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Math Help - orthonormal basis

  1. #1
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    orthonormal basis

    Hey,

    I would like to show that the sequence \{ \delta_{k} \}_{k=1}^{\infty}=(0,0,0,\ldots,1,0,0,\ldots) (the k'th entry is 1) is an orthonormal basis for the space

    l^{2}(\mathbb{N})=\Big\{\{x_{k}\}_{k=1}^{\infty} \vert x_{k}\in \mathbb{C} \forall k\in \mathbb{N}, \sum_{k \in \mathbb{N}}|x_{k}|^{2} < \infty \Big\}

    I know that to show this I must:

    1) Show that \{ \delta_{k} \}_{k=1}^{\infty} is a basis
    for l^{2}(\mathbb{N})
    2) Show that <\delta_{k},\delta_{k}>=1,\forall k
    3) Show that <\delta_{k},\delta_{k}>=0, k\neq j

    My questions is only concerning point 1): I have already shown that \{ \delta_{k} \}_{k=1}^{\infty}=(0,0,0,\ldots,1,0,0,\ldots) is a basis for l^{1}(\mathbb{N}). Can the sequence be a basis for both spaces? If yes, would it be acceptable to use the method used in the case of l^{1}(\mathbb{N})?

    Thanks.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by surjective View Post
    Hey,

    I would like to show that the sequence \{ \delta_{k} \}_{k=1}^{\infty}=(0,0,0,\ldots,1,0,0,\ldots) (the k'th entry is 1) is an orthonormal basis for the space

    l^{2}(\mathbb{N})=\Big\{\{x_{k}\}_{k=1}^{\infty} \vert x_{k}\in \mathbb{C} \forall k\in \mathbb{N}, \sum_{k \in \mathbb{N}}|x_{k}|^{2} < \infty \Big\}

    I know that to show this I must:

    1) Show that \{ \delta_{k} \}_{k=1}^{\infty} is a basis
    for l^{2}(\mathbb{N})
    2) Show that <\delta_{k},\delta_{k}>=1,\forall k
    3) Show that <\delta_{k},\delta_{k}>=0, k\neq j

    My questions is only concerning point 1): I have already shown that \{ \delta_{k} \}_{k=1}^{\infty}=(0,0,0,\ldots,1,0,0,\ldots) is a basis for l^{1}(\mathbb{N}). Can the sequence be a basis for both spaces? If yes, would it be acceptable to use the method used in the case of l^{1}(\mathbb{N})?

    Thanks.
    What is the inner product on the space?
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  3. #3
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    orthonormal basis

    Well, the inner product

    < \{ x_{k}\}_{k=1}^{\infty}, \{ y_{k}\}_{k=1}^{\infty} >= \sum_{k \in \mathbb{N}}x_{k}\overline{y_{k}}

    is defined on l^{2}(\mathbb{N}) and makes it a hilbert-space. Right?
    Last edited by surjective; April 6th 2010 at 03:47 PM.
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  4. #4
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    Quote Originally Posted by surjective View Post
    My questions is only concerning point 1): I have already shown that \{ \delta_{k} \}_{k=1}^{\infty}=(0,0,0,\ldots,1,0,0,\ldots) is a basis for l^{1}(\mathbb{N}). Can the sequence be a basis for both spaces? If yes, would it be acceptable to use the method used in the case of l^{1}(\mathbb{N})?
    Quote Originally Posted by surjective View Post
    Well, the inner product

    < \{ x_{k}\}_{k=1}^{\infty}, \{ y_{k}\}_{k=1}^{\infty} >= \sum_{k \in \mathbb{N}}x_{k}\overline{y_{k}}

    is defined on l^{2}(\mathbb{N}) and makes it a hilbert-space. Right?
    To answer both of your questions: yes.

    The fact that it is a basis does not depend on the norm of the space. Think about it, what part of x=\sum \lambda_i x_i involves a norm?

    All of the little L spaces are the same vector space, so once you have a basis, it is a basis for all of them.
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  5. #5
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    Orthonormal basis

    Thank you very much.
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