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Thread: Orthonormal Sequence

  1. #1
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    Orthonormal Sequence

    Let ({e}_{k } be an orthonormal sequence in a Hilbert space H, and let M=span({e}_{k }). show that for any x\in H we have a x \in \bar{M} if and only if x can be represented by \sum_{k = 1}^\infty {a}_{k} {e}_{k} with the coefficient {a}_{k} = <x, {e}_{k}.
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    By definition, $\displaystyle span(e_k) $ is the set of all x that can be written as a linear combination of $\displaystyle e_k $. So $\displaystyle x \in M$ if and only if $\displaystyle x = \sum_{k = 1}^{\infty} a_k e_k $ for some $\displaystyle a_k$. I believe all you need to show is how to derive a_k.

    $\displaystyle <x, e_k> = <\sum_{j = 1}^{\infty} a_j e_k, e_k > = \sum_{j = 1}^{\infty} a_j <e_j, e_k> = a_k $
    where the last equality holds because $\displaystyle <e_j, e_k> $ is 1 if j = k and 0 otherwise since $\displaystyle (e_k) $ is an orthonormal sequence.
    Last edited by nehme007; May 30th 2011 at 08:42 PM.
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  3. #3
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    Quote Originally Posted by nehme007 View Post
    By definition, $\displaystyle span(e_k) $ is the set of all x that can be written as a linear combination of $\displaystyle e_k $. So $\displaystyle x \in M$ if and only if $\displaystyle x = \sum_{k = 1}^{\infty} a_k e_k $ for some $\displaystyle a_k$.
    I suspect that this is not completely correct. I think that M is supposed to be the set of all finite linear combinations of the $\displaystyle e_k.$ This is not a closed set (if the space is infinite-dimensional), and the point of the question is to show that its closure $\displaystyle \overline{M}$ is given by infinite sums $\displaystyle \sum_{k = 1}^\infty {a}_{k} {e}_{k}.$
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  4. #4
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    Quote Originally Posted by Opalg View Post
    I suspect that this is not completely correct. I think that M is supposed to be the set of all finite linear combinations of the $\displaystyle e_k.$ This is not a closed set (if the space is infinite-dimensional), and the point of the question is to show that its closure $\displaystyle \overline{M}$ is given by infinite sums $\displaystyle \sum_{k = 1}^\infty {a}_{k} {e}_{k}.$
    Please can you help me to show that $\displaystyle \overline{M}$ is given by infinite sums $\displaystyle \sum_{k = 1}^\infty {a}_{k} {e}_{k}.$. This question is really confusing me
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