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Math Help - Orthogonal Complements

  1. #1
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    Orthogonal Complements

    U \subseteq R^n and U^\perp is it's orthogonal complement.

    Is there a case where

    {(U^\perp)}^\perp \neq U


    Thanks.
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  2. #2
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    Quote Originally Posted by jayshizwiz View Post
    U \subseteq R^n and U^\perp is it's orthogonal complement.

    Is there a case where

    {(U^\perp)}^\perp \neq U


    Thanks.

    Not in finite dimensional cases.

    Tonio
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  3. #3
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    Thanks.

    But is there a case where

    ((U^{\perp}){^\perp}){^\perp} \neq U{^\perp}{

    It would seem to me that I would reach the same conclusion as before...
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  4. #4
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    U need not be a subspace, so choose a subset. The orthogonal complement must be a subspace, so the orthogonal complement of the orthogonal complement would be the subspace spanned by U, which will not be U if U is not a subspace.
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  5. #5
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    I don't exactly understand what you wrote. Is there a difference between the first case i presented and this new one? Thanks.
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  6. #6
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    Yes.
    In your first case, you chose U \subseteq R^n such that U may or may not be a subspace. By definition of orthogonal complement, U^\perp is a subspace. So {(U^\perp)}^\perp is also a subspace (of R^n), and we know U \subseteq U^\perp and in fact U^\perp is the subspace spanned by U.
    For your second case, you chose to start with U^\perp which is a subspace of R^n, so {({(U^\perp)}^\perp)}^\perp is the subspace spanned by U^\perp, but that implies equality based on the definition of spanned.
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