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Math Help - Hilbert space proof

  1. #1
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    Hilbert space proof

    let F be a closed subspace of a hilbert space H. Let P:H->F be defined by ||x-P(x)||<_ ||x-y|| for every y in F. In other words P(x) is orthogonal projection of x on F.


    Show that orthogonal projection of x on the orthogonal complement of F is = to the identity operator - orthogonal projection of x on F.

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  2. #2
    Super Member Rebesques's Avatar
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    Re: Hilbert space proof

    You can invoke

    \langle x-P_F(x),u\rangle=0,\forall u\in F

    to show that

    \langle x-P_F(x),x'\rangle=0, x'\in F^{\perp} unique.

    Defining now P_{F^{\perp}}(x)=x', you will have proven what is required.
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