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Math Help - How to prove that something is an orthonormal basis of L2(N)?

  1. #1
    Ase
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    How to prove that something is an orthonormal basis of L2(N)?

    Hi

    I am trying to prove the following, but I am getting absolutely nowhere:

    {δ_k }_(k=1)^∞ is an orthonormal basis for L^2 (N),where δ_k is the sequence in L^2 (N) whose kth entry is 1.

    Thanks in advance
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  2. #2
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    Quote Originally Posted by Ase View Post
    Hi

    I am trying to prove the following, but I am getting absolutely nowhere:

    {δ_k }_(k=1)^∞ is an orthonormal basis for L^2 (N),where δ_k is the sequence in L^2 (N) whose kth entry is 1.

    Thanks in advance
    [/FONT]
    and all other entries are 0.

    The inner product defined on L^2(N), the space of square summable sequences (also called l^2), is <{a_n},{b_n}>= \sum a_nb_n so you need to show three things:

    1) That \{\delta_k\}_{k=1}^\infty is a basis for L^2(N)

    2) That <\delta_k, \delta_k>= 1 for all k

    3) That <\delta_k, \delta_j>= 0 if k\ne j
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  3. #3
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    u(i).u(j)=0 if i<>j
    u(i).u(j)=1 if i=j
    ||u(i)||=1
     x.u(i)=x(i)
    <br />
x.u(i)=0 => x=0
    GTM 233 page 332
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  4. #4
    Ase
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    Thanks a lot HallsofIvy. I will get at it right away and hopefully crack it.
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  5. #5
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    Basis

    Hey,

    The sequence

    \{\delta_{k}\}_{k=1}^{\infty}=(0,0,0,\ldots,1,0,0,  \ldots)

    is also a basis for the normed vector space l^{1}(\mathbb{N}).

    Couldn't you show that for any u \in l^{1}(\mathbb{N}) there exists unique scalars c_{k} \in \mathbb{C} such that

    u=\sum_{k=1}^{\infty} c_{k}\delta_{k} ?
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