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

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

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

    3) That $\displaystyle <\delta_k, \delta_j>= 0$ if $\displaystyle k\ne j$
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  3. #3
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    $\displaystyle u(i).u(j)=0 $ if $\displaystyle i<>j$
    $\displaystyle u(i).u(j)=1 $ if $\displaystyle i=j$
    $\displaystyle ||u(i)||=1 $
    $\displaystyle x.u(i)=x(i)$
    $\displaystyle
    x.u(i)=0 => x=0$
    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

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

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

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

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