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Math Help - Orthogonal Complement

  1. #1
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    Orthogonal Complement

    What is Y^{\perp} if Y= span{e_1,e_2, \cdots , e_n} \subset \mathbb{l}^2, where e_j=(\delta_{ij}) ?

    e_1 = (1,0,0,0, \ldots)
    e_2 = (0,1,0,0, \ldots) etc.
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  2. #2
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    It depends: Is your n a fixed integer or not? If it's not, then the orthogonal complement is \{0\} because \{e_n : n \in \mathbb{N}\} is dense in l^2. If, on the other hand, n is fixed, then Y is a finite dimensional subspace of l^2 and so is closed, therefore you have Y \oplus Y^{\perp}=l^2. (It should be clear from here who Y^{\perp} is in this last case, since to represent every element of l^2 you need all the elements in your sequence wich are greater or equal than the given n).
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  3. #3
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    Hi Jose.

    Thank you for your message! Very helpful man.
    So in the second case assuming n is fixed at say n=3.
    How would I write the Y^{\perp} in set notation?

    say x = (\xi_j) \in l^2 is it then:

    Y^{\perp} = \{ x = (\xi_j) \vert \sum^{j=4}^{\infty} < \infty \} ?

    or should I express interms of e_n 's.
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  4. #4
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    For example you have \{e1, e2, e3\} and you want to express a sequence (x_j)_{j \geq 1} in l^2, then what you are missing is the terms (x_i)_{i>3}, so that Y^{\perp} should be all sequences of the form (y_i)_{i>3}, and a basis for Y^{\perp} should be \{e_j : j>3 \}
    Last edited by Jose27; May 31st 2009 at 01:47 PM.
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