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Math Help - Orthogonal Complement 1

  1. #1
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    Orthogonal Complement 1

    Dear Colleagues,

    Could you please help me in solving the following problem:
    Show that Y=\{x| \ x=(x_{j})\in \ell^{2}, x_{2n}=0, n=1,2,3,...\} is a closed subspace of \ell^{2} and find Y^{\bot}.
    What is Y^{\bot} if Y=span\{e_{1},...,e_{n}\}\subset \ell^{2}, where e_{j}=(\delta _{jk})?


    Regards,

    Raed.
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  2. #2
    Super Member girdav's Avatar
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    We define a inner product in l^2 by \displaystyle \langle (x_j),(y_j)\rangle := \sum_{j=0}^{+\infty}\overline{x_j}y_j. By linearity, to be orthogonal to Y is equivalent to by orthogonal to each e_i, i=1,\ldots, n.
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  3. #3
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    Thank you very much for your reply but how can I prove that Y with zero even terms is closed and what its orthogonal complement.
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  4. #4
    Super Member girdav's Avatar
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    Let f : (x_0,\ldots,x_n,\ldots)\mapsto (x_0,x_2,\ldots,x_{2n},\ldots ). Show that f is linear and continuous.
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  5. #5
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    Thank you very much for your reply. But why this function.
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  6. #6
    Super Member girdav's Avatar
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    What about \ker f ?
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