpha_n): \exists m_0, \exists n_0:\alpha_{m_0}>0,\alpha_{n_0}<0\} \cup \{(\alpha_n): \exists m_0,\exists n_0>m_0:\alpha_{n_0}>\alpha_{m_0}\}

    But how do I show all this rigorously??? Does the reverse inclusion hold, as well?

    And I have no intuition about whether it's closed or not...
    Last edited by Mimi89; February 20th 2011 at 08:47 AM.
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  2. #2
    Junior Member
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    Dec 2009
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    P.S.: a more technical question: do questions like this belong to the analysis or algebra forum? As they're an amalgamation of both...
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